Gravitational conformal invariance and coupling constants ... v.527.pdf · Gravitational conformal invariance and coupling constants in Kaluza–Klein theory F. Darabia,b,c,P.S.Wessona

Physics Letters B 527 (2002) 1–8

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Gravitational conformal invariance and coupling constants inKaluza–Klein theory

F. Darabia,b,c, P.S. Wessona

a Department of Physics, University of Waterloo, Waterloo, ON, N2L 3G1, Canadab Department of Physics, Tarbiyat Moallem University of Tabriz, 51745-406, Tabriz, Iran

c Research Institute for Fundamental Sciences, Tabriz, Iran

Received 20 December 2000; received in revised form 6 June 2001; accepted 21 December 2001

Editor: J. Frieman

Abstract

We introduce a generalized gravitational conformal invariance in the context of non-compactified 5D Kaluza–Klein theory. Itis done by assuming the 4D metric to be dependent on the extra non-compactified dimension. It is then shown that the conformalinvariance in 5D is broken by taking an absolute cosmological scaleR0 over which the 4D metric is assumed to be dependentweakly on the 5th dimension. This is equivalent to Deser’s model for the breakdown of the conformal invariance in 4D by takinga constant cosmological mass termµ2 ∼ R−2

0 in the theory. We set the scalar field to its background cosmological value leadingto Einstein equation with the gravitational constantGN and a small cosmological constant. A dual Einstein equation is alsointroduced in which the matter is coupled to the higher-dimensional geometry by the couplingG−1

N . Relevant interpretations ofthe results are also discussed. 2002 Elsevier Science B.V. All rights reserved.

PACS: 04.20.-q; 04.50.+h; 04.90.+e

Keywords: Conformal invariance; Coupling constants; Higher dimension

1. Introduction

The theory of conformal invariance has been play-ing a particularly important role in the investigationof gravitational models since Weyl, who introducedthe notion of conformal rescaling of the metric ten-sor. Afterwards, it was promoted to the conformaltransformations in scalar–tensor theories, in which an-

E-mail addresses: [email protected] (F. Darabi),[email protected] (P.S. Wesson).

other transformation on the scalar field was requiredto represent the conformal invariance in modern grav-itational models. There is an open possibility that thegravitational coupling of matter may have its originin an invariance breaking effect of this conformal in-variance. In fact, since the ordinary coupling of mat-ter to gravity is a dimensional coupling (mediated bythe gravitational constant), the local conformal trans-formations which could change the strength of this di-mensional coupling, by affecting the local standardsof length and time, are expected to play a key role. Ina system which includes matter, conformal invariance

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2 F. Darabi, P.S. Wesson / Physics Letters B 527 (2002) 1–8

requires the vanishing of the trace of the stress tensorin the absence of dimensional parameters. However,in the presence of dimensional parameters, the confor-mal invariance can be also established for a large classof theories [1] if the dimensional parameters are con-formally transformed according to their dimensions.One general feature of conformally invariant theoriesis, therefore, the presence of varying dimensional cou-pling constants. In particular, one can say that the in-troduction of a constant dimensional parameter into aconformally-invariant theory breaks the conformal in-variance in the sense that a preferred conformal frameis singled out, namely that in which the dimensionalparameters have the assumed (constant) configuration.The determination of the corresponding preferred con-formal frame depends on the nature of the problem athand. In a conformally-invariant gravitational model,the symmetry breaking may be considered as a cos-mological effect. This means that one breaks the con-formal symmetry by defining a preferred conformalframe in terms of the large-scale characteristics of cos-mic matter distributed in a universe with finite scalefactor R0. In this way, the breakdown of conformalsymmetry becomes a framework in which one canlook for the origin of the gravitational coupling of mat-ter, both classical [2] and quantum [3], at large cosmo-logical scales.

The purpose of this Letter is to show that one maylook for the origins of both conformal invariance andits breakdown, leading to gravitational couplings, in a5-dimensional Kaluza–Klein type gravity theory [4].In this popular non-compactified approach to Kaluza–Klein gravity, known as “Space–Time-Matter” theory(STM), the gravitational field is unified with its sourcethrough a new type of 5D manifold in which space andtime are augmented by an extra non-compactified di-mension which induces 4D matter. Unlike the usualKaluza–Klein theory in which a cyclic symmetry as-sociated with the extra dimension is assumed, the newapproach removes the cyclic condition and derivativesof the metric with respect to the extra coordinate areretained. This induces non-trivial matter on the hy-persurface ofl = constant. This theory basically in-volves writing the Einstein field equations with matteras a subset of the Kaluza–Klein field equations with-out matter [4], a procedure which is guaranteed by anold theorem of differential geometry due to Campbell[8].

This view is also inherent in the membrane theorywhich is mathematically equivalent, since the canoni-cal metric in STM theory [4] is basically the warp met-ric of Randall–Sundrum [5]. In this membrane theorythe large non-compactified dimension are not in con-flict with observation if the Standard Model fields areconfined to a 3-brane in the extra dimensions [6]. Infact, the only reason to compactify the extra dimen-sion is to reproduce 4D Newtonian gravity at longdistances, and if gravity is somehow trapped into the3-brane then 4D gravity can be reproduced even ifthe extra dimensions are infinitely large [6]. Gener-ically, physical phenomena in the non-compact sce-narios have to show higher-dimensional character. So,one can test them by detecting the departures from thefour-dimensional physics, such as the appearance ofthe extra force [10]. Even, if these theories with largeextra dimensions cannot correctly describe our presentuniverse their potential for describing the early uni-verse is obvious since there is strong evidence that theearly universe underwent a phase where it was five-dimensional [11].

We show that in the context of STM theory,i.e. RAB = 0 in 5D, one may find a generalizedconformally-invariant gravitational model. The well-known conformally-invariant model of Deser [2] in4D is shown to be a special case when we drop thedependence of the 4D metric on the extra dimension.Moreover, we show that the breakdown of conformalinvariance which was introduced in [2] by an adhoc non-conformal invariant term inserted into theaction naturally emerges here by (i) assuming a weak(cosmological) dependence of the 4D metric on the 5thdimension1 and (ii) approximating the scalar field withits cosmological background value using the well-known cosmological coincidence usually referred toMach or Wheeler.

This geometric approach to the subject of con-formal invariance and its breakdown in gravitationalmodels accounts properly for coupling of the gravita-tional field with its source in 5D gravity. It also givesan explanation for the origin of a small cosmologi-cal constant emerging from non-compactified extra di-

1 This assumption is reasonable since 4D general relativity isknown to be in a very good agreement with present observations.

F. Darabi, P.S. Wesson / Physics Letters B 527 (2002) 1–8 3

mension. This subject is the most recent interest in the-ories with large extra dimensions [6].

The Letter is organized as follows: in Section 2,we briefly review the conformal invariant gravitationalmodel and its breakdown in 4 dimensions due toDeser [2]. In Section 3, we introduce a generalizedconformal invariant gravitational model in 5 dimen-sions. In Section 4, we study the breakdown of confor-mal invariance in 5 dimensions and discuss on somerelevant interpretations. The Letter ends with a con-clusion.

2. Breakdown of conformal invariance in 4D

In this section we briefly revisit the standard workin 4D conformal invariance due to Deser [2]. Considerthe action functional

(1)S[φ] = 1

2

∫d4x

√−g

(gαβ∂αφ∂βφ + 1

6Rφ2

),

which describes a system consisting of a real scalarfield φ non-minimally coupled to gravity through thescalar curvatureR. Variations with respect toφ andgαβ lead to the equations

(2)

( − 1

6R

)φ = 0,

(3)Gαβ = 6φ−2ταβ(φ),

whereGαβ = Rαβ − 12gαβR is the Einstein tensor and

ταβ(φ)= −[∇αφ∇βφ − 1

2gαβ∇γ φ∇γ φ

]

(4)− 1

6(gαβ − ∇α∇β)φ

2,

with ∇α denoting the covariant derivative. Taking thetrace of (3) gives

(5)

( − 1

6R

)φ = 0,

which is consistent with Eq. (2). This is a consequenceof the conformal symmetry of action (1) under theconformal transformations

(6)

φ → φ = Ω−1(x)φ, gαβ → gαβ = Ω2(x)gαβ,

where the conformal factorΩ(x) is an arbitrary,positive and smooth function of space–time. Adding

a matter sourceSm independent ofφ to the action (1)in the form

(7)S = S[φ] + Sm,

yields the field equations

(8)

( − 1

6R

)φ = 0,

(9)Gαβ = 6φ−2[ταβ(φ)+ Tαβ],

where Tαβ is the matter energy–momentum tensor.The following algebraic requirement

(10)T = 0,

then emerges as a consequence of comparing the traceof (9) with (8) which implies that only traceless mattercan couple consistently to such gravity models.

We may break the conformal symmetry by addinga dimensional mass term−1

2

∫d4x

√−gµ2φ2, withµ being a constant mass parameter, to the action (7).This leads to the field equations

(11)

( − 1

6R +µ2

)φ = 0,

(12)Gαβ + 3µ2gαβ = 6φ−2[ταβ(φ)+ Tαβ]

and we obtain as a result of comparing the trace of (12)with (11)

(13)µ2φ2 = T .

Now, the basic input is to consider the invariancebreaking as a cosmological effect. This would meanthat one may takeµ−1 as the length scale character-izing the typical size of the universeR0 andT as theaverage density of the large scale distribution of mat-ter T ∼ MR−3

0 , whereM is the mass of the universe.This leads, as a consequence of (13) to the estimationof the constant background value ofφ

(14)φ−2 ∼ R−20

(M/R3

0

)−1 ∼ R0/M ∼ GN,

where the well-known empirical cosmological relationGNM/R0 ∼ 1 (due to Mach or Wheeler) has beenused. In order to well-justify the results we will ap-proximate the correspondenceφ−2 ∼ GN with φ−2 ≈8π6 GN . This estimation for the constant background

value of the scalar field is usually considered in Brans–Dicke type scalar–tensor gravity theories. Insertingthis background value ofφ into the field Eq. (11) leads


to the following set of Einstein equations

(15)Gαβ + 3µ2gαβ = 6φ−2Tαβ ≈ 8πGNTαβ,

with a correct coupling constant 8πGN , and a term3µ2 which is interpreted as the cosmological con-stantΛ of the order ofR−2

0 . The field Eq. (11) forφ contains no new information. This is because it isnot an independent equation, namely it is the trace ofEinstein equations (15). One may easily check that us-ing φ = 0 andT = µ2φ2, Eq. (11) and the trace ofEq. (15) result in the same equation as−1

6R+µ2 = 0.

3. 5D gravity and generalized conformalinvariance

Consider the 5D metric given by

(16)dS2 = gAB dxA dxB = Gφ2gαβ dxα dxβ + dl2

where the 5D line interval is written as the sum of a4D part relevant to scalar–tensor theory and an extrapart due to the 5th dimension. The capital Latin indicesA,B, . . . run over 0, 1, 2, 3, 4, Greek indicesα,β, . . .run over 0, 1, 2, 3, and five-dimensional quantities aredenoted by hats. A constantG is also introduced toleaveGφ2 dimensionless. We proceed keepinggαβ =gαβ(x

α, l) andφ = φ(xα) as in modern Kaluza–Kleintheory [4]. The metric is general, since we have onlyused 4 of the available 5 coordinate degree of freedomto set the electromagnetic potentials,g4α to zero. Oncewe use the 5th coordinate degree of freedom, wemay choose a background value for the fieldφ. Thenthe metric will be generally in the form of Randall–Sundrum warp metric [5], or the canonical metric inSTM theory [4].

The corresponding Christoffel symbols are ob-tained

Γ αβγ = Γ α

βγ + φ−1(δαγ ∇βφ + δαβ∇γ φ − gβγ ∇αφ),

Γ αβα = Γ α

βα + 4φ−1∇βφ,

Γ 4βγ = −1

2∂4gβγ ,

Γ α4α = 1

2gαβ∂4gαβ,

Γ αβ4 = 1

2gαδ∂4gδβ ,

(17)Γ 4α4 = Γ α

44 = Γ 444 = 0,

where gαβ = Gφ2gαβ . The 5D Ricci tensor can bewritten in terms of the 4D one plus other terms

Rαβ =Rαβ − 2φ−1∇α∇βφ + 4φ−2∇αφ∇βφ

− φ−2[φφ + ∇αφ∇αφ]gαβ

(18)

+ 1

2Gφ2

[gγ δ∂4gδα∂4gβγ − 1

2gλδ∂4gαβ∂4gλδ

− ∂24gαβ

].

The field equationsRAB = 0 then give

Rαβ = 2φ−1∇α∇βφ − 4φ−2∇αφ∇βφ

+ φ−2[φφ + ∇αφ∇αφ]gαβ

(19)

− 1

2Gφ2

[gγ δgδαgβγ − 1

2gλδgαβ gλδ − gαβ

],

R4α = ∇α

(kαβ − δαβk

) = 0

(20)with kαβ = 1

2gαδ ˙gδβ = 1

2gαδgδβ ,

(21)R44 = 2(k − 4kαβk

βα

) = 0,

where an overdot denotes differentiation with respectto 5th coordinatel (see [7]). Eq. (19) may lead toa set of 10 Einstein equations. Eq. (20) which havethe form of conservation law may also lead to aset of 4 Gauss–Codazzi equations for the extrinsiccurvaturekαβ of a 4D hypersurfaceΣl foliating in 5thdimension. Finally, Eq. (21) is one equation for thescalar combinations of the extrinsic curvature. TheRicci scalar for the space–time part is obtained bycontracting Eq. (19) with the metricgαβ

R = 6φ−1φ − 1

2Gφ2

(22)

×[gαβgγ δgδαgβγ − 1

2gαβgλδgαβ gλδ

− gαβ gαβ

].

Combining Eqs. (19) and (22) we obtain the Einstein-like equations with Einstein tensorGαβ in the left-hand side and some terms of scalar field together with4D metric and their covariant derivatives in the right-hand side as follows

(23)Gαβ = 6φ−2ταβ(φ)+ 1

2Gφ2

[Tαβ − 1

2T gαβ

],


where

ταβ(φ)= −2

3∇αφ∇βφ + 1

6gαβ∇γ φ∇γ φ

(24)− 1

3φφgαβ + 1

3φ∇α∇βφ

and

(25)Tαβ = gγ δgδαgβγ − 1

2gλδgαβ gλδ − gαβ .

It is easy to show that the tensorταβ in Eq. (24) isexactly the same one in Eq. (4). The field equationfor the scalar field may be obtained by contractingEq. (23) withgαβ or gαβ as

(26)

( − 1

6R + 1

12Gφ2T

)φ = 0.

We notice that Eq. (26) has a dynamical mass term112Gφ2T with the dimension of(mass)2. In the pres-ence of dimensional parameters, the conformal invari-ance can be established for a large class of theories [1]if the dimensional parameters are conformally trans-formed according to their dimensions. In this regard,Eq. (26), although modified by the mass term com-pared to (5), but is still invariant under the generalizedconformal transformations

φ → φ = Ω−1(x, l)φ,

(27)gαβ → gαβ = Ω2(x, l)gαβ.

This is simply because the 5D metric (16) is invari-ant under the above conformal transformations. Obvi-ously, the following combination

Gαβ ≡ Rαβ − 1

2gγ λRγ λgαβ = Rαβ − 1

2gγλRγ λgαβ

is invariant under (27) due to the invariance of themetric gαβ . Therefore, Eq. (23) which arises as aresult of Gαβ = Rαβ − 1

2gγλRγ λgαβ = 0 is invari-

ant under (27). And Eq. (26) as a consequence ofgαβGαβ = 0 or gαβGαβ = 0 is invariant under (27)as well, regardless of which metric is used to con-traction since the right-hand side is zero. Note that al-though the initiall-independent scalar fieldφ trans-forms to anl-dependent oneφ, but the l-dependentfunctionΩ−1(x, l) will not appear in the transformedscalar field equation because the metric also trans-forms in such a way that the functionΩ−1(x, l) isfactored out throughout the transformed equation ren-dering the initial scalar field equation. Therefore, by

pure 5D approach we are able to introduce a general-ized conformal invariant gravitational model definedby Eqs. (23), (26) and (27) subject to the subsidiaryEqs. (20) and (21).

4. Breakdown of conformal invariance in 5D

Now, we are in a position to compare Eqs. (26),(23) with the corresponding Eqs. (11), (12). By thiscomparison it turns out that we are able to revisit thebreakdown of conformal invariance in 4D by a 5Dapproach since we have derived the field Eqs. (26),(23) which can be identified with (11), (12) in thebroken phase of the conformal invariance in 4D.

To this end, we take a dimensional analysis. Thedimension ofTαβ or T will no doubt be(length)−2.Now, we assume the cosmological effectgαβ ∼ 1

R0which fixes a very slow variation ofgαβ over theabsolute cosmological scaleR0. This assumption leadsto the breakdown of the conformal invariance since itmeans that we have fixed our standard of length bythe scale of the universe and that (comparing Eq. (26)with (11) and usingGφ2 ∼ 1) T may be identifiedwith 12µ2 which is a constant mass term breakingthe conformal invariance. Now, we put the aboveidentification into the Einstein-like equation (23). Wethen have

(28)Gαβ + 3µ2gαβ = 6φ−2[ταβ(φ)+ 1

12φ2Tαβ

],

which, comparing with Eq. (13), leads to the identifi-cation

(29)Tαβ = 1

12φ2Tαβ,

which is the desired result in the context of inducedmatter theory since the matter energy–momentumtensorTαβ is dynamically induced by the scalar fieldφ and higher dimension, namelyTαβ . Taking the traceof (29) we find

(30)T = 1

12φ2T ,

and by takingT = 12µ2 we obtain the Eq. (13).Now, according to (30) we may discuss on the back-ground valueφ corresponding to the absolute cosmo-logical scaleR0. We have already fixedT (or µ2) bycosmological considerations, namelyT ∼ R−2

0 . This


was achieved by the 5th coordinate degree of free-dom throughgαβ ∼ R−1

0 . The 5th coordinate degreeof freedom accounts for the scalar field in the generalmetric (16). Thus, (see Eq. (14) and the following dis-cussion) we may take a background valueφ, using thiscoordinate degree of freedom, as

(31)φ−2 ≈ 8π

6GN,

which identifiesG with 8π6 GN such thatGφ2 ≈ 1.

This condition reduces the general metric (16) tothe one which is mathematically equivalent to thewarp metric of Randall–Sundrum [5], or the canonicalmetric of STM theory [4]. If we now insert thisconstant background valueφ−2 into Eq. (28) anduse (29) we find

(32)Gαβ + 3µ2gαβ ≈ 8πGNTαβ,

in which

(33)Tαβ = 1

16πG−1

N Tαβ.

Eq. (32) is the well-known Einstein equation in thebroken phase of the conformal invariance with acosmological constantΛ = 3µ2 and a coupling ofmatter to gravity,GN . The scalar field Eq. (26) isthe trace of Einstein equations, so its informationis already included in them (see the discussion inSection 2).

Now, the relevance of 5D approach manifests. It re-lates the current upper bound value of the cosmologi-cal constantΛ ∼ R−2

0 to a geometric phenomenon inwhich the cosmological constant is generated by thevery slow variation of 4D metric with respect to 5thdimension.2 Moreover, it unifies the origins of the mat-ter and the cosmological constant in that they appearas “two manifestations of higher-dimensional geome-try”.

2 In a recent work of Arkani-Hamed et al. [6], a small effectivecosmological constant is emerged from a large extra dimension in anon-compactified approach to 5D Kaluza–Klein gravity. Also, in acompactified model of Kaluza–Klein cosmology [12], smallness ofthe cosmological constant is related to smallness of the compactifieddimension. Therefore, it seems that the subject of cosmologicalconstant in higher-dimensional (at least in 5D) models is inevitablyinvolved with extra dimension.

The traditional Einstein equation (32) may alterna-tively be written in its pure geometric form

(34)Gαβ + 3µ2gαβ ≈ 1

2Tαβ,

in which the coupling constantGN is removed fromtheory. To say, although the Einstein tensorGαβ

couples to the matterTαβ by GN but the matteritself couples byG−1

N to the geometryTαβ (33) andso the couplingGN is removed. In this level, theappearance ofGN in the traditional Einstein equationseems to be a mathematical tool only for dimensionalconsistency. However, in the physical level Eqs. (32)and (33) exhibit an interesting phenomenon, withvarying GN , in that if GN decreases with timeleading to a weakly coupling of gravityGαβ to thematterTαβ (32), the matter itself will then be coupledstrongly to the hidden geometryTαβ (33). Regardingthe present small value ofGN we find an strongcoupling of matterTαβ to the higher-dimensionalgeometry Tαβ . This strong coupling may accountfor non-observability of the 5th dimension. In otherwords, the effects of the 5th dimension may be hiddenbehind this strong coupling and what we observeas the matter may bethe manifestation of a weakeffect of 5th dimension which is strengthened by astrong coupling G−1

N . This means that going backin time in GN varying scenarios we will encounterwith an eraGN ∼ 1 in whichTαβ may decouple fromTαβ leading to a naked geometry of 5th dimensionwithout the concept of matter, as indicated in Eq. (34).In conclusion it may be said that two Eqs. (32)and (33) definedual weak-strong regimes, in 5Dapproach to coupling constants, and that Eq. (33)defines adual-Einstein equation coupling matter tohigher dimension.

It is worth noting that the conformal invariancein 4D may be easily recovered in this 5D approachby restricting the 4D metricgαβ to be independentof 5th dimension (simply by assuming Kaluza–Kleincompactification condition for higher dimension). Therelevant field equations in this choice are

Rαβ = 2φ−1∇α∇βφ − 4φ−2∇αφ∇βφ

(35)+ φ−2[φφ + ∇αφ∇αφ]gαβ,

where by taking the trace of (35) and combining itwith (35) we obtain the conformal invariant Eqs. (2)and (3). The origin of this conformal invariance in 4D


turns out to be the invariance of the 4D part of 5Dmetric

(36)gAB

(xA

) =(Gφ2(xα)gαβ(x

α) 00 1

),

under the conformal transformations (6).

Conclusion

A key feature of any fundamental theory consis-tent with a given symmetry is that its breakdownwould lead to effects which can have various man-ifestations of physical importance. Therefore, in thecase of conformal symmetry in gravitational models,one would expect that the corresponding cosmologi-cal invariance breaking would have important effectsgenerating the gravitational coupling and the cosmo-logical constant. In this Letter we have introduced ageneralized conformally-invariant gravitational modelof 5D gravity theoryRAB = 0, with 4D part that is de-pendent on the extra dimension. The conformal invari-ance in 4D then becomes a special case when we takethe 4D metric to be independent of the extra dimen-sion. Moreover, we have shown that the cosmologicalbreakdown of conformal symmetry in a conformally-invariant gravitational model in 4D may be naturallyderived in this context if we assume a weak (cosmo-logical) dependence of the 4D metric on the higher di-mension and use the cosmological coincidence due toMach or Wheeler [9] to approximate the scalar field byits cosmological background value. This approach tothe issue of couplings and parameters in gravity leadsto a geometric interpretation for the small cosmologi-cal constantΛ. Moreover, a dual couplingG−1

N is in-troduced by which the matter couples strongly to thegeometric effects of higher dimension through adualEinstein equation, and non-observability of higher di-mension is then justified.

We also mention to the generality of the 5Dconformal invariance. In Deser’s model the conformalsymmetry is broken once aconstant mass term isintroduced. However, in 5D approach adynamicalmass term is appeared without breaking the conformalsymmetry. This generalized symmetry is broken whenwe take a preferred conformal frame by introducingan absolute length scale R0 through gαβ ∼ R−1

0 . Inother words, what we call the conformal invariance

in Deser’s model is not really a conformal invariance;it is just scale invariance which is a special case ofconformal invariance. This is because the dimensionalconstant mass term could not transform conformally.3

Some remarks should be reminded about the choiceof the metric (16), its stability and (possible) con-flict with observation. We have taken (16) becausewe would expect to obtain a 5D metric of Randall–Sundrum or canonical type which seems to be able todescribe the universe after the breakdown of confor-mal invariance. Referring to a previous Letter [13] inwhich a conformally related metric to (16) was takenand stabilized, quantum mechanically, close tol = 0constant hypersurface, we may consider a same stabi-lization procedure with the metric (16). This is becausethe key features in [13] to stabilize the non-compactdimension, close tol = 0, namely “a weak dependenceof gαβ(xα, l) on l” and “l-independence of the scalarfield” are admitted in the present paper. In this regard,we may live in a 5D universe but very close to the 4Dreference hypersurfacel = 0, so that the evolution ofthe scalar fieldφ (varyingGN ) or the slowly varyingmetric gαβ ∼ R−1

0 , does not affect practically this sta-bilized hypersurface.

As we have mentioned in the introduction, referringto many papers in membrane theory, the large non-compactified dimension are not in conflict with obser-vations. The same applies for STM theory if we con-fine our universe very close to thel = 0 reference hy-persurface in which case the size of 5th dimension hasno observable effect. In particular, we have argued inthis Letter that the strong couplingG−1

N may accountfor the non-observability of the 5th large dimension,so that the effects of this dimension may be hidden be-hind this strong coupling.

There is a natural question in the context of inducedmatter theory about its possible connection to quan-tum theory. This is because we can induce the mat-ter geometrically from the 5th dimension whereas weknow the matter has a underlying quantum structure.Therefore, it deserves to pay attention to this issue.First, it is well-known that the existence of a dimen-

3 The conformal invariance is more general than scale invariancewhich is used in Deser’s model. If scale invariance is characterizedby vanishing of the trace of the energy–momentum tensor, confor-mal invariance implies scale invariance in the absence of dimen-sional parameters in the theory.


sional gravitational constantGN is the main sourceof non-renormalizability of quantum gravity. On theother hand, the quantum theory approach to the tradi-tional Einstein equation suffers from the problem thatthe left-hand side is geometry and the right-hand sideis the matter. Eq. (34), however, as a pure geometricEinstein equation is free ofGN . Moreover, both sidesof this equation has the geometric structure. Perhaps, itis helpful to study the 4D quantum gravity in this puregeometric 5D approach. Second, in a study of 4D con-formal invariance in QFT in [3], the following equa-tion like our scalar field Eq. (26) is obtained(

− 1

6+ φ−2Sα

α ω)φ = 0

in which Sαα ω is the trace of the tensorSαβω

describing the distribution of matter due to localquantum effects. It is therefore very appealing to thinkabout how the higher-dimensional effects in 5D mayplay the role of quantum effects in 4D.

Acknowledgements

F. Darabi would like to thanks B. Mashhoon, W.N.Sajko and L. de Menezes for useful discussions. Theauthors also thank the referee for useful comments.This work has been supported by the Research Insti-tute for Fundamental Sciences, Tabriz, Iran.

References

[1] J.D. Bekenstein, A. Meisel, Phys. Rev. D 26 (1980) 1313.[2] S. Deser, Ann. Phys. 59 (1970) 248.[3] H. Salehi, Int. J. Theor. Phys. 37 (1998) 1253.

[4] P.S. Wesson, B. Mashhoon, H. Liu, W.N. Sajko, Phys. Lett.B 456 (1999) 34;H. Liu, B. Mashhoon, Ann. Phys. 4 (1995) 565;B. Mashhoon, H. Liu, P.S. Wesson, Phys. Lett. B 331 (1994)305;W.N. Sajko, Phys. Rev. D 60 (1999) 1040381;P.S. Wesson, Space, Time, Matter: Modern Kaluza–KleinTheory, World Scientific, Singapore, 1999;J.M. Overduin, P.S. Wesson, Phys. Rep. 283 (1997) 303;J.E. Lidsey, C. Romero, R. Tavakol, S. Rippl, Class. QuantumGrav. 14 (1997) 865.

[5] L. Randall, R. Sundrum, Phys. Rev. Lett. 83 (1999) 4690;L. Randall, R. Sundrum, Phys. Rev. Lett. 83 (1999) 3370.

[6] N. Arkani-Hamed, S. Dimopoulos, N. Kaloper, R. Sundrum,Phys. Lett. B 480 (2000) 193;N. Arkani-Hamed, S. Dimopoulos, G. Dvali, N. Kaloper, Phys.Rev. Lett. 84 (2000) 586;M. Gogberashvili, Mod. Phys. Lett. A 14 (1999) 2025;M. Gogberashvili, hep-ph/9908347.

[7] P.S. Wesson, J. Ponce de Leon, J. Math. Phys. 33 (1992) 3883;W.N. Sajko, P.S. Wesson, J. Math. Phys. 39 (1998) 2193.

[8] J.E. Campbell, A Course of Differential Geometry, ClarendonPress, Oxford, 1926.

[9] W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation, Freeman,San Francisco, 1973.

[10] D. Youm, Phys. Rev. D 62 (2000) 084002;P.S. Wesson, B. Mashhoon, H. Liu, W.N. Sajko, Phys. Lett.B 456 (1999) 34;P.S. Wesson, B. Mashhoon, H. Liu, Mod. Phys. Lett. A 12(1997) 2309;B. Mashhoon, Phys. Lett. B 331 (1994) 305;B. Mashhoon, Gen. Rel. Grav. 30 (1998) 555;D. Kalligas, P.S. Wesson, C.W.F. Everitt, Astrophys. J. 439(1994) 548;Y.M. Cho, D.H. Park, Gen. Rel. Grav. 23 (1991) 741.

[11] P. Horava, E. Witten, Nucl. Phys. B 475 (1996) 94;E. Witten, Nucl. Phys. B 471 (1996) 135;J.E. Lidsey, D. Wands, E.J. Copeland, Phys. Rep. 337 (2000)343.

[12] F. Darabi, H.R. Sepangi, Class. Quantum Grav. 16 (1999)1565.

[13] F. Darabi, W.N. Sajko, P.S. Wesson, Class. Quantum Grav. 17(2000) 4357.



Cosmology with self-adjusting vacuum energy density froma renormalization group fixed point

A. Bonannoa,b, M. Reuterc

a Osservatorio Astrofisico,Via S.Sofia 78, I-95123 Catania, Italyb INFN Sezione di Catania, Corso Italia 57, I-95129 Catania, Italy

c Institut für Physik, Universität Mainz, Staudingerweg 7, D-55099 Mainz, Germany

Received 27 June 2001; received in revised form 12 December 2001; accepted 28 December 2001

Editor: J. Frieman

Abstract

Cosmologies with a time dependent Newton constant and cosmological constant are investigated. The scale dependence ofG andΛ is governed by a set of renormalization group equations which is coupled to Einstein’s equation in a consistent way.The existence of an infrared attractive renormalization group fixed point is postulated, and the cosmological implications ofthis assumption are explored. It turns out that in the late Universe the vacuum energy density is automatically adjusted so asto equal precisely the matter energy density, and that the deceleration parameter approachesq = −1/4. This scenario mightexplain the data from recent observations of high redshift type Ia supernovae and the cosmic microwave background radiationwithout introducing a quintessence field. 2002 Published by Elsevier Science B.V.

1. Introduction

Recent astronomical observations of high redshifttype Ia supernovae performed by two groups [1–3] aswell as the power spectrum of the cosmic microwavebackground radiation obtained by the BOOMERANG[4] and MAXIMA-1 [5] experiments seem to indicatethat at present the Universe is in a state of acceleratedexpansion. If one analyzes these data within theFriedmann–Robertson–Walker (FRW) standard modelof cosmology their most natural interpretation is thatthe Universe is spatially flat and that the (baryonicplus dark) matter densityρ is about one third of thecritical densityρcrit. Most interestingly, the dominant

E-mail address: [email protected] (A. Bonanno).

contribution to the energy density is provided by thecosmological constantΛ. The vacuum energy density

(1.1)ρΛ ≡Λ/(8πG)is about twice as large asρ, i.e., about two thirds of thecritical density. WithΩM ≡ ρ/ρcrit, ΩΛ ≡ ρΛ/ρcritandΩtot ≡ΩM +ΩΛ:

(1.2)ΩM ≈ 1/3, ΩΛ ≈ 2/3, Ωtot ≈ 1.

This implies that the deceleration parameterq is ap-proximately−1/2. While originally the cosmologicalconstant problem [6] was related to the question whyΛ is so unnaturally small, the discovery of the impor-tant role played byρΛ has shifted the emphasis towardthe “coincidence problem”, the question whyρ andρΛhappen to be of the same order of magnitude preciselyat this very moment [7].

0370-2693/02/$ – see front matter 2002 Published by Elsevier Science B.V.PII: S0370-2693(01)01522-2


10 A. Bonanno, M. Reuter / Physics Letters B 527 (2002) 9–17

In an attempt at resolving this naturalness problemthe quintessence models [8,9] have been proposedin which the cosmological constant becomes a timedependent quantity [10]. Their dynamics is arrangedin such a way thatρΛ automatically adjusts itselfrelative to the value ofρ. In most of the quintessencemodels the vacuum energy densityρΛ is carried by ascalar field which has to be introduced on an ad hocbasis [11].

In this Letter we shall describe a different scenariowhich provides a natural explanation for the approx-imate equality ofρ andρΛ and for the smallness ofthe cosmological constant. We are going to set up avery general framework for cosmologies in which bothNewton’s constant and the cosmological constant aretime dependent. Within this framework, we shall for-mulate a single hypothesis whose consequences willbe analyzed and which will turn out to imply thatρandρΛ are approximately equal in the late Universe.In a nutshell, the hypothesis is that there exists an in-frared (IR) attractive fixed point for the renormaliza-tion group (RG) flow of the (dimensionless) Newtonconstant and cosmological constant, respectively. Ouranalysis is at purely phenomenological level in thesense that it is not necessary to know the details of thephysics which is responsible for this postulated fixedpoint.

Before we formulate our hypothesis in detail wefirst describe the kinematic framework we are goingto employ. It is the same framework which we used inRef. [12], henceforth referred to as (I), for an analysisof the quantum gravity effects in the early Universe(Planck era).

2. The kinematic framework

We consider homogeneous and isotropic cosmolo-gies described by a standard Robertson–Walker met-ric containing the scale factora(t) and the parame-ter K = 0,±1 which distinguishes the three typesof maximally symmetric 3-spaces of constant cos-mological timet . The dynamics is governed by Ein-stein’s equationRµν − 1

2gµνR = −Λgµν + 8πGTµνwith a conserved energy–momentum tensorTµν =diag(−ρ,p,p,p) for which we assume the equationof statep(t) = wρ(t) wherew > −1 is an arbitraryconstant. Now we perform a “RG improvement” [13–

15] of Einstein’s equation by replacingG → G(t)

andΛ→ Λ(t) where the time dependence ofG andΛ is such that the integrability of the field equationsis maintained. This leads to the following system ofequations:

(2.1a)

(a

a

)2

+ Ka2

= 1

3Λ+ 8π

3Gρ,

(2.1b)ρ + 3(1+w)(a

a

)ρ = 0,

(2.1c)Λ+ 8πρG= 0,

(2.1d)G(t)≡G(k = k(t)), Λ(t)≡Λ(

k = k(t)).Eq. (2.1a) is the standard Friedmann equation with atime dependentΛ andG, and Eq. (2.1b) expressesthe conservation ofT µν . Eq. (2.1c) is a novel consis-tency condition which is dictated by Bianchi’s iden-tity. It guarantees that the RHS of Einstein’s equa-tion has vanishing covariant divergence. The systemof Eqs. (2.1a)–(2.1c) has already been studied in theliterature [16,17]. Our crucial new ingredient [12]are the RG Eqs. (2.1d). Their meaning is as fol-lows.

We describe gravitational phenomena at a typicaldistance scale ≡ k−1 in terms of a scale dependenteffective actionΓk[gµν] which should be thought ofas a Wilsonian coarse grained free energy functional.The mass parameterk is a IR cutoff in the sense thatΓk encapsulates the effect of all metric fluctuationswith momenta larger thank while those with smallermomenta are not yet “integrated out”. When evaluatedat tree level,Γk describes all processes involving asingle characteristic momentumk with all loop effectsincluded.

In [13],Γk has been identified with the effective av-erage action [18] for Euclidean quantum gravity andan exact functional RG equation for thek-dependenceof Γk has been derived. Nonperturbative solutionswere obtained within the “Einstein–Hilbert trunca-tion” which assumesΓk to be of the form

(2.2)

Γk = (16πG(k)

)−1∫d4x

√g

−R(g)+ 2Λ(k).

The RG equations yield an explicit answer for thek-dependence of the running Newton constantG(k)and the running cosmological constantΛ(k). Withinthe Einstein–Hilbert approximation, the renormaliza-

A. Bonanno, M. Reuter / Physics Letters B 527 (2002) 9–17 11

tion effects are strong only ifk is close to the PlanckmassmPl. In (I) we argued that they are important foran understanding of the Planck era immediately afterthe big bang. However, there are indications [19] thatquantum Einstein gravity, because of its inherent IRdivergences, is subject to strong renormalization ef-fects also at very large distances. In cosmology thoseeffects would be relevant to the Universe at late times.It has been speculated that they might lead to a dy-namical relaxation ofΛ, thus solving the cosmologicalconstant problem [19]. An analysis of such IR effectsin the framework of the effective average action is notavailable yet. It would require truncations which aremuch more complicated than (2.2) and which containnonlocal invariants [20], for instance.

Nevertheless, in order to describe the idea of the“RG improvement” let us assume that we actuallyknow the functionsG(k) andΛ(k) for all values ofk,in particular fork→ 0, i.e., in the IR. The idea is toexpress the mass parameterk in terms of the physicallyrelevant cutoff scale. In (I) we argued that, in leadingorder, the correct cutoff identification in a Robertson–Walker spacetime is

(2.3)k(t)= ξ/t,whereξ > 0 is an a priori unknown constant. Inserting(2.3) intoG(k) andΛ(k)we obtain thetime dependentquantitiesG(t) ≡ G(k = ξ/t) and Λ(t) ≡ Λ(k =ξ/t). This is precisely what is meant by the last twoequations of the system (2.1), Eq. (2.1d). (See (I) forfurther details.)

The cutoff identification (2.3) applies in the case ofperfect homogeneity and isotropy for whichkcosmo≡k(t) = ξ/t is the only relevant scale. Allowing for(large, nonlinear) density perturbationsδρ(x, t) of atypical wave lengthλpert we introduce a new scalekpert = 2π/λpert into the problem. Similarly, immers-ing a localized matter distribution (a massive body) oftotal massM into the cosmological fluid gives rise tothe scalekM =M. In the situations of interest the lattertwo mass scales are much larger than the cosmologicalone:kpert, kM kcosmo.

In a situation with more than one possible physicalcutoff scale the general theory of the effective averageaction [18] implies that the relevant action isΓk wherek is the largest one of the various competing scales.Hence in order to describe the physics of densityperturbations of the localized matter distribution one

has to useΓk at k = kpert and k = kM, respectively,rather than atk = kcosmo. In this case one needs toknow G(k) andΛ(k) for k nearkpert, kM kcosmo,i.e., in a very different regime.

In the present paper we are interested only in thelarge scale dynamics of the Universe. In the next sec-tion we are going to formulate a hypothesis on the run-ning of G(k), Λ(k) for k nearkcosmo. The only as-sumption which we make about the RG flow ofG andΛ at “non-cosmological” scalesk kcosmo (e.g. fork ≈ kpert, kM) is that at those scales thek-dependenceis very weak or zero so that standard gravity is recov-ered at sub-cosmological scales.

At this point we emphasize that it is not impor-tant for the present discussion which physical mech-anism actually causes thek- or t-dependence ofGandΛ. In particular, we also cover the possibility ofan entirelyclassical origin of this running. In fact,it has been pointed out [21,22] thatG andΛ natu-rally acquire a scale dependence if one starts from adensity distribution which is inhomogeneous at smalldistances and then performs spatial averaging over3-volumes of increasing linear extension= k−1. Theclassical dynamics of the averaged quantities leadsto a nontrivial RG flow ofG andΛ. Since the Uni-verse is certainly not homogeneous at small distancescales, knowledge of this RG flow is important ifone wants to parametrize observational data obtainedat those scales in terms of a homogeneous FRWmodel; it was argued that this classical scale depen-dence might resolve the controversy about the value ofthe Hubble constant [21]. Another intriguing result isthat, after spatial averaging, the backreaction of longwavelength scalar and tensor cosmological perturba-tions amounts to an effective negative energy densitywhich counteracts any pre-existing cosmological con-stant [22].

3. The fixed point hypothesis

The above remarks complete our motivation ofthe system of Eqs. (2.1). We shall now formulatea hypothesis about the RG behavior ofG and Λwhose dynamical origin is left open. Introducingdimensionless quantitiesg(k) ≡ k2G(k) and λ(k) ≡Λ(k)/k2 the hypothesis is that fork→ 0 bothg andλrun into an IR attractive non-Gaussian fixed point, i.e.,


that for a wide range of initial conditions

(3.1)limk→0g(k)= gIR∗ , lim

k→0λ(k)= λIR∗ ,

wheregIR∗ andλIR∗ are strictly positive. While there areencouraging indications pointing toward the existenceof this fixed point [19], a rigorous proof would be aformidable task, however, probably comparable to aproof of confinement in QCD. In the following weexplore the cosmological implications of (3.1) which,as we shall see, provide further evidence for the fixedpoint hypothesis from the phenomenological side.

The postulated fixed point is the IR counterpart ofthe UV attractive non-Gaussian fixed point which isknown to exist in the Einstein–Hilbert truncation ofpure quantum gravity [13,23,24]. For a large class oftrajectories [25],

(3.2)limk→∞g(k)= g

UV∗ , limk→∞λ(k)= λ

UV∗ .

More generally, we assume that the exact cosmologi-cally relevant RG trajectory in(g,λ)-space smoothlyinterpolates between(gUV∗ , λUV∗ ) for k → ∞ and(gIR∗ , λIR∗ ) for k → 0. The UV fixed point is impor-tant for the very early Universe(t → 0) while theIR fixed point determines the cosmology at late times(t→ ∞).

It is reassuring to note that a similar crossover be-tween two nontrivial RG fixed points has actually beenshown to exist in 2-dimensional Liouville quantumgravity [26]. Its RG trajectory connects two conformalfield theories with central charges 25− c and 26− c,respectively, wherec is the central charge of the mattersystem.

In the vicinity of either of the two fixed points theevolution of the dimensionfulG andΛ is approxi-mately given by

(3.3)G(k)= g∗k2, Λ(k)= λ∗k2.

Here and in the following the fixed point values are de-notedg∗ andλ∗ if the corresponding formula is validboth at the UV and at the IR fixed point. From (3.3)with (2.3) we obtain the time dependent Newton con-stant and cosmological constant:

(3.4)G(t)= g∗ξ−2t2, Λ(t)= λ∗ξ2

t2.

The power laws (3.4) are valid both fort 0 and, withdifferent coefficients, fort → ∞. By assumption, the

time dependence ofG andΛ at intermediate times isgiven by smooth functionsG(t) andΛ(t) which inter-polate between the UV and IR power laws. If we usethese functionsG(t) andΛ(t) in the coupled system(2.1), its solution gives us the scale factora(t) and thedensityρ(t) of the “RG improved cosmology”.

To be precise, our hypothesis about the RG tra-jectory k → (G(k),Λ(k)) consists of two parts. Fork kcosmo we assume the validity of the IR fixedpoint behavior (3.3). Fork kcosmo the assumptionis thatG andΛ depend onk only extremely weakly orarek-independent. In this manner we recover standardgravity withG,Λ= const at the length scales smallerthan the cosmological scale∝ t . In particularG andΛ are essentially constant atkpert andkM so that thedynamics of localized matter distribution remains un-changed and there is no conflict with the classical testsof general relativity. Stated differently, the hypothesisis that the nontrivial running is due to quantum fluctu-ations with momenta betweenkcosmoandkmax where1/kmax is the length scale characterizing the largestlocalized structures in the Universe of which we knowfor sure that standard gravity applies.

4. Cosmological solutions in the fixed point regime

It is important to note that the system (2.1) isactually overdetermined: it consists of 5 equationsfor the 4 unknownsa,ρ,G and Λ. This leads tonontrivial consistency conditions for admissible RGtrajectories and cutoff identifications. (See (I) for adetailed discussion.) In (I) we showed that ifG(t) andΛ(t) are given by (3.4) the system (2.1) has indeeda consistent solution providedξ assumes a specificvalue. Quite generally the consistency conditions havethe very welcome feature of fixing the ambiguities inthe modeling of the cutoff (hereξ ) to some extent.

For the case of a spatially flat Universe(K = 0) theconsistency condition reads

(4.1)ξ2 = 8

3(1+w)2λ∗ .If it is satisfied, the system (2.1) with (3.4) has thefollowing almost unique solution

(4.2a)

a(t)=[(

3

8

)2

(1+w)4g∗λ∗M]1/(3+3w)

t4/(3+3w),


(4.2b)ρ(t)= 8

9π(1+w)4g∗λ∗1

t4,

(4.2c)G(t)= 3

8(1+w)2g∗λ∗t2,

(4.2d)Λ(t)= 8

3(1+w)21

t2.

Apart from the parameterw and the productg∗λ∗,the solution (4.2) depends only on a single constantof integration, M, whose value affects only theoverall scale ofa(t). Numerically it equals 8πρ(t)×[a(t)]3+3w ≡ M which, like in standard cosmology, isa conserved quantity for the system (2.1).

The solution (4.2) has several very interesting andattractive features. Introducing the critical density

(4.3)ρcrit(t)≡ 3

8πG(t)

(a

a

)2

,

we find for any value ofw, g∗λ∗, andM thatρcrit(t)=2ρ(t) andρΛ(t)= ρ(t). Hence

(4.4)ρ = ρΛ = 1

2ρcrit.

Thus the total energy densityρtot ≡ ρ + ρΛ equalsprecisely the critical one:ρtot(t) = ρcrit(t). This lat-ter equality does not come as a surprise because alsothe RG improved Friedmann equation can be broughtto the form

(4.5)K = a2[ρtot/ρcrit − 1]so thatρtot = ρcrit holds true for any solution withK = 0. On the other hand, the exact equality ofthe matter energy densityρ and the vacuum en-ergy densityρΛ is a nontrivial prediction of thefixed point solution. In terms of the relative densi-ties,

(4.6)ΩM =ΩΛ = 1

2, Ωtot = 1.

Also the Hubble parameter of the solution (4.2)

(4.7)H ≡ aa

= 4

3+ 3w

1

t

and its deceleration parameter

(4.8)q ≡ −a aa2

= 3w− 1

4are independent ofg∗, λ∗ andM. It can be shown thatthe standard formula forq in terms of the relative den-sities continues to be correct for the improved system

(2.1) with an arbitrary RG solution (2.1d):

(4.9)q = 1

2(3w+ 1)ΩM −ΩΛ.

Clearly (4.9) is satisfied by (4.8) with (4.6).Another interesting feature of the fixed point solu-

tion is that it yields a universal, time independent valueof the “Machian” quantityρGt2 [27]

(4.10)ρ(t)G(t)t2 = 1

3π(1+w)2 .

Since the 1-parameter family of cosmologies (withparameterM) described by (4.2) is the most generalsolution of the system (2.1) with the fixed point run-ning (3.4) we conclude thatevery complete solutionof (2.1), valid for all t ∈ (0,∞), approaches one ofthe solutions (4.2) either fort 0 or for t → ∞, de-pending on whether the fixed point is UV or IR forthe trajectory considered. Thus the fixed point solu-tion (4.2) is an attractor in the space of the functions(a,ρ,G,Λ).

Note that the productG(t)Λ(t) = G(k)Λ(k) =g∗λ∗ is constant in the vicinity of any fixed point.Its actual value is characteristic of this fixed point.While, for pure gravity,gUV∗ λUV∗ = O(1) at the UVfixed point of [12], the hypothetical IR fixed pointof the coupled gravity-matter system hasgIR∗ λIR∗ =O(10−120). It is important to understand that thesmallness of this number does not pose any finetuningproblem as in the standard situation. In fact, in ourapproach bothgUV∗ λUV∗ and gIR∗ λIR∗ are fixed andwell-defined numbers which, at least in principle, canbe computed from the RG equation. However, apartfrom being a difficult task technically, their actualdetermination is possible only once we know thecomplete system of all matter fields in the Universe.The number 10−120 reflects specific properties of thismatter system coupled to gravity rather than an initialcondition.

In the following we focus on the possibility of an IRattractive fixed point which governs the cosmologicalevolution for t → ∞. Whether or not there exists inaddition an UV fixed point approached fort 0 isnot important in this context. Assuming the existenceof the IR fixed point and the validity of Eqs. (2.1) weare led to conclude that the late Universe, for whichthe RG trajectory is already sufficiently close to thefixed point, is described by the power laws (4.2). This


leads to the unambiguous prediction thatΩM =ΩΛ =1/2 for every value ofw. Moreover, if we make theadditional assumption that the late Universe is matterdominated (w = 0), Eq. (4.8) yieldsa ∝ t4/3 with thedeceleration parameter−1/4. Hence, near the fixedpoint,

(4.11)ΩM =ΩΛ = 1

2, q = −1

4(w = 0).

Without any further input from the RG equations(which would require detailed knowledge of the mattersector) we cannot assess at which timetFP the fixedpoint behavior sets in. AttFP a generic solution, arisingfrom arbitrary initial conditions, starts looking like(4.2). However, it is very intriguing that the prediction(4.11), valid for t > tFP, is quite close to the values(1.2) favored by the recent observations.1 In particular,the fixed point structure provides a natural explanationfor the mysterious equality (or approximate equality)of ρ andρΛ. This success supports the idea thatthepresent-day Universe is in the, or at least close to theIR fixed point regime.

The deviation of the observed values (1.2) fromΩM = ΩΛ = 1/2 could be due to the fact that thefixed point behavior is not fully developed yet sothat the Universe still has some way to go beforethe finer quantitative details of the solution (4.2)are realized. However, given the large observationaluncertainties [1] it is also well possible that moreprecise observations will lead to modified values ofΩM andΩΛ which are closer toΩM =ΩΛ = 1/2.

A further testable prediction of the fixed pointhypothesis is the time variation of Newton’s constant.From (4.2) we obtain

(4.12)G

G= 2

t= 1

2(3+ 3w)H(t).

The experimental upper bound from laboratory andSolar system experiments for the present-day valueof this quantity [28] is of the order of|G/G| (1011 yr)−1. Hence even the technology availabletoday is not very far away from being able to verify orfalsify (4.12). One should bear in mind, however, thattheG in Eq. (4.12) refers to a different length scale

1 Note that the values (1.2) are still afflicted with large error bars[1]. In the (ΩM ,ΩΛ)-plane, the values (4.11) lie within the ellipsecorresponding to the 2σ confidence region.

than the one measured in Solar system experiments,say.

In this context it is interesting to remark that re-cently a Brans–Dicke theory with a quadratic self-coupling of the Brans–Dicke field has been con-structed [29] which admits a solution very similar toour fixed point solution (4.2) and which predicts thesame time dependence of Newton’s constant.

Up to now we discussed the spatially flat Universeonly. The(K = 0)-solution (4.2) exists for every valueof the parameterw. The situation is different when wenow look for solutions withK = ±1 correspondingto spatially curved Universes. In (I) we have shownthat consistent solutions to the system (2.1), (3.4)with K = +1 or K = −1 exist only if w = +1/3,i.e., for a radiation dominated Universe. Assuming thevalidity of (2.1) and of the fixed point hypothesis, andexcluding the possibility ofw = +1/3 for t → ∞,we see that the Universe can fall into the basin ofattraction induced by the IR fixed point only if itis spatially flat, i.e., ifK = 0. By Eq. (4.5) this isequivalent toρtot = ρcrit, as it would be in standardcosmology.

It is important to stress that the fixed-point scenariosets in at scales that are much larger than the character-istic scales of local gravitational interactions. One can-not expect that any simple scaling law, or cutoff iden-tification, can be used in a general gravitating systemwhen we are probably far from any fixed point in theG andΛ evolution. Instead we assume thatG andΛcan be RG evolved only through the large scale evolu-tion of the Universe, which would then provide a nat-ural scaling law and a meaningful cutoff identificationby means of the cosmological time. In fact at muchsmaller scales than galaxy cluster scales neither wecan establish the flow equations, nor we could guessa meaningful identification of the cutoff since several“crossover” regions can be present, which would driveus away from the fixed point law and would make anyidentification of the cutoff problematic.

It is nevertheless possible to describe the evolu-tion of weak, localized density perturbationsδρ ρ

within our framework in a consistent way. The reasonis that, at the linearized level, the effective cutoff scaleis still given bykcosmorather than the scales associateswith δρ.

Let us consider a “fundamental observer” describ-ing the cosmological fluid flow lines. Its 4-velocity is


uµ = dxµ/dτ , uµuµ = −1 whereτ is the proper timealong the fluid flow lines. The projection tensor in thetangent 3-space orthogonal touµ is hµν = gµν+uµuνwith hµνhνσ = hµσ and hµνuν = 0. The energy–momentum tensor then readsT µν = ρuµuν + phµν ,whereρ = ρ + δρ(xµ) andp = p + δp(xµ) beingρandp the pressure and the density of the unperturbedUniverse. From the Bianchi identities and the conser-vation law∇νT µν = 0 we have the following propa-gation equations

(4.13)uν(∇νΛ+ 8πρ∇νG)= 0,

(4.14)hµν(−∇νΛ+ 8πp∇νG)= 0,

for the uµ direction and for the tangent orthogonal3-space, respectively, together with

(4.15)GΛ= λ∗g∗coming from the fixed point behavior (3.3). From(4.14) we thus see that in the late Universe, which isof interest for us, whenp = 0 is a good equation ofstate, there are no space gradients of the cosmologicalconstant up to first order perturbation theory. In factsince δp, δρ and the space gradients ofG and Λare assumed to be of the same order,δphνµ∇νGis of second order. Therefore, by differentiation of(4.15) and subsequent projection onto the 3-space,one concludes that also the space gradients ofG arenegligible in this approximation. This result impliesthat the description of local gravitational interactionsdepends only on the global, large scale, time evolutionofG andΛ and it does not introduce additional effectscoming from the space gradients ofG and Λ thatcould in principle appear in the description of localdeviations from homogeneity. One can then discussthe standard scenario of structure formation with thelarge scale structure evolution provided by the solution(4.2).

We also emphasize that the standard experimentalvalue of Newton’s constant,Gexp, does not coincidewith the valueG(k = ξ/t0) which is relevant forcosmology today, i.e., fort = t0. Gexp is measured(today) atkexp ∝ −1 where the length ≡ sol is atypical solar system length scale, say. Thus, in terms ofthe running Newton constant,Gexp =G(k = ξ ′/sol),sincesol t0, and since in presence of several scales

the relevant cutoff is always the larger one.2 It is onlythe cosmological quantityG(k = ξ/t) which grows∝ t2 in the fixed point regime, notGexp. This remarkentails that at2-growth of the cosmological Newtonconstant in the recent past does not ruin the predictionsabout primordial nucleosynthesis which requires thatG(k = ξ/tnucl) coincides withGexp rather precisely.In fact, at the timet = tnucl of nucleosynthesis thecosmological Newton constant was indeedG(k =ξ/tnucl) ≈ Gexp sincectnucl and sol are of the sameorder of magnitude (a few light minutes).

5. Discussion and conclusion

In this Letter we modified Einstein’s equation for aRobertson–Walker spacetime by allowing for a scale-,and hence, time-dependent Newton’s constant andcosmological constant. The scale dependence ofG

andΛ follows from a renormalization group whichcould be of either classical or quantum origin. We pos-tulated that the RG flow at large distances is governedby an IR fixed point and we investigated the cosmolog-ical implications of this assumption. It turned out that,in the fixed point regime, the vacuum energy densityρΛ equals precisely the matter densityρ and that theydecrease proportional to 1/t4, while Newton’s con-stant increases∝ t2. Assuming that the present Uni-verse is in that regime, this scenario leads to a nat-ural resolution of the coincidence problem (“Why isρ/ρΛ = O(1) today?”) and of the cosmological con-stant problem in its original form (“Why isΛ sosmall?”). It predicts that the universe is spatially flat.

Obviously cosmologies of the type found here arevery attractive from the phenomenological point ofview. This success provides a strong motivation forfurther attempts at actually proving the existence ofthe postulated IR fixed point and the validity of theimproved system of cosmological evolution equations,Eqs. (2.1).

In the present paper we assumed the existenceof a cosmologically relevant IR fixed point, whilein (I) we investigated the consequences of a UVfixed point for the Planck era directly after the bigbang. The UV fixed point has been shown to ex-

2 See Ref. [14] for a detailed discussion of this point.


ist in (the Einstein–Hilbert truncation of) pure quan-tum gravity. The assumption that the matter contentsof the Universe is such that there exists both theUV and the IR fixed point leads to a particularlysymmetric cosmological scenario: the Universe be-gins and ends at two different attractors, attractive fort 0 andt → ∞, respectively, and its evolution be-tween them is a kind of crossover between two fixedpoints.

Before closing let us see what happens if we relaxour hypothesis to some extent. Up to now we assumedthe IR fixed point to be attractive in all directionsin the space of coupling constants. It might be thatactually there are also unstable directions so that fort → ∞ the Universe is eventually driven away fromit. Nevertheless, if it stays for a sufficiently long timeclose to the fixed point, the solution (4.2) might still bea rather accurate description of the present Universeeven though its ultimate fate fort → ∞ cannot bepredicted then.

As a more radical step, let us give up our idea thatthe t-dependence ofG(t) andΛ(t) arises from someRG trajectoryG(k) andΛ(k) by identifyingk ≡ k(t).We retain however the three differential Eqs. (2.1a),(2.1b), and (2.1c). The system (2.1a)–(2.1c) without(2.1d) is underdetermined so that an additional con-dition on a,ρ,G andΛ may be imposed. Withoutproviding a physical explanation, it was assumed inRef. [17] that Newton’s constant varies according toa power lawG ∝ tn with an arbitrary, not necessar-ily integer exponentn. With this additional conditionthe system (2.1a)–(2.1c) withK = 0 has the following2-parameter family of solutions (the parameters areMandC > 0):

(5.1a)

a(t)=[

3(1+w)22(n+ 2)

MC]1/(3+3w)

t(n+2)/(3+3w),

(5.1b)ρ(t)= (n+ 2)

12π(1+w)2C1

tn+2,

(5.1c)G(t)= Ctn,(5.1d)Λ(t)= n(n+ 2)

3(1+w)21

t2.

The solution (4.2) resulting from the fixed pointhypothesis corresponds to the special casen = 2 andC = 3(1+w)2g∗λ∗/8. Note that the exponentn= 2 isan unambiguous prediction of the fixed point scenario.

It is obtained even if we use an identification ofk interms oft which is different fromk = ξ/t . The reasonis that (3.3) impliesGΛ = g∗λ∗ = const for everyfunction k = k(t), but according to (5.1) the productGΛ is constant only ifn= 2. For the cosmology (5.1)one easily computes

ΩM = 2

n+ 2, ΩΛ = n

n+ 2,

(5.2)Ωtot = 1, q = 1+ 3w− nn+ 2

.

For n = 2, ρ andρΛ are no longer equal:ΩΛ/ΩM =n/2. It is amusing to note that settingn= 4 andw = 0yields

ΩM = 1

3, ΩΛ = 2

3, q = −1

2(5.3)(n= 4,w= 0)

which equals quite precisely the values (1.2) favoredby the present experimental data.

Once more we see that a more accurate experimen-tal determination ofΩM andΩΛ is highly desirable. Incase the values ultimately stabilize nearΩM =ΩΛ =1/2 this would be an important step toward confirm-ing the fixed point scenario. If instead they remainclose to their present values (1.2) it might be worth-while to reconsider the more general cosmologies ofthe type (5.1) withn = 4. However, at present thereseems to be no theoretical argument which would sin-gle outG ∝ tn and n = 4. While the classical cos-mological tests related to the early Universe (nucle-osynthesis) are most probably insensitive to the mod-ifications caused by the IR fixed point, measurementsof G/G at different length scales are another impor-tant test for the validity of the fixed point hypothe-sis.

Acknowledgements

M.R. would like to thank the Department of Theo-retical Physics, University of Catania, the Departmentof Theoretical Physics, University of Trieste, and theAstrophysical Observatory of Catania for their hos-pitality while this work was in progress. He also ac-knowledges the financial support by INFN, MURST,and by a NATO traveling grant.


References

[1] S. Perlmutter et al., Astrophys. J. 517 (1999) 565.[2] A. Riess et al., Astron. J. 117 (1999) 707.[3] For a review, see A. Riess, astro-ph/0005229.[4] P. de Barnardis et al., Nature 404 (2000) 955;

A.E. Lange et al., astro-ph/0005004.[5] S. Hanany et al., astro-ph/0005123.[6] S. Weinberg, Rev. Mod. Phys. 61 (1989) 1, astro-ph/9610044.[7] V. Sahni, A. Starobinsky, astro-ph/9904398;

N. Straumann, astro-ph/9908342.[8] I. Zlatev, L. Wang, P.J. Steinhardt, Phys. Rev. Lett. 82 (1998)

896;C. Armendariz-Picon, V. Mukhanov, P.J. Steinhardt, Phys. Rev.Lett. 85 (2000) 4438;C. Armendariz-Picon, V. Mukhanov, P.J. Steinhardt, Phys. Rev.D 63 (2001) 103510;P.J. Steinhardt, L. Wang, I. Zlatev, Phys. Rev. D 59 (1999)123504;P. Brax, J. Martin, A. Riazuelo, Phys. Rev. D 62 (2000)103505.

[9] A. Hebecker, C. Wetterich, hep-ph/0003287;A. Hebecker, C. Wetterich, hep-ph/0008205.

[10] M. Reuter, C. Wetterich, Phys. Lett. B 188 (1987) 38.[11] For a recent review, see R. Brandenberger, hep-ph/9910410.[12] A. Bonanno, M. Reuter, hep-th/0106133.[13] M. Reuter, Phys. Rev. D 57 (1998) 971, hep-th/9605030;

For a brief introduction, see M. Reuter, in: Annual Report2000 of the International School in Physics and Mathematics,Tbilisi, Georgia, hep-th/0012069.

[14] A. Bonanno, M. Reuter, Phys. Rev. D 62 (2000) 043008, hep-th/0002196.

[15] A. Bonanno, M. Reuter, Phys. Rev. D 60 (1999) 084011, gr-qc/9811026.

[16] A. Beesham, Nuovo Cimento 96B (1986) 17;A. Beesham, Int. J. Theor. Phys. 25 (1986) 1295;A.-M.M. Abdel-Rahman, Gen. Rel. Grav. 22 (1990) 655;

M.S. Berman, Phys. Rev. D 43 (1991) 1075;M.S. Berman, Gen. Rel. Grav. 23 (1991) 465;R.F. Sistero, Gen. Rel. Grav. 23 (1991) 1265;T. Singh, A. Beesham, Gen. Rel. Grav. 32 (2000) 607;A. Arbab, A. Beesham, Gen. Rel. Grav. 32 (2000) 615;A. Arbab, gr-qc/9909044.

[17] D. Kalligas, P. Wesson, C.W.F. Everitt, Gen. Rel. Grav. 24(1992) 351.

[18] For a recent review, see J. Berges, N. Tetradis, C. Wetterich,hep-th/0005122.

[19] N.C. Tsamis, R.P. Woodard, Phys. Lett. B 301 (1993) 351;N.C. Tsamis, R.P. Woodard, Ann. Phys. (N.Y.) 238 (1995) 1;N.C. Tsamis, R.P. Woodard, Nucl. Phys. B 474 (1996) 235;I. Antoniadis, E. Mottola, Phys. Rev. D 45 (1992) 2013.

[20] C. Wetterich, Gen. Rel. Grav. 30 (1998) 159.[21] M. Carfora, K. Piotrkowska, Phys. Rev. D 53 (1995) 4393, and

references therein.[22] V. Mukhanov, L.R.W. Abramo, R. Brandenberger, Phys. Rev.

Lett. 78 (1997) 1624;

L.R.W. Abramo, R. Brandenberger, V. Mukhanov, Phys. Rev.D 56 (1997) 3248;R. Brandenberger, hep-th/0004016;W. Unruh, astro-ph/9802323.

[23] W. Souma, Prog. Theor. Phys. 102 (1999) 181.[24] O. Lauscher, M. Reuter, hep-th/0108049, and in preparation.[25] M. Reuter, F. Saueressig, in preparation.[26] M. Reuter, C. Wetterich, Nucl. Phys. B 506 (1997) 483;

A. Chamseddine, M. Reuter, Nucl. Phys. B 317 (1989) 757.[27] H. Bondi, J. Samuel, gr-qc/9607009;

For a comprehensive account, see J. Barbour, H. Pfister (Eds.),Mach’s Principle, Birkhäuser, Boston, 1995.

[28] G.T. Gillies, Rep. Prog. Phys. 60 (1997) 151.[29] O. Bertolami, P.J. Martins, gr-qc/9910056;

O. Bertolami, J. Mourao, J. Perez-Mercader, Phys. Lett. B 311(1993) 27;O. Bertolami, J. Garcia-Bellido, Int. J. Mod. Phys. D 5 (1996)363.



Late decaying axino as CDM and its lifetime bound

Hang Bae Kima, Jihn E. Kimb

a Department of Physics, Lancaster University, Lancaster LA1 4YB, UKb Department of Physics and Center for Theoretical Physics, Seoul National University, Seoul 151-747, South Korea

Received 13 November 2001; received in revised form 19 December 2001; accepted 22 December 2001

Editor: T. Yanagida

Abstract

The axino with mass in the GeV region can be cold dark matter (CDM) in the galactic halo. However, if R-parity is broken, forexample by the bilinear termsµα , then axino (a) can decay toν+γ . In this case, the most stringent bound on the axino lifetimecomes from the diffuse photon background and we obtain that the axino lifetime should be greater than 3.9×1024Ωah [s] whichamounts to a very small bilinear R-parity violation, i.e.,µα < 1 keV. This invalidates the atmospheric neutrino mass generationthrough bilinear R-violating terms within the context of axino CDM. 2002 Elsevier Science B.V. All rights reserved.

PACS: 95.35.+d; 98.35.Gi; 11.30.Pb

Keywords: Dark matter; Axino; R-parity violation; Detection of CDM

The observed rotation curve of the halo stars [1]requires to fill the galactic halo with cold dark matter(CDM) such as axion [2–4], the lightest supersymmet-ric particle (LSP) [5], the axino LSP [6,7], and wim-plzilla [8]. The axion is motivated from the solution ofthe strong CP problema la Peccei and Quinn (PQ) [9],which is required to be very light [10]. The LSP is mo-tivated fromR-parity conservation in the supersym-metric solution of the gauge hierarchy problem, whereR-parity is defined as(−1)3B+L−2S . For the LSP to beCDM, its mass is around 100 GeV [11]. The wimpzillais 1012-13 GeV stable particle.

The possibility of axino dark matter, which is ourinterest in this paper, has been suggested from time to

E-mail addresses: [email protected] (H.B. Kim),[email protected] (J.E. Kim).

time as the hot DM, the warm DM and the cold DMpossibilities [12]. Theoretically, it arises in SUSY the-ories with a spontaneously broken PQ symmetry. Insupergravity it is necessary to have a period of inflationto dilute the very weakly interacting gravitinos. But itis thermally produced in significant numbers even af-ter the inflation, which requires a low reheating tem-perature, 109 GeV [13]. A similar study for axinorequires a much lower reheating temperature of orderTeV in which case O (GeV) axino mass is allowed.Then axino can be a DM candidate [6,7]. Here, we fo-cus our attention on this CDM axino. Since there is noreliable constraint on the axino mass [7,14] , the axinois assumed to be the LSP. Then, the most importantquestion is whether the LSP (axino) is absolutely sta-ble (due to R-parity conservation) or unstable. Sincethe stable axino case has been extensively studied [7],we restrict our attention on the R-violating case.



H.B. Kim, J.E. Kim / Physics Letters B 527 (2002) 18–22 19

If a CDM candidate is proposed, it is of utmostimportance to devise a scheme to prove its existenceexperimentally as in the cases of the very light axion[15,16] and the LSP [5,11]. The axino CDM lacksthis kind of possible detection mechanism due toits extremely weak interaction strength if R-parity isconserved. However, if R-parity is broken, it may bepossible to detect its decay products. Therefore, it isworthwhile to study possible decay mechanisms ofCDM axino. The detection possibility by the axinodecay relies on the axino lifetime around> 1013 s afterthe galaxy formation era.

The R-parity conservation seems to be an attractiveproposal for proton stability. However, R-parity isnot dictated from any deep theoretical principle. Forexample, if there exists anSU(3) × SU(2) × U(1)singlet superfieldN which is needed from the see-sawmechanismLαNH2, whereLα is the lepton doublet oftheα-th family andHi is the Higgs doublet (i = 1,2),we may write a renormalizable superpotential,WN =mN2 + fN3. From the coupling to the observablesector fields for the Dirac mass coupling,N is requiredto carry −1 unit of the R-parity quantum number.But this R-parity is broken by theN3 term. Also,Nobtains a vacuum expectation value−2m/3f whichis too large for neutrino phenomenology. Namely,R-parity conservation is not guaranteeda priori atthe SM level. Thus, if a singletN is introduced,one should impose (at least an approximate) R-parityconservation, namely we should imposef = 0 in thisexample. If R-parity is broken, it must be done so veryweakly. In this paper, for simplicity of the discussion,we restrict our attention on the bilinear R-violatingterms,

(1)µαLαH2,

whereα = 1,2,3.µα is bounded by the eV order neu-trino mass. Without an explicit statement, this boundapplies to the heaviest SM neutrino, presumably thetau neutrino. With the R-parity violation, theντ massarises from the see-saw type diagram with an inter-mediate zino line with two insertions of the R-parityviolating 〈ν〉. Also, it can get a contribution from theintermediateH 0

2 line with two insertions ofµ3. Thesegive similar conclusions and we discuss theµ3 casefor an explicit illustration. Then,µ3 is bounded as

(2)|µ3| M1/2H0

2 ,TeV[MeV],

whereMH02 ,TeV is neutral Higgs mass in units of TeV.

In this paper, we introduce dimensionless numbers: fora small couplingε, ε−n represents it in units of 10−n,for MeV order massesmMeV in units of MeV, for GeVorder massesmGeV in units of GeV, for large massm[n] in units ofn GeV, and for super large massF12 inunits of 1012 GeV.

The bilinear R-violating parameters have been ex-tensively discussed in regards to the neutrino oscilla-tion [17] within the above bound (2). In this Letterwe will draw a conclusion that this is not consistentwith the CDM axino, already from the observed dif-fuse gamma ray background.

With the bilinear R-parity violation, we expect thefollowing decay modes of the axino

a → ν + γ (or l+l−), a → ν + a,

(3)a → τ+ + π−, etc.,

wherema >mτ +mπ is assumed.To estimate the partial decay widths, let us assume

the following R violating axino interaction

(4)

La-decay= ε0φaψ, or iε1αem

FaFµνaγ5[γµ, γν]ψ,

whereFa is the axion decay constant (including thedivision by the domain wall numberNDW), Fµν is thefield strength of a spin-1 fieldAµ, φ is a scalar fieldandψ is a fermion field (Dirac or Majorana field).Then, the lifetime of axino becomes

(5)

τa = nψm−1a,GeV

(1.32ε2

0,−11P0, or

2.57× 10−5 ε21F

−2a,12m

2a,GeVP1

)−1 [s],

where nψ = 1,2, respectively, for the Dirac andMajoranaψ , we neglectedmφ , andP0,1 are phasespace factors. For a massive final fermion,P−1

0,1 =(1 − m2

ψ/m2a)−1(1 + mψ/ma)

−2. Let us proceed todiscuss several possibilities of O(GeV) axino decay.

Firstly, for the a → ν + a decay, we note that theaxion multiplet couples to the standard model chi-ral fields, below the PQ symmetry breaking scale, asexp(iQA/Fa)WQ, whereWQ carries−Q units of thePeccei–Quinn (PQ) charge, andA is the axion super-multiplet. The heavy quark axion models [2] do notallow these tree level couplings since the SM fields

20 H.B. Kim, J.E. Kim / Physics Letters B 527 (2002) 18–22

are neutral under the PQ symmetry, but the lepton-coupling type models [3] can lead to this kind ofcouplings. Our interest is on the superpotential forthe axino–neutrino coupling,mννν exp(2iQνA/Fa),where Qν is the PQ charge of the neutrino, andµαLαH2e

iQA/Fa . SettingQν = 1/2, Q = 1/2, weobtain W = mνννe

iA/Fa and µαLαH2eiA/Fa from

which we obtain the relevant terms for the axino de-cay −(mνA

2/2F 2a )νν and −(µαA2/2F 2

a )LαH2, re-spectively. The R-parity violation by the vacuum ex-pectation value of a sneutrino,vν ≡ 〈ν〉, or byµα al-low the following Yukawa couplings

(6)La→νa =(mνvν

F 2a

, orµαv2

F 2a

)aνa,

from which we estimateε0 10−33mν,eVvν,GeV/F2a,12

and 10−25µα,MeVv2,[100]/F 2a,12, respectively, andv2 =

〈H 02 〉. Thus, in view of Eq. (5)ε0 is too small (i.e.,

τa ∼ 1028 s) and the decay modea → ν+ a is not im-portant cosmologically.

Second, the decaya → τ + π+ occurs throughthe bilinear R-parity violating termµαLαH2 withα = 3, given in Eq. (1), which allows the mixing ofτ andH−

1 . Then, the coupling(mτ/Fa)τ+τ a and the

Yukawa coupling∼ (md/v1)qdcH1 give an effective

interaction (4) with

(7)ε0 = mτfπm2π

Fav1

1

M2τ

*M2RPV

1

M2H−

1

,

where*M2RPV is theτ–H−

1 mixing parameter. The ef-fective interaction (4) withψ = τ andφ = π+ arisesfrom the tree diagram with theτ − H− mixing in-sertion in the intermediate scalar propagator with thefour external fermions,a, τ, d, andu. In estimatingε0,we used the PCAC relation in obtaining the matrix el-ement〈0|dRuL|π+〉 ∼ fπm

2π/md . The superpartner

masses for the gauge hierarchy solution are around100 GeV. The bound on the stau-charged Higgs mix-ing parameter*M2

RPV is bounded from the tau neu-trino mass bound,mν < 1 eV. For the R-parity violat-ing bilinear couplingµ3L3H2, the mixing parameter isestimated as*M2

RPV = 2µ∗3µ. Using Eq. (2), we ob-

tain*M2RPV< 2µTeVM

1/2H0

2 ,TeV[GeV2]. Then, we esti-

mateε0 < 3.71× 10−25(cosβ)−1F−1a,12µTeVM

−2H−

1 ,[100]×M−2

τ ,[100]M1/2H0

2,TeVfor theτπ decay mode, and the ax-

ino lifetime must satisfy a bound

(8)τa (0.959× 1027 s

) M4τ ,TeVM

4H−

1 ,TeV

ma,GeVµ2TeVMH0

2,TeV

P−10 ,

where the MSSM parameter tanβ = v2/v1, and weassume thatP0 is nonzero, i.e., the O(GeV) axino hasmassma > 1.92 GeV.

Third, we note that the interactionντ τ a, arisingfrom stau intermediate state and R-parity violatinginsertion of neutrino–B/W mixing, is not important.

Finally, we note that the decaya → ν+γ (or l+l−)occurs through the anomaly term [7],

(9)Laγ ν = icaγ γ αem

16πFa

c′µαµ

ναγ5[γ µ, γ ν

]a Fµν,

where caγ γ is the axion–photon–photon couplingwhich depends on models [18], andFµν is the pho-ton field strength, and the photino–neutrinomixing pa-rameterc′µα/µ has been introduced. Note thatc′ isO(< 1) andµα is O(<MeV). It turns out that this pho-ton mode constitutes the most important contributionin the axino decay. The gluon anomaly term can beconsidered, but it is not important since we must con-sider intermediate gluino and squark lines. From theinteraction Eq. (9), we estimate

(10)ε1 = 1.99× 10−7caγ γ c′µα,MeVµ

−1[100],

giving the axino lifetime

τa (9.8× 1017 s

)c−2aγ γ c

′−2µ−2α,MeVµ

2[100](11)× F 2

a,12m−3a,GeV.

The RHS of Eq. (11) can fall in the cosmologicallyinteresting scale forFa slightly smaller than 1012 GeVandma = O(GeV).

This leads us to the estimation of the cosmologicalabundance of axino. The decoupling temperature ofaxino is of order the PQ symmetry breaking scale [7,12]. Thus, for an O(GeV) axino, inflation must endbelow the PQ symmetry breaking scale so that axinosproduced at the decoupling temperature is sufficientlydiluted. However, if the reheating temperature afterinflation were high enough, a significant number ofaxinos would have been reproduced thermally and canconstitute cold dark matter. Here, we are interestedin this thermally produced axinos after inflation. Inthis scenario, the number density depends on axino

H.B. Kim, J.E. Kim / Physics Letters B 527 (2002) 18–22 21

mass and reheating temperatureTR . For O(GeV) axinoin R-conserving theories the reheating temperaturebound is TR 100–1000 GeV from the conditionthat the thermally produced axinos do not exceedthe critical energy density as estimated in Ref. [7].On the other hand, if there exists R-violating terms,then the lightest neutralino decays tol−l+ν within 1s for µα/µ < 10−6, which occurs through the dia-gram χ0

1 → W∓χ± and the mixing ofχ± and l∓.Namely, a neutralino predominantly decays to SMparticles, which is harmless at a later epoch. Thus,to produce O(GeV) axino copiously in R-violatingtheories, reheating temperature must be raised.

In the present estimation of reheating temperatureTR , we ignore the gluino decay to axino since R-parityis broken. Taking into account the photon thermaliza-tion after the decay of MSSM SUSY particles ande+e− annihilation, the axino energy density at presentis ρa(Tγ ) = ma(2π2/45)(43/11)T 3

γ YTPa(TR), where

the thermal production of axinoY TPa

is calculatedjust from the scattering processes. Thus, represent-ing ρa = Ωa× (the critical energy density), we have

maYTPa

(0.72 eV)(Ωah2/0.2). ForFa ∼ 1011 GeV,

Ωa ∼ 0.3 and O(GeV) axino,Y TPa

∼ 0.5 × 10−9 andwe can readTR at around 200 GeV from Fig. 1 ofRef. [7]. In this region, if one considers the gluinodecay in R-conserving theories,TR should be a fac-tor ∼ 2 smaller. The neutralino decay in R-conservingcase is not important in this region, but can be veryimportant forTR < 100 GeV [7].

The present dark matter density in the galactic halorequires a significant amount of dark matter. In thepresent case, there is the other candidate for dark mat-ter, the axion. For the axion CDM, we are restrictedto Fa ∼ 1012 GeV. But with the CDM axino,Fa canbe lower as far asFa > 109 GeV. If the axino lifetimeis of order the age of the universe, then there remainsa significant number of axinos which can be detectedin experiments in search of proton decay. The interest-ing decay mode for axino detection is theνγ mode.

For 103 s < τa < trec, where trec is the time atthe recombination, there can be an allowed regionτa > tmin so that the decay products are not copiousenough to dissociate the light nuclei. After the time ofrecombination, photons are not effective to scatter offthe neutral particles, and hencetmin trec. Since we

are considering the axino lifetime> 103 s, the latedecaying axino is safe from destroying light nuclei.

If axino decays toν + γ , the underground neutrinodetectors can detect the photon. The photon energy(≡ Eγ ) of 10 GeV, i.e., axino mass of 20 GeV,is the boundary for using different search types forthe Cherenkov rings. IfEγ < 10 GeV, the Comptonscattering on an atomic electron kicks out a highenergy electron whose Cherenkov radiation can bedetected. IfEγ > 10 GeV, thene+e− pair productionoff nucleus dominates and the Cherenkov rings fromthese pair can be detected.

From the super-K detector, one establishes theproton lifetime bound of 1033 seconds [19]. Thesedetectors use baryons in water,nB = NA/cm3 whereNA = 6.023 × 1023. On the other hand axino asCDM now has the local number density of orderna = 0.3m−1

a,GeV/cm3, giving the rationa/nB ∼ 5 ×10−25m−1

a,GeV. If we require the detection rate of axinodecay the same as that of proton decay, we obtain(τpna/τanp) 1. Thus, axinos are detectable at therate of counting proton decay debris with protonlifetime of 1033 seconds [19] if

(12)τa 1.6× 1016

ma,GeV[s].

Since we have not observed this kind of events, weobtainτa > 1.6× 1016m−1

a,GeV [s].However, the most stringent bound comes from the

diffuse gamma ray background. The classical studyon this effect has been published more than 20 yearsago [20]. The observed fluxFγ is bounded by [21,22]

(13)dFγ

dΩ

(10−4 ∼ 10−6)E−1

GeV [cm−2 sr−1 s−1],whereE is the decay photon energy at present. Thefigure 10−4 is for the conservative bound applicableto the whole observed range ofE. ForE = 1 MeV ∼10 GeV where we are interested, 10−6 gives a good fitto the data.

On the other hand, the decay of axinos produces thediffuse photon flux at present, for 0E ma/2

(14)dFγ

dΩ= 3na

8π

(E

E0

)3/2

e−(E/E0)3/2,

whereE0 = (ma/2)(τa/t0)2/3 andt0 is the age of theuniverse (∼ 4× 1017 s).

22 H.B. Kim, J.E. Kim / Physics Letters B 527 (2002) 18–22

For trec < τa < t0, most axinos have decayedand the flux has a peak atE = E0 with a valuena/8πe. Assuming the critical axino density, we havea condition that the photons from axinos decay do notexceed the observed flux,

(15)(τa/t0)2/3 < 1.4× 10−7Ωah

2,

which is inconsistent with the conditionτa > trec(∼ 1013 s).

For τa > t0, axino decays are increasing at presentand the maximum flux is(3na/8π)(t0/τa)e−t0/τa atE =ma/2. Comparing this with Eq. (13), we obtain

(16)τa,sec> 3.9× 1027Ωah.

For the region satisfied by Eq. (16), the bound onµα from Eq. (11) is very stringent, i.e., less thanO(10 eV), hence the idea for neutrino mass generationvia bilinear R-parity violation is not consistent withCDM axino. A very conservative upper bound onµαis 1 keV.

In conclusion, we searched for the detection possi-bility of axinos as CDM withTR ∼ 200 GeV. The dif-fuse gamma ray background gives a very strong boundon bilinear R-parity violating parameterµα . Even ifµα is of order keV, it can be detected by diffuse gammaray background observation. On the other hand, withO(keV) µα SUSY generation of neutrino oscillationparameters through bilinear R-parity violation is notachievable with CDM axino.

Acknowledgements

One of us (J.E.K.) has benefitted from very usefulcomments by E.J. Chun and S.B. Kim. This work issupported in part by the BK21 program of Ministry ofEducation, and by the Center for High Energy Physics,Kyungpook National University.

References

[1] M. Persic, P. Salucci, F. Stel, Mon. Not. R. Astron. Soc. 281(1986) 27;For a recent review, see: E. Battaner, E. Florido, Fund. CosmicPhys. 21 (2000) 1.

[2] J.E. Kim, Phys. Rev. Lett. 43 (1979) 103;M.A. Shifman, A.I. Vainshtein, V.I. Zakharov, Nucl. Phys.B 166 (1980) 493.

[3] M. Dine, W. Fischler, M. Srednicki, Phys. Lett. B 104 (1981)199;A. Zhitnitsky, Sov. J. Nucl. Phys. 31 (1980) 260.

[4] J. Preskill, M.B. Wise, Wilczek, Phys. Lett. B 120 (1983) 127;L.F. Abbott, P. Sikivie, Phys. Lett. B 120 (1983) 133;M.B. Dine, W. Fischler, Phys. Lett. B 120 (1983) 137.

[5] J. Silk, M. Srednicki, Phys. Rev. Lett. 53 (1984) 624.[6] L. Covi, J.E. Kim, L. Roszkowski, Phys. Rev. Lett. 82 (1999)

4180.[7] L. Covi, H.B. Kim, J.E. Kim, Roszkowski, JHEP 0105 (2001)

033.[8] D.J.H. Chung, E.W. Kolb, A. Riotto, Phys. Rev. D 59 (1999)

023501.[9] R.D. Peccei, H.R. Quinn, Phys. Rev. Lett. 38 (1977) 1440.

[10] J.E. Kim, Phys. Rep. 150 (1987) 1.[11] A. Bottino, F. Donato, N. Fornengo, S. Scopel, Phys. Rev. D 63

(2001) 125003, and references therein.[12] J.E. Kim, A. Masiero, D.V. Nanopoulos, Phys. Lett. B 139

(1984) 346;K. Rajagopal, M.S. Turner, F. Wilczek, Nucl. Phys. B 358(1991) 447;E.J. Chun, H.B. Kim, J.E. Kim, Phys. Rev. Lett. 72 (1994)1956;H.B. Kim, J.E. Kim, Nucl. Phys. B 433 (1995) 421.

[13] J. Ellis, J.E. Kim, V. Nanopoulos, Phys. Lett. B 145 (1984) 181.[14] T. Goto, M. Yamaguchi, Phys. Lett. B 276 (1992) 103;

E.J. Chun, J.E. Kim, H.P. Nilles, Phys. Lett. 287 (1992) 123;E.J. Chun, A. Lukas, Phys. Lett. 357 (1995) 43.

[15] P. Sikivie, Phys. Rev. Lett. 51 (1983) 1415.[16] J. Hong, J.E. Kim, Phys. Lett. B 265 (1991) 197.[17] E.J. Chun, Phys. Lett. B 454 (1999) 304.[18] J.E. Kim, Phys. Rev. D 58 (1998) 055006.[19] For thee+π0(νK+) mode, the 90% C.L. bound is 1.6(0.67)×

1033 yr. See: Super-Kamiokande Collaboration, Phys. Rev.Lett. 81 (1998) 3319;Super-Kamiokande Collaboration, Phys. Rev. Lett. 83 (1999)1529.

[20] A. Dolgov, Ya.B. Zel’dovich, Rev. Mod. Phys. 53 (1981) 1;M.S. Turner, in: R.J. Cense, E. Ma, A. Roberts (Eds.),Neutrino-81, Univ. Hawaii Press, Honolulu, 1981, p. 95.

[21] M.T. Ressell, M.S. Turner, preprint Fermilab-PUB-89-214-A(1989);E.W. Kolb, M.S. Turner, in: Early Universe, Addison–Wesley,New York, 1990.

[22] M. Kawasaki, T. Yanagida, Phys. Lett. B 399 (1997) 45;T. Asaka, J. Hashiba, M. Kawasaki, T. Yanagida, Phys. Rev.D 58 (1998) 083509;K. Choi, E.J. Chun, H.B. Kim, Phys. Rev. D 58 (1998) 046003.



New results on a search for a 33.9 MeV/c2 neutral particle fromπ+ decay in the NOMAD experiment

NOMAD Collaboration

P. Astiern, D. Autieroh, A. Baldisseris, M. Baldo-Ceolinm, M. Bannern,G. Bassompierrea, K. Benslamai, N. Bessons, I. Bird h,i , B. Blumenfeldb, F. Bobisutm,

J. Bouchezs, S. Boydu, A. Buenoc,y, S. Bunyatovf, L. Camillerih, A. Cardinij,P.W. Cattaneoo, V. Cavasinniq, A. Cervera-Villanuevah,w, G. Collazuolm,

G. Confortov, C. Contao, M. Contalbrigom, R. Cousinsj, D. Danielsc, H. Degaudenzii,T. Del Preteq, A. De Santoh, T. Dignanc, L. Di Lella h, E. do Couto e Silvah,J. Dumarchezn, M. Ellis u, T. Fazioa, G.J. Feldmanc, R. Ferrario, D. Ferrèreh,

V. Flaminioq, M. Fraternalio, J.-M. Gaillarda, E. Ganglerh,n, A. Geisere,h, D. Gepperte,D. Gibinm, S.N. Gninenkoh,l , A. Godleyu, M.C. Gonzalez-Garciaw,

J.-J. Gomez-Cadenash,w, J. Gossets, C. Gößlinge, M. Gouanèrea, A. Granth,G. Grazianig, A. Guglielmim, C. Hagners, J. Hernandow, D. Hubbardc, P. Hurstc,

N. Hyettk, E. Iacopinig, C. Josephi, F. Jugeti, M.M. Kirsanovl, O. Klimov f,J. Kokkonenh, A.V. Kovzelevl,o, N.V. Krasnikovl, A. Krasnoperova,f , S. Lacapraram,

C. Lachaudn, B. Lakic x, A. Lanzao, L. La Rotondad, M. Lavederm,A. Letessier-Selvonn, J.-M. Levyn, L. Linssenh, A. Ljubicic x, J. Longb, A. Lupi g,

A. Marchionnig, F. Martelliv, X. Méchains, J.-P. Mendiburua, J.-P. Meyers,M. Mezzettom, S.R. Mishrac,t, G.F. Moorheadk, D. Naumovf, P. Nédéleca,

Yu. Nefedovf, C. Nguyen-Maui, D. Orestanor, F. Pastorer, L.S. Peaku,E. Pennacchiov, H. Pessarda, R. Pettih,o, A. Placcih, G. Poleselloo, D. Pollmanne,

A. Polyarushl, B. Popovf,n, C. Poulsenk, P. Rathouits, J. Ricoy, C. Rodah,q,A. Rubbiah,y, F. Salvatoreo, K. Schahmanechen, B. Schmidte,h, M. Seviork, D. Silloua,

F.J.P. Solerh,u, G. Sozzii, D. Steeleb,i , U. Stieglerh, M. Stipcevic x, Th. Stolarczyks,M. Tareb-Reyesi, G.N. Taylork, V. Tereshchenkof, A.N. Toropinl, A.-M. Touchardn,

S.N. Toveyk, M.-T. Trani, E. Tsesmelish, J. Ulrichsu, L. Vacavanti,M. Valdata-Nappid,p, V. Valuevf,j , F. Vannuccin, K.E. Varvellu, M. Veltri v, V. Vercesio,

G. Vidal-Sitjesh,w, J.-M. Vieirai, T. Vinogradovaj, F.V. Weberc,h, T. Weissee,F.F. Wilsonh, L.J. Wintonk, B.D. Yabsleyu, H. Zaccones, K. Zubere, P. Zucconm



24 NOMAD Collaboration / Physics Letters B 527 (2002) 23–28

a LAPP, Annecy, Franceb Johns Hopkins University, Baltimore, MD, USA

c Harvard University, Cambridge, MA, USAd University of Calabria and INFN, Cosenza, Italy

e Dortmund University, Dortmund, Germanyf JINR, Dubna, Russia

g University of Florence and INFN, Florence, Italyh CERN, Geneva, Switzerland

i University of Lausanne, Lausanne, Switzerlandj UCLA, Los Angeles, CA, USA

k University of Melbourne, Melbourne, Australial Inst. Nucl. Research, INR, Moscow, Russia

m University of Padova and INFN, Padova, Italyn LPNHE, University of Paris VI and VII, Paris, France

o University of Pavia and INFN, Pavia, Italyp University of Perugia and INFN, Perugia, Italy

q University of Pisa and INFN, Pisa, Italyr Roma Tre University and INFN, Rome, Italy

s DAPNIA, CEA, Saclay, Francet University of South Carolina, Columbia, SC, USA

u University of Sydney, Sydney, Australiav University of Urbino, Urbino, and INFN Florence, Italy

w IFIC, Valencia, Spainx Rudjer Boškovic Institute, Zagreb, Croatia

y ETH Zürich, Zürich, Switzerland

Received 11 December 2001; accepted 18 December 2001

Editor: L. Montanet

Abstract

We report on a direct search in NOMAD for a new 33.9 MeV/c2 neutral particle (X) produced in pion decay in flight,π → µX followed by the decayX → νe+e−. Both decays are postulated to occur to explain the time anomaly observed by theKARMEN experiment. From the analysis of the data collected during the 1996–1998 runs with 4.1 × 1019 protons on target,a single candidate event consistent with background expectations was found. The search is sensitive to a pion branching ratioBR(π → µX) > 3.7× 10−15, significantly smaller than previous experimental limits. 2002 Elsevier Science B.V. All rightsreserved.

Keywords: Neutrino mixing; Neutrino decay

The Karmen Collaboration at the ISIS spallationneutron facility of the Rutherford Appleton Labora-tory has reported on an anomaly in the time distri-bution of neutrino interactions from muon decay atrest [1]. The anomaly is an enhancement in the timedistribution of νe and νµ induced events which was

E-mail address: [email protected] (S.N. Gninenko).

expected to be well described by the single expo-nential from muon decay at rest. This anomaly, seenin the KARMEN1 data, is not statistically signifi-cant in the KARMEN2 data. However, a substantialeffect still persists in the combined data from bothexperiments [2]. The KARMEN Collaboration inter-preted the anomaly as being due to an exotic decayof π+-mesons into a muon and a 33.9 MeV/c2 newfermionX. They reported values for the pion branch-

NOMAD Collaboration / Physics Letters B 527 (2002) 23–28 25

ing ratio BR(π+ → µ+ + X)BR(X → visible) as afunction of theX-lifetime, τX , needed to explain thiseffect with a branching ratioBR(π+ → µ+ + X) assmall as 10−16 [1]. TheX → visible decay is presum-ably associated with theX → νe+e− mode favouredby the energy distribution measured in the KARMENdetector. This and other hypotheses explaining theanomaly have been recently extensively investigatedboth theoretically [3,5] and experimentally [6,7,9].

Barger et al. [3] associated the new particle with a33.9 MeV/c2 isosinglet neutrino dominantly mixingwith ντ . This interpretation has been studied byNOMAD [8] resulting in a smallX lifetime windowbetween 10−2 s and the Big Bang Nucleosynthesis(BBN) lower limit of Ref. [4] of 0.1 s left untested.

The new X particle has been directly searchedfor at PSI by looking for a peak in the momentumspectrum of muons from this two-body pion decay.The result gives a limit on the branching ratio ofBR(π+ → µ+ + X) < 6.0 × 10−10 at 95% CL [6].A result on the search for theX → e+e−ν decay inflight has also been recently reported by the NuTeVCollaboration [9].

In this Letter we have studied the decayX →νe+e− as a possible manifestation of the presence ofX’s in the neutrino beam. The occurrence ofX →νe+e− decays would appear as an excess of isolatede+e− pairs in NOMAD above those expected fromstandard neutrino interactions. The present analysis aswell as the experimental signature of the signal eventsare similar to those of our published heavy neutrinosearch [8].

An essentially pureνµ beam is produced fromdecays of secondary pions in a 290 m long evacuateddecay tunnel. The pions are generated in a berylliumtarget irradiated by 450 GeV protons from the CERNSPS. The NOMAD detector is located 835 m from thetarget.

The detector is described in Ref. [11]. It consists ofa number of sub-detectors most of which are locatedinside a 0.4 T dipole magnet with a volume of 7.5 ×3.5 × 3.5 m3: an active target of drift chambers (DC)with a mass of 2.7 tons (mainly carbon), an averagedensity of 0.1 g/cm3 and a total thickness of aboutone radiation length (∼1.0X0) followed by a transitionradiation detector (TRD), a preshower detector (PRS),and a lead-glass electromagnetic calorimeter (ECAL).A hadron calorimeter (HCAL) and two muon stations

are located just after the magnet coils. A plane of vetoscintillation counters,V , in front of the magnet andtwo planes of scintillation countersT1 andT2 locatedbefore and after the TRD were used to form theV T1T2trigger for neutrino interactions or decays in the DCtarget.

The electron identification in NOMAD is pro-vided mainly by the TRD which has an efficiency ofmore than 90% for isolated electrons of momentum1–50 GeV/c for a charged pion rejection factor greaterthan 103 [12].

The flux and energy spectrum ofπ ’s producedin the Be target by primary protons were calculatedusing the Monte Carlo simulation described in [13].TheX flux was calculated assuming a branching ratioBR(π → µX) = 1 and mX = 33.9 MeV/c2 as afunction of theX lifetime. Note that due to the smallQ value (≈ 5 keV) in π → µX and the high valueof the pionγ -factor (γ ≈ 103) theX particle followstheπ+ direction. Once theX flux is known, the nextstep consists in calculating thee+e− energy spectrumbased on theX → νe+e− decay model used for theheavy neutrino search [8]. The decay electrons andpositrons were tracked through the DC target takinginto account the emission of photons, their conversionand multiple scattering in the target. The details of theNOMAD simulation and reconstruction are describedelsewhere [8,13].

The search forX → νe+e− described in this Letteruses the full data sample collected with theV T1T2trigger [11] during the years 1996–1998. The datacorrespond to a total number of protons on target(pots) of 4.1× 1019.

The following initial selection criteria were appliedto identify isolatede+e− pairs: (i) the presence of twotracks with at least one identified as an electron andforming a vertex within the DC fiducial volume; (ii) noother tracks incompatible with conversions of photonsemitted by the primarye+e− pair; (iii) the total energyof the pair must be greater than 4 GeV and its invariantmassme+e− must be lower than 95 MeV/c2 to removebackground from pairs of particles other thane+e−;(iv) no additional significant activity in the ECAL,HCAL and muon stations.

Only 207 events passed these criteria. At thenext step we used a collinearity variableC ≡ 1 −cosΘνe+e− , whereΘνe+e− is the angle between theaverage neutrino beam direction and the total momen-


Fig. 1. The final (1− cosΘνe+e− ) distribution for the data and MC.The dashed histogram represents the distribution expected fromsignal events.

tum of the reconstructede+e− pair. A cut on this vari-able, used very effectively in our previous search [8],allowed a strong background suppression. A MC sim-ulation ofX → νe+e− decays shows (see Fig. 1) thattheX → νe+e− events haveC < 2 × 10−5. The con-tribution to the broadening of the signal distributionfrom the angular divergence of the pion beam is muchsmaller than the expected resolution in the variableC.In order to avoid biases in the determination of selec-tion criteria, a blind analysis was performed. Events ina signal box defined byC < 2 × 10−5 were excludedfrom the analysis of the data until the validity of thebackground estimate in this region was demonstrated.This was done by verifying that the MC simulationof standard processes reproduced the data outside thebox.

The accuracy of the collinearity determination ob-tained with MC simulations was checked using aνµCC data sample with ane+e− pair from a pho-ton converted in the DC target at a large ( 100 cm)distance from the primary vertex. Fig. 2 shows the(1− cosΘe+e−) distribution of such events in the dataand simulation, whereΘe+e− is the angle between thee+e− pair momentum and the line joining the primaryvertex to the conversion point. The two distributionsare in reasonable agreement at all energies studied.This validates the resolution in the variableC (a fewmrad inΘνe+e− ) predicted by the MC program.

The reconstruction efficiency for theX → νe+e−decay in the NOMAD fiducial volume was calculated

Fig. 2. The (1− cosΘe+e− ) distribution for e+e− pairs fromphotons converted in the DC target at large distances from theprimary vertex for the data and MC.

from the MC simulation as a function ofe+e− energyin the range 4 to 50 GeV. The MC simulation was usedto correct the data for acceptance losses, experimentalresolution and reconstruction efficiencies. Two meth-ods using both data and MC simulation samples ofreconstructedπ0’s and γ ’s, described in detail else-where [8], have been used to verify the reliability ofthe simulation and to estimate the systematic uncer-tainties in thee+e− pair efficiency reconstruction inthe energy range predicted by the simulation.

It was found that the two methods agree quite wellin the low energy region and yield a correction factorclose to 1. However, in the high energy region thee+e− efficiency correction factor varied from 0.7 ±0.04 to 0.4± 0.03 depending on thee+e− energy.

The largest contribution to the background is ex-pected from neutrino interactions in the material up-stream of the decay volume yielding a singleπ0 withlittle hadronic activity in the final state. Because ofthe large mass of this upstream material the studyof this background would require the simulation of avery large number of events resulting in a prohibitivelylarge amount of computer time. Consequently, onlyabout 10% of the required statistics forνµCC(NC)

inelastic reactions were simulated, while other back-ground components, such asνeCC, coherentπ0 pro-duction, quasi-elastic reactions andνµe scatteringwere simulated with statistics comparable to the num-ber of events expected in the data from these reactions.

The distribution of the variableC for the sum of allthe MC samples is shown in Fig. 1. The plot covers the

NOMAD Collaboration / Physics Letters B 527 (2002) 23–28 27

regionC < 5× 10−4, which is 25 times larger than thesize of the signal box. NoνµCC(NC) event is foundin this region. The data outside the box, also shown inFig. 1, are consistent with the MC prediction (19 evensobserved and 20± 4 events predicted). The estimateof background fromνµCC(NC) in the signal box isbased on the observation that there areno physicalreasons for this background to be other than flat in theregionC < 5× 10−4.

Two independent methods based on the MC and onthe data themselves, described in detail elsewhere [8],were used for the background estimation in the signalregion. The final background estimates with the twomethods areNMC

bkg = 2.5+0.9−0.8(stat) ± 0.6(syst) events

from the MC andNDatabkg = 2.4+0.9

−0.9(stat) ± 0.7(syst)events from the data. The systematic error includes theuncertainties in the number ofpots (5%) and in thecoherentπ0 production cross section (25%). The to-tal systematic uncertainty was calculated by adding allerrors in quadrature. It was decided to use the back-ground estimate extracted from the data themselves.

Upon opening the signal box we have found oneevent that passes our selection criteria. This is consis-tent with the expected background and hence no evi-dence for the decayπ+ → µ+X has been found. Wecan then determine the 90% CL upper limit for the cor-responding value of BR(π → µX)BR(X → νe+e−)

from the 90% CL upper limit for the expected num-ber of signal events,Nup

X→νe+e− . Using the frequen-tist approach of Ref. [14] and taking into account theuncertainties in the background estimate we obtainN

upX→νe+e− = 2.1 events. The probability to obtain an

upper limit of 2.1 or lower is 29% given the back-ground estimates.

The final 90% CL exclusion region in the BR(π →µX)BR(X → νe+e−) vs τX plane is shown in Fig. 3together with the NuTeV [9] and PSI [6] exclusion re-gions forτX as well as the NOMAD lower limit [8]and BBN upper limit [10] obtained for the scenario ofRef. [3]. Our result is sensitive to a pion branching ra-tio BR(π → µX) > 3.7×10−15 which is significantlysmaller than the previous limit from the NuTeV exper-iment [9].

We gratefully acknowledge the CERN SPS accel-erator and beam-line staff for the magnificent perfor-mance of the neutrino beam. The experiment was sup-ported by the following funding agencies: AustralianResearch Council (ARC) and Department of Indus-

Fig. 3. The NOMAD 90% CL upper limit as a function of theX lifetime together with the NuTeV [9] and PSI [6] limits. TheNOMAD lower limit [8] and BBN upper limit [10] onX lifetimefor the interpretation ofX as a sterile neutrino mixing with theτneutrino are also shown.

try, Science, and Resources (DISR), Australia; InstitutNational de Physique Nucléaire et Physique des Par-ticules (IN2P3), Commissariat à l’Energie Atomique(CEA), France; Bundesministerium für Bildung undForschung (BMBF, contract 05 6DO52), Germany;Istituto Nazionale di Fisica Nucleare (INFN), Italy;Russian Foundation for Basic Research, Institute forNuclear Research of the Russian Academy of Sci-ences, Russia; Fonds National Suisse de la RechercheScientifique, Switzerland; Department of Energy, Na-tional Science Foundation (grant PHY-9526278), theSloan and the Cottrell Foundations, USA.

References

[1] KARMEN Collaboration, B. Armbruster et al., Phys. Lett.B 348 (1995) 19;KARMEN Collaboration, B. Zeitnitz, WIN ’99, Cape Town,ZAR (2000).

[2] K. Eitel et al., Nucl. Phys. B, Proc. Suppl. 91 (2001) 191.[3] V. Barger, R.J.N. Phillips, S. Sarkar, Phys. Lett. B 352 (1995)

365;V. Barger, R.J.N. Phillips, S. Sarkar, Phys. Lett. B 356 (1995)671, Erratum.

[4] A.D. Dolgov et al., Nucl. Phys. B 580 (2000) 331.[5] J. Govaerts, J. Deutsch, P.M. Van Hove, Phys. Lett. B 389

(1996) 700;


D. Choudhury, S. Sarkar, Phys. Lett. B 374 (1996) 87;D. Choudhury et al., Phys. Rev. D 61 (2000) 095009;M. Kachelrieß, hep-ph/0001160;I. Golman, R. Mohapatra, S. Nussinov, Phys. Lett. B 481(2000) 151;S.N. Gninenko, N.V. Krasnikov, Phys. Lett. B 434 (1998) 163;J.M. Levy, F. Vannucci, hep-ph/0003194;A. Lukas, A. Romanino, hep-ph/0004130.

[6] M. Daum et al., Phys. Rev. Lett. 85 (2000) 1815;M. Daum et al., Phys. Lett. B 361 (1995) 179.

[7] R. Bilger et al., Phys. Lett. B 363 (1995) 41;D.A. Bryman, T. Numao, Phys. Rev. D 53 (1996) 558;R. Bilger et al., Phys. Lett. B 446 (1999) 363.

[8] NOMAD Collaboration, P. Astier et al., Phys. Lett. B 506(2001) 27.

[9] NuTeV Collaboration, J.A. Formaggio et al., Phys. Rev.Lett. 84 (2000) 4043.

[10] A.D. Dolgov et al., Nucl. Phys. B 590 (2000) 562.[11] NOMAD Collaboration, J. Altegoer et al., Nucl. Instrum.

Methods A 404 (1998) 96.[12] G. Bassompierre et al., Nucl. Instrum. Methods A 403 (1998)

363;G. Bassompierre et al., Nucl. Instrum. Methods A 411 (1998)63.

[13] NOMAD Collaboration, P. Astier et al., Prediction of neutrinofluxes in the NOMAD experiment, in preparation.

[14] G.J. Feldman, R.D. Cousins, Phys. Rev. D 57 (1998) 3873.



Study of the W+W−γ process and limits on anomalous quarticgauge boson couplings at LEP

L3 Collaboration

P. Achardt, O. Adrianiq, M. Aguilar-Benitezx, J. Alcarazx,r, G. Alemanniv, J. Allabyr,A. Aloisio ab, M.G. Alviggi ab, H. Anderhubau, V.P. Andreevf,ag, F. Anselmoi,

A. Arefievaa, T. Azemoonc, T. Aziz j,r, P. Bagnaiaal, A. Bajox, G. Baksayp, L. Baksayy,S.V. Baldewb, S. Banerjeej, Sw. Banerjeed, A. Barczykau,as, R. Barillèrer, P. Bartaliniv,

M. Basilei, N. Batalovaar, R. Battistonaf, A. Bayv, F. Becattiniq, U. Beckern,F. Behnerau, L. Bellucciq, R. Berbecoc, J. Berdugox, P. Bergesn, B. Bertucciaf,

B.L. Betevau, M. Biasiniaf, M. Biglietti ab, A. Bilandau, J.J. Blaisingd, S.C. Blythah,G.J. Bobbinkb, A. Böhma, L. Boldizsarm, B. Borgiaal, S. Bottaiq, D. Bourilkovau,M. Bourquint, S. Braccinit, J.G. Bransonan, F. Brochud, A. Buijsaq, J.D. Burgern,

W.J. Burgeraf, X.D. Cain, M. Capelln, G. Cara Romeoi, G. Carlinoab, A. Cartacciq,J. Casausx, F. Cavallarial, N. Cavalloai, C. Cecchiaf, M. Cerradax, M. Chamizot,Y.H. Changaw, M. Chemarinw, A. Chenaw, G. Cheng, G.M. Cheng, H.F. Chenu,

H.S. Cheng, G. Chiefariab, L. Cifarelli am, F. Cindoloi, I. Claren, R. Clareak,G. Coignetd, N. Colinox, S. Costantinial, B. de la Cruzx, S. Cucciarelliaf,

J.A. van Dalenad, R. de Asmundisab, P. Déglont, J. Debreczenim, A. Degréd,K. Deitersas, D. della Volpeab, E. Delmeiret, P. Denesaj, F. DeNotaristefanial,

A. De Salvoau, M. Diemozal, M. Dierckxsensb, D. van Dierendonckb, C. Dionisial,M. Dittmarau,r, A. Doriaab, M.T. Dovak,5, D. Duchesneaud, P. Duinkerb, B. Echenardt,

A. Eliner, H. El Mamouniw, A. Englerah, F.J. Epplingn, A. Ewersa, P. Extermannt,M.A. Falaganx, S. Falcianoal, A. Favaraae, J. Fayw, O. Fedinag, M. Felciniau,

T. Fergusonah, H. Fesefeldta, E. Fiandriniaf, J.H. Fieldt, F. Filthautad, P.H. Fishern,W. Fisheraj, I. Fiskan, G. Forconin, K. Freudenreichau, C. Furettaz, Yu. Galaktionovaa,n,S.N. Gangulij , P. Garcia-Abiae,r, M. Gataullinae, S. Gentileal, S. Giagual, Z.F. Gongu,

G. Grenierw, O. Grimmau, M.W. Gruenewaldh,a, M. Guidaam, R. van Gulikb,V.K. Guptaaj, A. Gurtuj, L.J. Gutayar, D. Haase, D. Hatzifotiadoui, T. Hebbekerh,a,A. Hervér, J. Hirschfelderah, H. Hoferau, M. Hohlmanny, G. Holznerau, S.R. Houaw,

Y. Hu ad, B.N. Jing, L.W. Jonesc, P. de Jongb, I. Josa-Mutuberríax, D. Käfera,M. Kauro, M.N. Kienzle-Focaccit, J.K. Kimap, J. Kirkbyr, W. Kittel ad,

A. Klimentovn,aa, A.C. Königad, M. Kopalar, V. Koutsenkon,aa, M. Kräberau,

0370-2693/02/$ – see front matter 2002 Published by Elsevier Science B.V.PII: S0370-2693(02)01167-X


30 L3 Collaboration / Physics Letters B 527 (2002) 29–38

R.W. Kraemerah, W. Krenza, A. Krügerat, A. Kunin n, P. Ladron de Guevarax,I. Laktinehw, G. Landiq, M. Lebeaur, A. Lebedevn, P. Lebrunw, P. Lecomteau,P. Lecoqr, P. Le Coultreau, J.M. Le Goffr, R. Leisteat, P. Levtchenkoag, C. Li u,

S. Likhodedat, C.H. Linaw, W.T. Lin aw, F.L. Lindeb, L. Listaab, Z.A. Liu g,W. Lohmannat, E. Longoal, Y.S. Lug, K. Lübelsmeyera, C. Lucial, L. Luminarial,

W. Lustermannau, W.G. Mau, L. Malgerit, A. Malinin aa, C. Mañax, D. Mangeolad,J. Mansaj, J.P. Martinw, F. Marzanoal, K. Mazumdarj, R.R. McNeilf , S. Meler,ab,

L. Merolaab, M. Meschiniq, W.J. Metzgerad, A. Mihul l, H. Milcentr, G. Mirabellial,J. Mnicha, G.B. Mohantyj, G.S. Muanzaw, A.J.M. Muijsb, B. Musicaran, M. Musyal,

S. Nagyp, S. Natalet, M. Napolitanoab, F. Nessi-Tedaldiau, H. Newmanae, T. Niessena,A. Nisatial, H. Nowakat, R. Ofierzynskiau, G. Organtinial, C. Palomaresr,

D. Pandoulasa, P. Paolucciab, R. Paramattial, G. Passalevaq, S. Patricelliab, T. Paulk,M. Pauluzziaf, C. Pausn, F. Paussau, M. Pedaceal, S. Pensottiz, D. Perret-Gallixd,B. Petersenad, D. Piccoloab, F. Pierellai, M. Pioppiaf, P.A. Pirouéaj, E. Pistolesiz,

V. Plyaskinaa, M. Pohlt, V. Pojidaevq, J. Pothierr, D.O. Prokofievar, D. Prokofievag,J. Quartieriam, G. Rahal-Callotau, M.A. Rahamanj, P. Raicsp, N. Rajaj, R. Ramelliau,

P.G. Rancoitaz, R. Ranieriq, A. Rasperezaat, P. Razisac, D. Renau, M. Rescignoal,S. Reucroftk, S. Riemannat, K. Rilesc, B.P. Roec, L. Romerox, A. Roscah,

S. Rosier-Leesd, S. Rotha, C. Rosenblecka, B. Rouxad, J.A. Rubior, G. Ruggieroq,H. Rykaczewskiau, A. Sakharovau, S. Saremif, S. Sarkaral, J. Salicior, E. Sanchezx,

M.P. Sandersad, C. Schäferr, V. Schegelskyag, S. Schmidt-Kaersta, D. Schmitza,H. Schopperav, D.J. Schotanusad, G. Schweringa, C. Sciaccaab, L. Servoliaf,S. Shevchenkoae, N. Shivarovao, V. Shoutkon, E. Shumilovaa, A. Shvorobae,

T. Siedenburga, D. Sonap, P. Spillantiniq, M. Steuern, D.P. Sticklandaj, B. Stoyanovao,A. Straessnerr, K. Sudhakarj, G. Sultanovao, L.Z. Sunu, S. Sushkovh, H. Suterau,

J.D. Swaink, Z. Szillasiy,3, X.W. Tangg, P. Tarjanp, L. Tauschere, L. Taylork,B. Tellili w, D. Teyssierw, C. Timmermansad, Samuel C.C. Tingn, S.M. Tingn,S.C. Tonwarj,r, J. Tóthm, C. Tully aj, K.L. Tungg, J. Ulbrichtau, E. Valenteal,

R.T. Van de Wallead, V. Veszpremiy, G. Vesztergombim, I. Vetlitskyaa, D. Vicinanzaam,P. Violini al, G. Viertelau, S. Villaak, M. Vivargentd, S. Vlachose, I. Vodopianovag,

H. Vogelah, H. Vogtat, I. Vorobievah,aa, A.A. Vorobyovag, M. Wadhwae, W. Wallraff a,X.L. Wangu, Z.M. Wangu, M. Webera, P. Wienemanna, H. Wilkensad, S. Wynhoffaj,L. Xia ae, Z.Z. Xuu, J. Yamamotoc, B.Z. Yangu, C.G. Yangg, H.J. Yangc, M. Yangg,S.C. Yehax, An. Zaliteag, Yu. Zaliteag, Z.P. Zhangu, J. Zhaou, G.Y. Zhug, R.Y. Zhuae,

H.L. Zhuangg, A. Zichichi i,r,s, G. Zilizi y,3, B. Zimmermannau, M.Z. Zöllera

a I. Physikalisches Institut, RWTH, D-52056 Aachen, FRG1

and III. Physikalisches Institut, RWTH, D-52056 Aachen, FRG1

b National Institute for High Energy Physics, NIKHEF, and University of Amsterdam, NL-1009 DB Amsterdam, The Netherlands

L3 Collaboration / Physics Letters B 527 (2002) 29–38 31

c University of Michigan, Ann Arbor, MI 48109, USAd Laboratoire d’Annecy-le-Vieux de Physique des Particules, LAPP, IN2P3-CNRS, BP 110, F-74941 Annecy-le-Vieux cedex, France

e Institute of Physics, University of Basel, CH-4056 Basel, Switzerlandf Louisiana State University, Baton Rouge, LA 70803, USA

g Institute of High Energy Physics, IHEP, 100039 Beijing, PR China6

h Humboldt University, D-10099 Berlin, FRG1

i University of Bologna and INFN, Sezione di Bologna, I-40126 Bologna, Italyj Tata Institute of Fundamental Research, Mumbai (Bombay) 400 005, India

k Northeastern University, Boston, MA 02115, USAl Institute of Atomic Physics and University of Bucharest, R-76900 Bucharest, Romania

m Central Research Institute for Physics of the Hungarian Academy of Sciences, H-1525 Budapest 114, Hungary2

n Massachusetts Institute of Technology, Cambridge, MA 02139, USAo Panjab University, Chandigarh 160 014, Indiap KLTE-ATOMKI, H-4010 Debrecen, Hungary3

q INFN, Sezione di Firenze and University of Florence, I-50125 Florence, Italyr European Laboratory for Particle Physics, CERN, CH-1211 Geneva 23, Switzerland

s World Laboratory, FBLJA Project, CH-1211 Geneva 23, Switzerlandt University of Geneva, CH-1211 Geneva 4, Switzerland

u Chinese University of Science and Technology, USTC, Hefei, Anhui 230 029, PR China6

v University of Lausanne, CH-1015 Lausanne, Switzerlandw Institut de Physique Nucléaire de Lyon, IN2P3-CNRS, Université Claude Bernard, F-69622 Villeurbanne, France

x Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas, CIEMAT, E-28040 Madrid, Spain4

y Florida Institute of Technology, Melbourne, FL 32901, USAz INFN, Sezione di Milano, I-20133 Milan, Italy

aa Institute of Theoretical and Experimental Physics, ITEP, Moscow, Russiaab INFN, Sezione di Napoli and University of Naples, I-80125 Naples, Italy

ac Department of Physics, University of Cyprus, Nicosia, Cyprusad University of Nijmegen and NIKHEF, NL-6525 ED Nijmegen, The Netherlands

aeCalifornia Institute of Technology, Pasadena, CA 91125, USAaf INFN, Sezione di Perugia and Università Degli Studi di Perugia, I-06100 Perugia, Italy

ag Nuclear Physics Institute, St. Petersburg, Russiaah Carnegie Mellon University, Pittsburgh, PA 15213, USA

ai INFN, Sezione di Napoli and University of Potenza, I-85100 Potenza, Italyaj Princeton University, Princeton, NJ 08544, USA

ak University of Californa, Riverside, CA 92521, USAal INFN, Sezione di Roma and University of Rome “La Sapienza”, I-00185 Rome, Italy

am University and INFN, Salerno, I-84100 Salerno, Italyan University of California, San Diego, CA 92093, USA

ao Bulgarian Academy of Sciences, Central Lab. of Mechatronics and Instrumentation, BU-1113 Sofia, Bulgariaap The Center for High Energy Physics, Kyungpook National University, 702-701 Taegu, South Korea

aq Utrecht University and NIKHEF, NL-3584 CB Utrecht, The Netherlandsar Purdue University, West Lafayette, IN 47907, USA

asPaul Scherrer Institut, PSI, CH-5232 Villigen, Switzerlandat DESY, D-15738 Zeuthen, FRG

au Eidgenössische Technische Hochschule, ETH Zürich, CH-8093 Zürich, Switzerlandav University of Hamburg, D-22761 Hamburg, FRG

aw National Central University, Chung-Li, Taiwan, ROCax Department of Physics, National Tsing Hua University, Taiwan, ROC

Received 6 November 2001; received in revised form 17 December 2001; accepted 7 January 2002

Editor: L Rolandi


Abstract

The process e+e− → W+W−γ is studied using the data collected by the L3 detector at LEP. New results, corresponding toan integrated luminosity of 427.4 pb−1 at centre-of-mass energies from 192 to 207 GeV, are presented.

The W+W−γ cross sections are measured to be in agreement with Standard Model expectations. No hints of anomalousquartic gauge boson couplings are observed. Limits at 95% confidence level are derived using also the process e+e− → ννγ γ . 2002 Published by Elsevier Science B.V.

1. Introduction

The increase of the LEP centre-of-mass energy wellabove the W boson pair-production threshold opensthe possibility of studying the triple boson productionprocess e+e− → W+W−γ . We report on the crosssection measurement of this inclusive process wherethe photon lies inside a defined phase space region.

The three boson final state gives access to quar-tic gauge boson couplings represented by four-bosoninteraction vertices as shown in Fig. 1a. At the LEPcentre-of-mass energies the contribution of four-bosonvertex diagrams, predicted by the Standard Model ofelectroweak interactions [1,2], are negligible with re-spect to the other competing diagrams, mainly initial-state radiation. The study of the W+W−γ process isthus sensitive to anomalous quartic gauge couplings(AQGC) in both the W+W−Zγ and W+W−γ γ ver-tices. The presence of AQGC would increase the crosssection and modify the photon energy spectrum of theW+W−γ process. This search is performed within thetheoretical framework of Refs. [3,4].

The existence of AQGC would also affect thee+e− → ννγ γ process via the W+W− fusion dia-gram, shown in Fig. 1b, containing the W+W−γ γ

1 Supported by the German Bundesministerium für Bildung,Wissenschaft, Forschung und Technologie.

2 Supported by the Hungarian OTKA fund under contractnumbers T019181, F023259 and T024011.

3 Also supported by the Hungarian OTKA fund under contractnumber T026178.

4 Supported also by the Comisión Interministerial de Ciencia yTecnología.

5 Also supported by CONICET and Universidad Nacional de LaPlata, CC 67, 1900 La Plata, Argentina.

6 Supported by the National Natural Science Foundation ofChina.

(a)

(b)

Fig. 1. Feynman diagrams containing four-boson vertices leading tothe (a) W+W−γ and (b)ννγ γ final states.

vertex [5]. The reaction e+e− → ννγ γ is dominatedby initial-state radiation whereas the quartic StandardModel contribution from the W+W− fusion is negli-gible at LEP. Also in this case the presence of AQGCwould enhance the production rate, especially for thehard tail of the photon energy distribution and for pho-tons produced at large angles with respect to the beamdirection.

The results are based on the high energy data sam-ple collected with the L3 detector [6]. Data at the


centre-of-mass energy of√

s = 189 GeV, correspond-ing to an integrated luminosity of 176.8 pb−1, werealready analysed [7] and are used in the AQGC analy-sis. In the following, particular emphasis is given tothe additional luminosity of 427.4 pb−1 recorded atcentre-of-mass energies ranging from

√s = 192 up to

207 GeV.The results derived on AQGC from the W+W−γ

andννγ γ channels are eventually combined.Studies of triple gauge bosons production and

AQGC were recently reported for both the W+W−γ

[8] and Zγ γ [9] final states.

2. Monte Carlo simulation

The dominant contributions to the W+W−γ finalstate come from the radiative graphs with photonsemitted by the incoming particles (ISR), by the decayproducts of the W bosons (FSR) or by the W’sthemselves (WSR).

In this Letter, the signal is defined as the phasespace region of the e+e− → W+W−γ process wherethe photon fulfills the following criteria:

• Eγ > 5 GeV, whereEγ is the energy of thephoton,

• 20 < θγ < 160, whereθγ is the angle betweenthe photon and the beam axis,

• αγ > 20, where αγ is the angle between thedirection of the photon and that of the closestcharged fermion.

These requirements, used to enhance the effect ofpossible AQGC, largely contribute to avoid infraredand collinear singularities in the calculation of thesignal cross section.

In order to study efficiencies, background contam-inations and AQGC effects, several Monte Carlo pro-grams are used.

The KORALW [10] generator, which does not in-clude either the quartic coupling diagrams or the WSR,performs initial state multi-photon radiation in thefull photon phase space. FSR from charged leptonsin the event up to double bremsstrahlung is includedusing the PHOTOS [11] package. The JETSET [12]Monte Carlo program, which includes photons in theparton shower, is used to model the fragmentation

and hadronization process. The KORALW program isused in the analysis for the determination of efficien-cies. PYTHIA [13] is used to simulate the backgroundprocesses: e+e− → Z/γ → qq(γ ), e+e− → ZZ →4f(γ ) and e+e− → Zee→ ffee(γ ).

The EEWWG [4] program is used to simulate theeffect of AQGC. It includesO(α) calculations for vis-ible photons but is lacking the simulation of photonscollinear to the beam pipe and of FSR. The net ef-fect of collinear photons, included by implementingthe EXCALIBUR [14] collinear radiator function, isto move the effective centre-of-mass energy towardslower values, reducing the expected signal cross sec-tion by about 18%.

Other Monte Carlo programs which include WSRor full O(α) corrections, such as YFSWW3 [15]and RACOONWW [16], are used to cross check thecalculations.

For the simulation of the e+e− → ννγ γ processin the framework of the Standard Model the KO-RALZ [17] Monte Carlo generator is used. NUNUGPV

[18] is also used to cross check the results, and foundto be in agreement with KORALZ. The effects ofAQGC are simulated using the EENUNUGGANO pro-gram [5]. The missing higher order corrections due toISR in EENUNUGGANO are also estimated by imple-menting the EXCALIBUR collinear radiator function.

The response of the L3 detector is modelled withthe GEANT [19] detector simulation program whichincludes the effects of energy loss, multiple scatteringand showering in the detector materials and in thebeam pipe. Time dependent detector inefficiencies aretaken into account in the simulation.

3. W+W−γ event selection and cross section

The selection of W+W−γ events follows two steps:first semileptonic W+W− → qqeν or qqµν, and fullyhadronic W+W− → qqqq events are selected [20],then a search for isolated photons is performed.

The photon identification in W+W− events is op-timized for each four-fermion final state. Photons areidentified from energy clusters in the electromagneticcalorimeter not associated with any track in the cen-tral detector and with low activity in the nearby regionof the hadron calorimeter. The profile of the showermust be consistent with that of an electromagnetic par-


Fig. 2. Distributions at the centre-of-mass energies√

s =189–207 GeV of (a) the photon energy, (b) the cosine of the an-gle of the photon to the beam axis and (c) the cosine of the angle ofthe photon to the closest charged lepton or jet.

ticle. Experimental cuts on photon energy and anglesare applied, reflecting the phase space definition of thesignal.

Fig. 2 shows the distributions ofEγ , θγ , andαγ forthe full data set, including the data at

√s = 189 GeV.

Here,αγ is defined as the angle of the photon withrespect to the closest identified lepton or hadronic jet.Good agreement between data and Standard Modelexpectations is observed. Fig. 3 shows the distributionsof Eγ for the data collected at

√s = 192–202 GeV and√

s = 205–207 GeV, respectively.Table 1 summarizes the selection yield. In total 86

W+W−γ candidate events are selected at√

s = 192–207 GeV. The Standard Model expectation, inside thespecified phase space region, is of 87.8± 0.8 events.

The quantityεWW, representing the selection ef-ficiencies for the W+W− → qqlν and qqqq decaymodes, ranges from 70 to 87%. The quantityεγ isthe photon identification efficiency inside the selectedphase space region. This efficiency takes into accountsmall effects of events migrating from outside the sig-nal region into the selected sample due to the finite de-

Fig. 3. Distributions of the photon energy for the semileptonicqqeν, qqµν and fully hadronic W+W−γ decay modes corre-sponding to the data collected at (a)

√s = 192–202 GeV and (b)√

s = 205–207 GeV. The cross-hatched area is the background com-ponent from WW, ZZ, Zee, and qq(γ ) events. The FSR distributionincludes the contribution of photons radiated off the charged fermi-ons and photons originating from isolated meson decays. Distribu-tions corresponding to non-zero values of the anomalous couplingan/Λ2 are shown as dashed lines.

tector resolution. Its value ranges from 52 to 80%, thelowest efficiencies being obtained in the fully hadronicsample where the high multiplicity makes the photonidentification more difficult. The overall selection effi-ciencyεWW × εγ is around 45% for all final states.

The W+W−γ cross sections are evaluated channelby channel and then combined according to the Stan-dard Model W boson branching fractions. The datasamples at

√s = 192–196 GeV,

√s = 200–202 GeV

and√

s = 205–207 GeV are respectively merged.They correspond to the luminosity averaged centre-of-mass energies and to the integrated luminosities listedin Table 1.

The results, including the published value at√

s =189 GeV [7], are:

σWWγ (188.6 GeV)

= 0.29± 0.08± 0.02 pb

(σSM = 0.233± 0.012 pb),


Table 1Number of observed events,Ndata, W+W− and W+W−γ selection efficiencies,εWW and εγ , expected total number of events,N

expTOT,

and background estimates,NexpBkgr, for the various decay channels according to the Standard Model prediction. The background estimates

include FSR and misidentified photons. All uncertainties come from Monte Carlo statistics. The average centre-of-mass energies,〈√s〉, and theintegrated luminosity of the three subsamples are also listed

〈√s〉 Channel Ndata εWW εγ NexpTOT N

expBkgr

194.4 GeV qqeνγ 4 0.748± 0.007 0.613± 0.023 3.76± 0.14 1.41± 0.09(113.4 pb−1) qqµνγ 6 0.731± 0.007 0.728± 0.026 4.52± 0.15 1.68± 0.09

qqqqγ 9 0.865± 0.004 0.537± 0.013 12.89± 0.34 5.22± 0.28


qqqqγ 18 0.836± 0.004 0.521± 0.012 13.71± 0.29 5.31± 0.28


qqqqγ 29 0.823± 0.005 0.540± 0.011 26.97± 0.50 12.11± 0.39

σWWγ (194.4 GeV)

= 0.23± 0.10± 0.02 pb

(σSM = 0.268± 0.013 pb),

σWWγ (200.2 GeV)

= 0.39± 0.12± 0.02 pb

(σSM = 0.305± 0.015 pb),

σWWγ (206.3 GeV)

= 0.33± 0.09± 0.02 pb

(σSM = 0.323± 0.016 pb),

where the first uncertainty is statistical and the secondsystematic. The measurements are in good agreementwith the Standard Model expectations,σSM, calculatedusing EEWWG and reported with a theoretical uncer-tainty of 5% [21]. Fig. 4 shows these results togetherwith the predicted total W+W−γ cross sections as afunction of the centre-of-mass energy.

The ratio between the measured cross sectionσmeas and the theoretical expectations is derived ateach centre-of-mass energy. These values are thencombined as:

R = σmeas

σSM= 1.09± 0.17± 0.09,

where the first uncertainty is statistical and the secondsystematic.

The systematic uncertainties arising in the inclusiveW-pair event selections [20] are propagated to the fi-nal measurement and correspond to an uncertainty of

Fig. 4. Measured cross section for the process e+e− → W+W−γ

compared to the Standard Model cross section as a function of thecentre-of-mass energy, as predicted by the EEWWG Monte Carlowithin phase-space requirements. The shaded band correspondsto a theoretical uncertainty of±5%. The three dash-dotted linescorrespond to the cross section for the indicated values of theanomalous couplingan/Λ2 (in GeV−2 units).

0.008 pb for all the energy points. Additional system-atic uncertainties due to the electromagnetic calorime-ter resolution and energy scale are found to be negligi-ble. The total systematic uncertainty is dominated bythe JETSETmodelling of photons from meson decays(π0, η). Its effect has been directly studied on data [7]comparing the photon rate in e+e− → Z → qq(γ )

events with Monte Carlo simulations. A correctionfactor of 1.2 ± 0.1 is applied to the rate of photonsin the Monte Carlo simulation and its uncertainty ispropagated. This uncertainty, fully correlated among


the data taking periods, amounts to 6% of the mea-sured cross section.

4. Determination of anomalous quartic gaugecouplings

4.1. Thee+e− → W+W−γ process

In the framework of Refs. [3,4], the Standard ModelLagrangian of electroweak interactions is extendedto include dimension-6 operators proportional to thethree AQGC:a0/Λ

2, ac/Λ2 and an/Λ

2, whereΛ2

represents the energy scale for new physics.The two parametersa0/Λ

2 andac/Λ2, which are

separately C and P conserving, generate anomalousW+W−γ γ and ZZγ γ vertices. The terman/Λ

2,which is CP violating, gives rise to an anomalous con-tribution to the W+W−Zγ vertex. Indirect limits ona0/Λ

2 andac/Λ2 were derived [22], but only the study

of W+W−γ events allows for a direct measurementof the anomalous couplingan/Λ

2. These couplingswould manifest themselves by modifying the energyspectrum of the photons and the total cross sectionas shown in Figs. 3 and 4, respectively. The effect in-creases with increasing centre-of-mass energy. Thesepredictions are obtained by reweighting the KORALW

Monte Carlo events by the ratio of the differential dis-tributions as calculated by the EEWWG and KORALW

programs [7].The derivation of AQGC is performed by fitting

both the shape and the normalization of the photonenergy spectrum in the range from 5 to 35 GeV. Eachof the AQGC is varied in turn fixing the other two tozero.

The combination of all data, including the results at√s = 189 GeV [7], gives:

a0/Λ2 = 0.000± 0.010 GeV−2,

ac/Λ2 = −0.013± 0.023 GeV−2,

an/Λ2 = −0.002± 0.076 GeV−2,

where systematic uncertainties are included.At the 95% confidence level, the AQGC are con-

strained to:

−0.017< a0/Λ2 < 0.017 GeV−2,

−0.052< ac/Λ2 < 0.026 GeV−2,

Fig. 5. Recoil mass spectrum of the acoplanar photon pair eventsselected at

√s = 192–207 GeV.

−0.14< an/Λ2 < 0.13 GeV−2.

All these results are in agreement with the StandardModel expectation. The sign of thea0 andac AQGC,obtained with the EEWWG reweighting, is reversedaccording to the discussions in Refs. [23,24].

4.2. Thee+e− → ννγ γ process

The sensitivity of the e+e− → ννγ γ process to thea0/Λ

2 andac/Λ2 AQGC, through the diagrams shown

in Fig. 1b, is also exploited. Events with an acoplanarmulti-photon signature are selected [25]. In this Letterwe report on results from data at

√s = 192–207 GeV.

Fig. 5 shows the two-photon recoil mass,Mrec,distribution, with the predicted AQGC signal for anon-zero anomalous couplinga0/Λ

2. The number ofselected events in the Z peak region, defined as 75<

Mrec < 110 GeV, is 43 in agreement with the StandardModel expectation of 47.6± 0.7.

The AQGC signal prediction is reliable only forrecoil masses lower than the mass of the Z boson as theinterference with the Standard Model processes is notincluded in the calculation. RequiringMrec < 75 GeV,no event is retained by the selection in agreement withthe Standard Model expectation of 0.35± 0.05 events.

A reweighting technique based on the full matrixelements as calculated by the EENUNUGGANO MonteCarlo, is used to derive the AQGC values. Includingthe results at

√s = 183 GeV and

√s = 189 GeV [7],


the 95% confidence level upper limits are:

−0.031< a0/Λ2 < 0.031 GeV−2,

−0.090< ac/Λ2 < 0.090 GeV−2,

where only one parameter is varied at a time.The dominant systematic uncertainty comes from

the theoretical uncertainty of 5% [21] in the calcula-tion of anomalous cross sections.

4.3. Combined results

The results obtained from the W+W−γ andννγ γ

processes are combined. No evidence of AQGC isfound and 95% confidence level limits are obtainedseparately on each coupling as:

−0.015< a0/Λ2 < 0.015 GeV−2,

−0.048< ac/Λ2 < 0.026 GeV−2,

−0.14< an/Λ2 < 0.13 GeV−2.

Appendix A

The results on the e+e− → W+W−γ cross sectionsare also expressed for a different phase space regiondefined by:

• Eγ > 5 GeV,• |cos(θγ )| < 0.95,• cos(αγ ) < 0.90,• Mff ′ = MW ± 2ΓW, whereMff ′ are the two fermi-

on-pair invariant masses.

The results read:

σWWγ (188.6 GeV)

= 0.20± 0.09± 0.01 pb

(σSM = 0.190± 0.010 pb),

σWWγ (194.4 GeV)

= 0.17± 0.10± 0.01 pb

(σSM = 0.219± 0.011 pb),

σWWγ (200.2 GeV)

= 0.43± 0.13± 0.02 pb

(σSM = 0.242± 0.012 pb),

σWWγ (206.3 GeV)

= 0.13± 0.08± 0.01 pb

(σSM = 0.259± 0.013 pb),

where the first uncertainty is statistical, the secondsystematic and the values in parentheses indicate theStandard Model predictions.

References

[1] S.L. Glashow, Nucl. Phys. 22 (1961) 579;S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264;A. Salam, in: N. Svartholm (Ed.), Elementary Particle Theory,Almquist and Wiksell, Stockholm, 1968, p. 367.

[2] M. Veltman, Nucl. Phys. B 7 (1968) 637;G.M. ’t Hooft, Nucl. Phys. B 35 (1971) 167;G.M. ’t Hooft, M. Veltman, Nucl. Phys. B 44 (1972) 189;G.M. ’t Hooft, M. Veltman, Nucl. Phys. B 50 (1972) 318.

[3] G. Bélanger, F. Boudjema, Nucl. Phys. B 288 (1992) 201.[4] J.W. Stirling, A. Werthenbach, Eur. Phys. J. C 14 (2000) 103.[5] J.W. Stirling, A. Werthenbach, Phys. Lett. B 466 (1999) 369.[6] B. Adeva et al., L3 Collaboration, Nucl. Instrum. Methods

A 289 (1990) 35;J.A. Bakken et al., Nucl. Instrum. Methods A 275 (1989) 81;O. Adriani et al., Nucl. Instrum. Methods A 302 (1991) 53;B. Adeva et al., Nucl. Instrum. Methods A 323 (1992) 109;K. Deiters et al., Nucl. Instrum. Methods A 323 (1992) 162;M. Chemarin et al., Nucl. Instrum. Methods A 349 (1994) 345;M. Acciarri et al., Nucl. Instrum. Methods A 351 (1994) 300;G. Basti et al., Nucl. Instrum. Methods A 374 (1996) 293;A. Adam et al., Nucl. Instrum. Methods A 383 (1996) 342.

[7] M. Acciarri et al., L3 Collaboration, Phys. Lett. B 490 (2000)187.

[8] G. Abbiendi et al., Opal Collaboration, Phys. Lett. B 471(1999) 293.

[9] M. Acciarri et al., L3 Collaboration, Phys. Lett. B 478 (2000)34;M. Acciarri et al., L3 Collaboration, Phys. Lett. B 505 (2001)47.

[10] KORALW version 1.33 is used:S. Jadach et al., Comput. Phys. Commun. 94 (1996) 216;S. Jadach et al., Phys. Lett. B 372 (1996) 289.

[11] E. Barberio, B. van Eijk, Z. Was, Comput. Phys. Commun. 79(1994) 291.

[12] JETSET version 7.409 is used,T. Sjöstrand, H.U. Bengtsson, Comput. Phys. Commun. 46(1987) 43.

[13] PYTHIA version 5.722 is used,T. Sjöstrand, Comput. Phys. Commun. 82 (1994) 74.

[14] F.A. Berends, R. Pittau, R. Kleiss, Comput. Phys. Commun. 85(1995) 437.

[15] YFSWW3 version 1.14 is used:S. Jadach et al., Phys. Rev. D 54 (1996) 5434;S. Jadach et al., Phys. Lett. B 417 (1998) 326;S. Jadach et al., Phys. Rev. D 61 (2000) 113010;S. Jadach et al., hep-ph/0007012.


[16] A. Denner et al., Phys. Lett. B 475 (2000) 127;A. Denner et al., Nucl. Phys. B 587 (2000) 67.

[17] KORALZ version 4.03 is used,S. Jadach et al., Comput. Phys. Commun. 79 (1994) 503.

[18] G. Montagna et al., Nucl. Phys. B 541 (1999) 31.[19] GEANT Version 3.15 is used:

R. Brun et al., GEANT 3, preprint CERN DD/EE/84-1, 1984,revised 1987;The GHEISHA program, H. Fesefeldt, RWTH Aachen reportPITHA 85/02, 1985, is used to simulate hadronic interactions.

[20] M. Acciarri et al., L3 Collaboration, Phys. Lett. B 496 (2000)19.

[21] J.W. Stirling, A. Werthenbach, private communication.[22] O.J.P. Eboli, M.C. Gonzales-Garcia, S.F. Novaes, Nucl. Phys.

B 411 (1994) 381.[23] A. Denner et al., Eur. Phys. J. C 20 (2001) 201.[24] G. Montagna et al., Phys. Lett. B 515 (2001) 197.[25] M. Acciarri et al., L3 Collaboration, Phys. Lett. B 444 (1998)

503;M. Acciarri et al., L3 Collaboration, Phys. Lett. B 470 (1999)268.



Search for a narrow state at 1.9 GeV in 3π+2π−π0 exclusiveevents innp annihilation

M. Agnelloa, M. Astruab, E. Bottab, T. Bressanib, D. Calvob, A. Feliciellob,A. Filippi b,1, F. Iazzia, S. Marcellob

a Dipartimento di Fisica, Politecnico di Torino, and INFN, Sezione di Torino, Torino, Italyb Dipartimento di Fisica Sperimentale, Università di Torino, and INFN, Sezione di Torino, Torino, Italy

Received 10 December 2001; accepted 8 January 2002

Editor: L. Montanet

Abstract

No evidence has been found for a narrow state at 1.911 GeV, recently reported by E687, in an analysis of the 3π+2π−π0

exclusive events produced innp annihilations in flight. 2002 Elsevier Science B.V. All rights reserved.

PACS: 13.75.Cs; 12.39.Mk; 14.40.Cs

Keywords: Narrow state; Resonance formation;np annihilation at low energy

In a recent paper [1] the E687 Collaboration re-ported evidence for a narrow dip in the mass spectrumof the final state 3π+3π− produced by diffractive pho-toproduction. The dip was interpreted as due to the de-structive interference of a continuum background witha narrow resonance at(1.911±0.04±0.01)GeV withΓ = (29± 11± 4) MeV. The quantum numbers as-signed to this state wereJPC = 1−−, G = +1 andI = 1.

If such a state exists, it could be visible also in other6π final states produced in other interactions, like forexampleNN , provided the claimed quantum numbercan be reached. We performed a series of mesonspectroscopy investigations [2–4] with the OBELIX

E-mail address: [email protected] (A. Filippi).1 Address: INFN Torino, Via P. Giuria, 1, 10125 Torino, Italy.

spectrometer installed on the M2-branch of the LEARcomplex at CERN. Most of the studies were performedby analysing exclusive final states constituted byπ

and/orK mesons produced in the annihilation ofp

at rest on protons. The upper limit of the invariantmass spectra was∼ 1.70 GeV, the maximum allowedby kinematics for a massive object recoiling against a“spectator” pion emitted in a two-body annihilation,according to the isobar model. Some experiments[5–7] were performed also withn’s annihilating inflight on protons (the isospin of the reaction beingfixed to 1), with momenta from∼ 50 MeV/c (Tn,Lab =1.3 MeV) to 405 MeV/c (Tn,Lab = 84 MeV), and thedata analysed in the frame of the isobar model aswell. The upper limit of the invariant mass spectra wasslightly increased (1.75 GeV).

With n’s in flight it could also be possible tostudy exclusive final states resulting from the decay



40 M. Agnello et al. / Physics Letters B 527 (2002) 39–42

of objects directly formed in the annihilation, in aso narrow mass range (1.88–1.92 GeV) that thispossibility was neglected in our previous analyses.However, this range, close to theNN threshold, isexactly the one where E687 reports evidence for anarrow state, and we have searched for a possibleconfirmation in our data.

We examined the 3π+2π−π0 exclusive final statein np annihilation. At low energy this final state canbe produced only by3S1 and 1P1 initial states, andthe former allows to form a state withJP = 1−. Theprobability of S- and P -wave annihilations in flightis energy dependent and we referred to the model byDover and Richard [8] to reproduce the different anni-hilation probability trends from the two initial states;a proper convolution by the corresponding theoreti-cal distribution was applied to the experimental spec-tra whenever an hypothesis on the initial state was set.According to the Dover–Richard model, whose valid-ity was confirmed, among others, by our data on se-lected two-bodynp annihilations [9–11], the overallfraction of S-wave annihilations, integrated over theavailablen momentum range, is(57 ± 3)%; 3/4 ofthem proceed from3S1 initial state, assuming a sta-tistical distribution between the different spin com-ponents ofS-wave. Therefore, the3S1 wave amountshould be large enough to allow the observation ofthe mentioned narrow state. In fact, normalising to theannihilation probability over the two allowed initialstates, always assuming a statistical population ofS-and P -wave spin sublevels, about 80% of annihila-tions into 3π+2π−π0 must proceed from3S1, whilethe remainder from1P1.

The OBELIX spectrometer, then beam and themethods and criteria for the analysis of the spec-tra were described in previous papers [5,6,12]. The3π+2π−π0 exclusive events were selected out of thefive prong bulk by means of a 1C kinematic fit at 10%C.L., requiring the total energy of the tracked particlesto be less than 1.8 GeV. Only events with correctly re-constructed antineutron momentum and a well definedannihilation vertex in the target were retained. A 4Ckinematic fit (1% C.L.) was applied as well in order todiscardnp → 3π+2π− contaminating events.

The invariant mass resolution for the exclusive3π+2π−π0 events was about 6 MeV (RMS), eval-uated from the missing mass distribution. The se-lected sample is affected by a 16% background, com-

ing mainly from reactions with more than oneπ0.We measured that theωπ+π+π− annihilation chan-nel contributes to the selected sample to a level of 6%;on the contrary, noηπ+π+π− events are observed.

Fig. 1 shows the distribution of the 6π exclusiveevents yield normalised to the total number of eventsin a given bin of total center of mass energy cor-responding to inclusive 2π+π− and 3π+2π− finalstates. We consider the inclusive final states distribu-tions as very similar to the 6π exclusive ones as far asacceptances and geometrical cuts are concerned. Theyare inherently structureless. The error on each bin ofthe experimental distribution in Fig. 1 is directly con-nected to the error on the antineutron momentum mea-surement, whose details are reported in Ref. [12].

The obtained distribution is flat, with no hint fora narrow structure with parameters as those reportedby Ref. [1]. In the hypothesis that the mechanism forthe photoproduction of the mentioned narrow state and

Fig. 1. Yield for np → 3π+2π−π0 events as a function of thetotal c.m. energy, normalised to the total number of inclusivenp → 2π+π−X and np → 3π+2π−X events. The dotted linecorresponds to the trend expected if a resonant state with theparameters and the weight (31%) as in Ref. [1] is added coherentlyto anS-wave like continuum background. The details on how thiscurve has been obtained are explained in Fig. 2. The solid line isthe fit to the data by means of theP -wave part of the uncorrelatedbackground and the coherent sum of theS-wave part of it anda Breit–Wigner function centered at 1.911 GeV, 29 MeV wide,with free weights and phase. The contribution of the Breit–Wignerfunction in this case is less than 1.5% (see text).

M. Agnello et al. / Physics Letters B 527 (2002) 39–42 41

the formation of a 1− state innp annihilation are thesame, we checked the effect of the interference of aresonant state as the one observed in Ref. [1] and abackground, supposing it continuum, incoherent andmainly produced with 1− quantum numbers.

This is of course a simplification. First of all innp annihilations a different component of the isospin1 multiplet (I3 = +1) is formed. Other annihilationreactions can occur, producing the same final state,as ω(η)π+π+π−; these intermediate states are ab-sent in 3π+3π− photoproduction. The removal ofωπ+π+π− events however doesn’t change substan-tially the shape of the energy spectrum.

Moreover, the background innp data might havea different composition, since it could even proceed

by P -wave annihilations. To this purpose the spectrumwas first fitted by a second order polynomial functionweighted by a mixture ofS- andP -wave probabilitydistributions, in order to estimate the trends of six pi-ons incoherent production from the two allowed ini-tial states. We could, therefore, separate the total back-ground into anS- and aP -wave component. The con-tribution fromS-wave was shown to be dominant, asexpected (about 90%).

We plotted the expected distribution in case 31%of the exclusive 6π events were due to a narrow reso-nance, described by a Breit–Wigner function withm =1.911 GeV andΓ = 29 MeV, following the method in-dicated in Ref. [1] (dotted curve in Fig. 1). How thiscurve has been obtained is explained in detail in Fig. 2.

Fig. 2. Steps to get the curve reported in Fig. 1: a Breit–Wigner amplitude, centered atm = 1.911 GeV (dotted line), 29 MeV wide (a), isconvoluted with the Dover–Richard’s function forS-wave annihilation, and properly normalised (b). As a consequence of the convolution, theposition of the peak is slightly shifted as compared to the resonance nominal mass. A fit of the experimental spectrum with polynomial functionsweighted by Dover–Richard’s distributions [8] allows to disentangle the contribution of theS-wave (dot-dashed curve in (c)) andP -wave (solidcurve in (c)) part of the continuum background. The dot-dashed curve in (d) shows the coherent sum of the function in (b) and the dot-dashedcurve in (c), with phase and relative weight as reported in Ref. [1]. Adding incoherently to this curve theP -wave contribution to the continuumbackground, one gets the solid curve in (d), reported in Fig. 1 in a narrower energy range (up to the dashed vertical line shown in (c) and (d)).

42 M. Agnello et al. / Physics Letters B 527 (2002) 39–42

The Breit–Wigner interferes with theS-wave part ofthe background with a relative phaseφ = 62. TheBreit–Wigner amplitude must as well be convolutedby the Dover–Richard’s distribution forS-wave anni-hilation, to reproduce correctly in our case the forma-tion of a 1− state as a function of the available energy.The P -wave part of the background was then addedincoherently to this amplitude.

However, the curve in Fig. 1 does not reproduce thedata. A fit with free weights for the resonance signaland the background and their relative phase showsthat, again, theS-wave term of the total amplitudeis dominant, and the maximum contribution from theresonant state is 1.5%: this figure can be understood asan upper limit for its presence in our data. The relativephase between the resonance and the polynomialbackground is found to be(189± 24).

Therefore, we cannot confirm that the dip at1.911 GeV observed by E687 is due to a narrow reso-nance centered at this energy, if the formation mecha-nism is the same in photoproduction as well as innp

annihilation, and if all the components of the isospin 1multiplet showed up in a similar way.

If these last hypotheses hold, several explanationscan be proposed to justify the disagreement betweenthe two observations. The first is the presence of apossible systematic experimental error on the absoluteenergy scale. An about 2% error would bring the realvalue of mass out of our very narrow mass range.Another one is an inadequacy of the analysis method,based on a simple fit by means of a Breit–Wignerfunction of the observed dip in order to describe theinterference phenomenon.

A final remark is that, if the observed dip cor-responded to a resonancebelow NN threshold, itwould confirm previous observations of anomaloustrends. We remind that the FENICE Experiment [13]

reported a dip in thee+e− → multihadron cross sec-tion, described by the presence of a narrow state at(1870± 10) MeV, with a decay width of∼ 10 MeV.Such a state explained also the anomalous trends of thenucleon form factor in the time-like region. An anom-alous behaviour of the elasticnp cross section at lowmomenta was also recently reported [14]. A possibleexplanation [15] was the presence of a narrow statebelow threshold, corresponding to a spin triplet con-figuration of thenp system.

References

[1] E687 Collaboration, P.L. Frabetti et al., Phys. Lett. B 514(2001) 240.

[2] OBELIX Collaboration, C. Cicalò et al., Phys. Lett. B 462(1999) 453.

[3] OBELIX Collaboration, A. Bertin et al., Phys. Lett. B 414(1997) 200.

[4] OBELIX Collaboration, A. Bertin et al., Phys. Lett. B 408(1997) 476.

[5] OBELIX Collaboration, A. Bertin et al., Phys. Rev. D 57(2000) 55.

[6] OBELIX Collaboration, A. Filippi et al., Phys. Lett. B 495(2000) 284.

[7] OBELIX Collaboration, A. Adamo et al., Nucl. Phys. B 558(1993) 13c.

[8] C.B. Dover et al., Prog. Part. Nucl. Phys. 29 (1992).[9] OBELIX Collaboration, A. Bertin et al., Phys. Lett. B 410

(1997) 344.[10] OBELIX Collaboration, A. Filippi et al., Nucl. Phys. A 655

(1999) 453.[11] OBELIX Collaboration, A. Filippi et al., Phys. Lett. B 471

(1999) 263.[12] M. Agnello et al., Nucl. Instrum. Methods A 399 (1997) 11.[13] FENICE Collaboration, A. Antonelli et al., Nucl. Phys. B 517

(1998) 3.[14] OBELIX Collaboration, F. Iazzi et al., Phys. Lett. B 475 (2000)

378.[15] A.E. Kudryavtsev, B.L. Druzjinin, ITEP-23-94.



First observation of radiative photons associated with the µ−transfer process from (tµ−) to 3He through an intermediate

(t 3He µ−) mesomolecule

T. Matsuzakia,∗, K. Nagaminea,b, K. Ishidaa, N. Kawamuraa, S.N. Nakamuraa,1,Y. Matsudaa, M. Tanasec,2, M. Katoc, K. Kurosawac, H. Sugaic, K. Kudod,

N. Takedad, G.H. Eatone

a Muon Science Laboratory, RIKEN (The Institute of Physical and Chemical Research), 2-1 Hirosawa, Wako, Saitama 351-0198, Japanb Meson Science Laboratory, Institute of Materials Structure Science, High Energy Accelerator Research Organization (KEK-MSL),

1-1 Oho, Tsukuba, Ibaraki 305-0801, Japanc Department of Radioisotopes, Japan Atomic Energy Research Institute (JAERI), 2-4 Shirane, Tokai, Ibaraki 319-1106, Japan

d Quantum Radiation Division, Electrotechnical Laboratory (ETL), Tsukuba, Ibaraki 305-8568, Japane ISIS Facility, Rutherford Appleton Laboratory (RAL), Chilton, Didcot, Oxon, OX11 0QX, UK

Received 6 April 2001; received in revised form 10 December 2001; accepted 10 December 2001

Editor: J.P. Schiffer

Abstract

For the first time, we have observed the 6.76 keV radiative transition photons associated with the µ− transfer process from(tµ−) atoms to3He nuclei through intermediate (t3Heµ−) mesomolecule formation in a solid T2 target. The radiative decaybranching ratio of the (t3Heµ−) mesomolecule and the muon transfer rate were determined and compared with theoreticalvalues. We also studied an accumulation process of3He atoms in a solid T2 target by observing the neutron decay ratesoriginating from t–t muon-catalyzed fusions. Their time dependence indicates a sudden3He bubble formation in the solid T2 atan atomic concentration of 130 ppm. 2002 Elsevier Science B.V. All rights reserved.

PACS: 36.10.Dr; 34.70.+e; 29.30.Kv; 25.30.-c

Keywords: Muon transfer; Mesomolecule formation; Radiative transition photon; Muon-catalyzed fusion;3He bubble formation in solid T2

* Corresponding author. Teiichiro Matsuzaki, Muon Science Laboratory, RIKEN (The Institute of Physical and Chemical Research), 2-1Hirosawa, Wako, Saitama, 351-0198 Japan. Tel: +81-48-467-9354; Fax: +81-48-462-4648

E-mail address: [email protected] (T. Matsuzaki).1 Present address: Department of Physics, Tohoku University, Aramaki, Aoba-ku, Sendai 980-8578, Japan.2 Present address: Department of Radiation Research for Environment and Resources, Takasaki Radiation Chemistry Research Establish-

ment, Japan Atomic Energy Research, Institute, Watanuki-cho, Takasaki, Gunma 370-1292, Japan.



44 T. Matsuzaki et al. / Physics Letters B 527 (2002) 43–49

A negative muon transfer process from muonic hy-drogen atoms to helium nuclei is one of the impor-tant subjects related to the muon catalyzed d–t fu-sion (µCF) process in a hydrogen target system con-taining a helium impurity. In the µCF process, nega-tive muons stopped in a D–T mixture induce sponta-neous d–t nuclear fusions without any additional de-vices, and behave as catalysts for repeated µCF cy-cles during the muon lifetime. The muon mass is 207times larger than the electron mass, and the size ofmuonic hydrogen atoms, (dµ−) and (tµ−), are, there-fore, smaller than those of electronic hydrogen atomsby the same amount. These small muonic atoms ofneutral charge can approach the other hydrogen nu-clei, d or t, and induce d–t nuclear fusions without ex-periencing a Coulomb repulsive force. After d–t fusionproducing anα-particle (4He) and a neutron (n), mostof the negative muons are liberated to participate in thenext µCF cycle: d+ t+µ− → α+n+µ− +17.6 MeV.However, a small fraction of the negative muons arecaptured by theα-particle in the reaction, thereby ter-minating the µCF cycle. This fraction is called theα-sticking probability, and is the most important factorin limiting the number of fusion neutrons from a singlenegative muon, which places a limit on energy produc-tion by applying the µCF phenomena [1]. In the actualµCF process, helium impurities gradually accumulatein the D–T target. The major components are3He nu-clei originating from the tritiumβ-decay; the other is4He nuclei of the d–t fusion product. The muon lossprocess due to muon transfer from muonic hydrogenatoms to the accumulated helium nuclei is an impor-tant problem to understand the actual µCF process inthe D–T mixture, while muon sticking to anα-particleis the major muon loss in the µCF cycle.

In 1986, we succeeded for the first time in ob-serving the 6.85 keV radiative transition photons as-sociated with the µ− transfer from (dµ−) atoms to4He nuclei through the intermediate (d4He µ−) meso-molecule at KEK-MSL, using a liquid-deuterium tar-get with 4He impurity dissolved by pressurizing theliquid-deuterium surface with the4He gas [2]. Themuon transfer process from (dµ−) atoms to4He nucleithrough the intermediate (d4He µ−) mesomolecule isdescribed as

(dµ−) + 4He→ (d4Heµ−)∗

(1)→ d+ (4Heµ−) + γ(6.85 keV).

The 6.85 keV photon corresponds to a radiationphoton originating from the transition from the excitedstate to the unbound ground state of the mesomole-cule. The 6.85 keV photon observation provides di-rect evidence of the theoretically predicted µ− transfermechanism through the intermediate (d4He µ−) me-somolecule. Aristov et al. have proposed a descrip-tion of the µ− transfer process from muonic hydro-gen atoms to helium nuclei via the formation of in-termediate mesomolecules, and calculated the energylevels and the formation rates of mesomolecules [3].Kravtsov et al. have calculated the photon energy spec-tra originating from deexcitations of the excited meso-molecules formed by hydrogen and helium isotopes,and predicted asymmetrically energy-broadened pho-ton line shapes, which reflect the potential energycurves of the transitional states [4]. The photon en-ergy, the asymmetric line shape and the transfer rateobtained in our experiment were in good agreementwith theoretical values [2]. On the other hand, the di-rect transfer probability from the (dµ−) atoms to the4He nuclei,(dµ−) + 4He→ d + (4Heµ−), was cal-culated to be small:∼ 105 s−1 [5]. We have contin-ued further experimental studies on the muon transfermechanism from hydrogen to helium in the systems ofd–3He, d–4He and p–4He at KEK-MSL [6]. The trans-fer rates and the radiative decay branching ratios wereobtained and compared with theoretical values [7–9].A particle-emitting decay mode of the excited meso-molecules has been proposed theoretically in additionto radiative photon emission, which has explained wellthe isotope dependence of the radiative decay branch-ing ratios of mesomolecules [6]. Several experimentshave also been performed at PSI for these systems toobtain the photon energy spectra, transfer rates and ra-diative decay branching ratios [10,11].

As for the t–3He system, the µ− transfer mecha-nism can be expressed similarly as

(tµ−) + 3He→ (t 3Heµ−)∗

(2)

→ t + (3Heµ−) + γ (radiative photon).

The muon transfer rates from (tµ−) atoms to3Henuclei accumulated in the D–T target and their temper-ature dependence were obtained by the LAMPF groupin their fusion neutron data analysis of d–t µCF studies[12]. However, so far, no direct experiment has been

T. Matsuzaki et al. / Physics Letters B 527 (2002) 43–49 45

carried out to investigate the µ− transfer mechanismin the t–3He system by observing the radiative pho-ton, in spite of its importance for understanding therealistic d–t µCF process with the existence of a small3He impurity. Theoretical studies on the t–3He systemhave been made to predict the transfer rate [13], the ra-diative photon spectrum [4] and the decay rate of the(t 3He µ−) mesomolecule [8,9].

We recently performed a t–t µCF experiment us-ing a solid T2 target as a part of the d–t µCF re-search program at the RIKEN-RAL Muon Facility.A preliminary result on the t–t µCF experiment hasbeen published [14]. In the experiment, we also suc-ceeded in observing radiative photons associated withthe µ− transfer process from (tµ−) atoms to3He nu-clei through the intermediate (t3He µ−) mesomole-cule, where the3He impurity originates from the tri-tium β-decay and accumulates in the solid T2 target.We obtained the radiative decay branching ratio of the(t 3He µ−) mesomolecule and the muon transfer ratefrom (tµ−) atoms to3He nuclei. In this Letter we re-port on the experimental results concerning the muontransfer process in the t–3He system.

We conducted the present experiment using thesame experimental set-up installed for a series of d–tµCF studies [15] at Port 1 of the RIKEN-RAL pulsedMuon Facility located at the Rutherford AppletonLaboratory in the UK. The T2 target gas was producedat the Department of Radioisotopes of JAERI [16],and installed in an in-situ tritium gas-handing system[17]. The isotope enrichment of T was 99.1% andthe remaining component was H of 0.9%. The targetgas was purified by passing it through a palladiumfilter to remove any3He impurity originating fromtritium β-decay just before the measurements. TheT2 gas was introduced to the target cell cooled bya helium-flow cryostat and solidified at 16 K [17].The solid T2 target with a volume of 0.55 cm3 wasformed by the T2 gas: a volume of 0.558 liter at STPand an inventory of 53.7 TBq (1450 Ci). The targetcell, made of cupro-nickel alloy, was a cylinder of∅14 mm×14 mm with a beryllium window of 0.5 mmthickness for a low-energy photon observation. Theinside surface of the cell was covered by a silverfoil of 0.2 mm thickness to absorb any muon beam-induced photon background from the target wall. Thetarget was positioned at the magnetic field centerof a superconducting Helmholtz coil. A magnetic

field of 2.4 T converged the incoming µ− beamtowards the target cell and prevented most of the µ–edecay electron background from reaching the photondetector, thereby reducing the photon background. Forphoton detection, a Si(Li) X-ray detector (70 mm2

× 3.5 mm) was placed perpendicular to the µ−beam at a distance of 13.3 cm from the target. Inorder to detect the t–t fusion neutron, two calibratedNE213 liquid scintillators (∅2 inch × 2 inch) werepositioned at a distance of 84 cm downstream of thetarget. Lead bricks of 5 cm thickness were placedin front of the neutron counters to eliminate the µ–edecay electron background. The µ–e decay electronsoriginating from the solid T2 target were detectedby segmented plastic scintillation counters located atthe backward and forward directions from the target.A decay µ− beam with a momentum of 54.5 MeV/cwas extracted by the superconducting muon channel[18] and injected to the T2 target. About 34 muonswere stopped in the target during every muon pulsewith a double-pulsed time-structure (100 ns totalpulse width and 230 ns pulse separation at 50 Hz).It should be emphasized here that a pulsed muonbeam is essential for a photon detection experimentunder a huge white-noise type radiation backgroundassociated with the Bremsstrahlung of tritiumβ-raydecay. A vital improvement of the signal-to-noiseratio in the delayed photon spectrum was obtained byopening the observation time window synchronouslywith the muon pulse. We used the solid T2 target inthe experiment so that almost all the quantity of T2gas could be collected in the target cell for achievingthe maximum experimental yields.3He removal fromthe solid T2 target was carried out by either a T2 gas-purification using a tritium gas-handling system, orby forming it through the T2 liquid phase from thegaseous T2. In the course of the d–t µCF experiments,we found that3He accumulated in the solid D–Ttarget, but did not do so in the liquid target, byobserving the time dependence of the fusion neutrondecay rates after3He removal [19]. Immediatelyfollowing solid T2 formation, data taking was started,and continued for 60 hours; the final3He atomicconcentration reached to 385 ppm in the target. Sincethe data taking was made in an event-by-event mode[20], all of the data of the photon, neutron and decayelectron could be analyzed off-line as a function oftime (τ ) after 3He removal, so that the data at the


Fig. 1. Typical delayed photon spectrum originating from µ−stopped in a solid T2 target for a time region from 0.08 to 2.08 µsafter a muon pulse. This is an integrated spectrum for a timeperiod from τ = 0 to τ = 60 hours after solidification, and theBremsstrahlung background from tritiumβ-ray being subtracted.

3He-free limit could be obtained by extrapolating toτ = 0.

A typical delayed photon spectrum is shown inFig. 1. We have clearly observed for the first time acharacteristic radiative photon at an energy of (6.76±0.06) keV with a line width of (0.89± 0.14) keV atFWHM. These values are in good agreement with thepredicted values of the photon spectrum in the t–3Hesystem [4]. The observed photon has an asymmetricline shape with a tail at the low energy side; this featurehas also been predicted by theory [4]. Therefore,this photon can be considered to originate from theradiative decay of the excited (t3He µ−) intermediatemesomolecule formed in the µ− transfer process from(tµ−) atoms to the accumulated3He nuclei in the solidT2 target. In addition to the radiative photon, we haveobserved Kα (8.2 keV) and Kβ (9.6 keV) muonicX-ray lines, which originate from the (αµ−) atomsformed by the µ− to 4He sticking in the t–t µCF cycles:t + t + µ− → (αµ−) + 2n + 11.3 MeV [14]. Thesolid curve in the figure is a typical fitting result usingsingle Gaussian line shapes for Kα and Kβ lines, andan asymmetric Gaussian line shape for the radiative

photon with different Gaussian line widths at the lowand high-energy sides.

The measured neutron shows a simple exponen-tial decay time spectrum with a single componentand a continuous recoil proton energy spectrum up to9 MeV. A quantitative analysis of the observed energyspectrum is complicated because it is overlapped bytwo neutron-energy components from the t–t µCF re-action [14]. On the other hand, the single componentof the neutron time spectrum, called the neutron disap-pearance rate, gives information about the muon trans-fer loss process from (tµ−) atoms to the accumulated3He nuclei, because the active muons contributing tothe t–t µCF cycle are taken away by the muon transferprocess and the neutron disappearance rate increasesaccording to the3He atomic concentration in the solidT2 target.

The neutron disappearance rate,λn(τ ), in the solidT2 at time τ after the 3He removal is describedas follows, by assuming that the total muon loss iscomposed of two major components of the muontransfer and muon sticking processes in the t–t µCFcycle:

(3)λn(τ ) = λ0 + Wφλc + φCHe(τ )λt 3Heµ,

whereλ0, W , φ, λc , CHe(τ ) andλt 3Heµ are the free-muon decay rate (0.455× 106 s−1), effective stickingprobability in the t–t µCF cycle, T2 target densitynormalized to the liquid-hydrogen density (4.25 ×1022 atoms/cm3), t–t fusion cycling rate independentof the 3He impurity, 3He atomic concentration inthe solid T2 target at timeτ and muon transfer ratefrom (tµ−) atoms to3He nuclei, respectively. We haveobserved a time-dependent change ofλn(τ ), as shownin Fig. 2. Theλn(τ ) for the solid T2 target shows threeinteresting features: (1) a linear increase forτ = 0–20 h, (2) a gradual decrease forτ = 20–35 h and (3)nearly constant forτ = 35–60 h. A similar behaviorwith a different time scale has been observed for theD–T solid target (Ct = 70%), as also shown in thefigure. These interesting phenomena are consideredto originate from the3He accumulation effect in thesolid hydrogen target. On the other hand, such atime dependence ofλn(τ ) has never been observedin a liquid D–T target. This fact means that the3Heaccumulates in the solid T2 and D–T targets, butdoes not do so in liquid targets, because the3He isreleased to the gaseous space in the target [19]. We


Fig. 2. Time-dependent change of the fusion neutron disappearancerate,λn(τ ), for a solid T2 target in 2-hour bins after solidification.Similar data for a solid target of D–T mixture (Ct = 70%) is alsoshown for a comparison.

can neglect here the3He concentration dissolved inthe liquid target proportional to the partial pressurein the gaseous space according to Henry’s law. Thecalculated3He atomic concentrations in the solid atτ = 20 h for T2 and τ = 30 h for D–T (Ct = 70%)are approximately 130 ppm. After exceeding thisatomic concentration, the observedλn(τ ) graduallydecreases, indicating a decrease in the effective3Heatomic number contributing to the muon transferprocess in the solid hydrogen. This suggests thatthe distributed3He atoms collect together at thisspecific atomic concentration and create3He bubblesat the interstitial sites of the solid hydrogen lattice.However, further experimental and theoretical studiesare required to clarify this phenomenon.

It should be mentioned here that the solid tritiumin a cylindrical cavity is known to form a uniform andstable distribution with a certain time constant due tothe sublimation effect induced by theβ-decay radi-ation heating [21]. In the d–t µCF experiments withsolid targets of high tritium concentrations [19], wemonitored the change of stopping muon numbers inthe target by the µ–e decay, and found that the D–Tsolid formed the stable distribution at 16 K within onehour after the solidification. In the present experiment,

we also confirmed that the stable distribution of T2solid target completed within one hour. On the con-trary, the observed changes ofλn, as shown in Fig. 2,occur with a longer time scale than expected from thesublimation effect, and the phenomena cannot be ex-plained by the sublimation effect.

The observed increase ofλn(τ ) due to the µ− trans-fer process at timeτ after3He removal is described as

(4)λn(τ ) − λn(0) = φCHe(τ )λt 3Heµ,

where λn(0) (0.882 ± 0.030 µs−1) is the neutrondisappearance rate atτ = 0. The CHe(τ ) is simplyexpressed asCHe(τ ) = CTλTτ , whereCT andλT arethe tritium concentration (CT = 0.991) of the solidT2 target and the tritium decay rate (1.54 × 10−4

day−1), respectively. We can therefore expect a linearincrease ofλn(τ ) against the timeτ after3He removal.By taking a linear increasing region of the observedλn(τ ) (τ = 0 to 20 hours) and assuming that all ofthe 3He atoms accumulate in the solid T2 target, themuon transfer rate,λt 3Heµ, has been obtained to be(4.6± 0.4) × 109 s−1 at 16 K.

The radiative photon yield from the (t3He µ−)mesomolecule,Yt 3Heµ, can be expressed as

(5)Yt 3Heµ(τ ) = εt 3Heµ

λn(τ ) − λn(0)

λn(τ ),

whereεt 3Heµ and(λn(τ ) − λn(0))/λn(τ ) are the ra-diative decay branching ratio of the (t3He µ−) meso-molecule and the muon transfer loss ratio at timeτ

after 3He removal, respectively. The radiative pho-ton yield, corrected for the detection efficiency of theSi(Li) detector, was normalized to the stopping muonnumber in the target. Our measurement showed a goodcorrelation between the time dependence of the ra-diative photon yield and that of the muon transferloss ratio determined by the neutron disappearancerate shown in Fig. 2. The ratio of the radiative pho-ton yield to the muon transfer loss ratio,εt 3Heµ inEq. (5), was calculated at every 4 hours period andwas reasonably constant within the statistical error(typically ±10%). By combining the averaged ratio(0.947± 0.033) with the systematic error (±0.06),the radiative decay branching ratio was obtained to be(0.95± 0.07).


The muon transfer process from (tµ−) atoms to theaccumulated3He nuclei are expressed as

(tµ−) + 3He

→ (t 3Heµ−)∗

(6)→ (3Heµ−) + t + γ radiative decay,

(7)→ (3Heµ−) + t + K.E. particle decay,

(8)→ (3Heµ−) + t + e− Auger emission decay.

Three decay modes of the excited (t3He µ−)∗ me-somolecule have been theoretically predicted. In theradiative decay mode, we have observed the character-istic radiative photons with an asymmetrically energy-broadened line shape to provide direct evidence of thepredicted transfer mechanism through the intermedi-ate (t3He µ−) mesomolecule [3]. The observed pho-ton energy of (6.76± 0.06) keV and the line width of(0.89± 0.14) keV at FWHM in the present work arein good agreement with the theoretical values, reflect-ing the potential energy curve of the (t3He µ−) me-somolecule [4]. By taking into account of the particledecay mode, the isotope dependence of radiative de-cay branching ratios has been explained well by thereduced-mass effect for the mesomolecules formed inthe d–3He, d–4He and p–4He systems [6]. As for theAuger emission decay mode, the calculated rates areabout 25% of the radiative decay rates of the meso-molecules [8].

The radiative decay branching ratio is an importantvalue to investigate the dissociation mode of theexcited (t3He µ−) mesomolecule. The obtained valueof (0.95± 0.07) can be compared with the theoreticalvalues, 0.63 [9] and 0.58 [8], and shows a dominanceof the radiative decay mode of the mesomolecule.

We have obtained the muon transfer rate from (tµ−)atoms to3He nuclei to be(4.6 ± 0.4) × 109 s−1 ata temperature of 16 K. The LAMPF group has cal-culated the transfer rates and the temperature depen-dence using the neutron data of their d–t µCF exper-iments. They have obtained a transfer rate of(0.9 ±0.2) × 109 s−1 at the lowest temperature of 100 K[12]. Although these two experimental values were ob-tained at different temperatures, they seem to be com-parable by taking into account the temperature depen-dence ofλt 3Heµ− : it increases rapidly as the tempera-ture decreases from 100 K to 16 K. From a theoreticalpoint of view, the present value may be compared with

the predicted value of 4.6 × 109 s−1 at ε = 0.004 eVfor the simple-approach approximation with electronscreening and averaged over the Maxwellian distrib-ution by Kravtsov et al. [13]. The obtained transferrate can be also considered to be the formation rateof the (t3He µ−) mesomolecule because the dissocia-tion rates of the mesomolecules are much higher, bytwo orders of magnitude:∼ 1011 s−1 [8,9].

In summary, we have observed for the first timethe 6.76 keV radiative photons associated with themuon transfer process from (tµ−) atoms to3He nucleithrough the intermediate (t3He µ−) molecular forma-tion in a solid T2 target. The observed features of thephoton energy spectrum are in good agreement withtheoretical predictions. We also have determined a ra-diative decay branching ratio of (0.95± 0.07) for the(t 3He µ−) mesomolecule and a muon transfer rate of(4.6 ± 0.4) × 109 s−1 at 16 K. These values will begood objectives for theoretical studies on the muontransfer mechanism from (tµ−) atoms to3He nuclei.In addition, we have also studied the3He accumula-tion process in the solid T2 target by observing the t–tµCF neutron disappearance rates. Their time depen-dence after3He removal seems to indicate a sudden3He bubble formation in the solid T2 at an atomic con-centration of 130 ppm.

Acknowledgements

The authors would like to acknowledge contribu-tions to the construction and operation of the µCF fa-cility at the RIKEN-RAL Muon Facility made by as-sociated staff at RAL. The contributions at the earlierstage of the construction made by Dr. H. Umezawa,Prof. H. Kudo and Mr. M. Hashimoto are also ac-knowledged. The authors would like to express theirsincere thanks to Professors the late M. Oda, A. Arima,S. Kobayashi and related persons at RIKEN and Drs.P.R. Williams, T.G. Walker, R.G.P. Voss, A.D. Taylor,W.G. Williams and T.A. Broome and associated staffat RAL for their continuous support and encourage-ment. Helpful discussions with Prof. M. Kamimura,Dr. J.S. Cohen and Dr. E. Hiyama are also acknowl-edged.


References

[1] K. Nagamine, T. Matsuzaki, K. Ishida et al., Muon Catal. Fus. 1(1987) 137;K. Nagamine, T. Matsuzaki, K. Ishida et al., Muon Catal.Fus. 5/6 (1990/91) 289.

[2] T. Matsuzaki, K. Ishida, K. Nagamine et al., Muon Catal. Fus. 2(1988) 217.

[3] Yu.A. Aristov, A.V. Kravtsov, N.P. Popov et al., Sov. J. Nucl.Phys. 33 (1981) 1066.

[4] A.V. Kravtsov, N.P. Popov, G.E. Solyakin et al., Phys. Lett.A 83 (1984) 379.

[5] S.S. Gershtein, Zh. Eksp. Teor. Fiz. 43 (1962) 706, Sov. Phys.JETP 16 (1963) 501.

[6] K. Ishida, S. Sakamoto, Y. Watanabe et al., Hyp. Interact. 82(1993) 111.

[7] Y. Kino, M. Kamimura, Hyp. Interact. 82 (1993) 195.[8] A.V. Kravtsov, A.I. Mikhailov, V.I. Savichev, Hyp. Interact. 82

(1993) 205;A.V. Kravtsov, A.I. Mikhailov, V.I. Savichev, Z. Phys. D 29(1994) 49.

[9] S.S. Gershtein, V.V. Gusev, Hyp. Interact. 82 (1993) 185.[10] S. Tresch, P. Ackerbauer, W.H. Breunlich et al., Hyp. Inter-

act. 101/102 (1996) 221.[11] B. Gartner, P. Ackerbauer, W.H. Breunlich et al., Hyp. Inter-

act. 101/102 (1996) 249;

B. Gartner, P. Ackerbauer, W.H. Breunlich et al., Phys. Rev.A 62 (2000) 012501.

[12] A.J. Caffery, J.N. Bradbury, S.E. Jones et al., Muon Catal.Fus. 1 (1987) 53.

[13] A.V. Kravtsov, A.I. Mikhailov, N.P. Popov, J. Phys. B 19(1986) 2579.

[14] T. Matsuzaki, K. Nagamine, K. Ishida et al., Hyp. Interact. 118(1999) 229.

[15] K. Ishida, K. Nagamine, T. Matsuzaki et al., Hyp. Interact. 118(1999) 203.

[16] H. Kudo, M. Fujie, M. Tanase et al., Appl. Radiat. Isot. 43(1992) 577.

[17] T. Matsuzaki, K. Nagamine, M. Tanase et al., Hyp. Inter-act. 119 (1999) 361;T. Matsuzaki, K. Nagamine, M. Tanase et al., Nucl. Instrum.Methods A (2001), in press.

[18] T. Matsuzaki, K. Ishida, K. Nagamine et al., Nucl. Instrum.Methods A 465 (2001) 365;K. Nagamine, T. Matsuzaki, K. Ishida et al., Hyp. Inter-act. 101/102 (1996) 521.

[19] N. Kawamura, K. Nagamine, T. Matsuzaki et al., Hyp. Inter-act. 118 (1999) 213;N. Kawamura, K. Nagamine, T. Matsuzaki et al., Phys. Lett.B 465 (1999) 74.

[20] S.N. Nakamura, M. Iwasaki, Nucl. Instrum. Methods A 388(1997) 220.

[21] J.K. Hoffer, L.R. Foreman, Phys. Rev. Lett. 60 (1988) 1310.



First observation of neutron–proton halo structure forthe 3.563 MeV 0+ state in6Li via 1H(6He,6Li)n reaction

Zhihong Li, Weiping Liu, Xixiang Bai, Youbao Wang, Gang Lian, Zhichang Li,Sheng Zeng

China Institute of Atomic Energy, P.O. Box 275(46), Beijing 102413, PR China

Received 15 October 2001; received in revised form 3 December 2001; accepted 21 December 2001

Editor: V. Metag

Abstract

The angular distributions of the charge exchange reaction1H(6He, 6Li)n were measured in reverse kinematics with asecondary6He beam at the energy of 4.17A MeV. The data were analyzed in the context of a microscopic calculation. It isshown that both the ground state of6He and the second excited state of6Li (3.563 MeV, 0+) have halo structure. 2002Elsevier Science B.V. All rights reserved.

PACS: 21.10.-k; 21.10.Ft; 21.10.Gv; 25.60.Lg

Keywords: 1H(6He,6Li)n reaction in inverse kinematics; Angular distribution; Microscopic optical potential; Nucleon density distribution;Proton–neutron halo structure

So far the prominent halo structure has been re-vealed in the ground states of some nuclei near theneutron drip line. As the nuclei near the neutron dripline have high isospin, Y. Suzuki et al. [1] raised aquestion: “What about the possibility that stable nu-clei have extended halolike structure in high isospinexcited states?” If the answer is positive, the isobaricanalog state (IAS) of the neutron halo nuclei shouldalso have halolike structure. Arai et al. [2] calculatedthe nucleon density distributions of the ground stateof 6He and of it is isobaric analog state, i.e., the3.563 MeV 0+ state of6Li with a fully microscopicthree-cluster model and predicted that the latter has

E-mail address: [email protected] (Z. Li).

a more conspicuous halolike structure formed by theneutron and the proton surrounding the a core as com-pared to the former.

In recent years, the1H(6He,6Li)n charge exchangereaction has been studied experimentally by twogroups at beam energies of 93A MeV and 41.6A MeV,respectively [3–6]. The aim of these experiments wasin search for the signature of halo structure throughthe measurement of the angular distributions at veryforward angles for the Gamow–Teller transition tothe ground state of6Li and for the Fermi transitionto the 3.563 MeV 0+ state of6Li. No signature ofhalo structure was found in these experiments, how-ever. A microscopic analysis indicated that the pres-ence or absence of a halo structure would not influ-ence the transition strength at 0 [5]. The halo ef-



Z. Li et al. / Physics Letters B 527 (2002) 50–54 51

Fig. 1. Experimental setup.

fect should appear in the region of large angles. Thusthe measurement of full angular distributions for the1H(6He,6Li)n reaction is necessary.

In this Letter, we briefly report our experiment ofthe1H(6He,6Li)n reaction in reverse kinematics witha secondary6He beam at the energy of 4.17AMeVand a microscopic analysis.

The experiment was carried out using the sec-ondary beam facility [7] of the HI-13 tandem accel-erator at China Institute of Atomic Energy, Beijing.The experimental setup was similar to the previouslyillustrated [8], as shown in Fig. 1. A7Li beam withenergy of 44 MeV from the tandem impinged on a D2gas cell at pressure of 1.5 atm, in which6He ions wereproduced via2H( 7Li, 6He)3He reaction. The front andrear windows of the gas cell were Havar foils, each inthickness of 1.9 mg/cm2. The6He beam with energyof 35.7 MeV was delivered through a careful tuningof the magnets’ currents. The typical purity of6Hebeam was about 90%. The main contaminants were7Li2+, 7Li3+, 4He2+ and6Li2+ ions. Because the re-action products to be detected were the6Li ions, the6Li2+ contaminants would make a severe disruptionto the measurement even though they were very few inthe secondary beam. An aluminum absorber in thick-ness of 45.9 mg/cm2 was used for the sake of elimi-nating the6Li2+ as well as7Li ions in the secondarybeam thoroughly. In the meantime the absorber de-

graded the6He energy down to 25 MeV. The beamwas then collimated by an aperture in diameter of3 mm and directed onto a secondary target placedon the focal plane. A polyethylene (CH2)n foil of1.5 mg/cm2 thick served as the secondary target tostudy the reaction of interest, and a carbon foil of1.8 mg/cm2 thick was used to measure the back-ground. The reaction products were detected and iden-tified using aE–E counter telescope consisting of a19.3 µm thick siliconE detector and a 45× 45 mm2

HamamatsuX–Y sensitive silicon detector (PSSD) inthickness of 300 µm. The PSSD enabled us to deter-mine both the remaining energy and emission anglesof the outgoing ions. The inverse kinematics of the1H(6He, 6Li)n reaction restricted the maximum emis-sion angle of6Li ions to about 12.8. The setup shownin Fig. 1 covered an angular region up to 14, thus thefull angular distribution can be measured. The over-all angular resolution was about 2 FWHM, whichmainly resulted from the6He angular straggling inpassing through the aluminum absorber.

The experimental setup also facilitated to determi-nate the accumulated6He events precisely becausethe6He themselves were recorded by the counter tele-scope simultaneously. However, it brought about aproblem of pulse pileup. In order to solve the prob-lem, the beam intensity on the target was kept at avery low level of 300–500 cps and the pulse pileuprejection trigger was used. The measurement for the(CH2)n target accumulated approximately 1.38× 108

6He events, while the background measurement withthe carbon target about 6.53× 107 6He events.

The scatter plot ofE vs. E (remaining energy)from a few runs with the (CH2)n target is shown inFig. 2. For the sake of saving CPU time, we set a cutatE = 1.8 MeV using software, all the events belowthe cut were scaled down by a factor of 100. The scat-ter plot ofEt vs. θlab for all the6Li events within twodimension gate in Fig. 2 is shown in the bottom panelof Fig. 3, whereEt is the total energy andθlab denotesthe laboratory emission angle converted from the po-sition data of PSSD. According to Monte Carlo sim-ulation with the reaction kinematics and experimentalsetup, the events were identified as two groups whichcorresponding to the ground and 3.563 MeV 0+ statesof 6Li, respectively. The projection onto the energyaxis for the6Li events ofθlab < 5 is shown in the toppanel of Fig. 3. A well separation between the ground

52 Z. Li et al. / Physics Letters B 527 (2002) 50–54

Fig. 2. Scatter plot ofE vs.E.

and 3.563 MeV 0+ states of6Li can be seen. In theregion ofθlab > 5, however, the events of two statescan only be identified with the scatter plot ofEt vs.θlab due to the overlap of their projections onto eitherthe energy or angle axis.

The angular distributions are shown in Fig. 4.The open circles and filled circles are the differentialcross sections with the6Li formed in its groundand 3.563 MeV 0+ states, respectively. In the caseof θlab > 5, two states are not well separated, andthus a correction for their mutual contributions wascarried out appropriately. The errors of cross sectionsare due to both the statistical uncertainties and theadditional ones from the above correction. As canclearly be seen from Fig. 4, the cross section leadingto the 3.563 MeV 0+ state of6Li is obviously largerthan that to the ground state aroundθc.m. = 90. Thepoor angular resolution is due to the enlargement ofthe emission angle uncertainties in the transformationfrom laboratory frame to center of mass frame. Thelines are the conventional DWBA calculations withthe zero-range approximation by the code KORP[9] in which the microscopic optical potentials wereadopted. The code has been successfully applied tothe low energy(n,p) and (p,n) reactions [9,10].The experimental angular distributions were fairlyreproduced for both the ground and 3.563 MeV 0+states by the calculations.

In the calculation of microscopic optical potentials,the first and second order mass operators in nuclearmatter were derived with Skyrme effective interactions

Fig. 3. Bottom panel: scatter plot ofEt vs. θlab for 6Li events. Toppanel: a projection for6Li events ofθlab < 5 onto the energy axis.The left and middle peaks correspond to the 3.563 MeV 0+ state,the angular ranges are 150 < θc.m. < 180 and 0 < θc.m. < 40 ,respectively. The right peak corresponds to the ground state,θc.m. < 27. The ground state events ofθc.m. > 160 are rejectedby the threshold ofE–E counter.

and the real and imaginary parts of the optical potentialfor finite nuclei were obtained by applying a localdensity approximation [11]. The proton and neutrondensity distributions of the 3.563 MeV 0+ state of6Liand the ground state of6He used in the calculations areshown in Fig. 5. As to the6Li ground state, the densitydistributions for both neutron and proton are assumedto be the same as the charge density distribution inRef. [12].

Fig. 6 shows the angular distributions of the3.563 MeV 0+ state of6Li calculated with different

Z. Li et al. / Physics Letters B 527 (2002) 50–54 53

Fig. 4. Angular distributions of1H(6He,6Li)n reactions.

Fig. 5. Neutron and proton density distributions for the 3.563 MeV0+ state of6Li and the ground state of6He used in the calculationof microscopic optical potentials. This figure is taken from Ref. [2].

nucleon density distributions. The dotted line resultsfrom the assumption of both 3.563 MeV 0+ state of6Li and the ground state of6He without halo, i.e., thedensity distribution of the former is the same as that ofthe ground state of6Li; and for the ground state of6He,the density distribution of neutron is similar to thatof proton. The dashed-dotted line is the case of6Hewith neutron halo and6Li without halo. The dashedline stands for the situation of6Li with neutron–proton

Fig. 6. DWBA calculations by different nucleon density distribu-tions.

halo and6He without halo. The solid line refers to thecalculation in which both6Li and 6He are assumed tohave halo structure with nucleon density distributionsas shown in Fig. 5. The comparison of the calculationswith different nucleon density distributions bears outthe existence of both the neutron–proton halo struc-ture for the 3.563 MeV 0+ state in6Li and the neutronhalo structure for6He ground state.

In summary, the angular distributions of1H(6He,6Li)n reaction connecting the ground state of6He andthe ground and 3.563 MeV 0+ states of6Li have beenmeasured using the secondary6He beam at energyof 4.17AMeV. The experimental data can be wellreproduced with the microscopic DWBA analysis if itis assumed that both the ground state of6He and thesecondary excited state of6Li have halo structure. Thepresent work reveals the proton–neutron halo structureof the secondary excited state of6Li predicted by Araiet al. [2] for the first time.

Acknowledgements

The authors would like to thank Drs. Q. Shen andC. Lin for their help in the theoretical analysis. Thiswork was funded by the Major State Basic ResearchDevelopment Program under Grant No. G200077400and the National Natural Science Foundation of China

54 Z. Li et al. / Physics Letters B 527 (2002) 50–54

under Grant Nos. 19735010, 19935030, 10025524 and10045002.

References

[1] Y. Suzuki, K. Yabana, Phys. Lett. B 272 (1991) 173.[2] K. Arai, Y. Suzuki, K. Varga, Phys. Rev. C 51 (1995) 2488.[3] M.D. Cortina-Gil et al., Phys. Lett. B 371 (1996) 14.[4] J.A. Brown et al., Phys. Rev. C 54 (1996) R2150.

[5] M.D. Cortina-Gil et al., Nucl. Phys. A 616 (1997) 215c.[6] M.D. Cortina-Gil et al., Nucl. Phys. A 641 (1998) 263.[7] X. Bai et al., Nucl. Phys. A 588 (1995) 273c.[8] W. Liu et al., Phys. Rev. Lett. 77 (1996) 611;

W. Liu et al., Nucl. Phys. A 616 (1997) 131c.[9] Z. Yu, Y. Zuo, CNIC-00736, NKU-0002 (1993).

[10] Q. Shen, J. Zhang, Phys. Rev. C 50 (1994) 2473.[11] Q. Shen et al., Z. Phys. A 303 (1981) 69.[12] G.C. Li, I. Sick, R.R. Whitney, M.R. Yearian, Nucl. Phys.

A 162 (1971) 583.



Singular character of critical points in nuclei

V. Wernera, P. von Brentanoa, R.F. Castenb, J. Joliea

a Institut für Kernphysik, Universität zu Köln, Germanyb WNSL, Yale University, New Haven, CT 06520-8124, USA

Received 22 June 2001; received in revised form 12 December 2001; accepted 3 January 2002

Editor: W. Haxton

Abstract

The concept of critical points in nuclear phase transitional regions is discussed from the standpoints ofQ-invariants, simpleobservables and wave function entropy. It is shown that these critical points very closely coincide with the turning points of thediscussed quantities, establishing the singular character of these points in nuclear phase transition regions between vibrationaland rotational nuclei, with a finite number of particles. 2002 Elsevier Science B.V. All rights reserved.

PACS: 21.60.-n; 21.60.Ev; 21.60.Fw

Keywords: Critical point symmetry; Phase transition; Shape transition; Quadrupole shape invariants; Wave function entropy; IBA

Nuclear structural evolution in transitional regionsis often thought of as a continuous variation of proper-ties, as a function of nucleon number, from one ideal-ized limit (e.g., vibrator, rotor) to another. The rapid-ity of structural change may vary across a transitionalsequence of nuclei, and different mass regions exhibitdifferent rates of change but, until recently, no individ-ual point along these evolutionary trajectories could besingled out with special observational properties.

In the last years, however, the concept of criticalpoints in shape/phase transition regions has been muchdiscussed [1–5]. While the concept itself is wellknown in nuclei (in the context of the coherent stateformalism [6,7] of the IBA model [8]), it is onlyvery recently that analytic descriptions of critical pointnuclei have been given [9,10]. This is a significant

E-mail address: [email protected] (V. Werner).

point since, historically, such nuclei have been themost difficult to treat: they exhibit competing degreesof freedom, and one has had to resort to numericalcalculations.

Two critical point symmetries, called E(5) andX(5), have been proposed [9,10], giving analyticexpressions for observables which are exactly at thecritical points of a vibrator to axially asymmetric(γ -soft) rotor transition region, and of a vibrator tosymmetric rotor transition region, respectively, for aninfinite number of nucleons. An important aspect ofthis is that, for the first time, one is able to associatespecial observational characteristics to a specific pointalong a trajectory from one structural limit to another.Recently [11], using the methods presented here, thewell-known O(6) limit of the IBA has also beenidentified as another, heretofore unrecognized, criticalpoint symmetry, for the transition between prolateand oblate nuclei. This is an important result since



56 V. Werner et al. / Physics Letters B 527 (2002) 55–61

the O(6) symmetry can be calculated in the IBA forfinite nucleon numbers, in contrast to the non-IBAsymmetries E(5) and X(5). So far only two examplesfor nuclei [12,13] which lie close to the X(5) and E(5)symmetries are known while, interestingly, there aremany examples for O(6) like nuclei. In the presentwork we will restrict our discussion to prolate nuclei.

To understand the evolution of structure in realnuclei, with a finite number of nucleons, it is importantto gather information about systematic changes ofobservables at or near such critical points. This aimcan be achieved by the use of a model that isable to describe limiting cases of nuclear structure—vibrators, rotors andγ -soft nuclei—and a large varietyof nuclei between these limits. Such a model is givenby the IBA, which—in the expansion of the coherentstate formalism—exhibits critical points as has beendiscussed in Refs. [6,7,14,15]. We stress that thecritical point descriptions X(5) and E(5) are definedin terms of a geometrical approach, not the IBA.Nevertheless the IBA provides a convenient tool tospan a range of structure, including phase transitions,and also to assess effects of finite particle numbers.

It is the purpose of this Letter to show, from sev-eral complementary theoretical approaches, that thereis independent evidence for the singular character ofthese critical points, and independent ways of identify-ing them in observables calculated in collective mod-els. To do so we bring together three major themes:the already mentioned study of phase transitional re-gions and critical point nuclei, the behavior of quadru-pole shape (Q)-invariants, and the study of chaos andentropy in nuclear systems. We show that the criticalpoints occur very near to the turning points (points ofsteepest descent or ascent) of theseQ-invariants—thatis, at the extrema of their first derivatives. The samebehavior will also be shown to hold for some moreeasily accessible observables.

To span the transition regions, it is convenient touse the IBA Hamiltonian in the following form

(1)H = a

[(1− ζ )nd − ζ

4NQ · Q

],

whereQ = s†d + d†s + χ[d†d](2) and we considerthe well known parameter space of the extendedconsistentQ formalism (ECQF) varyingζ between 0and 1, andχ from 0 to −√

7/2 = −1.32, whilea isa scaling factor. This parametrization is equivalent to

Fig. 1. Symmetry triangle of the IBA model. The U(5) ↔ O(6) legis characterized byχ = 0 and varyingζ , while the U(5) ↔ SU(3)

transition region hasχ = −√7/2 and ζ is varied. The dashed

line indicates the phase transitional region where critical points arefound.

the more commonly encountered (equivalent) ECQF[16,17] form of H , which includes the parametersεandκ .

Fig. 1 illustrates the three dynamical symmetries ofthe IBA in terms of a triangle. With the Hamiltonianof Eq. (1) it is easy to calculate the structure for anypoint in the triangle. Forζ = 0 one obtains a U(5)structure (for anyχ ), and ζ = 1, χ = −√

7/2 givesSU(3). Thus, a U(5) ↔ SU(3) transition region isdefined byχ = −√

7/2 andζ varying from 0 to 1,while a U(5) ↔ O(6) region hasχ = 0 andζ varyingfrom 0 to 1.

One can use the coherent state formalism [6,7] ofthe IBA model to identify the critical points in theECQF space. In this approach, the energy functionalfor the ECQF Hamiltonian is given by

E(ζ,χ,β, γ )

= Nβ2(1− ζ(χ2−3)

4N−4Nζ+ζ

)1+ β2

− N(N − 1)ζ

4N − 4Nζ + ζ

×(

4β2 − 4√

2/7χβ3 cos 3γ + 2

7χ2β4

)

(2)× (1+ β2)−2

.

The variation of ζ changes the structure betweenthe vibrator limit and rotational nuclei—both axially

V. Werner et al. / Physics Letters B 527 (2002) 55–61 57

symmetric and axially asymmetric—which are thetransitions we will focus on. Critical points inζ arefound whereE becomes flat atβ = 0. These points,which we refer to asζc, can be derived by evaluatingthe condition

(3)

∣∣∣∣∂2E(ζc)

∂β2

∣∣∣∣β=0

= 0.

On the transition path from U(5) to O(6) (forχ =0) exactly one critical point is found, namely, wherea second, deformed, minimum inβ of the energyfunctional emerges.

The situation becomes more complicated for tran-sitions withχ = 0. In these cases, the spherical min-imum is joined by a deformed minimum and bothminima coexist in a very close parameter range inζ ,converging to one point when approachingχ = 0.Thus, in general there exist three critical points, whichis illustrated in Fig. 2 forN = 10 bosons for the lim-iting case ofχ = −√

7/2. The thick lines in Fig. 2give points in the(ζ,β) plane, which are local min-ima of the energy functional (2). The shaded area isthe parameter range ofζ , where two local minima ofthe energy functional coexist. The lower dashed linegives the criticalζ value where a deformed minimumappears, while the upper dashed line gives the criticalpoint in ζ where the spherical minimum disappearsand only the deformed minimum is left. The dottedline gives the criticalζ value where two coexisting

Fig. 2. The thick lines represent the locus in the (ζ ,β) parameterspace where the energy functional of the coherent state formalismhas a local minimum. The thick line atβ = 0 extends downwardsto ζ = 0. The results are shown for the case ofN = 10 bosons.Dashed lines mark criticalζ values where one minimum disappears(the spherical one at and above the larger value, the deformed oneat and below the lower value). Only in the shaded area two minimacoexist. The dotted line marks the criticalζ value where these twominima are equally deep.

minima are equally deep. The parameter region in be-tween is small for any boson number.

Thus, as it is the aim of this work to identify thecritical points in observables, and we do not expectto be able to distinguish between these three points(close lying inζ ) in real nuclei, we restrict ourselvesto the critical point given by condition (3) where thespherical minimum disappears, and which is given by

(4)ζc = 4N

8N − 8+ χ2N→∞−→ 0.5.

The χ dependence is just a finiteN effect, and thusit is convenient to vary only the parameterζ for theinvestigation of phase transitions between vibrationaland rotational nuclei. Additionally we note that thechoice of our parametrization has the convenientfeature that in the largeN limit we getζc = 0.5.

While, due to their physical meaning, the endpointsof the line of critical points betweenχ = 0 andχ =−√

7/2 in Fig. 1 can be approximately related to thenon-IBA symmetries E(5) and X(5), we see that amuch richer structure shows up in the IBA, wherecritical points occur over the whole transitional regionbetween these legs of the symmetry triangle.

Since we are interested in obtaining signatures forcritical points in observables including matrix ele-ments, we now survey the behavior ofQ-invariants[18,19] in the transition regions. Recently, the conceptof Q-invariants has been re-investigated in the frame-work of the IBA model and theQ-phonon approach[20,21], and the behavior of these moments across thegamut of nuclear collective structures has been elu-cidated [22–24]. These invariants represent quadraticand higher order moments of the quadrupole operator.The invariants are denotedqn andKn ≡ qn/q

n/22 , and

are defined by expressions of the generic type

(5)qn ∼ 〈Ψ0|Q1 · Q2 · · ·Qn|Ψ0〉,where Ψ0 is the ground state wave function, andwhere intermediate angular momentum couplings inthe operator are omitted for simplicity.

For the IBA [8], theQ-invariants have been eval-uated over the entire symmetry triangle of Fig. 1. Toshow the extreme cases, we first focus on the twotransition paths U(5) ↔ SU(3) (χ = −√

7/2) andU(5) ↔ O(6) (χ = 0). We note that the invariantsq2,K3, K4, and σγ ≡ K6 − K2

3 represent, respectively,


Fig. 3. Behavior ofq2, K4 andσγ , and their first derivatives with respect toζ , for the U(5) ↔ SU(3) transition region, calculated forN = 10bosons.

the quadrupole deformation, the triaxiality, the soft-ness of the nuclear shape inβ , and inγ .

We first study the U(5) ↔ SU(3) transition andobtain the results shown forN = 10 in the top rowof Fig. 3 for q2,K4 andσγ . Each of these exhibits arapidly changing behavior which has a turning pointζt near ζ = 0.5. To investigate this in more detail,the second row of Fig. 3 shows the first derivativeswith respect toζ . Again there is a striking consistencyof behavior: the first derivative has an extremum atessentially the same point for each invariant.

Specifically, the turning points (the zeros of the sec-ond derivatives) are:ζt = 0.54 for q2; ζt = 0.53 forK4; and ζt = 0.52 for σγ . In the coherent state for-malism, for N = 10, one obtainsζc = 0.54 for theU(5) ↔ SU(3) case. This is very close to the turningpoints in q2,K4 and σγ : that is ζt ∼ ζc. This corre-spondence between the turning points and the criticalpoints is the main result of this work. The small differ-ences probably represent a finite boson number effect.

This identification of a special point along thestructural evolution from vibrator to rotor is apparenteven in the simplest observables as well. In Fig. 4we show the behavior of the structural observablesR4/2 ≡ E(4+

1 )/E(2+1 ) andB(E2 : 2+

1 → 0+1 ) for the

U(5) ↔ SU(3) transition, again forN = 10. Clearly,as seen in the first derivative plots in the second row,both quantities exhibit their steepest rates of changenear the critical points. Here, the first derivative has anextremum atζt = 0.54 for bothR4/2 and theB(E2)

Fig. 4. Similar to Fig. 3 (forN = 10) for the observablesR4/2 and

B(E2 : 2+1 → 0+

1 ) for the U(5) ↔ SU(3) transition region.

value. In this latter case, this result is not surprisingsince thisB(E2) value andq2 are directly related.

The existence of three critical points on the U(5) ↔SU(3) transition path seems not to be reflected in theQ-invariants, which may be explained by the verycompact parameter region inζ where these criticalpoints occur, while the peaks in the derivatives havea certain width. Also note that fluctuations, resultingfrom the limited numerical accuracy of the PHINT

code used for these calculations, have been smoothedby the use of splines. Thus, perhaps the three criticalpoints just cannot be resolved in the observables dueto numerical truncations.


Fig. 5. Similar to Fig. 3 (forN = 10), forq2 andK4, for the U(5) ↔ O(6) transition region (left panels), and forq2 in the O(6) ↔ SU(3) (notehere with respect toχ ) transition region (right).

Returning to theQ-invariants, similar results applyin the U(5)→O(6) region. Fig. 5 (left panels) showsthis for q2 and K4. In this case the turning points(determined from the rates of change), are:ζt = 0.60for q2 andζt = 0.56 forK4. From Eq. (2), the coherentstate formalism givesζc = 0.56 forN = 10. Again theζt and ζc values obtained from the behavior of theQ-invariants and from the coherent state formalismare quite close. Lastly, we note that the rate of changeof q2 andK4 in the U(5) ↔ O(6) case is much lessthan in the first order U(5)→SU(3) transition region.For example,(dq2/dζ )max ∼ 800 for U(5) ↔ SU(3)

while it is only ∼200 for U(5) ↔ O(6). Also, thewidths of the first derivative curves are much wider(corresponding to a more gradual structural evolution)in the U(5) ↔ O(6) case.

Using the IBA, it is also possible to investigateinternal paths in the symmetry triangle. In particular,internal straight line trajectories, starting from U(5),will correspond toχ values between 0 and−√

7/2,allowing a full mapping of transitional trajectories. Weillustrate such results by showing the change of thefirst derivative of the shape invariantK4 for variousvalues ofχ in Fig. 6. The minima of the derivativesfollow the line of critical points that is also givenin the coherent state formalism, with only a smallχ

dependence.Finally, in regard toQ-invariants, we look at the

O(6) ↔ SU(3) transitional region. The right panels of

Fig. 6. First derivative ofK4 (for N = 10), but for various valuesof the parameterχ . A peak indicating a phase transition occurs forevery value ofχ . The dependence of its position onχ is a finiteN

effect.

Fig. 5 show the behavior ofq2 and its derivative. Notethat the shape is qualitatively different than in the othertransition regions, showing a gradually asymptoticcurve and a first derivative againstχ (the appropriatevariable for this region) which is monotonic. Nocritical point is definable in this region ofχ values,except when O(6) itself is reached (see Ref. [11]).

Another theme in nuclear structure recently hasbeen the study of order and chaos for different struc-tures. It was shown in Ref. [25] that nuclear systemsdisplay ordered spectra at and near the three symme-try limits of the IBA, but that there is a rapid onset ofchaotic behavior away from these benchmark regions.(See Fig. 1 of Ref. [25] but note that the symmetry tri-


angle is differently defined therein.) Recently, Cejnarand Jolie [26,27] have developed the concept of wavefunction entropy as an alternate (and physically intu-itive) way of studying the relative complexity of nu-clear wave functions. Basically, the entropy of a stateis a measure of its spreading within a given basis. Notethat this is not the same as the chaoticity (which is ba-sis invariant) since a wave function may have high en-tropy in one basis [e.g., U(5)] and low entropy in an-other [e.g., SU(3)].

Now that we showed a visible effect of criticalpoints in various observables, it is interesting to seewhether effects of a phase transition can also be seen inthe wave functions and thus the wave function entropy.A rise of the wave function entropy can be expected inmoving from one limit to another, but the question iswhether it also appears in a close region with turningpoints which coincide with the turning points of thepreviously mentioned observables. Thus, we define[26] a quantity, calledWB

Ψ , for a stateΨ , that can bewritten in the basisB asΨ = ∑n

iB aiB |ΨB〉, as

(6)WBΨ ≡ −

n∑iB=1

|aiB |2 ln∣∣aBiB

∣∣2,wheren is the number of basis vectors. IfΨ coincideswith a basis vector, thenWB

Ψ = 0. If Ψ is uniformlyspread out over the basisB, thenWB

Ψ ≈ lnn.A physically intuitive expression of the entropy is

the quantity [27]

(7)nBeffΨ ≡ expWBΨ

which expresses a kind of “effective number” of wavefunction components. For a “pure” stateΨ , nBeffΨ

= 1

and for a fully de-localized statenBeffΨ≈ n.

To properly normalize the entropies we define theentropy ratio

(8)rB ≡ expWBΨ − 1

exp〈WGOE〉 − 1

relative to that for the Gaussian Orthogonal Ensemble[27]. The ratiorBΨ varies from 0 for a pure (localizedin the basisB) state to∼1 for a highly mixed state(see Ref. [27] for a more detailed discussion of thisnormalization).

We show the results in Fig. 7 forrB0+

1and its

derivative as a function of the order parameterζ

Fig. 7. The entropy ratio (forN = 10) for the 0+1 state (top row) inthe three transition regions, plotted againstζ and given, for eachregion, in two bases as indicated (e.g., U(5) and SU(3) for theU(5) ↔ SU(3) transition). The lower panels give the derivative ofthe entropy ratio againstζ in the appropriate basis.

for the U(5) ↔ SU(3) and U(5) ↔ O(6) transitionregions (all forN = 10). The entropy ratio for theground state undergoes a very rapid change nearζc forboth transition regions. We note that for larger bosonnumbersN the transition becomes much sharper (seeFig. 6 in [11]). For the U(5) ↔ SU(3) and U(5) ↔O(6) phase transitions, it is easy to read the turningpoints,ζt , values from the derivative plots, obtainingζt = 0.52 andζt = 0.59 (in a U(5) basis), respectively,compared to values ofζc = 0.54 and ζc = 0.56from the coherent state formalism. We note that thesteepness of the entropy functions againstζ increaseswith boson numberN , as pointed out in Ref. [28]. Thisalso holds true for the observables studied above.

To conclude, from the behavior of several ratherdifferent quantities, theQ-invariants, the simple ob-servablesR4/2 andB(E2 : 2+

1 → 0+1 ), and the wave

function entropy, we have shown that critical pointsof the phase transitional regions U(5) ↔ SU(3) andU(5) ↔ O(6) are reflected in the behavior of theseobservables along these evolutionary trajectories. Thisresult was obtained for finite boson numbers, making itpossible to investigate effects of valence particle num-ber on the singularities.

Acknowledgements

We are grateful to N.V. Zamfir, F. Iachello, J. Eberthand K. Heyde for useful discussions, and to P. Cejnar


for the entropy calculations. Work supported by theUS DOE under Grant number DE-FG02-91ER40609and by the DFG under Project number Br 799/10-1and by NATO Research Grant No. 950668. One of us[R.F.C.] is grateful to the Institut für Kernphysik inKöln for support.

References

[1] A. Wolf et al., Phys. Rev. C 49 (1994) 802.[2] R.F. Casten, N.V. Zamfir, D.S. Brenner, Phys. Rev. Lett. 71

(1993) 227.[3] F. Iachello, N.V. Zamfir, R.F. Casten, Phys. Rev. Lett. 81

(1998) 1191.[4] R.F. Casten, D. Kusnezov, N.V. Zamfir, Phys. Rev. Lett. 82

(1999) 5000.[5] J. Jolie, P. Cejnar, J. Dobes, Phys. Rev. C 60 (1999) 061303.[6] A.E.L. Dieperink, O. Scholten, F. Iachello, Phys. Rev. Lett. 44

(1980) 1747.[7] J.N. Ginocchio, M.W. Kirson, Phys. Rev. Lett. 44 (1980) 1744.[8] F. Iachello, A. Arima, The Interacting Boson Model, Cam-

bridge Univ. Press, Cambridge, 1987.

[9] F. Iachello, Phys. Rev. Lett. 87 (2001) 052502.[10] F. Iachello, Phys. Rev. Lett. 85 (2000) 3580.[11] J. Jolie, R.F. Casten, P. von Brentano, V. Werner, Phys. Rev.

Lett. 87 (2001) 162501.[12] R.F. Casten, N.V. Zamfir, Phys. Rev. Lett. 87 (2001) 052503.[13] R.F. Casten, N.V. Zamfir, Phys. Rev. Lett. 85 (2000) 3584.[14] E. Lopez-Moreno, O. Castanos, Phys. Rev. C 54 (1996) 2374.[15] E. Lopez-Moreno, O. Castanos, Rev. Mex. Fis. 44 (1998) 48.[16] D.D. Warner, R.F. Casten, Phys. Rev. Lett. 48 (1982) 1385.[17] P.O. Lipas, B.P. Toivonon, D.D. Warner, Phys. Lett. B 155

(1985) 295.[18] D. Cline, Ann. Rev. Nucl. Part. Sci. 36 (1986) 683.[19] K. Kumar, Phys. Rev. Lett. 28 (1972) 249.[20] G. Siems et al., Phys. Lett. B 320 (1994) 1.[21] T. Otsuka, K.-H. Kim, Phys. Rev. C 50 (1994) 1768.[22] R.V. Jolos et al., Nucl. Phys. A 618 (1997) 126.[23] Yu.V. Palchikov, P. von Brentano, R.V. Jolos, Phys. Rev. C 57

(1998) 3026.[24] V. Werner et al., Phys. Rev. C 61 (2000) 021301.[25] Y. Al-hassid, N. Whelan, Phys. Rev. Lett. 67 (1991) 816.[26] P. Cejnar, J. Jolie, Phys. Lett. B 420 (1998) 241.[27] P. Cejnar, J. Jolie, Phys. Rev. E 58 (1998) 387.[28] P. Cejnar, J. Jolie, Phys. Rev. E 61 (2000) 6237.



Bridging two ways of describing final-state interactions inA(e, e′p) reactions

D. Debruyne, J. Ryckebusch, S. Janssen, T. Van Cauteren

Department of Subatomic and Radiation Physics, Ghent University, Proeftuinstraat 86, B-9000 Gent, Belgium

Received 8 August 2001; received in revised form 21 October 2001; accepted 24 December 2001

Editor: J.-P. Blaizot

Abstract

We outline a relativistic and unfactorized framework to treat the final-state interactions in quasielasticA(e, e′p) reactionsfor four-momentum transfersQ2 0.3 (GeV/c)2. The model, which relies on the eikonal approximation, can be used incombination with optical potentials, as well as with the Glauber multiple-scattering method. We argue that such a model canbridge the gap between a typical “low” and “high-energy” description of final-state interactions, in a reasonably smooth fashion.This argument is made on the basis of calculated structure functions, polarization observables and nuclear transparencies forthe target nuclei12C and16O. 2002 Elsevier Science B.V. All rights reserved.

PACS: 24.10.-i; 24.10.Jv; 25.30.Fj

Keywords: Relativistic eikonal approximation; Glauber theory; ExclusiveA(e, e′p) reactions; Nuclear transparency

1. Introduction

At intermediate values of the four-momentumtrans-fer, here loosely defined asQ2 0.5 (GeV/c)2, theexclusive electroinducede+A → e′+(A−1)∗+ p re-action offers great opportunities to study the propertiesof bound nucleons in a regime where one expects thatboth hadronic and partonic degrees of freedom mayplay a role. One such example is the study of the short-range structure of nuclei. These studies are meant toprovide insight into the origin of the large-momentacomponents in the nucleus. Amongst other things,constituent-quark models for the nucleon predict mea-surable medium modifications of the bound nucle-

E-mail address: [email protected] (J. Ryckebusch).

on’s properties. At present, high-resolution doublepolarization experiments ofA( e, e′ p ) reactions areputting these predictions to stringent tests [1]. Anothermedium-dependent effect, which has attracted a lotof attention in recent years, is the color transparency(CT) phenomenon. ForA(e, e′p) processes, CT pre-dicts that, at sufficiently high values ofQ2, the struckproton may interact in an anomalously weak mannerwith the spectator nucleons in the target nucleus [2].

For all of the aforementioned physics issues, the in-terpretation of theA(e, e′p) measurements very muchdepends on the availability of realistic models todescribe the final-state interactions (FSI) which theejected proton is subject to. There are basically twoclasses of models to treat the FSI effects in electroin-duced proton knockout. At lower energies, most theo-retical A(e, e′p) investigations are performed within



D. Debruyne et al. / Physics Letters B 527 (2002) 62–68 63

the context of the so-called distorted-wave impulseapproximation (DWIA), where the scattering wave-function of the struck nucleon is calculated in a po-tential model [3]. The parameters in these optical po-tentials, which are available in both relativistic andnon-relativistic forms, are obtained from global fits toelastic proton–nucleus scattering data. The DWIA cal-culations typically rely on partial-wave expansions ofthe exact solutions to the scattering problem, a methodwhich becomes increasingly cumbersome at higherenergies. To make matters worse, global parametriza-tions of optical potentials are usually not available forproton kinetic energies beyond 1 GeV. In this energyregime the Glauber model [4], which is a multiple-scattering extension of the eikonal approximation,offers a valid alternative for describing final-state in-teractions. In such a framework, the effects of FSIare calculated directly from the elementary proton–nucleon scattering data through the introduction of aprofile function [5–9]. Several non-relativistic stud-ies have formally investigated the applicability ofthe Glauber model for describingA(e, e′p) reac-tions at higher energies and momentum transfers.These investigations were often hampered by thelack of high-qualityA(e, e′p) data to compare themodel calculations with. Recently, the first high-quality data for16O(e, e′p) cross sections, separatedstructure functions and polarization observables atQ2 = 0.8 (GeV/c)2 became available [10].

The purpose of this Letter is to investigate whetherthe optical potential and the Glauber method for de-scribing final-state interactions lead to comparable re-sults in an energy regime where both methods ap-pear applicable. An observation which may pointtowards inconsistencies in the description of FSI ef-fects inA(e, e′p) processes at “low” and “high” en-ergies, is the apparentQ2 evolution of the extractedspectroscopic factors [11]. Whereas numerous optical-potential analyses ofA(e, e′p) measurements at lowQ2 have systematically produced values which repre-sent 50–70% of the sum-rule strength, it has recentlybeen suggested that in order to describe the data athigher Q2 within the context of the Glauber model,substantially higher values are required [11,12].

We propose a relativistic formalism for computingA(e, e′p) observables at medium energies. The for-malism is developed in such a way that it can be usedin combination with either optical potentials or the

Glauber method without affecting any other ingredientof the model. Results of optical potential and Glauberlike calculations of structure functions and polariza-tion observables for the target nuclei16O and12C arepresented and compared. In addition, results of rela-tivistic and unfactorized nuclear transparency calcula-tions for the12C(e, e′p) reaction are presented.

2. Formalism

In the one-photon-exchange approximation, thecross section for a process in which an electronimpinges on a nucleus and induces the knockout ofa single nucleon with momentumkf , leaving theresidual nucleus in a certain discrete state, can bewritten in the following form [13](

d5σ

dε′ dΩe′ dΩf

)

= MpMA−1kf

8π3MA

f −1rec σM

(1)× [vLRL + vT RT + vT T RT T + vT LRT L

],

wherefrec is the hadronic recoil factor, andσM is theMott cross section. The electron kinematical factorsvi and the structure functionsRi are defined in theusual manner [13]. Remark that in our model calcu-lations an unfactorized expression for the differentialcross section is adopted. This means that the off-shellelectron–proton coupling is not separated from the nu-clear dynamics. Although the factorized approach haslong been abandoned in the description of low-ener-gy A(e, e′p) reactions, it is still widely used when itcomes to describing high-energyA(e, e′p) processes.

In our model, the relativistic bound-state wavefunc-tions are calculated within the context of a mean-fieldapproximation to theσ–ω model [14,15]. Assumingspherical symmetry, the following type of solutions tothe Dirac eigenvalue problem result

(2)ψα( r ) ≡ ψnκmt ( r ) =[

ıGnκt (r)/rYκmηt

−Fnκt (r)/rY−κmηt

],

wheren denotes the principal,κ andm the general-ized angular momentum andt the isospin quantumnumbers. TheY±κm are the well-known spin-sphericalharmonics and determine the angular and spin parts

64 D. Debruyne et al. / Physics Letters B 527 (2002) 62–68

of the wavefunction. In solving the relativistic bound-state problem, we have adopted the values for the cou-pling constants and meson masses of Ref. [15].

In the relativistic eikonal approximation, the scat-tering wave function for a nucleon subject to a scalar(Vs) and a vector potential (Vv) reads

ψ(+)

kf ,s=

√E + M

2M

[1

1E+M+Vs−Vv

σ · p]

(3)× eıkf ·reıS( r )χ 12ms

,

where the eikonal phaseS(b, z) is defined as

ıS(b, z)

= −ıM

K

z∫−∞

dz′

(4)

× [Vc(b, z′) + Vso(b, z′)

[σ · (b × K) − ıKz′]],with r ≡ (b, z) and K ≡ 1

2(q + kf ). The centralVcand spin-orbit potentialVso occurring in the aboveexpression are determined byVs and Vv and theirderivatives. In general, strength from the incidentbeam is drained into other inelastic channels. Withinthe context of a DWIA approach, the inelasticities arecommonly implemented through the use of a complexoptical potential which is gauged against elasticpA

scattering data. In the numerical calculations, we haveused the global relativistic optical potentials of Cooperet al. [16]. By fitting proton elastic scattering datain the energy range of 20–1040 MeV, Cooper et al.obtained a set of energy-dependent potentials for thetarget nuclei12C, 16O, 40Ca,90Zr and208Pb. In whatfollows, we refer to calculations on the basis of Eq. (3)as the optical model eikonal approximation (OMEA).

For proton kinetic energiesTp 1 GeV, the useof optical potentials appears no longer justifiable inview of the highly inelastic character of the elementaryproton–nucleon scattering process. Here, a way out isoffered by an extension of the eikonal method, namelythe Glauber multiple-scattering method, which is usu-ally adopted in its non-relativistic version. Here, wepropose the use of a relativized version which allowsus to write the wavefunction of the escaping proton as

(5)ψ(+)

kf ,s=

√E + M

2MS

[1

1E+M

σ · p]

eıkf ·rχ 12ms

.

This expression for the relativistic scattering wavefunction is derived in similar manner as for the non-relativistic (NR) case where the wave function adoptsthe well-known form

(6)ψ(+),NRkf ,s

= Seıkf ·rχ 12ms

.

The operatorS defines the action of the subsequentcollisions that the ejectile undergoes with the specta-tor nucleons

(7)S( r, r2, . . . , rA) =A∏

j=2

[1− Γ (b − bj )θ(z − zj )

],

whereθ(z − zj ) ensures that the hit proton only inter-acts with other nucleons if they are localized in its for-ward propagation path. The profile functionΓ (kf , b)

for central elasticpN scattering reads

(8)Γ (kf , b) = σ totpN(1− iεpN)

4πβ2pN

exp

(− b2

2β2pN

).

The parameters in Eq. (8) can be taken directly fromnucleon–nucleonscattering measurements and includethe totalpN cross sectionsσ tot

pN , the slope parametersβpN and the ratios of the real to imaginary part of thescattering amplitudeεpN . TheA(e, e′p) calculationson the basis of the scattering state of Eq. (5) are here-after referred to as the relativistic multiple-scatteringGlauber approximation (RMSGA).

3. Results

We first compare our relativistic calculations to re-cent quasielastic16O(e, e′p) data from JLAB. In thesehigh-resolution experiments atQ2 = 0.8 (GeV/c)2

differential cross sections, separated structure func-tions and polarization observables were obtained [10].The variation in missing momentum was achievedby varying the detection angle of the ejected protonwith respect to the direction of the momentum trans-fer (“quasiperpendicular kinematics”). Hence, it wasonly possible to isolate the combinationRL+T T ≡RL + vT T

vLRT T . In Fig. 1 we display the different struc-

ture functions against the missing momentumpm. Thedifferent curves all use the same bound-state wave-functions and electron–proton coupling but differ inthe way the FSI are treated. In Table 1 we display the


Fig. 1. Separated structure functions for16O(e, e′p) in quasiperpendicular kinematics atε = 2.4 GeV,q = 1 GeV/c andω = 0.439 GeV. Thesolid and dashed curves denote the eikonal (OMEA) and Glauber (RMSGA) calculations. The data are from Ref. [10].

Table 1The spectroscopic factors as derived from the16O(e, e′p) results contained in Fig. 1 through aχ2 fitting procedure

CC1 operator CC2 operatorRPWIA OMEA RMSGA RPWIA OMEA RMSGA

3/2− (Ex = 6.3 MeV) 0.59 0.96 0.96 0.61 1.00 1.001/2− (g.s.) 0.53 0.79 0.80 0.53 0.82 0.82

spectroscopic factors which are obtained from aχ2 fitfrom the calculations to the data. Although the Glauberand the optical-potential framework provide an intrin-sically very different treatment of the final-state in-teractions, they lead to almost identical spectroscopicfactors atQ2 = 0.8 (GeV/c)2. Another striking ob-servation from Fig. 1 is that both types of calcula-tions produce almost identical results for theRT andRL+T T structure functions. In theRT L response forexcitation of the 6.3 MeV 3/2− state the differencesare somewhat larger. It is worth mentioning here thatof all structure functions, theRT L one has been iden-tified as being most sensitive to changes in the currentoperator and relativistic corrections [17–20].

A quantity which is particularly sensitive to FSIeffects is the induced polarizationPn

(9)Pn = σ(siN = ↑) − σ(si

N = ↓)

σ (siN = ↑) + σ(si

N = ↓),

wheresiN denotes the spin orientation of the ejectile in

the direction orthogonal to the reaction plane. In the12C(e, e′ p) experiment of Woo and collaborators [21],the quantityPn was determined at quasifree kinemat-ics for an energy and momentum transfer of (ω,q) =(294 MeV, 756 MeV/c). The results of the12C(e, e′ p )measurements are shown in Fig. 2, along with our the-oretical results. The fair agreement of the Glauber re-


Fig. 2. Induced polarization for the12C(e, e′ p )11B reaction in quasiperpendicular kinematics atε = 579 MeV, q = 756 MeV/c andω = 294 MeV. The data are from Ref. [21].

sults with the data is striking. It turns out that the ap-plicability of the RMSGA method is wider than onewould naively expect. We believe that the extendedrange of validity of the RMSGA method observedhere, is (partly) caused by the relativistic and unfac-torized treatment of the Glauber method. In Ref. [22]we have demonstrated that the effect of dynamical rel-ativity (i.e., the effect of the lower components of thewavefunctions) can be significant at lowQ2.

We now turn to the study of the nuclear trans-parency in the quasielastic12C(e, e′p) reaction in awide Q2 range of 0.3 Q2 20 (GeV/c)2. Theresults of our calculations are contained in Fig. 3.We have performed calculations within the relativisticGlauber framework and the eikonal model with the op-tical potentials from Ref. [16]. The optical potential re-sults are limited to kinetic energies belowTp = 1 GeV.In the Glauber model, we have also performed calcu-lations which include the effect of short-range correla-tions (SRC). Each of these calculations was done withthe CC1 and CC2 current operator. The transparen-cies calculated within the Glauber framework exhibitsome fluctuations. The magnitude of these fluctuations

mark the intrinsic uncertainties on the computed trans-parencies caused by the error bars on the measured el-ementarypN scattering parameters. Most publishedGlauber calculations for the nuclear transparencies donot exhibit these fluctuations, but use (smooth) globalfits to determine the energy dependence of the elemen-tarypN scattering data. In our numerical calculations,we use the listed experimentalpN results.

The measurements in Refs. [23–25] were per-formed in certain regions of the phase space, dictatedby the requirement that quasielastic conditions shouldbe met. We have constrained our calculations to thesame segment of the phase space. In general, the ex-perimental transparencyTexp is defined as

(10)Texp=∫43k d k ∫

4E dE(

d5σdε′ dΩe′ dΩf

)exp

cA

∫43k

∫4E

dE(

d5σdε′ dΩe′ dΩf

)PWIA

.

The A-dependent factorcA renormalizes the non-relativistic plane–wave impulse approximation(PWIA) predictions for corrections induced by SRC.For the 12C(e, e′p) process, a correction factor of0.901 ± 0.024 was adopted in Refs. [23–25]. As


Fig. 3. Nuclear transparency for12C(e, e′p) as a function ofQ2. The curves denote calculations within the relativistic Glauber framework(RMSGA) and the eikonal model with optical potentials (OMEA). Glauber results with and without inclusion of SRC effects are presented.The calculations in the upper (lower) panel employ the CC1 (CC2) current operator. The curves assume full occupancy of thes- andp-shelllevels in12C. The data are from BATES [31] (triangles), SLAC [23,24] (squares) and from JLAB [25] (circles).

the implementation of short-range correlations can bedone in numerous ways, we have removed this factorand rescaled the data accordingly.

At lower values ofQ2 there are substantial devi-ations between the transparencies computed with theCC1 and the CC2 current operator. At higher valuesof Q2 [Q2 3–4 (GeV/c)2], where the differencesbetween the CC1 and CC2 predictions are negligible,the predicted transparencies tend to underestimate themeasured transparencies, even when assuming full oc-cupancy for the single-particle levels in12C. This ap-parent shortcoming can be cured by introducing the ef-fect of short-range correlations. They are implementedthrough the introduction of a central (or, Jastrow) cor-relation functiong(| r1 − r2|) in the two-body densi-ty components which are part of the Glauber cal-culations. In practice, this procedure amounts to thefollowing replacement in the matrix elements that de-termine the rescattering effects

(11)ψα( r1)ψβ( r2) → ψα( r1)ψβ( r2)g(| r1 − r2|

).

In line with the findings of other studies [6,12,26–28] we observe that short-range correlations in-

crease the calculated transparencies by about 10%. Wehave used the central correlation functiongGD(r) froma nuclear-matter calculation of Gearhart and Dick-hoff [29]. Amongst different other candidates, this cor-relation function emerged as the preferred choice in ananalysis of12C(e, e′pp) data [30]. Being a rather hardcorrelation function, the effect of introducinggGD(r)

on the computed values of the transparencies is max-imized. We have also evaluated the role of relativis-tic effects on the computed transparencies. In general,these effects are rather small. For example, the cou-pling between the lower component in the bound andscattering state marginally affects the predictions forthe transparencies. For some specific observables, liketheRT L structure function in Fig. 1, on the other hand,the relativistic effects are substantial.

For four-momentum transfers aboutQ2 ≈1 (GeV/c)2 it appears legitimate to directly comparethe optical potential and the Glauber calculations. Itis obvious from Fig. 3 that the OMEA curves ex-hibit a behaviour very similar to the correlated Glauberresults. This observation may suggest that the in-mediumpN cross sections are modestly reduced com-


pared to the on-shell values and that the major part ofthis effect can be modeled through the introduction ofSRC mechanisms. As for the OMEA results, it can beargued that the SRC effects, which belong to the classof medium effects, are already effectively incorporatedin the formalism. After all, the optical potentials areobtained from a global fit to proton–nucleus scatteringdata.

The apparent consistency between the OMEA andthe correlated RMSGA predictions is an interestingresult. Indeed, it demonstrates that the low and thehigh Q2 regime can be bridged in a satisfactory man-ner. This feature has it consequences for the appar-entQ2 evolution of the spectroscopic factors extractedfrom 12C(e, e′p) [11]. As suggested by the authors ofRef. [11], a consistent analysis of all12C(e, e′p) databetween 0.1 Q2 10 (GeV/c)12 could much im-prove insight into this matter. A consistent treatmentwould at least allow to separate genuine physical ef-fects (contributions from meson exchange,4-isobars,SRC etc.) from model-dependent uncertainties (likegauge ambiguities and problems related to the treat-ment of the FSI). Such an analysis should preferablybe carried out in a framework that is able to describeboth the low and highQ2 (e, e′p) data without any in-consistencies in some intermediate-energy range. Wefeel that the framework presented here, is an initialstep in this direction.

4. Conclusion

Summarizing, we have outlined a relativistic andunfactorized framework for computingA(e, e′p) ob-servables at intermediate and high four-momentumtransfersQ2. The model is based on the eikonal ap-proximation and can accommodate both relativisticoptical potentials and a Glauber approach, which aretwo substantially different techniques to deal withfinal-state interactions. We have shown that optical-potential and Glauber predictions are reasonably con-sistent at intermediate values ofQ2. Indeed, atQ2 val-ues of about 0.8 (GeV/c)2, which is a regime in which

both approaches for dealing with the FSI’s appear jus-tified, comparable results for the transparencies andstructure functions are obtained.

References

[1] S. Dieterich et al., Phys. Lett. B 500 (2001) 47.[2] L. Frankfurt, G. Miller, M. Strikman, Annu. Rev. Nucl. Part.

Sci. 45 (1994) 501.[3] A. Picklesimer, J. Van Orden, S. Wallace, Phys. Rev. C 32

(1985) 1312.[4] R. Glauber, G. Matthiae, Nucl. Phys. B 21 (1970) 135.[5] S. Jeschonnek, T. Donnelly, Phys. Rev. C 59 (1999) 2676.[6] L. Frankfurt, E. Moniz, M. Sargsyan, M. Strikman, Phys. Rev.

C 51 (1995) 3435.[7] O. Benhar, S. Fantoni, N. Nikolaev, J. Speth, A. Usmani,

B. Zakharov, Z. Phys. A 355 (1996) 267.[8] N. Nikolaev, A. Szcurek, J. Speth, J. Wambach, B. Zakharov,

V. Zoller, Nucl. Phys. A 582 (1995) 665.[9] C. Ciofi degli Atti, D. Treleani, Phys. Rev. C 60 (1999) 024602.

[10] J. Gao et al., Phys. Rev. Lett. 84 (2000) 3265.[11] L. Lapikas, G. van der Steenhoven, L. Frankfurt, M. Strikman,

M. Zhalov, Phys. Rev. C 61 (2000) 064325.[12] L. Frankfurt, M. Strikman, M. Zhalov, Phys. Lett. B 503 (2001)

73.[13] A. Raskin, T. Donnelly, Adv. Nucl. Phys. 191 (1989) 78.[14] C. Horowitz, B. Serot, Nucl. Phys. A 368 (1981) 503.[15] B. Serot, J. Walecka, Adv. Nucl. Phys. 16 (1986) 1.[16] E. Cooper, S. Hama, B. Clarck, R. Mercer, Phys. Rev. C 47

(1993) 297.[17] J. Udias, J. Caballero, E. Moya de Guerra, J. Amaro, T. Don-

nelly, Phys. Rev. Lett. 83 (1999) 5451.[18] S. Gardner, J. Piekarewicz, Phys. Rev. C 50 (1994) 2822.[19] S. Jeschonnek, T. Donnelly, Phys. Rev. C 57 (1998) 2438.[20] J. Udias, J. Vignote, Phys. Rev. C 62 (2000) 034302.[21] R. Woo et al., Phys. Rev. Lett. 80 (1998) 456.[22] D. Debruyne, J. Ryckebusch, W. Van Nespen, S. Janssen, Phys.

Rev. C 62 (2000) 024611.[23] N. Makins et al., Phys. Rev. Lett. 72 (1994) 1986.[24] T. O’Neill et al., Phys. Lett. B 351 (1995) 87.[25] D. Abbott et al., Phys. Rev. Lett. 80 (1998) 5072.[26] A. Kohama, K. Yazaki, R. Seki, Nucl. Phys. A 551 (1993) 687.[27] O. Benhar, A. Fabrocini, S. Fantoni, V. Pandharipande, I. Sick,

Phys. Rev. Lett. 69 (1992) 881.[28] V. Pandharipande, S. Pieper, Phys. Rev. C 45 (1992) 791.[29] C. Gearheart, PhD thesis, Washington University, St. Louis,

1994, unpublished.[30] K. Blomqvist et al., Phys. Lett. B 421 (1998) 71.[31] G. Garino et al., Phys. Rev. C 45 (1992) 780.



Deeply bound pionic atoms from the(γ,p) reaction in nuclei

S. Hirenzakia,b, E. Oseta

a Departamento de Física Teórica and IFIC, Centro Mixto Universidad de Valencia-CSIC, Ap. Correos 22085, E-46071 Valencia, Spainb Department of Physics, Nara Women’s University, Nara 630-8506, Japan

Received 7 September 2001; received in revised form 5 December 2001; accepted 5 January 2002


Abstract

We study the(γ,p) reaction on208Pb leading to207Pb with a bound pion attached to it in the lowest 1s or 2p pionic levels.The reaction can be made recoilless to optimize the production cross section but we must choose a bit higher photon energyto overcome the Coulomb barrier in the proton emission. The cross sections obtained are easily measurable and can be largerthan 50 per cent of the background from inclusive(γ,p). This makes it a clear case for the detection of the pionic atom signals,converting this reaction into a practical tool to produce deeply bound pionic atoms. 2002 Elsevier Science B.V. All rightsreserved.

1. Introduction

In the past decade a search, both theoretical andexperimental, for deeply bound states was conducted,which lead to the successful detection of these statesin Pb isotopes in [1,2]. A review of the methods pro-posed and early attempts prior to the detection at GSIcan be found in [3]. Out of many reactions proposedit was envisaged in [3] that two reactions stood bet-ter chances, the(d, 3He) reaction [4] and the radia-tive capture in low energy pion scattering(π−, γ ) [5],both of them leaving a bound pion in the nucleus. Thefirst reaction offers a special characteristic, very dearin production of bound particles in nuclei, which is itsrecoilless nature for some kinematics. The second re-action is also nearly recoilless and has the additionaladvantage of being a coherent reaction, hence benefit-ing from an extraA2 factor in the cross section. This

E-mail address: [email protected] (E. Oset).

extra advantage is, however, counterbalanced by factthat the pions are secondary beams and hence one hassmaller fluxes than with primary beams like thed inthe first reaction. The first reaction is the one that ledto the successful detection of the pionic states, whilethe second one, carried out at TRIUMF [6], offered amuch less clear evidence because lack of enough res-olution in the photon detector. Yet, it proved a worthmethod to be considered in the future when better res-olution is achieved since it produces the pionic statesin the ground state of stable nuclei, like208Pb. The(d, 3He) reaction leads instead both to ground and ex-cited states ofA-odd nuclei where the atomic and nu-clear levels mix.

The experience on the history of these reactionsalso stresses the importance of having a recoillessreaction and using primary beams. Another reactionwhich fulfills these conditions is the(γ,p) reactionwhere a particular kinematics can be chosen to makethe reaction recoilless in the production of the pionicatom. A suggestion to use this reaction for the produc-



70 S. Hirenzaki, E. Oset / Physics Letters B 527 (2002) 69–72

Fig. 1. Diagram for the(γ,p) reactions to form pionic bound stateson 207Pb.

tion of omega bound states in nuclei has been donein [7], although the large width predicted for thesestates would most probably prevent the observationof clean peaks. In the present case we already knowfrom experiment that the widths are smaller than theseparation of the levels. The success of the reactionis then tied to the magnitude of the cross sections forthe reaction and the ratio of signal to the backgroundcoming from other reactions where no pions are pro-duced but a proton is detected. Indeed, considerationof the background is important since the signal forthe bound pion production is the detection of a protonwith an energy equal to the one of the photon minusthe pion mass (and the binding energies). However,the same protons can be obtained from the inclusive(γ,p) process in which the proton has collisions andloses energy.

In the present work we perform calculations for thereaction

γ + 208Pb→ p + ( 207Pb· π−b

)which is depicted diagrammatically in Fig. 1. Theelementary reaction isγ n → π−p and the final protonis emitted leaving a nuclear state of207Pb with a boundpion tied to it.

Since the reaction is made practically recoilless, theelementary amplitude for the process is extremely sim-ple since only the Kroll Ruderman term contributes.Thus the nuclear amplitude for the process of Fig. 1 is

given by

−iT =∫

d3rfπNN

mπ

e√

2

(2MN

2MN − mπ

)σ · ε 1√

V

(1)× eik·r 1√

Ve−i p·rΦ∗

nlm( r )φJLM( r ),

whereΦ andφ are the pion and neutron wave functionrespectively,k, p the photon and proton momenta, andε the photon polarization.

In Eq. (1) we have used a plane wave for the proton,but it is clear that a distorted wave should be used andwe shall do that below. In addition we sum and averagethe cross section over final and initial polarizations.We show explicitly below the results obtained with thephoton polarization given byε− = (i − ij )/

√2.

We shall look at protons going in the forwarddirection. The matrix element ofT , removing thevolumeV in the denominator, becomesT ′ given by

T ′ = 2πδM−1/2,mfπNN

mπ2e

2MN

2MN − mπ

×∑lp

∞∫0

b db

∞∫−∞

dz R∗nl(r)RNL(r)

× Yl,m

(z

r

)YL,M−1/2

(z

r

)×C(L,1/2, J ;M − 1/2,1/2)

(2)× eikz(2lp + 1)(−i)lp ˜jlp(pr)Plp

(z

r

),

whereR andR are the radial wave functions of thepion and bound neutron states, andY the spherical har-monics removing theeimϕ factor. The proton distortedwave ˜jlp (pr) is obtained for each proton partial waveby solving the Schrödinger equation with the appropri-ate boundary condition shown in Eq. (12) in Ref. [8].We take the proton–nucleus optical potential from [9]and it is given by

(3)U(r) = V + iW

1+ exp[(r − R)/a] ,

where R and a are the radius and diffuseness pa-rameters taken to be 7.35 and 0.65 fm for Pb, re-spectively. The energy dependent potential strengthV andW are shown in Fig. 2.10 (for the real part)and Fig. 4.6 (for the imaginary part) in Ref. [9]. Wealso add the Coulomb potential with finite size of the

S. Hirenzaki, E. Oset / Physics Letters B 527 (2002) 69–72 71

nucleus, which is the same one used for theπ− withopposite sign.

The cross section for the process is then given by

d2σ

dΩ dEp

= pMN

2mπ

1

(2π)3

1

2k

(4)

×∑

(n−1⊗π)

Γ[k + M

( 208Pb) − Ep

− M( 207Pb· π−

b

)]2

+ (Γ /2)2−1∑∑

|T ′|2,whereΓ is the width of the pionic atom state.

By numerical calculations, we find that the effectof the Coulomb barrier for the emitted proton islarge for low energy protons, suppressing the signalssignificantly. On the other hand, if we increase theproton kinetic energy in the final states, which isequivalent to increasing the incident photon energy,this moves us away from the recoilless condition of thereaction. Thus, we evaluate the pionic atom formationspectra for several incident photon energies in order tofind the optimal kinematical condition. We concludethat we have the largest signals at an incident photonenergyk = 170 MeV, while the recoilless conditionappears atk = 155 MeV. The results for the doubledifferential cross section of the reaction are shownin Fig. 2. As we expect for the nearly recoillesskinematics, the substitutional states make the largestpeaks and the dominant contributions come from the2p pionic state with neutronp3/2 andp1/2 hole states.We can also see the pionic 1s state formation atTp =29∼ 30 MeV. The formation cross section is reducedby around the 30% from the plane wave approximationfor the largest signal due to the distortion effectson the emitted proton and is estimated to be about20 µb sr−1 MeV−1 for the largest signal.

As for the background for the present reaction itcan be easily estimated using experimental results al-ready available from the study of the inclusive(γ,p)reaction in nuclei [10]. There we can see results in208Pb atk = 227 MeV and 390 MeV. The cross sec-tions go down toEp = 40 MeV and hence it is easy toextrapolate the results smoothly toEp = 30 MeV thatwe have in the present case. The differential cross sec-tions for a proton angle of 52 are 50 µb sr−1 MeV−1

and 100 µb sr−1 MeV−1 for k = 227 MeV andk =

Fig. 2. Expected spectra of the208Pb(γ,p)207Pb· π−b reaction at

the photon energyk = 170 MeV as a function of the emitted protonenergy. A convolution with the experimental resolution of 50 keVFWHM is implemented in the results. For a resolution of 200 keVthe figure is similar, but the strength at the peak of thep-wave statesis reduced by about 20%.

390 MeV, respectively. Extrapolating from these twodata tok = 170 MeV we obtain a cross section of32 µb sr−1 MeV−1 for the background of inclusive(γ,p) in our reaction at this angle. This result couldbe also obtained theoretically using the approach of[11] where a Monte Carlo simulation was performedwith the probabilities for the primary nuclear stepsevaluated with a microscopic many body calculationin [12]. This inclusive(γ,p) cross section is aboutdouble the one obtained for the peaks of the signalsof the pionic atoms in an average. However, at smallangles, where we have evaluated the pionic atom for-mation cross sections, the background should be evensmaller, and this is the case in the theoretical model of[11], because kinematically it is not possible to havecontribution to the process with just one collision.

This situation is particularly rewarding in view todistinguish these states experimentally. On the otherhand, although our calculations do not deem it neces-sary, it should be possible to further increase the ratioof signal to background using delayed fission of pionabsorption fragments, a technique used with success inthe production ofΛ hypernuclei in [13] and tentativelysuggested for the present reaction [14]. The set up forthe experiment is suited to present experimental facil-

72 S. Hirenzaki, E. Oset / Physics Letters B 527 (2002) 69–72

ities, particularly those of low energies, and the crosssections predicted are of the order of those presentlybeing measured with high precision in the(γ,p) re-action [10], which makes this reaction a very practicalmethod to produce deeply bound pionic atoms.

Acknowledgements

One of us, S.H. wishes to acknowledge the hospi-tality of the University of Valencia where this workwas done and financial support from the FundacionBBV. We would like to thank A. Margarian for use-ful discussions and encouragement to perform thesecalculations. This work is also partly supported byDGICYT contract number BFM2000-1326 and by theGrants-in-Aid for Scientific Research of the JapanMinistry of Education, Culture, Sports, Science andTechnology (No. 11440073 and No. 11694082).

References

[1] T. Yamazaki et al., Z. Phys. A 355 (1996) 219;

T. Yamazaki et al., Phys. Lett. B 418 (1998) 246;H. Gilg et al., Phys. Rev. C 62 (2000) 025201;K. Itahashi et al., Phys. Rev. C 62 (2000) 025202.

[2] H. Geissel et al., Nucl. Phys. A 663 (2000) 206c.[3] E. Oset, J. Nieves, Hyperfine Interactions 103 (1996) 285.[4] H. Toki, S. Hirenzaki, T. Yamazaki, Nucl. Phys. A 530 (1991)

679;S. Hirenzaki, H. Toki, T. Yamazaki, Phys. Rev. C 44 (1991)2472.

[5] J. Nieves, E. Oset, Phys. Lett. B 282 (1992) 24.[6] K.J. Raywood et al., Phys. Rev. C 55 (1997) 2492.[7] E. Marco, W. Weise, Phys. Lett. B 502 (2001) 59.[8] J. Nieves, E. Oset, Phys. Rev. C 47 (1993) 1478.[9] C. Mahaux, P.F. Bortignon, R.A. Broglia, C.H. Dasso, Phys.

Rep. 120 (1985) 1.[10] J. Arends et al., Nucl. Phys. A 526 (1991) 479.[11] R.C. Carrasco, M.J. Vicente Vacas, E. Oset, Nucl. Phys. A 570

(1994) 701.[12] R.C. Carrasco, E. Oset, Nucl. Phys. A 536 (1992) 445.[13] H. Ohm et al., Phys. Rev. C 55 (1997) 3062;

F. Garibaldi et al., 7th International Conference on Mesons andLight Nuclei 98, Pruhonice, Prague, Czech Republic, Mesonsand light nuclei ’98* 216–223.

[14] A.T. Margarian, Nucl. Instrum. Methods A 357 (1995) 495.



Q2-dependence of backward pion multiplicity inneutrino–nucleus interactions

O. Benhara, S. Fantonib, G.I. Lykasovc,1, U. Sukhatmed

a INFN, Sezione Roma 1, I-00185 Rome, Italyb International School for Advanced Studies (SISSA), I-34014 Trieste, Italy

c Joint Institute for Nuclear Research, Dubna 141980, Moscow Region, Russiad Department of Physics, University of Illinois, Chicago, IL 60607, USA



Abstract

The production of pions emitted backward in inelastic neutrino–nucleus interactions is analyzed within the impulseapproximation in the framework of the dual parton model. We focus on theQ2-dependence of the multiplicity of negativepions, normalized to the total cross section of the reactionν + A → µ + X. The inclusion of planar (one-Reggeon exchange)and cylindrical (one-Pomeron exchange) graphs leads to a multiplicity that decreases asQ2 increases, in agreement with recentmeasurements carried out at CERN by the NOMAD Collaboration. A realistic treatment of the high momentum tail of thenucleon distribution in a nucleus also allows for a satisfactory description of the semi-inclusive spectrum of backward pions. 2002 Published by Elsevier Science B.V.

PACS: 13.60.Le; 25.30.Fj; 25.30.Rw

Over the past several years, backward hadron pro-duction in inelastic lepton–nucleus scattering has beenextensively investigated, both experimentally [1–3]and theoretically [4–8], with the aim of pinning downthe dominant reaction mechanism and extracting in-formation on the underlying dynamics.

The analysis carried out on Refs. [6,7] focusedon nuclear effects ine + A → e′ + p + X reac-tion, while in Ref. [8], hereafter referred to as I, we

E-mail address: [email protected] (O. Benhar).1 Supported by the Russian Foundation for Fundamental Re-

search under grant 99-02-17727.

studied the spectrum of pions emitted in the processν + A → µ + π + X. The results in I show that themain contribution to this spectrum comes from scat-tering off nucleons carrying high momentum. The cal-culations were performed within the impulse approx-imation (IA) scheme, using the Quark Gluon StringModel (QGSM) [9,10] to describe the interaction ver-tex. This model is similar to the Dual Parton Model(DPM) developed in Refs. [11,12], in that both ap-proaches are aimed at implementing the dual topo-logical unitarization scheme. The relativistic invariantsemi-inclusive spectrum of pions emitted strictly back-ward was obtained including only the cylindrical (i.e.,one-Pomeron exchange) graph, while the contribution



74 O. Benhar et al. / Physics Letters B 527 (2002) 73–79

of the planar (i.e., one-Reggeon exchange) diagramwas neglected.

In this Letter we continue the investigation of semi-inclusive pion production, started in I. We extend ouranalysis to pions emitted in the whole backward hemi-sphere, with respect to the direction of the incomingbeam, and use QGSM to consistently include both thecylindrical and planar graphs in the calculation of thespectrum integrated over the transverse pion momen-tum. Our results are compared to the new data from theNOMAD Collaboration [3], which measured the spec-trum of π emitted in the backward hemisphere in thereactionν +C →µ+π +X and theQ2-dependenceof the pion multiplicity,〈nπ 〉, normalized to the totalcross section of the reactionν +C →µ+X.

Our study focuses on the kinematical region oflarge pion momentum (pπ > 0.3 GeV/c), where theeffects of final state interactions (FSI) leading topion absorption associated with production of baryonresonances are expected to be small [13,14] and the IAcan be safely used.

The relativistic invariant semi-inclusive spectrumof pions produced in the process+A → ′ +π +X,in which the incoming lepton is scattered with energyE′ into the solid angledΩ , is defined as

(1)ρA→′πX ≡Eπdσ

d3pπ dΩ dE′ ,

whereEπ andpπ are the total energy and three mo-mentum of the produced pion, respectively. Within theframework of IA the quantity appearing on the right-hand side of the above equation can be rewritten inconvolution form, in terms of the semi-inclusive spec-tra of pions produced in lepton–proton and lepton–neutron processesρp→′πX andρn→′πX , accordingto (see, e.g., Refs. [6,8,15])

ρA→′πX(x,Q2; z,pπt

)=

∫x,zy

dy d2kt fA(y, kt)

(2)

×[Z

Aρp→′πX

(x

y,Q2; z

y,pπt − z

ykt

)+ N

Aρn→′πX

(x

y,Q2; z

y,pπt − z

ykt

)],

where Z and N are the number of protons andneutrons,A =Z+N andQ2 = −q2, q being the four-

momentum tranferred by the lepton. The relativisticinvariant variablesx andz are defined as [2,3]

(3)x = MA

m

Q2

2(PA · pν), z = MA

m

(pπ · pν)

(PA · pν),

wherePA andpν are the four-momenta of the nucleusand the initial neutrino, respectively, andMA andm are the nucleus and nucleon masses. The abovevariables are best suited to analyze the data of Ref. [3],in which the backward direction is defined withrespect to the beam direction. Note that they do notcoincide withx andz employed in I, whose definitioncan be recovered by replacing the four-momentum ofthe incoming neutrino,pν , with the four momentumtransferq in Eq. (3).

The nucleon distribution functionfA(y, kt) can bewritten [6]

fA(y, kt)

(4)=∫

dk0dkz S(k)yδ

(y − MA

m

(k · pν)

(PA · pν)

),

whereS(k) is the relativistic invariant function de-scribing the nuclear vertex with an outgoing virtualnucleon. The functionfA(y) resulting from the trans-verse momentum integration offA(y, kt ) can be ob-tained by approximatingS(k) with the nonrelativis-tic spectral functionP(k,E) yielding the probabilityof finding a nucleon with momentumk and removalenergyE = m − k0 in the target nucleus [16]. How-ever, due to the limited range of momentum and re-moval energy covered by nonrelativistic calculationsof P(k,E) (typically |k| < kmax ∼ 0.7–0.8 GeV/cand (m − k0) < 0.6 GeV (see, e.g., Ref. [16]), thisprocedure can only be used in the regiony < y0 ∼1.7–1.85. An alternative approach to obtainfA(y) atlargery, based on the calculation of the overlap of therelativistic invariant phase-space available to quarksbelonging to strongly correlated nucleons, has beenproposed in Ref. [6]. A similar procedure has beenalso used in Ref. [13] to obtain the quark distributionin deuteron at largey.

The analysis of backward pion production requiresthe full nucleon distribution function given by Eq. (4).We assume that it can be written in the factorized form

(5)fA(y, kt)= fA(y)gA(kt ),

O. Benhar et al. / Physics Letters B 527 (2002) 73–79 75

Fig. 1. Planar (a) and cylindrical (b) graphs contributing to the reactionν +A → µ+ h+X. Diagrams (a) and (b) describe processes in whichthe incoming neutrino interacts with a valence quark or a seaqq pair, respectively.

with the functiongA chosen of the Gaussian form

(6)gA(kt )= 1

π 〈k2t 〉

e−k2t /〈k2

t 〉,

where〈k2t 〉 is the average value of the squared nucleon

transverse momentum.The second ingredient entering the calculation of

the spectrum of Eq. (2) is the elementary semi-inclusive spectrum. It can be calculated within theQGSM, based on the 1/N expansion,N being thenumber of flavors or colors, developed in [10]. Ac-cording to Ref. [9], the elementary production processcan be described in terms of planar and cylindricalgraphs in thes-channel, as shown in Fig. 1 for thecase of neutrino interactions associated with the ex-change of aW -boson. The planar graph of Fig. 1(a)describes neutrino scattering off a valence quark.According to [5,8] its contribution to the processν +N →µ+ π +X reads

FNP

(x1,Q

2; z1,p1t)

(7)

= z1φ1(x1,Q

2)[1

3Duu→π

(z1

1− x1,p1t

)+ 2

3Dud→π

(z1

1− x1,p1t

)],

with

φ1(x1,Q

2)(8)= G2mE

π

x1

1− x1

m2W

m2W +Q2

dv(x1,Q

2),

whereG is the Fermi coupling constant,E is the en-ergy of the incomingν, mW is theW -boson mass andx1 ≡ x/y =Q2/2(k ·pν) is the Bjorken variable asso-ciated with a nucleon bound in a nucleus. In Eq. (7),dv is thed-quark distribution inside a proton or a neu-tron,Duu→π andDud→π are the fragmentation func-tions of uu or ud diquarks into positive or negativepions,p1t = pt − (z/y)kt is the transverse momentumof a pion produced on a nucleon carrying transversemomentumkt andz1 = z/y.

According to Refs. [9,10] the planar graph ofFig. 1(a) corresponds to one-Reggeon exchange inthe t-channel, leading to the asymptotic behaviorW

αR(0)−1X , whereWX is the squared invariant mass of

the undetected debris andαR(0)= 1/2 is the Reggeonintercept. Hence, from

(9)WX = (k + q)2 = k2 +Q2 (1− x1)

x1,

where x1 = Q2/2(k · q), it follows that the Reggebehavior of the planar graph of Fig. 1(a) at moderateand largeQ2 is given by

(10)W−1/2X ∼

√x1

Q2(1− x1).

The fact that theQ2-dependence of the graph ofFig. 1(a) is determined mainly by this Regge asymp-totic is a consequence of the weakQ2-dependence ex-hibited by the calculatedFN

P .The second contribution to the spectrum, coming

from the cylindrical graph of Fig. 1(b) can also be


obtained within the approach of Ref. [10]. Accordingto I, it can be written the form

FNC

(x1,Q

2; z1,p1t)

= z1φ2(Q2)[L1

(x1,Q

2; z1,p1t)

+ L1(x1,Q

2; z1,p1t) +L2

(x1,Q

2; z1,p1t)(11)+ L2

(x1,Q

2; z1,p1t)],

with

(12)φ2(Q2) =mE

G2

π

m2W

Q2 +m2W

,

L1 =1−x1∫z1

dy

y

∫d2kt

(13)

×[uv

(y, kt;Q2)Du→π

(z1

y,p1t − z1

ykt

)+ dv

(y, kt ;Q2)Dd→π

(z1

y,p1t − z1

ykt

)],

L2 =1−x1∫z1

dy

y

∫d2kt

(14)

×[

4

3fud

(y, kt;Q2)Dud→π

(z1

y,p1t − z1

ykt

)+ 1

3

(fuu

(y, kt ;Q2)

×Duu→π

(z1

y,p1t − z1

ykt

)+ fdd

(y, kt;Q2)

×Ddd→π

(z1

y,p1t − z1

ykt

))],

(15)L1 = ds(x1,Q

2)Du→π (z1,p1t )

and

L2 =1−x1∫z1

dy

y

∫d2kt ds

(y, kt ;Q2)

(16)×Dd→π

(z1

y,p1t − z1

ykt

).

In the above equationsuv is the distribution of thevalenceu-quark,fuu, fud and fdd are the distribu-tions ofuu-, ud- anddd-diquarks inside the nucleon,Du→π ,Dd→π are the fragmentation functions of theu- andd-quark into a pion andDdd→π is the fragmen-tation function of thedd-diquarkdd into a pion. Theexplicit expression of the quark and diquark distribu-tions and fragmentation functions, obtained within theapproach of Ref. [10], are given in I.

According to Refs. [9,10] the diquark distributionfqq(y) coincides with the distribution of the corre-sponding valence quark evaluated at 1− y, i.e., withqv(1 − y), if qv(y) is normalized to unity. The maincontribution to the pion spectrum is coming from tar-get fragmentation, i.e., the kinematical region cor-responding to large values ofz1 = z/y. Within theapproaches of Refs. [9,10] and [11,12] the main con-tribution of sea quarks to the hadron spectrum is as-sociated withn-Pomeron exchange processes (withn 2), which are not taken into acount in this analy-sis as they provide a vanishingly small contributionat largey [10–12]. We only include sea quarks pro-duced from gluon decay, as shown in Fig. 1(b). In ourapproach the gluon is seen as a nucleon constituent,in addition to the three valence quarks. Therefore,we have used an average value of the gluon fraction〈yg〉 and normalized the valence quark distributionin a nucleon to 1− 〈yg〉, the value〈yg〉 0.15–0.2being taken from Ref. [19]. This procedure amountsto settingfqq(y) = qv(1 − 〈yg〉 − y). Our calcula-tions show that the contribution of fragmentation ofsea quarks and antiquarks (see the cylindrical graph ofFig. 1(b)), described by the termsL1 andL2, is muchsmaller than that coming from fragmentation of va-lence quarks and diquarks, described by the termsL1andL2 of Eq. (11).

We assume a factorized form, similar to that ofEq. (5), for the quark and diquark distributions andfragmentation functions. Thekt dependence of thequark distribution is again chosen of the Gaussianform

(17)gq(kt )= 1

〈k2qt 〉π

e−k2t /〈k2

qt 〉,

where〈k2qt 〉 is the average value of the squared quark

transverse momentum. For the fragmentation function


we have used

(18)gq(qq)→h(kt )= γ

πe−γ k2

t .

The cylindrical graph of Fig. 1(b) corresponds toone-Pomeron exchange in thet-channel, whose as-ymptotic behavior isWαP (0)−1

X , where for the super-critical Pomeron [9,10] the value of the exponent isgiven by∆ ≡ αP (0) − 1 0.1–0.15 [17,18]. Hence,the Pomeron asymptotic of the cylindrical graph ofFig. 1(b) turns out to be(k2 + Q2(1 − x1)/x1)

∆.TheQ2-dependence of the cylindrical graph is dom-inated by the supercritical Pomeron behavior, asFN

C

of Eq. (11) depends weakly uponQ2.In conclusion, the relativistic invariant spectrum of

pions produced inν +N → µ+π +X processes canbe written in the form

ρνN→µπX ≡ z d3σ

dx1dz1dp1t

= FNP

(x1,Q

2; z1,p1t)

×((k2 +Q2(1− x1)/x1)

s0

)−1/2

+ FNC

(x1,Q

2; z1,p1t)

(19)×((k2 +Q2(1− x1)/x1)

s0

)∆

,

wheres0 = 1 (GeV/c)2 is a parameter usually intro-duced in Regge theory in order to get the correct di-mensions. SubstitutingρνN→µπX of Eq. (19) and thenucleon distributionfA(y, kt ) into Eq. (2) one can cal-culate the relativistic invariant spectrum of pions pro-duced in the reactionν +A → µ+ π +X.

The multiplicity of pions normalized to the crosssection of the processν +A → µ+X, σ , defined as

〈nπ 〉 ≡ 〈nπ 〉σ

(20)= 1

σ

xmax∫xmin

dx

zmax∫zmin

dz

z

pπ max∫0

d2pπt ρνA→µπX,

can also be split into two parts, corresponding to theplanar, or one-Reggeon exchange, diagram (Fig. 1(a))and cylindrical, or one-Pomeron exchange diagram(Fig. 1(b)). While at fixedx1 the first term decreasesas Q2 increases, the second increases with a slopedictated by the value of∆. The Q2-dependence

Fig. 2. Q2-dependence of the normalized multiplicity of pionsproduced in the backward hemisphere in theν+C → µ−+π−+X

reaction. The solid line shows the results of the full calculation,including both graphs of Fig. 1. The dashed and dash-dot linescorrespond to the separated contributions of the cylindrical graphof Fig. 1(b) and the planar graph of Fig. 1(a), respectively. Thedots show the results obtained settingP (k,E) ≡ 0 in the highenergy–momentum domain, not covered by the nonrelativisticcalculation of Ref. [16]. The experimental data are taken fromRef. [3].

of 〈nπ 〉 resulting from our approach is shown inFig. 2, together with experimental data taken fromRef. [3].

The NOMAD Collaboration carried out a studyof inelastic ν–C interactions in which the negativepions emitted in the whole backward hemisphere,with respect to the incoming neutrino beam, weredetected [3]. The data of Fig. 2 show the multiplicity ofnegative pions carrying momenta in the range 0.35<pπ < 0.8 GeV/c, measured in a kinematical setupin which WX increases asQ2 increases. Theoreticalcalculations have been performed applying the samekinematical conditions.

The results obtained including both diagrams isrepresented by the solid line, while the dashed anddash-dot lines correspond to the separated contribu-tions of the cylindrical graph of Fig. 1(b) and the pla-nar graph of Fig. 1(a), respectively.

The planar diagram provides the main contributionat small Q2, while the cylindrcal one dominatesat Q2 > 10 (GeV/c)2. The decrease of the planargraph contribution to〈nπ 〉 and the increase of thecylindrical graph contribution with increasingQ2 areboth consequences of the kinematical setup of Ref. [3],


in which increasingQ2 leads to an increase of thesquared invariant massWX (see Eqs. (9) and (19)). Itclearly appears that both diagrams have to be includedto explain the observedQ2-dependence. The fact thatthe solid line lies somewhat below the experimentaldata atQ2 > 2 (GeV/c)2 is likely to be ascribed to thecontribution of secondary rescattering effects, whichare not taken into account in our approach.

The dotted line has been obtained settingP(k,E) ≡ 0 in the domain of large energy and largemomentum, not covered by the nonrelativistic calcula-tions of Ref. [16]. Comparison between the solid anddotted line shows that the dominant contribution to〈nπ 〉 comes from the high momentum tail of the nu-cleon distribution.

Numerical calculations have been carried out us-ing values of the average squared transverse mo-mentum, entering Eq. (6), in the range〈k2

t 〉 = 0.12–0.14 (GeV/c)2. These values correspond to an aver-age nucleon momentum∼ 0.4 GeV/c, which seemsto be reasonable for processes dominated by the highmomentum tail of the nucleon distribution. The asso-ciated ambiguity in the results is always within the ex-perimental errors.

The other two parameters entering our calculationsare the average quark and diquark transverse momen-tum and the slope of the Gaussian describing thekt -dependence of the fragmentation functions. Theirvalues have been taken from Ref. [20], where a modi-fied QGSM, explicitly including the transverse motionof quarks and diquarks, has been developed.

Besides theQ2 dependence of the negative pionmultiplicity, the NOMAD Collaboration measured thesemi-inclusive spectrum of negative pions, defined as

(21)2π

σEπ

dσ

d3pπ

≡ 1

σ

Eπ

pπ

dσ

dp2π

,

in the kinematical region corresponding to pions emit-ted in the backward hemisphere withp2

π >

0.05 (GeV/c)2 [3]. In Fig. 3 we compare thep2π -de-

penden ce of the experimental spectrum to that result-ing from our approach. It clearly appears that the con-tribution of the cylindrical graph dominates the spec-trum and that the inclusion of the high momentum tailof the nucleon distribution, corresponding top > 0.4GeV/c, is needed to describe the data.

In this Letter we have analyzed inelastic neutrino–nucleus processes in which negative pions are emit-

Fig. 3.p2-dependence of the spectrum of backward pions producedin the reactionν+C → µ− +π− +X. The solid curve correspondsto the full calculation whereas the dashed line has been obtainedincluding the cylindrical graph of Fig. 1(b) only. The dash-dotline shows the results obtained settingP (k,E) ≡ 0 in the highenergy–momentum domain, not covered by the nonrelativisticcalculation of Ref. [16]. The experimental data are taken fromRef. [3].

ted in the backward emishpere. The main conclusionsof our work, concerning both the reaction mechanismand role of nuclear structure, can be summarized asfollows. The dominant contribution to the reactioncomes from target fragmentation, as shown by the cal-culation of the two topological graphs of Fig. 1 withinthe dual topological unitarization scheme. Two kindsof the elementary processes have to be included: neu-trino scattering off valence quarks, corresponding toone-Reggeon exchange in thet-channel (planar graph,Fig. 1(a)), and neutrino scattering off sea quarks, cor-responding to one-Pomeron exchange in thet-channel(cylindrical graph, Fig. 1(b)). In the kinematical setupof Ref. [3] the contribution of the planar graph tothe analyzed pion multiplicity decreases as

√1/Q2

as Q2 and WX increase, while the contribution ofthe cylindrical graph increases asQ2∆, with ∆ inthe range 0.1–0.15, asQ2 andWX increase. The re-sulting Q2-dependence of the normalized multiplic-ity of pions emitted in the backward hemisphere inthe reactionν + C → µ− + π− + X, turns out tobe in fair agreement with the experimental data ofRef. [3]. As for the role of nuclear structure, the dom-inant contribution is coming from the high momen-tum tail of nucleon distribution, which can be de-scribed in terms of overlaps of distributions of three-


quark colorless objects [6]. Including both graphs ofFig. 1, our approach also provides a satisfactory de-scription of thep2-dependence of the measured pionspectrum.

Acknowledgements

We are indebted to A.B. Kaidalov and M. Veltrifor many helpful discussions. This work was partlysupported by the US Department of Energy.

References

[1] P.V. Degtyarenko et al., Phys. Rev. C 50 (1994) R541.[2] BEBC WA 59 Collaboration, E. Matsinos et al., Z. Phys. C 44

(1989) 79.[3] NOMAD Collaboration, P. Astier et al., Nucl. Phys. B 609

(2001) 255.[4] C.E. Carlson, K.E. Lassila, U.P. Sukhatme, Phys. Lett. B 263

(1991) 277.[5] G.D. Bosveld, A.E.I. Dieperink, A.G. Tenner, Phys. Rev. C 49

(1993) 2379.[6] O. Benhar, S. Fantoni, G.I. Lykasov, N.V. Slavin, Phys. Rev.

C 55 (1997) 244.[7] O. Benhar, S. Fantoni, G.I. Lykasov, N.V. Slavin, Phys. Lett.

B 415 (1997) 311.

[8] O. Benhar, S. Fantoni, G.I. Lykasov, Eur. Phys. J. A 7 (2000)415.

[9] A.B. Kaidalov, Sov. J. Nucl. Phys. 33 (1981) 733.[10] A.B. Kaidalov, Phys. Lett. B 116 (1982) 459;

A.B. Kaidalov, Sov. J. Nucl. Phys. 33 (1981) 733;A.B. Kaidalov, Sov. J. Nucl. Phys. 45 (1987) 902.

[11] A. Capella, U. Sukhatme, C.-I. Tan, J. Tran Thanh Van, Phys.Lett. B 81 (1979) 68;A. Capella, U. Sukhatme, C.-I. Tan, J. Tran Thanh Van, Z.Phys. 63 (1979) 329.

[12] A. Capella, U. Sukhatme, C.-I. Tan, J. Tran Thanh Van, Phys.Rep. 236 (1994) 223.

[13] L.L. Frankfurt, M.I. Strikman, Phys. Rep. 160 (1988) 236.[14] N.S. Amelin, G.I. Lykasov, Sov. J. Nucl. Phys. 33 (1981) 100.[15] L.L. Frankfurt, M.I. Strikman, Phys. Rep. 76 (1981) 215.[16] O. Benhar, A. Fabrocini, S. Fantoni, Nucl. Phys. A 505 (1989)

267.[17] A. Capella, A. Kaidalov, C. Merino, J. Tran Thanh Van, Phys.

Lett. B 337 (1994) 358;A. Capella, A. Kaidalov, C. Merino, J. Tran Thanh Van, Phys.Lett. B 343 (1995) 403.

[18] A. Capella, E.G. Ferreiro, C.A. Salgado, A.B. Kaidalov, Phys.Rev. D 63 (2001) 054010.

[19] NM Collaboration, M. Arneodo et al., Phys. Lett. B 309 (1993)222.

[20] G.I. Lykasov, M.N. Sergeenko, Z. Phys. C 52 (1991) 635;

G.I. Lykasov, M.N. Sergeenko, Z. Phys. C 56 (1992) 697;G.I. Lykasov, M.N. Sergeenko, Z. Phys. C 70 (1996) 455.



Transverse energy dependence of J/Psi suppressionin Au + Au collisions at RHIC energy

A.K. Chaudhuri

Variable Energy Cyclotron Centre, 1/AF, Bidhan Nagar, Calcutta 700 064, India

Received 26 June 2001; received in revised form 23 December 2001; accepted 4 January 2002


Abstract

Prediction for transverse energy dependence ofJ/ψ to Drell–Yan ratio in Au+Au collisions at RHIC energy was obtainedin a model which assume 100% absorption ofJ/ψ above a threshold density. The threshold density was obtained by fittingthe NA50 data onJ/ψ suppression in Pb+Pb collisions at SPS energy. At RHIC energy, hard processes may be important.Prediction ofJ/ψ suppression with and without hard processes were obtained. With hard processes included,J/ψ ’s are stronglysuppressed. 2002 Elsevier Science B.V. All rights reserved.

PACS: 25.75.-q; 25.75.Dw

In relativistic heavy ion collisionsJ/ψ suppressionhas been recognized as an important tool to identifythe possible phase transition to quark–gluon plasma.Because of the large mass of the charm quarks,cc

pairs are produced on a short time scale. Their tightbinding also make them immune to final state in-teractions. Their evolution probes the state of matterin the early stage of the collision. Matsui and Satz[1] predicted that in presence of quark–gluon plasma(QGP), binding ofcc pairs intoJ/ψ meson will behindered, leading to the so-calledJ/ψ suppression inheavy ion collisions [1]. Over the years several groupshave measured theJ/ψ yield in heavy ion collisions(for a review of the data and the interpretations see[2,3]). In brief, experimental data do show suppres-

E-mail address: [email protected] (A.K. Chaudhuri).

sion. However, this could be attributed to the con-ventional nuclear absorption, also present in pA col-lisions.

The latest data obtained by the NA50 Collabora-tion [4] on J/ψ production in Pb+Pb collisions at158 A GeV is the first indication of anomalous mecha-nism of charmonium suppression, which goes beyondthe conventional suppression in nuclear environment.The ratio ofJ/ψ yield to that of Drell–Yan pairs de-creases faster withET in the most central collisionsthan in the less central ones. It has been suggested thatthe resulting pattern can be understood in a deconfine-ment scenario in terms of successive melting of char-monium bound states [4].

In a recent paper Blaizot et al. [5] showed thatthe data can be understood as an effect of transverseenergy fluctuations in central heavy ion collisions.Introducing a factorε =ET /ET (b) and assuming that



A.K. Chaudhuri / Physics Letters B 527 (2002) 80–84 81

the suppression is 100% above a threshold density(a parameter in the model) and smearing the thresholddensity (at the expense of another parameter) best fit tothe data was obtained. Capella et al. [6] analysed thedata in the comover approach. There also, the comoverdensity has to be modified by the factorε. Introductionof this adhoc factorε can be justified in a model basedon excited nucleons represented by strings [7].

At a fixed impact parameter, the transverse energyas well as the number of NN collisions fluctuate.The fluctuations in the number of NN collisions werenot taken into account in the calculations of Blaizotet al. [5] or in the calculations of Capella et al. [6].We have analysed [8] the NA50 data, extending themodel of Blaizot et al. [5] to include the fluctuationsin number of NN collisions at fixed impact parameter.The ET distribution was obtained in the geometricmodel, which includes these fluctuations. It was shownthat with a single parameter, the threshold density,above which theJ/ψ suppression is assumed to be100%, good fit to the data can be obtained. In thepresent Letter we have applied the model to predict theET dependence ofJ/ψ to Drell–Yan ratio at RHICenergy. With RHIC being operational, it is hoped thatthe prediction will help to plan experiments to detectQGP. In the following, we will present in brief themodel. The details can be found in [8].

In [8] ET distribution for Pb+Pb collision wasobtained in the geometric model [9,10]. In this model,ET distribution of AA collisions is written in termsof ET distribution in NN collisions. One also assumethat the Gamma distribution with parametersα andβ describe theET -distribution in NN collisions.ETdistribution for Pb+Pb could be fitted well withα = 3.46± 0.19 andβ = 0.379± 0.021. In Fig. 1(a)the experimental data along with the fit are shown.Parametric values ofα and β indicate that averageET produced in individual NN collisions isβ/α ∼0.1 GeV. This is to be contrasted with the averageET ∼ 1 GeV produced in other AA collisions [10].

As mentioned in the beginning, we have assumedthat above a threshold densitync, the charmoniumsuppression is 100% effective [5]. Charmonium pro-duction cross-section at impact parameterb is writtenas

(1)d2σJ/ψ

d2b= σJ/ψ

∫d2s T eff

A (s)T effB (s − b)S(b, s),

Fig. 1. (a) Transverse energy distribution in Pb+ Pb collisions,(b) J/ψ to Drell–Yan ratio in Pb+ Pb collisions as a function oftransverse energy.

whereT effA,B is the effective nuclear thickness function,

(2)

T eff(s)=∞∫

−∞dzρ(s, z)exp

(−σabs

∞∫z

dz′ ρ(s, z′)),

with σabs as the cross-section forJ/ψ absorptionby nucleons. The exponential factor is the nuclearabsorption survival probability, the probability forthe cc pair to avoid nuclear absorption and form aJ/ψ . S(b, s) is the anomalous part of the suppression.Blaizot et al. [5] assumed thatJ/ψ suppressionis 100% effective above a threshold density (nc),a parameter in the model. Accordingly, the anomaloussuppression part was written as

(3)S(b, s)=Θ(nc − εnp(b, s)

),

wherenp is the density of participant nucleons in theimpact parameter space,

np(b, s)= TA(s)[1− e−σNNTB(b−s)]

(4)+ [TA ↔ TB ]and ε = ET /ET (b) = ET /nβ/α is the modificationfactor which takes into account the transverse energy

82 A.K. Chaudhuri / Physics Letters B 527 (2002) 80–84

fluctuations at fixed impact parameter [5]. This modi-fication makes sense only whennp is assumed to beproportional to the energy density. Implicitly it wasalso assumed that theET fluctuations are strongly cor-related in different rapidity gaps. The assumption wasessential as NA50 Collaboration measuredET in the1.1–2.3 pseudorapidity window while theJ/ψ ’s weremeasured in the rapidity window 2.82< y < 3.92 [4].Strong correlation betweenET fluctuations in differ-ent rapidity windows is explained in the geometricmodel [8].

We calculate theJ/ψ production as a function oftransverse energy, at an impact parameterb as

(5)dσJ/ψ

dET=

∞∑n=1

Pn(b,ET )d2σJ/ψ

d2b,

wherePn(b,ET ) is the probability to obtainET inn NN collisions, expression for which can be foundin [8].

The Drell–Yan production was calculated similarly,replacing charmonium cross section in Eq. (5) by theDrell–Yan cross-section,

(6)d2σDY

d2b= σDY

∫d2s TA(s)TB(s − b).

In Fig. 1(b) we have compared the theoretical char-monium production cross-section with NA50 experi-mental data. The normalization factorσJ/ψ/σDY wastaken to be 53.5. The solid curve is obtained withσabs= 6.4 mb, andnc = 3.8 fm−2. Very good descrip-tion of the data from 40 GeV onward is obtained. Themodel reproduced the 2nd drop around 100 GeV, (theknee of theET distribution). It may be noted that ifthe fluctuations in the NN collisions were neglected,equivalent description is obtained with threshold den-sity nc = 3.75 fm2, with smearing of theΘ functionat the expense of another parameter. It is evident thatin this model, the smearing is done by fluctuating NNcollisions. Theoretical calculations predict more sup-pressions below 40 GeV, a feature evident in othermodels also. It is possible to fit the entireET range,reducing theJ/ψ–nucleon absorption cross-section.Recent data [11] on theJ/ψ cross section in pA colli-sions point to a smaller value ofσabs∼ 4 to 5 mb. Thedashed line in Fig. 1(b), corresponds toσabs= 4 mbandnc = 3.42 fm−2. However, it may be mentionedthat σabs= 4 mb does not allow a good fit to the pAand S-U data [12].

Present model can be used to predictET depen-dence ofJ/ψ to Drell–Yan ratio at RHIC energy. Re-cent PHOBOS experiment [13] showed that for centralcollisions, total multiplicity is larger by 70% at RHICthan at SPS. We assume thatET is correspondinglyincreased [15]. Accordingly, we rescale theET distri-bution for Pb+Pb collisions and assume that it rep-resent the experimentalET distribution for Au+Aucollisions at RHIC (small mass difference between Auand Pb is neglected). At RHIC energy the so-calledhard component which is proportional to number of bi-nary collisions appear. Model-dependent calculationsindicate that the hard component grows from 22% to37% as the energy changes from

√s = 56 GeV to

130 GeV [14]. However, we choose to ignore the hardcomponent inET -distribution. The multiplicity dis-tribution obtained by the PHOBOS Collaboration, inthe rapidity range 3< |η| < 4.5 could be fitted wellwith or without this hard component. Indeed, it ap-pears that the data are better fitted without the hardcomponent [14]. Global distribution, e.g., multiplicityor transverse energy distributions are not sensitive tothe hard component. In Fig. 2(a), filled circles repre-sent the “experimental”ET distribution for Au+Aucollisions at RHIC, obtained by scaling theET distri-bution in Pb+Pb collisions at SPS. The solid line isa fit to the “experimental”ET -distribution in the geo-metric model, obtained withα = 3.09 andβ = 0.495.Nucleon–nucleon inelastic cross section (σNN) was as-sumed to be 41 mb at RHIC, instead of 32 mb at SPS[15]. Fitted values ofα and β are interesting. Withthese values averageET produced in individual NNcollisions at RHIC isβ/α = 0.16 GeV. This is to becompared with the value 0.1 GeV for Pb+Pb colli-sions at SPS. AverageET produced in individual NNcollisions at RHIC is increased by 60%, compared toSPS energy. The apparent inconsistency is resolved ifwe remember thatσNN is increased by 30% from SPSto RHIC energy.

While theET distribution at RHIC may not be sen-sitive to the hard scattering component, theJ/ψ sup-pression will be. In the modelJ/ψ suppression is100% if thenpET /ET (b) exceeds the threshold den-sity nc. Without the hard component, transverse den-sity in Au+Au collisions will be nearly same as thatin Pb+Pb collisions at SPS. At RHIC,ET (b) is in-creased, individual NN collisions produces moreET .Then anomalous suppression will set at largerET

A.K. Chaudhuri / Physics Letters B 527 (2002) 80–84 83

Fig. 2. (a) ExpectedET distribution in Au+ Au collisions atRHIC energy. The solid line is a fit in the geometric model (seetext), (b) expectedET dependence ofJ/ψ to Drell–Yan ratio inAu+ Au collisions at RHIC energy. The dotted (solid) lines are thepredictions with (without) the hard scattering component, for twosets of parameters. The dashed curve is the prediction of Blaizotet al. [15].

(threshold density being same). However, with thehard component, the transverse density will be mod-ified. Forf fraction of hard scattering transverse den-sity can be written as [15]

(7)np(b, s)→ (1− f )np(b, s)+ 2f nhardp (b, s)

with nhardp (b, s) = σNNTA(s)TB(b − s). With hard

component, transverse density is increased, as a result,anomalous suppression will set in at lowerET .

In Fig. 2(b), we have presented theJ/ψ to Drell–Yan ratio obtained in the present model for Au+Aucollision at RHIC energy. We have presented theresults for the two sets of parameters: (A)σabs =6.4 mb, nc = 3.8 fm−2 and (B) σabs= 4 mb, nc =3.44 fm−2. The solid lines are the prediction forJ/ψ suppression neglecting the hard component inthe transverse density, for the two sets of parameters(A) and (B), respectively. The dotted lines are theprediction including the hard component. We findthat without the hard component,J/ψ suppression

at RHIC is similar to that obtained at SPS energy.The 2nd drop which occurred at 100 GeV in Pb+Pbcollisions at SPS energy, now sets in around 180 GeV(knee of theET distribution being around that energy).As mentioned earlier, at RHIC,ET (b) is increased,and without the hard component the transverse densityis nearly same as it was in Pb+Pb collisions at SPS.Anomalous suppression then occurs at largerET . Asit was for Pb+Pb collisions at SPS, the two sets givenearly same suppression forET beyond 100 GeV. It isalso an indication that anomalous suppression occur atlargerET .

If the transverse density is modified to includethe hard component,J/ψ ’s are strongly suppressed.In Fig. 2(b), the dotted lines presents the resultsobtained with 37% [14] hard component. Nearly samesuppression is obtained for set A and B. At knee,suppression is 6 times greater than correspondingsuppression without the hard scattering component.Very strong suppression wash out the 2nd drop,which was clearly visible at SPS, or in the predictionwithout the hard component. With hard component,as mentioned earlier, transverse density is increasedand anomalous suppression sets in earlier. Blaizotet al. [15] also predicted theET dependence of theJ/ψ suppression. In Fig. 2(b), their result is shown(the dashed line). As mentioned earlier, fluctuationsin number of NN collisions at fixed impact parameterwas neglected. Also the model forET production wasnot microscopic. However, at largeET , our predictionagrees closely with theirs.

To summarize, prediction forJ/ψ suppression inAu+Au collisions at RHIC energy was obtained in amodel which assume 100% absorption ofJ/ψ abovea threshold density. Transverse energy fluctuations aswell as fluctuations in the number of NN collisions atfixed impact parameter were taken into account. Thethreshold density was obtained from the analysis ofNA50 data onJ/ψ suppression in Pb+Pb collisionsat SPS energy. At RHIC energy hard processes are im-portant. Prediction ofJ/ψ suppression, with and with-out the hard processes differ considerably. Withouthard processes, predictedJ/ψ suppression at RHICenergy is similar to that obtained at SPS energy. The2nd drop which occur atET ∼ 100 GeV at SPS energymoves upward to 180 GeV. Inclusion of hard processesmodifies the transverse density resulting in consider-able larger suppression. Very large suppression washes

84 A.K. Chaudhuri / Physics Letters B 527 (2002) 80–84

out the 2nd drop, which was visible at SPS energy orin the prediction without the hard processes.

References

[1] T. Matsui, H. Satz, Phys. Lett. B 178 (1986) 416.[2] R. Vogt, Phys. Rep. 310 (1999) 197.[3] C. Gerschel, J. Hufner, Annu. Rev. Nucl. Part. Sci. 49 (1999)

255.[4] NA50 Collaboration, Phys. Lett. B 477 (2000) 28.[5] J.P. Blaizot, P.M. Dinh, J.-Y. Ollitrault, Phys. Rev. Lett. 85

(2000) 4012.[6] A. Capella, E.G. Ferreiro, A.B. Kaidalov, Phys. Rev. Lett. 85

(2000) 2080, hep-ph/0002300.

[7] J. Hufner, B. Kopeliovich, A. Polleri, nucl-th/0012003.[8] A.K. Chaudhuri, Phys. Rev. C 64 (2001) 054903, hep-ph/

0102038.[9] A.K. Chaudhuri, Nucl. Phys. A 515 (1990) 736.

[10] A.K. Chaudhuri, Phys. Rev. C 47 (1993) 2875.[11] M.J. Leitch et al., E866/NuSea Collaboration, Phys. Rev.

Lett. 84 (2000) 3256, nucl-ex/9909007.[12] D. Kharzeev, C. Lourenco, M. Nardi, H. Satz, Z. Phys. C 74

(1997) 307, hep-ph/9612217.[13] B.B. Back et al., PHOBOS Collaboration, Phys. Rev. Lett. 85

(2000) 3100.[14] D. Kharzeev, M. Nardi, Phys. Lett. B 507 (2001) 121, nucl-ph/

0012025.[15] P.M. Dinh, J.P. Blaizot, J.-Y. Ollitrault, nucl-th/0103083.



Gluon shadowing and hadron production at RHIC

Shi-yuan Lia, Xin-Nian Wanga,b

a Department of Physics, Shandong University, 250100 Jinan, PR Chinab Nuclear Science Division, Mailstop 70-319, Lawrence Berkeley Laboratory, Berkeley, CA 94720, USA

Received 6 November 2001; accepted 7 January 2002

Editor: W. Haxton

Abstract

Hadron multiplicity in the central rapidity region of high-energy heavy-ion collisions is investigated within a two-componentmini-jet model which consists of soft and semi-hard particle production. The hard contribution from mini-jets is reevaluatedusing the latest parameterization of parton distributions and nuclear shadowing. The energy dependence of the experimentaldata from RHIC requires a strong nuclear shadowing of the gluon distribution in this model. The centrality dependence of thehadron multiplicity at

√s = 130 GeV is reproduced well with the impact-parameter dependent parton shadowing. However,

energy variation of the centrality dependence is needed to distinguish different particle production mechanisms such as theparton saturation model. 2002 Elsevier Science B.V. All rights reserved.

PACS: 25.75.-q; 12.38.Bx; 12.38.Mh; 24.85.+p

Formation of quark–gluon plasma (QGP) in high-energy heavy-ion collisions hinges crucially on the ini-tial condition that is reached in the earliest stage ofthe violent nuclear interaction. Though many proposedsignals can provide more direct measurements of theinitial parton density, they must compliment results in-ferred indirectly from the measurement of final hadronmultiplicity using either simple scenarios such as theBjorken model [1] or other dynamic models. There-fore, global observables such as the rapidity density ofhadron multiplicity can provide an important link ofa puzzle that can eventually lead one to a more com-plete picture of the dynamics of heavy-ion collisionsand formation of QGP. Furthermore, the study of en-

E-mail address: [email protected] (X.-N. Wang).

ergy and centrality dependence of central rapidity den-sity [2] can also provide important constraints on mod-els of initial entropy production and shed lights on theinitial parton distributions in nuclei. For example, theavailable RHIC experimental data [3–7] can alreadyrule out the simple two-component model without nu-clear modification of the parton distributions in nu-clei [2]. In this Letter, we will study within the two-component model how the RHIC data constrain theunknown nuclear shadowing of the gluon distributionin nuclei and how to further distinguish such a conven-tional parton production mechanism from other novelphysics such as parton saturation [8,9].

Mini-jet production in a two-component model haslong been proposed to explain the energy dependenceof total cross section [10,11] and particle production[12,13] in high-energy hadron collisions. It has also



86 S.-Y. Li, X.-N. Wang / Physics Letters B 527 (2002) 85–91

been proposed [14,15] and incorporated in the HI-JING model [16,17] to describe initial parton produc-tion in high-energy heavy-ion collisions. In this sim-ple two-component model, one assumes that eventsof nucleon–nucleon collisions at high energy can bedivided into those with and without hard or semi-hard processes of jet production. The soft and hardprocesses are separated by a cut-off scalep0. Whilethe cross section of soft interactionσsoft is considerednonperturbative and thus noncalculable, the jet pro-duction cross sectionσjet is assumed to be given byperturbative QCD (pQCD) for transverse momentumtransferpT > p0. The two parameters,σsoft andp0,are determined phenomenologically by fitting the ex-perimental data of totalp+p(p) cross sections withinthe two-component model [10–13,16,17].

The cut-off scalep0, separating nonperturbativeand pQCD components, could in principle depend onboth energy and nuclear size. Using Duke–Owens pa-rameterization [18] of parton distributions in nucleons,it was found in the HIJING [16] model that an energyindependent cut-off scalep0 = 2 GeV/c is sufficientto reproduce the experimental data on total cross sec-tions and the hadron multiplicity inp + p(p) colli-sions, assuming that the soft cross sectionσsoft is alsoconstant. Since then, analysis of recent experimentaldata from deep-inelastic scattering (DIS) of lepton andnucleon at HERA indicated [19] that gluon distribu-tion inside a nucleon is much larger than the DO pa-rameterization at smallx. Many new parameteriza-tions of the parton distributions have become avail-able. Using the Gluck–Reya–Vogt (GRV) parameter-ization [19] of parton distributions and following thesame procedure as in the original HIJING [16], we findthat one can no longer fit the experimentalp + p(p)

data using a constant cut-off scalep0 within the two-component model. One has to assume an energy de-pendent cut-off scalep0(

√s ). Because of the rapid in-

crease of gluon distribution at smallx, we find that thecut-off p0(

√s ) has to increase slightly with energy in

order to fit the experimental data.Shown in Fig. 1 is the calculated central rapidity

density,

(1)dNch

dη= 〈n〉s + 〈n〉h

σjet(s)

σin(s),

for p + p(p) collisions as a function of energy√s,

where 〈n〉s = 1.6 and 〈n〉h = 2.2 represent particle

Fig. 1. Charged particle rapidity densityper participating nucleonpair versus the c.m. energy. The RHIC data [3,4] (filled circle andup-triangle) for the 6% most central Au+ Au are compared topp andpp data (open symbols) [20–22] and the NA49 Pb+ Pb(central 5%) data [23] (filled square). The two-component mini-jetmodel with and without shadowing is also shown. The shaded areafor central Au+ Au collisions corresponds to the range of gluonshadowing parametersg = 0.24–0.28 (Eq. (9)).

production from soft interaction and jet hadronization,respectively. The jet cross section in lowest order ofpQCD is given by

σjet =K

s/4∫

p20

dp2T dy1dy2

(2)

× 1

2

∑a,b,c,d

x1x2fa(x1)fb(x2)dσab→cd

dt,

with the GRV parameterization [19] of parton distrib-utionsfa(x), aK-factor of 2 and an energy-dependentcut-off scale

p0(√s )= 3.91− 3.34 log(log

√s )

+ 0.98 log2(log√s )

(3)+ 0.23 log3(log√s ).

Assuming eikonalization of hard and soft processes,the total inelasticp + p(p) cross section in this two-component model is [16],

(4)σin =∫d2b

[1− e−(σjet+σsoft)TNN (b)

],

S.-Y. Li, X.-N. Wang / Physics Letters B 527 (2002) 85–91 87

whereTNN(b) is the nucleon–nucleon overlap func-tion andσsoft = 57 mb represents the inclusive softcross section. Notice that the energy-dependence ofp0(

√s ) is quite weak, ranging fromp0 = 1.7 GeV/c

at√s = 20 GeV to 3.5 GeV/c at

√s = 5 TeV.

To extrapolate the two-component model to nuclearcollisions, one assumes that multiple mini-jet produc-tion is incoherent and thus is proportional to the num-ber of binary collisionsNbinary. The soft interaction ishowever coherent and proportional to the number ofparticipant nucleonsNpart according to the woundednucleon model [24]. Assuming no final state effects onmultiplicity from jet hadronization, the rapidity den-sity of hadron multiplicity in heavy-ion collisions isthen,

(5)dNch

dη= 1

2〈Npart〉〈n〉s + 〈n〉h〈Nbinary〉

σAAjet (s)

σin,

whereσAAjet (s) is the averaged inclusive jet cross sec-tion perNN in AA collisions. The average number ofparticipant nucleons and number of binary collisionsfor given impact-parameters can be estimated usingHIJING Monte Carlo simulation. If one assumes thatthe jet production cross sectionσAAjet (s) is the same asin p+p collisions, the resultant energy dependence ofthe multiplicity density in central nuclear collisions ismuch stronger than the RHIC data as shown in Fig. 1.Therefore, one has to consider nuclear effects of jetproduction in heavy-ion collisions.

In high-energy nuclear collisions, multiple mini-jetproduction can occur within the same transverse area.If there are more than one pair of mini-jet productionwithin the transverse area given by the jet’s intrinsicsizeπ/p2

0, jet production within this area might notbe independent any more [16]. If such a criteria isused for independent jet production within one unitof rapidity, one can then obtain a cut-off scalep0 ina so-called final state saturation model [25],

(6)p0 ≈ 0.187A0.136(√s )0.205,

that also depends on nuclear size for central heavy-ion collisions. This cut-off scale, though increasingwith nuclear size, ranges from 0.7 GeV/c at

√s =

20 GeV to 2.2 GeV/c at√s = 5 TeV for central

Au + Au collisions, which is much smaller than whatwe have obtained in Eq. (3) by fittingp + p(p) data.Therefore, if we apply the two-component model to

heavy-ion collisions with the same cut-off scale inEq. (3) as determined inp + p(p) collisions, thecriteria for independent jet production will never beviolated. Instead, we will assume the cut-off scale tobe independent of nuclear size in this Letter.

In principle, jets produced in the early stage ofheavy-ion collisions will also suffer final state inter-action and induced gluon bremsstrahlung. For an en-ergetic jet, this will lead to induced parton energyloss [26,27] and the suppression of large transversemomentum hadrons [28]. Such a jet quenching effectcould also lead to increased total hadron multiplicity[28] due to the soft gluons from the bremsstrahlung.However, a recent study [29] of parton energy lossin a thermal environment found that the effective en-ergy loss is significantly reduced for less energetic par-tons due to detailed balance by thermal absorption.Thus, only large energy jets lose significant energy viagluon bremsstrahlung. Since the production rates ofthese large energy jets are very small at the RHIC en-ergy, their contributions to the total hadron multiplic-ity via jet quenching should also be small. Similarlywe also assume that parton thermalization during theearly stage contributes little to the final hadron multi-plicity.

One important nuclear effect we have to considerin our two-component model is the nuclear shadowingof parton distributions or the depletion of effectiveparton distributions in nuclei at smallx. Such a nuclearshadowing effect in jet production can be taken intoaccount by assuming modified parton distributions innuclei,

(7)f Aa(x,Q2) =ARAa

(x,Q2)f Na (

x,Q2).Using the experimental data from DIS off nucleartargets and unmodified DGLAP evolution equations,one can parameterizeRAa (x,Q

2) for different partonsand nuclei [31,32]. Recent new data [30] howeverindicate that the simple parameterization for nuclearshadowing used in HIJING [16] is too strong for heavynuclei. In this Letter, we will use the following newparameterization,

RAq (x)= 1.0+ 1.19 log1/6A(x3 − 1.2x2 + 0.21x

)− sq

(A1/3 − 1

)0.6(1− 3.5

√x )

(8)× exp(−x2/0.01

),


Fig. 2. Ratio of nuclear structure functions as measured in DIS.Solid lines are the new HIJING parameterization (Eq. (8)), dashedlines are the HKM parameterization [32] and dot-dashed lines arethe old HIJING parameterization [16]. The data are from Ref. [30].

with sq = 0.1 for all quark distributions as shown inFig. 2 (solid lines) in comparison with the experimen-tal data [30]. Also shown in Fig. 2 are parameteriza-tions (dashed lines) by Hirai, Kumano and Miyama(HKM) [32] and the old HIJING parameterization[16]. The shadowing in the old HIJING parameteri-zation (dot-dashed lines) is apparently too strong forheavy nuclei. The HKM parameterizations also takeinto account constraints by momentum sum rules, assimilarly in the original parameterizations by Eskola,Kolhinen and Salgado (EKS) [31]. For the purpose ofthis Letter, one can neglect the scale dependence of theshadowing.

The nuclear shadowing for gluons is somewhatconstrained by the momentum sum rules in the HKMparameterization. However, the constraint is not verystrong, leaving a lot of room for large variation ofgluon shadowing. Shown in Fig. 3 are the shad-owing factors for gluon distribution from EKS andHKM parameterizations. They both have strong anti-shadowing aroundx ∼ 0.1. The stronger anti-shadow-ing in EKS parameterization is due to additional con-straints by theQ2 dependence ofF2(Sn)/F2(C), as-suming the same unmodified DGLAP evolution equa-

Fig. 3. Ratios of gluon distributions in different nuclei given by thenew HIJING (Eq. (9)) (solid line, the shaded area corresponds tosg = 0.24–0.28), old HIJING [16] (dot-dashed), HKM [32] (dashed)and EKS [31] (dotted) parameterization.

tion for parton distributions of a nucleon. Since gluon–gluon scattering dominate the jet production crosssection, such a strong gluon anti-shadowing leads tolarger jet cross section and thus larger hadron multi-plicity than in the case of no shadowing at the RHICenergies. Such a scenario within the two-componentmodel is clearly inconsistent with the experimentaldata. We therefore propose the following parameter-ization for gluon shadowing,

RAg (x)= 1.0+ 1.19 log1/6A(x3 − 1.2x2 + 0.21x

)− sg

(A1/3 − 1

)0.6(1− 1.5x0.35)(9)× exp

(−x2/0.004),

with sg = 0.24–0.28. This is shown in Fig. 3 as thesolid lines. The hadron multiplicity density in the two-component model using the above gluon shadowingis shown in Fig. 1. The shaded area correspondsto the variation ofsg = 0.24–0.28. The RHIC datathus indicate that such a strong gluon shadowingis required within the two-component model. If oneassumes the same gluon shadowing as the quarks inEq. (8), the resultantdN/dη is only slightly smallerthan the one without shadowing. Such a constraint ongluon shadowing is indirect and model dependent. It


Fig. 4. The charged hadron central rapidity density per participantnucleon pair as a function of the averaged number of participantsfrom the two-component model (shaded lines), two-parameter fit(Eq. (11)) (dot-dashed lines) and parton saturation model [9] ascompared to experimental data [3,5,20,21].

is important to study directly the gluon shadowing inother processes inAA or pA collisions.

To take into account the impact-parameter depen-dence of the shadowing, we simply replace the shad-owing parameterssa in Eqs. (8) and (9) by

(10)sa(b)= sa5

3

(1− b2/R2

A

),

whereRA = 1.12A1/3 is the nuclear size. With thisimpact-parameter dependence, the calculated jet crosssectionσAAjet (s)/σin will also depend on the centralityof heavy-ion collisions, decreasing from peripheral tocentral collisions. One can then calculate the central-ity dependence of the hadron multiplicity density. Theresults are shown in Fig. 4 as a function ofNpart forAu+Au collisions at

√s = 56, 130 and 200 GeV. The

shaded areas again correspond to the variation of thegluon shadowing parametersg = 0.24–0.28. Withinstatistical and systematic errors, the two-componentmini-jet model with impact-parameter dependent par-ton shadowing describes the PHOBOS and PHENIXdata [3,5] at

√s = 130 GeV well.

To illustrate the effect of the impact-parameterdependence of the parton shadowing, we compare theresults with a two-component parameterization,

(11)dNch

dη= 1

2〈Npart〉ns + 〈Nbinary〉nh,

shown as dot-dashed lines, where the two parame-ters, ns and nh, fixed at each energy by values ofdNAA

ch /dη in p + p collisions and the most centralAu + Au collisions, are assumed to be independent ofthe centrality. The increase of 2dNch/dη/〈Npart〉 with〈Npart〉 is driven only by the centrality dependence of〈Nbinary〉/〈Npart〉 in this two-parameter fit. Comparingto such a two-parameter fit, the two-component mini-jet model has a flatter centrality dependence at high en-ergies because the effective jet cross section decreasesfrom peripheral to central collisions due to the impact-parameter dependence of parton shadowing. The bet-ter agreement between the experimental data and thetwo-component mini-jet model at

√s = 130 GeV is

another indication of strong nuclear shadowing of thegluon distribution in mini-jet production.

Similar centrality dependencies are also predictedby other models [25,33], in particular the initial stateparton saturation model [8,9]. It is based on the nonlin-ear Yang–Mills field dynamics [14,34] assuming thatnonlinear gluon interaction below a saturation scaleQ2s ∼ αs xGA(x,Q

2s )/πR

2A leads to a classical behav-

ior of the gluonic field inside a large nucleus, whereGA(x,Q

2s ) is the gluon distribution atx = 2Qs/

√s.

Assuming particle production in high-energy heavy-ion collisions is dominated by gluon production fromthe classical gluon field, one has a simple form [9] forthe charged hadron rapidity density atη= 0,

2

〈Npart〉dNch

dη

(12)= c

(s

s0

)λ/2[log

(Q2

0s

Λ2QCD

)+ λ

2log

(s

s0

)],

with c ≈ 0.82 [8]. This is shown in Fig. 4 as solidlines. Here,ΛQCD = 0.2 GeV, λ = 0.25 and thecentrality dependence of the saturation scaleQ2

0s at√s0 = 130 GeV is taken from Ref. [8].Comparing the two model results in Fig. 4, one

notices that the saturation and two-component modelagree with each other in most regions of centrality ex-cept very peripheral and very central collisions. In cen-tral collisions, results of saturation model tend to beflatter than the two-component model. In this region,there are still strong fluctuations in parton productionin the two-component model through the fluctuationof Nbinary whileNpart is limited by its maximum valueof 2A. That is whydNch/dη/〈Npart〉 continues to in-


crease with〈Npart〉 in the central region. Such a fluctu-ation is not currently taken into account in the satura-tion model calculation. More accurate measurementswith small errors (less than 5%) will help to distin-guish these two different behaviors. For peripheral col-lisions, saturation model results fall off more rapidlythan the mini-jet results. However, the experimentalerrors are very big in this region because of large un-certainties related to the determination of the num-ber of participants. Therefore, it will be very usefulto have light-ion collisions at the same energy to mapout the nuclear dependence of the hadron multiplicityin this region. An alternative is to study the ratios ofhadron multiplicity of heavy-ion collisions at two dif-ferent energies as a function of centrality. In this case,the errors associated with the determination of central-ity will mostly cancel. Shown in Fig. 5 are the ratiosof hadron multiplicity at three different energies as afunction of the averaged number of participants pre-dicted by saturation and two-component model. Onenotices that while the results from saturation modelhave the same centrality dependence at all three en-ergies the two-component model predicts slightly dif-ferent behavior at different energies, indicating the en-ergy dependence of the mini-jet component. So the ra-tios given by the saturation model are almost indepen-dent of centrality. On the other hand, two-componentmodel predicts noticeable centrality dependence of the

Fig. 5. The ratios of charged hadron multiplicity density in Au+ Aucollisions at different energies as a function of the averaged numberof participants shown with PHOBOS data [3,4].

ratios. This is especially true for the ratio between col-lisions at

√s = 200 and 56 GeV.

It is interesting to point out that in the saturationmodel that assumes a particle production mechanismdominated by coherent mini-jet production below thesaturation scaleQs , the value ofQs determined inRefs. [8,9] is much smaller than the cut-offp0 in thetwo-component model constrained by thep + p(p)

data. As demonstrated in this Letter, the number ofmini-jet production below such scale is still very largeand should contribute to the final hadron multiplicity.

In summary, we have studied the energy andcentrality dependence of the central rapidity densityof hadron multiplicity in heavy-ion collisions at RHICenergies within a two-component mini-jet model.As a consequence of the latest parameterization ofparton distributions [19] which have a higher gluondensity than the old parameterization [18] used inprevious studies [16], the cut-off scale that separatessoft and hard processes is found to increase slightlywith energy in order to fit thep + p(p) data. Thecut-off scale, however, is still large enough that theindependent jet production picture is still valid. With anew parameterization of nuclear shadowing of partondistributions in nuclei, we also found that RHICdata require a strong shadowing of gluon distribution.Using this strong gluon shadowing with an assumedimpact-parameter dependence, the predicted centralitydependence of the hadron multiplicity agrees wellwith the recent RHIC results. We have also comparedour results with the parton saturation model [8,9]. Wepoint out that in order to differentiate the two modelsone needs more accurate experimental data in boththe most central and peripheral regions of centralityor study the centrality dependence of the ratios atdifferent colliding energies.

Acknowledgements

The authors thank M. Baker, K. Eskola, D. Khar-zeev and P. Steinberg for comments and discussions.This work was supported by the Director, Office ofEnergy Research, Office of High Energy and NuclearPhysics, Division of Nuclear Physics, and by theOffice of Basic Energy Science, Division of NuclearScience, of the US Department of Energy under


Contract No. DE-AC03-76SF00098 and in part byNSFC under project 19928511 and 10075031.

References

[1] J.D. Bjorken, Phys. Rev. D 27 (1983) 140.[2] X.-N. Wang, M. Gyulassy, Phys. Rev. Lett. 86 (2001) 3496,

nucl-th/0008014.[3] B.B. Back et al., PHOBOS Collaboration, Phys. Rev. Lett. 85

(2000) 3100, hep-ex/0007036;B.B. Back et al., PHOBOS Collaboration, nucl-ex/0105011.

[4] B.B. Back et al., PHOBOS Collaboration, nucl-ex/0108009.[5] K. Adcox et al., PHENIX Collaboration, Phys. Rev. Lett. 86

(2001) 3500, nucl-ex/0012008.[6] C. Adler et al., STAR Collaboration, Phys. Rev. Lett. 87 (2001)

112303, nucl-ex/0106004.[7] I.G. Bearden et al., BRAHMS Collaborations, nucl-ex/

0108016.[8] D. Kharzeev, M. Nardi, Phys. Lett. B 507 (2001) 121, nucl-

th/0012025.[9] D. Kharzeev, E. Levin, nucl-th/0108006.

[10] T.K. Gaisser, F. Halzen, Phys. Rev. Lett. 54 (1985) 1754.[11] G. Pancheri, Y.N. Srivastava, Phys. Lett. B 182 (1986) 199.[12] W.R. Chen, R.C. Hwa, Phys. Rev. D 39 (1989) 179.[13] X.-N. Wang, Phys. Rev. D 43 (1991) 104.[14] J.P. Blaizot, A.H. Mueller, Nucl. Phys. B 289 (1987) 847.[15] K. Kajantie, P.V. Landshoff, J. Lindfors, Phys. Rev. Lett. 59

(1987) 2527;K.J. Eskola, K. Kajantie, J. Lindfors, Nucl. Phys. B 323 (1989)37.

[16] X.-N. Wang, M. Gyulassy, Phys. Rev. D 44 (1991) 3501;X.-N. Wang, M. Gyulassy, Comput. Phys. Commun. 83 (1994)307, nucl-th/9502021.

[17] X.-N. Wang, Phys. Rep. 280 (1997) 287, hep-ph/9605214.[18] D.W. Duke, J.F. Owens, Phys. Rev. D 30 (1984) 49.[19] M. Gluck, E. Reya, A. Vogt, Z. Phys. C 67 (1995) 433.[20] G.J. Alner et al., UA5 Collaboration, Z. Phys. C 33 (1986) 1.

[21] W. Thome et al., Aachen–CERN–Heidelberg–Munich Collab-oration, Nucl. Phys. B 129 (1977) 365.

[22] J. Whitmore, Phys. Rep. 10 (1974) 273.[23] H. Appelshauser et al., NA49 Collaboration, Phys. Rev.

Lett. 82 (1999) 2471, nucl-ex/9810014;C. Hohne, NA49 Collaboration, Nucl. Phys. A 661 (1999)485c.

[24] A. Bialas, M. Bleszynski, W. Czyz, Nucl. Phys. B 111 (1976)461.

[25] K.J. Eskola, K. Kajantie, P.V. Ruuskanen, K. Tuominen, Nucl.Phys. B 570 (2000) 379, hep-ph/9909456;K.J. Eskola, K. Kajantie, K. Tuominen, hep-ph/0106330.

[26] M. Gyulassy, X.-N. Wang, Nucl. Phys. B 420 (1994) 583, nucl-th/9306003.

[27] R. Baier, Y.L. Dokshitzer, S. Peigne, D. Schiff, Phys. Lett.B 345 (1995) 277, hep-ph/9411409.

[28] X.-N. Wang, M. Gyulassy, Phys. Rev. Lett. 68 (1992) 1480.

[29] E. Wang, X.-N. Wang, Phys. Rev. Lett. 87 (2001) 142301, nucl-th/0106043.

[30] M. Arneodo et al., New Muon Collaboration, Nucl. Phys.B 481 (1996) 3.

[31] K.J. Eskola, V.J. Kolhinen, P.V. Ruuskanen, Nucl. Phys. B 535(1998) 351, hep-ph/9802350;K.J. Eskola, V.J. Kolhinen, C.A. Salgado, Eur. Phys. J. C 9(1999) 61, hep-ph/9807297.

[32] M. Hirai, S. Kumano, M. Miyama, Phys. Rev. D 64 (2001)034003, hep-ph/0103208.

[33] A. Capella, D. Sousa, Phys. Lett. B 511 (2001) 185, nucl-th/0101023;N. Armesto, C. Pajares, D. Sousa, hep-ph/0104269;J. Dias de Deus, R. Ugoccioni, Phys. Lett. B 494 (2000) 53,hep-ph/0009288;H.J. Pirner, F. Yuan, Phys. Lett. B 512 (2001) 297, hep-ph/0101115.

[34] L. McLerran, R. Venugopalan, Phys. Rev. D 49 (1994) 2233,hep-ph/9309289;A. Krasnitz, R. Venugopalan, Phys. Rev. Lett. 84 (2000) 4309,hep-ph/9909203.



Analysis of the first RHIC results in the string fusion model

N. Armestoa, C. Pajaresb, D. Sousac

a Departamento de Física, Módulo C2, Planta baja, Campus de Rabanales, Universidad de Córdoba, E-14071 Córdoba, Spainb Departamento de Física de Partículas, Universidade de Santiago de Compostela, E-15706 Santiago de Compostela, Spain

c Laboratoire de Physique Théorique, Université de Paris XI, Batiment 210, F-91405 Orsay Cedex, France

Received 27 April 2001; received in revised form 22 October 2001; accepted 24 December 2001


Abstract

First results from RHIC on charged multiplicities, evolution of multiplicities with centrality, particle ratios and transversemomentum distributions in central and minimum bias collisions, are analyzed in a string model which includes hard collisions,collectivity in the initial state considered as string fusion, and rescattering of the produced secondaries. Multiplicities and theirevolution with centrality are successfully reproduced. Transverse momentum distributions in the model show a largerpT -tailthan experimental data, disagreement which grows with increasing centrality. Discrepancies with particle ratios appear and areexamined comparing with previous features of the model at SPS. 2002 Published by Elsevier Science B.V.

With the first collisions at the Relativistic HeavyIon Collider (RHIC) at BNL in June 2000, the study ofnuclear collisions has entered the truly ultrarelativis-tic domain. While there exist predictions from manymodels [1], now experiments have presented results[2–13] on several aspects of data, most of them corre-sponding to AuAu collisions at 130 GeV per nucleonin the center of mass. So it comes the time to exam-ine the ability of models for ultrarelativistic heavy ioncollisions, fitted to describe nuclear data at the muchlower energies of the Super Proton Synchrotron (SPS)at CERN and nucleon data in the range of energies go-ing from SPS to TeVatron at FNAL, to describe thenew situation, and whether the evidences of Quark–Gluon Plasma (QGP) already obtained at SPS are ver-

E-mail addresses: [email protected] (N. Armesto),[email protected] (C. Pajares), [email protected](D. Sousa).

ified or not [14]. The aim of this letter is to comparethe results of the String Fusion Model (SFM) [15,16]with some of the first RHIC data. Other comparisonscan be found in [17,18].1 After a very brief model de-scription, charged multiplicities at midpseudorapidityin central collisions, evolution of charged multiplic-ities at midpseudorapidity with centrality, transversemomentum distributions of charged particles at differ-ent centralities and ratios of different particles will becompared with available data coming from the experi-ments. Finally some conclusions will be summarized.

An exhaustive description of the model can befound in [16]. Its main features are the following. El-ementary inelastic collisions (binary nucleon–nucleon

1 In [18] a model which, like ours, contains multipomeronexchange, a hard component and rescattering of secondaries, butno string fusion, is shown to be able to reproduce the experimentaldata [3] on elliptic flow.



N. Armesto et al. / Physics Letters B 527 (2002) 92–98 93

collisions) are considered as collisions between par-tons from nucleons of the projectile and the target,distributed in the transverse plane of the global col-lision. Some of these elementary collisions are takenas hard ones, and proceed as gluon–gluon→ gluon–gluon through PYTHIA [19] with GRV 94 LO partondensity functions (pdfs) [20] and EKS98 modificationof pdfs inside nuclei [21], with subsequent radiationand fragmentation performed by ARIADNE [22] andJETSET [19]. Those collisions not being consideredhard produce soft strings in pairs. These strings are al-lowed to fuse if their parent partons are close enoughin impact parameter [15]; as the number of stringsincreases with increasing energy, atomic number andcentrality, this mechanism accordingly grows in im-portance. Fragmentation of soft strings is performedusing the tunneling mechanism for mass and trans-verse momentum distributions, while longitudinal mo-menta are simulated by an invariant area law. The mainconsequences of string fusion are a reduction of mul-tiplicities in the central rapidity region and an increasein heavy particle production. The produced particlesare allowed to rescatter (between themselves and withspectators nucleons) using a very naive model with noproper space–time evolution, whose consequences area small multiplicity reduction, an increase in strangeand multistrange baryons and nucleon annihilation.Some comments are in order at this point. First, par-tons which generate both soft and hard strings can bevalence quarks and diquarks, and sea quarks and an-tiquarks, so the number of soft strings is not simplyproportional to the number of wounded nucleons buthas some proportionality, increasing with increasingenergy, centrality and nuclear size, on the number ofbinary nucleon–nucleon collisions.2 Besides, only fu-sion of two strings in considered in the actual versionof the model, and hard strings are not fused. Finally,the rescattering model is simplistic and has been in-cluded just to estimate the effects that such kind of

2 Usually the soft contribution is taken as proportional to thenumber of wounded nucleons, while the contribution proportional tothe number of binary nucleon–nucleon collisions is considered hard.Let us stress that this is a misleading (model dependent) statement:some proportionality with the number of binary nucleon–nucleoncollisions is demanded by a basic requirement of the theory asunitarity, and has nothing to do with the soft or hard origin of thesebinary nucleon–nucleon collisions.

Fig. 1. Results of the model for the pseudorapidity distributionof charged particles for central (5%) PbPb collisions at 17.3 GeVper nucleon in the center of mass (dashed-dotted line), and central(6%) AuAu collisions at 56 (dotted line), 130 (dashed line) and200 (solid line) GeV per nucleon in the center of mass, comparedwith experimental data at SPS from NA49 [2,23] (black square)and WA98 [24] (black, upward pointing triangle), and at RHICfrom PHOBOS [2,13] (black, downward pointing triangle for56 GeV, open circle for 130 GeV and black circle for 200 GeV),BRAHMS [6] (open square) and PHENIX [4] (open triangle).

physics could have and to tune the parameters of themodel as an initial condition for a more sophisticatedevolution; thus, results depending strongly on it shouldbe taken with great caution. All these aspects will becommented more extensively when the comparisonwith experimental data is performed.

In Fig. 1 results of the model (unless otherwisestated, results of the model correspond to its defaultversion with the mentioned pdfs and string fusionand rescattering, see [16]) for the pseudorapiditydistribution of charged particles in central collisions atSPS and RHIC are compared with experimental data.For central AuAu collisions at 130 and 200 GeV pernucleon in the center of mass, the model successfullyreproduce the data (the ratio of multiplicities at 200and 130 GeV is 1.08 in the model, slightly smallerthan the experimental value 1.14± 0.05 measured byPHOBOS [13]), while at 56 GeV it overestimates thePHOBOS results [2]. Nevertheless, the situation atthese energies is not clear: WA98 results [24] at SPS

94 N. Armesto et al. / Physics Letters B 527 (2002) 92–98

lie above the PHOBOS data at 56 GeV, and far aboveNA49 data [23] (as extracted in [2]) at SPS; NA49results on multiplicities in central PbPb collisions atSPS are in agreement with those from WA97 [25]. Soit is difficult to conclude anything definitive on theevolution from SPS to RHIC, of multiplicities withincreasing energy in the model.

Recently it has been proposed [26] that the evolu-tion of multiplicities with centrality can be used as atool to discriminate among several models for multi-particle production in high-energy nuclear collisions.In this way, models which consider saturation [27]of either the number of partons in the wave functionof the projectile and target or in the number of par-tons produced in the collision [28], show a constantor slightly decreasing behavior of the multiplicity perparticipant (wounded) nucleon with increasing num-ber of participants.3 On the other hand, models whichconsider some proportionality with the number of bi-nary nucleon–nucleon collisions based on the AGKcancellation [30], being this proportionality alreadypresent in the soft component [16,31–33] or only inthe hard component [34], show a behavior, with themultiplicity per participant increasing with increasingnumber of participants, qualitatively or quantitativelycompatible with data. The results of our model forthe 75% more central collisions at SPS and RHIC areshown in Fig. 2 and compared with experimental data.It can be seen that the model underestimates WA98data at SPS, while it overestimates those from NA49,as could be expected from the discussion about Fig. 1,but the qualitative behavior seems correct. At RHICthe agreement with data is quite satisfactory. It canbe seen that the inclusion of rescattering results in aslight decrease of multiplicities, while the influence ofstring fusion is relatively small at SPS but very im-portant at RHIC and crucial for the agreement withexperimental data. In our model it is this latter mech-anism the one which plays the role of shadowing cor-rections in [31,32,34], parton saturation in [28,29] orstring percolation [36] in [33]. Concerning the limita-tion of fusion of just soft strings in groups of two, letus point out that it seems to be compensated at RHICwith the choice of the fusion strength, while the non-

3 Other proposals which include saturation [29] show an increas-ing behavior compatible with data.

Fig. 2. Pseudorapidity density of charged particles atη = 0 dividedby one half the number of participant nucleons, versus the numberof participant nucleons, in PbPb collisions at 17.3 GeV per nucleonin the center of mass (multiplied by 1/2, lower curves and symbols)and in AuAu collisions at 130 GeV per nucleon in the center ofmass (upper curves and symbols); also the experimental numberfor pp collisions at 130 GeV per nucleon is given [35], filledsquare. Experimental data are from PHENIX [4] (filled triangles),PHOBOS [2] (open triangle), WA98 [24] (filled circles) and NA49[2,23] (open circle). Curves are results of the model for the 75%more central events, without fusion or rescattering (dotted lines),with fusion (dashed lines) and with fusion and rescattering (solidlines).

inclusion of fusion of hard strings is unimportant, asthey amount for just 1% of the total number of ele-mentary inelastic collisions. This is no longer the casefor the future Large Hadron Collider (LHC) at CERN,situation for which we present the results of the modelin Fig. 3 (results with rescattering are not presentedbecause this mechanism is too CPU-time consumingat LHC energies for large nuclei). Here, the fusion ofjust two strings has reached its limit, so multiplicitiesare not so strongly damped as at RHIC, and fusion ofmore than two strings (and of hard strings, which nowamount for 32% of the total number of elementary in-elastic collisions), or even a phase transition like per-colation [36], have to be introduced in the model.

Let us now turn to the transverse momentumspectrum. Preliminary measurements [7,9] show thatthe spectrum in AuAu collisions at 130 GeV per


Fig. 3. The same as in Fig. 2, but for PbPb collisions at 5.5 TeV pernucleon in the center of mass.

nucleon in the center of mass falls with increasingpT faster than predictions from models [34] whichreproduce thepT -distributions in pp collisions at200 GeV in the center of mass; this discrepancy growswith increasing centrality. A possible explanation isjet quenching [37], i.e., the energy loss of highenergy partons in a hot medium containing free colorcharges. So, there has been a great debate on theexplanation of the absence of jet quenching at SPSand its presence at RHIC [38], and its interpretationas a QGP signature. In our model we find quite thesame feature as in [34], see Fig. 4, namely, an excessof particles with highpT compared with experimentaldata, excess which becomes less pronounced whengoing from central to minimum bias collisions. Ourmodel correctly reproduces multiplicities and theirevolution with centrality at this energy (as seen inFigs. 1 and 2), and thepT -spectrum in pp collisionsat SPS and inpp collisions at SppS at CERN andTeVatron, and the increase of〈pT 〉 with energy andmultiplicity (see [16]); we have also checked thatthis is neither an effect of pdfs or of their nuclearmodifications, nor of rescattering, whose influenceon the pT -spectrum is tiny, see [16] and Fig. 4; infact, from the studies in [16] it can be concluded thatthe transverse momentum enhancement in collisionsbetween nuclei compared to those between nucleons

is due in the model both to the hard contribution whichbecomes more important with an increasing number ofelementary collisions, and, above all, to the transversemomentum broadening of the partons at the ends ofthe strings introduced in the model and responsibleof the increase of〈pT 〉 with increasing multiplicity,while string fusion has a very small effect. It is alsoremarkable that the discrepancy with the experimentaldata appears in a model like ours, which for thecollisions studied at RHIC produces only 1% of hardelementary collisions, and in a model like that of [34],in which most of particle production at RHIC energiescomes from the hard contribution.4 So, it really lookslike an effect which diminishes the number of highpT partons, leading them to the lowpT region. Jetquenching [37,38] seems a good candidate to explainthis experimental finding, but it should be taken intoaccount that it also leads to the appearance of moreparticles at lowpT and η; thus, the simultaneouscomparison of the evolution of both multiplicitiesand transverse momentum distributions with centralityshould be a crucial test for this mechanism.5 Onewould think that the presence of saturation of lowtransverse momentum partons [27,28] would make thecomparison with experimental data even worse: thelow pT region of the spectrum, populated of poorlyresolved partons, would be damped due to partonfusion and the spectrum become flatter than withoutsaturation. Quite the same would occur in percolationof strings [36]: soft strings have a larger transversedimension than hard partons and would fuse moreeasily, and fused strings with higher string tension

4 Possible differences in thepT -spectrum in nucleon–nucleoncollisions between our model and those based on hard scatteringslike HIJING [34] should become visible at LHC, where the resultsare not so tightly constrained by the existing experimental dataat SPS, SppS and TeVatron: in our model the contribution fromhard scatterings will be smaller and thus we expect less high-pT

particles.5 In [39] the evolution of〈p〉/〈π−〉 versuspT with centrality is

proposed as a test of jet quenching; the increase of this ratio withincreasingpT observed by PHENIX [7] is reproduced with a softexponential component proportional to the number of participantsplus a quenched perturbative distribution proportional to the numberof binary collisions. In our model, the corresponding increase due tothe soft part would be stronger than in [39] due to string fusion andto the fact that this component is, in our case, proportional to thenumber of both wounded nucleons and binary collisions.


Fig. 4. Transverse momentum spectrum (1/(2πpT )dN

/(dηdpT )|η=0 versus pT ) of charged particles atη = 0 inAuAu collisions at 130 GeV per nucleon in the center of mass, forcentral collisions (5%, solid and dashed lines and filled circles) andfor minimum bias collisions (92%, multiplied by 0.01, dotted anddashed-dotted lines and open circles). Data are from PHENIX [7];solid and dotted lines are results of the model with string fusion,dashed and dashed-dotted lines with string fusion and rescattering.

would produce particles with higherpT than ordinarystrings, so the meanpT would increase with atomicsize or centrality [40], contrary to what data apparentlyshow.6

Finally, in Table 1 model results for different parti-cle ratios are shown and compared with published ex-perimental data [6,7,9–12,42]. For completeness, letus indicate the results in the model for the ratios/, +/ −, K+/K−, p/π− and K−/π− at η ∼ 0, forwhich we get 0.85|0.87|0.87, 0.60|0.92|0.88,1.08|1.03|1.04, 0.02|0.07|0.04 and 0.08|0.12|0.16, re-spectively, without string fusion or rescattering|withstring fusion|with string fusion and rescattering.7 Theresults in the model have been obtained in the cor-

6 A recent analysis [41] shows that nevertheless it is possibleto simultaneously explain the evolution with centrality of bothmultiplicity distributions and transverse momentum spectra in avery crude realization of the percolating string approach.

7 These results can be compared with preliminary, not yetpublished results: 0.73±0.03, 0.82±0.08, 1.12±0.01±0.06, 0.08and 0.15, respectively, presented by STAR at QM2001 [9].

responding pseudorapidity regions, for AuAu colli-sions at 130 GeV per nucleon in the center of masswith a centrality of 10% and for particles withpT >

0.2 GeV/c. Each experiment applies different central-ity and kinematical cuts for the different ratios, but acommon conclusion of all of them is that ratios arevery weakly dependent on centrality of the collisionandpT of the particles, so this should not seriously af-fect the comparison. From these results it can be seenthat the model overestimates antibaryon production,a feature already present at SPS, see [16], but stringfusion is needed to increase the strangeness and an-tibaryon yield, which is badly underestimated, see thecomparison with SPS data in [16], if this mechanismis not included (in the ratios at central rapidities anddue to the lack of stopping at RHIC energies, see be-low, and to the fact that string fusion creates on av-erage the same amount of baryons and antibaryons,this feature is mainly visible in those involving multi-strange baryons or inp/π−). This discrepancy is lesspronounced for ’s than for’s, and for’s than fornucleons, and is more pronounced in the central regionof (pseudo)rapidity. As stated in the brief model de-scription, our rescattering model is simplistic, and can-not be expected to produce correct quantitative results,only the trend which it shows should be considered. Soall that we can conclude is that for the ratios at RHIC,similar problems appear than those already present atSPS.8 As a last comment, a preliminary, non-correctedfor hyperon decay, measurement of the p–p yield atmidpseudorapidity by BRAHMS [6], gives 8–10 for acentrality of 6% (a value 4–6 has been extracted [42]from preliminary STAR data for the same centrality),while in our model we get a lower value∼ 2; this maysuggest that the problem in thep/p ratio lies not onlyin a p excess, but also in some lack of stopping in themodel.

In conclusion, we have compared the results of theSFM with some of the first RHIC data. At RHIC,charged multiplicities in the central region for centralcollisions and their evolution with centrality are suc-cessfully reproduced, suggesting the presence of somemechanism, like string fusion, which moderates the in-

8 Apparently, the antibaryon-to-baryon ratios measured at RHICfavor [9] a coalescence model [1,43], see [16] for a comparison ofour results at full RHIC energy and those coming from other models.


Table 1Different particle ratios in central (10%) AuAu collisions at 130 GeV per nucleon in the center of mass in the model without string fusionor rescattering (NF), with string fusion (F) and with string fusion and rescattering (FR) for particles withpT > 0.2 GeV/c, compared withexperimental data [6,7,9–12,42]. For the centrality criteria and kinematical cuts in the different experiments and ratios, see the experimentalreferences and comments in the text

Ratio NF F FR BRAHMS PHENIX PHOBOS STAR

p/p 0.81 0.85 0.80 0.64± 0.04 0.64± 0.01 0.60± 0.04 0.65± 0.01(η ∼ 0) ±0.06 (y ∼ 0) ±0.07 ±0.06 ±0.07

p/p 0.38 0.50 0.38 0.41± 0.04(y ∼ 2) ±0.06

K−/K+ 0.92 0.97 0.96 0.91± 0.07(η ∼ 0) ±0.06

π−/π+ 1.02 1.02 1.01 1.00± 0.01(η ∼ 0) ±0.02

crease of multiplicities with increasing centrality; Onthe other hand and in view of the SPS data, it is dif-ficult to obtain clear conclusions from the behavior ofmultiplicities in the transition from SPS to RHIC. Re-sults on particle ratios show, when compared to ex-perimental data, similar problems of antibaryon ex-cess previously found at SPS, and are probably relatedto the oversimplification of the model of rescatteringand to problems with data at SPS, see [16]. Finally, inthe SFM thepT -spectrum at RHIC is flatter than indata and this problem gets worse with increasing cen-trality, a feature which also appears in other models[34,38] in which the contribution of hard elementarycollisions is much larger than in ours. At first sight, itlooks improbable that parton saturation or percolationof strings could improve the comparison with thepT -distributions (but see [41]). So, from our point of viewthese data are most striking and, if confirmed, maybea good candidate for a signature of non-conventionalphysics appearing in heavy ion collisions at RHIC.Although the results of the model on features whichshould depend strongly on the evolution of the sys-tem (particle ratios andpT -spectrum if jet quench-ing is present) cannot be considered satisfactory, theagreement with multiplicities and their evolution withcentrality, which are usually assumed not to vary toomuch during evolution [28,29], gives us some confi-dence in the ability of the model to describe the initialcondition, to be used for further evolution, in a colli-sion between heavy ions at high energies.

Acknowledgements

We express our gratitude to A. Capella, K.J. Eskolaand R. Ugoccioni for useful discussions, and to F.Messer for providing us the preliminary PHENIX datain Fig. 4. N.A. and C.P. acknowledge financial supportby CICYT of Spain under contract AEN99-0589-C02.N.A. and D.S. also thank Universidad de Córdoba andFundación Barrié de la Maza of Spain, respectively,for financial support. N.A. thanks Departamento deFísica de Partículas of the Universidade de Santiagode Compostela for stays during which part of this workwas completed. Laboratoire de Physique Théorique isUnité Mixte de Recherche—CNRS—UMR no. 8627.

References

[1] S.A. Bass et al., Nucl. Phys. A 661 (1999) 205c;N. Armesto, C. Pajares, Int. J. Mod. Phys. A 15 (2000) 2019.

[2] B.B. Back et al., PHOBOS Collaboration, Phys. Rev. Lett. 85(2000) 3100;B.B. Back et al., PHOBOS Collaboration, Phys. Rev. Lett. 87(2001) 102303.

[3] K.H. Ackermann et al., STAR Collaboration, Phys. Rev.Lett. 86 (2001) 402.

[4] K. Adcox et al., PHENIX Collaboration, Phys. Rev. Lett. 86(2001) 3500.

[5] K. Adcox et al., PHENIX Collaboration, Phys. Rev. Lett. 87(2001) 052301.

[6] F. Videbaek et al., BRAHMS Collaboration, in: Proc. of ‘QuarkMatter 2001’ Brookhaven, USA, January 14–20, 2001.

[7] W.A. Zajc et al., PHENIX Collaboration, in: Proc. of ‘QuarkMatter 2001’ Brookhaven, USA, January 14–20, 2001, nucl-ex/0106001;


F. Messer et al., in: Proc. of ‘Quark Matter 2001’ Brookhaven,USA, January 14–20, 2001.

[8] G. Roland et al., PHOBOS Collaboration, in: Proc. of ‘QuarkMatter 2001’ Brookhaven, USA, January 14–20, 2001.

[9] J.W. Harris et al., STAR Collaboration, in: Proc. of ‘QuarkMatter 2001’ Brookhaven, USA, January 14–20, 2001.

[10] B.B. Back et al., PHOBOS Collaboration, Phys. Rev. Lett. 87(2001) 102301.

[11] C. Adler et al., STAR Collaboration, Phys. Rev. Lett. 86 (2001)4778;C. Adler et al., STAR Collaboration, Phys. Rev. Lett. 87 (2001)112303.

[12] I.G. Bearden et al., BRAMS Collaboration, Phys. Rev. Lett. 87(2001) 112305.

[13] B.B. Back et al., PHOBOS Collaboration, nucl-ex/0108009.[14] CERN Press Release, February 10th 2000;

U. Heinz, M. Jacob, nucl-th/0002042;M. Gyulassy, Prog. Theor. Phys. Suppl. 140 (2000) 68;D. Zschiesche et al., nucl-th/0101047.

[15] N.S. Amelin, M.A. Braun, C. Pajares, Phys. Lett. B 306 (1993)312;N.S. Amelin, M.A. Braun, C. Pajares, Z. Phys. C 63 (1994)507.

[16] N.S. Amelin, N. Armesto, C. Pajares, D. Sousa, PreprintUCOFIS 1/01, US-FT/2-01 and LPT Orsay 01-15, hep-ph/0103060, Eur. Phys. J. C, in press.

[17] K.J. Eskola, in: Proc. of ‘Quark Matter 2001’, Brookhaven,USA, January 14–20, 2001, Preprint JYFL-5/01, hep-ph/0104058, and references therein;Z. Lin et al., Phys. Rev. C 64 (2001) 011902;D.E. Kahana, S.H. Kahana, nucl-th/0010043.

[18] E.E. Zabrodin, C. Fuchs, L.V. Bravina, A. Faessler, Phys. Lett.B 508 (2001) 184.

[19] T. Sjöstrand, Comput. Phys. Commun. 82 (1994) 74.[20] M. Glück, E. Reya, A. Vogt, Z. Phys. C 67 (1995) 433.[21] K.J. Eskola, V.J. Kolhinen, P.V. Ruuskanen, Nucl. Phys. B 535

(1998) 351;K.J. Eskola, V.J. Kolhinen, C.A. Salgado, Eur. Phys. J. C 9(1999) 61.

[22] L. Lönnblad, Comput. Phys. Commun. 71 (1992) 15.[23] H. Appelshäuser et al., NA49 Collaboration, Phys. Rev.

Lett. 82 (1999) 2471;P.G. Jones et al., Nucl. Phys. A 610 (1996) 188c.

[24] M.M. Aggarwal et al., WA98 Collaboration, Eur. Phys. J. C 18(2001) 651.

[25] F. Antinori et al., WA97 Collaboration, Nucl. Phys. A 661(1999) 130c.

[26] X.-N. Wang, M. Gyulassy, Phys. Rev. Lett. 86 (2001) 3496.[27] J.-P. Blaizot, A.H. Mueller, Nucl. Phys. B 289 (1987) 847;

L.V. Gribov, E.M. Levin, M.G. Ryskin, Phys. Rep. 100 (1983)1;

A.H. Mueller, J.-W. Qiu, Nucl. Phys. B 268 (1986) 427;J. Jalilian-Marian, A. Kovner, L. McLerran, H. Weigert, Phys.Rev. D 55 (1997) 5414;A.H. Mueller, Nucl. Phys. B 558 (1999) 285.

[28] K.J. Eskola, K. Kajantie, P.V. Ruuskanen, K. Tuominen, Nucl.Phys. B 570 (2000) 379;

K.J. Eskola, K. Kajantie, K. Tuominen, Phys. Lett. B 497(2001) 39;K.J. Eskola, P.V. Ruuskanen, S.S. Räsänen, K. Tuominen,Preprint JYFL-3/01, hep-ph/0104010.

[29] D. Kharzeev, M. Nardi, Phys. Lett. B 507 (2001) 121;H.-J. Pirner, F. Yuan, Phys. Lett. B 512 (2001) 297.

[30] V.A. Abramovsky, V.N. Gribov, O.V. Kancheli, Sov. J. Nucl.Phys. 18 (1974) 308.

[31] A. Capella, D. Sousa, Phys. Lett. B 511 (2001) 185;A. Capella, A.B. Kaidalov, J. Tran Thanh Van, Heavy IonPhys. 9 (1999) 169.

[32] S. Bondarenko, E. Gotsman, E.M. Levin, U. Maor, Nucl.Phys. A 683 (2001) 649, Preprint TAUP-2663-2001, hep-ph/0101060.

[33] J. Dias de Deus, R. Ugoccioni, Phys. Lett. B 491 (2000) 253;J. Dias de Deus, R. Ugoccioni, Phys. Lett. B 494 (2000) 53.

[34] X.-N. Wang, M. Gyulassy, Comput. Phys. Commun. 83 (1994)307.

[35] G.J. Alner et al., UA5 Collaboration, Z. Phys. C 33 (1986) 1.[36] N. Armesto, M.A. Braun, E.G. Ferreiro, C. Pajares, Phys. Rev.

Lett. 77 (1996) 3736;M. Nardi, H. Satz, Phys. Lett. B 442 (1998) 14;H. Satz, Nucl. Phys. A 642 (1998) 130;J. Dias de Deus, R. Ugoccioni, A. Rodrigues, Eur. Phys. J. C 16(2000) 537.

[37] R. Baier, D. Schiff, B.G. Zakharov, Annu. Rev. Nucl. Part.Sci. 50 (2000) 37, and references therein;M. Gyulassy, P. Lévai, I. Vitev, Nucl. Phys. B 571 (2000) 197;M. Gyulassy, P. Lévai, I. Vitev, Phys. Rev. Lett. 85 (2000)5535.

[38] X.-N. Wang, Phys. Rev. Lett. 81 (1998) 2655, in: Proc. of‘Quark Matter 2001’, Brookhaven, USA, January 14–20, 2001,nucl-th/0105053;P. Lévai et al., in: Proc. of ‘Quark Matter 2001’, Brookhaven,USA, January 14–20, 2001, nucl-th/0104035;A. Drees, in: Proc. of ‘Quark Matter 2001’, Brookhaven, USA,January 14–20, 2001, nucl-ex/0105019.

[39] I. Vitev, M. Gyulassy, nucl-th/0104066.[40] M.A. Braun, C. Pajares, Phys. Rev. Lett. 85 (2000) 4864.[41] M.A. Braun, F. del Moral, C. Pajares, hep-ph/0105263.[42] N. Xu, M. Kaneta, in: Proc. of ‘Quark Matter 2001’,

Brookhaven, USA, January 14–20, 2001, nucl-ex/0104021.[43] J. Zimányi, T.S. Biró, T. Csörgö, P. Lévai, Phys. Lett. B 472

(2000) 243.



Low lying S = −1 excited baryons and chiral symmetry

E. Oseta, A. Ramosb, C. Bennholdc

a Departamento de Física Teórica and IFIC, Centro Mixto Universidad de Valencia-CSIC, Institutos de Investigación de Paterna,Aptd. 22085, 46071 Valencia, Spain

b Departament d’Estructura i Constituents de la Matèria, Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spainc Center for Nuclear Studies and Department of Physics, The George Washington University, Washington, DC 20052, USA

Received 23 August 2001; received in revised form 4 December 2001; accepted 27 December 2001


Abstract

The s-wave meson–baryon interaction in theS = −1 sector is studied by means of coupled-channels, using the lowest-orderchiral Lagrangian and the N/D method to implement unitarity. The loops are regularized using dimensional renormalization.In addition to the previously studiedΛ(1405), employing this chiral approach leads to the dynamical generation of two mores-wave hyperon resonances, theΛ(1670) andΣ(1620) states. We make comparisons with experimental data and look forpoles in the complex plane obtaining the couplings of the resonances to the different final states. This allows us to identify theΛ(1405) and theΛ(1670) resonances withKN andKΞ quasibound states, respectively. 2002 Elsevier Science B.V. Allrights reserved.

1. Introduction

The low-energyK−N scattering and transition tocoupled channels is one of the cases of successful ap-plication of chiral dynamics in the baryon sector. Thestudies of [1] and [2] showed that one could obtainan excellent description of the low-energy data start-ing from chiral Lagrangians and using the multichan-nel Lippman–Schwinger equation to account for mul-tiple scattering and unitarity in coupled channels. Byincluding all open channels above threshold and fit-ting a few chiral parameters of the second-order La-grangian one could obtain a good agreement with thedata at low energies. This line of work was contin-ued in [3], where all coupled channels were included

E-mail address: [email protected] (E. Oset).

that could be arranged from the octet of pseudoscalarGoldstone bosons and the baryon ground state octet.In Ref. [3] it was demonstrated that using the Bethe–Salpeter equation (BSE) with coupled channels andusing the lowest-order chiral Lagrangians, togetherwith one cut off to regularize the intermediate meson–baryon loops, a good description of all low-energydata was obtained. One of the novel features with re-spect to other approaches using the BSE is that thelowest-order meson–baryon amplitudes, playing therole of a potential, could be factorized on shell inthe BSE, and thus the set of coupled-channels inte-gral equations became a simple set of algebraic equa-tions, thus technically simplifying the problem. Thejustification of this procedure is seen in a more gen-eral way in the treatment of meson–meson interactionsusing chiral Lagrangians and the N/D method in [4].One uses dispersion relations and shows that neglect-



100 E. Oset et al. / Physics Letters B 527 (2002) 99–105

ing the effects of the left-hand singularity (also shownto be small there) one needs only the on-shell scatter-ing matrix from the lowest-order Lagrangian, and theeventual effects of higher-order Lagrangians are ac-counted for in terms of subtractions in the dispersionintegrals. The N/D method has also been recently ap-plied to study pion–nucleon dynamics in Ref. [5].

The work of Ref. [3] was reanalyzed recently [6]from the point of view of the N/D method and disper-sion relations, leading formally to the same algebraicequations found in [3]. There are also technical nov-elties in the regularization of the loop function, whichis done using dimensional regularization in Ref. [6],while it was regularized with a cut off in Ref. [3].

One of the common findings shared by all thetheoretical approaches is the dynamical generation oftheΛ(1405) resonance which appears with the rightwidth, and at the correct position, with the choiceof a cut off of natural size. This natural generationfrom the interaction of the meson–baryon system withthe lowest-order Lagrangian allows us to identify thatstate as a quasibound meson–baryon state. This wouldexplain why ordinary quark models have had so manyproblems explaining this resonance [7].

In ordinary quark models theΛ(1405) resonancewould mostly be a SU(3) singlet ofJP = 1/2− andthere would be an associated octet of s-wave excitedJP = 1/2− baryons that would include theN∗(1535),theΛ(1670), theΣ(1620) and aΞ∗ state. In the chiralapproach one would also expect the appearance ofsuch a nonet of resonances. In fact, it appears naturallyin the approach of Ref. [3], with a degenerate octet,when setting all the masses of the octet of stablebaryons equal on one side and the masses of the octetof pseudoscalar mesons equal on the other side. Yet,to obtain this result it is essential that the coupledchannels do not omit any of the channels that can beconstructed from the octet of pseudoscalar mesons andthe octet of stable baryons.

The lowest-order Lagrangian involving the octet ofpseudoscalar mesons and the 1/2+ baryons is givenin [8–11]. At lowest order in momentum, that we willkeep in our study, the interaction Lagrangian reads

(1)

L(B)1 =

⟨Biγ µ 1

4f 2

[(Φ∂µΦ − ∂µΦΦ)B

−B(Φ∂µΦ − ∂µΦΦ)]⟩,

where Φ and B are the SU(3) matrices for themesons and baryons, respectively, and the symbol〈〉stands for the trace of the resulting SU(3) matrix. TheLagrangian of Eq. (1) leads to a common structureof the typeuγ u(kµ + k′

µ)u for the different channels,whereu, u are the Dirac spinors andk, k′ the momentaof the incoming and outgoing mesons.

We take theK−p state and all those that couple toit within the chiral scheme, namelyK 0n, π0Λ,π0Σ0,π+Σ−, π−Σ+, ηΛ, ηΣ0,K0Ξ0 andK+Ξ−. Hencewe have a problem with ten coupled channels.

The lowest-order amplitudes for these channels areeasily evaluated from Eq. (1) and are given by

(2)Vij = −Cij 1

4f 2 u(pi)γµu(pj )(kjµ + kiµ),

wherepj ,pi(kj , ki) are the initial, final momenta ofthe baryons (mesons). For low energies one can writethis amplitude as

Vij = −Cij 1

4f 2 (2√s −MBi −MBj)

(3)×(MBi +E

2MBi

)1/2(MBj +E′

2MBj

)1/2

,

and the matrixCij , which is symmetric, is given in [3].Note that the use of physical masses in Eq. (3) is in-

troducing effectively some contributions of higher or-ders in the chiral counting. In the standard chiral ap-proach one would be using the average mass of theoctets in the chiral limit and higher-order Lagrangiansinvolving SU(3) breaking terms would generate themass differences. By introducing the physical massesone guarantees that the phase space for the reactions,thresholds and unitarity in coupled channels are re-spected from the beginning. We also use in our ap-proach an average value for the pseudoscalar mesondecay constant,f = 1.15fπ , as done in [3].

We shall construct the amplitudes using the isospinformalism for which we must use average massesfor the K (K0,K+), K (K−, K 0), N (p,n), π(π+,π0,π−), Σ (Σ+,Σ0,Σ−) and Ξ (Ξ−,Ξ0)

states. The isospin states are given in [3].We have fourI = 0 channels,KN,πΣ , ηΛ and

KΞ , while there are fiveI = 1 channels,KN,πΣ,πΛ,ηΣ andKΞ . The transition matrix elements inisospin formalism read like Eq. (3) substituting theCijcoefficients byDij for I = 0 and byFij for I = 1, withtheDij andFij coefficients given in [3].

E. Oset et al. / Physics Letters B 527 (2002) 99–105 101

In [6], using the N/D method of [4] for this par-ticular case it was proved that the scattering ampli-tude could be written by means of the algebraic matrixequation

(4)T = [1− VG]−1V

with V the matrix of Eq. (3) evaluated on shell, orequivalently

(5)T = V + VGTwith G a diagonal matrix given by

Gl = i∫

d4q

(2π)4Ml

El(q )1

k0 + p0 − q0 −El(q )+ iε× 1

q2 −m2l + iε

=qmax∫

d3q

(2π)31

2ωl(q )Ml

El(q )(6)× 1

p0 + k0 −ωl(q )−El(q )+ iεwhich depends onp0 + k0 = √

s andqmax.One can see that Eq. (5) is just the Bethe–Salpeter

equation but with theV matrix factorized on shell,which allows one to extract the scattering matrixTtrivially, as seen in Eq. (4).

The analytical expression forGl can be obtainedfrom [12] using a cut off and from [6] using dimen-sional regularization,

Gl = i2Ml

∫d4q

(2π)41

(P − q)2 −M2l + iε

× 1

q2 −m2l + iε

= 2Ml

16π2

al(µ)+ ln

M2l

µ2+ m2

l −M2l + s

2slnm2l

M2l

(7)

+ ql√s

[ln

(s − (

M2l −m2

l

) + 2ql√s)

+ ln(s + (

M2l −m2

l

) + 2ql√s)

− ln(−s + (

M2l −m2

l

) + 2ql√s)

− ln(−s − (

M2l −m2

l

)

+ 2ql√s)]

,

which has been rewritten in a convenient way to showhow the imaginary part ofGl is generated and howone can go to the unphysical Riemann sheets in orderto identify the poles. The dimensional regularizationscheme is preferable if one goes to higher energieswhere the on-shell momentum of the intermediatestates is not reasonably smaller than the cut off.

The coupled set of Eq. (4) were solved in [3] usinga cut off momentum of 630 MeV in all channels.Changes in the cut off can be accommodated interms of changes inµ, the regularization scale in thedimensional regularization formula forGl , or in thesubtraction constantal . In order to obtain the sameresults as in [3] at low energies, we setµ equal tothe cut off momentum of 630 MeV (in all channels)and then find the values of the subtraction constantsal such as to haveGl with the same value with thedimensional regularization formula (Eq. (7)) and thecut off formula (Eq. (6)) at theKN threshold. Thisdetermines the values

aKN = −1.84, aπΣ = −2.00,

aπΛ = −1.83, aηΛ = −2.25,

(8)aηΣ = −2.38, aKΞ = −2.52.

In this way we guarantee that we obtain the sameresults at low energies as in [3] and we find indeed thatthis is the case when we repeat the calculation with thenewGl of Eq. (7). Then we extend the results at higherenergies, looking for the eventual appearance of newresonances.

The solid lines of Figs. 1 and 2 show the results forthe real and imaginary parts of theI = 0 amplitudesfor KN → KN and KN → πΣ , respectively. Bothchannels clearly display the signal from theΛ(1670)resonance, although large background contributionsare present in the amplitudes as well.

The normalization of the amplitudes shown inFigs. 1 and 2 is different from the one of Eq. (4).We shall call TM the plotted amplitudes and therelationship to our former amplitudes is given by

(9)TM,ij = −Tij√MiMjpipj

4π√s

.

The normalization ofTM is particularly suited toanalyze the data in terms of the speed plot [14]. One


Fig. 1. KN → KN amplitude in theI = 0 channel. Data are takenfrom Ref. [13].

has the amplitude written as

(10)TM,ij (W)= T BGM,ij (W)−

xΓ/2eiφ

W −MR + iΓ /2,

with x = √ΓiΓj/Γ , whereΓ ,Γi ,Γj are, respectively,

the total width and the partial decay widths of theresonance into thei, j channels.

The speed is defined as

(11)Spij (W)=∣∣∣∣dTM,ijdW

∣∣∣∣ xΓ/2

(W −MR)2 +Γ 2/4,

where the second equality assumes that the back-ground is smoothly dependent on the energy and doesnot contribute significantly to the derivative.

In Fig. 3 we show the obtained speedSpij (W)for different transitions,KN → KN (solid line),KN → ηΛ (dotted line) andKN → πΣ (dashedline). As is evident from the plots in Fig. 3, thebackground induced by the already opened meson–baryon channels in the region of interest is quitesmooth, since an approximate Breit–Wigner shape is

Fig. 2. KN → πΣ amplitude in theI = 0 channel. Data are takenfrom Ref. [13].

obtained from the derivative of the amplitudes. On theother hand, the resonance region we study does indeedlie above the two-pion threshold for bothΛ andΣproduction. Such threshold openings could show up insome form as a non-smooth background contribution.However, there is no empirical evidence that any of thes-channel hyperon resonances under investigation herecouple strongly to the two-pion channel, hence, wedo not expect the Breit–Wigner shape of Fig. 3 to bemodified by the inclusion of extra inelastic channels.

The study of the speed plots shown in Fig. 3 allowsus to obtain the energyMR = 1708 MeV, the totalwidth Γ = 40 MeV, and the branching ratios,BKN =48%, BηΛ = 45%, andBπΣ = 7%. Experimentally,one hasMR = 1660–1680 MeV,Γ = 25–50 MeV,BKN = 15–25%,BηΛ = 15–35%, andBπΣ = 15–35% [15]. We seem to overestimate theKN andηΛbranching ratios and underestimate theπΣ one.

Comparison of our results with the experimentaldata in Figs. 1 and 2 shows qualitative but notquantitative agreement. This is not surprising since no


Fig. 3. Speed plot for the amplitudesKN → KN (solid line),KN → ηΛ (dotted line) andKN → πΣ (dashed line).

parameters have been fitted to these data, but ratherwe have chosen the low-energy parametrization ofour theory which contained only one free parameter,the cut off to regularize the loops. We now exploitthe freedom that the theory has by changing theparametersal . However, this must be accomplishedin a way that does not ruin the very good agreementwith the low-energy data, found in Ref. [3]. We findthat the results at low energies are very insensitive tochanges in theaKΞ parameter, but they are sensitive tochanges inaKN , aπΣ andaηΛ. On the other hand, theposition of theΛ(1670) resonance is quite sensitiveto changes in theaKΞ parameter and only moderatelysensitive toaKN , aπΣ andaηΛ. This allows us to finetune the parameteraKΞ (without changing the otheralparameters) in order to better reproduce the position ofthe resonance found by experiment while maintainingthe agreement found at low energies. Figs. 1 and 2 alsodisplay the results usingaKΞ = −2.67 (dashed lines).We see that a change of 6% in this parameter movesthe position of the resonance by 28 MeV and it agreesbetter with experiment. The values of the resonancemass, width and branching ratios obtained now areMR = 1679 MeV, the total widthΓ = 40 MeV, andthe branching ratiosBKN = 61%, BηΛ = 30%, andBπΣ = 9%.

Comparison of the theoretical results forKN →KN with the data shows agreement for the imaginarypart within errors, while our prediction for the realpart below the resonance differs from the data by

what appears to be a large constant background term.This discrepancy needs to be looked at with someperspective. The contribution of the real part to thecross section from the experimental data is negligibleand is only 10 percent in the theoretical case. On theother hand, our results around

√s = 1440 MeV, the

KN threshold, are in good agreement with the datafor theK−p andK−n scattering lengths, which wouldsuggest some discrepancy at low energies between thedata shown in Fig. 1 [13] and those of [16,17].

In Fig. 2 we display theI = 0 KN → πΣ ampli-tude. The theoretical amplitude shows the resonancefeatures with the same pattern as the experiment, bothfor the real as for the imaginary parts. Yet there is dis-agreement with the data in the imaginary part, againwith an apparent background missing for the theoreti-cal prediction. Once again, the discrepancy looks puz-zling since up to

√s = 1460 MeV, and even beyond

where the s-wave is still dominant, the agreement ofthe present model with the experimental cross sectionsfor K−p→ π−Σ+,π+Σ−,π0Σ0 is very good [3].

We should note that the errors plotted in Figs. 1and 2 correspond to the reasonable guesses of Ref. [18],but there are actual deviations between the data of [13,19,20]. Due to the large background in the experimen-tal analysis, the interference effects with the resonanceare more apparent, leading to a larger branching ratioto theπΣ channel than the theory predicts.

We can also compare the results of the model withthe recent data on theK−p → ηΛ reaction [21],which improve on the older experiments [22]. Theshape of the results and the position of the peak thatwe obtain agree well with the data for a parameteraKΞ = −2.67 but we get a strength at the peakof σ = 2.7 mb, about a factor of two larger thanthe latest experimental value ofσ = 1.4 mb. Thisreflects the fact that our predictedKN branching ratiooverestimates the experimental value.

We have studied the reactionsK−p → K+Ξ−and K−p → K0Ξ0, which take place at energiesbeyond

√s = 1.815 GeV, hence above the position

of theΛ(1670) resonance. Around a laboratoryK−momentum of 1.6 GeV/c, and usingaKΞ = −2.67,our model predicts a cross section of 0.17 mb for thereactionK−p→K+Ξ−, which compares favourablywith the experimental value of 0.16–0.18 mb [23,24].For the reactionK−p → K0Ξ0 we find a crosssection of 0.24 mb at 1.6 GeV/c, which overestimates


by almost a factor of three the experimental value of0.08–0.1 mb [24].

In theI = 1 channel we find only rough agreementwith the data in theKN → KN and KN → πΣ

amplitudes, but, just like the data, we find no evidenceof a resonance signal that would allow us to identifythe Σ(1620) resonance. Clearly, the absence of asignal even in some of the experimental amplitudeshas lead to classifying theΣ(1620) as only a 2-starresonance. However, the absence of such a resonancewould be somewhat surprising since we expect to getan octet of meson–baryon resonances and so far onlya singlet and theI = 0 part of the octet (eventuallymixed between themselves) have appeared. Since wedo not see this state in the amplitudes at real energieswe look for a pole in the complex plane. We go directlyto the second Riemann sheet, which we take in ourcase as the one where the momenta of the channelswhich are open at energyW , with Re(z) = W , aretaken negative inGl .

Near the poles the amplitudes that we are analyzingbehave as

(12)Tij gigj

z− zR , TM,ij xΓ/2eiφ′

z− zR .

Thus, the residues of theTij matrix give the productof the coupling of the resonance to thei, j channels,while the residues of theTM,ij give one half of theproduct of the two partial decay widths. The first oneof Eq. (12) determines the coupling of the resonanceto different final states, which are well defined even ifthese states are closed in the decay of the resonance.

The search of the poles leads us, usingaKΞ =−2.52, to the valuesMR = 1708+ i21 MeV, Γ =42 MeV,BKN = 47%,BηΛ = 47%, andBπΣ = 6%,in remarkable agreement with the values obtainedfrom the speed plot. ForaKΞ = −2.67, we obtainMR = 1680+ i20 MeV, Γ = 40 MeV,BKN = 54%,BηΛ = 38%, andBπΣ = 8%.

The couplings obtained for theΛ(1670) resonance,usingaKΞ = −2.52, are

|gKN | = 0.51, |gπΣ | = 0.052,

(13)|gηΛ| = 1.0, |gKΞ | = 11,

and, usingaKΞ = −2.67, we obtain

|gKN | = 0.61, |gπΣ | = 0.073,

(14)|gηΛ| = 1.1, |gKΞ | = 12.

It is also interesting to display the results of thecomplex plane search for theΛ(1405) resonance. Wefind

(15)MR = (1426+ i16)MeV (Γ = 32 MeV),

|gKN | = 7.4, |gπΣ | = 2.3,

(16)|gηΛ| = 2.0, |gKΞ | = 0.12,

with only theπΣ channel open for the decay.We have also performed a search in theI = 1

channel and we indeed find a pole at

(17)MR = (1579+ i264)MeV (Γ ∼ 528 MeV),

from the model withaKΞ = −2.67. The couplingsobtained are

|gKN | = 2.6, |gπΣ | = 7.2, |gπΛ| = 4.2,

(18)|gηΣ | = 3.5, |gKΞ | = 12.

We find that the agreement with the PDG [15] isquite good for the case of theΛ(1405). For the case oftheΛ(1670) we find a good agreement with the totalwidth, but the partial decay widths to theKN andηΛchannels is somewhat overpredicted while the partialdecay width to theπΣ channel that we obtain is abit too small. For the case of theΣ(1620) resonancewe find a very large width which may be the reasonwhy this state does not provide a clearer signal in thescattering amplitudes.

The analysis of the couplings is very interesting. Inthe case of theΛ(1405) state the coupling to theKNchannel is found to be very large, while the coupling tothe other channels is very small. This would allow usto identify this resonance as a quasiboundKN statein the present approach. Similarly, we find that theΛ(1670) resonance has a large coupling to theKΞchannel and unusually small couplings to the otherfinal states. This is responsible for the small width ofthe resonance in spite of the large phase space open fordecay into the different channels. The large couplingto theKΞ channel allows identifying this state asa KΞ quasibound state in the present approach. Bycontrast, theΣ(1620) resonance has couplings ofnormal size to all channels, and, given the large phasespace available, it has a sizable decay width into anyof the channels and hence a considerably larger totalwidth.

In summary, we have demonstrated that the chiralapproach to theKN and the other coupled channels,


which proved so successful at low energies, extrap-olates smoothly to higher energies and provides thebasic features of the scattering amplitudes, generatingthe resonances which would complete the states of thenonet of theJP = 1/2− excited states. The qualita-tive description of the data without adjusting any pa-rameters is telling us that the basic information on thedynamics of these processes is contained in the chiralLagrangians. There is still some freedom left with thechiral symmetry breaking terms. In our formulationthey would go into theal subtraction constants, andthe use of different decay constants for each meson, bymeans of which one could obtain a better descriptionof the data. However, before proceeding in this direc-tion, and eventually introduce further chiral symmetrybreaking terms, it would be important to sort out theapparent discrepancies between different sets of data.The analysis of the poles and the couplings of the res-onances to the different channels lead us to identifythe strong coupling of theΛ(1405) resonance to theKN state and the large coupling of theΛ(1670) res-onance to theKΞ state, allowing us to classify theseresonances as quasibound states ofKN andKΞ , re-spectively.

Acknowledgements

E.O. and C.B. wish to acknowledge the hospitalityof the University of Barcelona and A.R. and C.B.that of the University of Valencia, where part of thiswork was done. We would also like to acknowledgesome useful discussions with J.A. Oller and U.G.Meissner. This work is also partly supported byDGICYT contract numbers BFM2000-1326, PB98-1247, by the EU TMR network Eurodaphne, contractno. ERBFMRX-CT98-0169, and by the US-DOEgrant DE-FG02-95ER-40907.

References

[1] N. Kaiser, P.B. Siegel, W. Weise, Nucl. Phys. A 594 (1995)325.

[2] N. Kaiser, T. Waas, W. Weise, Nucl. Phys. A 612 (1997) 297.[3] E. Oset, A. Ramos, Nucl. Phys. A 635 (1998) 99.[4] J.A. Oller, E. Oset, Phys. Rev. D 60 (1999) 074023.[5] J.A. Oller, U.G. Meissner, Nucl. Phys. A 673 (2000) 311.[6] J.A. Oller, U.G. Meissner, Phys. Lett. B 500 (2001) 263.[7] N. Isgur, G. Karl, Phys. Rev. D 18 (1978) 4187;

N. Isgur, G. Karl, Phys. Rev. D 20 (1979) 1191;S. Capstick, W. Roberts, Phys. Rev. D 49 (1994) 4570.

[8] A. Pich, Rep. Prog. Phys. 58 (1995) 563.[9] G. Ecker, Prog. Part. Nucl. Phys. 35 (1995) 1.

[10] V. Bernard, N. Kaiser, U.G. Meissner, Int. J. Mod. Phys. E 4(1995) 193.

[11] U.G. Meissner, Rep. Prog. Phys. 56 (1993) 903.[12] J.A. Oller, E. Oset, J.R. Peláez, Phys. Rev. D 59 (1999)

074001.[13] G.P. Gopal et al., Nucl. Phys. B 119 (1977) 362.[14] G. Hoehler,πN -Newsletter 9 (1993) 1.[15] D.E. Groom et al., Eur. Phys. J. C 15 (2000) 1.[16] A.D. Martin, Nucl. Phys. B 179 (1981) 33.[17] M. Iwasaki et al., Phys. Rev. Lett. 78 (1997) 3067.[18] M.Th. Keil, G. Penner, U. Mosel, Phys. Rev. C 63 (2001)

045202.[19] W. Langbein, F. Wagner, Nucl. Phys. B 47 (1972) 477.[20] M. Alston-Garnjost et al., Phys. Rev. D 18 (1978) 182.[21] B. Nefkens, in: D. Drechsel, L. Tiator (Eds.), Proc. of the

Workshop on The Physics of Excited Nucleons, NSTAR2001,World Scientific, 2001, p. 427;D.M. Manley, Proc. of the 9th International Symposium onMeson–Nucleon Physics and the Structure of the Nucleon,πN -Newsletter 16 (2001) 68.

[22] G.W. London et al., Nuc. Phys. B 85 (1975) 289;D.F. Baxter et al., Nuc. Phys. B 67 (1973) 125;R. Rader et al., Nuovo Cimento 16A (1973) 178;D. Berley et al., Phys. Rev. Lett. 15 (1965) 641.

[23] J. Griselin et al., Nuc. Phys. B 93 (1975) 189.[24] A. de Bellefon et al., Nuovo Cimento 7A (1972) 567.



Proton decay in the semi-simple unification

Masaaki Fujii, T. Watari

Department of Physics, University of Tokyo, Tokyo 113-0033, Japan


Editor: T. Yanagida

Abstract

Semi-simple unification is one of models which naturally solve two difficulties in the supersymmetric grand unificationtheory: doublet–triplet splitting problem and suppression of dimension 5 proton decay. We analyzed the dimension 6 protondecay of this model using perturbative analysis at the next-to-leading order. The life time of proton is 3× 1034–1035 yearsfor wide range of SUSY breaking parameters, and there is an intriguing possibility of observing proton decay signals in thenext-generation waterCerenkov detectors such as Hyper-Kamiokande and TITAND. Several uncertainties in this prediction arealso discussed. 2002 Elsevier Science B.V. All rights reserved.

1. Introduction

Supersymmetric (SUSY) SU(5)GUT grand unifica-tion theories (GUTs) are supported by the approxi-mate SU(5)GUT unification at around 1016 GeV of thethree gauge coupling constants of the minimal super-symmetric standard model (MSSM) [1]. However, theconceptual beauties of the GUTs [2] and such a phe-nomenological success are not more than indirect ev-idences, and it would be the proton decay signals thatmakes us believe the GUTs in nature.

The minimal SU(5)GUT model predicts proton de-cay through dimension 5 operators [3], and is nowalmost excluded [4] by experimental bounds such asτ (p→K+ν) 6.7× 1032 yr. (90% C.L.) [5]. There-fore, we have to analyze the proton decay in an ex-tended model in which those operators are suppressedor absent. Predictions on the proton decay through di-

E-mail address: [email protected](T. Watari).

mension 6 operators severely depend on how a modelis extended;1 the life time of the proton depends on thefourth power of the mass of SU(5)GUT-off-diagonalgauge boson (GUT gauge boson), and hence on thedetailed spectrum of the model around the GUT scale.Therefore, an analysis on the proton decay has to bebased on a phenomenologically reliable model of theGUTs.

The doublet–triplet splitting problem [7] and sup-pression of the dimension 5 proton decay operatorshad been the two major obstacles in model buildingsof the GUTs. The semi-simple unification [8–10] is amodel that solves these two problems in a natural way.In this Letter, we calculate the proton decay rate inthis model. The proton decay is relatively fast in thismodel, whose reason will be clear in the text. We re-strict ourselves in parameter region of the model wherea perturbative analysis is valid. As a result of a full

1 For example, in some type of model [6], the dimension 6operators are not induced by the GUT gauge boson exchange.



M. Fujii, T. Watari / Physics Letters B 527 (2002) 106–114 107

next-to-leading order analysis [11], we found that themass of the GUT gauge boson can be determined, andthat the resulting life time of proton does not dependon SUSY breaking parameters so much: the life timeis τ (3–10) × 1034 yr. Various sources of uncer-tainties in this prediction are summarized at the endof this letter. This result means that the proton decayis generically detectable in the next-generation waterCerenkov detectors such as Hyper-Kamiokande andTITAND [12,13].

2. Brief review of the semi-simple unificationmodel

We briefly review the semi-simple unification mo-del that uses SU(5)GUT × U(3)H gauge group. Thisgauge group is directly broken down to the SU(3)C ×SU(2)L × U(1)Y of the MSSM. Quark and lepton(5∗ + 10) and Higgs (Hi(5) + Hi(5∗)) supermulti-plets are singlets of the U(3)H gauge group and trans-form under the SU(5)GUT as in the standard SU(5)GUTmodel. Some more fields are required for the GUTsymmetry breaking:Xαβ (α,β = 1,2,3) transform-ing as (1, adj.= 8 + 1) under the SU(5)GUT × U(3)Hgauge group, andQαi (i = 1, . . . ,5)+Qα6 and Qi

α

(i = 1, . . . ,5)+ Q6α transforming as (5∗ + 1, 3) and

(5 + 1, 3∗). Indicesα andβ are for the U(3)H and ifor the SU(5)GUT. Xαβ is also expressed asXc(tc)αβ(c= 0,1–8), whereta (a = 1–8) are Gell’mann matri-ces2 andt0 ≡ 13×3/

√6. Superpotential is given as [10]

W = √2λ3H Qi

αXa(ta)

αβQ

βi

+ √2λ′

3HQ6

αXa(ta)

αβQ

β6

+ √2λ1H Qi

αX0(t0)

αβQ

βi

+ √2λ′

1HQ6

αX0(t0)

αβQ

β6 − √

2λ1Hv2Xαα

+h′ Hi QiαQ

α6 + hQ6

αQαiH

i

(1)+ y1010 · 10 ·H + y5∗5∗ · 10 · H + · · · ,where the parameterv is of order of the GUT scale,y10, y5∗ are Yukawa coupling constants of the quarksand leptons, andλ3H, λ

′3H, λ1H, λ

′1H, h

′, h are dimen-sionless coupling constants. One can see that the above

2 A normalization condition tr(ta tb) = δab/2 is understood.Note that the normalization of the followingt0 is determined so thatit also satisfies tr(t0t0)= 1/2.

Table 1Charge assignment of theZ4 R-symmetry is given.1 denotes a righthanded neutrino

Fields 5∗, 10, 1 H , H Xαβ Qi, Qi Q6 Q6

Z4 R charge 1 0 2 0 2 −2

superpotential hasZ4 R symmetry under a charge as-signment given in Table 1, and this symmetry for-bids enormous mass termW = HH for the Higgsdoublets.3 The bifundamental representationQαi andQiα acquire vacuum expectation value,〈Qαi〉 = vδαi

and〈Qiα〉 = vδiα , because of the second and the third

line in (1). Then, the mass terms of the colored Higgsmultiplets arise from the fourth line in (1) in the GUT-breaking vacuum. TheZ4 R-symmetry is not brokeneven after the GUT symmetry is broken. One can alsosee that thisZ4 R-symmetry forbids the dangerous di-mension 5 proton decay operatorsW = 10 ·10 ·10 ·5∗.

3. Naive estimation

At tree level, the gauge coupling constants of theSU(3)C ×SU(2)L×U(1)Y are given in terms of thoseof the SU(5)GUT × U(3)H as:

(2)

(1

α3≡ 1

αC

)= 1

αGUT+ 1

α3H,

(3)

(1

α2≡ 1

αL

)= 1

αGUT,

and

(4)

(1

α1≡ 3/5

αY

)= 1

αGUT+ 2/5

α1H,

whereαC,αL,αY ,αGUT, α3H andα1H are fine struc-ture constants of the three MSSM gauge groups,SU(5)GUT, SU(3)H and U(1)H,4 respectively. The ap-proximate SU(5)GUT relation and deviation form it(Fig. 1) are naturally explained through the aboveequations if 1/αGUT ∼ 24 and 1/α3H 1, 1/α1H 2.5. At the same time, we notice that the “GUT

3 µ-term can be obtained through the Giudice–Masiero mecha-nism [14].

4 The normalization of the U(1)H-generator is determined so thatQ, Q have charge±1/

√6.

108 M. Fujii, T. Watari / Physics Letters B 527 (2002) 106–114

Fig. 1. Approximate SU(5)GUT relation between the three MSSMgauge coupling constants and deviation from it. 1σ error bar ofthe QCD coupling are also described. SUSY threshold correc-tions are calculated using the spectrum of mSUGRA model withm0 = 250 GeV,M1/2 = 500 GeV,A0 = 0 and tanβ = 10. The signof µ-term is taken to be negative.

scale”, an energy scale at which Eqs. (2)–(4) hold,lies lower than theα1–α2 unification scaleM1−2 andhigher than theα2–α3 unification scaleM2−3. There-fore, the decay rate of proton is expected to be en-hanced compared with the rate using theM1−2 as theGUT gauge boson mass.

At one-loop order, the gauge coupling of the U(1)Hruns5 asymptotic non-free:

(5)∂

∂ lnµ

(1

α1H(µ)

)= − 6

2π.

There are two important remarks here. First, the cut-off scaleΛ of this model exists below the Planckscale; 1/α1H should be positive even at theΛ. Theconstraint 1/α1H 2.5 at the “GUT scale” allows theΛ to be higher than the “GUT scale” by one order ofmagnitude or a little bit more, and this is much enoughto justify the field theoretical description of the GUTsymmetry breaking and to accommodate all the GUTspectrum below the cut-off scaleΛ. ThisΛ lies around1017 GeV or a little higher, though it may be below1018 GeV. Secondly, the IR-free (asymptotic non-free)behavior of the U(1)H coupling leads to

(6)1

α3H 1

α1H

5 The one-loopβ-function of the SU(3)H coupling is 0.

at the “GUT scale” under an assumption

(7)1

α3H(Λ) 1

α1H(Λ).

This U(3)H-relation atΛ is quite natural if there isU(3)H-structure in a fundamental theory [15]. Then,as a consequence of the relation Eq. (6), we notice thatthe “GUT scale” is closer to theM2−3 rather than totheM1−2 since(1/α1 − 1/α2) > (1/α3 − 1/α2) there,and hence the proton decay is relatively fast.

4. Threshold corrections at the GUT scale

In the analysis at the next-to-leading order, one-loop threshold corrections of the GUT model are alsotaken into account. The three MSSM gauge couplingconstants just below the GUT scale are expressed interms of the gauge coupling constants and variousmasses in the spectrum of the GUT model, includingthe mass of the GUT gauge boson,Mv . Particlespectrum around the GUT scale is summarized inTable 2. Explicit expressions of the MSSM gaugecoupling constants are given as follows:

1

α3(µ)= 1

αGUT(Λ)+ 1

α3H(Λ)+ −3

2πln

(Λ

µ

)

+ 1

2πln

(Λ2

McMc

)− 4

2πln

(Λ

Mv

)

(8)+ 6

2πln

(M8v

M8c

),

(9)

1

α2(µ)= 1

αGUT(Λ)+ 1

2πln

(Λ

µ

)− 6

2πln

(Λ

Mv

),

1

α1(µ)= 1

αGUT(Λ)+ 2/5

α1H(Λ)+ 33/5

2πln

(Λ

µ

)

(10)+ 2/5

2πln

(Λ2

McMc

)− 10

2πln

(Λ

Mv

),

whereµ is a renormalization point, which is takento be just below the GUT scale,Mv , Mc , Mc, M8v,M8c are masses of particles around the GUT scale(see Table 2) andαGUT,3H,1H(Λ) are fine structureconstants of the gauge groups SU(5)GUT, SU(3)H andU(1)H, respectively, at the cut-off scaleΛ.

In general, it is impossible to determine theMvif GUT models have more than three parameters.


Table 2Summary of the particle spectrum around the GUT scale. The first line denotes the representation under the MSSM gauge group. In the secondline, m. vect. denotesN = 1 massive vector multiplet andχ + χ† a pair ofN = 1 chiral and anti-chiral multiplet. Mass of each multiplet isgiven in terms of gauge couplings and parameters in the superpotential (1) in the fourth line, and given in the third line is the expression of themass used in the text. Multiplets with massesM1v andM1c , M8v andM8c can be regarded asN = 2-SUSY partner with each other in theN = 2-SUSY limit (see also Appendix A)

(3,2)−56 (3,1)−

13 (3,1)−

13 (1,1)0 (1,1)0 (adj.,1)0 (adj.,1)0

m. vect. χ + χ† χ + χ† m. vect. χ + χ† m. vect. χ + χ†

Mv Mc Mc M1v M1c M8v M8c

= √2gGUTv = hv = h′v =

√2(g2

1H + 2g2GUT/5

)v = √

2λ1Hv =√

2(g2

3H + g2GUT

)v = √

2λ3Hv

However, it is not necessarily the case in the semi-simple unification model. Threshold corrections inEqs. (8)–(10) is simplified considerably under two rea-sonable assumptions. One is the U(3)H-relation Eq. (7)and the other isN = 2-SUSY-relation:

(11)g1H λ1H(∼ λ′

1H

), g3H λ3H

(∼ λ′3H

).

Under the latter condition, a large threshold correctionfrom the massive SU(3)H vector multiplet6 is almostcanceled by itsN = 2-partner, the SU(3)C-adj. chiralmultiplets, sinceM8v M8c. Now that the thresholdcorrections form the SU(3)C-adj. multiplets decouplefrom Eqs. (8)–(10), we are left only with two thresh-old corrections: one from the massive vector multipletof the GUT gauge boson and the other from coloredHiggs chiral multiplets. Then, one can easily see thatthree combinations,

1

αGUT(Λ), ln

(Λ

Mv

)and

(12)1

α3H(Λ)+ 1

2πln

(Λ2

McMc

),

are determined in terms of the values to be put in theLHSs and deviation from the U(3)H-relation and theN = 2-relation, once the cut-off scaleΛ is fixed.

In particular, the GUT gauge boson mass is givenby

Mv =√µ3

Λexp

(−2π

24

(2

α3+ 3

α2− 5

α1

)(µ)

)

6 Note thatM8v ∼ 10×Mv , and hence the threshold correctionis large.

(13)

×√M8v

M8cexp

(−2π

12

(1

α1H− 1

α3H

)(Λ)

).

The last two factors show how the result is changeddue to the deviation from the assumptions we made.Λ−1/2-dependence is a direct consequence of the one-loop running of theα1H in Eq. (5), and this negativepower dependence implies that this gauge boson massis generically light. The life time of proton throughthis GUT gauge boson exchange is given in terms oftheMv as [11]

τ(p→ π0e+

) 0.61× 1035 ×

(Mv

1016 GeV

)4

(14)×(

1

24αGUT(Mv)

)2(0.15 (GeV)2

|W |)2

yr.,

whereW is a hadron matrix element calculated withlattice quenched QCD [16].

5. Threshold corrections at the weak scale andtwo-loop running

In order to determine the precise value of the GUTgauge boson mass by using (13), we must accuratelydetermine the three MSSM gauge coupling constantsat the GUT scale. For this purpose, we take full one-loop threshold corrections at the weak scale into ac-count for the three gauge coupling constants and top-and bottom-Yukawa coupling constants by followingthe method in Ref. [17], and use two-loop renormal-ization group (RG) equations. For illustration, let usbriefly review the procedures which we adopt in thisLetter. The conventions of SUSY breaking parameters


and of the sign of theµ-term are the same as those inRef. [17].

The SUSY threshold corrections to fix theDR cou-pling α3(MZ) is very simple and the result is as fol-lows:

(15)α3(MZ)= α3(MZ)MS

1−+α3,

where

+α3 = α3(MZ)MS

2π

[1

2− 2

3ln

(mt

2

)− 2 ln

(mg

MZ

)

(16)− 1

6

∑q

2∑i=1

ln

(mqi

MZ

)].

Here,MZ = 91.188 GeV is theZ-boson pole massand we takeα3(MZ)MS = 0.118(2) [20]. The summa-tion with q runs over all the six squark flavors, andthe constant contribution 1/2 in Eq. (16) is necessarywhen the coupling is translated from theMS schemeto theDR scheme.

Because of the breaking effects of the SU(2)Lgauge group, the determinations of theDR gauge cou-pling constantsαY (MZ) andαL(MZ) are much morecomplicated. First, we calculate theDR electromag-netic coupling constant,α(MZ). The explicit formulais given by

(17)α(MZ)= αem

1−+α, αem = 1

137.036,

where

+α = 0.0682± 0.0007

− αem

2π

[−7 ln

(MW

MZ

)+ 16

9ln

(mt

MZ

)

+ 1

3ln

(mH+

MZ

)+ 4

9

∑u

2∑i=1

ln

(mui

MZ

)

+ 1

9

∑d

2∑i=1

ln

(mdi

MZ

)

(18)

+ 1

3

∑e

2∑i=1

ln

(mei

MZ

)+ 4

3

2∑i=1

ln

(mχ+

MZ

)].

Here,∑u denotes a sum overu, c, t , and similarly

for∑d and

∑e . The numerical values appearing in

the above expression includes the two-loop QED andQCD corrections in Ref. [18], as well as the five-flavorcontributions in Ref. [19].

Next, we need to fix theDR weak mixing angleθewto derive theDR gauge coupling constants,αY (MZ)andαL(MZ). The formula to get theDR weak mixingangle is given by

(19)cos2(θew)sin2(θew)= πα(MZ)√2M2

ZGµ(1−+r),

(20)+r = ρΠTWW (0)

M2W

−ReΠTZZ(M

2Z)

M2Z

+ δVB,

whereMW = 80.419 GeV is theW -boson pole mass,ρ is defined asρ ≡ M2

W/(cos2(θew)M2Z), Gµ =

1.16639×10−5 GeV−2 is the Fermi constant, andδVBdenotes the non-universal vertex and box diagram cor-rections. The explicit formulae to calculate the quanti-ties given in the above expressions and theDR Yukawacoupling constants are all given in Ref. [17].

Taking αem, α3(MZ)MS, the quark and leptonmasses, and SUSY particle masses as inputs, wecalculate all the three gauge coupling constants andtop- and bottom-Yukawa coupling constants inDRscheme at theZ-boson pole mass with full one-loop threshold corrections. With these values and treelevel tau-Yukawa coupling, we use the two-loop RGequations to obtain the gauge coupling constants at theGUT scale. For the Yukawa coupling constants, we useone-loop RG equations.

In this Letter, we adopt the central values given inRef. [20] for the masses of vector bosons, quarks andleptons.7 As for the mass spectrum of the SUSY par-ticles and light Higgs particle, we take the values cal-culated by theSOFTSUSY code [21] with mSUGRAboundary conditions for demonstration.8 By usingthese input values with mSUGRA boundary condi-tions, we also confirm that the unification-scale cor-rectionεg

(21)α3(M1−2)= α1(M1−2)(1+ 2εg),

at theα1–α2 unification scaleM1−2 is quite consistentwith the result given in Ref. [17].

7 Neutrino masses are set to be zero.8 We greatly thank K. Suzuki for this task.


Fig. 2. Contour plots of the proton life time inm0–M1/2 plane forµ < 0 cases. SUSY threshold corrections are calculated using mSUGRAsparticle spectrum with universal boundary conditionsm0,M1/2, A0 at the GUT scale. As for(tanβ, A0[GeV]), we take them to be(10,0),(10,−300), (30,0), (30,−300), respectively, as you can see from each figure. Solid lines correspond to the contours of the proton life time,5× 1034, 7× 1034, 1035, 2× 1035 yr. from in to out, respectively. Some of them are explicitly denoted in each figure.

6. Conclusion

Now, we can estimate the proton life time forvarious SUSY particle spectra. We neglect, for themoment, possible two uncertainties expressed by thelast two factors in (13) coming from the deviation fromN = 2-relation and U(3)H-relation. Effects of suchdeviations are discussed later. Here, we also set thecut-off scale to be 1017 GeV; in most part of SUSYbreaking parameter space, the three gauge couplingconstants unify approximately at around 1016 GeVand hence the cut-off scaleΛ is expected to be no lessthan 1017 GeV. Therefore, we obtain a conservative

upper bound of the proton life time, using the lowestcut-off scale (see (13)).

We plot the contours of the life time of proton inm0–M1/2 plane, wherem0 andM1/2 are the universalsoft scalar mass and gaugino mass at the GUT scale,respectively. In Fig. 2, we show contours of the protonlife time for µ < 0 cases with several choices ofA0 (= 0, − 300 GeV), the universalA-term at theGUT scale, and tanβ (= 10, 30). The contour plotsfor µ> 0 cases are given in Fig. 3.

As we can see from these contours, the proton lifetime is in the range 3× 1034–1035 yr. in most part ofthe parameter space regardless of choices of tanβ , A0


Fig. 3. Contour plots of the proton life time inm0–M1/2 plane forµ> 0 cases. Other conventions are the same as those in Fig. 2.

and sign ofµ. We find the minimum of the protonlife time is no less than 3× 1034 yr. in whole para-meter space, which is well above the current experi-mental limit by the Super-Kamiokande, 5.0× 1033 yr.(90% C.L.) [13,22]. The thick gray contour lines cor-responding to the life time of proton 7× 1034 yr. rep-resent the 3σ discovery limit of the 1 Mt (fiducial vol-ume) detector after ten years running [12,13].

Therefore, in the semi-simple unification model,we have an intriguing possibility to confirm the ex-istence of the GUT in nature by observing the pro-ton decay in the next-generation Mt waterCerenkovdetectors, such as Hyper-Kamiokande [12] and TI-TAND [13]. In the optimistic cases with some en-hancement factors of the decay rate of proton (see be-low), we have a chance to detect the proton decay alsoin UNO [23] (∼ 500 kt fiducial volume) experiment.

Although we set the cut-off scaleΛ to be 1017 GeVin calculating the GUT gauge boson mass to obtainthe conservative lower bound of the proton decay rate,the actual cut-off scale may be a little more higher.In that case, the rate is enhanced by(Λ/1017 GeV)2.Another possible enhancement of the decay rate ariseswhen there are SU(5)GUT-charged particles at anintermediate scale. Existence of such particles arehighly motivated in the semi-simple unification model;5 + 5∗ representations are required at the TeV scalewhen the discreteZ4 R-symmetry is gauged sincethe discrete gauge anomalyZ4R-[SU(5)GUT]2 shouldbe canceled [24]. In this case, the gauge couplingconstantαGUT is stronger as a result of the RG flowwith new particles, and the decay rate is enhanced

by 1.6. Although one might suspect that there is aone-loop threshold correction from a possible masssplitting between triplets and doublets in5 + 5∗,and that the GUT gauge boson mass would be alsochanged, the GUT gauge boson mass is actuallystable against this correction, since Eq. (13) is anexpression from which the threshold corrections fromthe colored Higgs multiplets decouple. The samething happens when the SUSY breaking is mediatedthrough gauge mediation because of the presence ofthe messenger sector, though the SUSY thresholdcorrection should be re-analyzed using the spectrumof the gauge mediated SUSY breaking in that case.

Finally, we summarize various uncertainties in thetheoretical prediction given above. The first uncer-tainty comes from possible violation of the U(3)H rela-tion. The violation|(1/α3H − 1/α1H)(Λ)| = 1/3 leadsto a change in the decay rate by×/ ÷ 0.5. The sec-ond uncertainty comes from an error bar of the ex-perimental values of the QCD coupling. This resultsin uncertainties by factor×/ ÷ 0.7 for 1σ error. Thecalculation of hadron matrix element in [16] has an er-ror W = −0.153(19) GeV2, which leads to a factor×/ ÷ 0.8. Another uncertainty comes from a possi-ble non-renormalization operators involving the〈QQ〉vacuum expectation value in the gauge kinetic func-tion of the SU(5)GUT.9 They generically modifies the

9 Such non-renormalizable terms in the gauge kinetic function isexpected to be suppressed when one considers a certain structure ofthe fundamental theory [15].


SU(5)GUT relation directly by〈QQ〉/Λ2 10−2 attree level. If it is the case, the possible change in the re-sult will be at most roughly the same amount as thosediscussed above.

There are two more sources of uncertainties whoseeffects we cannot estimate. First, if one considers anexotic situation in which unknown non-renormalizableoperators are relevant in the Wilsonian RG equations,then the perturbative analysis we adopted in this Let-ter is not adequate since we omitted such effects.Secondly, we cannot estimate anything without theN = 2-SUSY relation. This is because the perturba-tive analysis above the GUT scale is no longer validwithout this relation, as is discussed in Appendix A.

Acknowledgements

Earlier part of this work was done in collaborationwith K. Kurosawa. The authors are grateful to Y. Shir-man for discussion, to K. Suzuki for generating spectraof SUSY particles, and to T. Yanagida for discussionand careful reading of this manuscript. M.F. and T.W.thank the Japan Society for the Promotion of Sciencefor financial support.

Appendix A. Role of approximate N = 2 SUSYrelation in perturbative analysis

The GUT-breaking sector of the semi-simple unifi-cation model has a multiplet structure ofN = 2 SUSY,and the interactions between them (the first—the thirdlines in Eq. (1)) are quite similar to theN = 2 gaugeinteractions with Fayet–IliopoulosF -term. Therefore,it is quite likely that this apparentN = 2 structure is aremnant of theN = 2 SUSY in a fundamental theory[15]. Then, the approximateN = 2 relation Eq. (11)at the cut-off scale would be a natural consequence.

The approximateN = 2-relation is not only ex-pected as above, but also almost required from anotherreason. The perturbation analysis performed in the textis no longer valid if it is not satisfied and that is thereason why we assumed this relation throughout thisLetter.

Let us suppose that the couplingsλ(′)3H andλ(′)1H inthe superpotential (1) are large compared withg3H andg1H. Then, those couplings become large extremely

fast through one-loop RG equations, and hence wehave to require thatαλ3H ≡ λ2

3H/(4π) and αλ1H ≡λ2

1H/(4π) are well below 2α3H and 2α1H, respectively.The same discussion also holds forλ′

3H andλ′1H. Now

what if those couplings are small compared with thecorresponding gauge couplings? In this case, we canneglect the last two terms in the following two-loopRG equations of the gauge couplings,

∂

∂ lnµ

(1

α3H

)

−α1H + 17α3H

2π2+

56(α

λ1H + 17αλ3H)

2π2

(A.1)+16(α

λ′1H + 17αλ

′3H)

2π2 ,

∂

∂ lnµ

(1

α1H

)

− 6

2π− α1H + 8α3H

2π2+

56(α

λ1H + 8αλ3H)

2π2

(A.2)+16(α

λ′1H + 8αλ

′3H)

2π2 .

Then,α3H becomes large quite rapidly andα1H be-comes large more faster than in the one-loop running.Thus, we require thatαλ

(′)3H andαλ

(′)1H are comparable

to the gauge couplings so that the two-loop effects arenegligible.

In the approximateN = 2-SUSY limit and onlyin this limit, α3H αλ(′)3H andα1H αλ(′)1H , anomalousdimensions of hyper multiplets,

(A.3)γQi =8αλ3H + αλ1H

6π− 8α3H + α1H

6π+ · · · ,

vanish at all order, and the RG flows of the gaugecouplings are one-loop exact. Then, in turn, all otherparameters in the superpotential, in particularh andh′,are stable against quantum corrections from the strongcouplingsα1H, α3H, αλ

(′)1H andαλ

(′)3H .

Values of the coupling constantsh and h′ them-selves are the possible obstruction left behind forthe perturbative analysis.10 They are obtained from

10 We already know that other coupling constants such asy10,5∗andgGUT are weak and they are stable under their RG equations.Their perturbation to the approximateN = 2-SUSY relation is alsosmall enough.


a ratio√McMc/Mv , which in turn is obtained from

Eqs. (8)–(10) in the way described in the text:

√hh′ = √

2gGUTeπα3H

(Λ)(Λ

µ

)

× exp

(2π

12

(− 4

α3+ 9

α2− 5

α1

)(µ)

)

(A.4)

×(M8v

M8c

)2

exp

(2π

6

(1

α1H− 1

α3H

)(Λ)

).

The value of the RHS of this equation varies fromsub-O(1) to O(1). Therefore, we can expect that theperturbative analysis performed in the text is validfor most part of the SUSY breaking parameter space,taking into account the uncertainties in the gaugecoupling constants.

References

[1] P. Langacker, M. Luo, Phys. Rev. D 44 (1991) 817.[2] H. Georgi, S.L. Glashow, Phys. Rev. Lett. 32 (1974) 438.[3] N. Sakai, T. Yanagida, Nucl. Phys. B 197 (1982) 533;

S. Weinberg, Phys. Rev. D 26 (1982) 287.[4] J. Hisano, T. Moroi, K. Tobe, T. Yanagida, Mod. Phys. Lett.

A 10 (1995) 2267, hep-ph/9411298;H. Murayama, A. Pierce, hep-ph/0108104.

[5] Y. Hayato et al., SuperKamiokande Collaboration, Phys. Rev.Lett. 83 (1999) 1529, hep-ex/9904020.

[6] E. Witten, Nucl. Phys. B 258 (1985) 75.[7] L. Maiani, in: Comptes Rendus de l’Ecole d’Etè de Physique

des Particules, Gif-sur-Yvette, 1979;S. Dimopoulos, H. Georgi, Nucl. Phys. B 150 (1981) 193;M. Sakai, Z. Phys. C 11 (1981) 153;E. Witten, Nucl. Phys. B 188 (1981) 573.

[8] T. Yanagida, Phys. Lett. B 344 (1995) 211, hep-ph/9409329.[9] J. Hisano, T. Yanagida, Mod. Phys. Lett. A 10 (1995) 3097,

hep-ph/9510277;T. Hotta, K.I. Izawa, T. Yanagida, Phys. Rev. D 53 (1996) 3913,hep-ph/9509201;T. Hotta, K.I. Izawa, T. Yanagida, Phys. Rev. D 54 (1996) 6970,hep-ph/9602439.

[10] K.I. Izawa, T. Yanagida, Prog. Theor. Phys. 97 (1997) 913, hep-ph/9703350.

[11] J. Hisano, H. Murayama, T. Yanagida, Nucl. Phys. B 402(1993) 46, hep-ph/9207279.

[12] M. Koshiba, Phys. Rep. 220 (1992) 229;K. Nakamura, Talk presented at Int. Workshop on NextGeneration Nucleon Decay and Neutrino Detector, 1999,SUNY at Stony Brook;K. Nakamura, Neutrino Oscillation and Their Origin, Univer-sal Academy Press, Tokyo, 2000, p. 359.

[13] Y. Suzuki et al., TITAND Working Group Collaboration, hep-ex/0110005.

[14] G.F. Giudice, A. Masiero, Phys. Lett. B 206 (1988) 480.[15] Y. Imamura, T. Watari, T. Yanagida, Phys. Rev. D 64 (2001)

065023, hep-ph/0103251;T. Watari, T. Yanagida, Phys. Lett. B 520 (2001) 322, hep-ph/0108057.

[16] S. Aoki et al., Phys. Rev. D 62 (2000) 014506.[17] D.M. Pierce, J.A. Bagger, K.T. Matchev, R.J. Zhang, Nucl.

Phys. B 491 (1997) 3, hep-ph/9606211.[18] S. Fanchiotti, B. Kniehl, A. Sirlin, Phys. Rev. D 48 (1993) 307,

hep-ph/9212285.[19] S. Eidelman, F. Jegerlehner, Z. Phys. C 67 (1995) 585, hep-

ph/9502298.[20] Particle data group, D.E. Groom et al., Particle Data Group

Collaboration, Eur. Phys. J. C 15 (2000) 1.[21] B.C. Allanach, hep-ph/0104145.[22] M. Shiozawa et al., Super-Kamiokande Collaboration, Phys.

Rev. Lett. 81 (1998) 3319, hep-ex/9806014.[23] C.K. Jung, hep-ex/0005046.[24] K. Kurosawa, N. Maru, T. Yanagida, Phys. Lett. B 512 (2001)

203, hep-ph/0105136.



(g − 2)µ from noncommutative geometry

N. Kersting

Lawrence Berkeley National Laboratory, Berkeley, CA, USA


Editor: H. Georgi

Abstract

This brief Letter demonstrates that effects from a noncommutative spacetime geometry will measurably affect the valueof (g − 2)µ inferred from the decay of the muon to an electron plus two neutrinos. If the scale of noncommutativity isO (TeV), the alteration of theV –A structure of the lepton–lepton–W vertex is sufficient to shift the inferred value of(g − 2)µ to one part in 108. This may account for the recently reported 2.6σ discrepancy between the BNL measurementaexpt= 11659202(14)(6) × 10−10 and the Standard Model predictionaSM = 11659159.6(6.7) × 10−10. 2002 Published byElsevier Science B.V.

1. Introduction

The measurement of the anomalous magnetic mo-ment of the muon,aµ ≡ (g − 2)µ, has undergone con-tinual refinement (for history and experimental details,see [1,2]) to the point whereaµ is now very preciselyknown [3]:

(1)aexptµ = 11659202(14)× 10−10.

The experimental technique employs muons trappedin a storage ring. A uniform magnetic fieldB is ap-plied perpendicular to the orbit of the muons; hencethe muon spin will precess. The signal is a discrep-ancy between the observed precession and cyclotron

This work was supported by the Director, Office of Science,Office of Basic Energy Services, of the US Department of Energyunder Contract DE-AC03-76SF0098.

E-mail address: [email protected](N. Kersting).

frequencies. Precession of the muon spin is deter-mined indirectly from the decayµ → eνeνµ. Elec-trons emerge from the decay vertex with a character-istic angular distribution which in the Standard Model(SM) has the following form in the rest frame of themuon

(2)dP(y,φ) = n(y)(1+A(y)cos(φ)

)dy d

(cos(φ)

),

whereφ is the angle between the momentum of theelectrone and the spin of the muon,y = 2pe/mµ mea-sures the fraction of the maximum available energywhich the electron carries, andn(y),A(y) are particu-lar functions which peak aty = 1. The detectors (posi-tioned along the perimeter of the ring) accept the pas-sage of only the highest energy electrons in order tomaximize the angular asymmetry in (2). In this way,the electron count rate is modulated at the frequencyaµeB/(2πmc).

The leading theoretical prediction ofaµ in the SMis aSM

µ = 11659159.6(6.7)× 10−10 [4] which leads to



116 N. Kersting / Physics Letters B 527 (2002) 115–118

(a) (b)

Fig. 1. (a) The muon decay to an electron plus two neutrinos. Each vertex receives a noncommutative loop correction (b) (withl = e,µ) whichupsets the electron’s angular distribution.

a 2.6σ deviation from the data

(3)aexptµ − aSM

µ = 43(16)× 10−10.

If this discrepancy persists as more data arrives andtheoretical uncertainties improve, then there is a clearsignal of new physics. Many proposals to accountfor this discrepancy have already appeared in theliterature.1

This Letter is a consideration of a novel effect onthe measurement ofaµ from noncommutative geom-etry, a theory in which the coordinates of spacetimebecome noncommuting operators:[xµ, xν] = iθµν .There is an extensive collection of papers devotedto both the theoretical foundations of noncommuta-tive geometry [6–11] and its phenomenology [12–15].The reader may consult the above references for amore thorough understanding of the noncommutativequantum field theory underlying the present calcula-tion. We will employ perturbation theory in leadingpowers of the dimensionful matrix of parametersθµνin accord with the work done in [15].

2. Preliminaries

Although aµ does receive a sizable contributionfrom noncommutative geometry, it is aconstant con-tribution [14], i.e., the interaction with the externalmagnetic fieldE ∼ Biθjkε

ijk is independent of themuon spin, and therefore the experiment describedabove is not sensitive to this perturbation ofaµ.

1 For a partial list, see [5].

The effect of noncommutative geometry on thismeasurement does however enter in the manner inwhich the muon spin is measured in its decay. Each oftheW -boson vertices in the decay diagram Fig. 1(a)receives corrections from noncommutative geometryat the one loop level, as shown in Fig. 1(b). One mightexpect such corrections to be negligible, but in fact theloop integral in Fig. 1(b) involvesθ -dependent verticeswhich lead to integrals of the form

(4)∫

d4k

16π2

eip·θ ·q

k4

for loop momenta much larger than the external mo-mentap, q . In the limit |θ | → 0 the integral (4)formally diverges so one has to renormalize care-fully (see [15] for a discussion of this point). Thegeneric size of the noncommutative contribution willbe α

16π2 |p2µθ | ln |p2

µθ | which for fast muons (pµ ≈3 GeV at BNL) and low scales of noncommutativ-ity (|θ | ≈ (1 TeV)−2) gives a suppression factor ofO(10−8) relative to the tree level decay diagram. Sincethe current deviation of the SM prediction from exper-iment in (3) is of this size, we see that noncommutativeeffects cannot be neglected on the basis of their mag-nitude.

More importantly, the appearance of the antisym-metric objectθµν in the decay amplitude leads to com-binations of the muon and electron spins and mo-menta which alter the modulation frequency of thedecay rate (2). Specifically, one anticipates factors of( pe · sµ)( pe · θ · sµ) which for electron momenta closeto their kinematical limit (i.e.,y = 1) behaves likecos(φ)sin(φ). In what follows we explicitly demon-strate these terms exist in the decay rate.

N. Kersting / Physics Letters B 527 (2002) 115–118 117

3. The calculation2

Define the muon decay amplitude

(5)M = GF√2ue

(CiOα

i

)v1u2

(C′jO′α

j

)uµ

involving the electron, muon, and neutrino(1,2)spinors and the most general set of operators at theinteraction vertices,Oi (i ⊂ S,P,A,V,T ) whichmay depend on momenta. The muon decay rate isproportional to the squared matrix element

|M|2 = G2F

2TeTµ,

Te ≡ tr(ue

(CiOα

i

)v1v1

(C∗jO

βj

)ue

),

(6)Tµ ≡ tr(u2(CkOk,α)uµuµ

(C∗l Ol,β

)u2

).

This is a product of two terms: the electron traceTeand the muon traceTµ. If θ were zero, all operatorswould be of the standardV –A form, and the traceswould be

Te(SM) = 4(qα

1pβe + q

β

1pαe − (q1 · pe)g

αβ

+ iqγ

1 pδeε

αβγ δ),

(7)

Tµ(SM) = 4(qα

2pβµ + q

β

2pαµ − (q2 · pµ)g

αβ

+ iqγ

2 pδµε

αβγ δ)

− 4m(qα

2 sβµ + q

β2 s

αµ − (q2 · sµ)gαβ

+ iqγ

2 sδµε

αβγ δ),

wherem is the muon mass and we neglect the mass ofthe electron in this and all that follows. The lowest-order contribution from noncommutative geometrywill be proportional to one power ofθ , so to extractit one calculates the contribution to|M|2 from eachway it is possible to change oneV –A operatorinto a noncommutative one, giving altogether twentyO(θ) terms in |M|2. To find the precise form ofthese operators, we next calculate the loop. In Fig. 2we show the loop with incoming charged leptonmomentump and outgoing neutrino momentumq .The loop amplitude is

Mloop =∫

d4k

(2π)4ue(q)

[−igγ γ (1− γ5)]

2 For an excellent treatment of the corresponding SMcalculation, see [16].

Fig. 2. Variables defined in the loop calculation.

× −igγ δ

(k − q)2 −m2W

i

/k −m

[−ieγ α] −igαβ

(p − k)2u(p)

(8)

× g[gηβ(q + k − 2p)δ + gβδ(q + p − 2k)η

+ gηδ(k + p − 2q)β]exp

[ik · θ · (p − q)

]

which becomes

ue(q)g2e

∫ (d4k(2π)4

Nη1 +N

η2 +N

η3

(k2−m2)(p−k)2((k−q)2−m2W )

× eik·θ ·(p−q))u(p),

Nη1 = (/q + /k − 2/p)(1− γ5)(/k +m)γ η,

Nη

2 = γ β(1− γ5)(/k +m)γ β(q + k − 2p)η,

(9)Nη3 = γ η(1− γ5)( /p + /k − 2/q).

Now using the on-shell conditionue(p)/p = mue(p)

and only retaining terms which coupleθµν to theoverall Dirac structure3 we arrive at

Nη1 → 2m/k(1+ γ5)γ

η − 2/kpη(1+ γ5),

Nη2 → mkη(1+ γ5) − 2kη/k(1− γ5),

(10)Nη

3 → mγ η(1− γ5)/k.

Of the above terms in the numerator, the dominant oneis the tensor piece ofNη

2 , i.e., the one proportional tokη/k, since it has the most powers ofk. To compute itseffect, we consider first the alteration of the electrontrace, keeping theV –A vertices of the muon traceintact. This tensor part of the electron traceTe is

Te = tr(/pe(1− γ5/se)γ

µγ αθµρ(pe − q1)ρ/q1γ

β(1− γ5))

+ tr(/pe(1− γ5/se)γ

α(1− γ5)/q1γµγ βθµρ

× (pe − q1)ρ)

(11)× g2e

16π2 ln∣∣m2

µθ∣∣

3 I.e., terms containingkη or /k, sinceθ needs to be contractedwith the electron or muon spin.

118 N. Kersting / Physics Letters B 527 (2002) 115–118

which, after some Dirac algebra, dotting into the SMmuon trace (7), and integration over the neutrinomomentaq1,2 (since these are not observed) gives

(12)|M|2 ⊃ G2Fg

2em6µ

64πln

∣∣m2µθ

∣∣(se · pe)(sµ · θpe).

The other half of the calculation, keeping the electrontrace fixed and insertingθ -dependent operators intothe muon trace, yields a very similar result. For highelectron momenta, the muon neutrino and electronantineutrino momenta are approximately opposite thatof the electron, forcing the spin of the electronto match the spin of the muon. In this case theproduct(se · pe)(sµ · θ · pe) becomes approximatelycos(φ)sin(φ) sincese ≈ sµ andθµν is antisymmetric.This upsets the cos(φ) angular dependence that theSM predicts in (2) potentially at the level of 1 partin 108.

4. Concluding remarks

It is interesting not only that noncommutativegeom-etry can account for the recent measurement ofaµ ifthe scale of noncommutativity is of the order of 1 TeV,but also that a noncommutative spacetime at this en-ergy can account forεK and possibly some of theCP violating observables inB-meson physics [15].The caveat however is thatθµν , being an intrinsicallydirectional object, is subject to being averaged away ifexperiments collect and average data over time scalesof days or longer due to the rotation of the Earth. In astorage ring such as the one at BNL, the circulation ofthe muons at their cyclotron frequency introduces anadditional averaging of the components ofθ , so someof the effects of noncommutative geometry are bound

to be projected away. Nonetheless, it is hoped that ex-perimenters will look for a time-varying effect in thedata foraµ which would be a definite positive signalof noncommutative geometry.

Acknowledgements

This work was supported by the Director, Officeof Science, Office of Basic Energy Services, of theUS Department of Energy under Contract DE-AC03-76SF0098.

References

[1] F.J.M. Farley, E. Picasso, in: T. Kinoshita (Ed.), QuantumElectrodynamics, World Scientific, Singapore, 1990.

[2] B. Lee Roberts, Int. J. Mod. Phys. A 15S1 (2000) 386.[3] H.N. Brown et al., Phys. Rev. Lett. 86 (2001) 2227.[4] A. Czarnecki, W.J. Marciano, Nucl. Phys. (Proc. Suppl.) B 76

(1999) 245.[5] http://phyppro1.phy.bnl.gov/g2muon/new_theory.html.[6] A. Connes, Noncommutative Geometry, Academic Press, New

York, 1994.[7] E. Witten, Nucl. Phys. B 268 (1986) 253;

N. Seiberg, E. Witten, JHEP 9909 (1999) 032.[8] J. Madore, An Introduction to Noncommutative Differential

Geometry and its Physical Applications, Cambridge Univ.Press, New York, 1999.

[9] M. Chaichian, A. Demichev, P. Presnajder, Nucl. Phys. B 567(2000) 360.

[10] S. Minwalla, M. Van Raamsdonk, N. Seiberg, JHEP 0002(2000) 020.

[11] M. Hayakawa, hep-th/9912167.[12] J.J.L. Hewett, F.J. Petriello, T.G. Rizzo, hep-ph/0010354.[13] M. Chaichian, M.M. Sheikh-Jabbari, A. Tureanu, Phys. Rev.

Lett. 86 (2001) 2716.[14] I.F. Riad, M.M. Sheikh-Jabbari, JHEP 0008 (2000) 045.[15] I. Hinchliffe, N. Kersting, hep-ph/0104137.[16] E.D. Commins, Weak Interactions, McGraw–Hill, New York,

1973.

http://phyppro1.phy.bnl.gov/g2muon/new_theory.html



A general-covariant concept of particles in curved background

Hrvoje Nikolic

Theoretical Physics Division, Rudjer Boškovic Institute, P.O.B. 180, HR-10002 Zagreb, Croatia


Editor: G.F. Giudice

Abstract

A local current of particle density for scalar fields in curved background is constructed. The current depends on the choiceof a two-point function. There is a choice that leads to local non-conservation of the current in a time-dependent gravitationalbackground, which describes local particle production consistent with the usual global description based on the Bogoliubovtransformation. Another choice, which might be the most natural one, leads to the local conservation of the current. 2002Published by Elsevier Science B.V.

PACS: 04.62.+v; 11.10.-z

Keywords: Current of particle density; Curved background; Particle production

One of the main problems regarding quantum fieldtheory in curved space–time is how to introduce theconcept of particles. The problem is related to the factthat particles areglobal objects in the conventional ap-proach [1,2], while covariance with respect to generalcoordinate transformations requires local objects. Thisled some experts to believe that only local operators infield theory were really meaningful and, consequently,that the concept of particles did not have any funda-mental meaning [3]. However, particles are what weobserve in experiments. Moreover, from the experi-mental point of view, nothing is more local than theconcept of particles. If we require that quantum fieldtheory describes the observed objects, then it shoulddescribe particles as local objects. In this Letter we

E-mail address: [email protected] (H. Nikolic).

show that the concept of particles can be introduced ina local and covariant manner.

Our approach is based on a similarity between thenumber of particles and charge. For complex fields,the total number of particles is the sum of the num-ber of particles and antiparticles, while the total chargeis the difference of these two numbers. The conceptof charge can be described in a local and covariantmanner because there exists a local vector current ofcharge density. We find that a similar vector currentexists for the number of particles as well. Neverthe-less, it appears that this local current is not unique,but depends on the choice of a two-point functionW(x,x ′). When a unique (or a preferred) vacuum|0〉 exists, thenW(x,x ′) is equal to the Wightmanfunction〈0|φ(x)φ(x ′)|0〉 and the current is conserved.When such a vacuum does not exist, then there is achoice ofW that leads to local non-conservation of thecurrent in a time-dependent gravitational background,

0370-2693/02/$ – see front matter 2002 Published by Elsevier Science B.V.PII: S0370-2693(02)01170-X


120 H. Nikolic / Physics Letters B 527 (2002) 119–124

which describes local particle production consistentwith the usual global description based on the Bogoli-ubov transformation. Another choice, which might bethe most natural one, leads to the local conservationof the current in an arbitrary curved background, pro-vided that other interactions are absent.

Let us start with a Hermitian scalar fieldφ in acurved background. The field satisfies the equation:

(1)(∇µ∂µ +m2 + ξR

)φ = 0,

whereR is the curvature. LetΣ be a spacelike Cauchyhypersurface. We define the scalar product

(2)(φ1, φ2)= i

∫Σ

dΣµ φ∗1

↔∂µ φ2,

where↔∂µ is the usual antisymmetric derivative. Ifφ1

andφ2 are solutions of (1), then (2) does not dependon Σ . It is convenient to choose coordinates(t,x)such thatt = constant onΣ . In these coordinates,the canonical commutation relations can be written as[φ(x),φ(x ′)]Σ = [∂0φ(x), ∂

′0φ(x

′)]Σ = 0 and

dΣ ′µ[φ(x), ∂ ′

µφ(x′)]Σ

= d3x ′ n0(x ′)[φ(x), ∂ ′

0φ(x′)]Σ

(3)= d3x ′ iδ3(x − x′).

The labelΣ denotes thatx andx ′ lie onΣ . The tildeon nµ = |g(3)|1/2nµ denotes that it is not a vector,while nµ is a unit vector normal toΣ .

Let us first construct the particle current assumingthat a unique (or a preferred) vacuum exists. The fieldφ can be expanded as

(4)φ(x)=∑k

akfk(x)+ a†kf

∗k (x),

where (fk, fk′) = −(f ∗k , f

∗k′) = δkk′ , (f ∗

k , fk′ ) =(fk, f

∗k′)= 0. The operators

(5)ak = (fk,φ), a†k = −(

f ∗k , φ

)satisfy the usual algebra of lowering and raisingoperators. This allows us to introduce the vacuum asa state with the propertyak|0〉 = 0 and the operator ofthe total number of particles as

(6)N =∑k

a†kak.

We also introduce the functionW(x,x ′), here definedas

(7)W(x,x ′)=∑k

fk(x)f∗k (x

′).

(Later we also study different definitions ofW(x,x ′).)For future reference, we derive some properties of thisfunction. For the fields described by (4), we find

(8)W(x,x ′)= 〈0|φ(x)φ(x ′)|0〉.From (7) we findW∗(x, x ′)=W(x ′, x). From the factthatfk andf ∗

k satisfy Eq. (1) we find

(9)(∇µ∂µ +m2 + ξR(x)

)W(x,x ′)= 0,

(10)(∇′µ∂ ′

µ +m2 + ξR(x ′))W(x,x ′)= 0.

From (8) and the canonical commutation relations wefind thatfk andf ∗

k are functions such that

(11)W(x,x ′)∣∣Σ

=W(x ′, x)∣∣Σ,

(12)n0∂ ′0

[W(x,x ′)−W(x ′, x)

]Σ

= iδ3(x − x′).

So far nothing has been new. However, a new wayof looking into the concept of particles emerges when(5) is put into (6) and (2) and (7) are used. This leadsto the remarkable result that (6) can be written in thecovariant form as

(13)N =∫Σ

dΣµ jµ(x),

where the vectorjµ(x) is defined as

(14)

jµ(x)=∫Σ

dΣ ′ ν 1

2

W(x,x ′)

↔∂µ

↔∂ ′ν φ(x)φ(x

′)+ h.c..

Obviously, the Hermitian operatorjµ(x) should beinterpreted as the local current of particle density. If(4) is used in (14), one can show thatjµ(x)|0〉 = 0.

Assuming that (4) is the usual plane-wave expan-sion in Minkowski space–time and that the(t,x) coor-dinates are the usual Lorentz coordinates, we find

(15)Jµ ≡∫

d3x jµ(x)=∑

k

kµ

ωka

†kak.

The quantityki/ωk is the 3-velocityvi , so we find therelationJ|nq〉 = vnq |nq〉, where|nq〉 is the state withnq particles with the momentumq . The relations of

H. Nikolic / Physics Letters B 527 (2002) 119–124 121

this paragraph support the interpretation ofjµ as theparticle current.

Note that althoughjµ(x) is a local operator, somenon-local features of the particle concept still remain,because (14) involves an integration overΣ on whichx lies. Sinceφ(x ′) satisfies (1) andW(x,x ′) satisfies(10), this integral does not depend onΣ . However, itdoes depend on the choice ofW(x,x ′). Note also thatthe separation betweenx andx ′ in (14) is spacelike,which softens the non-local features becauseW(x,x ′)decreases rapidly with spacelike separation. As can beexplicitly seen with the usual plane-wave modes inMinkowski space–time,W(x,x ′) is negligible whenthe spacelike separation is much larger than the Comp-ton wavelengthm−1.

Using (1) and (9), we find that the current (14)possesses another remarkable property:

(16)∇µjµ(x)= 0.

This covariant conservation law means that back-ground gravitational field does not produce particles,provided that a unique (or a preferred) vacuum exists.

To explore the analogy with the current of charge,we generalize the analysis to complex fields, with theexpansion

φ(x)=∑k

akfk(x)+ b†kf

∗k (x),

(17)φ†(x)=∑k

a†kf

∗k (x)+ bkfk(x).

We introduce two global quantities:

(18)N(±) =∑k

a†kak ± b

†kbk.

Here N(+) is the total number of particles, whileN(−) is the total charge. In a similar way we find thecovariant expression

(19)N(±) =∫Σ

dΣµ j(±)µ (x),

where

j (±)µ (x)=

∫Σ

dΣ ′ ν 1

2

W(x,x ′)

↔∂µ

↔∂ ′ν

(20)

× [φ†(x)φ(x ′)± φ(x)φ†(x ′)

] + h.c..

The operatorj (+)µ (x) is the generalization of (14) with

similar properties, including the non-local features.On the other hand, the apparent non-local featuresof j

(−)µ (x) really do not exist, because, by using

the canonical commutation relations, (11) and (12),j(−)µ (x) can be written as

j (−)µ (x)= iφ†(x)

↔∂µ φ(x)

(21)+∫Σ

dΣ ′ ν W(x, x ′)↔∂µ

↔∂ ′ν W(x ′, x),

so all non-local features are contained in the secondterm that does not depend onφ. Using (17), (11) and(12), one can show that the quantity−〈0|iφ†(x)×↔∂µ φ(x)|0〉 is equal to the second term, which revealsthat the second term represents the subtraction of theinfinite vacuum value of the operator represented bythe first term. In other words,j (−)

µ is the usual normalordered operator of the charge current.

In a way similar to the case of Hermitian field, wefind the propertiesj (±)

µ |0〉 = 0 and∇µj(±)µ = 0, while

the generalization of (15) is

(22)J (±)µ ≡

∫d3x j(±)

µ (x)=∑

k

kµ

ωk

(a

†kak ± b

†kbk

).

Let us now study the case in which a non-grav-itational interaction is also present. In this case, theequation of motion is

(23)(∇µ∂µ +m2 + ξR

)φ = J,

whereJ (x) is a local operator containingφ and/orother dynamical quantum fields. Since it describes theinteraction, it does not contain terms linear in quantumfields. (For example,J (x) may be the self-interactionoperator −λφ3(x).) We propose that even in thisgeneral case the currents of particles and of chargeare given by the expressions (14) and (20), whereW(x,x ′) is the same function as before, satisfyingthe “free” equations (9) and (10). As we show below,such an ansatz leads to particle production consistentwith the conventional approach to particle productioncaused by a non-gravitational interaction.

Note also that our ansatz forW makes (21) correcteven in the caseJ = 0, because the interaction doesnot modify the canonical commutation relations. Thisimplies that the charge is always conserved, provided


that the Lagrangian possesses a global U(1) symmetry.On the other hand, using (9) and (23), we find

∇µjµ(x)

(24)

=∫Σ

dΣ ′ ν 1

2

W(x,x ′)

↔∂ ′ν J (x)φ(x

′)+ h.c.,

and similarly for j (+)µ . Note that onlyx (not x ′)

appears as the argument ofJ on the right-hand sideof (24), implying thatJ plays a strictly local role inparticle production.

Let us now show that our covariant description(24) of particle production is consistent with theconventional approach to particle production causedby a non-gravitational interaction. LetΣ(t) denotesome foliation of space–time into Cauchy spacelikehypersurfaces. The total mean number of particles atthe timet in a state|ψ〉 is

(25)N (t) = 〈ψ|N(t)|ψ〉,where

(26)N(t) =∫

Σ(t)

dΣµ jµ(t,x).

Eq. (25) is written in the Heisenberg picture. How-ever, matrix elements do not depend on picture. Weintroduce the interaction picture, where the interactionHamiltonian is the part of the Hamiltonian that gener-ates the right-hand side of (23). The state|ψ〉 becomesa time-dependent state|ψ(t)〉, the time evolution ofwhich is determined by the interaction Hamiltonian.In this picture the fieldφ satisfies the free equation (1),so the expansion (4) can be used. Since we have pro-posed thatW satisfies the free equations (9) and (10),this implies thatN(t) becomes

∑k a

†kak in the inter-

action picture. Therefore, (25) can be written as

(27)N (t) =∑k

⟨ψ(t)

∣∣a†kak

∣∣ψ(t)⟩,which is the usual formula that describes particleproduction caused by a quantum non-gravitationalinteraction.

Let us now show that (14) can also describe par-ticle production by a classical time-dependent curvedbackground, provided that we take a different choicefor the functionW(x,x ′). When the particle produc-tion is described by a Bogoliubov transformation, then

the preferred modesfk in (4) do not exist. Instead,one introduces a new set of functionsul(x) for eachtime t , such thatul(x) are positive-frequency modesat that time. This means that the modesul possess anextra time dependence, i.e., they become functions ofthe formul(x; t). These functions do not satisfy (1).However, the functionsul(x; τ ) satisfy (1), providedthatτ is kept fixed when the derivative∂µ acts onul .To describe the local particle production, we take

(28)W(x,x ′)=∑l

ul(x; t)u∗l (x

′; t ′),

instead of (7). Sinceul(x; t) do not satisfy (1), thefunction (28) does not satisfy (9). Instead, we have

(29)

(∇µ∂µ +m2 + ξR(x))W(x,x ′)≡ −K(x,x ′) = 0.

Using (1) and (29) in (14), we find a relation similar to(24):

∇µjµ(x)

(30)

=∫Σ

dΣ ′ ν 1

2

K(x,x ′)

↔∂ ′ν φ(x)φ(x

′)+ h.c..

This local description of the particle production isconsistent with the usual global description based onthe Bogoliubov transformation. This is because (26)and (14) with (28) and (4) lead to

(31)N(t)=∑l

A†l (t)Al(t),

where

(32)Al(t)=∑k

α∗lk(t)ak − β∗

lk(t)a†k ,

(33)αlk(t)= (fk, ul), βlk(t)= −(f ∗k , ul).

The time dependence of the Bogoliubov coefficientsαlk(t) andβlk(t) is related to the extra time depen-dence of the modesul(x; t). If we assume that thechange of the average number of particles is slow, i.e.,that ∂tAl(t) ≈ 0, ∂tul(τ,x; t)|τ=t ≈ 0, then the Bo-goliubov coefficients (33) are equal to the usual Bo-goliubov coefficients. This approximation is nothingelse but the adiabatic approximation, which is a usualpart of the convential description of particle produc-tion [4].

H. Nikolic / Physics Letters B 527 (2002) 119–124 123

In general, there is no universal natural choice forthe modesul(x; t). In particular, in a given space–time, the choice of the natural modesul(x; t) maydepend on the observer. If different observers (thatuse different coordinates) use different modes for thechoice of (28), then the coordinate transformationalone does not describe how the particle currentis seen by different observers. In this sense, theparticle current is not really covariant. There are alsoother problems related to the case in which differentobservers use different modes [5].

Since the covariance was our original aim, it is de-sirable to find a universal natural choice ofW(x,x ′),such that, in Minkowski space–time, it reduces to theusual plane-wave expansion (7). Such a choice exists.This is the functionG+(x, x ′) satisfying (9) and (10),calculated in a well known way from the FeynmanpropagatorGF(x, x

′) [2,6]. The Feynman propaga-torGF(x, x

′), calculated using the Schwinger–DeWittmethod, is unique, provided that a geodesic connect-ing x andx ′ is chosen. Whenx andx ′ are sufficientlyclose to one another, then there is only one such geo-desic. In this case, the adiabatic expansion [2,6] of theFeynman propagator can be used. For practical cal-culations, the adiabatic expansion may be sufficientbecauseG+(x, x ′) decreases rapidly with spacelikeseparation, so the contributions from large spacelikeseparations may be negligible.

To define the exact unique particle current, we needa natural generalization ofGF(x, x

′) to the case withmore than one geodesic connectingx andx ′. The mostnatural choice is the two-point functionGF(x, x

′)defined as the average over all geodesics connectingx andx ′. Assuming that there areN such geodesics,this average is

(34)GF(x, x′)=N−1

N∑a=1

GF(x, x′;σa),

where GF(x, x′;σa) is the Feynman propagator

GF(x, x′) calculated with respect to the geodesicσa .

Eq. (34) can be generalized even to the case with acontinuous set of geodesics connectingx andx ′. Thisinvolves a technical subtlety related to the definitionof measure on the set of all geodesics, which we shallexplain in detail elsewhere.

Since the functionG+(x, x ′) calculated fromGF(x, x

′) always satisfies (9) and (10), it follows that

the corresponding particle current is conserved anddoes not depend on the choice ofΣ , provided thatthe field satisfies (1). Note that this definition of parti-cles does not always correspond to the quantities de-tected by “particle detectors” of the Unruh–DeWitttype [7,8]. Instead, this number of particles is deter-mined by a well-defined Hermitian operator that doesnot require a model of a particle detector, just as isthe case for all other observables in quantum mechan-ics. Moreover, since (14) does not require a choice ofrepresentation of field algebra, the definition of parti-cles based onGF does not require the choice of rep-resentation either. This allows us to treat the particlesin the framework of algebraic quantum field theory incurved background [9]. Furthermore, as noted by Un-ruh [7], only one definition of particles can correspondto the real world, in the sense that their stress-energycontributes to the gravitational field. Since the defini-tion of particles based onGF is universal, unique andreally covariant, it might be that these are the parti-cles that correspond to the real world. Besides, as weshall discuss elsewhere, it seems that these particlesmight correspond to the objects detected by real de-tectors (such as a Wilson chamber or a Geiger–Müllercounter) in real experiments.

In this Letter we have constructed the operatordescribing the local current of particle density, whichallows us to treat the concept of particles in quantumfield theory in a local and generally covariant manner.This operator is not unique, but depends on the choiceof the two-point functionW(x,x ′). Different choicescorrespond to different definitions of particles. Inparticular, various choices based on (28) correspondto particle production by the gravitational field. Thislocal description of particle production is consistentwith the conventional global description based on theBogoliubov transformation. Similarly, various choicesbased on (7) give a local description of particle contentin various inequivalent representations of field algebra.

We have seen that a particularly interesting choiceof W(x,x ′) is that based on the Feynman propagatorGF(x, x

′), because this choice might correspond to themost natural universal definition of particles for quan-tum field theory in curved background. It is tempt-ing to interpret these particles as real particles. If thisinterpretation is correct, then classical gravitationalbackgrounds do not produce real particles. There arealso other indications that classical gravitational back-


grounds might not produce particles [10–12]. Even ifthe particles based onGF do not correspond to realphysical particles in general, it is interesting to askabout the physical meaning of this time-independentHermitian observable that corresponds to physical par-ticles at least in Minkowski space–time.

Note finally that our definition of the particle cur-rent can be generalized to the case of scalar andspinor fields in classical electromagnetic backgrounds.Again, a two-point function can be chosen such thatthe particle production is described in a way consistentwith the Bogoliubov-transformation method, but therequirement of gauge invariance leads to a conservedcurrent, indicating that classical electromagnetic back-grounds also might not produce particles. Only thequantized electromagnetic field (having a role simi-lar to J (x) in (23)) may cause particle production inthis formalism, which is in agreement with some otherresults [12]. This will be discussed in more detail else-where.

Acknowledgements

This work was supported by the Ministry of Sci-ence and Technology of the Republic of Croatia underContract No. 00980102.

References

[1] S.A. Fulling, Phys. Rev. D 7 (1973) 2850.[2] N.D. Birrell, P.C.W. Davies, Quantum Fields in Curved Space,

Cambridge Press, New York, 1982.[3] P.C.W. Davies, in: S.M. Christensen (Ed.), Quantum Theory of

Gravity, Hilger, Bristol, 1984.[4] L. Parker, Phys. Rev. 183 (1969) 1057.[5] H. Nikoli c, Mod. Phys. Lett. A 16 (2001) 579.[6] B.S. DeWitt, Phys. Rep. 19 (1975) 295.[7] W.G. Unruh, Phys. Rev. D 14 (1976) 870.[8] B.S. DeWitt, in: S.W. Hawking, W. Israel (Eds.), General

Relativity: an Einstein Centenary Survey, Cambridge Univ.Press, 1979.

[9] R.M. Wald, Quantum Field Theory in Curved Spacetime andBlack Hole Thermodynamics, University of Chicago Press,Chicago, 1994.

[10] T. Padmanabhan, Phys. Rev. Lett. 64 (1990) 2471.[11] V.A. Belinski, Phys. Lett. A 209 (1995) 13.[12] H. Nikolic, hep-th/0103053;

H. Nikolic, hep-th/0103251;H. Nikolic, hep-ph/0105176.



Noncommutative field theory in formalism of first quantization

A.Ya. Dymarsky

ITEP and MSU, Moscow, Russia

Received 17 October 2001; received in revised form 21 December 2001; accepted 4 January 2002

Editor: L. Alvarez-Gaumé

Abstract

We present a first-quantized formulation of the quadratic noncommutative field theory in the background of abelian (gauge)field. Even in this simple case the Hamiltonian of a propagating particle depends non-trivially on the momentum (since externalfields depend on location of the Landau orbit) so that one cannot integrate out momentum to obtain a local theory in the second-order formalism. The cases of scalar and spinning particles are considered. A representation for exact propagators is found andthe result is applied to description of the Schwinger-type processes (pair-production in homogenous external field). 2002Published by Elsevier Science B.V.

1. Introduction

The noncommutative (NC) fields theory have been extensively studied recently. However, most of considerationwere based on the second-quantizes field theory. Here we are going to study the NC theory within the firstquantization approach.

More concretely we claim that the unique difference between ordinary and NC theories in the formalism of firstquantization is that the external fields in the NC case depend on “shifted” coordinatesqµ = xµ − 1

2θµνpν , which

are the counterparts of coordinates of the Landau orbit for the charged particle in the magnetic field. Herepµ arethe momenta,xµ are the coordinates of the particle andθµν is the parameter of noncommutativity (the constantantisymmetric tensor). We call the coordinatesqµ noncommutative despite they arec-numbers, since their Poissonbracket, is non-zero so that they generate along withpµ the algebra

qµ, qν = iθµν, pµ,pν = 0, qµ,pν = iδµν .

At the same time,xµ are the ordinary coordinates with canonical Poisson brackets

xµ, xν = 0, pµ,pν = 0, xµ,pν = iδµν .

We consider the theories quadratic in quantum fields, since only such theories have first-quantized formulation.

E-mail address: [email protected] (A.Ya. Dymarsky).



126 A.Ya. Dymarsky / Physics Letters B 527 (2002) 125–130

This Letter is organized as follows. We begin with verifying our claim in the case of scalar particle in the nextsection. In Section 3 we apply our results to the pair-production process. Section 4 constructs the first-quantizedtheory of spinning particle. At last we end up with concluding remarks in Section 5.

2. Scalar particle

In this section we consider correspondence between the first and second quantized description of the of scalarparticle in the example of the scalar QED with external currents. The main statement is that the first-quantizedtheory with the action (in the first order formalism)

(1)S1 =∫dt(px − (p −A(q))2 − J (q)+m2), qµ = xµ − 1

2θµνpν,

corresponds to the second quantized theory with the action

(2)S2 =∫dDx

(Dµφ

+ ∗Dµφ −m2φ+ ∗ φ)+ J ∗ φ+ ∗ φ, Dµ = ∂µ − iAµ ∗ .Here∗ is the Moyal product [1] with the parameter of noncommutativityθµν . We omit the 1-dimensional metric

along the trajectory from the action (1). It means that we already fixed the gauge:t is the natural parameter alongthe trajectory andt ∈ [0, T ], whereT is the proper time (the length of trajectory). Then, of the whole path integralover 1-dimensional metric, we are left with an ordinary integral over the proper time with the measuredT (fordetails see [2]).

We prove the correspondence between the theories identifying the exact propagators in the first and secondquantized theories. The propagators will be obtained as perturbative series in both cases. The exact propagator forthe first-quantized theory is given by the formula

(3)G(x,y)=∞∫

0

dT

x(T )=x∫x(0)=y

Dx(t)Dp(t) eiS .

We demonstrate by manifest checking that the perturbative series in powers of external fields from propagator (3)with action (1) coincides with the field series. In other words, we obtain the Feynman rules for the field theory (2)from the particle theory (1). (The perturbative theory in the formalism of first quantization is described in [3]. Therelativistic case and the correspondence to the field perturbation theory were discussed in [4].)

First we represent the path integral (3) as the limit of ordinary integrals

(4)G(x,y)≡∞∫

0

dT eim2T G(x, y,T ), G(x, y,T )≡ lim

N→∞

(N−1∏i=1

dxi G(xi+1, xi)T/N

)G(x1, y)T/N .

G(x, y)T is as usual defined up to the first order inT only

G(x,y)T ≡∫

dDp eip(x−y)

(2π)D(e−ip2T + i

(Aµ(x

′)pµ +Aµ(y′)pµ −Aµ(x

′)Aµ(y′)− J (x ′)

)T

),

(5)x ′µ = xµ − 1

2θµνpν, y ′µ = yµ − 1

2θµνpν.

The pointsx ′ andy ′ are split in such a way that in the limitθµν → 0, the expression (5) becomes the ordinary oneknown in the first-quantized scalar QED.

Note that the parameter of noncommutativityθµν is contained only in terms, describing the interaction withexternal fields. Therefore, the free propagator obtained from (3) with the action (1) coincides with the ordinary

A.Ya. Dymarsky / Physics Letters B 527 (2002) 125–130 127

one. This is why in the NC theory the internal lines remain ordinary. This fact is in coincidence with the fieldtheory result.

Now we begin to consider the vertex part of Feynman diagrams. To this end, we need the following key formula,which could be proved from the definition of the Moyal product,

(6)〈x|f ∗ |y〉 =∫

dDp

(2π)Dei(p(x−y))f

(xµ + α(y − x)µ − 1

2θµνpν

)for arbitraryα ∈ [0,1]. Note that the dependence onα vanishes in (6) after the integration overp. The crucial pointhere is thatθµν is constant and (6) has the very simple form. Yet another useful formula, which follows from theprevious one, is

(7)〈x|(f ∗ g) ∗ |y〉 =∫

dDp

(2π)Dei(p(x−y))f

(xµ − 1

2θµνpν

)g

(yµ − 1

2θµνpν

).

It is evident now that the vertex term (formula (5) without the terme−ip2T which describes free propagation)corresponds to the NC theory∫

dDp

(2π)Dei(p(x−y))i

(Aµ(x

′)pµ +Aµ(y′)pµ −Aµ(x

′)A(y ′)µ − J (x ′))T

(8)= −i〈x|2i(Aµ∗)∂µ + i(∂µAµ) ∗ +(Aµ ∗Aµ) ∗ +J ∗ |y〉.It is also obvious now how to construct for arbitrary scalar theory the first-quantized formulation from the

second-quantized one.

3. Schwinger type processes

One cannot calculate the exact propagator for arbitrary external field. However, when the external field ishomogenous it is possible. In this section we calculate the probability for the pair production (the imaginary part ofthe propagator) in this the background in the formalism of first quantization and compare our result with the fieldtheory calculation.

Let us consider a particle on the noncommutative plane in the classical external fieldAµ(q) = 12Bνµq

ν withhomogenous field’s strengthBµν = const andJ = 0. In this case, one can easy calculate the probability of pairproduction since the action (1) is quadratic in all variables. It is convenient to change variables fromp,q to thecanonicalπ,y

(9)πµ = pνKνµy

µ = xν(K−1)µ

ν, Kµ

ν = δµν + 1

4Bσµθ

σν,

wherexµ = qµ + 12θ

µνpν . In the new variables, the action can be rewritten as

(10)S =∫dt(πµy

µ − (πµ − Aµ(y))2 +m2), A(y)µ =Aµ(Ky)= 1

2BσµK

σν y

ν.

It is remarkable that this action corresponds to an ordinary particle in external field with the strength

(11)Fµν = ∂Aν

∂yµ− ∂Aµ

∂yν.

Now one can calculate the probability of pair creation in the usual way. The result (gauge invariant quantity) willdepend only onFµν . At the same time,Fµν changes under gauge transformations

(12)Aµ → g ∗Aµ ∗ g+ − i∂µg ∗ g+, g+ ∗ g = 1.


The solution of this puzzle is that the NC strength

(13)Fµν = ∂Aν

∂qµ− ∂Aµ

∂qν− i[Aµ,Aν]∗

is still gauge invariant for homogenous fields. Moreover (a little surprise),Fµν andFµν are equal to each other

(14)Fµν = Fµν = Bµν + 1

4θσρBσµBρν.

Therefore, the probability is given by the standard formula (see [5]) but with noncommutative field strengthFµν .For example, in the 4-dimensional space, when the magnetic field vanishes (Fi,j = 0, i, j = 1,3), the probabilityis

(15)w = e2| E|28π3

∞∑n=1

(−1)n+1

n2exp

(−πnm2

|e E|), eEi = F0i .

This result certainly coincides with the field theory calculation (see [1,6]).

4. Spinning particle

Now we are going to construct the exact propagator for the spinning particle in the NC case similar to what wedid below.

First, we rederive the analogue of formula (3) for the spinning particle and calculate the first correction inexternal fields to the exact propagator in the ordinary (commutative) theory. In this section we work in the Euclideanspace for simplicity. We start from another representation for the propagator of scalar particle (see [2])

(16)G(p2,p1)=∞∫

0

dT e−mT

(2π)2D

∫Dx(t) eip2x(T )δ(x2 − 1)e

∫Axe−ip1x(0).

In further consideration we mainly follow the approach of spin quantization developed in [7–9] (see also [10]). Inaccordance with their approach, in order to quantize the spin one needs to add a termeiSspin to (16). Note that thedelta-functionδ(x2−1) keeps the vectorxµ in the sphere. Thus, in fact, in (16) one sums over the trajectories lyingon the sphere (this fact is important). The termSspin is the integral of a special external 1-form connected with theSO(D) group, along such a trajectory. Note that this sphere is the configuration space for the particle’s velocitynµ = xµ (or the phase space for the spin), not for the particle. In order to distinguish these spaces, we denote thespace-time trajectory viax(t), and “spin” space trajectory viaη(t). We also use the convenient variablesnµ = ηµ,Dn=Dηδ(η2 − 1). Then the exact propagator for spinning particle has the form

G(p2,p1)=∞∫

0

dT e−mT

(2π)2D

∫Dη(t) eip2η(T )δ(η2 − 1)eiSspine

∫Aµη

µ

e−ip1η(0)

(17)=∞∫

0

dT e−mT

(2π)2D

∫η(0)=0

Dη(t) eip2∫ T

0 η dtδ(η2 − 1)eiSspine∫Aµη

µ

δ(p2 − p1).

In this case, the perturbative theory is more complicated than in the scalar case. This is why we first demonstratein detail how the vertex terms emerge from (17). The first correction inAµ is

A.Ya. Dymarsky / Physics Letters B 527 (2002) 125–130 129

G1(p2,p1)= limt→0

∫Γ

dT dt e−mT

(2π)2D

∫Dη(t) eip2ηT Aµ(ηt )(ηt ′ − ηt )

µδ(η2 − 1)eiSspine−ip1η0,

(18)t ′ = t +t, Γ : T ∈ [0,∞], t ∈ [0, T ].In order to transform it to the standard form, we use the following important trick: we multiply (18) by unity

(19)1=∫

dη′t ′ dk2

(2π)Ddη′

t dk1

(2π)Deik2(ηt ′−η′

t ′ )eik1(ηt−η′t ),

and rewrite it as

G1(p2,p1)= limt→0

∫Γ

dT dt e−mT

(2π)2Ddk1dk2dη

′t dη

′t ′

(2π)2D

(20)

×∫Dηe

i(∫p2η+η′

t ′ (p2−k2)+k2ηt ′ )Aµ(η′t )(η

′t ′ − ηt )

µei(−k1η′t+∫k1η+η0(k1−p1))δ(η2 − 1)eiSspin.

We have to removeeik2(ηt ′−η′t ) from formula (20), analogously we removee−ip2T from (8) before taking the

continuum limitt → 0. Only after this, the first correction acquires the form corresponding to QED

(21)G1(p2,p1)=∫dk1dk2dηG

0(p2, k2)eik2ηAµ(η)γ

µe−ik1ηG0(k1,p1).

The important moment is that we remove here the integral∫Dηδ(η2 − 1)gamma-matricesγ µ (see the works

[7] and [10]).Now we demonstrate how to deform the propagator (17) in the NC case. The main difficulty is that in formula

(17) we sum over the trajectories in the special “spin” space, when the particle propagates straightforwardly(p = const). However, the terms which depend onqµ need to be integrated over the trajectories in the “space-time” phase space. This is why if we want to change the argument of external fields in (17), similarly to (1), wehave to add in (17) the sum over trajectories in such a space. We can do this in the following way: represent thepath integral as the continuum limit of ordinary integrals similarly we did with (3). After that, multiply (17) byunity

(22)1=∫

dx dp

(2π)Deip(η−x)

for all η(t). It is evident that the dependencies of external fields onη or on x are equivalent. Then, we integrateout the delta-function and pass from integrating overη to integrating overn. (Note that in a similar way one canobtain (3) with the commutative (θµν = 0) action (1) from (16), by adding new degrees of freedom.) After that,we change the arguments of all external fields fromxµ to qµ. Since we work with propagator in the momentumrepresentation, the boundary conditions allow us to change the integration over variablesxµ to the integration overvariablesqµ. Finally, one obtains

(23)G=∞∫

0

dT e−mT∫Dq(t)Dp(t)Dn(t) ei

∫ T0(p dq+ 1

2pθ dp)ei∫ T

0 pndte∫ T

0 Aµ(q)nµ dt eiSspin[n(t)].

The perturbation series for this formula corresponds to the NCQED. For example, the first correction to thepropagator has the form

G1(p2,p1)=∫dk1dk2dx G

0(p2, k2)eik2qAµ(q)γ

µei(−k1q+k1θ(k2−k1))G0(k1,p1)

(24)=∫dk1dk2dq G

0(p2,k2)eik2q ∗Aµ(q)γ

µ ∗ e−ik1qG0(k1,p1).


It is obvious that we have to adde∫ T

0 J (q) dt to ((23) in order to obtain the first-quantized formulation of NCQEDwith external current

(25)S2 =∫dDx

(ψ ∗ (Dµγ

µ + J ) ∗ψ).5. Concluding remarks

We demonstrated that quadratic noncommutative field theory can be described in terms of particles similarly tothe ordinary case. However, the particle action depends on the momentum nontrivially and the theory is non-local.This description is also available in the case of arbitrary (non-constant) parameter of noncommutativity, but in thiscase the analogue of formula (6) and thus the results (1) and (23) become very complicated.

We also constructed the exact propagators in the case of scalar and spinning particle in the background ofclassical abelian gauge field and current. The result is applied to the pair-production processes.

Acknowledgements

Author is grateful to A. Gorsky and K. Selivanov for initiating this work and discussions, to A. Alexandrov,D. Melnikov and A. Solovyov for useful advices and especially to A. Mironov and A. Morozov for careful readingthe manuscript and support.

This work was partly supported by the Russian President’s grant 00-15-99296, INTAS grant 01-334 and RFBRgrant 01-02-17682-a.

References

[1] L. Alvarez-Gaumé, J.L.F. Barbon, hep-th/0006209.[2] A.M. Polyakov, Gauge Fields and Strings, Harwood Academic, Chur, 1987.[3] R.P. Feynman, A.R. Hibbs, Quantum Mechanics and Path Integrals, McGraw–Hill, New York, 1965.[4] Yu.M. Makeenko, A.A. Migdal, Nucl. Phys. B 188 (1981) 269.[5] J. Schwinger, Phys. Rev. 82 (1951) 914.[6] N. Chair, M.M. Sheikh-Jabbari, hep-th/0009037.[7] A.Yu. Alekseev, S.L. Shatashvili, Mod. Phys. Lett. A 3 (16) (1988).[8] A.Yu. Alekseev, L. Faddeev, S. Shatashvili, J. Geom. Phys. 5 (1989) 391.[9] H.B. Nielsen, D. Rohrlich, Nucl. Phys. B 299 (1988).

[10] A.M. Polyakov, Mod. Phys. Lett. A 3 (1988) 325.



Supersymmetric D2–anti-D2 strings

Dongsu Baka, Nobuyoshi Ohtab

a Physics Department, University of Seoul, Seoul 130-743, South Koreab Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan


Editor: T. Yanagida

Abstract

We consider the flat supersymmetric D2 and anti-D2 system, which follows from ordinary noncommutative D2–anti-D2branes by turning on an appropriate worldvolume electric field describing dissolved fundamental strings. We study the stringsstretched between D2 and anti-D2 branes and show explicitly that the would-be tachyonic states become massless. We computethe string spectrum and clarify the induced noncommutativity on the worldvolume. The results are compared with the matrixtheory description of the worldvolume gauge theories. 2002 Elsevier Science B.V. All rights reserved.

1. Introduction

Recently an interesting class of 1/4 BPS brane configurations has been constructed including the supertubesand the supersymmetric brane–antibrane system [1–5]. The supertubes are tubular configurations embedded in flat10-dimensional space and self supported from collapse by the contribution of angular momentum produced by theirown worldvolume gauge fields [1]. Many tubes separated parallelly are again 1/4 BPS and there is no static forcebetween them, which is shown in the matrix theory description [2,4] or the supergravity analysis [3]. The tubesinvolve acritical electric field1 as well as a magnetic field, which respectively correspond to dissolved fundamentalstrings and D0-branes. The tubes have no net D2-brane charge. Instead they carry nonvanishing dipole componentsof D2 charges [3,4].

The elliptic deformation of tubes discussed in Ref. [5] still preserves 1/4 supersymmetries and, in the limitwhere the ellipse becomes two parallel lines, the system becomes a flat D2 and anti-D2 system which issupersymmetric. (One may actually show that there are many other supersymmetric solutions with various shapeslike hyperbola, for example.) As long as theE-fields are critical andB-fields come with opposite sign, thesupersymmetries satisfied by the brane turn out to be the same as those of the antibrane. By the matrix theorydescription of the worldvolume theory, one can indeed prove that the would-be tachyons in the ordinary D2 andanti-D2 system withB-fields disappear in the supersymmetric case [5]. Thus the system is stable against decay.In the corresponding supergravity solutions, the D2 and anti-D2 can easily be identified separately when they are

E-mail addresses: [email protected] (D. Bak), [email protected] (N. Ohta).1 The critical value of the electric field here does not mean the tensionless limit due to the nonvanishing magnetic components.



132 D. Bak, N. Ohta / Physics Letters B 527 (2002) 131–141

separated. One might ask what happens when D2 and anti-D2 are brought together and become coincident. Clearlyin this limit the dipole moment of the D2 charges becomes zero and massless modes are expected to appear betweenD2 and anti-D2. When D2 and anti-D2 are on top of each other, should we view that they disappear to nothing? Inthe matrix model description, however, nothing particular happens other than the massless degrees of freedom andthe corresponding nonabelian structure due to brane and antibrane remains.

In this Letter, we would like to probe the coincident limit of the supersymmetric D2 and anti-D2 system bystudying the worldsheet CFT of strings connecting the D2 and anti-D2. WithoutE-field the tachyonic modesof D2–anti-D2 strings are computed in Ref. [6] and shown to agree with those of the matrix theory description[6–8] in the zero slope limit. We begin with a generic value ofE-field and show how would-be tachyonic modesdisappear in the string spectrum when theE-field approach to the critical value. The limit we are taking is notthat straightforward; the coordinate fieldsX become singular in their overall factors. However, the modes and theircorrelation functions do not share these apparent singularity, so the worldsheet CFT is well defined in the limit inspite of the apparent coordinate-like singularity. We shall clarify the noncommutativity arising in the low-energyworldvolume theories by the analysis of the commutators of the coordinate fields. We also compute the fluctuationspectra from the matrix theory and show that a consistent picture emerges as a consequence.

In Section 2, we study the mode expansion and spectrum of the D2–anti-D2 strings as well as D0–D2 stringspectrum. In Section 3, we investigate the noncommutativity induced by the background gauge field on D2 andanti-D2 and discuss the Seiberg–Witten limit. We compare the above results to those of the matrix theory analysisfocused on the fluctuation spectra. Last section comprises conclusions and remarks.

2. D2–anti-D2 strings

We are interested in the D2–D2 system extended to 1, 2 directions, which is supersymmetric due to the presenceof the specific backgroundE- andB-field [5]. For the supersymmetric configurations, the electric component, e.g.,E = B20 should be critical and the magnetic partB = B12 should come with opposite signs on D2 and anti-D2. Webegin with generic values of electric components and take the critical limit, i.e.,E = 1/(2πα′). Also we takeB tohave the same magnitude on the D2 and anti-D2. SinceB comes with opposite signatures on the D2 and anti-D2,only D0-branes (notD0) are induced on them with the same densities. The same magnitude condition ofB-fieldmay be relaxed as far as they are nonvanishing on the D2 and anti-D2.

Let us examine the open string spectrum between the D2 and anti-D2. The background field atσ = 0 isspecifically taken as

(1)B(0) = 1

2πα′

(0 0 −e0 0 −be b 0

), gij = ηij ,

andB(π) atσ = π is obtained by reversing the sign ofb. The boundary condition is given as

(2)gij ∂σXj + 2πα′Bij ∂tXj = 0.

In our case of D2 atσ = 0 and anti-D2 atσ = π , this is written as

∂σX0 + e∂tX2 = 0,

∂σX1 ∓ b∂tX2 = 0,

(3)∂σX2 + e∂tX0 ± b∂tX1 = 0, atσ =

0,π.

The boundary conditions for fermions are similar and the mode oscillators can be easily obtained once the bosonicpart is found.

D. Bak, N. Ohta / Physics Letters B 527 (2002) 131–141 133

With the specified boundary conditions, we solve the equation of motion and find

X0 = i√

2α′1− e2

∑n=0

an

ne−int cosnσ + 1√

1− e2

(2α′p0t + x0)

− e√

α′1− e2

∑(cn+νn+ ν e

−i(n+ν)t−i πν2 + d−n−νn+ ν e

i(n+ν)t+i πν2)

sin

[(n+ ν)σ − πν

2

],

X1 = √α′∑(

cn+νn+ ν e

−i(n+ν)t−i πν2 + d−n−νn+ ν e

i(n+ν)t+i πν2)

cos

[(n+ ν)σ − πν

2

]+ x1,

(4)

X2 = − e√

2α′1− e2

∑n=0

an

ne−int sinnσ − e√

1− e22α′p0

(σ − π

2

)+ x2

+ i√

α′1− e2

∑(cn+νn+ ν e

−i(n+ν)t−i πν2 − d−n−νn+ ν e

i(n+ν)t+i πν2)

cos

[(n+ ν)σ − πν

2

],

whereν is defined by

(5)tanπν

2= b√

1− e2(0 ν < 1).

Here cn+ν ’s are complex oscillator and the reality condition of the mode expansion says thatan = a†−n and

cn+ν = d†−n−ν .

This result shows thatν = 1 for e= 1. This implies that the level becomes integer, which is in accordance withthe supersymmetry restoration in the limite= 1. Furthermore if one takes the limite→ 0, Eq. (4) reduces to thosein Ref. [6] for the ordinary noncommutative D2 and anti-D2 system which is of course unstable. Though thee→ 1limit for X0 andX2 looks singular in Eq. (4), there is no singularity in the worldsheet CFT. In fact, the modeexpansions perfectly make sense in the combinations(

∂σX0 + e∂tX2)/√1− e2,

(e∂tX

0 + ∂σX2)/√1− e2,

(6)(e∂σX

0 + ∂tX2)/√1− e2,(∂tX

0 + e∂σX2)/√1− e2,

even in the limit. This is just like considering, in CFT,∂zX, which has good conformal property. Alternatively, onemay explicitly compute the correlation functions〈Xi(z)Xj (z′)〉 and verify that they are nonsingular in the limite→ 1.

Upon quantization, we posit the nonvanishing commutation relations between oscillators to be

[am,an] = −mδm+n (m = 0),

(7)[x0,p0]= −i, [cm+ν, dn−ν] = (m+ ν)δm+n.

The justification of the quantization here is relegated to the appendix. One thing to note is that we have not yetspecified the commutation relation betweenxi ’s and these will be fixed in the next section by requiring the equal-time commutators ofXi(σ) to commute with each other for 0< σ < π . The corresponding Virasoro generatorscan be found as

(8)Lm = 1

2

∑n

(−am−nan + cm−n+νdn−ν + dn−νcm−n+ν )

with L0 being the Hamiltonian of the system.


We define vacuum bycn+ν |0〉 = 0 for n 0 anddn−ν |0〉 = 0 for n > 0. Hence the corresponding vacuumenergy fromL0 becomes

(9)Eν = −1

2

∞∑n=1

n+ 1

2

∞∑n=1

(n− ν)+ 1

2

∞∑n=1

(n− (1− ν))= + 1

24+(

1

24− 1

8(2ν − 1)2

).

Note that this is the contribution to the vacuum energy fromX0, X1, X2.The vacuum energy for the Ramond sector is trivial; the bosonic one is canceled by the fermionic contribution

and the total is zero. In the Neveu–Schwarz sector, the vacuum energy from the NS-fermions associated with 0,1,2components is given by the substitutionν → |ν − 1/2|. In the “light-cone” gauge, the contributions froman andone transverse oscillator cancel with each other. Summing the contributions of the rest of oscillators, we find theground state energy

Etotalν =

(1

24− 1

8(2ν − 1)2

)−(

1

24− 1

8(2|ν − 1/2| − 1)2

)− 6

24− 6

48

(10)= −1

4− |ν − 1/2|

2.

For ν = 0, this is consistent with the vacuum energy of NS sector without background field.The ground state energy (10) gives two lower states with energies

E1 = −1

2ν, for 1 ν 1

2,

(11)E0 = 1

2(ν − 1), for

1

2 ν 0.

When ν = 0, |E0〉 gives the true ground state and|E1〉 is the first excited state, but the energy changes whenν is increased. Forν 1

2, |E1〉 becomes the true ground state and|E0〉 is the first excited state. Forν = 0 andD2–D2, the ground state|E0〉 is projected out by GSO projection and|E1〉 is kept. However, our D2–D2 systemhas opposite GSO projection, and the state withE1 is projected out and|E0〉 is kept, giving tachyonic state. Thisstate becomes massless forν = 1, giving a stable system.

Let us consider what would be the spectrum at lower levels. The ground state is denoted by| − ν/2〉.Corresponding to the mode oscillatorscn+ν and d−n−ν , we have fermionic oscillatorsψn+ν− 1

2and ψ−n−ν− 1

2.

Forν 12, ψ−ν+ 1

2,ψν− 3

2and transverse oscillatorsψs− 1

2(s = 3, . . . ,8) give lower states (which remain after GSO

projection)

(12)∣∣(ν − 1)/2

⟩≡ ψ−ν+ 12| − ν/2〉, ∣∣3(1− ν)/2⟩≡ψν− 3

2| − ν/2〉, ∣∣(1− ν)/2⟩s ≡ψs− 1

2| − ν/2〉.

Forν < 1, the first state is the ground state discussed above and gives tachyonic one. All these states give 8 masslessfor ν = 1, and the level is degenerate with Ramond sector, in accordance with the restoration of supersymmetry.In Ref. [5], it was argued that 1/4 supersymmetry is restored. Our result of 8 massless bosonic states is consistentwith this claim because the representation of 8 supercharges contains 24 states in total, half of which are bosonic.

It is interesting that the limitν = 1 can be achieved just by sendinge→ 1 butb finite. In the absence of electricbackground,b must be sent to infinity in order to achieve supersymmetry. However, that is incompatible with theSeiberg–Witten limit to be discussed in the next section.

The physical picture of the system can be most easily understood by combining T-duality [5,9,10] and Lorentzboost. Let us make T-duality in theX2-direction, and then make the Lorentz boost in the same direction by

(13)

(X0 ′X2 ′

)=(

coshβ sinhβsinhβ coshβ

)(X0

X2

), coshβ = 1√

1− e2, sinhβ = e√

1− e2.


The boundary conditions (3) are then cast into

∂σX0 ′ = 0,

∂σ(√

1− e2X1 ∓ bX2 ′)= 0,

(14)∂t(bX1 ±

√1− e2X2 ′)= 0, atσ =

0,π.

This means that the system consists of two D1-branes, one tilted in the(X1,X2 ′) plane by the angleπ2 ν at σ = 0and another by−π

2 ν at σ = π . There is no supersymmetry in the tilted two D1-branes. In the limitν → 1,however, this system reduces to boosted parallel branes, and supersymmetry is restored. The number of restoredsupersymmetry cannot be determined by simply looking at these boundary conditions, as is usual for rotated braneconfiguration. It was determined 1/4 by Killing spinor analysis [5].

We would like to comment upon the 0–2 string spectra at this point. We consider D0 atσ = 0 and D2 atσ = π ,for which the boundary conditions become

(15)

∂σX

0 = 0,∂tX

2 = 0,∂tX

1 = 0, atσ = 0,

∂σX

0 + e∂tX2 = 0,∂σX

1 − b∂tX2 = 0,∂σX

2 + e∂tX0 + b∂tX1 = 0, atσ = π.For this case, we find

X0 = i√

2α′(1− e2)

∑n=0

an

ne−int cosnσ + 1√

1− e2

(2α′p0t + x0)

+ e√

α′1− e2

∑(cn+ν ′

n+ ν′ e−i(n+ν ′)t + d−n−ν ′

n+ ν′ ei(n+ν ′)t

)cos(n+ ν′)σ,

X1 = √α′∑(

cn+ν ′

n+ ν′ e−i(n+ν ′)t + d−n−ν ′


)sin(n+ ν′)σ + x1,

(16)

X2 = − e√

2α′1− e2

∑n=0

an

ne−int sinnσ − e√

1− e22α′p0

(σ − π

2

)+ x2

+ i√

α′1− e2

∑(cn+ν ′

n+ ν′ e−i(n+ν ′)t − d−n−ν ′


)sin(n+ ν′)σ,

whereν′ is defined by

(17)e2iπν ′ = −√

1− e2 + ib√1− e2 − ib = eiπ(ν+1) (1/2 ν′ < 1).

Then the expressions for the commutation relation between oscillators, the Hamiltonian, the ground state, theexcited spectrum take the same forms as before. So the ground states after GSO projection give 8 massless bosonicdegrees again but with the newly definedν′.

3. Noncommutativity and Seiberg–Witten limit

We shall compute here the noncommutativity induced due to the background gauge fields of the D2–anti-D2system and investigate the Seiberg–Witten decoupling limit involved with the system.


For this let us first compute the equal-time commutators ofXi(σ). Using the commutation relations in (7), wehave

(18)[X1(σ ),X2(σ ′)

]= [x1, x2]− 2iα′

√1− e2

∑n

1

n+ ν cos

((n+ ν)σ − πν

2

)cos

((n+ ν)σ ′ − πν

2

).

To evaluate this, we use the identities∞∑

n=−∞

cosnθ

n+ a = π

sinaπcos(a(2m+ 1)π − aθ) for 2mπ θ 2(m+ 1)π,

(19)∞∑

n=−∞

sinnθ

n+ a =

πsinaπ sin

(a(2m+ 1)π − aθ) for 2mπ < θ < 2(m+ 1)π,

0 for θ =mπ,with an integerm anda = 0. One then finds that

(20)∑n

1

n+ ν cos

((n+ ν)σ − πν

2

)cos

((n+ ν)σ ′ − πν

2

)= π cot

νπ

2

1 if 0< σ + σ ′ < 2π,cosνπ if σ = σ ′ = 0,π.

By requiring the bulk contribution to be zero, we fix the commutator[x1, x2] as

(21)[x1, x2]= iα′π

b.

The commutation relation becomes

(22)[X1(σ ),X2(σ ′)

]∣∣σ=σ ′=0,π = i 2πα′b

1+ b2 − e2,

while vanishing in the bulk.For the 0–2 commutator, we get

[X0(σ ),X2(σ ′)

]= 1√e2 − 1

[x0, x2]+ i 2α′e

1− e2 (σ′ − π/2)+ i 2α′e

1− e2

∑n=0

1

ncosnσ sinnσ ′

(23)+ i 2α′e1− e2

∑n

1

n+ ν sin

((n+ ν)σ − πν

2

)cos

((n+ ν)σ ′ − πν

2

).

In addition to (19), we shall use also the identity

(24)∑n=0

sinnθ

n=π − θ for 0< θ < 2π,−π − θ for − 2π < θ < 0,0 for θ = 0.

The straightforward evaluation leads to

(25)[X0(σ ),X2(σ ′)

]∣∣σ=σ ′=0 = −[X0(σ ),X2(σ ′)

]∣∣σ=σ ′=π = −i 2πα′e

1+ b2 − e2 ,

where [x0, x2] = 0 is chosen such that there is again no bulk contribution to the commutator. Finally, with[x0, x1] = 0, [X0(σ ),X1(σ ′)] = 0 including boundaries. Hence we see that the end points of the string becomenoncommutative. In particular, the contribution of[X1(π),X2(π)] has the opposite signature to the one given inRef. [11]. This is because they consider strings ending on D-branes with the sameb while we are considering herestrings from D2 withb ending on anti-D2 with−b.

The result can be neatly written as

(26)[Xi(σ),Xj (σ ′)

]= iΘij,(σ )εσ,σ ′,


whereΘij,(σ ) is the Seiberg–Witten expression of noncommutativity

(27)Θij,(σ ) = 2πα′(

1

g + 2πα′B(σ)

)ijA

= −(2πα′)2(

1

g + 2πα′B(σ)B(σ)

1

g− 2πα′B(σ)

)ij,

where the backgroundB(σ) is specified in Eq. (1) and is nonvanishing only atσ = 0,π , andεσ,σ ′ is defined to be±1 for σ = 0,π .

To discuss the Seiberg–Witten decoupling limit, we restoregij = diag(−|g00|, g11, g22) and take the limit whereg11, g22 ∼ ε andα′ ∼ √

ε while fixingB02,B12 andg00 [5]. The critical condition becomese2 = |g00|g22, for whichthe system is supersymmetric. The noncommutativity at the end points of the open string becomes

[X1(0),X2(0)

]= [X1(π),X2(π)

]= i 2πα′bg11g22

(1− e2

|g00|g22

)+ b2,

(28)[X0(0),X2(0)

]= −[X0(π),X2(π)]= −i 2πα′e

|g00|g22(1− e2

|g00|g22+ b2

g11g22

) .

Now using the critical condition and taking the limit in whichg11, g22 ∼ ε andα′ ∼ √ε while fixingB02, B12 and

g00, we obtain

(29)[X1(0),X2(0)

]= [X1(π),X2(π)

]= i

B,

with B = b/(2πα′) and all other coordinate commutators vanish especially at the end points.Thus we end up only with the spatial noncommutativity in the Seiberg–Witten limit [5]. This implies that

the worldvolume theory in the zero slope limit may be described by the noncommutative gauge theory withspatial noncommutativity[x, y] = iθ . In (29), it looks like that the noncommutativities on both branes have thesame signature. But, taking into account of the fact that the two ends of open strings are oppositely charged,the noncommutativity geometry on the D2 is withθ = 1/B while on the anti-D2 withθ = −1/B. This suggests theeffective low-energy theory on this system is a noncommutative super Yang–Mills theory from D2-branes and thatwith opposite noncommutativity from anti-D2-branes, together with bifundamental scalars from the open stringsbetween them. For the field theory description of the worldvolume theory, the use of different noncommutativitieson D2 and anti-D2 is inconvenient and not conventional. Instead, one may use the description whereθ = 1/B forboth D2 and anti-D2 and the effects of opposite noncommutativity on the anti-D2 is described by the backgroundmagnetic field. Here the background magnetic field is not decoupled in general, but theU(2) noncommutativegauge symmetries are still in effect. As shown in Refs. [6–8], the background magnetic field then makes thespectrum tachyonic for the ordinary noncommutative D2 and anti-D2. In our case, the would-be tachyonic degreesdisappear, in spite of the presence of the same background magnetic field, and the massless continuum spectraappear as the above analysis of the string spectrum indicates. To study the details, one has to look at the interactionamplitudes obtained from the worldsheet, which goes beyond the scope of this short note. The worldvolumegauge theory has been obtained in the matrix theory description [5] and we shall use this for further check ofthe continuum spectra of the fluctuation.


4. Comparison to the matrix theory description

In the matrix theory description [5], it was shown that the supersymmetric D2 anti-D2 system is described, inthe gaugeA0 = Y , by the background2

(30)X =(x 00 x

), Y =

(y 00 −y

)

with

(31)x + iy = √2θ

∞∑n=0

√n+ 1|n〉〈n+ 1|.

This satisfies[x, y] = iθ with the noncommutativity scaleθ being 2πα′/b in the previous sections. The dynamicsof strings connecting D2 and anti-D2 is described by the off-diagonal fluctuations of the above matrices. Hence weshall turn on these off diagonal components by

(32)X =(x H1H

†1 x

), Y =

(y H2H

†2 −y

), Xs =

(0 HsH

†s 0

),

where the indexs = 3,4, . . . ,9 refers to the transverse matrix coordinates includingZ =X3. The linear fluctuationsare governed by the equations of motion

(33)D0D0XJ + [XI , [XI ,XJ ]

]= 0

together with the Gauss law[XI,D0XI ] = 0. In the gaugeA0 = Y again, we find that the Gauss law leads to

(34)θ∂yH1 − 2iyH2 − θ2∂2yH2 − 4iθH1 − 2iθy∂yH1 = 0,

where we have used[x, ] = iθ∂y and transformed all the matrix variables to ordinary functions using Weyl–Moyalmapping. Consequently, the variablesH ’s appearing in the above equation are now ordinary functions with all theproducts here ordinary. Similarly from the other components, one obtains

H1 − 4iyH1 − θ∂yH2 = 0, H2 − 2iyH2 − θ2∂2yH2 − 4iθH1 − 2iθy∂yH1 = 0,

(35)Hs − 4iyHs − θ2∂2yHs = 0.

Combining (34) and (35), one finds thatH2 is related toH1 by

(36)H2 = θ∂yH1,

whileH1 andHs satisfies

(37)H − 4iyH − θ2∂2yH = 0.

Hence there are eight independent degrees are present in the fluctuations. For the case of the harmonic timedependenceH = h(x, y)e−iωt , the equation becomes

(38)−∂2yh+ 4ω

θ2

(y − ω

4

)h= 0

which corresponds to the Airy equation. The matrix Hamiltonian is proportional toω2H 2 and clearly nonnegativedefinite for the fluctuation.

2 Unlike the notation in [5], we use hereY as the second worldvolume matrix coordinate andZ as a transverse direction.


Since the spectrum is continuous, we conclude that the degrees involved are massless. However, the equationis not a free wave equation but involves a peculiar background proportional toy that is not enough to make thespectrum discrete. Hence the results here are consistent with the string theory analysis especially in the number ofthe massless degrees. Of course, there are no tachyons in the spectrum as found from the string spectrum. Finally,we note that a multiplication of an arbitrary function ofx only to h still solves the equation without affectingω.This property will disappear if one considers nonlinear corrections to the equations of motion.

5. Conclusions

In this Letter, we have considered the strings connecting D2 and anti-D2 that are supersymmetric due tothe backgroundE- andB-fields. Beginning with generic values ofE-field, we have shown that the tachyonsdisappear in the critical limit corresponding to the supersymmetric configurations. Although there appear apparentsingularities in the coordinate fields, the worldsheet CFT is well defined. In the limit, the ground states after GSOprojection become massless, which is consistent with the restoration of the supersymmetries. For the low-energydescription implied by the worldsheet CFT, one has to compute the three and four point amplitudes to extract theinteraction terms in the zero slope limit. Instead of going this direction, we investigate the matrix theory fluctuationspectra corresponding to the D2 and anti-D2 strings and found that the massless degrees indeed appear in spite ofthe presence of the background fields. Consequently a coherent picture from both descriptions emerges.

The original supertube has a circular geometry and the elliptic deformation preserves again a quarter of thesupersymmetries. Probing these tubes by strings are of interest though the analysis are expected to be more involvedthan those presented here. One may ask how the radial size and the backgroundE- andB-fields are related to thedisappearance of tachyonic modes or how the 0-tube strings behave. These kinds of information ought to be helpfulin resolving the dynamical issues involved with the appearance of tubes out of F1 and D0’s or what governs theshapes [4,12].

The dynamics we are ultimately interested in are the dynamical processes in which an initial collection ofD0’s and F1 flows into the tubular branes or the supersymmetric D2 and anti-D2 systems or vice versa. Suchdeformations are large in the sense that we call only local bounded-energy fluctuations small. Along these largedeformation including the change of the radius,E- andB-field, it is particularly of interest to know how thestability of the systems are affected.

Acknowledgements

We would like to thank Jin-Ho Cho, Pillial Oh, Hyeonjoon Shin and Piljin Yi for discussions. The work ofD.B. was supported in part by KOSEF 1998 Interdisciplinary Research Grant 98-07-02-07-01-5. That of N.O. wassupported in part by a Grant-in-Aid for Scientific Research No. 12640270, and by a Grant-in-Aid on the PriorityArea: Supersymmetry and Unified Theory of Elementary Particles.

Appendix A. Quantization of the system

We justify the quantization appearing in (7) by the following reverse procedure. Note first that we do not yetspecify our system by a specified action. We define our system by the following first order form

(A.1)S =∫dt

(− i

2

∑n=0

1

nana−n − x0p0 + i

∑n

1

n+ ν (cn+νd−n−ν −L0)

),


where arbitrary time dependence of the oscillator variables are to be understood. Upon quantization, the equal timecommutation relations in (7) follows. In connection with the original system, we defineXi such that the oscillatorvariables in (A.1) replaces the corresponding oscillator variables absorbing the time dependent phase factors of (4).Then the Heisenberg equation of motion ofXi implies thatXi satisfies the desired free wave equation. Furthermorethe time development ofXi is consistent with the boundary conditions in (3) on shell. If one turns the action in(A.1) into second order form and rewrite oscillator variables in terms ofXi , one then can show in fact that thesecond order action is proportional toT2

∫dt dσ(X · X− ∂σX · ∂σX) up to boundary terms that may be relevant in

specifying the boundary conditions. This completes the definition of our system of the D2–anti-D2 strings.Another way to derive the commutation relations of the oscillators is to use the “canonical” momenta defined

by

(A.2)Pi = 1

2πα′(gij ∂tX

j + 2πα′B(σ)ij ∂σXj),

and impose the equal-time commutation relations

(A.3)[Xi(σ, t),Pj (σ

′, t)]= iδij δ(σ − σ ′),

with the delta function specified by the Neumann boundary conditions. Or again reversing the procedure, themomenta defined above satisfy the equal time commutation relations (A.3) provided the commutation relations ofthe oscillator variables in (7) holds.

In this procedure, care must be taken of the fact that the background changes depending onσ , as specified inEq. (1). In fact as far as the boundary value ofBij (σ ) is specified as in Eq. (1), how to extend it inside bulk doesnot matter. One may prove this fact by the explicit computation using the commutation relation.

For the sake of illustration, we here present a proof only for the casei = j = 1. First note that

(A.4)[X1(σ ), X1(σ ′)

]= 2iα′∑n

cos

[(n+ ν)σ − πν

2

]cos

[(n+ ν)σ ′ − πν

2

],

where we have used the commutation relations. Using∑n sinnθ = 0, one may rearrange the above as

(A.5)[X1(σ ), X1(σ ′)

]= iα′∑n

[cosn(σ + σ ′)cosν(σ + σ ′ − π)+ cosn(σ − σ ′)cosν(σ − σ ′)

].

Now we use the fact that

(A.6)∑n

cosnx = 2π∑k

δ(x − 2kπ),

whereδ(x) is the usual delta function defined as∫∞−∞ δ(x − x0)f (x)= f (x0) for any continuous functionf . One

then obtains

(A.7)[X1(σ ), X1(σ ′)

]= 2πα′i[δ(σ − σ ′)+ (

δ(σ + σ ′)+ δ(σ + σ ′ − 2π))cosνπ

]for 0 σ,σ ′ π . By a similar computation, one has

(A.8)[X1(σ ), ∂σ ′X2(σ ′)

]= 4πα′ib1+ b2 − e2

(−δ(σ + σ ′)+ δ(σ + σ ′ − 2π))

again for 0 σ , σ ′ π . Combining these two results, one finally gets

(A.9)[X1(σ ),P1(σ

′)]= i[δ(σ − σ ′)+ δ(σ + σ ′)+ δ(σ + σ ′ − 2π)

]= iδ(σ − σ ′)for 0 σ,σ ′ π . This completes the proof fori = j = 1 case and the remaining can be proved in a similar way.

Finally using again (7) and regulating appropriately, one can show that the momenta commutes with each otherat equal time, i.e.,[Pi(σ, t),Pj (σ ′, t)] = 0.


References

[1] D. Mateos, P.K. Townsend, Phys. Rev. Lett. 87 (2001) 011602, hep-th/0103030.[2] D. Bak, K. Lee, Phys. Lett. B 509 (2001) 168, hep-th/0103148.[3] R. Emparan, D. Mateos, P.K. Townsend, JHEP 0107 (2001) 011, hep-th/0106012.[4] D. Bak, S.W. Kim, hep-th/0108207.[5] D. Bak, A. Karch, hep-th/0110039.[6] P. Kraus, A. Rajaraman, S.H. Shenker, Nucl. Phys. B 598 (2001) 169, hep-th/0010016.[7] M. Li, Nucl. Phys. B 602 (2001) 201, hep-th/0010058.[8] G. Mandal, S.R. Wadia, Nucl. Phys. B 599 (2001) 137, hep-th/0011094.[9] J.H. Cho, P. Oh, Phys. Rev. D 64 (2001) 106010, hep-th/0105095.

[10] N. Ohta, D. Tomino, Prog. Theor. Phys. 105 (2001) 287, hep-th/0009021.[11] C.S. Chu, P.M. Ho, Nucl. Phys. B 550 (1999) 151, hep-th/9812219.[12] I. Bena, hep-th/0111156.



Relativistic massive vector condensation

Francesco Sannino, Wolfgang Schäfer

NORDITA, Blegdamsvej 17, DK-2100 Copenhagen, Denmark


Editor: H. Georgi

Abstract

At sufficiently high chemical potential massive relativistic spin-one fields condense. This phenomenon leads to thespontaneous breaking of rotational invariance while linking it to the breaking of internal symmetries. We study the relevantfeatures of the phase transition and the properties of the generated Goldstone excitations. The interplay between the internalsymmetry of the vector fields and their Lorentz properties is studied. We predict that new phases set in when vectors condensein Quantum Chromodynamics like theories. 2002 Elsevier Science B.V. All rights reserved.

1. Introduction

The Quantum Chromodynamics (QCD) phase dia-gram as a function of temperature, chemical potentialand the number of light flavors is very rich and highlystructured. QCD should behave as a color supercon-ductor for a sufficiently large quark chemical poten-tial [1]. This leads to new phenomenological appli-cations associated with the description of quark stars,neutron star interiors and the physics near the core ofcollapsing stars [1–3]. Much is known about the phasestructure of QCD at nonzero temperature through acombination of perturbation theory and lattice simu-lations while the phase structure at nonzero chemicalpotential and for large numbers of flavors has been lessextensively explored [4]. Standard importance sam-pling methods employed in lattice simulations fail atnonzero chemical potential forNc = 3 since the fermi-

E-mail addresses:[email protected] (F. Sannino),[email protected] (W. Schäfer).

onic determinant is complex. ForNc = 2, the situationis very different since the quarks are in a pseudorealrepresentation of the gauge group and lattice simula-tions can be performed [5]

At nonzero chemical potential Lorentz invarianceis explicitly broken down to the rotational subgroupSO(3) and higher-spin fields can condense, thus poten-tially enriching the phase diagram structure of QCDand QCD-like theories. Indeed, in [6] it has been sug-gested that some of the lightest massive vectors atnonzero baryon chemical potential may condense. Re-cent lattice simulations seem to support these predic-tions [7]. In the context of the Electroweak theory vec-tor condensation in the presence of a strong externalmagnetic field has also been studied in [8].

In this Letter we explore the condensation of rel-ativistic massive vectors when introducing a nonzerochemical potential via an effective Lagrangian ap-proach. In order for the vectors to couple to the chemi-cal potential they must transform under a given globalsymmetry group. In view of the possible physical ap-plications we consider the vectors to belong to the ad-



F. Sannino, W. Schäfer / Physics Letters B 527 (2002) 142–148 143

joint representation of theSU(2) group. We choose thechemical potential to lie along one of the generator’sdirections explicitly breakingSU(2) to U(1) × Z2.When the vector field condenses the rotationalSO(3)invariance breaks spontaneously together with part ofthe flavor (i.e., global) symmetry of the theory. Thecondensate locks together space and flavor symme-tries. Following the standard mean-field approach weidentify our order parameter with the vector field it-self.

In order to elucidate the mechanism we constructthe simplest effective Lagrangian at zero chemical po-tential. This consists of a standard kinetic type termand a tree level potential constructed as a series ex-pansion in the order parameter. We truncate our po-tential at the fourth order in the fields while retainingall the terms allowed by Lorentz andSU(2) symme-try. The introduction of the potential term allows us todescribe the phase transition in some detail while ourresults will be general.

We prove that the number of gapless states emerg-ing when the flavor×rotational symmetry breaks spon-taneously is insensitive to the choice of the coefficientsin the effective potential. However, the dispersion re-lations of the gapless excitations are heavily affectedby the choice of the potential. Here a crucial roleis played by a mismatch between the symmetries ofthe non-derivative terms and the full Lagrangian. Thefull vector Lagrangian (i.e., kinetic potential) alwayspossesses theZ2 × U(1) × SO(3) (flavor×rotational)symmetry which at high enough chemical potentialbreaks spontaneously toZ2 × SO(2). The gaplessmodes are linked to the 3 broken generators. One stateis an SO(2) scalar while the other two states consti-

tute a vector ofSO(2). The scalar Goldstone possesseslinear dispersion relations independently of the choiceof the vector potential. However, theSO(2) vectorstate can have either quadratic or linear dispersion re-lations. In particular (as summarized in the Table 1),the quadratic ones occur when the potential parame-ters are such that it possesses an enhancedSO(6) sym-metry. When condensing the vector breaks theSO(6)symmetry of the potential down toSO(5) while thekinetic-type term still has only aU(1) × SO(3) in-variance. In this case the potential has 5 flat direc-tions (i.e., 5 null curvatures) and we would count in thespectrum 2-gapless vectors and anSO(2) scalar. How-ever, the reduced symmetry of the kinetic term pre-vents the emergence of two independent gapless vec-tors while turning the linear dispersion relations of onevector state into quadratic ones. This result is also inagreement with the Chadha–Nielsen counting scheme.However, since the Goldstone modes with quadraticdispersion relations must be counted twice relativelyto the ones with linear dispersion relations we saturatethe number of broken generators associated with thesymmetries of just the potential term which is largerthan the ones associated with the symmetries of thefull Lagrangian. It is certainly important to investigate,in the future, if the choice of the parameter space lead-ing to anSO(6) symmetry for the potential term is sta-ble against quantum corrections.

We note that our Lagrangian approach allows ananalytical simple analysis of the relativistic vectorcondensation at nonzero chemical potential. The ro-bustness of our approach, based entirely on symmetryconsiderations, and its predictions is tested against thegeneral theorem on the number of Goldstone modes

Table 1Summary of the relevant symmetries and spectrum of the Goldstone modes for the parameter choicesλ > λ′ andλ′ = 0 of the fourth-orderpotential. We also indicate in the first row the symmetries (after the arrows) which are not broken by the vector condensate

λ > λ′ λ′ = 0

Lagrangian Z2 × U(1) × SO(3) −→ Z2 × SO(2) Z2 × U(1) × SO(3) −→ Z2 × SO(2)Potential Z2 × U(1) × SO(3) −→ Z2 × SO(2) Z2 × SO(6) −→ Z2 × SO(5)

# of broken generators:Lagrangian 3 3Potential 3 5

Spectrum of Goldstone modes:Type I (E ∝ | p |) 1 SO(2)-scalar+ 1 SO(2)-vector 1SO(2)-scalarType II (E ∝ | p |2) – 1 SO(2)-vector

144 F. Sannino, W. Schäfer / Physics Letters B 527 (2002) 142–148

by Nielsen and Chadha [9]. However, the Nielsen andChadha theorem alone (which relies only on Lorentzbreaking) cannot capture all of the relevant features re-lated to the chemical potential phase transition. In par-ticular, in this Letter, as explained above, we demon-strate the existence of an intimate relation between thevelocities of the gapless modes and the symmetries ofpart of the theory associated with the non-derivativeterms in the action. Our results are not limited to thevector condensation and shed new light on the super-fluid phenomenon and more generally on the nature ofthe phase transition at nonzero chemical potential.

There are also a number of relevant physical ap-plications of topical interest. More specifically, ourmodel Lagrangian describes the physics of 2-colorQCD at nonzero baryon chemical potential as well as3-color QCD at finite-isospin chemical potential. In-deed, in recent lattice simulations for 2-color QCDwith even number of flavors at nonzero matter density[7] massive vector condensation was observed callingfor such a detailed investigation. Here we show thatthe phase transition is very rich. Indeed, we suggestthat by studying the dispersion relations of the gap-less excitations lattice studies will be able to pin pointthe symmetries of the non-derivative terms of the vec-tor theory which encode information of the underlyingstrongly interacting dynamics.

Finally, in this Letter we also predict that vectorcondensation can be the leading mechanism for anovel superfluid phase transition for 2-color QCD withone flavor which was not considered at all in [6] orany other place in literature. Indeed, in this case theglobal symmetry isSU(2) and it remains intact at zerochemical potential and no Goldstone bosons emerge.However, the massive spectrum of the theory containsan SU(2) spin-one massive multiplet. At sufficientlyhigh nonzero chemical baryon potential the vectorcondensation will lead to a new type of superfluidphase transition. Clearly, our model is also applicableto QCD with 3 color at nonzero-isospin chemicalpotential [10–12].

This Letter is structured as follows. In the nextsection we introduce a simple effective Lagrangian forvectors which allows us to uncover the main featuresrelated to the condensation of relativistic massive spin-one fields. We conclude and point out some physicalapplications in Section 3. In particular, we suggestthat new phases should set in for QCD like theories.

Finally, we extend our simple model toD − 1 numberof space dimensions (with EuclideanSO(D) Lorentzsymmetry).

2. Vector condensation at nonzero chemicalpotential

There are different ways to introduce vector fieldsat the level of the effective Lagrangian (for example,the hidden local gauge symmetry of Ref. [13], or theantisymmetric tensor field of Ref. [14]) and they areall equivalent at tree-level. We choose to introduce themassive vector fields following the method outlinedin [15–18]. This method also allows a straightforwardgeneralization of our model to an arbitrary number ofspace dimensions. We adopt the following model La-grangian for describing relativistic spin-one fields (in3+ 1 dimensions) belonging to the adjoint representa-tion of SU(2)

L= −1

4FaµνF

aµν + m2

2Aa

µAaµ

(1)− λ

4

(Aa

µAaµ

)2 + λ′

4

(Aa

µAaν

)2,

with Faµν = ∂µA

aν −∂νA

aµ, a = 1,2,3, and metric con-

ventionηµν = diag(+,−,−,−). Herem is the treelevel mass term andλ andλ′ are positive dimension-less coefficients withλ > λ′. Other possible choices ofthe parameters do not guarantee stability of the poten-tial and will not be considered in the following.

The effect of a nonzero chemical potential associ-ated to a given conserved charge—related to the gen-erator (sayB)—can be readily included [6] by modi-fying the derivatives acting on the vector fields:

(2)∂νAρ → ∂νAρ − i[Bν,Aρ],with Bν = µδν0B ≡ VνB, where V = (µ, 0). Thevector kinetic term modifies according to:

Tr[FρνF

ρν] → Tr

[FρνF

ρν] − 4i Tr

[Fρν

[Bρ,Aν

]]− 2 Tr

[[Bρ,Aν

][Bρ,Aν

](3)− [

Bρ,Aν

][Bν,Aρ

]].


The terms due to the kinetic term, after integration byparts, yield [6]

(4)

Lkinetic = 1

2Aa

ρ

δab

[gρν − ∂ρ∂ν

]− 4iγab

[gρνV ∂ − V ρ∂ν + V ν∂ρ

2

]

+ 2χab

[V · Vgρν − V ρV ν

]Ab

ν

with

γab = Tr[T a

[B,T b

]],

(5)χab = Tr[[B,T a

][B,T b

]].

ForB = T 3 we have

γab = − i

2εab3,

(6)χ11 = χ22 = −1

2, χ33 = 0.

The chemical potential induces a “magnetic-type”mass term for the vectors at tree-level. The symmetriesof the vector Lagrangian are more easily understoodusing the following Euclidean notation:

(7)ϕaM = (

A1M,A2

M

), ψM = A3

M,

with AM = (iA0, A) and with metric signature(+,+,+,+). In these variables the potential reads:

VVector= m2

2

[| ϕ0|2 + ψ2M

] + m2 − µ2

2| ϕI |2

+ λ

4

[| ϕM |2 + ψ2M

]2

(8)− λ′

4

[ ϕM · ϕN + ψMψN

]2

with I = X,Y,Z while M,N = 0,X,Y,Z and re-peated indices are summed over. At zero chemical po-tential VVector is invariant under theSO(4) Lorentztransformations while only theSO(3) symmetry ismanifest at nonzeroµ. The SU(2) symmetry is alsoexplicitly broken, at nonzero chemical potential, toU(1).

For the reader’s convenience we summarize thequadratic term in the fields Lagrangian in the newvariables.

Lkinetic = 1

2ψM

[δMNE − ∂M∂N

]ψN

+ 1

2ϕaM

[δMNE − ∂M∂N

]ϕaN

− ϕaMεab3

[δMNV ∂ − VM∂N + VN∂M

2

]ϕbN

(9)− ϕaM [VV δMN − VMVN ]ϕa

N .

According to the value of the chemical potential wedistinguish two phases.

2.1. The symmetric phase:µ m

Here theSO(4) Lorentz andSU(2) symmetries areexplicitly broken toSO(3) andU(1) respectively bythe presence of the chemical potential but no conden-sation happens (i.e.,〈 ϕM〉 = 〈ψM 〉 = 0). The curva-tures of the potential in the vicinity of the vacuum atµ m are:

(10)M2ϕa

0= M2

ψM= m2, M2

ϕaI

= m2 −µ2.

The dispersion relations obtained by diagonalizing the12 (4 space-time× 3 SU(2) states) by 12 quadraticmatrix leads to 3 physical vectors (i.e., each of the fol-lowing states has 3 components) with the followingdispersion relations:

Eϕ∓ = ±µ +√

p 2 + m2,

(11)Eψ =√

p 2 + m2.

This shows that when approachingµ = m the 3 physi-cal components associated withEϕ+ become masslesssignaling an instability. Indeed, we now show that forvalues of the chemical potential larger than the criticalvalueµc = m, a vector type condensation sets in.

2.2. The spin–flavor broken phase:µ>m

In this phase the global minimum of the potential isfor 〈ϕa

0〉 = 〈ψM 〉 = 0, while we can choose the vev tolie in the (spin–flavor) direction:

(12)⟨ϕ1X

⟩ =√

µ2 − m2

λ− λ′ .

We have a manifold of equivalent vacua which areobtained rotating the chosen one under aZ2 ×U(1)×SO(3) transformation. The choice of the vacuumpartially locks together the Lorentz group and theinternal symmetry while leaving unbroken only the


subgroupZ2 × SO(2). Two generators associated tothe Lorentz rotations are now spontaneously brokentogether with theU(1) generator.

To compute the full dispersion relations of thetheory we first need to provide the curvatures of thepotential evaluated on the new vacuum:

M2ψ0

= M2ψY

= M2ψZ

= m2 + λµ2 − m2

λ− λ′ ,

(13)M2ψX

= µ2,

M2ϕ1

0= µ2, M2

ϕ1X

= 2(µ2 − m2),

(14)M2ϕ1Y

= M2ϕ1Z

= 0,

M2ϕ2

0= m2 + λ

µ2 − m2

λ − λ′ , M2ϕ2X

= 0,

(15)M2ϕ2Y

= M2ϕ2Z

= λ′ µ2 −m2

λ − λ′ .

In general three states have null curvature (specificallyM2

ϕ2X

= M2ϕ1Y

= M2ϕ1Z

= 0), however, for the special

caseλ′ = 0 we have 5 zero curvature states. To explainthis behavior we note that forλ′ = 0 the potentialpossessesSO(6) global symmetry which breaks toSO(5) when the vector field condenses. The associated5 states would correspond to the ordinary Goldstonemodes in the absence of an explicit Lorentz breaking.Using these curvatures we compute the dispersionrelations. Four of theϕ states have (2 per each sign)the following dispersion relations:

E 2ϕ±V

= p2 + ∆2 ∓√

4µ2 p 2 + ∆4

(16)with ∆2 = 2µ2 + λ′

2

(µ2 − m2)

λ − λ′ ,

and to each sign we associate a vector (with twocomponents) with respect toSO(2). In the limit ofsmall momenta for the gapless mode:

E2ϕ+V

= λ′

2∆2

µ2 −m2

(λ− λ′)p 2 + 2µ4

∆6 | p |4 +O(p6)

= v2ϕ−V

p 2 + · · · ,where we denoted byvϕ−

Vthe superfluid velocity as-

sociated to the stateϕ−V . For the last twoϕ physical

Fig. 1. Mass gapsE(0) as a function of the chemical potentialµ. The phase transition occurs forµ/m = 1. The curves wereobtained for the valuesλ = 1, λ′ = 0.33 of the potential coefficients.For µ/m < 1, all the gaps are threefold degenerate, and show thesplitting of the different charge states for nonzeroµ. Forµ/m > 1the modes with a nonvanishing gap split further. The solid linesare twofold degenerate (SO(2)-vectors), while the dashed linescorrespond toSO(2)-scalar states.

modes (scalars ofSO(2)) we show directly the disper-sion relations as a momentum expansion.

E2ϕ+S

= µ2 − m2

3µ2 − m2p 2 +O

(p4) = v2

ϕ−S

p 2 + · · · ,(17)E2

ϕ−S

= 2(3µ2 − m2) + γ p 2 +O

(p4),

where the subscriptsV,S stand for vector and scalarrespectively, andγ > 0. Finally the threeψ type state

dispersion relations areEψI =√

p 2 + M2ψI

, with MψI

defined in Eq. (13). In Fig. (1) we show the mass gapsEi( p = 0) for all the states.

A relevant result is that the velocities of our gaplessmodes can be expressed directly in terms of thecurvatures in the directions orthogonal to the gaplessmodes as follows:

(18)v2ϕ+V

=M2

ϕ2Y

M2ϕ2Y

+ 4µ2, v2

ϕ+S

=M2

ϕ1X

M2ϕ1X

+ 4µ2.

We find that the number of gapless states emergingwhen the flavor×rotational symmetry spontaneouslybreaks is insensitive to the choice of the coefficients inthe effective potential. However, the dispersion rela-tions depend crucially on the symmetries of the poten-tial. The full vector Lagrangian (i.e., kinetic potential)possesses theZ2 × U(1) × SO(3) (flavor×rotational)


symmetry which breaks, above the critical chemicalpotential, when the vector condenses toZ2 × SO(2).The gapless modes are associated to the 3 broken gen-erators. One state is a scalar ofSO(2) while the othertwo fields constitute a vector ofSO(2). The scalarGoldstone possesses linear dispersion relations inde-pendently of the choice of the vector potential. This isthe Goldstone associated to theU(1) breaking. How-ever, for the dispersion relations of theSO(2) vec-tor state we have either quadratic or linear dispersionrelations. The quadratic ones occur when the poten-tial possesses an enhancedSO(6) symmetry (i.e., forλ′ = 0). When condensing the vector breaks the poten-tial symmetries fromSO(6) to SO(5) while the deriv-ative part of the Lagrangian is still invariant under thecontinuousU(1) × SO(3) only. In this case the poten-tial has 5 flat directions (i.e., 5 null curvatures (occur-ring for λ′ = 0)) and we would count in the spectrum2-massless vectors and a scalar ofSO(2). However, thereduced symmetry of the derivative terms prevents theemergence of another gapless vector type mode andinstead turns the linear dispersion relations of one ofthe vector states into quadratic ones.

It is important to observe that at the second orderphase transition point, all the velocities of the gap-less modes vanish regardless of the value of the cou-pling constants. This is so since at the second or-der phase transition point also the potential curvatures(which at zero density would correspond to the phys-ical masses ofσ -like fields) in the directions orthogo-nal to the spontaneously broken generators ones van-ish. This behavior is a natural consequence of the ex-tended conformal symmetry of potential at this point.However, at nonzero chemical potential part of the the-ory is not conformal and the information encoded inthe conformal part of the potential (as for the enhancedglobal symmetry case) is nicely and efficiently trans-ferred to the momentum dependence of the disper-sion relations via the velocities of the gapless modes(see Eq. (18)). Clearly, the continuity of the disper-sion relations is still ensured. Our result is in agree-ment with the Nielsen–Chadha counting scheme, andallows for a number of relevant predictions which canbe tested, as we shall discuss, by lattice simulations.It is amusing to note that, forλ′ = 0, the Nielsen–Chadha counting rule yields a number which is largerthan the number of broken generators of the full the-ory. To the best of our knowledge this is the first time

that such a phenomenon has been discussed in the lit-erature.

3. Conclusions and physical applications

We summarized our findings in the Table 1 andconclude our work by considering some possiblephysical applications of massive relativistic vectorcondensation at high chemical potential.

Acknowledgements

It is a pleasure for us to thank M.P. Lombardo, H.B.Nielsen, and P. Olesen for helpful discussions. We arealso indebt to J. Schechter for discussions and carefulreading of our manuscript. The work of F.S. is sup-ported by the Marie-Curie Fellowship under contractMCFI-2001-00181, while W.S. acknowledges supportby DAAD.

Appendix A. 2-color QCD

Two-color QCD at nonzero chemical potential re-cently attracted a great flurry of interest since lat-tice simulations can be reliably performed at nonzeroquark chemical potential. In [6] vector condensation atnonzero quark chemical potential has been predictedfor two-color QCD with an even numbers of flavors.The predictions are supported [7] by lattice studies.Here we provided a detailed study of the vector con-densation phenomenon at nonzero chemical potentialwhile predicting new features related to the physics ofthe phase transition.

A direct application of our results is for QCD withtwo colors and one light flavor. This theory has globalquantum symmetry groupSU(2). The extra classicalaxial UA(1) symmetry is broken by the Adler–Bell–Jackiw anomaly. At zero chemical potential Lorentzinvariance cannot be broken and the simplest bilinearcondensate (see Ref. [6] for conventions) is of thetype:

(A.1)εc1c2εαβQIα,c1

QJβ,c2

EIJ ,

with E = 2iT 2 the antisymmetric matrix in the flavorspace andQI

αc is a Weyl spinor withα = 1,2 the


spinorial index, c = 1,2 the color andI = 1,2the flavor indices. The generators ofSU(2) in thefundamental representation areT a = τa/2 with τa thestandard Pauli matrices. It is easy to check thatSU(2)remains unbroken since we haveT aT

E + ET a = 0for all of the SU(2) generators. So we have noGoldstone bosons at all in the theory. However, we dohave physical vectors (i.e., spin-1 fields) and massivescalar states at zero density. These states should becomparable in mass (recall theη′ versus the octet ofordinary vector bosons in 3-color QCD). The massivespin-one fields (i.e., the hadrons) must transformaccording to the regular representation ofSU(2) andcarry nontrivial baryon number which is proportionalto T 3.

Our analysis can be immediately used to predictthat when turning on a nonzero baryon chemicalpotential a superfluid phase transition will set in forµ larger then the mass of the vector fields. This isso since we always have at least one gapless modewith linear dispersion relations. We also predict thespontaneous breaking of rotational invariance and theemergence of a vector state (i.e., doublet ofSO(2))with either linear or quadratic dispersion relations. Itis straightforward to extend and apply our analysis tothe case of different number of flavors and to ordinaryQCD at finite-isospin chemical potential. [6].

Appendix B. Extra-dimensions

Finally, we extend our model to anSU(2) chargedvector in D − 1 number of space dimensions (withSO(D) Euclidean Lorentz symmetry) with an associ-ated nonzero chemical potential. In this case the chem-ical potential first breaks explicitlySO(D) to the spa-tial SO(D − 1). At sufficiently high chemical poten-tial SO(D − 1) breaks spontaneously toSO(D − 2).The gapless states consist of a scalar field under theSO(D − 2) transformations with linear dispersion re-

lations and one vector ofSO(D − 2) with either linearor quadratic dispersion relations.

References

[1] For recent reviews and a rather complete list of references, see:K. Rajagopal, F. Wilczek, hep-ph/0011333;

M.G. Alford, hep-ph/0102047;S.D.H. Hsu, hep-ph/0003140;D.K. Hong, Acta Phys. Polon. B 32 (2001) 1253, hep-ph/0101025.

[2] R. Ouyed, F. Sannino, astro-ph/0103022;R. Ouyed, F. Sannino, Phys. Lett. B 511 (2001) 66, hep-ph/0103168.

[3] D.K. Hong, S.D. Hsu, F. Sannino, Phys. Lett. B 516 (2001)362, hep-ph/0107017.

[4] M.P. Lombardo, hep-ph/0103141.[5] For a recent review and a rather complete list of references,

see: S. Hands, hep-lat/0109034.[6] J.T. Lenaghan, F. Sannino, K. Splittorff, hep-ph/0107099,

accepted for publication in Phys. Rev. D.[7] M.P. Lombardo, Talk given at the GISELDA meeting, Firenze,

October 2001.[8] J. Ambjorn, P. Olesen, Phys. Lett. B 218 (1989) 67;

J. Ambjorn, P. Olesen, Phys. Lett. B 220 (1989) 659, Erratum;J. Ambjorn, P. Olesen, Phys. Lett. B 257 (1991) 201;J. Ambjorn, P. Olesen, Nucl. Phys. B 330 (1990) 193.

[9] H.B. Nielsen, S. Chadha, Nucl. Phys. B 105 (1976) 445.[10] K. Splittorff, D.T. Son, M.A. Stephanov, Phys. Rev. D 64

(2001) 016003, hep-ph/0012274.[11] V.A. Miransky, I.A. Shovkovy, hep-ph/0108178.[12] T. Schafer, D.T. Son, M.A. Stephanov, D. Toublan, J.J. Ver-

baarschot, Phys. Lett. B 522 (2001) 67, hep-ph/0108210.[13] M. Bando, T. Kugo, K. Yamawaki, Phys. Rep. 164 (1988) 217.[14] G. Ecker, J. Kambor, D. Wyler, Nucl. Phys. B 394 (1993) 101.[15] Ö. Kaymakcalan, S. Rajeev, J. Schechter, Phys. Rev. D 30

(1984) 594;J. Schechter, Phys. Rev. D 34 (1986) 868;P. Jain, R. Johnson, U.-G. Meissner, N.W. Park, J. Schechter,Phys. Rev. D 37 (1988) 3252.

[16] Ö. Kaymakcalan, J. Schechter, Phys. Rev. D 31 (1985) 1109.[17] T. Appelquist, P.S. Rodrigues da Silva, F. Sannino, Phys. Rev.

D 60 (1999) 116007, hep-ph/9906555.[18] Z. Duan, P.S. Rodrigues da Silva, F. Sannino, Nucl. Phys.

B 592 (2001) 371, hep-ph/0001303.



Aharonov–Bohm effect in noncommutative spaces

M. Chaichiana, P. Prešnajdera,b, M.M. Sheikh-Jabbaric, A. Tureanua

a High Energy Physics Division, Department of Physics, University of Helsinki and Helsinki Institute of Physics,P.O. Box 64, FIN-00014 Helsinki, Finland

b Department of Theoretical Physics, Comenius University, Mlynská dolina, SK-84248 Bratislava, Slovakiac The Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, Trieste, Italy

Received 9 July 2001; received in revised form 13 November 2001; accepted 8 January 2002


Abstract

The Aharonov–Bohm effect on the noncommutative plane is considered. Developing the path integral formulation of quantummechanics, we find the propagation amplitude for a particle in a noncommutative space. We show that the corresponding shiftin the phase of the particle propagator due to the magnetic field of a thin solenoid receives certain gauge invariant correctionsbecause of the noncommutativity. Evaluating the numerical value for this correction, an upper bound for the noncommutativityparameter is obtained. 2002 Published by Elsevier Science B.V.

PACS: 11.15.-q; 11.30.Er; 11.25.Sq

1. Introduction

Besides the string theory interests [1], recently the-ories on noncommutative space–time have received alot of attention. In this way both problems of quan-tum mechanics (QM) and field theories on noncom-mutative spaces have their own excitements; for theQM side see [2–4]. Inferred from the string theory,where noncommutative geometry appears naturally intheories with antisymmetric background tensor, thenoncommutative space–time (but with commutativemomenta) can be realized as a space where coordinateand momentum operators satisfy the commutation re-lations[xµ, xν

] = iθµν,[pµ, pν

] = 0,

E-mail address: [email protected] (M. Chaichian).

(1.1)[xµ, pν

] = ihδµν.

Here θµν is an antisymmetric tensor of dimensionof (length)2. Also from general arguments based onQM and classical gravity, one can deduce the inher-ent noncommutativity of space–time due to the impos-sibility of measuring space–time points with infiniteaccuracy [5]. We note that a space–time noncommu-tativity, θ0i = 0, may lead to some problems with uni-tarity and causality [6,7]. Such problems do not oc-cur for the QM on a noncommutative space with ausual (commutative) time coordinate, so we shall as-sume that there exists a frame in whichθ0i = 0 and weshall restrict ourselves to this particular frame.

Given the noncommutative space (1.1), which isalso a natural extension of the usual QM, one shouldstudy its physical consequences. Comparing thesenoncommutative results with the present experimental



150 M. Chaichian et al. / Physics Letters B 527 (2002) 149–154

data one can find an upper bound onθ . It appears thatthe most natural places to trace the noncommutativityeffects are simple QM systems, such as the hydrogenatom or systems in external magnetic field [4,8].The former has been considered in [4] and the shiftin the spectrum, and in particular, the modificationsto Lamb-shift, due to noncommutativity have beendiscussed there. As one expects (1.1) breaks therotational symmetry of the hydrogen atom spectrumand as a result we would face a “polarized Lamb-shift” [4]. In this work we shall study the other system,the Aharonov–Bohm effect. The physical significanceof the Aharonov–Bohm effect resides in the fact thatit is the place to check the noncommutative gaugeinvariance, which is a deformed version of the usualgauge freedom [1,3,9]. We will come back to this pointlater.

In order to study the Aharonov–Bohm effect oneshould develop the proper QM setup for the noncom-mutative case. As the Hilbert space is assumed to bethe same for the commutative case and its noncom-mutative extension, it is enough to give the Hamil-tonian. Once we have the Hamiltonian, the dynamicsof the states is given by the usual Schrödinger equa-tion, H |ψ〉 = ih ∂

∂t|ψ〉. Because of the noncommuta-

tivity of the coordinates, the coordinate basis does notexist and the very concept of wave function,〈x|ψ〉,fails. However, the usual momentum space descriptionis still valid.

To handle the NCQM, one can also use a moreunusual approach to QM, using the operator valued“wave functions”. In the usual QM because of theWeyl–Moyal correspondence [3,9] there is a one-to-one correspondence between such operators and theusual wave functions so that the usual algebra ofthe functions is now applicable to them. However, inthe noncommutative case, instead of the usual productbetween functions, the Weyl–Moyal correspondenceyields the-product:

(f g)(x)

= exp

i

2θµν∂xµ∂yν

f (x)g(y)

∣∣∣∣x=y

(1.2)= f (x)g(x)+ i

2θµν∂µf ∂νg + O

(θ2),

between the “wave functions”. We should remind that,although always a function corresponds to any opera-

tor valued wave function, the argument of these func-tions cannot be treated as the space coordinates. Ac-cording to this point of view the probability ampli-tudes are given by the square of the norm of the op-erator valued wave functions.

Having discussed the NCQM kinematics, we shouldthen give the proper Hamiltonian for the noncommuta-tive systems. As in the usual QM, this can be done us-ing the nonrelativistic limit of the corresponding fieldtheory. The difference between the commutative andnoncommutative field theories are only in the interac-tion terms1 [9,10], and this will lead to some newθ de-pendent interaction potentials. For the electromagneticinteractions the corresponding field theory is NCQED.As discussed in [4,9] for the electromagnetic interac-tion the extraθ dependence of the Hamiltonian, in thefirst order inθ , always can be obtained assigning anelectric dipole moment,

(1.3)die = e

2hθ ij pj ,

to the charged particle. This can be understood intu-itively noting thatf (x) g(x) = f (xi + i

2θij ∂j )g(x).Since we believe that the effect of noncommutativityin nature, if it is there, should be very small, one cantrust the perturbation inθ .

It turns out that to study the Aharonov–Bohm effectit is more convenient to formulate the problem via pathintegral. So, first we construct the proper definitionof the path integral and transition amplitude in thenoncommutative case and then we evaluate the extrashift in the Aharonov–Bohm experiment interferencepattern which comes about due to noncommutativityin the quasi classical approximation, i.e., leadingorder inh and first order inθ .

2. Aharonov–Bohm effect on a noncommutativeplane

The Aharonov–Bohm effect concerns the shift ofthe interference pattern in the double-slit experiment,due to the presence of a thin long solenoid put justbetween the two slits [12]. Although the magnetic

1 Of course this is not quite true, and for the field theories onnoncommutative spaces with nontrivial topology, such as cylinderand torus, one should treat the problem more carefully [17].

M. Chaichian et al. / Physics Letters B 527 (2002) 149–154 151

field B is present only inside the solenoid, the cor-responding Schrödinger equation depends explicitlyon the magnetic potentialA (nonvanishing outside thesolenoid). Therefore, the wave function depends onA

and consequently the interference pattern shifts. Theshift in the phase of the particles propagator,δφ0, isgauge invariant itself and can be expressed in nonlo-cal terms ofB. In the quasi-classical approximation,δφ0 = e

hcΦ, whereΦ = Bπρ2 is the magnetic flux

through the solenoid of radiusρ. This effect has beenconfirmed experimentally [13].

Below, we present the quasi-classical approach tothe Aharonov–Bohm effect on a NC-plane for a thin,but of finite radius solenoid. However, first we need toformulate a noncommutative path integral.

Since the very concept of the wave function in thenoncommutative case is a problematic one, in orderto study the noncommutative Aharonov–Bohm effect,first we present the noncommutative formulation ofpath integral QM.2 Then, by means of path integrals,we find the propagator and hence the desired noncom-mutative corrections to the Aharonov–Bohm phase.

The Hilbert spaceH of quantum mechanics on anoncommutative space is formed by the normalizablefunctions Ψ (x) with finite norm, belonging to anoncommutative algebra of functionsA on R

2. Thewave function is an element fromH, normalized tounity. However, we remind that wave functions arejust symbols; the physical meaning is contained onlyin some smeared values of them, e.g., by a coherentstate [11]. InA, we can introduce the scalar productas:

(ψ,φ) =∫

d3x ψ(x) φ(x)

=∫

d3x ψ(x)φ(x)

(2.1)=∫

d3k¯ψ(k)φ(k),

where ψ(k) and φ(k) are the corresponding Fouriertransforms. Here, we have used the well-known factthat in the integrals containing as integrand a-productof two functions, their-product can be replaced by astandard one. The operatorsPi andXi acting inH and

2 The path integral formulation of aq-deformed harmonicoscillator was investigated in [14,15].

satisfying the commutation relations (1.1) are given as:

PiΨ (x)= −i∂iΨ (x), XiΨ (x) = xi Ψ (x).

(2.2)

Along the arguments of [4], the problem of a particlemoving in an external magnetic field on a noncommu-tative plane is specified by the Hamiltonian:

H = 1

2(Pi +Ai)

2

(2.3)= 1

2(Pi +Ai) (Pi +Ai).

We note that the transition amplitude(Ψf ,e−iHtΨi)

is invariant under the noncommutative gauge transfor-mations defined by

Ψ (x) → U(x) Ψ (x),

Ai(x) → U(x) Ai(x) U−1(x)− iU(x) ∂iU

−1(x)

whereU(x) ≡ (e)iλ(x), for real functionsλ(x), andthe (e) is defined by the usual Taylor expansion,with all products of λ’s replaced by the ones.Then, one can easily show thatU−1 = (e)−iλ(x)

satisfiesU−1 U = 1. We point out the non-Abeliancharacter of the above gauge transformations, due tothe noncommutativity of the space. Consequently, thefield strength is given by a non-Abelian formula, too:

(2.4)Fij (x)= ∂[iAj ](x)+ (A[i Aj ])(x).

Moreover, one can easily see that

(2.5)Pi +Ai → U(x) (Pi +Ai) U−1(x).

In quantum mechanics, the exponents of the opera-tors (e.g., e−iHt ) often do not correspond to local oper-ators. However, they can be conveniently representedby bi-local kernels. This is true in the noncommutativeframe, also. It can be easily seen that to any operatorK = K(Pi,Xi) = K(−i∂i, xi), cf. (2.2), we can as-sign a kernel (a bi-local symbol)K(x, y) ∈ A ⊗ A,defined by:

(2.6)K(x, y) =∫

d3q

(2π)3

(Keiqx)e−iqy

(we omit the symbol⊗ for the direct product). We notethat straightforwardly the-product defined betweentwo functions (Weyl symbols) can be generalized tothe kernels which are functions in two variables. The


action ofK in terms of kernel is

(KΦ)(x)=∫

d3yK(x, y) Φ(y)

=∫

d3yK(x, y)Φ(y).

For a product of two operators, one can use either thestandard formula for the kernel composition

(GK)(x, y)=∫

d3zG(x, z) K(z, y)

(2.7)=∫

d3zG(x, z)K(z, y)

or use the formula

(2.8)(GK)(x, y)=∫

d3q

(2π)3

(Geiqx)(K†eiqy

).

The proof of (2.8) is straightforward.The kernel corresponding to the operator e−iHt will

be denoted byKt (x, y) and called propagator:

(2.9)Kt (x, y)=∫

d3q

(2π)3

(e−iHteiqx)e−iqy.

From the product formula (2.7) and the identitye−iHt1e−iHt2 = e−iH(t1+t2), the usual composition lawfollows:

(2.10)Kt1+t2(x, y)=∫

d3zKt1(x, z)Kt2(z, y).

Iterating this formulaN times and taking the limitN → ∞, we arrive by standard arguments at the pathintegral representation of the propagator:

Kt (x, y)

= limN→∞

∫d3xN−1 · · ·d3x1

(2.11)×Kε(x, xN−1) · · ·Kε(x2, x1)Kε(x1, y),

with ε = t/N . We stress that there is no need to use-product between twoKε ’s.

The formula for the gauge transformation of thepropagator follows directly from Eq. (2.5). In fact, (2.5)implies:

(2.12)e−iHt → U(x) e−iHt U−1(x)

(as operators), so that

Kt (x, y)

→ U(x) e−iHt U−1(y)

=∫

d3q

(2π)3

(eiλ

(e−iHteiqx)) (

eiλeiqy)

(2.13)= U(x) Kt (x, y) U−1(y).

This is exactly the expected formula (here, the-product cannot be omitted).

As the next step, we shall calculate the short-timepropagatorKε(x, y) entering (2.11) to first orders inεandθ . Using the Hamiltonian (2.3) and (2.9) we have

Kε(x, y)

=∫

d3p

(2π)3

([1− iε

2(Pi +Ai)

2 + · · ·]eipx

)e−ipy

=∫

d3p

(2π)3eip(x−y)−iεHe(p,x),

wherex = 12(x+y) and the effective Hamiltonian,He,

is given as:3

He ∼= 1

2

(Πi +Ai(x)

)2,

(2.14)Πi = pi − 1

2θjk

(∂jAi(x)

)pk.

The symbol∼= means equality in the first order inεand θ . The above effective Hamiltonian can also beobtained if we assign an electric dipole moment,Eq. (1.3), to electron. Performing thed3p integration,we obtain the effective Lagrangian:

L∼= 1

2ViVi − ViAi(x),

(2.15)Vi = vi + 1

2θji∂jAk(x)vk.

The formula forKε(x, y) then reads:

(2.16)Kε(x, y) ∼= ei∫dtL(x(t), ˙x(t)),

with the effective action calculated for a linear path,starting atxi(0) = xi and terminating atxi(ε) = yi ,i.e., vi = (yi − xi)/ε and Ai(x) = Ai

( x+y2

). Up to

terms linear inθ , the Lagrangian, with all physicalconstants included, becomes:

L= L0 − em

4hcθ ·

[vi

(v × ∇Ai

) − e

mcvi

( A× ∇Ai

)],

(2.17)

3 We note thatA A = A2 +O(θ2).

M. Chaichian et al. / Physics Letters B 527 (2002) 149–154 153

whereL0 = m2 v2 − e

cv · A and the vectorθ is defined

as θi = εijkθjk. Thus, the total shift of phase forthe Aharonov–Bohm effect, including the contributiondue to noncommutativity, will be:

(2.18)δφtotal = δφ0 + δφNCθ ,

whereδφ0 = ehc

∮dr · A = e

hc

∫ B · d S = ehcΦ (Φ be-

ing the magnetic flux through the surface bounded bythe closed path) is the usual (commutative) phase shiftand

(2.19)

δφNCθ = em

4h2cθ ·

∮dxi

[(v × ∇Ai

)

− e

mc

( A× ∇Ai

)]

represents the noncommutative corrections.4

For a finite-radius solenoid, the vector potentialAentering (2.17)–(2.19) is given by:

(2.20)A = 1

2Bρ2

rn, r > ρ,

where B is the constant magnetic field inside thesolenoid,ρ is the radius of the solenoid andn is theunit vector orthogonal tor .

The expression for the correctionδφNCθ to the usual

Aharonov–Bohm phase due to noncommutativity canbe explicitly obtained from (2.19) and (2.20). Inan analogous way as in the usual Aharonov–Bohmcase [12], the calculation can be done by taking theclosed classical path (what is valid according to theexperimental setup), which starts from the source andreaches the point on the screen by passing through oneof the two slits and returns to the source point throughthe other slit.

An estimation for the upper bound on the para-meter of noncommutativityθ can be made using theavailable experimental data on the Aharonov–Bohmeffect [13]. For the purpose of this estimation, we tookθ along the magnetic field of the solenoid, for in thiscase the effect is the largest; the integration path wastaken to be circular, although this is not significant andthe result would be general. Then, the contribution tothe shift coming from noncommutativity, relative to

4 After our work had appeared in the hep archive, similar resultswere presented in [16].

the usual shift of phase, will be:

(2.21)δφNC

θ

δφ0∼ θ

λeR

v

c− δφ0

θ

S,

whereR is the radius of the approximate path,S =πR2 is the area of the surface bounded by the closedpath andλe is the Compton wavelength of the electron.We should point out the fact that, comparing the twoterms of the noncommutative correction, it appearsthat the energy-dependent term prevails over the otherone (by 5 orders of magnitude). Fitting the ratio (2.21)into the accuracy bound of the experiment [13], weobtain:

(2.22)√θ 106 GeV−1,

which corresponds to a relatively large scale of 1 Å.Such a value emerges due to the large error of 20%in the experimental test [13] of the Aharonov–Bohmeffect.

3. Concluding remarks

In this work we have studied the Aharonov–Bohmeffect for the noncommutative case. In order to obtainthe transition amplitudes, and hence finding the shiftin the interference pattern, we worked out the pathintegral formulation. Using this formulation we havealso required the transition amplitudes to be invariantunder the noncommutative gauge transformations. Inthis way we have found the result in thequasi-classical approach up to first order inθ . However,besides the quasi-classical result one can solve theSchrödinger equation for this case explicitly, thoughfor the operator valued wave functions [17].

Acknowledgements

The financial support of the Academy of Finlandunder the Project No. 163394 is greatly acknowledged.P.P.’s work was partially supported by VEGA project1/7069/20. The work of M.M.Sh.-J. was partiallysupported by the EC contract No. ERBFMRX-CT 96-0090.


References

[1] N. Seiberg, E. Witten, JHEP 9909 (1999) 032, hep-th/9908142.[2] H. Snyder, Phys. Rev. 71 (1947) 38.[3] L. Alvarez-Gaumé, S.R. Wadia, Phys. Lett. B 501 (2001) 319,

hep-th/0006219.[4] M. Chaichian, M.M. Sheikh-Jabbari, A. Tureanu, Phys. Rev.

Lett. 86 (2001) 2716, hep-th/0010175.[5] S. Doplicher, K. Fredenhagen, J.E. Roberts, Commun. Math.

Phys. 172 (1995) 187.[6] J. Gomis, T. Mehen, Nucl. Phys. B 591 (2000) 265, hep-

th/0005129.[7] M. Chaichian, A. Demichev, P. Prešnajder, A. Tureanu, Eur.

Phys. J. C 20 (2001) 767, hep-th/0007156.[8] V.P. Nair, A.P. Polychronakos, Phys. Lett. B 505 (2001) 267,

hep-th/0011172.[9] I.F. Riad, M.M. Sheikh-Jabbari, JHEP 0008, 045, hep-

th/0008132.

[10] A. Micu, M.M. Sheikh-Jabbari, JHEP 01 (2001) 025, hep-th/0008057.

[11] M. Chaichian, A. Demichev, P. Prešnajder, Nucl. Phys. B 567(2000) 360, hep-th/9812180;M. Chaichian, A. Demichev, P. Prešnajder, J. Math. Phys. 41(2000) 1647, hep-th/9904132.

[12] R.P. Feynmann, R.B. Leighton, M. Sands, The Feynmanlectures on physics, Vol. II, Addison–Wesley, Reading, MA,1964;B.R. Holstein, Topics in Advanced Quantum Mechanics,Addison–Wesley, Redwood, 1992.

[13] A. Tonomura et al., Phys. Rev. Lett. 48 (1982) 1443.[14] L. Baulieu, E.G. Floratos, Phys. Lett. B 258 (1991) 171.[15] M. Chaichian, A. Demichev, Phys. Lett. B 320 (1994) 273.[16] J. Gamboa, M. Loewe, J.C. Rojas, hep-th/0101081.[17] M. Chaichian, A. Demichev, P. Prešnajder, M.M. Sheikh-

Jabbari, A. Tureanu, hep-th/0101209.



Dynamical fermion algorithm for variable actions

I. Montvay

Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, D-22603 Hamburg, Germany

Received 8 November 2001; received in revised form 3 January 2002; accepted 5 January 2002

Editor: P.V. Landshoff

Abstract

A new version of the two-step multi-boson algorithm is developed with different fermion actions in the multi-boson and noisyMetropolis steps. 2002 Elsevier Science B.V. All rights reserved.

1. Introduction

Multi-boson algorithms for dynamical fermions[1–4] offer promising possibilities for numerical sim-ulations of QCD and other similar theories [5]. Thetwo-step multi-boson (TSMB) algorithm [3] has beensuccessfully applied, for instance, in supersymmetricYang–Mills theories [6] and for simulations of QCDwith SU(2) colour at high densities [7]. In the presentLetter I shall consider a TSMB algorithm allowing fordifferent actions in the two steps.

For convenience of the reader let us first shortlysummarize the main features of TSMB. Let us con-sider the case of an arbitrary number of identicalfermion flavoursNf and assume the existence of a her-mitean fermion matrixQ, which has the determinantdet(Q) appearing in the effective action for the gaugefield after the integration over the fermionic variablesin the path integral. Multi-boson algorithms are based

E-mail address: [email protected] (I. Montvay).

on the representation [1]

(1)∣∣det

(Q

)∣∣Nf = det

(Q2)Nf /2 1

detPn(Q2).

Here the polynomialPn satisfies

(2)limn→∞Pn(x)= x−Nf /2

in an intervalx ∈ [ε,λ] covering the spectrum ofQ2.For the multi-boson representation of the determinantone uses the roots of the polynomialrj (j = 1, . . . , n)

(3)Pn

(Q2)= r0

n∏j=1

(Q2− rj

).

Assuming that the roots occur in complex conjugatepairs, one can introduce the equivalent forms

Pn

(Q2)= r0

n∏j=1

[(Q±µj

)2+ ν2j

](4)= r0

n∏j=1

(Q− ρ∗j

)(Q− ρj

),

whererj ≡ (µj + iνj )2 andρj ≡ µj + iνj . With thehelp of complex boson (pseudofermion) fieldsΦjx



156 I. Montvay / Physics Letters B 527 (2002) 155–160

one can write∣∣det(Q

)∣∣Nf

∝n∏

j=1

det[(

Q− ρ∗j)(

Q− ρj

)]−1

∝∫[dΦ]exp

−

n∑j=1

∑xy

Φ+jy

[(Q− ρ∗j

)× (

Q− ρj

)]yx

Φjx

.

(5)

Since for a finite polynomial of ordern the approxi-mation in (2) is not exact, one has to extrapolate theresults ton→∞.

The difficulty for small fermion masses in largephysical volumes is that thecondition number λ/ε be-comes very large (104–106) and very high ordersn=O(103) are needed for a good approximation. Thisrequires large storage and the autocorrelation of thegauge configurations becomes very bad since the au-tocorrelation length is a fast increasing function ofn.One can achieve substantial improvements on boththese problems by introducing a two-step polynomialapproximation:

(6)limn2→∞

P (1)n1

(x)P (2)n2

(x)= x−Nf /2, x ∈ [ε,λ].The multi-boson representation is only used here forthe first polynomialP (1)

n1 which provides a first crudeapproximation and hence the ordern1 can remainrelatively low. The correction factorP (2)

n2 is realizedin a stochasticnoisy Metropolis correction step witha global accept-reject condition during the updatingprocess. In order to obtain an exact algorithm one hasto consider in this case the limitn2→∞.

In the two-step approximation scheme forNf

flavours of fermions the absolute value of the deter-minant is represented as

(7)∣∣det

(Q

)∣∣Nf 1

detP (1)n1 (Q2) detP (2)

n2 (Q2).

After an update sweep over the gauge field a globalaccept-reject step is introduced in order to reach thedistribution of gauge field variables[U ] correspond-ing to the right-hand side of (7). This can be done sto-chastically by generating a random vectorη according

to the normalized Gaussian distribution

(8)e−η†P

(2)n2 (Q[U ]2)η∫ [dη]e−η†P

(2)n2 (Q[U ]2)η

,

and accepting the change[U ]→ [U ′] with probability

(9)PA

([U ′] ← [U ])=min1,A

(η; [U ′] ← [U ]),

where

A(η; [U ′] ← [U ])= exp

−η†P (2)

n2

(Q[U ′]2)η+ η†P (2)

n2

(Q[U ]2)η

.

(10)

The Gaussian noise vectorη can be obtainedfrom η′ distributed according to the simple Gaussiandistribution

(11)e−η′†η′∫ [dη′]e−η′†η′

by setting it equal to

(12)η= P (2)n2

(Q[U ]2)− 1

2 η′.

In order to obtain the inverse square root on theright-hand side of (12), one can proceed with apolynomial approximation

(13)P (3)n3

(x) P (2)n2

(x)−12 , x ∈ [0, λ].

This is a relatively easy approximation because

P(2)n2 (x)− 1

2 is not singular atx = 0, in contrast to thefunctionx−Nf /2. A practical way to obtainP (3) is touse some approximation scheme for the inverse squareroot. A simple possibility is to use a Newton iteration

P(3)k+1=

1

2

(P

(3)k +

1

P(3)k P (2)

), k = 0,1,2, . . . .

(14)

The second term on the right hand side can be evalu-ated by a polynomial approximation as forP (2) in (6)with Nf = 0 and P (1) → P

(3)k P (2). The iteration

in (14) is fast converging and allows for an iterativeprocedure stopped by a prescribed precision. A start-ing polynomial P (3)

0 can be obtained, for instance,from P (2) in (6) with Nf →−1

2Nf .The TSMB algorithm becomes exact only in the

limit of infinitely high polynomial order:n2 → ∞

I. Montvay / Physics Letters B 527 (2002) 155–160 157

in (6). Instead of investigating the dependence of ex-pectation values onn2 by performing several simula-tions, it is better to fix some relatively high ordern2for the simulation and perform another correction inthe “measurement” of expectation values by still finerpolynomials. This is done byreweighting the configu-rations.

The reweighting for generalNf is based on a

polynomial approximationP (4)n4 which satisfies

limn4→∞

P (1)n1

(x)P (2)n2

(x)P (4)n4

(x)= x−Nf /2,

(15)x ∈ [ε′, λ].The interval[ε′, λ] can be chosen by convenience, forinstance, such thatε′ = 0, λ = λmax, whereλmax isan absolute upper bound of the eigenvalues ofQ2.In practice, instead ofε′ = 0, it is more effective totake ε′ > 0 and determine the eigenvalues belowε′and the corresponding correction factors explicitly [6].With properly chosen[ε′, λ] the limit n4 → ∞ isexact on an arbitrary set of gauge configurations. Forthe evaluation ofP (4)

n4 one can usen4-independentrecursive relations [8], which can be stopped byobserving the required precision of the result. Afterreweighting, the expectation value of a quantityO isgiven by

(16)〈O〉 =⟨O exp

η†[1−P

(4)n4 (Q2)]η⟩

U,η⟨exp

η†[1− P

(4)n4 (Q2)]η⟩

U,η

,

whereη is a simple Gaussian noise likeη′ in (11).Here 〈· · ·〉U,η denotes an expectation value on thegauge field sequence, which is obtained in the two-step process described above, and on a sequence ofindependentη’s.

The necessity and importance of reweighting thegauge field configurations depends on the conditionnumberλ/ε, which grows roughly as the squared in-verse of the fermion mass in lattice units. For relativelyheavy fermions one can choosen2 high enough, suchthat the effect of reweighting is negligible comparedto the statistical errors. This can be checked on a sub-sample of statistically independent configurations andthen the systematic errors due to the finiteness ofn2are under control. The reweighting becomes necessaryonly for very light fermions as, for example, in [7].In cases if reweighting becomes important the com-putational work for obtaining the reweighting factors

is comparable for the calculation of the inverse ofQ2

on the vectorsη in (16). This is typically less than theoff-line calculations performed, for instance, for deter-mining some correlators and their matrix elements. Ofcourse, if the reweighting has some effect, then it hasto be taken into account in the process of estimatingstatistical errors (see [9]).

2. Variable multi-boson update step

The TSMB algorithm is based on the fact thatthe change of the fermion determinant in the updatesequence can be approximated by a substantially lowerorder polynomial than would be required for the finalprecision of the simulation. More generally, one canalso allow for the fermion action in the multi-bosonstep to differ from the true action describing the modelto be simulated. (This has been done in a special casein Ref. [10] where the update step is performed withthe pure gauge action.) Of course, the correction stepsbecome in general more involved because one has tocorrect for the difference of the two actions, too.

Let us denote the auxiliary hermitean fermion ac-tion in the multi-boson step in general byQ0. Insteadof (7) the fermion determinant is now represented by∣∣det

(Q

)∣∣Nf

(17) 1

detP (1)n1

(Q20) detP (2)

n2(Q2

0) detP (2)n2 (Q2)

.

The main difference compared to (7) is that here anadditional polynomial (P (2)) appears which is neededin order to compensate for the use of a different actionQ0 in the first polynomialP (1). The polynomialsappearing in (17) have to satisfy, withR(···) 1 andR (···) 1, the relations

x−Nf /2= 1P (1)(x)R (1)(x)

= 1

P (2)(x)R(2)(x),

(18)P (1)(x)P (2)(x)= R (12)(x).

HereR(2)(x) 1 andR (12)(x) 1 have to be muchbetter approximations thanR (1)(x) 1.

The factor (detP (1))−1 in (17) is generated bythe multi-boson update step using the actionQ0. Theremaining factor, which has to be reproduced by the


noisy Metropolis step, can be written, for instance, as

1

detP (2)(Q20) detP (2)(Q2)

(19)= 1

det(P (2)(Q2

0)12 detP (2)(Q2)P (2)(Q2

0)12) .

Note that here and in what follows the roles ofP (2)

andP (2) can also be exchanged.Using the form in (19) one can write the detailed

balance condition for the acceptance probabilityPA

in (9) as

PA([U ′] ← [U ])PA([U ]← [U ′])

= det(P (2)[U ] 12 P (2)[U ]P (2)[U ] 12 )

det(P (2)[U ′] 12 P (2)[U ′]P (2)[U ′] 12 )

=∫ [dη]exp

(−η†P (2)[U ′] 12 P (2)[U ′]P (2)[U ′] 12 η)∫ [dη]exp

(−η†P (2)[U ] 12 P (2)[U ]P (2)[U ] 12 η) .

(20)

Here the gauge field dependence of the polynomialsis displayed, which reflects the dependence of thefermion actions on the gauge field. The detailedbalance condition can be satisfied similarly to (9) with

A(η; [U ′] ← [U ])

(21)

= exp−η†P (2)[U ′] 12 P (2)[U ′]P (2)[U ′] 12 η

+ η†P (2)[U ] 12 P (2)[U ]P (2)[U ] 12 η.

The required Gaussian can now be obtained from asimple one in (11) by

(22)η = P (2)[U ]− 12 P (2)[U ]− 1

2 η′.

For the evaluation of (21) we need the polynomialapproximations

P (3)n3

(x) P (2)n2

(x)−12 ,

(23)P (3)n3

(x) P (2)n2

(x)−12 , x ∈ [0, λ],

and

P (−3)n−3

(x) P (3)n3

(x)−1 P (2)n2

(x)12 ,

P (−3)n−3

(x) P (3)n3

(x)−1 P (2)n2

(x)12 , x ∈ [0, λ].

(24)

The measurement correction can be performed inan analogous manner as before. For this we needthe polynomial approximationsP (4)(x) and P (4)(x)

corresponding to (18):

x−Nf /2= 1

P (2)(x)P (4)(x)R(24)(x),

(25)P (1)(x)P (2)(x)P (4)(x)= R (124)(x).

The final precision is given now by the quality of theapproximationsR(24)(x) 1 andR (124)(x) 1.

3. An application

Apart from the doubling of the number of poly-nomials in the noisy Metropolis correction step thepresent algorithm is entirely analogous to normalTSMB. As a first application let us here consider thecase where the two hermitean actionsQ0 andQ arethe same Wilson fermion action, except for the valuesof the parametersβ (the gauge coupling) andκ (thehopping parameter).

Let us denote the parameters in the multi-bosonstep by (β0, κ0) and in the noisy Metropolis stepby (β,κ). An interesting question is how for fixedparameters (β,κ) the change of (β0, κ0) does influencethe behaviour of the algorithm. To see this I performeda test run with two flavours of Wilson fermions on an83 × 16 lattice at parameters (β = 5.28, κ = 0.160).(This corresponds to a point on theNt = 4 cross-overline of [11].) The main polynomial parameters were asfollows: [ε,λ] = [0.00875,2.8], n1= 24,n2= 70.

The change of the acceptance rate in the noisyMetropolis step for changingβ0, respectively,κ0 isshown by Fig. 1. As one can see, the acceptanceremains quite good in a relatively broad range ofparameters. This is partly due to the fact that thedistribution of the exponent in the noisy Metropolisstep becomes substantially broader when the multi-boson update parameters (β0, κ0) differ from (β,κ)(see Fig. 2).

Changing (β0, κ0) can be used to introduce morerandomness in the update process. In fact, since thefinal distribution of gauge configurations does notdepend on the parameters of the multi-boson update,during a run one can randomly change (β0, κ0). Thisimproves the autocorrelations. As an example, the

I. Montvay / Physics Letters B 527 (2002) 155–160 159

Fig. 1. The average acceptance of the noisy Metropolis step at (β = 5.28,κ = 0.16). On left (right): changingβ0 (κ0) in the multi-boson updatestep.

Fig. 2. The distribution of the exponent in the noisy Metropolis step at (β = 5.28,κ = 0.16). The parameters in the multi-boson update step: onthe left (right)β0 = 5.28,κ0= 0.160 (β0= 5.28,κ0= 0.157). The vertical lines show the mean value of the distributions (and zero).


measured integrated autocorrelation of the plaquettein the original TSMB run with the above polynomialparameters isτplaq

int = 2.9(3) × 105 MVM (Matrix-Vector Multiplications by the even-odd preconditionedWilson–Dirac matrix). This is decreased toτplaq

int =1.2(3)× 105 MVM if (for fixed κ0= κ) β0 is changedrandomly during the gauge update in the intervalβ0 ∈[5.16,5.40]. For counting the number of MVM’s inan update cycle the following approximate formula isused:

NMVM 6(n1NB +NG)

(26)+ 2NC(n2+ n3+ n3+ n−3).

Here NB is the number of boson field updates,NG

the gauge field updates andNC the number of noisyMetropolis accept-reject steps. The autocorrelationswere determined in independent runs which were atleast as long as hundred times the measured autocor-relation. In these test runs with relatively heavy quarksthe polynomial ordern2 is high enough and the effectof reweighting in (16) is much smaller than the statis-tical errors. Therefore, the autocorrelations were de-termined without taking into account the reweightingfactors.

The decrease of the plaquette autocorrelation asan effect of updating with a “wrong” action is atthe first sight against intuition. This unexpected effectcan be understood as due to partly compensating thedominance of the bosonic contributions against thepure gauge contributions in the effective gauge action.As a consequence of the gauge field part in (21), thefluctuation of the gauge fields within the natural bandof fluctuations is intensified. At the same time there issome decrease of the acceptance, too, but in total theeffect of the amplified fluctuations is more important.

An interesting question is the volume dependenceof the effect of action variations in the update. Ingeneral, the allowed action changes may be expectedto decrease in larger lattice volumes. However, onehas to observe that the range with good acceptance inFig. 1 is much larger than the range which would beallowed for entirely updating with different parameters

and performing the necessary reweighting afterwardson the equilibrium configurations. The main reason isthat the distance between two noisy Metropolis steps istypically less than 1% of the plaquette autocorrelationdistance. In addition, on larger volumes the number of

boson fieldsn1 is also larger and the dominance of thebosonic part in the effective gauge action is stronger.Therefore, the question of the volume dependence isnot easy to answer and can be best done by performingactual test runs on larger volumes [12].

Another interesting possibility is to apply TSMBfor variable actions in numerical simulations withimproved fermion actions. Using a simplified versionof the particular improved action in the multi-bosonupdate step facilitates the implementation of TSMBalgorithms and may also improve the autocorrelations.

Acknowledgement

I thank Hartmut Wittig and Claus Gebert for acritical reading of the manuscript.

References

[1] M. Lüscher, Nucl. Phys. B 418 (1994) 637.[2] A. Borici, Ph. de Forcrand, Nucl. Phys. B 454 (1995) 645.[3] I. Montvay, Nucl. Phys. B 466 (1996) 259.[4] R. Frezzotti, K. Jansen, Phys. Lett. B 402 (1997) 328.[5] I. Montvay, Nucl. Phys. Proc. Suppl. 83 (2000) 188.[6] I. Campos et al., DESY-Münster Collaboration, Eur. Phys. J.

C 11 (1999) 507.[7] S. Hands et al., DESY-Swansea Collaboration, Eur. Phys. J.

C 17 (2000) 285.[8] I. Montvay, Comput. Phys. Commun. 109 (1998) 144;

I. Montvay, in: Numerical Challenges in Lattice QuantumChromodynamics, Wuppertal 1999, Springer, 2000, p. 153;I. Montvay, hep-lat/9911014.

[9] R. Frezzotti et al., ALPHA Collaboration, Comput. Phys.Commun. 136 (2001) 1.

[10] F. Knechtli, A. Hasenfratz, Phys. Rev. D 63 (2001) 114502.[11] T. Blum et al., Phys. Rev. D 50 (1994) 3377.[12] F. Farchioni, C. Gebert, I. Montvay, L. Scorzato, in preparation.



Measurement ofe+e− → π+π− cross-sectionwith CMD-2 aroundρ-meson

R.R. Akhmetshina, E.V. Anashkina, A.B. Arbuzovb, V.M. Aulchenkoa,c,V.Sh. Banzarova, L.M. Barkova, S.E. Barua, N.S. Bashtovoya, A.E. Bondara,c,D.V. Bondareva, A.V. Bragina, D.V. Chernyaka, S. Dhawane, S.I. Eidelmana,c,

G.V. Fedotovicha,c, N.I. Gabysheva, D.A. Gorbacheva,c, A.A. Grebenuka,D.N. Grigorieva,c, V.W. Hughese, F.V. Ignatova,c, S.V. Karpova, V.F. Kazanina,B.I. Khazina,c, I.A. Koopa,c, P.P. Krokovnya, E.A. Kuraevb, L.M. Kurdadzea,

A.S. Kuzmina,c, I.B. Logashenkoa,d, P.A. Lukina, A.P. Lysenkoa, K.Yu. Mikhailov a,J.P. Millerd, A.I. Milstein a,c, I.N. Nesterenkoa, V.S. Okhapkina, A.A. Polunina,

A.S. Popova, T.A. Purlatza, B.L. Robertsd, N.I. Roota, A.A. Rubana, N.M. Ryskulova,A.G. Shamova, Yu.M. Shatunova, B.A. Shwartza,c, A.L. Sibidanova, V.A. Sidorova,A.N. Skrinskya, V.P. Smakhtina, I.G. Snopkova, E.P. Solodova,c, P.Yu. Stepanova,

A.I. Sukhanova, J.A. Thompsonf, V.M. Titov a, Yu.Y. Yudina, S.G. Zvereva

a Budker Institute of Nuclear Physics, Novosibirsk, 630090, Russiab Joint Institute of Nuclear Research, Dubna, 141980, Russiac Novosibirsk State University, Novosibirsk, 630090, Russia

d Boston University, Boston, MA 02215, USAe Yale University, New Haven, CT 06511, USA

f University of Pittsburgh, Pittsburgh, PA 15260, USA

Received 14 December 2001; received in revised form 21 December 2001; accepted 8 January 2002

Editor: L. Montanet

Abstract

The cross-section of the processe+e− → π+π− has been measured using about 11 4000 events collected by the CMD-2detector at the VEPP-2Me+e− collider in the center-of-mass energy range from 0.61 to 0.96 GeV. Results of the pionform factor determination with a 0.6% systematic uncertainty are presented. The following values of theρ- andω-mesonparameters were found:Mρ = (776.09 ± 0.81) MeV, Γρ = (144.46 ± 1.55) MeV, Γ (ρ → e+e−) = (6.86 ± 0.12) keV,Br(ω → π+π−) = (1.33± 0.25)%. Implications for the hadronic contribution to the muon anomalous magnetic moment arediscussed. 2002 Elsevier Science B.V. All rights reserved.

E-mail address: [email protected] (S.I. Eidelman).



162 R.R. Akhmetshin et al. / Physics Letters B 527 (2002) 161–172

1. Introduction

The cross-section of the processe+e− → π+π− isusually expressed in terms of the pion electromagneticform factorFπ(s):

(1)σe+e−→π+π− = πα2

3sβ3π

∣∣Fπ(s)∣∣2,wheres is the center-of-mass (c.m.) energy squared,mπ is the pion mass andβπ =√

1− 4m2π/s is the pion

velocity in the c.m. frame.The pion form factor measurement is crucial for

a number of physics problems. Detailed experimentaldata in the time-like region allow a determination ofthe parameters of theρ(770) meson and its radialexcitations. Extrapolation of the energy dependence ofthe pion form factor to the points = 0 provides a valueof the pion electromagnetic radius.

In the energy range belowφ(1020) the processe+e− → π+π− gives the dominant contribution to thequantityR(s) defined as

R = σ(e+e− → hadrons)

σ (e+e− →µ+µ−).

R(s) is an important measurable quantity widelyused for various QCD tests as well as calculationsof the dispersion integrals. For such applications, athigh energiesR(s) is usually calculated within theperturbative QCD frame, while for the low energyrange the direct measurement of thee+e− → hadronscross-section is necessary.

Particularly, knowledge ofR(s) with high accu-racy is required for the evaluation of the hadronic con-tribution ahad

µ to the anomalous magnetic moment ofthe muon(g − 2)/2 (see [1] and references therein).The ultimate goal of the experiment E821 [2] run-ning in Brookhaven National Laboratory is to measurethe muon anomalous magnetic moment with a rela-tive precision of 0.35 ppm. Within the Standard Model(SM) the uncertainty of the theoretical value of theleading orderaµ is dominated by the uncertainty ofthe hadronic contributionahad

µ calculated via the dis-persion integral

ahadµ =

(αmµ

3π

)2 ∞∫4m2

π

R(s)K(s)

s2 ds

(2)= m2µ

12π3

∞∫4m2

π

σ (s)K(s)

sds,

whereK(s) is the QED kernel andσ(s) is the cross-section of e+e− → hadrons. The precision of theahadµ calculation depends on the approach used and

varies from 1.34 ppm based one+e− data only [3]to 0.53 ppm if in additionτ -lepton decay data aswell as perturbative QCD and QCD sum rules areextensively used [4]. The major contribution to itsuncertainty comes from the systematic error of theR(s) measurement at low energies (s < 2 GeV2),which, in turn, is dominated by the systematic errorof the measured cross sectione+e− → π+π− .

Assuming conservation of the vector current (CVC)and isospin symmetry, the spectral function of theτ− → π−π0ντ decay can be related to the isovectorpart of the pion form factor [5]. The detailed mea-surement of the spectral functions was provided byALEPH [6], OPAL [7] and CLEO-II [8]. The compar-ison of the pion form factor measured ate+e− collid-ers with the spectral function of theτ− → π−π0ντdecay provides a test of CVC. If CVC holds withhigh accuracy,τ -lepton decay data can be also usedto improve the accuracy of the calculations mentionedabove [9,10].

E821 has recently published the result of its mea-surement ofaµ with an accuracy of 1.3 ppm [11].The measured value ofaµ is 2.6 standard deviationshigher than the SM prediction of [4].1 This observa-tion makes new high precision measurements of thee+e− → hadrons cross-section and particularly of thepion form factor extremely important.

Since early 70s the VEPP-2Me+e− collider hasbeen running in the Budker Institute of NuclearPhysics in the c.m. energy range 360–1400 MeV. Themost precise pion form factor data were obtained inlate 70s–early 80s by CMD and OLYA detectors [13].Their accuracy was limited by systematic errors of the

1 Recent progress in estimating the light-by-light scatteringcontribution to ahad

µ [12] implies that the difference betweenexperiment and theory reduces to about 1.5 standard deviations.

R.R. Akhmetshin et al. / Physics Letters B 527 (2002) 161–172 163

experiments, varying from 2 to 15% over the VEPP-2M energy range.

The CMD-2 detector installed in 1991 is a generalpurpose detector consisting of the drift chamber, theproportional Z-chamber, the barrel CsI calorimeter,the endcap BGO calorimeter installed in 1996, andthe muon range system. The drift chamber, Z-chamberand the endcap calorimeters are placed inside a thinsuperconducting solenoid with a field of 1 T. Moredetail on the detector can be found elsewhere [14].

Though the collider luminosity has not consider-ably increased since the previous work in the 1980s,the present detector design is significantly improved,resulting in a smaller systematic error of the pion formfactor in this measurement. Particularly, the followingadvantages of CMD-2 should be mentioned comparedto previous detectors: simultaneous measurement ofthe particle momentum and energy deposition whichsimplified particle identification and helped to reducebackground; high precision determination of the fidu-cial volume with the help of the Z-chamber; a thinnerbeam pipe which reduced the nuclear interactions ofpions.

During 1994–1995 a detailed scan of the c.m. en-ergy range 610–960 MeV was performed. Since thepion form factor changes relatively fast in this energyrange dominated by theρ-meson, the systematic errordue to an uncertainty in the energy measurement canbe significant. To reduce this contribution to a negligi-ble level, the beam energy was measured with the helpof the resonance depolarization technique with an ac-curacy of 140 keV for almost all energy points [15].This Letter presents the final analysis of the data takenin 1994–1995 with the integrated luminosity of about310 nb−1 and a systematic uncertainty of the cross sec-tion of 0.6%. Preliminary results with a systematic un-certainty of 1.4% were published in [16], based on thesame data sample.

2. Data analysis

The data were collected at 43 points with c.m.energy ranging from 610 to 960 MeV in 10 MeVenergy steps, except for the narrow energy range neartheω-meson, where the energy steps were 2–6 MeV.

From more than 4× 107 triggers recorded, about3 × 105 events were selected ascollinear, with a

signature of two particles of opposite charge andnearly back-to-back momenta originating from theinteraction point. The following selection criteria wereused:

(1) Two tracks of opposite charge originating fromthe interaction region are reconstructed in the driftchamber.

(2) The distance from the vertex to the beam axis,ρ,is less than 0.3 cm and thez-coordinate of thevertex (along the beam axis) is within−15< z <

15 cm.(3) The average momentum of the two particles(p1+

p2)/2 is between 200 and 600 MeV/c.(4) The difference between the azimuthal angles (in

the plane perpendicular to the beam axis) of twoparticles|ϕ| = |π − |ϕ1 − ϕ2||< 0.15.

(5) The difference between the polar angles (the anglebetween the momentum and the beam axis) of twoparticles|Θ| = |Θ1 − (π −Θ2)|< 0.25.

(6) The average polar angle of two particlesΘavr =[Θ1 + (π − Θ2)]/2 is within 1.1 < Θavr <

(π − 1.1). This criterion determines the fiducialvolume.

The selected sample of collinear events containse+e− → e+e−, e+e− → π+π−, e+e− → µ+µ−events (below referred to as beam originating) as wellas the background of cosmic particles which passnear the interaction region and are misidentified ascollinear events. The number of cosmic backgroundeventsNcosmic was determined by the analysis of thespatial distribution of the vertex. Both distributions ofthe longitudinal coordinate (z) and the distance fromthe beam axis (ρ) are peaked around zero for thebeam originating events, but are very broad, almostflat for the cosmic background events. Typicalρ- andz-distributions are shown in Fig. 1.

The momenta ofe, µ, π from the beam originatingevents are rather close (the difference is comparableto the momentum resolution of the drift chamber), sothe overlap of the momentum distributions is large.Therefore, separation ofe, µ andπ by their momentais impossible at

√s > 600 MeV.

On the contrary, the energy deposition of theparticles in the calorimeter [17] is quite different fore, µ and π . The typical energy deposition of twoparticles (E+ vs.E−) for experimental events selected


Fig. 1. Spatial distribution of the vertex. The left plot shows the distribution of the distance from the vertex to the beam axis (ρ), the rightplot presents the distribution of the distance between the vertex and the center of the detector along the beam axis (z). The open histogramcorresponds to all collinear events, the filled one shows the subset of background events.

Fig. 2. Energy deposition of collinear events for the beam energy of400 MeV.

at the beam energy of 400 MeV is shown in Fig. 2.The high deposition spot corresponds toe+e− pairs,where both particles leave almost all their energy inthe calorimeter. The low deposition spot representsµ+µ− pairs, cosmic muons and thoseπ+π− pairs,in which both particles interact as minimum-ionizing.The long tails correspond toπ+π− pairs in which oneor both particles undergo nuclear interactions insidethe calorimeter.

Therefore, the energy deposition of the particleswas used for the separation of the beam originatingevents. Since the overlap of the distributions fore+e−andπ+π− pairs is small, this approach gives stableresults with a small systematic error. However,µ+µ−and π+π− pairs cannot be separated well by theirenergy deposition. To avoid this problem, the numberof µ+µ− pairs was derived from the number ofe+e− pairs according to QED, taking into accountradiative corrections and detection efficiencies. Sincein this energy range the number ofµ+µ− pairs issmall compared to that ofπ+π−, the systematic errorcaused by the corresponding calculation is negligible(less than 0.03%).

The separation was based on the minimization ofthe following unbinned likelihood function:

(3)

L= −∑

events

ln

(∑a

Na · fa(E+,E−))+

∑a

Na,

wherea is the event type (a = ee, µµ, ππ , cosmic),Na is the number of events of the typea andfa(E

+,E−) is the probability density function (p.d.f.)for a typea event to have energy depositionsE+ andE−. It was assumed thatE+ andE− are not corre-lated for events of the same type, so the p.d.f. can befactorized as

fa(E+,E−)= f+

a

(E+) · f−

a

(E−),


wheref±a (E) is the p.d.f. fore±, µ±, π± and cosmic

muons to have energy deposition equal toE. Thisassumption is not entirely correct since there aresmall correlations betweenE+ andE− because of thedependence of the calorimeter thickness on the polarangle as well as the c.m. energy shift due to the initialstate radiation. The first effect was corrected for whilethe second one was studied with the help of simulationand was shown to be negligible (below 0.1%).

For e+e−, µ+µ− pairs and cosmic events theenergy deposition does not depend on the particlecharge, while the energy depositions forπ+ andπ−are different. Thereforef+

a ≡ f−a for a = ee,µµ and

cosmic, but in case of pions the corresponding p.d.f.’sare different.

As was mentioned before, the ratioNµµ/Nee wasfixed during minimization according to the QEDcalculation

Nµµ

Nee

= σµµ · (1+ δµµ)εµµ

σee · (1+ δee)εee,

whereσ is the Born cross-section,δ is the radiativecorrection andε is the detection efficiency which in-cludes acceptance as well as reconstruction and trig-ger efficiencies. The number of cosmic eventsNcosmicdetermined separately was fixed during the minimiza-tion while its fluctuation was added in quadrature tothe fluctuation ofNππ .

To obtain the specific form of p.d.f.’s, the energydeposition ofe, µ andπ in the CMD-2 calorimeterwas studied. For electrons (positrons) and cosmicmuons it can be obtained with the help of the dataitself. Particles of positive charge with large enoughenergy deposition are almost 100% positrons andthey were used to tag electrons. Events with a largevalue of ρ are mostly cosmic muons. Such taggedparticles were used to determine the energy depositionof electrons and cosmic muons, respectively.

The simulation was used to obtain the energy de-position of muons frome+e− → µ+µ−. In the en-ergy range under study these muons interact purely asminimum-ionizing particles which are well describedby the simulation.

On the contrary, the simulation of the interactionof low energy pions inside the calorimeter is notreliable enough. In addition, there is no good methodto tag pions frome+e− → π+π− events. Therefore,the p.d.f.’s forπ+ and π− were obtained from the

analysis of the energy deposition of pions comingfrom the φ(1020) → π+π−π0 decay. From a largedata sample collected by CMD-2 around theφ-mesonpeak, about 105 φ(1020)→ 3π events were selectedwith practically no background. Pions found in theseevents cover the whole interesting range of momentaand angles. The energy deposition of these pions wasanalyzed and the parameterization forπ+ and π−p.d.f.’s was derived.

Finally, to simplify the final error calculation, thelikelihood function (3) was rewritten to have thefollowing global fit parameters:

(Nee +Nµµ),Nππ

Nee +Nµµ

,

instead ofNee andNππ (with Nµµ/Nee andNcosmicfixed). The likelihood function has some other fit pa-rameters characterizing the p.d.f.’s for different typesof particles, such as the mean energy, energy resolu-tion, the asymmetry of some distributions etc. Moredetail about the energy deposition of different types ofcollinear events and the p.d.f. parameterization can befound in [16].

After the separation procedure the following num-ber of events was obtained for three described aboveclasses:Nee + Nµµ = 180038, Nππ = 113824 andNcosmic= 17390.

Special studies were performed to estimate thesystematic error of the separation procedure. Thedominant effect was produced by the small non-uniformity of the calorimeter calibration. Due to theforward–backward asymmetry of thee+e− → e+e−cross-section, the calorimeter calibration error leadsto a small difference betweene+ and e− energydepositions. The corresponding error was found tobe less than 0.2%. Several different functional formswere used to parameterize p.d.f.’s ofe’s and π ’sand the final cross section was stable within 0.1%for different selections. The existing variation of thecalorimeter response between calibrations leads tosmall variations of the energy resolution which couldalso influence the results. The estimated contributionof this effect is below 0.1%. As a final test, the largeamount ofe+e− → e+e−(γ ), µ+µ−(γ ), π+π−(γ )events was generated in a proper proportion withthe help of full detector simulation at several energypoints covering the whole energy range. After that thesimulated events were subject to the same separation


Fig. 3. Difference between the initial and the reconstructed valuesof the form factor for the simulated data set.

procedure as for the data. Results are shown in Fig. 3.The reconstructed value is always consistent with theinput one and the average difference is below 0.1%or consistent with zero. Thus, the systematic errorbecause of the event separation taking into account theabove effects is estimated to be 0.2%.

3. Form factor calculation

The pion form factor was calculated as:

|Fπ |2 = Nππ

Nee +Nµµ

× σee · (1+ δee)εee + σµµ · (1+ δµµ)εµµ

σππ · (1+ δππ)(1+N)(1+D)εππ(4)−3π ,

where the ratioNππ/(Nee +Nµµ) was obtained fromthe minimization of (3),σ are the correspondingBorn cross-sections,δ are the radiative corrections,ε are the detection efficiencies,D and N arethe corrections for the pion losses caused by decaysin flight and nuclear interactions, respectively, and3π is the correction for misidentification ofω →π+π−π0 events as thee+e− → π+π−. In the caseof the e+e− → π+π− process,σππ corresponds topoint-like pions.

Radiative corrections were calculated according toRefs. [18,19] in which the accuracy of the obtainedformulae was 0.2%.2 The improved precision of the

2 We corrected a misprint in formula (2.5) of Ref. [19], alsonoted by the authors of Ref. [21].

calculations compared to previous works [20] comesfrom taking into account the radiation of the addi-tional photon in a narrow cone along the direction ofelectrons and positrons. The radiative corrections werecalculated by the Monte Carlo integration of the dif-ferential cross-sections imposing all selection criteria.Analysis was repeated for three various sets of selec-tion criteria corresponding to notably different radia-tive corrections. The obtained form factor values wereconsistent with each other. We estimate the uncertaintyof the form factor because of the radiative correctionsto be 0.4% dominated by the accuracy of the ratio(1+ δee)/(1+ δππ ).

The radiative corrections for the processe+e− →π+π− include the effects of both initial (ISR) and fi-nal state radiation (FSR) and do not include the vac-uum polarization terms (both leptonic and hadronic)since the latter are considered to be an intrinsic partof the hadronic cross-section and corresponding formfactor (1). However, for various applications basedon dispersion relations and involving the total cross-section ofe+e− → hadrons, the radiation by final pi-ons is no longer a radiative correction, so thatπ+π−γwith a photon radiated by one of the final pions shouldbe considered as one of the possible hadronic finalstates contributing to the total cross-section. There-fore, the bare cross-sectione+e− → π+π−(γ ), belowreferred to asσ 0

ππ(γ ), was also calculated as

σ 0ππ(γ ) =

πα2

3sβ3π

∣∣Fπ(s)∣∣2 · ∣∣1−Π(s)∣∣2

(5)×(

1+ α

πΛ(s)

).

The factor|1 − Π(s)|2 with a polarization operatorΠ(s) excludes the effect of leptonic and hadronic vac-uum polarization, so that one obtains the bare cross-section required for various applications.3 A correc-tion for the final state radiationΛ(s) was calculatedbased on [19], where only the effects of FSR, inte-grated over the whole allowed kinematic region, were

3 Note that the definition of the cross-section above is differentfrom that in [16] where the FSR cross-section was not taken intoaccount and only the correction for the leptonic vacuum polarizationwas applied. Therefore, their comparison is meaningless.


taken into account. The result

Λ(s)= 1+ β2π

βπ

4 Li2

(1− βπ

1+ βπ

)+ 2 Li2

(−1− βπ

1+ βπ

)

−[3 ln

(2

1+ βπ

)+ 2 lnβπ

]Lβπ

− 3 ln

(4

1− β2π

)− 4 lnβπ

(6)

+ 1

β3π

[5

4

(1+ β2

π

)2 − 2

]Lβπ + 3

2

1+ β2π

β2π

,

whereLβπ = ln 1+βπ1−βπ , Li2(z) = − ∫ z0 dx

xln(1 − x),

coincides with [21,22].Two different trigger settings were used during data

taking. For energies below 810 MeV only a chargedtrigger was used in which the positive decision wasbased on the information from the tracking systemonly. The efficiency of the charged trigger was mea-sured to be higher than 99% and, more important,equal for different types of collinear events in theenergy range under consideration. Therefore, the un-certainties related to the trigger efficiency cancel in(4). Above 810 MeV the additional requirement thatthe energy deposition in the calorimeter be at least20 MeV was applied for the trigger. The efficiencyof this requirement was measured to be 99.5% fore+e− → π+π−, 99.2% fore+e− → µ+µ− and 100%for e+e− → e+e− events, and was included in thecorresponding detection efficiency. These correctionsgive a negligible contribution to the systematic uncer-tainty.

The reconstruction efficiency was measured usingthe experimental data themselves. It was found tobe within the 98–100% range at all energies and,within the statistical accuracy, nearly the same forall types of collinear events. Therefore, it cancelsin (4). The systematic error of the cancellation wasestimated to be better than 0.2%. It is worth notingthat such a cancellation allows a determination ofthe form factor with better precision than that of theluminosity.

The fiducial volume (detection solid angle) isdetermined by selecting events with the average polarangleΘavr = (Θ1 + π −Θ2)/2 in the range betweenΘmin and(π − Θmin) with Θmin = 1.1. The value ofΘavr is determined by the CMD-2 Z-chamber [23]which has a spatial resolution along the beam axis

better than 0.6 mm. That corresponds to a systematicerror in the form factor of about 0.2%. To test thisestimate, the pion form factor was also determinedat Θmin = 1.0 radian. The difference between theform factor values determined at two values ofΘminaveraged over the c.m. energy range was found to be(0.1± 0.3)%, consistent with zero.

The correctionD for the pion losses causedby decays in flight was calculated with the helpof simulation. Its value, varying from 0.2% at 600MeV to 0.03% around theφ(1020)-meson, turnedout to be small because of the small size of thedecay volume and the fact that the maximum decayangle is small enough and of the same magnitudeas the angular resolution of the drift chamber. Thiseffect gives a negligible contribution to the systematicuncertainty.

The correctionN for the pion losses caused bynuclear interactions inside the wall of the beam pipeor the drift chamber was calculated with the helpof FLUKA-based simulation [24]. The value of thecorrection is slowly changing from 1.7% at 600 MeVto 0.8% at theφ(1020)-meson energy. The systematicerror of 0.2% for theN calculation was estimatedfrom the uncertainty of the nuclear cross-sections usedin FLUKA [25].

There is also a small correction for the lossesof e+e− → e+e− events, caused by interactions ofelectrons with material of the beam pipe and the driftchamber. It was taken into account by reducingN by0.15% according to the simulation.

Use of the resonance depolarization for the absolutebeam energy calibration reduced the systematic uncer-tainty of the form factor due to the beam energy mea-surement to 0.1%.

In the narrow energy range around theω(782)-meson there is a small background ofe+e− →π+π−π0 events, misidentified ase+e− → π+π−.The corresponding correction3π was calculatedfrom the simulation. It reaches its maximum valueof about 1% at theω(782)-meson energy and dropsfast to nearly zero at the energies outside theω-meson.

The measured values of the pion form factor as wellas those of the baree+e− → π+π−(γ ) cross-sectionobtained from (5) are shown in Table 1. Only statisti-cal errors are given. The main sources of the system-atic error are listed in Table 2. The overall system-


Table 1The measured values of the pion form factor and bare cross-sectione+e− → π+π−(γ ). Only statistical errors are shown. Thesystematic error is estimated to be 0.6%

2E (MeV) |Fπ |2 σ0ππ(γ )

, nb

610.50 8.00 ± 1.13 327.8 ± 46.2620.50 9.71 ± 0.72 390.6 ± 29.1630.50 10.82 ± 0.73 426.9 ± 28.6640.51 11.02 ± 0.72 426.2 ± 27.9650.49 12.66 ± 0.83 480.3 ± 31.6660.50 13.40 ± 0.75 498.8 ± 27.8670.50 15.36 ± 0.82 560.5 ± 30.0680.59 19.49 ± 0.90 697.2 ± 32.3690.43 20.49 ± 0.73 718.6 ± 25.5700.52 23.81 ± 0.62 818.1 ± 21.2710.47 27.52 ± 0.99 926.1 ± 33.4720.25 31.80 ± 0.82 1048.1 ± 27.0730.24 34.34 ± 1.18 1107.2 ± 38.1740.20 37.82 ± 1.18 1191.9 ± 37.1750.28 42.33 ± 1.18 1302.9 ± 36.2760.18 43.44 ± 1.21 1306.9 ± 36.5764.17 43.23 ± 1.03 1289.6 ± 30.6770.11 44.23 ± 1.14 1304.6 ± 33.5774.38 43.19 ± 1.15 1265.7 ± 33.7778.17 45.41 ± 1.29 1321.6 ± 37.7780.17 42.90 ± 1.21 1234.9 ± 34.8782.23 36.59 ± 0.66 1026.6 ± 18.5784.24 34.30 ± 0.97 935.9 ± 26.5786.04 30.01 ± 1.07 809.5 ± 28.9790.10 31.96 ± 1.11 859.3 ± 30.0794.14 29.81 ± 0.85 799.6 ± 22.7800.02 29.40 ± 0.68 782.7 ± 18.1810.14 25.60 ± 0.55 669.7 ± 14.4820.02 24.09 ± 0.78 619.0 ± 20.0829.97 20.52 ± 0.73 517.5 ± 18.5839.10 16.75 ± 0.70 415.4 ± 17.3849.24 14.16 ± 0.70 344.8 ± 17.0859.60 14.47 ± 0.67 345.7 ± 15.9869.50 11.10 ± 0.46 260.7 ± 10.7879.84 10.13 ± 0.79 233.6 ± 18.1889.72 8.44 ± 0.34 191.1 ± 7.7900.04 7.74 ± 0.29 172.1 ± 6.4910.02 6.82 ± 0.31 149.1 ± 6.8919.56 6.03 ± 0.30 129.7 ± 6.5930.11 5.70 ± 0.36 120.3 ± 7.7942.19 5.16 ± 0.25 106.8 ± 5.1951.84 4.56 ± 0.23 92.9 ± 4.8961.52 4.29 ± 0.23 86.2 ± 4.6

atic error obtained by summing individual contribu-tions in quadrature is about 0.6%. The values of thepion form factor and the bare cross-section in Table 1supersede our preliminary results presented in Table 3of Ref. [16].

Table 2Main sources of the systematic errors

Source Contribution (%)

Event separation 0.2Radiative corrections 0.4Detection efficiency 0.2Fiducial volume 0.2Correction for pion losses 0.2Beam energy determination 0.1

Total 0.6

4. Fit to data

The parameterization of the pion form factor in theenergy range under study should include contributionsfrom theρ(770), ω(782) andρ(1450) resonances. Itwas assumed that the only mechanism for theω →π+π− decay is ρ–ω mixing. Following [26], werepresent the wave function of theω(782)-meson as

|ω〉 = |ω0〉 + ε|ρ0〉 ,where|ω0〉 and|ρ0〉 are the pure isoscalar and isovec-tor, respectively, andε is theρ–ω mixing parameter.Then, in the energy region close to theρ(770)- andω(782)-meson masses, the form factor can be writtenas

Fπ(s)=[

Fρ

s −M2ρ

+ εFω

s −M2ω

][Fρ

−M2ρ

+ εFω

−M2ω

]−1

(7)≈ − M2ρ

s −M2ρ

[1+ ε

Fω(M2ω −M2

ρ)s

FρM2ω(s −M2

ω)

],

where we keep only the terms linear inε. ThequantitiesMω andMρ are complex and contain thecorresponding widths.

Combining together the contributions from theρ(770)- andρ(1450)-mesons, and including that fromtheω(782)-meson as in (7), we write the pion formfactor:

Fπ(s)=(

BWGSρ(770)(s) ·

(1+ δ

s

M2ω

BWω(s)

)

(8)+β · BWGSρ(1450)(s)

)(1+ β)−1,

where parametersδ andβ describe the contributionsof the ω(782)- and ρ(1450)-mesons relative to thedominant one of theρ(770)-meson. For theρ(770)


andρ(1450) the GS parameterization is used [27]:

(9)BWGSρ(Mρ)

= M2ρ

(1+ d · Γρ/Mρ

)M2

ρ − s + f (s)− iMρΓρ(s),

where

f (s)= ΓρM2

ρ

p3π (M

2ρ)

(10)

×[p2π (s)

(h(s)− h

(M2

ρ

))

+ (M2ρ − s

)p2π

(M2

ρ

) dhds

∣∣∣∣s=M2

ρ

],

(11)h(s)= 2

π

pπ(s)√s

ln

√s + 2pπ(s)

2mπ

,

andd is chosen to satisfy BWGSρ(Mρ)

(0)= 1:

d = 3

π

m2π

p2π (M

2ρ)

lnMρ + 2pπ(M2

ρ)

2mπ

(12)+ Mρ

2πpπ(M2ρ)

− m2πMρ

πp3π(M

2ρ),

whereMρ, Γρ are theρ(770)-meson mass and width,andpπ(s) is the pion momentum at the squared c.m.energys. For the energy dependence of theρ(770)width, the P-wave phase space is taken:

(13)Γρ(s)= Γρ

[pπ(s)

pπ (M2ρ)

]3[M2ρ

s

]1/2

.

For theω(782)-meson contribution the simple Breit–Wigner parameterization with a constant width is used.

To obtain theρ(770)-meson leptonic widthΓ (ρ →e+e−), the well-known VDM relations were used[28]:

ΓV→e+e− = 4πα2

3M3V

g2V γ and

(14)ΓV→π+π− = g2Vππ

6π

p3π

(M2

V

)M2

V

.

Assuming

gργ gρππ = M2ρ

(1+ d · Γρ/Mρ

)(1+ β)

and

(15)Γρ→π+π− = Γρ,

the following result was obtained:

(16)Γρ→e+e− = 2α2p3π (M

2ρ)

9MρΓρ

(1+ d · Γρ/Mρ)2

(1+ β)2.

Similarly, assuming (14) and

(17)gωγ gωππ = δ ·M2ω · |BWGS

ρ(770)(M2ω)|

(1+ β),

the following expression is obtained for the branchingratio of theω → π+π− decay:

Br(ω → π+π−)

(18)

= 2α2p3π (M

2ω)

9MωΓω→e+e−Γω

∣∣BWGSρ(770)

(M2

ω

)∣∣2 |δ|2(1+ β)2

.

It should be mentioned that our parameterizationof theω(782)-meson contribution is slightly differentfrom that in [6,13]:(1+ δ · s/M2

ω ·BWω(s)) instead of(1+ δ · BWω(s))/(1+ δ). Fits to either parameteriza-tion give the same result.

To estimate the model dependence of the obtainedparameters, another model of the form factor parame-terization was considered—the hidden local symme-try (HLS) model [28,29]. In this model theρ(770)-meson appears as a dynamical gauge boson of a hiddenlocal symmetry in the non-linear chiral Lagrangian.The ρ(1450) contribution is not taken into account,replaced by the non-resonant couplingγπ+π−. Thismodel introduces a real parametera related to thisnon-resonant coupling. The original parameterizationof the pion form factor was modified in a similar wayto take into account theω(782)-meson contribution.

5. Results and discussion

5.1. ρ(770)- and ω(782)-meson parameters

There are several parameters of both models whosevalues have to be taken from other measurements[30,31]. Their values were allowed to fluctuate withinthe stated experimental errors. The following valuesof theω(782)-meson parameters were taken from theCMD-2 experiment [31]:Mω = (782.71±0.08)MeV,Γω = (8.68 ± 0.24) MeV, Γωee = (0.595± 0.017)keV. Parameters of theρ(1450) were taken from [30]:


Fig. 4. Fit of the CMD-2 pion form factor data to the GS model.

Mρ(1450) = (1465± 25) MeV andΓρ(1450) = (310±60) MeV.

Results of the fit in the GS model are shown inFig. 4. Parameters of GS and HLS models obtainedfrom the fit are in good agreement with each otherand are shown in Table 3. The first error is statisti-cal, the second one is systematic. Two effects weretaken into account in the estimation of the latter: thesystematic uncertainty of the form factor measure-ment of 0.6% (Table 2) and the contribution of theρ(1700) resonance missing in the adopted GS para-meterization. The parameterβ effectively describesthe overall contribution of theρ(1450) andρ(1700)and its value is strongly correlated with theρ(1450)

width. To extract theρ(1450) contribution, a fit of thedata should be performed in a broader energy range.As a result, the value ofβ is strongly model depen-dent and is not well-defined in the energy range un-der study. For this reason, its systematic error is notshown.

In Table 4 the final results obtained in the GS modelare compared to the world average values [30]. Thevalues of theρ(770) mass and width shown thereare based on the previous measurements at VEPP-2M [13], ALEPH [6] and CLEO [8]. While the massof the ρ(770) obtained in this Letter is in very goodagreement with the previous measurements, our valueof the width is 2.7 standard deviations lower. Thisdifference can partly be explained by the difference ofour parameterization compared to the previous works.Particularly, the value ofΓρ is correlated with thevalue of argδ. In previous papers the parameterδwas assumed to be real. Fixing argδ = 0 increasesour value ofΓρ by 2 MeV. Such model uncertaintiesas the effect of the complex phase ofδ and thedifference between the results of GS and HLS fitswere not included into the systematic error of the fitparameters.

The leptonic width of theρ(770) is in good agree-ment with the result of [13] quoted by [30]. Our valueof the branching ratioω → π+π− is 1.6 standard de-viations lower than the world average(2.1 ± 0.4)%based on the two most precise measurements frome+e− experiments [13,32]. Again, a different para-meterization of the form factor was used in previousworks. Also note that the parameters of theω(782)-meson, such as mass, width and the leptonic width,have changed, that also affects the extracted value ofthe branching ratio. Our fit to the old data [13] gives

Table 3Results from fits to|Fπ (s)|2 for the GS and HLS models

Parameter GS model HLS model

Mρ (MeV) 776.09± 0.64± 0.50 775.23± 0.61± 0.50Γρ (MeV) 144.46± 1.33± 0.80 143.88± 1.44± 0.80Γ (ρ → e+e−) (keV) 6.86± 0.11± 0.05 6.84± 0.12± 0.05Br(ω→ π+π−) (%) 1.33± 0.24± 0.05 1.32± 0.24± 0.05|δ| (1.57± 0.15± 0.05)× 10−3 (1.57± 0.15± 0.05)× 10−3

argδ 12.6 ± 3.7 ± 0.2 13.0 ± 3.7 ± 0.2β (GS) −0.0695± 0.0053 –a (HLS) – 2.336± 0.016± 0.007χ2/ν 0.92 0.94


Table 4Comparison of the fit to|Fπ(s)|2 in the GS model to the worldaverage values

Parameter GS model World average

Mρ (MeV) 776.09± 0.64± 0.50 775.7± 0.7Γρ (MeV) 144.46± 1.33± 0.80 150.4± 1.6Γ (ρ → e+e−) (keV) 6.86± 0.11± 0.05 6.77± 0.32Br(ω→ π+π−) (%) 1.33± 0.24± 0.05 2.1± 0.4

the value Br(ω → π+π−)= (2.00± 0.34)%, still 1.6standard deviations above our present result.

5.2. Hadronic contribution to the muon anomalousmagnetic moment

Let us estimate the implication of our results foraππµ , the contribution from the annihilation into twopions, which dominates the hadronic contribution to(g − 2)/2. To this end we compare its value in theenergy range studied in this Letter and calculated fromCMD-2 data only to that based on the previouse+e−measurements [13,32]. Table 5 presents results of theaππµ calculations performed using formula (2) and thedirect integration of the experimental data over theenergy range studied in this Letter. As was explainedabove, the bare cross-sectionσ 0

ππ(γ ) was used in thecalculation. The method is straightforward and hasbeen described elsewhere [3]. The first line of theTable 5 (Old data) gives the result based on the dataof OLYA, CMD and DM1 while the second one (Newdata) is obtained from the CMD-2 data only. The thirdline (Old + new data) presents the weighted averageof these two estimates. The assumption about thecomplete independence of the old and new data usedin the averaging procedure seems to be well justified.For convenience, we list separately statistical andsystematic uncertainties in the second column whilethe third one gives the total error obtained by addingthem in quadrature. One can see that the estimatebased on the CMD-2 data is in good agreement withthat coming from the old data. It is worth noting thatthe statistical error of our measurement is slightlylarger than the systematic uncertainty. Because of thesmall systematic error of the new data, the uncertaintyof the new result foraππµ is almost three times betterthan the previous one. As a result, the combinedvalue based on both old and new data is completelydominated by the CMD-2 measurement.

Table 5Contributions of theππ channel to(g − 2)/2

aππµ , 10−10 Total error, 10−10

Old data 374.8 ± 4.1 ± 8.5 9.4New data 368.1 ± 2.6 ± 2.2 3.4Old + new data 368.9 ± 2.2 ± 2.3 3.2

The example above only illustrates the importanceof the improved accuracy. At the present time theanalysis of theππ data as well as other hadronicfinal states in the whole energy range accessible toCMD-2 is in progress. Independent information is alsoavailable [33] or expected in close future from otherexperiments studying low energye+e− annihilation[34]. When all the above mentioned data are takeninto account, one can expect a significant improvementof the overall error ofahad

µ (by a factor of about 2)compared to the previous one based on thee+e− dataonly [3].

The completion of the analysis of theππ datawill also provide a possibility of the precise test ofthe CVC based relation between the cross-section ofe+e− → π+π− and the spectral function in the decayτ− → π−π0ντ . The solution of the problem of thepossible deviation betweene+e− and τ -lepton data[8,10] should also involve a thorough investigation ofthe effects of isospin breaking corrections as well asadditional radiative corrections inτ decays [22,35,36].

6. Conclusion

The following values of theρ- and ω-mesonparameters have been obtained with the Gounaris–Sakurai fit to the formfactor data:

Mρ = (776.09± 0.64± 0.50) MeV,

Γρ = (144.46± 1.33± 0.80) MeV,

Γ(ρ → e+e−)= (6.86± 0.11± 0.05) keV,

Br(ω → π+π−)= (1.33± 0.24± 0.05)%,

argδ = 12.6 ± 3.7 ± 0.2.The measurement presented in this Letter supersedesthe preliminary result [16], obtained from the same


data. Analysis of the much larger data sample col-lected by CMD-2 in 1996 (the 370–540 MeV energyrange), 1997 (the 1040–1380 MeV energy range) and1998 (the second scan of the 370–960 MeV energyrange) is in progress.

Acknowledgements

The authors are grateful to the staff of VEPP-2Mfor excellent performance of the collider, to all engi-neers and technicians who contributed to the CMD-2experiment. We acknowledge numerous useful discus-sions with M. Benayoun, A. Höcker, F. Jegerlehner,W.J. Marciano, K.V. Melnikov and G.N. Shestakov.

This work is supported in part by grants DOEDEFG0291ER40646, INTAS 96-0624, IntegrationA0100, NSF PHY-9722600, NSF PHY-0100468 andRFBR-98-02-17851.

References

[1] T. Kinoshita, B. Nizic, Y. Okamoto, Phys. Rev. D 31 (1985)2108.

[2] R.M. Carey et al., Phys. Rev. Lett. 82 (1999) 1632.[3] S. Eidelman, F. Fegerlehner, Z. Phys. C 67 (1995) 585.[4] M. Davier, A. Höcker, Phys. Lett. B 435 (1998) 427.[5] Y.S. Tsai, Phys. Rev. D 4 (1971) 2821;

H.B. Thacker, J.J. Sakurai, Phys. Lett. B 36 (1971) 103.[6] R. Barate et al., Z. Phys. C 76 (1997) 15.[7] K. Ackerstaff et al., Eur. Phys. J. C 7 (1999) 571.[8] S. Anderson et al., Phys. Rev. D 61 (2000) 112002.[9] R. Alemany, M. Davier, A. Höcker, Eur. Phys. J. C 2 (1998)

123.[10] S. Eidelman, Nucl. Phys. B (Proc. Suppl.) 98 (2001) 281.[11] H.N. Brown et al., Phys. Rev. Lett. 86 (2001) 2227.[12] M. Knecht, A. Nyffeler, hep-ph/0111058;

M. Hayakawa, T. Kinoshita, hep-ph/0112102;

I. Blokland, A. Czarnecki, K. Melnikov, hep-ph/0112117.[13] L.M. Barkov et al., Nucl. Phys. B 256 (1985) 365.[14] R.R. Akhmetshin et al., Preprint BudkerINP 99-11, Novosi-

birsk, 1999;E.V. Anashkin et al., ICFA Instrumentation Bulletin 5 (1988)18.

[15] A.P. Lysenko et al., Nucl. Instrum. Methods A 359 (1995) 419.[16] R.R. Akhmetshin et al., Preprint INP 99-10, Novosibirsk, 1999,

hep-ex/9904027.[17] V.M. Aulchenko et al., Nucl. Instrum. Methods A 336 (1993)

53.[18] A.B. Arbuzov et al., JHEP 10 (1997) 001.[19] A.B. Arbuzov et al., JHEP 10 (1997) 006.[20] F.A. Berends, R. Kleiss, Nucl. Phys. B 177 (1981) 237;

W. Beenakker, F.A. Berends, S.C. van der Marck, Nucl. Phys.B 349 (1991) 323.

[21] A. Hoefer, J. Gluza, F. Jegerlehner, Preprint DESY 00-163,2001, hep-ph/0107154.

[22] K. Melnikov, Int. J. Mod. Phys. A 16 (2001) 4591.[23] E.V. Anashkin et al., Nucl. Instrum. Methods A 323 (1992)

178.[24] P.A. Aarnio et al., Technical report TIS-RP-190, CERN, 1987,

1990.[25] A. Fasso et al., in: Proceedings of the Conference on Advanced

Monte Carlo, Lisbon, 2000, p. 955;A.M. Makhov, Preprint BudkerINP 92-66, Novosibirsk, 1992.

[26] R. Feynman, Photon–hadron Interactions, Benjamin, 1972,Chapter 6.

[27] G.J. Gounaris, J.J. Sakurai, Phys. Rev. Lett. 21 (1968) 244.[28] M. Benayoun et al., Eur. Phys. J. C 2 (1998) 269.[29] M. Bando et al., Phys. Rev. Lett. 54 (1985) 1215.[30] D.E. Groom et al., Eur. Phys. J. C 15 (2000) 1, and 2001 off-

year partial update,http://pdg.lbl.gov.[31] R.R. Akhmetshin et al., Phys. Lett. B 476 (2000) 33.[32] A. Quenzer et al., Phys. Lett. B 76 (1978) 512.[33] J.Z. Bai et al., Phys. Rev. Lett. 84 (2000) 594;

J.Z. Bai et al., hep-ex/0102003.[34] A. Aloisio et al., hep-ex/0107023;

E.P. Solodov, hep-ex/0107027.[35] H. Czyz, J.H. Kühn, Eur. Phys. J. C 18 (2001) 497.[36] V. Cirigliano, G. Ecker, H. Neufeld, Phys. Lett. B 513 (2001)

361.

http://pdg.lbl.gov



Measurement ofD0 production in neutrinocharged-current interactions

CHORUS Collaboration

A. Kayis-Topaksu, G. Onengüt

Çukurova University, Adana, Turkey

R. van Dantzig, M. de Jong, O. Melzer, R.G.C. Oldeman1, E. Pesen, J.L. Visschers

NIKHEF, Amsterdam, The Netherlands

M. Güler2, M. Serin-Zeyrek, R. Sever, P. Tolun, M.T. Zeyrek

METU, Ankara, Turkey

N. Armenise, M.G. Catanesi, M. De Serio, M. Ieva, M.T. Muciaccia, E. Radicioni,S. Simone

Università di Bari and INFN, Bari, Italy

A. Bülte, K. Winter

Humboldt Universität, Berlin, Germany 3

R. El-Aidi, B. Van de Vyver4, P. Vilain5, G. Wilquet5

Inter-University Institute for High Energies (ULB-VUB) Brussels, Belgium

B. Saitta6

Università di Cagliari and INFN, Cagliari, Italy

E. Di Capua

Università di Ferrara and INFN, Ferrara, Italy



174 CHORUS Collaboration / Physics Letters B 527 (2002) 173–181

S. Ogawa, H. Shibuya

Toho University, Funabashi, Japan

A. Artamonov7, M. Chizhov8, M. Doucet9, D. Kolev8, H. Meinhard, J. Panman,I.M. Papadopoulos, S. Ricciardi10, A. Rozanov11, R. Tsenov8, J.W.E. Uiterwijk,

P. Zucchelli12

CERN, Geneva, Switzerland

J. Goldberg

Technion, Haifa, Israel

M. Chikawa

Kinki University, Higashiosaka, Japan

E. Arik

Bogazici University, Istanbul, Turkey

J.S. Song, C.S. Yoon

Gyeongsang National University, Jinju, South Korea

K. Kodama, N. Ushida

Aichi University of Education, Kariya, Japan

S. Aoki, T. Hara

Kobe University, Kobe, Japan

T. Delbar, D. Favart, G. Grégoire, S. Kalinin, I. Maklioueva

Université Catholique de Louvain, Louvain-la-Neuve, Belgium

P. Gorbunov6, V. Khovansky, V. Shamanov, I. Tsukerman

Institute for Theoretical and Experimental Physics, Moscow, Russian Federation

N. Bruski, D. Frekers

Westfälische Wilhelms-Universität, Münster, Germany 3

K. Hoshino, M. Komatsu, M. Miyanishi, M. Nakamura, T. Nakano, K. Narita, K. Niu,K. Niwa, N. Nonaka, O. Sato, T. Toshito

Nagoya University, Nagoya, Japan

CHORUS Collaboration / Physics Letters B 527 (2002) 173–181 175

S. Buontempo, L. Castagneto, A.G. Cocco, N. D’Ambrosio, G. De Lellis, F. Di Capua,G. De Rosa, A. Ereditato, G. Fiorillo, T. Kawamura6, M. Messina, P. Migliozzi,

V. Palladino, P. Strolin, V. Tioukov

Università Federico II and INFN, Naples, Italy

K. Nakamura, T. Okusawa

Osaka City University, Osaka, Japan

U. Dore, P.F. Loverre, L. Ludovici, P. Righini, G. Rosa, R. Santacesaria, A. Satta,F.R. Spada

Università La Sapienza and INFN, Rome, Italy

E. Barbuto, C. Bozza, G. Grella, G. Romano, C. Sirignano, S. Sorrentino

Università di Salerno and INFN, Salerno, Italy

Y. Sato, I. Tezuka

Utsunomiya University, Utsunomiya, Japan


Editor: L. Montanet

Abstract

During the years 1994–1997, the emulsion target of the CHORUS detector was exposed to the Wide Band Neutrino Beamfrom the CERN-SPS. About 170 000 neutrino interactions were successfully located in the emulsion. Improvements in theautomatic emulsion scanning systems and application of different criteria allowed the sample of located events to be used forstudies of charm production. We present a measurement of the production rate ofD0 mesons based on a sample of 25 693locatedνµ charged-current (CC) interactions analysed so far. After reconstruction of the event topology in the vertex region,283 D0 decays were observed with an estimated background of 9.2 K0 and Λ decays. The ratio of cross-section ofD0

E-mail address: [email protected] (B. Saitta).1 Now at University of Pennsylvania, Philadelphia, USA.2 Now at Nagoya University, Nagoya, Japan.3 Supported by the German Bundesministerium für Bildung und Forschung under contract numbers 05 6BU11P and 05 7MS12P.4 Fonds voor Wetenschappelijk Onderzoek, Belgium.5 Fonds National de la Recherche Scientifique, Belgium.6 Now at CERN, 1211 Geneva 23, Switzerland.7 On leave of absence from ITEP, Moscow.8 On leave of absence from St. Kliment Ohridski University of Sofia, Bulgaria.9 Now at DESY, Hamburg, Germany.

10 Now at Royal Holloway College, University of London, Egham, UK.11 Now at CPPM CNRS-IN2P3, Marseille, France.12 On leave of absence from INFN, Ferrara, Italy.


production andνµ CC interactions is found to be (1.99± 0.13(stat.) ± 0.17(syst.)) × 10−2 at 27 GeV averageνµ energy. 2002 Elsevier Science B.V. All rights reserved.

1. Introduction

Charm production in neutrino charged-current (CC)interactions has been studied in several experiments,in particular, CDHS [1], CCFR [2], CHARM [3],CHARM-II [4], NOMAD [5] and NuTeV [6] bymeans of electronic detectors and through the analysisof dimuon events. In those events, the leading muonis interpreted as originating from the neutrino ver-tex and the other, of opposite charge, as the productof the charmed particle semileptonic decay. Experi-ments of this type, however, suffer from significantbackground (∼ 30%) in which the second muon orig-inates from an undetected decay in flight of a pion ora kaon rather than from a charm decay. Moreover, thetype of charmed particle and its decay topology cannotbe identified in these experiments. Nevertheless, theyhave provided measurements of the strange quark con-tent of the nucleon as well as an estimate of the charmquark mass.

A much lower level of background can be achievedusing an emulsion target which provides a sub-micronspatial resolution, and hence, the topological identifi-cation of charmed hadron decays. However, the statis-tics accumulated in this way, in particular, in the E531experiment at FNAL [7], was limited. Only recently,with the development of automatic scanning devicesof much higher speed within CHORUS have studiesof charm production with high statistics become pos-sible.

In this Letter the first results onD0 productionobtained from a sample of the CHORUS data arereported.

2. The experimental apparatus

The CHORUS detector [8] is a hybrid setup thatcombines a nuclear emulsion target with variouselectronic detectors. The nuclear emulsion is usedas target for neutrino interactions, allowing three-dimensional reconstruction of short-lived particles likethe τ lepton and any charmed hadron. The emulsion

target, which is segmented into four stacks, has anoverall mass of 770 kg, each of the stacks consistingof eight modules of 36 plates of size 36× 72 cm2.Each plate has a 90 µm plastic support coated on bothsides with a 350 µm emulsion layer [9]. Each stackis followed by three interface emulsion sheets with a90 µm emulsion layer on both sides of an 800 µm thickplastic base and by a set of scintillating fibre trackerplanes. The interface sheets and the fibre trackersprovide accurate predictions of particle trajectoriesinto the emulsion stack for the location of the vertexpositions. The accuracy of the fibre tracker predictionis about 150 µm in position and 2 mrad in the trackangle.

The emulsion scanning has been performed by fullyautomatic microscopes equipped with CCD camerasand a read-out system, calledTrack Selector [10]. Inorder to recognize track segments in an emulsion,a series of tomographic images are taken by focusingat different depths in the emulsion thickness. Thedigitized images are shifted according to the predictedtrack angle and then added. The presence of alignedgrains forming a track is detected as a local peak ofthe grey level of the summed image. The track findingefficiency of the track selector is higher than 98% fortrack slopes less than 400 mrad.

The electronic detectors downstream of the emul-sion target include a hadron spectrometer which mea-sures the bending of charged particles in an air-coremagnet, a calorimeter where the energy and direc-tion of showers are measured and a muon spectrom-eter which determines the charge and momentum ofmuons.

3. Data collection

The West Area Neutrino Facility (WANF) at CERNprovides a beam of 27 GeV average energy consistingmainly of νµ with a 5% νµ contamination. Duringthe four years of operation the emulsion target wasexposed to the beam with an integrated intensity whichcorresponds to 5.06×1019 protons on target. The data


from the electronic detectors were analysed and theset of events possibly originating from the emulsionstacks was identified.

Since the CHORUS experiment was designed pri-marily to search forνµ → ντ oscillation, the data se-lection for the first phase of the analysis was optimizedfor the detection of aτ decaying into a single chargedparticle. The selection criteria are described in [11]. Itshould be noted that among the events containing onereconstructed muon track of negative charge (‘one-mu’ sample), only those with a muon momentum lessthan 30 GeV/c were selected for emulsion scanning.This cut has a marginal effect on theτ signal but playsa more important role in the present study of charmproduction. In order to correct for the effect of thiscut, about 3000νµ CC events with a muon momentumgreater than 30 GeV/c were located and analysed.

The emulsion scanning of a one-mu event starts atthe impact point position of the reconstructed muontrack in the interface sheets. The track segments foundin these sheets are then used to predict with highaccuracy the position and angle of the muon trackat the exit of the emulsion stack. The track is thenfollowed from one plate to the next using the segmentsin the most upstream 100 µm part of each plate. The

Table 1Data flow of the one-mu sample

Number of events

Vertex predicted in emulsion 713 000pµ < 30 GeV/c and angular selections 477 600Scanned 355 395Located 143 742Analysed with netscan 22415+ 3278a

a Muon momentum larger than 30 GeV/c.

interaction vertex is assumed to be located if the trackis not observed in two consecutive plates, the firstof which is defined as the ‘vertex plate’. This platemay contain the primary or the decay vertex, or both.The efficiency of this scan-back procedure has only aweak dependence on the momentum and angle (up to400 mrad) of the particle.

The data flow of the one-mu sample is summarizedin Table 1.

To perform the search for charm decays, a newmethod called ‘netscan’ is applied. This method origi-nally developed for the DONUT experiment [12] con-sists in recording all track segments within a largeangular acceptance in a volume surrounding the lo-cated vertex position. In our application, the scan vol-ume is 1.5 mm wide in each transverse direction and6.3 mm long, corresponding to eight plates, i.e., thevertex plate, one plate upstream and six downstream(see Fig. 1). The parameters (positions, slopes, pulseheights, etc.) of all track segments with angles below400 mrad found in this volume are stored in a database.Typically, five thousand track segments are recordedper event. With the scanning systems used, the netscanof one event takes about 11 min. The results presentedhere are based on a sample of 25 693 events analysedwith this method.

4. Reconstruction and selection of decaytopologies

The offline reconstruction program first aims to se-lect from the large number of recorded track segments

Fig. 1. The netscan volume with a sketch of aD0 decay. The ellipses represent track segments in 100 µm emulsion layers in which automaticscanning is performed. Offline programs reconstruct the tracks and associate them to common vertices.


only those belonging to the neutrino interaction understudy. It consists of the following steps:

– A first plate-to-plate alignment is performed bycomparing the pattern of segments in a plate withthe corresponding pattern in the next upstreamplate.

– Each segment of a plate is extrapolated to thenext plate where a matching segment is looked forwithin about 4 µm (3σ of alignment resolution) inposition and 20 mrad in angle. If none is found, theextrapolation is tried one plate further upstream.

– After connection of all matched segments, a sec-ond and more accurate alignment of the plates isperformed using tracks passing through the entirevolume, mainly from muons associated with theneutrino beam or charged particle beams in thesame experimental area. After this alignment, theresidual of the segment positions with respect tothe fitted track is measured to be about 0.45 µm.

– At this stage, about 400 tracks remain in the fidu-cial volume. The majority of them can be recog-nized as due to low energy (typically less than100 MeV) particles (mainly Compton electronsandδ rays) and can be discarded. Moreover, tracksare selected on the basis of theχ2 of the fit to astraight line.

– The final step is the rejection of the tracks passingthrough the scanning volume. After this filtering,the mean number of tracks originating in the scanvolume is about 40.

The program then tries to associate these tracksto common vertices. A detailed description of thealgorithms to reconstruct the vertices is given in [13].In short, a track is attached to a vertex if the distanceof the vertex point to the track line (called hereafterimpact parameter) is less than 10 µm. At the end ofthe procedure, one defines a primary vertex (and itsassociated tracks) and possibly one or more secondaryvertices to which ‘daughter’ tracks are attached.

To select interesting decay topologies while pre-serving a good efficiency forD0 detection, the follow-ing selection is applied:

– The primary muon track and at least one of thedaughter tracks are detected in more than oneplate and the direction measured in emulsion

Table 2Results of visual inspection (eye-scan) of candidates

Accepted events Rejected events

V2 226 Low momentum 174V4 57 Hadron int. 68C1 121 γ -conversion 42C3 124 δ-ray 2C5 7 Other 30

Total 535 316

matches that reconstructed in the fibre trackersystem.

– The impact parameter to the vertex of at least oneof the daughter tracks is larger than a value whichis determined on the basis of the resolution.13

– The impact parameter must also be smaller than400 µm. This cut mainly rejects spurious tracksnot related to the neutrino interaction.

From the present sample of 25 693 analysed events,these criteria select 851 events which are visuallyinspected (eye-scan) to confirm the decay topology.A secondary vertex is accepted as a decay if the num-ber of prongs is consistent with charge conservationand no other activity (Auger electron orblob) is ob-served. The result of the visual inspection is given inTable 2. The purity of the automatic selection is foundto be 63%.

The observable decay topologies are classified asodd-prong decays of a charged particle (mainlyD+,D+

s , Λ+c ) or even-prong decays of a neutral particle

(mainly D0). These are denoted in Table 2 as V2or V4 for neutral and C1, C3 or C5 for charged decaysaccording to the multiplicity.

The rejected sample consists mainly of hadronic in-teractions, delta rays or gamma conversions (∼ 35%)and of low momentum tracks which, due to multiplescattering, appear as tracks with a large impact para-meter (∼ 55%). The remaining 10% consists either offalse vertices, being reconstructed using one or morebackground tracks, or of vertices with a parent tracknot connected to the primary vertex.

13 The impact parameter is required to be greater than√32 + (2σ dx)2 µm; σ =

√0.003052 + (0.0194θ)2 is a parame-

trization of the angular error anddx is the distance of the vertexto the most upstream daughter track segment.


The confirmedD0 sample (V2 and V4) contains283 candidates. For 34 of these events (25 V2 and9 V4) the muon momentum is greater than 30 GeV/c.

5. Reconstruction efficiency and backgroundevaluation

Efficiencies and backgrounds were evaluated witha GEANT3 [14] based simulation of the experiment.Large samples of neutrino interactions were generatedaccording to the beam spectrum using the JETTA[15] generator derived from LEPTO [16] and JETSET[17]. Quasi-elastic reactions were generated with theRESQUE [18] package with a rate of 8.5% relative todeep inelastic scattering reactions.

The simulated response of the CHORUS electronicdetectors is processed through the same reconstructionprogram used for the data. The tracks in emulsionand the performance of the track selector are alsosimulated in order to evaluate the efficiency of thescanning procedure.

The ratio of reconstruction and location efficiencyof events with aD0 in the final state to that ofall νµ CC events is found to be 0.87 ± 0.01. Anadditional 4% uncertainty on this ratio is estimated byvarying the conditions of the simulation.

In order to evaluate the netscan efficiency, realisticconditions of track densities need to be reproduced.This was achieved by merging the emulsion data of thesimulated events with real netscan data which do nothave a reconstructed vertex but contain tracks whichstop or pass through the netscan fiducial volume,representing the real background. The combined dataare passed through the same netscan reconstructionand selection programs used for real data.

The mean efficiencies of the different steps of thenetscan analysis are presented in Table 3, separatelyfor V2 and V4D0 decay topologies.

In Table 3:

– εgeo is the geometrical acceptance of the netscanvolume.

– εnet reflects the performance of the filtering andvertexing algorithms discussed in Section 4.

– εsel measures the efficiency of the specific se-lection criteria applied in this analysis. Amongthese criteria the most sensitive one is the re-

Table 3Efficiencies of the netscan analysis forD0 decays into V2 and V4topologies. The errors quoted are statistical only

V2(%) V4 (%)

εgeo 96.6± 0.2 96.3± 0.5εnet 88.5± 0.4 95.2± 0.6εsel 68.6± 0.6 76.4± 1.1

Combined 58.6± 0.7 70.1± 1.3

quirement that at least one of the decay daugh-ters matches the angles of a track reconstructedin the fibre tracker. This matching efficiency canbe directly measured from the ratio of observedV2 events with one matched prong to those withtwo matched prongs and is found to be(70.5 ±2.6)%, in good agreement with the simulation re-sult (70.7± 2.0)%.

The systematic uncertainties of the netscan efficien-cies arise mainly from the choice of the event gener-ator and from variations in the emulsion data quality.By comparing the results obtained with samples gen-erated with different structure functions (EHLQ [19],GRV [20]) and different charm fragmentation func-tions (Peterson model [21], Bowler model [22]), weestimate the first source of systematic error to be 4.6%.The second contribution is estimated to be 2% bymerging the generated events with different sets ofspurious netscan data corresponding to different trackdensities and different alignment accuracies. The av-erage efficiency of theD0 decay search is found tobe (58.6 ± 5.1)% for V2 and(70.1 ± 5.2)% for V4,where the errors combine statistical and systematic un-certainties in quadrature. The combined detection ef-ficiency (including the factor 0.87 quoted above) isshown in Fig. 2 as a function of neutrino energy. Byprocessing with the same chain of programsνµ CC in-teractions with noD0 in the final state, the backgroundrate is evaluated to be(3.6 ± 1.0) × 10−4 per locatedCC event. In the present sample of 25 693 events thiscorresponds to 9.2 ± 2.6 background events, mainlyK0

s andΛ0 decays. No strong energy dependence ofthis background is observed.

6. Results and conclusion

To estimate theD0 production rate inνµ CCinteractions, an additional weight factor needs to be


Fig. 2. Detection efficiency as a function ofEν for V2 (left) and V4 (right) topologies. The error bars are statistical uncertainties in thesimulation.

applied to the 34D0 events (25 V2 and 9 V4) observedin the sample withPµ > 30 GeV/c, since only a subsetof this category of events has been analysed with thenetscan procedure so far. This factor was found to beequal to 2.6 and was evaluated from the measuredratio of 0.381± 0.001 between CC events withPµ >

30 GeV/c and Pµ < 30 GeV/c and the size of thesamples (see Table 1) included in the current analysis.

Taking into account the estimated efficiencies andbackground described in Section 5, one obtains anaverage ratio of

σ(D0)

σ (CC)= (

1.99± 0.13(stat.) ± 0.17(syst.)) × 10−2.

This result is in good agreement with that of E531 [7](dashed crosses in Fig. 3) based on a statistically lesssignificant sample. The topological ratio V4/V2 isfound to be(23.1 ± 4.0) × 10−2 in agreement withthe world average value

(20.1+2.7

−1.9

) × 10−2 [23].In past years, analyses of events with opposite-sign

dimuons in the final state in electronic detectors haveprovided a large amount of information. In particular,the charm (neutral and charged) production rate as afunction of energy has been determined and a valuefor mc, the effective mass of the charm quark, has beenestimated within the formalism of slow rescaling [24].

Among the other experiments, NOMAD, whichwas exposed to the same neutrino beam as CHO-RUS, has given a measurement of charm production

Fig. 3. D0 production rate as a function of neutrino energy. Theresults of this analysis are shown as solid lines and comparedwith those of the E531 experiment (dashed lines). The data pointsof E531 have been scaled with their measurement of theD0 rate [7]compared to the total charm production rate. The curve shows a fitbased on the slow rescaling model [24] to NOMAD charm data [5]multiplied by the (D0/charm) cross-section ratio measured in thepresent experiment.

extending to low energies. Using the NOMAD [5]results on the total charm production and on themuonic branching ratio of charmed particles,Bc =


0.095+0.007+0.014−0.007−0.013, the value of 0.53± 0.11 has been

obtained for the ratioσ(D0)/σ (charm) in the energyrange of the experiment (〈Eνµ〉 = 27 GeV).

The slow rescaling model withmc = 1.3 GeV/c2

which fits the NOMAD data is superimposed to thedata points of Fig. 3 and it agrees well with theenergy behaviour of the cross-section ratio measuredin emulsion experiments.

A second phase of analysis of the CHORUS datahas started, with improved reconstruction codes andscanning systems. In about one year, a sample of3000 charm events will be collected. A detailed andessentially background-free study of all the charmproduction processes will then become possible.

Acknowledgements

We gratefully acknowledge the help and support ofour numerous technical collaborators who contributedto the detector construction and operation. We thankthe neutrino beam staff for their competent assistance,ensuring the excellent performance of the facility. Theaccumulation of a large data sample in this experimentwas also made possible thanks to the efforts of thecrew operating the CERN PS and SPS. The generaltechnical support from EP (ECP) and IT Divisions isgratefully acknowledged.

The experiment has been made possible by grantsfrom our funding agencies: the Institut Interuniver-sitaire des Sciences Nucleaires and the Interuniver-sitair Instituut voor Kernwetenschappen (Belgium),The Israel Science foundation (Grant 328/94) andthe Technion Vice President Fund for the Promo-tion of Research (Israel), CERN (Geneva, Switzer-land), the German Bundesministerium fur Bildungund Forschung (Grant 057MS12P(0)) (Germany), theInstitute of Theoretical and Experimental Physics(Moscow, Russia), the Instituto Nazionale di FisicaNucleare (Italy), the Promotion and Mutual Aid Cor-poration for Private Schools of Japan and Japan Soci-

ety for the Promotion of Science (Japan), the KoreaResearch Foundation Grant (KRF-99-005-D00004)(Republic of Korea), the Foundation for Fundamen-tal Research on Matter FOM and the National Scien-tific Research Organization NWO (The Netherlands)

and the Scientific and Technical Research Council ofTurkey (Turkey).

References

[1] H. Abramowicz et al., CDHS Collaboration, Z. Phys. C 15(1982) 19.

[2] S.A. Rabinowitz et al., CCFR Collaboration, Phys. Rev. Lett.70 (1993) 134.

[3] M. Jonker et al., CHARM Collaboration, Phys. Lett. B 107(1981) 241.

[4] P. Vilain et al., CHARM II Collaboration, Eur. Phys. J. C 11(1999) 19.

[5] P. Astier et al., NOMAD Collaboration, Phys. Lett. B 486(2000) 35.

[6] M. Goncharov et al., NuTeV Collaboration, hep-ex/0102049.[7] N. Ushida et al., E531 Collaboration, Phys. Lett. B 206 (1988)

375.[8] E. Eskut et al., CHORUS Collaboration, Nucl. Instrum. Meth-

ods A 401 (1997) 7.[9] S. Aoki et al., Nucl. Instrum. Methods A 447 (2000) 361.

[10] T. Nakano, Ph.D. Thesis, Nagoya University, Japan, 1997.[11] E. Eskut et al., CHORUS Collaboration, Phys. Lett. B 503

(2001) 1.[12] K. Kodama et al., DONUT Collaboration, Phys. Lett. B 504

(2001) 218.[13] M. Güler, Ph.D. Thesis, M.E.T.U., Turkey, 2000.[14] GEANT 3.21, CERN program library long write up W5013.[15] P. Zucchelli, Ph.D. Thesis, Università di Ferrara, Italy, 1995.[16] G. Ingelman, Preprint TSL/ISV 92-0065, Uppsala University,

Sweden, 1992.[17] T. Sjöstrand, Comput. Phys. Commun. 82 (1994) 74.[18] S. Ricciardi, Ph.D. Thesis, Università Ferrara, Italy, 1996.[19] E. Eichten, I. Hinchliffe, K. Lane, C. Quigg, Rev. Mod.

Phys. 56 (1984) 579.[20] M. Glück, E. Reya, A. Vogt, Z. Phys. C 67 (1995) 433.[21] C. Peterson et al., Phys. Rev. D 27 (1983) 105.[22] M.G. Bowler, Z. Phys. C 11 (1981) 169.[23] Particle Data Group, Eur. Phys. J. 15 (2000).[24] R.M. Barnett, Phys. Rev. Lett. 36 (1976) 1163;

H. Georgi, H.D. Politzer, Phys. Rev. D 14 (1976) 1829.



Search for neutrinolessββ decay in134Xe

R. Bernabeia, P. Bellia, F. Cappellaa, R. Cerullia, F. Montecchiaa, A. Incicchitti b,D. Prosperib, C.J. Daic

a Dip. di Fisica, Università di Roma “Tor Vergata” and INFN, sez. Roma 2, I-00133 Rome, Italyb Dip. di Fisica, Università di Roma “La Sapienza” and INFN, sez. Roma, I-00185 Rome, Italy

c IHEP, Chinese Academy, P.O. Box 918/3, Beijing 100039, China


Editor: L. Rolandi

Abstract

New results have been obtained by investigating the neutrinolessββ decay in134Xe deep underground at the Gran SassoNational Laboratory of INFN by means of the 6.5 kg liquid xenon setup of the DAMA experiment. New limits on the half-lives of theβ−β−0ν(0+ → 0+) andβ−β−0ν(0+ → 2+) decay modes have been obtained. They are:T1/2 > 5.8 × 1022 yr

(90% C.L.) andT1/2 > 2.6 × 1022 yr (90% C.L.), respectively. TheT1/2 limit for β−β−0ν(0+ → 0+) process is about threeorders of magnitude higher than the one previously available, while theT1/2 limit for β−β−0ν(0+ → 2+) has been set for thefirst time. 2002 Elsevier Science B.V. All rights reserved.

1. Introduction

It has been pointed out that the neutrinoless dou-ble beta decay—which violates the lepton numberconservation—has a great potential to effectivelysearch for neutrino mass, right-handed admixture inthe weak interactions, neutrino coupling constant withMajorons, etc., as sign of a possible physics beyondthe standard model [1,2]. Moreover, at present, theββ0ν decay is considered also as a powerful test fordifferent extensions of the standard model (includingseveral SUSY models), which could offer not onlycomplementary but—in some cases—competitive re-

E-mail address: [email protected] (R. Bernabei).

sults than other running or forthcoming acceleratorand non-accelerator experiments [2–4].

Since the xenon—both in liquid or in gaseousphase—is a good detection medium, the present ex-periment has exploited the active source technique1 tosearch forβ−β−0ν decay processes in134Xe; the rel-evant decay scheme is shown in Fig. 1.

A previous result for this isotope has been obtainedin Ref. [6], where the following limit on the halflifeof the β−β−0ν(0+ → 0+) decay was established:T1/2 > 8.2× 1019 yr at 68% C.L.

1 This technique was originally suggested in Ref. [5] and assures,for example, higher detection efficiencies than the passive sourceapproach.



R. Bernabei et al. / Physics Letters B 527 (2002) 182–186 183

Fig. 1. Scheme of the134Xe ββ decay in the ground state and in the first excited states of134Ba.

2. The experimental setup

The low background liquid xenon (LXe) DAMAsetup and its performances have been described inRefs. [7,8].

In particular, the inner vessel (containing the liq-uid xenon) is made by low activity OFHC copper( 100 µBq/kg for U/Th and 310 µBq/kg for potas-sium).

The LXe scintillation light is collected throughthree windows (10 cm in diameter) made of specialcultured crystal quartz (total transmission of the LXeultraviolet light≈ 80%) by three EMI photomultipli-ers (PMTs) with MgF2 windows. The PMTs are work-ing in coincidence and their quantum efficiency fornormal incidence has a flat behaviour around the LXescintillation wavelength (175 nm); depending on PMT,its value ranges between 18% and 32%.

The passive shield of the setup consists of a lowradioactive copper (10 cm) surrounding the PMTs fol-lowed by 2 cm of steel (thermo-insulation vessel thick-ness), 5–10 cm of low radioactive copper, 15 cm of lowradioactive lead,≈ 1 mm of cadmium and≈ 10 cm ofpolyethylene. The environmental radon near the exter-nal surface of the thermo-insulation vessel is removedby high purity (HP) nitrogen gas, which is continu-ously flushed inside a sealed plexiglass box wrappingthe whole shield.

The inner vessel contains 6.5 kg (i.e., 2 lvolume) of liquid Xe. It has been previously filledwith Kr-free xenon enriched in129Xe at 99.5% [9],while more recently it has been filled with Kr-freexenon enriched in136Xe at 68.8% [8]. Here the datacollected in this latter condition during 6843.8 h havebeen analysed focusing the attention on the presence

of 134Xe at level of 17.1% [10] (this can be comparedwith the134Xe natural abundance: 10.4%).

3. Data analysis and results

The response functions of the detector for theβ−β−0ν(0+ → 0+) and for theβ−β−0ν(0+ → 2+)

decay modes of134Xe have been simulated by usingEGS4 [11] and an event generator code [12]. Herethe angular and energy distributions of the outgoingelectrons have been estimated under the hypothesisthat the right-handed currents can be neglected. Theseresponse functions are given in Fig. 2.

We recall that to estimate the halflife,T1/2, ofa given ββ decay process the standard formula is:T1/2 = N ·T

Ndln2, whereN is the number of atoms of

134Xe, T is the running time andNd is the numberof signal events which can be ascribed to the processsearched for on the basis of the experimental data.

The experimental data, considered here, have beencollected during 6843.8 hours, giving a useful sta-tistics for the 134Xe isotope of 0.87 kg yr. Fig. 3shows the experimental energy distribution (solid his-togram) in the 0.65–1.10 MeV energy interval, wherethe peak of the decay mode searched for is expected(see Fig. 2). Since there is no peak evidence,Nd canbe preliminarily estimated by using the so-called “oneσ approach”: the maximum number of events whichcan be hidden by background fluctuations at one sigmalevel is estimated for the considered decay mode asthe square root of the number of background eventsin the given energy window. In spite of its simplic-ity, this method gives the right scale of the sensitiv-ity of the experiment. In particular, 181 counts (see

184 R. Bernabei et al. / Physics Letters B 527 (2002) 182–186

Fig. 2. Expected energy distributions for theβ−β−0ν(0+ → 0+) and theβ−β−0ν(0+ → 2+) decay modes in134Xe. In the latter case the firstpeak corresponds to events where the de-excitationγ escapes, while the higher energy peak corresponds to the case when it is fully containedin the detector.

Fig. 3. The continuous histogram represents the experimental data;the running time is 6843.8 hours. The dashed and dotted curves arethe expected distributions for theβ−β−0ν(0+ → 0+) and for theβ−β−0ν(0+ → 2+) decay modes, respectively, calculated forT1/2equal to the corresponding 90% C.L. limit given in the text. The twocurves are largely overlapped. The solid curve is the best fit one canobtain when assumingNd = 0.

Fig. 3) are present in the energy window 750–910 keV(±1σ around the peak at 830 keV of the expectedenergy distributions; see Fig. 2). In this energy win-dow the 66% and the 31% of the signal is expected forthe β−β−0ν(0+ → 0+) and forβ−β−0ν(0+ → 2+)

decay modes, respectively (see Fig. 2). Thus, we ob-tained: (i) for theβ−β−0ν(0+ → 0+) decay modeNd <

√181/0.66 (68% C.L.) andT1/2 > 1.3×1023 yr

(68% C.L.); (ii) for the β−β−0ν(0+ → 2+) decaymodeNd <

√181/0.31 (68% C.L.) andT1/2 > 6.1×

1022 yr (68% C.L.).Then, the number of eventsNd , which can be as-

cribed to the considered decay process, has been de-termined by using two further different methods. Asfirst, we considered the approach (widely used in lit-erature) based on the standard least squares proce-dure, where the experimental energy distribution inthe neighborhood of the peak searched for is fitted bythe sum of contributions due to the background (seebelow) and to the signal peak being sought. The be-haviour of the background in the considered energyinterval has been described by the simple formula:2

FP1P2P3(x) = P1 · eP2·x + P3. Thus,Nd has been cal-culated by minimizing (with respect to theP1, P2,P3 and Nd ) the function:Z2 = ∑

k(FP1P2P3(Ek) +Nd ·Mk −Nk)

2/Nk where the sum is performed in the650–1100 keV energy range (the corresponding effi-ciencies values are 0.977 for theβ−β−0ν(0+ → 0+)

process and 0.464 for theβ−β−0ν(0+ → 2+) one).Here: (i)Ek is the mean energy of thekth energy bin

2 Other possible parametrizations give similar or slightly morestringent final results.

R. Bernabei et al. / Physics Letters B 527 (2002) 182–186 185

(here 50 keV each one); (ii)Nd ·Mk are the counts ex-pected for the considered decay mode in thekth energybin as evaluated by the Monte Carlo code; (iii)Nk arethe measured counts in the given running period andin the kth energy bin. TheZ2 function has aχ2 pro-file; the minimization program uses the well-knownMINUIT package [13]. The obtained results are con-sistent with the absence of signal; in fact,−12± 35events are ascribed to theβ−β−0ν(0+ → 0+) processand−25± 77 to theβ−β−0ν(0+ → 2+) one, givingthe following 90% C.L. limits:Nd < 46 events andNd < 102 events, respectively. These limits have beencalculated according to the now-standard Feldman andCousins procedure [14]. Consequently, the following90% C.L. lower limits have been obtained:T1/2 >

5.8× 1022 yr for theβ−β−0ν(0+ → 0+) decay modeandT1/2 > 2.6 × 1022 yr for theβ−β−0ν(0+ → 2+)

one. In Fig. 3 the continuous histogram represents theexperimental data. The dashed and dotted curves arethe expected distributions for theβ−β−0ν(0+ → 0+)

and for theβ−β−0ν(0+ → 2+) decay modes, respec-tively, calculated forT1/2 equal to the correspond-ing 90% C.L. limit given above. The two curves arelargely overlapped. The solid curve is the best fit onecan obtain when assumingNd = 0.

In the second analysis a two-steps procedure is con-sidered: (i) the best estimate of the background is ex-trapolated from a near region excluding an appropri-ate interval including the signal searched for; (ii) theresiduals in the region where the signal is searchedfor are, then, estimated andNd is determined. For thispurpose we consider the experimental energy distribu-tion shown in Fig. 4; it ranges from 540 keV (roughlyjust above the energy threshold used in these measure-ments) up to about 2000 keV. Thus, the behaviour ofthe background has been fitted by the sum of an ex-ponential plus a straight line considering all the filledcircle data in Fig. 4, but not the open circle data whichrefer to the (690–990) keV energy interval where mostof the peak contribution is expected. In the inset ofFig. 4 the residuals calculated for the open circle dataare given; their integral sum is(2 ± 19) counts, thuscompatible with zero. It confirms the absence of sig-nal at the given sensitivity. The upper limit onNd at90% C.L. is, therefore, given byNd < 33/ε, whereε is the efficiency in the (690–990) keV energy win-dow for the considered process. It is equal to 92%for theβ−β−0ν(0+ → 0+) decay mode and 43% for

Fig. 4. Experimental data (filled and open circles) with superim-posed (continuous line) the behaviour of the background obtainedby fitting only the filled circle data. In the inset the residualsof the open circle data are shown; the dashed and dotted curves(there practically indistinguishable) are the expected distributionsfor theβ−β−0ν(0+ → 0+) and for theβ−β−0ν(0+ → 2+) decaymodes, respectively, calculated forT1/2 equal to the corresponding90% C.L. limits.

the β−β−0ν(0+ → 2+) one. In conclusion, the ob-tained limits (90% C.L.) are:T1/2 > 7.6 × 1022 yrfor the β−β−0ν(0+ → 0+) decay mode andT1/2 >

3.5×1022 yr for theβ−β−0ν(0+ → 2+) decay mode.These values are slightly more stringent than the onesobtained with the previous approach. For safety, wewill consider the more cautious values we obtainedand, consequently, according to the theoretical modelfor the nuclear matrix elements given in Ref. [15],we quote for the effective neutrino mass,〈mν〉, the90% C.L. limit: 〈mν〉 < 17 eV.

4. Conclusions

In this Letter the data collected during 6843.8 hoursby the LXe DAMA detector, filled with isotopicallyenriched xenon, have been presented. The experimenthas been carried out in the Gran Sasso NationalLaboratory of the INFN.

New experimental limits have been obtained forthe halflives of theβ−β−0ν(0+ → 0+) and of the

186 R. Bernabei et al. / Physics Letters B 527 (2002) 182–186

β−β−0ν(0+ → 2+) decay modes in134Xe. They havebeen cautiously estimated to be:T1/2 > 5.8× 1022 yrand T1/2 > 2.6 × 1022 yr (90% C.L.), respectively.The first one improves the previously available limitof about three orders of magnitude, while the secondhas been set for the first time.

Acknowledgements

It is a pleasure to thank Prof. Fiorini and theMiBeta group for providing us the xenon used in thisexperiment.

We thank the Gran Sasso Laboratory for supportand Mr. A. Bussolotti and A. Mattei for qualifiedtechnical help.

References

[1] V.I. Tretyak, Yu.G. Zdesenko, At. Data Nucl. Data Tables 61(1995) 43.

[2] H.V. Klapdor-Kleingrothaus et al., J. Phys. G 24 (1998) 483.[3] V.A. Bednyakov et al., Mod. Phys. Lett. A 12 (4) (1997) 233.[4] M. Hirsch, H.V. Klapdor-Kleingrothaus, Nucl. Phys. B (Proc.

Suppl.) A 52 (1997) 257.

[5] G.F. Dell’Antonio, E. Fiorini, Suppl. Nuovo Cimento 17(1960) 132.

[6] A.S. Barabash et al., Phys. Lett. B 223 (1989) 273.[7] R. Bernabei et al., ROM2F/2001-09 and INFN/AE-01/02

available as on-line preprint atwww.lngs.infn.it, Nucl. Instrum.Methods A, to appear.

[8] R. Bernabei et al., ROM2F/2001-26 and INFN/AE-01/19available as on-line preprint atwww.lngs.infn.it, submitted forpublication.

[9] R. Bernabei et al., Phys. Lett. B 387 (1996) 222;R. Bernabei et al., Phys. Lett. B 389 (1996) 783;R. Bernabei et al., Nuovo Cimento C 19 (1996) 537;R. Bernabei et al., Astropart. Phys. 5 (1996) 217;R. Bernabei et al., Phys. Lett. B 436 (1998) 379;R. Bernabei et al., Phys. Lett. B 465 (1999) 315;R. Bernabei et al., New J. Phys. 2 (2000) 15.1;R. Bernabei et al., Phys. Rev. D 61 (2000) 117301;R. Bernabei et al., Phys. Lett. B 493 (2000) 12;R. Bernabei et al., Eur. Phys. J. C 11 (2001) 1.

[10] M. Balata, ISRIM measurements on 2000, private communica-tion.

[11] W.R. Nelson et al., SLAC-265, UC-32 (E/I/A).[12] F. Cappella, Thesis, Univ. Roma “Tor Vergata” A.A.

2000/2001.[13] CN/ASD Group, MINUIT user guide, Program Library D506

CERN (1993).[14] G.J. Feldman, R.D. Cousins, Phys. Rev. D 57 (1998) 3873.

[15] K. Muto et al., Z. Phys. A 334 (1989) 187.

www.lngs.infn.it

www.lngs.infn.it



Strong coupled-channel effects in the barrier distributionsof 16,18O+ 58Ni

R.F. Simões1, D.S. Monteiro, L.K. Ono, A.M. Jacob, J.M.B. Shorto,N. Added, E. Crema

Instituto de Fisica DA, Departamento de Fisica Nuclear, Universidade de São Paulo, 05508-900 São Paulo, Brazil

Received 15 October 2001; received in revised form 3 December 2001; accepted 21 December 2001

Editor: V. Metag

Abstract

Quasi-elastic barrier distributions for the16,18O + 58Ni systems have been deduced from high precision measurements oftheir quasi-elastic excitation functions. The distribution obtained with this method for16O+58Ni is in complete agreement withthat deduced from fusion measurements. Contrary to present beliefs, these light systems present complex barrier distributionsthat reveal important couplings of inelastic and transfer channels to fusion. Coupled channel calculations are in good agreementwith our experimental data. 2002 Elsevier Science B.V. All rights reserved.

Keywords: Nuclear reactions;58Ni(16O, X), E = 30–50 MeV;58Ni(18O, X), E = 30–50 MeV; Measured large-angle elastic and inelasticscattering; Measured large-angle transfer reactions; Fusion barrier distributions

A remarkable characteristic of heavy ion reactionsat energies near the Coulomb barrier is the strongcoupling that sometimes occurs between the intrin-sic degrees of freedom and the relative motion. Con-sequently, fusion dynamics can be strongly affectedby intrinsic excitation of the interacting nuclei or nu-cleon transfer between them, with consequent largeenhancement of the fusion cross section [1–4]. Inother words, complex fusion barrier distributions canbe generated. Rowley, Satchler and Stelson [5] pro-posed an elegant and precise method to disclose thechannels which are responsible for the fusion cross

E-mail address: [email protected] (E. Crema).1 The work presented here is part of the PhD-project of

R.F. Simões and was supported by FAPESP.

section enhancement. It is a question of measuringthe fusion barrier distribution, which is extracted di-rectly from the experimental fusion excitation func-tion by Dfus ≡ d2[Eσfus(E)]/dE2 [6–12]. Since theinelastic barrier widths are proportional toZ1Z2βR,we expect that only heavy systems will present in-elastic (rotational or vibrational) couplings as isolatedpeaks in their fusion barrier distribution. In fact, allavailable experimental data have confirmed this pre-diction [13]. However, for lack of experimental data,the ability of Dfus to reveal the influence of transferchannels on the fusion of light systems is unknown.Recently, it has been proposed that the experimen-tal quasi-elastic excitation function can also providea barrier distribution when submitted to the opera-tion Dqe = −d[σqe(E)/σruth(E)]/dE, which shouldbe equivalent toDfus [14,15]. Since then, there have



188 R.F. Simões et al. / Physics Letters B 527 (2002) 187–192

been studies dedicated to a comparison of the barriersobtained with these two methods [15–17]. Their re-sults reveal that, in fact, the two distributions are iden-tical for those systems without strong coupling of thereaction channel to the fusion channel. On the otherhand, when a target or projectile excitation producesa peak in the barrier distribution, it is more evident inDfus than inDqe. Up to now, only heavy systems havebeen the object of these comparative studies. Thereare no experimental data available for light systems(Z1Z2 ≈ 224), mainly in those cases for which thetransfer channel is strongly coupled to fusion.

In order to study the effect of transfer channels onlight system fusion we have compared the fusion bar-rier distributions of16,18O + 58Ni. A high precisionquasi-elastic excitation function has been measuredfor each system atθlab = 161. As all transfer chan-nels in16O + 58Ni have large negativeQ-values, weexpected that fusion at energies near the Coulomb bar-rier should be weakly influenced by transfer couplingsand that the mean barrier should be a good referencefor the calculation with a one-dimensional barrier pen-etration model (BPM). On the other hand, the one andtwo neutron stripping in the18O + 58Ni system haveQ1n = +0.956 MeV andQ2n = +8.20 MeV. If bothprocesses were to produce barriers in the distribution,the large difference of theirQ-values would allow usto distinguish them. The comparison of these differ-ent neutron transfers would be an opportunity for dis-cussing the still open question: is multi-neutron trans-fer a simultaneous or sequential process? Finally, sincethe fusion barrier distribution for16O + 58Ni was de-duced directly from the fusion excitation function byKeeley et al. [18], it would be of great interest to com-pare, for the first time,Dfus andDqe for a light system.

The experiment was carried out with16O and18Obeams from the 8 UD Pelletron Accelerator at theUniversidade de São Paulo. The accelerated beamshad energies in the range 30–48 MeV, with steps of0.5 MeV over most of the energy range, and intensi-ties in the range 10–100 pnA. A 90 analyzing mag-net defined the beam energy with an uncertainty of≈ 40 keV. Before taking data, the analyzing magnetwas properly recycled, and during the measurementsthe energy was decreased only. The self-supportingtarget was 80 µg/cm2 of isotopically enriched58Ni(99.9%). The identification of the scattered nuclei atθlab = 161 was extracted fromE–E spectra, where

E was provided by the energy loss in a gas propor-tional counter which was backed by a silicon surface-barrier detector measuring the residual energy of thedetected nuclei. The resolution inZ allowed the un-ambiguous identification ofZ = 6,7,8 present in thespectra. However, as expected in that case, the en-ergy resolution wasn’t sufficient to resolving the elas-tic scattering and inelastic scattering from the low-est excited states of the target. Three silicon surface-barrier detector, located at angles±30 and−45 withrespect to the beam direction, were used to monitor thebeam direction and for normalization purposes. Threequasi-elastic excitation functions were calculated withthe three monitors and their results coincide withinthe experimental uncertainties. The statistical uncer-tainties associated with the measurements are lessthan 1%, except at the highest energies (45–48 MeV)where they are≈ 3%.

In the data analysis, quasi-elastic scattering wasdefined as the sum of all elastic, inelastic and transferevents. When transfer processes were observed, theZ = 6 channel was always the most probable. Athigher energies,Z = 7 events are also present in the18O + 58Ni spectra. The inelastic excitations as wellas neutron transfer processes were not resolved inthe spectra, but they were included by summing thecounts in the residual energy range of−5 MeV and+8 MeV around the elastic peak. AllECM valueswere corrected by the centrifugal potential energyat 161 [15]. The measured quasi-elastic excitationfunctions are shown in Fig. 1(a), as a function ofECM/VB, whereVB is the height of the average barrierextracted from the barrier distribution. The valuesare 31.55 MeV for16O + 58Ni, and 30.75 MeV for18O + 58Ni. The distributions are similar for nearlythe entire energy range investigated, but there areimportant differences near and below the Coulombbarrier. This could be a sign that, at these energies,a larger quasi-elastic flux is deviated to fusion in the18O+ 58Ni system.

As usual, the experimental barrier distributionswere extracted using a point-difference approxima-tion, with energy steps ofElab = 2.0 MeV, whichare shown in Fig. 1(b), where a barrier penetrationmodel calculation (BPM) for the system16O + 58Niis also plotted. This simple calculation is unable to ex-plain the data. As theoretically expected, the total ar-eas of both distributions are≈ 1. However, it is sur-

R.F. Simões et al. / Physics Letters B 527 (2002) 187–192 189

Fig. 1. (a) The measured excitation functions for quasi-elasticscattering atθlab = 161 relative to Rutherford scattering. Theerror bars shown correspond to the statistical uncertainties. (b) Thecorresponding experimental distributionsDqe(E). The dotted curveis a barrier penetration model calculation for the system16O+58Ni.

prising that such light systems present such differentbarrier distributions. While the low energy part of the16O + 58Ni distribution is well adjusted by BPM, the18O + 58Ni distribution is flattened below the meanpeak which is much less pronounced. This barrier dis-tribution is very similar to those of much heavier sys-tems,16O+ 186W for example [15]. In this latter case,the explanation was the large prolate deformation ofthe 186W nucleus. Obviously, this is not the case for58Ni and we must search for other kinds of couplings.

Another very interesting result comes from thequalitative comparison presented in Fig. 2, where ourdistributionDqe for 16O + 58Ni is exhibited togetherwith Dfus obtained by Keeley et al. [18] for the samesystem. This is the first time that this kind of compari-son is done for a light system. The agreement betweenthe two representations is impressive. All character-istics of Dfus can be equally identified inDqe, eventwo weak structures located at≈ 1 MeV and≈ 2 MeV

Fig. 2. Comparison of the experimental barrier distributions for16O + 58Ni deduced from fusion and quasi-elastic excitationfunctions. The fusion data were measured by Keeley et al. [18].

above the Coulomb barrier. The only discrepancy be-tween them is the amplitude of the structure located at≈ 5 MeV above the main barrier. Although this peakis more pronounced inDfus, it is not completely ab-sent inDqe, as in the16O + 144Sm case [14]. It hasbeen shown by ECIS as well as by the eigenchannelmodel calculations that the second peak inDfus is dueto couplings of the lowest octupole- and quadrupole-phonon states of144Sm. Nevertheless, these two the-oretical calculations predicted a second peak also inDqe, which was not confirmed by the data. In order tofit the experimental barrier distribution, it was neces-sary to include a surface imaginary potential for ab-sorbing the flow of the direct channels not directly in-cluded in the coupling matrix [15,19]. However, in oursystem16O+ 58Ni, the second peak inDqe is present.The error bars ofDfus make a more accurate compar-ison between their amplitudes difficult. The remark-able agreement ofDfus andDqe over most of the en-ergy range investigated gives confidence in the analy-sis presented below and suggests that these two barrierrepresentations are more nearly equal in light systemsthan in heavy ones. It would be interesting to confirmthis hypothesis by measuring other light systems, be-causeDqe has the advantage of smaller errors at higherenergies.


In order to reveal the origin of the barriers observedin the experimental distributions, calculations havebeen done within the coupled-channels approach,using the CCFULL code [20]. This code employs theisocentrifugal approximation, uses the incoming waveboundary condition inside the barrier, takes account ofthe finite excitation energies of the coupled modes, andincludes the effects of inelastic nonlinear coupling toall orders. The program includes one transfer couplingbetween the ground states and uses the macroscopicform factor given byFtrans(r) ≡ Ft dVN/dr, whereFt is the coupling strength. The nuclear potentialused was a proximity potential prediction with smallvariations to fit the main peak of the distribution:V0 = 53 MeV; r0 = 1.29 fm and 1.28 fm; anda0 = 0.49 fm and 0.51 fm for16O + 58Ni and18O + 58Ni, respectively. They were deep enough toavoid reflection effects at the highest energies. Thecalculations for both systems with no couplings areshown in Figs. 3 and 4 by the thin lines. We havebegun calculating the influence of16O excitation onthe 16O + 58Ni distribution. In agreement with otherstudies [21], we found that in our system the couplingto the 3− octupolar vibration state of the16O producesonly an energy shift in the entire distribution withoutchanging its shape.

On the other hand, the target excitations are able togenerate new barriers in the distribution. Couplings tothe lowest vibrational states 2+ (1.454 MeV) and 3−(4.475 MeV) of58Ni generate the barrier distributionthat has the same shape as the experimental, except forthe second peak at 36 MeV, as shown by the dashedline in Fig. 3. It must be stressed that the 2+ stateplays the major role in this result. The deformationparameters used (β2 = 0.1828 andβ3 = 0.190) wereobtained from Refs. [22,23]. Several combinations ofexcited states of the target and projectile were triedbut the best fit was obtained including only the 2+and 3− vibrational couplings discussed above. Thenwe investigated the effect of some transfer couplingon barrier distributions. The best result was obtainedfor one α-particle stripping (Q = −3.792 MeV andFt = 0.28) coupled together with the lowest 2+vibrational state of the58Ni with the experimentalvalue β2 = 0.1828. The result is the excellent fitplotted with the full line in Fig. 3. The inclusionof the 3− state has little influence on this fit. Ourresult is consistent with our experimental data which

Fig. 3. The experimental barrier distributions for16O + 58Nideduced from quasi-elastic excitation functions. The curves are theresults of coupled-channel calculations (see the text).

Fig. 4. The experimental quasi-elastic excitation function for18O + 58Ni. The lines correspond to cc-calculations and are dis-cussed in the text.

present theα-particle stripping as the most probabledirect transfer. Besides, theFt value is consistent withtheoretical predictions [26].

R.F. Simões et al. / Physics Letters B 527 (2002) 187–192 191

Therefore, our result differs from the16O+ 144Smcase. In the16O+ 58Ni system, the equivalent theoret-ical calculation, including the same vibrational statesof the target and one transfer channel, does not pre-dict the amplitude of the second peak observed in theexperimentalDfus. On the contrary, the theoretical dis-tribution is very close to the experimentalDqe, as canbe seen in Fig. 2. Of course, the inevitable large er-ror bars ofDfus in the high energy region makes amore accurate comparison difficult, but it is impres-sive that in the light system the coupled-channel cal-culation reproduces quite well the quasi-elastic distri-bution. This could indicate that in light systems thedirect reaction channels not explicitly included in thecoupled-channel calculation do not produce an impor-tant reflected flux at energies above the Coulomb bar-rier. We have already measured other light systems andthey confirm this result [24].

Now, we turn our attention to the more complexbarrier distribution of the18O + 58Ni system, whichis very different from that of the neighboring system16O + 58Ni. The presence of two weakly bound neu-trons in the projectile (compared to16O) generated fu-sion barriers in the energy region below the Coulombbarrier and reduced the amplitude of the main peak.So, the obvious processes that we may invoke to ex-plain the data would be the coupling to one- and two-neutron transfer. We started with one-neutron transfer(Q = 0.956 MeV); the result of the coupled-channelcalculation withFt = 0.33 is the dotted line in Fig. 4.This Ft value is also consistent with Ref. [26]. Thecomparison with the bare barrier distribution showsthat this coupling alone has a strong influence on thefusion process. In addition, when one neutron strip-ping is coupled with the lowest 2+ vibrational exci-tation of 58Ni the quality of the fit is improved in thehigh energy part of the distribution as well as in itslower part. This can be seen by the full line in Fig. 4,which is the best fit that we have achieved with one-neutron transfer. We were unable to fit the data by in-cluding the projectile vibrations.

Two-neutron transfer with its very largeQ-value(8.2 MeV), compared to the Coulomb barrier, is diffi-cult to be treated correctly in the calculation. Even so,we tried to fit the data with two-neutron transfer as thecoupled-channel in the calculation. A fit was achievedonly if the parameters of the bare potential are changedto unreasonable values, which do not fit the elastic ex-

citation function. Therefore, the fit obtained with two-neutron transfer is artificial and we must be aware ofthe ambiguities in the potential choices. So, within alllimitations of the calculation employed here, the reac-tion mechanisms that generate the barriers observed inthe experimental distribution of18O + 58Ni would beone-neutron stripping and the 2+ vibration of the tar-get. The lowest positiveQ-value transfer mechanismwould be more easily coupled to the fusion process inthis system, in other words the more adiabatic transferprocess would be an easy path to fusion of18O+ 58Niat sub-barrier energies. If this conclusion is correct, itwould be evidence for the possibility of a neutron-pairbreakup before fusion occurs. This would also indicatethat, in this case, sequential neutron transfer is a possi-ble reaction mechanism that could improve still morethe fit to the experimental barrier distribution at lowerenergies. Meanwhile, a more precise coupled channelcalculation is needed to confirm this result. This hy-pothesis is also supported by a precise coupled chan-nel calculation that was done by Rowley, Thompsonand Nagarajan [25] in order to decide between sequen-tial or simultaneous neutron transfer. The shape of thebarrier distribution in18O+ 58Ni is similar to that pre-dicted for sequential neutron transfer.

In conclusion, the precisely measured quasi-elasticexcitation function for the systems16,18O + 58Niproduced very different fusion barrier distributionswhich reveal important coupled-channels effects, de-spite their lowZ1Z2 values. The experimental barrierdistributions extracted from fusion and quasi-elasticmeasurements for the system16O+ 58Ni are very sim-ilar and, using a simplified coupled channel calcula-tion, a very good fit was obtained by coupling the low-est 2+ vibrational state of the58Ni and oneα-particlestripping. Finally, the more surprising results were thecomplex fusion barrier distribution obtained for the18O + 58Ni and its explanation by one neutron strip-ping coupled with the 2+ vibrational excitation of the58Ni. However, more precise calculations are need toconfirm these conclusions.

Acknowledgements

The authors wish to thank Dr. W.A. Seale and Dr.J.R. Lubian for useful discussions. This work is a partof the PhD Thesis of R.F. Simões, was supported by


the Fundação de Amparo à Pesquisa do Estado de SãoPaulo (FAPESP).

References

[1] W. Reisdorf, J. Phys. G 20 (1994) 1297.[2] M. Beckerman, Rep. Prog. Phys. 51 (1988) 1047.[3] S.G. Steadman, M.J. Rhoades-Brown, Annu. Rev. Nucl. Sci. 36

(1986) 649.[4] C.H. Dasso et al., Nucl. Phys. A 405 (1983) 381;

C.H. Dasso et al., Nucl. Phys. A 407 (1983) 221.[5] N. Rowley et al., Phys. Lett. B 254 (1991) 25.[6] J.X. Wei et al., Phys. Rev. Lett. 67 (1991) 3368.[7] R.C. Lemmon et al., Phys. Lett. B 316 (1993) 32.[8] J.R. Leigh et al., Phys. Rev. C 47 (1993) 437.[9] C.R. Morton et al., Phys. Rev. Lett. 72 (1994) 4074.

[10] C.R. Morton et al., Phys. Rev. C 52 (1995) 243.

[11] J.R. Leigh et al., Phys. Rev. C 52 (1995) 3151.[12] A.M. Stefanini et al., Phys. Rev. Lett. 74 (1995) 864.[13] M. Dasgupta et al., Nucl. Phys. A 630 (1998) 78c, and

references therein.[14] H. Timmers et al., Nucl. Phys. A 584 (1995) 190.[15] N. Rowley et al., Phys. Lett. B 373 (1996) 23.[16] H. Timmers et al., Nucl. Phys. A 633 (1998) 421;

H. Timmers et al., Phys. Lett. B 399 (1997) 35.[17] O.A. Capurro et al., Phys. Rev. C 61 (2000) 037603.[18] N. Keeley et al., Nucl. Phys. A 628 (1998) 1.[19] A.T. Krupa et al., Nucl. Phys. A 560 (1993) 845.[20] K. Hagino et al., Comput. Phys. Commun. 123 (1999) 143.[21] K. Hagino et al., Phys. Rev. Lett. 79 (1997) 2014.[22] J.K. Tuli, Nucl. Data Sheets 56 (1989) 683.[23] R. Spear, At. Data Nucl. Data Tables 42 (1989) 55.[24] E. Crema et al., to be published.[25] N. Rowley et al., Phys. Lett. B 282 (1992) 276.[26] C.H. Dasso, G. Pollarolo, Phys. Lett. B 155 (1985) 223.



Radiativeφ → f0(980)γ decay in light cone QCD sum rules

T.M. Aliev a, A. Özpinecib, M. Savcıa

a Physics Department, Middle East Technical University, 06531 Ankara, Turkeyb The Abdus Salam International Centre for Theoretical Physics, I-34100 Trieste, Italy

Received 13 November 2001; received in revised form 13 December 2001; accepted 7 January 2002


Abstract

The light cone QCD sum rules method is used to calculate the transition form factor for the radiativeφ → f0γ decay,assuming that the quark content of thef0 meson is puress state. The branching ratio is estimated to beB(φ → f0γ ) =3.5 × (1 ± 0.3) × 10−4. A comparison of our prediction on branching ratio with the theoretical results and experimental dataexisting in literature is presented. 2002 Elsevier Science B.V. All rights reserved.

PACS: 11.55.Hx; 13.40.Hq; 14.40.Ev

1. Introduction

According to the quark model, mesons are inter-preted as pureqq states. Scalar mesons constitute aremarkable exception to this systematization and theirnature is not well established yet [1–4].

In the naiveqq picture, one can treat the isoscalarf0(980) either as the meson that exists mostly asnonstrange and almost degenerate with the isovectora0(980) or as mainlyss, in analogy to the puressvector mesonφ(1020).

In order to understand the content of thef0 mesonseveral alternatives have been suggested, such as, theanalysis of thef0 → 2γ decay [5,6]; study of the ra-tio Γ (a0 → f0γ )/Γ (φ → f0γ ) [7]. However, amongthese, the(φ → f0γ ) decay occupies a special place,

E-mail addresses: [email protected] (T.M. Aliev),[email protected] (A. Özpineci), [email protected](M. Savcı).

since the branching ratio expected of this decay, is es-sentially dependent on the content off0. For example,B(φ → f0γ ) is as high as∼ 10−4 if it were composedof qqqq , and∼ 10−5 if f0 were a puress state.

It has been known for a long time thatf0(980)couples significantly through itsss content, from itsdetection as a peak in theJ/ψ → φf0 [8] andDs →πf0 [9] decays, as discussed in [10] and [11] (see also[12]). For this reason, in this work we assume thatquark content of bothφ andf0 mesons are puress. Inthe present Letter we analyze the radiativeφ → f0γ

decay in framework of the light cone QCD sum rules(about light cone QCD sum rules and its applications,see, for example, [13]). Note also that theφ → f0γ

decay is analyzed in framework of the 3-point sumrules in [14]. In order to calculate the transition formfactor describing theφ → f0γ decay in light coneQCD sum rules, we consider the following correlator

(1)Πµ = i

∫d4x eipx

⟨0∣∣T

J s(x)J φµ (0)

∣∣0⟩γ,



194 T.M. Aliev et al. / Physics Letters B 527 (2002) 193–198

where J s = ss and Jφµ = sγµs are interpolating

currents forf0 andφ mesons, respectively, andγ isthe background electromagnetic field (for more aboutexternal field technique in QCD see [15,16]).

The physical part of the correlator can be obtainedby inserting a complete set of one meson states intothe correlator,

(2)Πµ =∑ 〈0|J s(x)|f0(p)〉〈f0(p)|φ(p1)〉γ 〈φ(p1)|J φ

µ (0)|0〉(p2−m2

f0

)(p2

1−m2φ

) ,

whereφ andf0 are the quantum numbers andp1 =p + q with q being the photon momentum.

The matrix element〈φ(p1)|J φµ (0)|0〉 in Eq. (1) is

defined as

(3)〈φ(p1)|J φµ (0)|0〉 =mφfφε

φµ,

where εφµ is the φ meson polarization vector. The

coupling of thef0(980) to the scalar currentJ s = ss

is defined in terms of a constantλf

(4)〈0|J s |f0(p)〉 =mf0λf .

The relevant matrix element describing the transi-tion φ → f0 induced by an external electromagneticcurrent can be parametrized in the following form:⟨f0(p)

∣∣φ(p1, ε

φ)⟩γ

(5)= eεµ[F1

(q2)(p1q)ε

φµ + F2

(q2)(εφq)

p1µ],

whereε is the photon polarization and we have used(εq)= 0. From gauge invariance we have

(6)F1(q2) = −F2

(q2),

and since the photon is real in the decay underconsideration, we need the values of the form factorsonly at the pointq2 = 0. Using Eq. (6) the matrixelement〈f0|φ〉γ takes the following gauge-invariantform,

(7)〈f0|φ〉γ = eεµF1(0)[(p1q)ε

φµ − (

εφq)p1µ

].

Using Eqs. (1)–(4) and (7), for the phenomenologicalpart of the correlator we have

Γphenµ = eF1(0)εν

[−(p1q)gµν + p1νqµ]

(8)× λf fφmf0mφ

(p2 −m2f0)(p2

1 −m2φ)

.

In order to construct the sum rule, calculation of thecorrelator from QCD side (theoretical part) is needed.

From Eq. (1) we get

(9)Πµ =∫

d4x eipx⟨0∣∣Tr

−γµSs(−x)Ss(x)∣∣0⟩

γ,

whereSs is the full propagator of the strange quark(see below). Theoretical part of the correlator containstwo pieces, perturbative and nonperturbative. Pertur-bative part corresponds to the case when photon is ra-diated from the freely propagating quarks. Its expres-sion can be obtained by making the following replace-ment in each one of the quark propagators in Eq. (9)

(10)

(Ss)abαβ → 2eeq

(dy Fµνy

νSfrees (x − y)γ µSfree

s (y))abαβ,

where the Fock–Schwinger gaugexµAµ(x) = 0 isused andSfree

s is the free s-quark propagatorSfrees (x)=

i/x/(2π2x4) and the remaining one is the full quarkpropagator.

The nonperturbative piece of the theoretical partcan be obtained from Eq. (9) by replacing each oneof the propagators with

(11)(Ss)abαβ = −1

4qaAiq

b(Ai)αβ,

whereAi is the full set of Dirac matrices and sum overAi is implied and the other quark propagator is the fullpropagator, involving perturbative and nonperturbativecontributions. In order to calculate perturbative andnonperturbative parts to the correlator function (1),expression of the s-quark propagator in external fieldis needed.

The complete light cone expansion of the lightquark operator in external field is presented in [16].The propagator receives contributions from the non-local operatorsqGq , qGGq , qqqq , whereG is thegluon field strength tensor. In the present work we con-sider operators with only one gluon field and neglectterms with two gluonsqGGq , and four quarksqqqqand formal neglect of these these terms can be justifiedon the basis of an expansion in conformal spin [17].In this approximation full propagator of the s-quark isgiven as

Ss(x)= i/x

2π2x4 − 〈ss〉12

(1+ x2

16m2

0

)

+ ims〈ss〉48

/x − im20ms

2732x2/x

T.M. Aliev et al. / Physics Letters B 527 (2002) 193–198 195

− igs

1∫0

dv

[/x

16π2x2Gµν(vx)σ

µν

(12)− i

4π2x2vxµGµνγ

ν

].

It follows from Eqs. (11) and (9) that in calculating theQCD part of the correlator, as is generally the case,we are left with the matrix elements of the gauge-invariant nonlocal operators, sandwiched in betweenthe photon and the vacuum states〈γ (q)|sAis|0〉.These matrix elements define the light cone photonwave functions. The photon wave functions up totwist-4 are [17,18]

〈γ (q)|q(x)σµνq(0)|0〉 = ieeq〈qq〉

×1∫

0

du eiqx(εµqν − ενqµ)

× [χφ(u)+ x2(g1(u)− g2(u)

)]+ [

(qx)(εµxν − ενxµ)

(13)+ (εx)(xµqν − xνqµ)]g2(u)

,

〈γ (q)|q(x)γµγ5q(0)|0〉

(14)= ef

4eqεαβρσ ε

βqρxσ

1∫0

du eiuqxψ(u).

The path-ordered gauge factorP exp(igs∫ 1

0 duxµ ×Aµ(ux)) is emitted since the Schwinger–Fock gaugexµAµ(x) = 0 is used. The functionsφ(u), ψ(u) arethe leading twist-2 photon wave functions, whileg1(u)

andg2(u) are the twist-4 photon wave functions. Notethat twist-3 photon wave functions are neglected inthe calculations, since their contributions are smalland change the result by 5%. In Eq. (13)χ is themagnetic susceptibility of the quark condensate andeq is the quark charge. The theoretical part is ob-tained by substituting photon wave functions and ex-pression for the s-quark propagators into Eq. (9). Thesum rules is obtained by equating the phenomeno-logical and theoretical parts of the correlator. In or-der to suppress higher states and continuum contri-bution (for more details see [19,20]) double Boreltransformations of the variablesp2

1 = p2 and p22 =

(p + q)2 are performed on both sides of the cor-relator, after which the following sum rule is ob-

tained

F1(0)= em2f0/M2

2 em2φ/M

21

es

λf0fφmf0mφ

×[

2χ〈ss〉φ(u0)− 3ms

2π2 (1+ γE)

]M2E0

(s20/M

2)

+ 1

24〈ss〉[−192g1(u0)+msφ(u0)〈ss〉

]+ 3ms

2π2

[M2

(γE + ln

M2

Λ2

)E0

(s0/M

2)

(15)+M2f(s0/M

2)],

wheres0 is the continuum threshold

E0(s0/M

2) = 1− e−s0/M2,

f(s0/M

2) =s0/M

2∫0

dy lnye−y,

which have been used to subtract continuum, and

u0 = M22

M21 +M2

2

, M2 = M21M

22

M21 +M2

2

,

whereM21 andM2

2 are the Borel parameters inφ andf0 channels, respectively,Λ is the QCD scale para-meter andγE is the Euler constant. Since the massesof φ andf0 are very close to each other we will setM2

1 = M22 ≡ 2M2, obviously from which it follows

thatu0 = 1/2.It is clear from Eq. (14) that the values ofλf0

and fφ are needed in order to determineF(0). Thecoupling of thef0(980) to the scalarss current isdetermined by the constantλf0 and in the two-pointQCD sum rules its value is found to beλf0 = (0.18±0.0015) GeV [14]. In further numerical analysis wewill use fφ = 0.234 GeV which is obtained from theexperimental analysis of theφ → e+e− decay [21].

Having the values ofλf0 and fφ , our next andfinal attempt is the calculation of transition form factorF1(0). As we can easily see from Eq. (15) the maininput parameters of the light cone QCD sum rules isthe photon wave function. It is known that the leadingphoton wave function receive only small correctionsfrom the higher conformal spin [17,19,22], so thatthey do not deviate much from the asymptotic form.The photon wave functions we use in our numerical


analysis are given as

φ(u) = 6u(1− u), ψ(u) = 1,

g1(u)= −1

8(1− u)(3− u).

Furthermore, the values of the input parameters thatare used in the numerical calculations are:f = 0.028GeV2, χ = −4.4 GeV−2 [23] (in [24] this quantityis predicted to have the valueχ = −3.3 GeV−2),〈ss(1 GeV)〉 = −0.8 × (0.243)3 GeV3 and the QCDscale parameter is taken asΛ = 0.2 GeV. The strangequark mass is chosen in the rangems = 0.125–0.16 GeV, obtained in the QCD sum rules approach[25]. The masses of theφ andf0 mesons aremφ =1.02 GeV,mf0 = 0.98 GeV. The transition form factoris a physical quantity and therefore it must be inde-pendent of the auxiliary continuum thresholds0 andand the Borel massM2 parameters. So our main con-cern is to find a region where the transition form fac-tor F1(0) is practically independent of the parameterss0 andM2. For this aim in Fig. 1 we present the de-pendence of the transition form factorF1(0) on theBorel parameterM2 at three different values of thecontinuum threshold:s0 = 2.0 GeV2, 2.2 GeV2 and2.4 GeV2. It follows from this figure that for the choiceof the continuum thresholds in the above-mentionedrange, the variation of the result on the transition formfactorF1(0) is about 10%. In other words, we can con-

clude thatF1(0) is practically independent of the con-tinuum threshold. Furthermore, we observe that when1.4 M2 2.0 GeV2, F1(0) is quite stable with re-spect to the variations of the Borel parameterM2. Asa result, one can directly read from this figure

F1(0)= (3.25± 0.20) GeV−1,

where the resulting error is due to the variations ins0andM2. The other sources of errors contributing to thenumerical analysis of the transition form factor comefrom the strange quark mass and the uncertainties invalues of various condensates. Hence, our final pre-diction on the transition form factor is

(16)F1(0)= (3.25± 0.50) GeV−1.

Using the matrix element (7) for the decay width ofthe considered process, we obtain

(17)Γ (φ → f0γ )= α∣∣F1(0)

∣∣2 (m2φ −m2

f0)3

24m3φ

.

Using the experimental valueΓtot(φ) = 4.458 MeV[21], and Eqs. (16) and (17), we get for the branchingratio

(18)B(φ → f0γ )= 3.5× (1.0± 0.3)× 10−4.

Our result on the branching ratio is obtained underthe assumption thatf0 meson is represented as a

Fig. 1. The dependence of the transition form factorF1(0) for the radiativeφ → f0γ decay onM2 at three different values of the continuumthresholds0 = 2.0 GeV2, 2.2 GeV2 and 2.4 GeV2.

T.M. Aliev et al. / Physics Letters B 527 (2002) 193–198 197

puress component. How does the result change if weassume thatφ andf0 mesons can be represented as amixing of ss andnn = (uu+ dd)/

√2 state, i.e.,

φ = cosα ss + sinα nn,

f0 = sinβ ss + cosβ nn?

Analysis of the processφ → π0γ and combinedanalysis of theφ → f0γ andf0 → 2γ decays showthat |α| 4 and two solutions are found forβ , i.e.,β = −48 ± 6 or β = 85 ± 5, respectively (see,for example, [5]). In other words, quark content ofφ meson is puress state, while inf0 meson theremight be sizablenn component. Obviously, whenF1(0) is calculated from QCD side, only sinβss com-ponent operates (see Eq. (1)) and, therefore, the de-cay widthΓ (φ → f0γ ), and hence the correspond-ing branching ratio, contains an extra factor sin2β . Ifβ = 85 ± 5, then prediction for the branching ratiogiven in Eq. (18) is practically unchanged, but whenβ = −48 ±6 B(φ → f0γ ) decreases by about a fac-tor of 2.

Finally, let us compare our prediction on branch-ing ratio with the existing theoretical results and ex-perimental data in the literature. Obviously, our re-sult is slightly larger compare to the 3-point QCDsum rule result which predictsB(φ → f0γ ) (2.7 ±1.1)×10−4 [14], and approximately three times largercompared to the prediction of the spectral QCD sumrules and chiral unitary approaches, whose predic-tions areB(φ → f0γ ) = 1.3× 10−4 [26] andB(φ →f0γ ) = 1.6 × 10−4 [27], respectively. It is interest-ing to note that this value of the branching ratio iscloser to our prediction when the mixing angle ischosen to beβ = −48 ± 6. Our result, which isgiven in Eq. (18), is larger compared to the predic-tions of [7,28], whose results areB(φ → f0γ )= 1.9×10−4 [28], andB(φ → f0γ ) = 1.35× 10−4 [7], re-spectively.

As the final words we would like to point outthat our prediction given in Eq. (18), is in a verygood agreement with the existing experimental resultB(φ → f0γ ) (3.4± 1.1)× 10−4 [21].

References

[1] L. Montanet, Rep. Prog. Phys. 46 (1983) 337;F.E. Close, Rep. Prog. Phys. 51 (1988) 833;

N.N. Achasov, Nucl. Phys. (Proc. Suppl.) B 21 (1991) 189;T. Barnes, hep-ph/0001326;V.V. Anisovich, hep-ph/0110326.

[2] N.A. Tornqvist, Phys. Rev. Lett. 49 (1982) 624;N.A. Tornqvist, Z. Phys. C 68 (1995) 647.

[3] N.A. Tornqvist, M. Ross, Phys. Rev. Lett. 76 (1996) 1575.[4] E. van Beveren et al., Z. Phys. C 30 (1986) 615;

E. van Beveren, G. Rupp, M.D. Scadron, Phys. Lett. B 495(2000) 300;E. van Beveren, G. Rupp, M.D. Scadron, Phys. Lett. B 509(2001) 365, Erratum.

[5] A.V. Anisovich, V.V. Anisovich, V.A. Nikonov, hep-ph/0011191.

[6] M. Boglione, M.R. Pennington, Phys. Rev. Lett. 79 (1997)1998.

[7] F.E. Close, N. Isgur, S. Kumano, Nucl. Phys. B 389 (1993)513;N. Brown, F.E. Close, in: L. Maiani, G. Pancheri, N. Paver(Eds.), The DAFNE Physics Handbook, INFN, Frascati, 1995.

[8] G. Gidal et al., MARK II Collaboration, Phys. Lett. B 107(1981) 153;A. Falvard et al., DM2 Collaboration, Phys. Rev. D 38 (1988)2706.

[9] J.C. Anjos et al., E691 Collaboration, Phys. Rev. Lett. 62(1989) 125;E.M. Aitala et al., E791 Collaboration, Phys. Rev. Lett. 86(2001) 125.

[10] K.L. Au, D. Morgan, M.R. Pennington, Phys. Rev. D 35 (1987)1633.

[11] D. Morgan, M.R. Pennington, Phys. Rev. D 48 (1993) 1185.[12] R. Delbourgo, D.-S. Liu, M.D. Scadron, Phys. Lett. B 446

(1999) 332.[13] P. Colangelo, A. Khodjamirian, in: M. Shifman (Ed.), At

the Frontier of Particle Physics, Handbook of QCD, WorldScientific, Singapore, 2001, p. 1495.

[14] F. De Fazio, M.R. Pennington, Phys. Lett. B 520 (2001) 78.[15] B.L. Ioffe, A.V. Smilga, Nucl. Phys. B 232 (1984) 109.[16] I.I. Balitsky, V.M. Braun, Nucl. Phys. B 311 (1988) 541.[17] V.M. Braun, I.E. Filyanov, Z. Phys. C 48 (1990) 239.[18] A. Ali, V.M. Braun, Phys. Lett. B 359 (1995) 223.[19] V.M. Belyaev, V.M. Braun, A. Khodjamirian, R. Rückl, Phys.

Rev. D 57 (1995) 6177.[20] T.M. Aliev, A. Özpineci, M. Savcı, Nucl. Phys. A 678 (2000)

443;T.M. Aliev, A. Özpineci, M. Savcı, Phys. Rev. D 62 (2000)053012.

[21] Particle Data Group, D.E. Groom et al., Eur. Phys. J. C 15(2000) 1.

[22] I.I. Balitsky, V.M. Braun, A.V. Kolesnichenko, Nucl. Phys.B 312 (1989) 509;V.M. Braun, I.E. Filyanov, Z. Phys. C 44 (1989) 157.

[23] V.M. Belyaev, Ya.I. Kogan, Yad. Fiz. 40 (1984) 1035, Sov. J.Nucl. Phys. 40 (1984) 659.

[24] I.I. Balitsky, A.V. Kolesnichenko, Yad. Fiz. 41 (1985) 282, Sov.J. Nucl. Phys. 41 (1985) 178.


[25] P. Colangelo, F. De Fazio, G. Nardulli, N. Paver, Phys. Lett.B 408 (1997) 340.

[26] S. Narison, Nucl. Phys. (Proc. Suppl.) 96 (2001) 244.

[27] E. Marco, S. Hirenzaki, E. Oset, H. Toki, Phys. Lett. B 470(1999) 20.

[28] J. Lucio, J. Pestiean, Phys. Rev. D 42 (1990) 3253.



On the effect ofθ13 on the determination of solar oscillationparameters at KamLAND

M.C. Gonzalez-Garciaa,b,c, C. Peña-Garayb

a Theory Division, CERN, CH-1211 Geneva 23, Switzerlandb Instituto de Física Corpuscular, Universitat de València, C.S.I.C. Edificio Institutos de Paterna, Apt 22085, 46071 València, Spain

c C.N. Yang Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, NY 11794-3840, USA



Abstract

If the solution to the solar neutrino puzzle falls in the LMA region, KamLAND should be able to measure with good precisionthe corresponding oscillation parameters after a few years of data taking. Assuming a positive signal, we study their expectedsensitivity to the solar parameters (θ12,m2

21) when considered in the framework of three-neutrino mixing after taking intoaccount our ignorance on the mixing angleθ13. We find a simple “scaling” dependence of the reconstructedθ12 range with thevalue ofθ13 while them2

12 range is practically unaffected. Our results show that the net effect is approximately equivalent toan uncertainty on the overall neutrino flux normalization of up to∼ 10%. 2002 Elsevier Science B.V. All rights reserved.

The Sudbury Neutrino Observatory (SNO) mea-surement on the charged current reaction for solar neu-trino absorption in deuterium [1] has provided an im-portant piece of information in the path to solve the so-lar neutrino problem (SNP) [2–6]. In particular, all thepost-SNO global analysis [7–11] have shown that theinclusion of the SNO results have further strengthenthe case for solar neutrino oscillations with large mix-ing angles with best fit in the region with largerm2,LMA. Unfortunately, the LMA region is broad in mix-ing and mass splitting due to the uncertainties in thesolar fluxes and the lack of detailed data in the low en-ergy range.

E-mail addresses:[email protected](M.C. Gonzalez-Garcia), [email protected] (C. Peña-Garay).

This situation will be improved in the very nearfuture, in particular, if LMA is the right solution tothe SNP. If nature has arranged things favourably,the SNO measurements of the day–night asymmetryand the neutral to charged current ratio could help toidentify and constrain the LMA solution [11]. Fur-thermore, the terrestrial experiment, KamLAND [12],should be able to identify the oscillation signal and sig-nificantly constrain the region of parameters for theLMA solution.

The KamLAND reactor neutrino experiment, whichis to start taking data very soon, is sensitive to theLMA region of the solar neutrino parameter space. Af-ter a few years of data taking, it should be capable ofeither excluding the entire LMA region or, not onlyestablishingνe ↔ νother oscillations, but also measur-ing the oscillation parameters(tan2 θ12,m2

21) with



200 M.C. Gonzalez-Garcia, C. Peña-Garay / Physics Letters B 527 (2002) 199–205

unprecedented precision [13–16]. Previous analysis ofthe attainable accuracy in the determination of the os-cillation parameters at KamLAND have studied theeffect of the time dependent fuel composition, theknowledge of the flux uncertainty, the role of thegeological neutrinos as well as the combined analy-sis of solar and KamLAND data. These studies havebeen performed in the simplest two-neutrino oscilla-tion scheme or equivalently for three-neutrino oscilla-tions assuming a small fixed value ofθ13 [13–16].

In this Letter, we revisit the problem of how theprecision to which KamLAND should be able tomeasure the solar oscillation parameters,m2

12 andθ12, is affected by our ignorance on the exact valueof the mixing angleθ13. At present our most preciseinformation on this parameter comes from the negativeresults from the CHOOZ reactor experiment [18],which, when combined with the results from theatmospheric neutrino experiments [19] results into a3σ upper bound sin2 θ13 0.06 [20,21]. To addressthis question we study how the reconstructed rangeof m2

12 andθ12 depends on the value of the mixingangleθ13. We conclude that the reconstructedm2

12is very mildly affected by the value ofθ13, whilethe θ12 range scales withθ13 in a simple way. Wedetermine the reconstructed region of solar parametersobtained from a given signal onceθ13 is left freeto vary below the present bound. We find that thenet effect is approximately equivalent to that of anuncertainty on the overall neutrino flux normalizationof up to ∼ 10% and it should be taken into accountonce enough statistics is accumulated.

KamLAND is a reactor neutrino experiment lo-cated at the old Kamiokande site in the Kamioka minein Japan. It is sensitive to theνe flux from some10+ reactors which are located “nearby”. The dis-tances from the different reactors to the experimen-tal site vary from slightly more than 80 km to over800 km, while the majority (roughly 80%) of the neu-trinos travel from 140 to 215 km. KamLAND “sees”the antineutrinos by detecting the total energy de-posited by recoil positrons, which are produced viaνe +p → e+ +n. The total visible energy correspondsto Ee+ + me, whereEe+ is the kinetic energy of thepositron andme the electron mass. The positron en-ergy, on the other hand, is related to the incoming anti-neutrino energyEe+ = Eν + 1.293 MeV up to correc-tions related to the recoil momentum of the daughter

neutron (1.293 MeV is the neutron–protonmass differ-ence). KamLAND is expected to measure the visibleenergy with a resolution which is expected to be betterthanσ(E)/E = 10%/

√E, for E in MeV [12,16].

The antineutrino spectrum which is to be measuredat KamLAND depends on the power output and fuelcomposition of each reactor (both change slightly asa function of time), and on the cross section forνe + p → e+ + n. For the results presented here wewill follow the flux and the cross section calculationsand the statistical procedure described in Ref. [17].We use one “KamLAND-year” as the amount of timeit takes KamLAND to see 800 events with visibleenergy above 1.22 MeV. This is roughly what isexpected after one year of running (assuming a fiducialvolume of 1 kton), if all reactors run at (constant)78% of their maximal power output [12]. We assumea constant chemical composition for the fuel of allreactors (explicitly, 53.8% of235U, 32.8% of239Pu,7.8% of238U, and 5.6% of241Pu, see [13,22]).

The shape of the energy spectrum of the incomingneutrinos can be derived from a phenomenologicalparametrisation, obtained in [23],

(1)dNνe

dEν∝ ea0+a1Eν+a2E

2ν ,

where the coefficientsai depend on the parent nucleus.The values ofai for the different isotopes we usedare tabulated in [15,23]. These expressions are verygood approximations of the (measured) reactor fluxfor values ofEν 2 MeV.

The cross section forνe + p → e+ + n has beencomputed including corrections related to the recoilmomentum of the daughter neutron in [24]. We usedthe hydrogen/carbon ratio,r = 1.87, from the pro-posed chemical mixture (isoparaffin and pseudoc-umene) [12]. It should be noted that the energy spec-trum of antineutrinos produced at nuclear reactors hasbeen measured with good accuracy at previous reac-tor neutrino experiments (see [12] for references). Forthis reason, we will first assume that the expected (un-oscillated) antineutrino energy spectrum is known pre-cisely. Some of the effects of uncertainties in the in-coming flux on the determination of oscillation para-meters have been studied in [15], and are supposedlysmall.

In order to simulate events at KamLAND, weneed to compute the expected energy spectrum for

M.C. Gonzalez-Garcia, C. Peña-Garay / Physics Letters B 527 (2002) 199–205 201

Table 1Reconstructed ranges for tan2 θ12 at 3σ (in the first octant) for differentθ13 (sin2 θ13 in parenthesis) cases for the simulated points listed in thefirst column. See text for details

Signal≡ (tan2 θ12,m221, θ13) Nev α θ13 = 0 θ13 = 12.6 (0.048) θ13 < 13.8 (0.057)

1≡ (0.37,3.7× 10−5,0) 1022 0.65 [0.31,0.43] [0.25,0.35] [0.24,0.43]2≡ (0.60,2.5× 10−5,0) 802 0.65 [0.48,1.00] [0.39,0.68] [0.38,1.00]3≡ (0.70,7.0× 10−5,0) 1335 0.56 [0.47,1.00] [0.35,0.57] [0.34,1.00]4≡ (0.25,3.0× 10−5,0) 1237 0.65 [0.21,0.30] [0.16,0.24] [0.15,0.30]5≡ (0.50,2.0× 10−4,0) 1321 0.44 [0.37,0.85] [0.25,0.44] [0.22,0.85]

the incoming reactor antineutrinos for different valuesof the neutrino oscillation parameters mass-squareddifferences and mixing angles. In the frameworkof three-neutrino mixing theνe survival probabilitydepends on the two relevant mass differences and thethree-mixing angles but it can be simplified if we takeinto account that:

• Matter effects are completely negligible at Kam-LAND-like baselines;

• As observed in [14], form2 3×10−4 eV2, thedetermination ofm2 is rather ambiguous. Thisis due to the fact that ifm2 is too large, theKamLAND energy resolution is not sufficientlyhigh to resolve the oscillation lengths associatedwith these values ofm2 and Eν . This allowsus to consider the higherm2

23 andm213 to be

averaged.

Thus, the relevant (energy dependent) electronantineutrino survival probability at KamLAND is

P(νe ↔ νe

)= sin4 θ13 + cos4 θ13

×[1−

∑i

fi sin2 2θ12sin2(

1.27m221Li

Eν

)],

(2)

whereLi is the distance of reactori to KamLAND inkm, Eν is in GeV andm2

12 is in eV2, while fi is thefraction of the total neutrino flux which comes fromreactori (see [12]).

From Eq. (3) we can easily derive the effectof the nonzeroθ13. The energy independent termcontains the factor sin4 θ13+cos4 θ13 while the energydependent term contains cos4 θ13. Thus the shapeof the spectrum (this is, the ratio of the energy

dependent versus the energy independent term) is onlymodified by a factor cos4 θ13/(sin4 θ13 + cos4 θ13) ∼1 − sin4 θ13. Given the present bound we concludethat θ13 does not affect significantly the shape of thespectrum which is the most relevant information inthe determination ofm2

12. Conversely, the overallspectrum normalization is scaled by cos4 θ13 ∼ 1 −2 sin2 θ13 and this factor introduces an non-negligibleeffect.

In order to quantify the effect of this term wehave simulated the KamLAND signal correspondingto some points in the parameter space (see Table 1).Following the approach in Ref. [17] our simulated datasets are analysed via a standardχ2 function,

χ2(θ12,m212, θ13

)

=Nbin∑j=1

(Nj (θ12,m212, θ13) − Tj (θ12,m2

12, θ13))2

(√

Nj )2

+ Nd.o.f.,

whereNj(θ12,m212, θ13) is the number of simulated

events in thej th energy bin which would correspondto the parametersθ12,m2

12, θ13 (see first column inTable 1 for the values of the 5 simulated points).Tj (θ12,m2

12, θ13) is the theoretical prediction for thenumber of events in thej th energy bin as a functionof the oscillation parameters.Nbin = 12 is the totalnumber of bins (binwidth is 0.5 MeV), and the addedconstant,Nd.o.f., is the number of degrees of freedom.This is included in order to estimate the statistical ca-pabilities of anaverageexperiment. An alternative op-tion would be not to include theNd.o.f. term but to in-clude random statistical fluctuations in the simulateddata as done in Ref. [13]. We have verified that our re-sults are not quantitatively affected by the choice ofsimulation procedure. The fit is first done for visibleenergies 1.22 < Evis < 7.22 MeV. Note that we as-


Table 2Same as Table 1 after removing the lower energy bins from the fit (therefore including only events with 2.72< Evis < 7.22 MeV)

Signal≡ (tan2 θ12,m221, θ13) Nev α θ13 = 0 θ13 = 12.6 (0.048) θ13 < 13.8 (0.057)

1≡ (0.37,3.7× 10−5,0) 756 0.65 [0.31,0.44] [0.25,0.36] [0.24,0.44]2≡ (0.60,2.5× 10−5,0) 695 0.72 [0.41,1.00] [0.35,1.00] [0.34,1.00]3≡ (0.70,7.0× 10−5,0) 1167 0.52 [0.44,1.00] [0.32,0.59] [0.31,1.00]4≡ (0.25,3.0× 10−5,0) 1004 0.65 [0.20,0.42] [0.15,0.68] [0.14,1.00]5≡ (0.50,2.0× 10−4,0) 1090 0.44 [0.36,1.00] [0.24,0.46] [0.22,1.00]

sume statistical errors only, and do not include back-ground induced events. This seems to be a reasonableassumption, given that KamLAND is capable of tag-ging theνe by looking for a delayedγ signal due tothe absorption of the recoil neutron. There still re-mains, however, the possibility of irreducible back-grounds from geological neutrinos in the lower energybins (Evis 2.6 MeV) [16,17]. To verify how this pos-sible background may affect the effect here studied wehave repeated the analysis discarding the three lowerenergy bins, i.e., considering only events with visibleenergies 2.72< Evis < 7.22 MeV.

We have generated the signal for the five points inparameter space listed in Table 1. For the sake of con-creteness we have chosen the five points with differ-ent values ofm2

12 and tan2 θ12 distributed within the3σ allowed LMA region from the present analysis ofthe solar data [11] and withθ13 = 0. For each of thesimulated points we obtained the reconstructed regionof parameters in the plane(m2

12, tan2 θ12) by find-ing the minimumχ2 and then calculating the confi-dence level (CL) for two degrees of freedom assumingthree KamLAND-years of simulated data. The numberof expected events corresponding to each of the simu-lated points with 1.22< Evis < 7.22 MeV is given inTable 1 and in Table 2 for 2.72< Evis < 7.22 MeV. Wehave repeated this procedure for the same five simu-lated signals under different assumptions for the valueof the reconstructedθ13.

In Fig. 1(a) we show the allowed regions in the(tan2 θ12,m2

12) plane assuming that we know a prio-ri that θ13 = 0 (which is the simulated value). In otherwords in our minimization procedure we fixθ13 = 0in Tj and the only fitted parameters are tan2 θ12and m2

12. This case corresponds to the usual two-neutrino analysis. Given ourχ2 prescription the bestfit reconstructed point corresponds exactly with thesimulated point. The shown regions correspond to 1σ ,

2σ and 3σ for 2 d.o.f. (χ2 = 2.30, 6.18 and 11.83,respectively). Similar regions are obtained ifθ13 is dif-ferent from zero but assumed to be known so that itsvalue in the simulated number of events and in the re-constructed one is kept to be the same and constant.In the fourth column in Table 1 we list the 3σ al-lowed range for tan2 θ12 for each of the five simu-lated points. Table 2 contains the corresponding re-sults from the “conservative” analysis in which thethree first bins have been removed. The main effect is adecrease in the statistics which translates into slightlylarger ranges. We only list the reconstructed range inthe first octant but, as shown in the figure, due to thenegligible matter effects, the results from KamLANDwill give us a degeneracy inθ12 and an equivalentrange is obtained in the second octant corresponding totan2 θ12 → 1/ tan2 θ12. Solar data will be able to selectthe allowed region in the first octant and we show the3σ contours of the LMA region from the latest analy-sis. As discussed in Ref. [17], if the mixing angle isfar enough from maximal mixing, the allowed regionis clearly separated from the mirror one.

In order to illustrate the effect of the unknownθ13we have repeated this exercise but now using a differ-ent value ofθ13 for the simulated point and the recon-structed ones. In Fig. 1(b) we show the reconstructedregions in(tan2 θ12,m2

12) corresponding to the samefive generated points (which are marked by stars inFig. 1) but usingθ13 = 12.6 (sin2 θ13 = 0.048) in thecalculation of the expected number of eventsTj . FromFig. 1 we see that best fit reconstructed points (whichare marked by squares in Fig. 1) as well as the allowedregions are shifted in mixing angle with respect to theones in Fig. 1(a) whilem2

12 remains practically un-affected. In Tables 1 and 2 we list the reconstructedranges of tan2 θ23 for this academic case.

The observed shift can be easily understood asfollows. From Eq. (3), the total number of events is


Fig. 1. (a) Regions of the(m221 × tan2 θ12)-parameter space al-

lowed by three KamLAND-years of simulated data at the 1σ , 2σand 3σ CL, for different input values ofm2

21 and tan2 θ12

(θ13 = 0) and assumingθ13 = 0. The stars indicate the best fitpoints (corresponding also to the simulated signals). (b) Regionsof the (m2

21 × tan2 θ12)-parameter space allowed by three Kam-LAND-years of simulated data at the 1σ , 2σ and 3σ CL forθ13 = 12.6 (sin2 θ13 = 0.048). The stars indicate the points usedto simulate the signal (with fixedθ13 = 0) while the squares indi-cate the reconstructed best fit point (with fixedθ13 = 12.6).

equal for differentθ13 with the condition

(3)1− α × sin2 2θ12 = cos4 θ13(1− α × sin2 2θ ′

12

),

whereα is a number coming from all the detailed in-tegration of the oscillating phase factor which dependsmainly onm2

12 and which, for completeness, we alsolist in Table 1. For example, for the simulated point (1),tan2 θ12 = 0.37,m2

12 = 3.7× 10−5 eV2 andθ13 = 0,the 3σ reconstructed range, 0.31 tan2 θ12 0.43,

for θ13 = 0 is shifted using the above relation withα = 0.65 to 0.25 tan2 θ12 0.35 for tan2 θ13 = 0.05which precisely coincides with the values listed in the5th column of Table 1. Strictly speaking this scalingis slightly violated due to the change of the spectralshape with the change fromθ12 to θ ′

12 which also wors-ens theχ2

min for the reconstructed point.Let us finally consider which are the allowed

regions in the parameter space(tan2 θ12,m221) taking

into account that we just know thatθ13 is belowsome limit. In order to do so one must integrateover θ13 in the allowed range, or what is equivalent,for each pair (tan2 θ12,m2

21) we must minimizeχ2(θ12,m2

21, tan2 θ13) with respect toθ13 (restrictedto be below the present bound). Notice that, belowthe bound, we used a flat probability distribution forθ13 to keep the analysis just KamLAND-dependent. Inthe future, the combined analysis of solar, atmosphericand CHOOZ results with KamLAMD data will allowus to include the probability distribution forθ13. InFig. 2 we shown the allowed regions for suchθ13-freeanalysis, where free means allowed to vary below its3σ limit, θ13 13.8 (sin2 θ13 0.057). As seen bycomparing Fig. 2 with Fig. 1 the reconstructed regionsare enlarged and roughly correspond to the overlap ofthe allowed regions for the different fixed values ofθ13. Given our signal generating procedure the best fitreconstructed point corresponds to the simulated pointin (tan2 θ12, m2

21) and θ13 = 0. The reconstructedranges can be read from the last column in Tables 1and 2. Also comparing the results in both tables onesees that the presence of this effect is not quantitativelyaffected by not including in the fit the data from thelowest energy bins. The main effect being a decreasein the statistics which translates into slightly largerranges.

We have studied the role ofθ13 in the hypotheticalcase of perfect knowledge of the overall flux normal-ization and fuel composition. Comparing our resultswith the expected degradation on the parameter de-termination associated with the uncertainty on thosetwo assumptions [15], we note that the reconstructedranges obtained in theθ13-free case are very close tothose obtained in the analysis with perfect knowledgeof θ13 but where the overall flux normalization is un-known by about∼ 10%. As mentioned above, the roleof θ13 is essentially the change of the normalization.This is a larger normalization error that the 3% ex-


Fig. 2. 1σ , 2σ and 3σ CL allowed regions of the(m221 ×

tan2 θ12)-parameter space whenθ13 is left free to vary in the allowedrange (θ13 < 13.8 (sin2 θ13 = 0.057)). The regions are obtainedfor three KamLAND-years of simulated data and for the same fivesimulated signals as Fig. 1(a). The stars indicate the best fit points(corresponding also to the simulated points).

pected one in the theoretical calculation of the fluxfrom the reactors (induced from theβ-spectroscopyexperiment at the Goesgen reactor [25]). We also findthat the uncertainty associated withθ13 has a larger im-pact on the determination of the mixing angleθ12 thanthe expected error associated with the fuel composi-tion although, unlike this last one, it does not affectthe determination ofm2

12.Let us point out, that obviously, in order for this

effect to become relevant it has to be larger than theexpected statistical uncertainty on the overall flux nor-malization. If we repeat this exercise assuming onlyone KamLAND-years of simulated data, we find verylittle difference between the results corresponding tofixed θ13 = 0 and the free-θ13 analysis, as expectedsince the expected number of events would be of theorder∼ 400 and the associated statistical uncertaintyfor the overall normalization would be comparablewith the maximum effect associated toθ13.

Summarizing, the KamLAND reactor neutrino ex-periment, which is to start taking data very soon, issensitive to the LMA region and should be able ofmeasuring the solar oscillation parameters(tan2 θ12,

m221) with unprecedented precision. In this Letter,

we have addressed the question of the degradation onthe determination of the oscillation parameters associ-ated with our ignorance of the exact value ofθ13. We

have shown that the determination ofm212 is practi-

cally unaffected because the effect ofθ13 on the shapeof the spectrum is very small. The dominant effect isa shift in the overall flux normalization which impliesthat the reconstructedθ12 range scales withθ13 in asimple way. As a consequence the allowed region ofsolar parameters obtained from KamLAND signal willbe broader inθ12. Comparing this effect with the onesfrom the expected uncertainties associated with thetheoretical error on the overall flux normalization andthe fuel composition, we find that, after enough sta-tistics is accumulated, the uncertainty associated withθ13 may become the dominant source of degradation inthe determination the mixing angleθ12 at KamLAND.

Acknowledgements

We thank A. de Gouvea for comments and sugges-tions. M.C.G.-G. is supported by the European UnionMarie Curie fellowship HPMF-CT-2000-00516. Thiswork was also supported by the Spanish DGICYT un-der grants PB98-0693 and PB97-1261, by the Gener-alitat Valenciana under grant GV99-3-1-01, by the Eu-ropean Commission RTN network HPRN-CT-2000-00148 and by the European Science Foundation net-work grant N. 86.

References

[1] Q.R. Ahmad et al., SNO Collaboration, Phys. Rev. Lett. 87(2001) 071301.

[2] B.T. Cleveland et al., Astrophys. J. 496 (1998) 505.[3] J.N. Abdurashitov et al., SAGE Collaboration, Phys. Rev. C 60

(1999) 055801.[4] W. Hampel et al., GALLEX Collaboration, Phys. Lett. B 447

(1999) 127.[5] E. Belloti, Talk at XIX International Conference on Neu-

trino Physics and Astrophysics, Sudbury, Canada, June 2000,http://nu2000.sno.laurentian.ca.

[6] S. Fukuda et al., Super-Kamiokande Collaboration, Phys. Rev.Lett. 86 (2001) 5651.

[7] J.N. Bahcall, M.C. Gonzalez-Garcia, C. Peña-Garay, JHEP0108 (2001) 014.

[8] G.L. Fogli, E. Lisi, D. Montanino, A. Palazzo, Phys. Rev. D 64(2001) 093007.

[9] A. Bandyopadhyay, S. Choubey, S. Goswami, K. Kar, Phys.Lett. B 519 (2001) 83.

[10] P.I. Krastev, A.Y. Smirnov, hep-ph/0108177.

http://nu2000.sno.laurentian.ca


[11] J.N. Bahcall, M.C. Gonzalez-Garcia, C. Peña-Garay, hep-ph/0111150.

[12] J. Busenitz et al., Proposal for US Participation in Kam-LAND, March 1999, unpublished, may be downloaded fromhttp://bfk0.lbl.gov/kamland/.

[13] V. Barger, D. Marfatia, B.P. Wood, Phys. Lett. B 498 (2001)53.

[14] R. Barbieri, A. Strumia, JHEP 0012 (2000) 016.[15] H. Murayama, A. Pierce, hep-ph/0012075.[16] K. Ishihara (for the KamLAND Collaboration), Talk at the

NuFACT’01 Workshop in Tsukuba, Japan, May 24–30, 2001,transparencies athttp://psux1.kek.jp/~nufact01/index.html.

[17] A. de Gouvea, C. Peña-Garay, Phys. Rev. D 64 (2001) 113011,hep-ph/0107186.

[18] M. Apollonio et al., CHOOZ Collaboration, Phys. Lett. B 466(1999) 415.

[19] Y. Fukuda et al., Phys. Lett. B 433 (1998) 9;Y. Fukuda et al., Phys. Lett. B 436 (1998) 33;Y. Fukuda et al., Phys. Lett. B 467 (1999) 185;Y. Fukuda et al., Phys. Rev. Lett. 82 (1999) 2644.

[20] M.C. Gonzalez-Garcia, M. Maltoni, C. Peña-Garay, J.W. Val-le, Phys. Rev. D 63 (2001) 033005, updated versionin C. Peña-Garay, Talk at the NuFACT’01 Workshopin Tsukuba, Japan, May 24–30, 2001, transparencies athttp://psux1.kek.jp/~nufact01/index.html.

[21] G.L. Fogli, E. Lisi, A. Marrone, D. Montanino, A. Palazzo,hep-ph/0104221.

[22] Y. Declais et al., Phys. Lett. B 338 (1995) 383.[23] P. Vogel, J. Engel, Phys. Rev. D 39 (1985) 3378.[24] P. Vogel, J.F. Beacom, Phys. Rev. D 60 (1999) 053003.[25] G. Zacek et al., Phys. Rev. D 34 (1986) 1918.

http://bfk0.lbl.gov/kamland/

http://psux1.kek.jp/~nufact01/index.html

http://psux1.kek.jp/~nufact01/index.html



Lepton flavor violating process in degenerate andinverse-hierarchical neutrino models

Atsushi Kageyama, Satoru Kaneko, Noriyuki Shimoyama, Morimitsu Tanimoto

Department of Physics, Niigata University, Ikarashi 2-8050, 950-2181 Niigata, Japan

Received 23 October 2001; received in revised form 2 December 2001; accepted 4 January 2002

Editor: T. Yanagida

Abstract

We have investigated the lepton flavor violation in the supersymmetric framework assuming the large mixing angle MSWsolution with the quasidegenerate and the inverse-hierarchical neutrino masses. In the case of the quasidegenerate neutrinos,the predicted branching ratio BR(µ → eγ ) strongly depends onmν andUe3. ForUe3 0.05 withmν 0.3 eV, the predictionis close to the present experimental upper bound if the right-handed Majorana neutrino masses are degenerate. On the otherhand, the prediction is larger than the experimental upper bound forUe3 0.05 in the case of the inverse-hierarchical neutrinomasses. 2002 Elsevier Science B.V. All rights reserved.

Super-Kamiokande has almost confirmed the neutrino oscillation in the atmospheric neutrinos, which favors theνµ → ντ process [1]. For the solar neutrinos [2,3], the recent data of the Super-Kamiokande and the SNO also favorthe neutrino oscillationνµ → νe with the large mixing angle (LMA) MSW solution [4,5]. These results mean thatneutrinos are massive, moreover, they indicate the bilarge flavor mixing in the lepton sector. The non-zero neutrinomasses clearly imply new physics beyond the standard model (SM), and the large flavor mixings suggest that theflavor structure in the lepton sector is very different from that in the quark sector.

If neutrinos are massive and mixed in the SM, there exists a source of the lepton flavor violation (LFV). However,due to the smallness of the neutrino masses, the predicted branching ratios for these processes are tiny [6] such asBR(µ→ eγ ) < 10−50.

On the other hand, in the supersymmetric framework the situation is completely different. The SUSY providesnew direct sources of flavor violation in the lepton sector, namely the possible presence of off-diagonal soft termsin the slepton mass matrices(m2

L)ij , (m2

eR)ij , and trilinear couplingsAe

ij . Strong bounds on these matrix elementscome from requiring branching ratios for LFV processes to be below the observed ratios. For the present thestrongest bound comes from theµ → eγ decay.

In order to avoid these dangerous off-diagonal terms, one often imposes the perfect universality of the(m2L)ij ,

(m2eR)ij , Ae

ij matrices, i.e., to take them proportional to the unit matrix. However, even under the universality

E-mail addresses: [email protected] (A. Kageyama), [email protected] (S. Kaneko),[email protected] (N. Shimoyama), [email protected] (M. Tanimoto).



A. Kageyama et al. / Physics Letters B 527 (2002) 206–214 207

assumption, radiative corrections generate off-diagonal soft terms due to the massive neutrinos. The flavor changingoperators giving rise to the non-diagonal neutrino mass matrices will contribute to the renormalization groupequations (RGEs) of the(m2

L)ij , (m2

eR)ij andAe

ij matrices and induces off-diagonal entries.Suppose that neutrino masses are produced by the see-saw mechanism [7], there are the right-handed neutrinos

above a scaleMR. Then neutrinos have the Yukawa coupling matrixYν with off-diagonal entries in the basis of thediagonal charged-lepton Yukawa couplings. The off-diagonal entries ofYν drive off-diagonal entries in the(m2

L)ij

andAeij matrices through the RGEs running [8].

Some authors [9–12] have already analyzed the effect of neutrino Yukawa couplings for the LFV focusingon the recent data of the Super-Kamiokande. Those analyses have been done assuming that neutrino masses arehierarchical ones, which is similar to quarks and charged-leptons. However, since the data of neutrino oscillationsonly indicate the differences of the mass squarem2

ij , the neutrinos may have the quasidegenerate spectrumm1 m2 m3 or the inverse-hierarchical onem1 m2 m3. These cases have been discussed qualitativelyneglectingUe3 in Ref. [12]. In our work, we present quantitative analyses of BR(µ → eγ ) including effect ofUe3 in the case of the quasidegenerate and inverse-hierarchical neutrino masses. It should be emphasized that themagnitude ofUe3 is one of important ingredients to predict the branching ratio.

In terms of the standard parametrization of the mixing matrix [13], the MNS matrixU (lepton mixing matrix)[14] are given as

(1)U =(

c13c12 c13s12 s13e−iφ

−c23s12 − s23s13c12eiφ c23c12 − s23s13s12e

iφ s23c13s23s12 − c23s13c12e

iφ −s23c12 − c23s13s12eiφ c23c13

),

where sij ≡ sinθij and cij ≡ cosθij are mixings in vacuum, andφ is the CP violating phase. Assuming thatoscillations need only accounting for the solar and the atmospheric neutrino data, we take the LMA-MSW solution,in which the magnitude of the MNS matrix elements are given in Ref. [15] and the neutrino mass scales are givenby the data

(2)m2atm= (1.5∼ 5)× 10−3 eV2, m2 = 2.5× 10−5 ∼ 1.6× 10−4 eV2.

In our calculation of the LFV effect, we take the typical valuesm2atm= 3× 10−3 eV2,m2 = 7× 10−5 eV2

ands23 = 1/√

2, s12 = 0.6, s13 0.2, taking account of the CHOOZ data [16]. The CP violating phase is neglectedfor simplicity.

First, let us consider the degenerate neutrino case. The neutrino masses are given as

(3)m1 ≡mν, m2 =mν + 1

2mν

m2, m3 = mν + 1

2mν

m2atm,

in terms of the experimental valuesm2atm andm2. Since the best bound on the neutrinoless double beta

decay obtained by Heidelberg–Moscow group gives [17]mee < 0.34 eV at 90% C.L., we takemν = 0.3 eV inthe following calculations. Themν dependence of our result is commented later.1

The see-saw mechanism leads to the tiny neutrino mass as followsY Tν M

−1R Yνv

2, wherev is the vacuumexpectation value of the relevant Higgs field andMR is the right-handed Majorana neutrino mass matrix. Then, wecan get the Yukawa coupling in the basis of the diagonal charged-lepton Yukawa couplings as [12]

(4)Yν = 1

v

(√MR1 0 00

√MR2 0

0 0√MR3

)R

(√m1 0 00

√m2 0

0 0√m3

)U†,

whereR is an orthogonal 3× 3 matrix and depends on the model. We takeR to be the unit matrix and degenerateright-handed Majorana massesMR1 = MR2 = MR3 ≡ MR . This assumption is derived logically, otherwise a big

1 If we take account of the non-zero Majorana phases, we can takemν larger than 0.34 eV.

208 A. Kageyama et al. / Physics Letters B 527 (2002) 206–214

conspiracy would be needed betweenYν andMR . Actually, this assumption is realized in some models as discussedlater. However, as the conspiracy betweenYν andMR is not excluded a priori, we also discuss the effect of thenon-degenerate right-handed Majonara neutrino masses.

Eq. (4) presents the Yukawa coupling at the electroweak scale. Since we need the Yukawa coupling at the GUTscale or the Planck scale, Eq. (4) should be modified by taking account of the effect of the RGEs [18–20]. ModifiedYukawa couplings at a scaleMR are given as

(5)Yν =√MR

v

(√m1 0 00

√m2 0

0 0√m3

)U†√IgIt

(1 0 00 1 00 0

√Iτ

),

with

(6)Ig = exp

[1

8π2

tR∫tZ

−cig2i dt

], It = exp

[1

8π2

tR∫tZ

y2t dt

], Iτ = exp

[1

8π2

tR∫tZ

y2τ dt

],

wheretR = lnMR andtZ = lnMZ . Here,gi ’s (i = 1–2) are gauge couplings andyt andyτ are Yukawa couplings.In practice, the degenerate neutrino masses with the LMA-MSW solution are predicted by the models with the

flavor symmetryS3R ×S3L [21] orO3R ×O3L [22],2 which gives the democratic mass matrices in the quark sector[23]. In those models,Ue3 is predicted to be∼ 0.05. We will show results for both cases ofUe3 = 0.05 and 0.2.

Second, let us consider the case of the inverse-hierarchical neutrino masses. The neutrino masses are given as

(7)m2 ≡√m2

atm, m1 =m2 − 1

2m2m2, m3 0,

wherem2 >m1 is taken in order to keep the MSW effect.The typical model of the inverse-hierarchy is the Zee model [24], in which the right-handed neutrinos do not

exist. However, one can also consider Yukawa textures which lead to the inverse mass hierarchy through the see-saw mechanism as seen in Eq. (4). The detailed analyses with definite models [25] will show in a further comingpaper. In this work, we takeR = I and degenerate right-handed Majorana massesMR1 = MR2 = MR3 ≡ MR

as well as the case of degenerate neutrino masses. We also discuss the case of the non-degenerate right-handedMajonara neutrino masses.

In the presence of non-zero neutrino Yukawa couplings, we can expect the LFV phenomena in the lepton sector.Within the framework of SUSY models, the flavor violation in neutrino Yukawa couplings induces the LFV inslepton masses even if we assume the universal scalar mass for all scalars at the GUT scale. In the present models,the LFV is generated in left-handed slepton masses since right-handed neutrinos couple to the left-handed leptonmultiplets. A RGE for the left-handed doublet slepton masses(m2

L) can be written as

µd(m2

L)ij

dµ=(µd(m2

L)ij

dµ

)MSSM

(8)+ 1

16π2

[m2

LY †ν Yν + Y †

ν Yνm2L

+ 2(Y †ν m

2νYν + m2

HuY †ν Yν +A†

νAν

)]ij,

wherem2ν

and m2Hu

are soft SUSY-breaking masses for right-handed sneutrinos (ν) and doublet Higgs (Hu),

respectively. Here(µd(m2L)ij /dµ)MSSM denotes the RGE in case of the minimal SUSY standard model (MSSM),

and the terms explicitly written are additional contributions in the presence of the neutrino Yukawa couplings. In

2 These models predicted the effective left-handed Majorana neutrino masses by the symmetry without introducing the see-saw mechanism.However, one can easily get the see-saw realization by introducing the right-handed neutrinosνR . Assuming the same transformation propertyof the flavor symmetry for bothνL andνR , one obtainsMR ∝ I andYν ∝ I .


a basis where the charged-lepton Yukawa couplings are diagonal, the term(µd(m2L)ij /dµ)MSSM does not provide

any flavor violations. Therefore the only source of LFV comes from the additional terms. In our analysis, wenumerically solve the RGEs, and then calculate the event rates for the LFV processes by using the complete formulain Ref. [9]. Here, in order to obtain an approximate estimation for the LFV masses, let us consider approximatesolution to the LFV mass terms (i = j ):

(9)(m2

L

)ij

− (6+ 2a20)m

20

16π2

(Y †ν Yν

)ij

lnMX

MR

,

where we assume a universal scalar mass(m0) for all scalars and a universalA-term(Af = a0m0Yf ) at the GUTscaleMX . It is noticed that large neutrino Yukawa couplings and large lepton mixings generate the large LFV inthe left-handed slepton masses.

The decay rates can be approximated as follows:

(10)Γ (ei → ejγ ) e2

16πm5

eiF∣∣(m2

L

)ij

∣∣2,whereF is a function of masses and mixings for SUSY particles.

Before showing numerical results, we present a qualitative discussion on(Y †ν Yν)21, which is a crucial quantity to

predict the branching ratio BR(µ→ eγ ). This is given in terms of neutrino masses and mixings at the electroweakscale as follows:

(11)(Y †ν Yν

)21 = MR

v2u

[Uµ2U

∗e2(m2 −m1)+Uµ3U

∗e3(m3 −m1)

],

wherevu ≡ v sinβ with v = 174 GeV is taken as an usual notation and a unitarity relation of the MNS matrixelements is used. Taking the three cases of the neutrino mass spectra, the degenerate, the inverse-hierarchical andthe normal hierarchical masses, one obtains

(Y †ν Yν

)21 MR

2√

2v2u

m2atm

mν

[1√2U∗e2

m2m2

atm+U∗

e3

](degenerate)

MR√2v2

u

√m2

atm

[1

2√

2U∗e2

m2m2

atm−U∗

e3

](inverse)

(12) MR√2v2

u

√m2

atm

[1√2U∗e2

√m2m2

atm+U∗

e3

](hierarchy),

where we take the maximal mixing for the atmospheric neutrinos. SinceUe2 1/√

2 for the bimaximal mixingmatrix, the first terms in the square brackets of the right-hand sides of Eqs. (12) can be estimated by puttingthe experimental data. We should take care the magnitude ofUe3 in the second terms to predict the branchingratios because the second term is the dominant one as far asUe3 0.01 (0.1) for the degenerate and the inverse-hierarchical cases (for the normal hierarchical case). For the case of the degenerate neutrino masses,(Y †

ν Yν)21depends on the unknown neutrino mass scalemν . As one takes the largermν , one predicts the smaller branchingratio. In our calculation, we takemν = 0.3 eV, which is close to the upper bound of the neutrinoless double betadecay experiment.

As seen in Eqs. (12), we expect that the case of the degenerate neutrino masses gives the smallest branchingratio BR(µ→ eγ ). The comparison in cases of the inverse-hierarchical masses and the normal hierarchical massesdepends on the magnitude and phase ofUe3. In the limit of Ue3 = 0, the case of the normal hierarchical massespredicts larger branching ratio. However, forUe3 0.2 the predicted branching ratios are almost same in bothcases.


Fig. 1. Predicted branching ratio BR(µ → eγ ) versus theleft-handed selectron mass for tanβ = 3, 10, 30 in the case of thedegenerate neutrino masses. HereMR = 1014 GeV andUe3 = 0.2are taken. The solid curves correspond toM2 = 150 GeV and thedashed ones toM2 = 300 GeV. A horizontal dotted line denotesthe experimental upper bound.

Fig. 2. Predicted branching ratio BR(µ → eγ ) in the case of thehierarchical neutrino masses. Parameters are taken as same as inFig. 1.

Let us present numerical calculations in the case of the degenerate neutrino masses assuming a universal scalarmass(m0) for all scalars anda0 = 0 as a universalA-term at the GUT scale (MX = 2 × 1016 GeV). We showBR(µ → eγ ) versus the left-handed selectron massmeL for each tanβ = 3, 10, 30 and a fixed wino massM2 atthe electroweak scale. In Fig. 1, the branching ratio is shown forM2 = 150, 300 GeV in the case ofUe3 = 0.2 withMR = 1014 GeV, in which the solid curves correspond toM2 = 150 GeV and the dashed ones toM2 = 300 GeV.The threshold of the selectron mass is determined by the recent LEP2 data [26] forM2 = 150 GeV, but determinedby the constraint that the left-handed slepton should be heavier than the neutralinos forM2 = 300 GeV. As the tanβincreases, the branching ratio increases because the decay amplitude from the SUSY diagrams is approximatelyproportional to tanβ [9]. It is found that the branching ratio is almost larger than the experimental upper bound inthe case ofM2 = 150 GeV. On the other hand, the predicted values are smaller than the experimental bound exceptfor tanβ = 30 in the case ofM2 = 300 GeV.

In order to compare the degenerate neutrino mass case with the hierarchical one (m3 m2 m1), we show theresults of the hierarchical case in Fig. 2, in which parameters are taken as same as in Fig. 1. It is remarked that thebranching ratio is suppressed in the degenerate case.

Since the typical models [21,22] predictUe3 0.05, we also show the branching ratio forUe3 = 0.05 in Fig. 3.In this case, our predictions lie under the experimental upper bound even in the case ofM2 = 150 GeV exceptfor tanβ = 30, however, it is not far away from the experimental bound. Therefore, we expect the observation ofµ → eγ in the near future.

Our predictions depend onMR strongly, because the magnitude of the neutrino Yukawa coupling is determinedby MR as seen in Eq. (5). IfMR reduces toMR = 1012 GeV, the branching ratio reduces to 10−4 roughly since itis proportional toM2

R . The numerical result is shown in Fig. 4. Thus, the branching ratio is not so large comparedwith the predictions in the case of the hierarchical neutrino masses [11].


Fig. 3. Predicted branching ratio BR(µ → eγ ) versus theleft-handed selectron mass for tanβ = 3, 10, 30 in the caseof the degenerate neutrino masses. HereMR = 1014 GeVand Ue3 = 0.05 are taken. The solid curves correspond toM2 = 150 GeV and the dashed ones toM2 = 300 GeV.

Fig. 4. Predicted branching ratio BR(µ → eγ ) versus theleft-handed selectron mass for tanβ = 3, 10, 30 in the case of thedegenerate neutrino masses. HereMR = 1012 GeV andUe3 = 0.2are taken. The solid curves correspond toM2 = 150 GeV and thedashed ones toM2 = 300 GeV.

It may be important to comment on the effect of the non-degenerate right-handed Majorana neutrino massessince our results depend on the degeneracy of the right-handed Majorana neutrino masses. We replace(Y †

ν Yν)21 inEq. (11) with

(13)(Y †ν Yν

)21 = MR

v2u

[Uµ2U

∗e2(ε2m2 − ε1m1)+Uµ3U

∗e3(m3 − ε1m1)

],

whereε1 = MR1/MR3, ε2 = MR2/MR3 andMR = MR3. The degenerate right-handed Majorana neutrino massescorrespond toε1 = ε2 = 1. If ε1 andε2 deviate from 1 (the non-degenerate case), the cancellation amongν1, ν2andν3 is weakened. In other words, the LFV is not so suppressed. In the caseε1 ε2 1, the result is similar tothe one in the case of hierarchical neutrino masses.

Next we show the results in the case of the inverse-hierarchical neutrino masses. As expected in Eq. (12), thebranching ratio is much larger than the one in the degenerate case. In Fig. 5, the branching ratio is shown forM2 = 150, 300 GeV in the case ofUe3 = 0.2 with MR = 1014 GeV. In Fig. 6, the branching ratio is shown forUe3 = 0.05 with MR = 1014 GeV. TheMR dependence is similar to the case of the degenerate neutrino masses.The predictions almost exceed the experimental bound as far asUe3 0.05, tanβ 10 andMR 1014 GeV.

These results also depend on the degeneracy of the right-handed Majorana neutrino masses. As seen in Eq. (13),the cancellation betweenν1 andν2 is weakened ifε1 = ε2. The predicted branching ratio depends onε1 andε2. Forexample, it could be larger than the one in the case of hierarchical neutrino masses ifε1 ε2 1 is realized in theright-handed Majorana neutrino mass matrix. Actually, the typical model [25] does not guarantee our assumptionMR = I , we need careful analyses, which will be presented in a further coming paper.


Fig. 5. Predicted branching ratio BR(µ → eγ ) versus theleft-handed selectron mass for tanβ = 3, 10, 30 in the case ofthe inverse-hierarchical neutrino masses. HereMR = 1014 GeVand Ue3 = 0.2 are taken. The solid curves correspond toM2 = 150 GeV and the dashed ones toM2 = 300 GeV.

Fig. 6. Predicted branching ratio BR(µ → eγ ) versus theleft-handed selectron mass for tanβ = 3, 10, 30 in the case ofthe inverse-hierarchical neutrino masses. HereMR = 1014 GeVand Ue3 = 0.05 are taken. The solid curves correspond toM2 = 150 GeV and the dashed ones toM2 = 300 GeV.

Summary

We have investigated the LFV effect in the supersymmetric framework assuming LMA-MSW solution withthe quasidegenerate and the inverse-hierarchical case of neutrino masses. We show the predicted branching ratioof µ → eγ for both cases. We expect the relation BR(degenerate) BR(inverse) < BR(hierarchy) if the right-handed Majorana neutrino masses are degenerate. In the case of the quasidegenerate neutrinos, the predictedbranching ratio strongly depends onMR, mν andUe3. ForUe3 0.05 with mν 0.3 eV, the prediction is closeto the present experimental upper bound. On the other hand, the prediction is larger than the experimental upperbound forUe3 0.05 in the case of the inverse-hierarchical neutrino masses. More analyses including the processτ →µγ will be present elsewhere.

Acknowledgements

We would like to thank Drs. J. Sato and H. Nakano for useful discussions. We also thank the organizers andparticipants of Summer Institute 2001 held at Yamanashi, Japan for helpful discussions. This research is supportedby the Grant-in-Aid for Science Research, Ministry of Education, Science and Culture, Japan (No. 10640274,No.12047220).

References

[1] Super-Kamiokande Collaboration, Y. Fukuda et al., Phys. Rev. Lett. 81 (1998) 1562;Super-Kamiokande Collaboration, Y. Fukuda et al., Phys. Rev. Lett. 82 (1999) 2644;Super-Kamiokande Collaboration, Y. Fukuda et al., Phys. Rev. Lett. 82 (1999) 5194.


[2] Super-Kamiokande Collaboration, S. Fukuda et al., Phys. Rev. Lett. 86 (2001) 5651;Super-Kamiokande Collaboration, S. Fukuda et al., Phys. Rev. Lett. 86 (2001) 5656.

[3] SNO Collaboration, Q.R. Ahmad et al., Phys. Rev. Lett. 87 (2001) 071301, nucl-ex/0106015.[4] L. Wolfenstein, Phys. Rev. D 17 (1978) 2369;

S.P. Mikheyev, A.Yu. Smirnov, Yad. Fiz. 42 (1985) 1441;E.W. Kolb, M.S. Turner, T.P. Walker, Phys. Lett. B 175 (1986) 478;S.P. Rosen, J.M. Gelb, Phys. Rev. D 34 (1986) 969;J.N. Bahcall, H.A. Bethe, Phys. Rev. Lett. 65 (1990) 2233;N. Hata, P. Langacker, Phys. Rev. D 50 (1994) 632;P.I. Krastev, A.Yu. Smirnov, Phys. Lett. B 338 (1994) 282.

[5] G.L. Fogli, E. Lisi, D. Montanino, A. Palazzo, Phys. Rev. D 64 (2001) 093007;J.N. Bahcall, M.C. Gonzalez-Garcia, C. Peña-Garay, JHEP 0108 (2001) 014;V. Barger, D. Marfatia, K. Whisnant, hep-ph/0106207;A. Bandyopadhyay, S. Choubey, S. Goswami, K. Kar, hep-ph/0106264.

[6] S.T. Petcov, Sov. J. Nucl. Phys. 25 (1977) 340;T.P. Cheng, L.F. Li, Phys. Rev. D 16 (1977) 1425;W. Marciano, H. Sandra, Phys. Lett. B 67 (1977) 303;B.W. Lee, R. Shrock, Phys. Rev. D 16 (1977) 1455.

[7] M. Gell-Mann, P. Ramond, R. Slansky, in: P. van Nieuwenhuizen, D. Freedman (Eds.), Supergravity, Proc. of the Workshop, Stony Brook,New York, 1979, North-Holland, Amsterdam, 1979, p. 315;T. Yanagida, in: O. Sawada, A. Sugamoto (Eds.), Proc. of the Workshop on the Unified Theories and Baryon Number in the Universe,Tsukuba, Japan, 1979, KEK Report, No. 79-18, Tsukuba, 1979, p. 95;R.N. Mohapatra, G. Senjanovic, Phys. Rev. Lett. 44 (1980) 912.

[8] F. Borzumati, A. Masiero, Phys. Rev. Lett. 57 (1986) 961.[9] J. Hisano, T. Moroi, K. Tobe, M. Yamaguchi, T. Yanagida, Phys. Lett. B 357 (1995) 579;

J. Hisano, T. Moroi, K. Tobe, M. Yamaguchi, Phys. Rev. D 53 (1996) 2442.[10] J. Hisano, D. Nomura, T. Yanagida, Phys. Lett. B 437 (1998) 351;

J. Hisano, D. Nomura, Phys. Rev. D 59 (1999) 116005;M.E. Gomez, G.K. Leontaris, S. Lola, J.D. Vergados, Phys. Rev. D 59 (1999) 116009;W. Buchmüller, D. Delepine, F. Vissani, Phys. Lett. B 459 (1999) 171;W. Buchmüller, D. Delepine, L.T. Handoko, Nucl. Phys. B 576 (2000) 445;J. Ellis, M.E. Gomez, G.K. Leontaris, S. Lola, D.V. Nanopoulos, Eur. Phys. J. C 14 (2000) 319;J.L. Feng, Y. Nir, Y. Shadmi, Phys. Rev. D 61 (2000) 113005;S. Baek, T. Goto, Y. Okada, K. Okumura, Phys. Rev. D 63 (2001) 051701.

[11] J. Sato, K. Tobe, T. Yanagida, Phys. Lett. B 498 (2001) 189;J. Sato, K. Tobe, Phys. Rev. D 63 (2001) 116010;S. Lavignac, I. Masina, C.A. Savoy, hep-ph/0106245.

[12] J.A. Casas, A. Ibarra, hep-ph/0103065;J.A. Casas, A. Ibarra, hep-ph/0109161.

[13] Particle Data Group, D.E. Groom et al., Eur. Phys. J. C 15 (2000) 1.[14] Z. Maki, M. Nakagawa, S. Sakata, Prog. Theor. Phys. 28 (1962) 870.[15] M. Fukugita, M. Tanimoto, Phys. Lett. B 515 (2001) 30.[16] CHOOZ Collaboration, M. Apollonio et al., Phys. Lett. B 420 (1998) 397.[17] Heidelberg–Moscow Collaboration, L. Baudis et al., Phys. Rev. Lett. 83 (1999) 41;

H.V. Klapdor-Kleingrothaus, hep-ph/0103074.[18] P.H. Chankowski, Z. Pluciennik, Phys. Lett. B 316 (1993) 312;

K.S. Babu, C.N. Leung, J. Pantaleone, Phys. Lett. B 319 (1993) 191;P.H. Chankowski, W. Krolikowski, S. Pokorski, hep-ph/9910231;J.A. Casas, J.R. Espinosa, A. Ibarra, I. Navarro, Phys. Lett. B 473 (2000) 109.

[19] M. Tanimoto, Phys. Lett. B 360 (1995) 41;J. Ellis, G.K. Leontaris, S. Lola, D.V. Nanopoulos, Eur. Phys. J. C 9 (1999) 389;J. Ellis, S. Lola, Phys. Lett. B 458 (1999) 310;J.A. Casas, J.R. Espinosa, A. Ibarra, I. Navarro, Nucl. Phys. B 556 (1999) 3;J.A. Casas, J.R. Espinosa, A. Ibarra, I. Navarro, JHEP 9909 (1999) 015;J.A. Casas, J.R. Espinosa, A. Ibarra, I. Navarro, Nucl. Phys. B 569 (2000) 82;M. Carena, J. Ellis, S. Lola, C.E.M. Wagner, Eur. Phys. J. C 12 (2000) 507.

[20] N. Haba, Y. Matsui, N. Okamura, M. Sugiura, Eur. Phys. J. C 10 (1999) 677;


N. Haba, Y. Matsui, N. Okamura, M. Sugiura, Prog. Theor. Phys. 103 (2000) 145;N. Haba, N. Okamura, Eur. Phys. J. C 14 (2000) 347;N. Haba, N. Okamura, M. Sugiura, Prog. Theor. Phys. 103 (2000) 367.

[21] M. Fukugita, M. Tanimoto, T. Yanagida, Phys. Rev. D 57 (1998) 4429;M. Tanimoto, Phys. Rev. D 59 (1999) 017304.

[22] M. Tanimoto, T. Watari, T. Yanagida, Phys. Lett. B 461 (1999) 345.[23] H. Harari, H. Haut, J. Weyers, Phys. Lett. B 78 (1978) 459;

Y. Koide, Phys. Rev. D 28 (1983) 252;Y. Koide, Phys. Rev. D 39 (1989) 1391;P. Kaus, S. Meshkov, Mod. Phys. Lett. A 3 (1988) 1251;M. Tanimoto, Phys. Rev. D 41 (1990) 1586;G.C. Branco, J.I. Silva-Marcos, M.N. Rebelo, Phys. Lett. B 237 (1990) 446;H. Fritzsch, J. Plankl, Phys. Lett. B 237 (1990) 451.

[24] A. Zee, Phys. Lett. B 93 (1980) 389;A. Zee, Phys. Lett. B 161 (1985) 141;L. Wolfenstein, Nucl. Phys. B 175 (1980) 92;S.T. Petcov, Phys. Lett. B 115 (1982) 401;C. Jarlskog, M. Matsuda, S. Skadhauge, M. Tanimoto, Phys. Lett. B 449 (1999) 240;P.H. Frampton, S. Glashow, Phys. Lett. B 461 (1999) 95.

[25] Q. Shafi, Z. Tavartkiladze, Phys. Lett. B 482 (2000) 145, hep-ph/0101350.[26] LEP2 SUSY Working Group,http://alephwww.cern.ch/~ganis/SUSYWG/SLEP/sleptons_2k01.html.

http://alephwww.cern.ch/~ganis/SUSYWG/SLEP/sleptons_2k01.html



Core structure and exactly solvable models in dilaton gravitycoupled to Maxwell and antisymmetric tensor fields

Konstantin G. Zloshchastieva,b

a Department of Physics, National University of Singapore, Singapore 117542, Singaporeb Department of Theoretical Physics, Dnepropetrovsk State University, Dnepropetrovsk 49050, Ukraine


Editor: T. Yanagida

Abstract

We consider theD-dimensional massive dilaton gravity coupled to Maxwell and antisymmetric tensor fields (EMATD).We derive the full separability of this theory in static case. This discloses the core structure of the theory and yields the simpleprocedure of how to generate integrability classes. As an example we take a certain new class, obtain the two-parametric familiesof dyonic solutions. It turns out that at some conditions they tend to theD-dimensional dyonic Reissner–Nordström–de-Sittersolutions but with “renormalized” dyonic charge plus a small logarithmic correction. The latter has the significant influence onthe global structure of the non-perturbed solution. We speculate on physical importance of the deduced integrability classes,in particular on their possible role in understanding of the problem of unknown dilaton potential in modern cosmological andlow-energy string models. 2002 Elsevier Science B.V. All rights reserved.

PACS: 04.20.Jb; 04.40.Nr; 04.50.+h

1. Introduction

We start with the following action

(1)S =∫

dDx√−g

[R − β

2(∂φ)2 +ΞF 2 +ΨF 2

(p) +Λ

],

whereR is the Ricci scalar,p =D − 2,F andF(p) are two- andp-forms, respectively,Ξ , Ψ andΛ are functionsof dilatonφ [1], andβ is some unspecified constant (not necessary positive). The models of such a kind appear inmodern cosmological and low-energy string and supergravity theories. We will be interested in static solutions ofthis system hence further we will work with the metric ansatz

(2)ds2 = −eU(r) dt2 + e−U(r) dr2 + eA(r) ds2(p,k),

E-mail address: [email protected] (K.G. Zloshchastiev).



216 K.G. Zloshchastiev / Physics Letters B 527 (2002) 215–225

whereds2(p,k)

is ap-dimensional maximally symmetrical space withk being−1, 0,+1 depending on geometry(Hp , Ep , Sp)—we are going to handle them simultaneously and uniformly. The Maxwell andp-form fields areassumed being in the form

(3)F =Qe−pA/2

Ξdt ∧ dr, F(p)M...N = Pe−pA/2εM...N

in an orthonormal frame, whereQ andP are electric and magnetic charges. Then the field equations become

(4)A′′ +A′(U ′ + pA′

2

)− 2

pΞ e−pA−U − 2

pΛe−U − 2k(p− 1)e−A−U = 0,

(5)βφ′′ + βφ′(U ′ + pA′

2

)+ Ξ,φ e

−pA−U +Λ,φ e−U = 0,

(6)A′′ + A′2

2+ βφ′2

p= 0,

where′ ≡ ∂r andΞ ≡ 2Q2Ξ−1 + p!P 2Ψ .

2. Separability and core structure of EMATD gravity

Now we will rule out the full separability of the static dyonic EMATD theory (the case without antisymmetrictensor has been considered in Ref. [2]). Due to that separability, we will be equipped with the straightforwardprocedure of generating of the numerous classes of integrability that are dyonic besidesΛ is non-zero in generalcase.

Applying the approach of Ref. [2] one can obtain the following system (which is similar to that from the EMDcase)

(7)2k(p− 1)+ 2eA

p

(Λ+ e−pAΞ

) + eU+2Y(

β

pA2,φ

− U,φ

A,φ

− p − 1

2

)= 0,

(8)2k(p− 1)+ 2eA

p

(Λ+ p

2βΛ,φA,φ

)+ eU+2Y

(1

A,φ

),φ

+ 2e−(p−1)A

p

(Ξ + p

2βΞ,φA,φ

)= 0,

(9)φ′ ± eY−A/2

A,φ

= 0, Y (φ)≡ −β

p

∫dφ

A,φ

+ Y0,

whereU(r) ≡ U(φ(r)) andA(r) ≡ A(φ(r)), the subscript “φ” stands for the derivative with respect to dilaton.This system is equivalent to the initial one but has much more capabilities. First of all, it is explicitly separable:U

is algebraically given by Eq. (8) so one can easily exclude it from Eq. (7) to receive the so-called class equation

(10)H,φ

A,φ

+(

β

pA2,φ

+ p− 1

2

)H + k(p − 1)+ eA

p

(Λ+ e−pAΞ

) = 0,

where

H ≡ 1

p(1/A,φ),φ

[kp(p − 1)+ eA

(Λ+ p

2βΛ,φA,φ

)+ e−(p−1)A

(Ξ + p

2βΞ,φA,φ

)],

and thus to come to the system of autonomous equations yieldingA, φ, U consecutively. Eq. (10) is a non-linearthird-order ODE with respect toA(φ) so the direct task (finding ofA at givenΞ ,Ψ ,Λ) is still hard to solve without

K.G. Zloshchastiev / Physics Letters B 527 (2002) 215–225 217

supplementary symmetries or assumptions. However, using this equation one may study the inverse problem, i.e.,the obtaining of theΞ–Ψ –Λ triplets corresponding to a concrete fixedA. Thus, with everyA it is associated theappropriate class of integrability determined by the equation above. It will help that the equation is a linear (atmost) second-order ODE with respect toΞ andΛ, besides, having only one equation it is much easier to studythe integrability classes numerically, e.g., to clarify whether they always have stable solutions, see Ref. [3] andreferences on appropriate methods therein.

Moreover, there is an exceptional class in this construction. IfA∼ φ thenU immediately disappears in Eq. (8),so the latter becomes a linear first-order ODE with respect toΛ andΞ . The linear class is of interest both by itselfand in connection with supergravity models, so below we will study it in more details. Then, as an example, wewill pick some concrete EMATD model to obtain its general-in-class solutions.

3. An example: linear class

Let us impose

(11)A= 4d1

pφ − lnd2,

with di being arbitrary constants. Then Eq. (9) yields

(12)φ =

4pd1

8d21 + pβ

ln

[√d2(8d2

1 +pβ)(r − r0)

16d21

], d2

1 + pβ

8= 0,

−i

√d2p

2β(r − r0), d1 = i

2

√pβ

2,

and Eq. (8) becomes the equation of integrability class

(13)e

4d1p φ

d2

(Λ+ 2d1

βΛ,φ

)+ e

4(1−p)d1p φ

d1−p2

(Ξ + 2d1

βΞ,φ

)+ kp(p− 1)= 0,

whereasU can be easily found from Eq. (7) which is the linear first-order ODE with respect toeU . It should benoted that the extended Lambda–Maxwell duality (discussed in Ref. [2] atΨ = 0) appears to be broken atD = 4;it is curious that electric–magnetic duality is also broken ifD = 4, therefore,D = 4 turns out to be a magic numberagain. Now it is time to take some concrete narrow physical system and obtain its solutions within the frameworksof the linear class.

3.1. String-inspired model: solutions

We choose the physically important model which was first integrated (atΛ= 0) by Gibbons and Maeda [4]:

(14)Ξ = −e− 4g2

p φ, Ψ = − 2

p!e− 4gp

p φ, β = 8

p,

whereg’s are coupling constants. WhenΨ vanishes theng2 = 1 corresponds to field theory limit of superstringmodel,g2 = √

1+ p/n corresponds to the toroidalT n reduction of (D + n)-spacetime toD-spacetime,g2 = 0 isa usual Einstein–Maxwell system;Λ is precisely unknown in string theory [1]. The models of such a type havebeen intensively studied in the caseΛ= 0 [5] but the progress in the models which contain both an antisymmetrictensor and a massive dilaton is still rather slow [6] despite their obvious importance. With the settings (14) Eq. (12)


becomes

(15)φ =

pd1

2(d21 + 1)

ln

[√d2(d

21 + 1)(r − r0)

2d21

], d2

1 + 1 = 0,

−i

√d2

4p(r − r0), d1 = i.

For further it is convenient to define the three polynomials:

π1 = d21 − 1, π2 = pd2

1 − g2d1 − 1, π3 = pd21 + gpd1 − 1.

The integration of the class equation above reveals the following cases (note that additionally each case may containthe multiple subcases determined by the combinations of parameters apart fromπi = 0 ones at which an initialU ,but notΛ, becomes singular):

(i) π1 = 0,π2 = 0,π3 = 0.Integrating Eq. (13) with (14) we see thatΛ must be

(16)Λ= a0e− 4

pd1φ + kd2

π1p(p − 1)e− 4d1

p φ − 2dp2 e−4d1φ

[Q2

π2(1+ g2d1)e

4g2p φ + P 2

π3(1− gpd1)e

− 4gpp φ

],

wherea0 is integration constant, correspondingA, φ are given by Eqs. (11), (15), andU is given by Eq. (7)provided (11) and (14):

eU = ce2

1−(p−1)d21

pd1φ + 4k(p− 1)d4

1e4φpd1

π1(pd21 − π1)

+ 4a0d21e

4d1p φ

d2p(pd21 + π1)

+ 8Q2d41d

p−12 e

− 4(π2−d21)

pd1φ

π2(π2 − d1(d1 + g2))

(17)+ 8P 2d41d

p−12 e

− 4(π3−d21)

pd1φ

π3(π3 − d1(d1 − gp)),

where c is another integration constant related to mass. As was alerted above, this case contains subcasespd2

1 ± π1 = 0, π2 − d1(d1 + g2) = 0, π3 − d1(d1 − gp) = 0 making the last equation, but not Eq. (16), singular.For the sake of brevity, we do not present them here.

(ii) π1 = 0,π2 = 0,π3 = 0.We choose the positive rootd1 = 1 then in the same way as above one can show thatΛ must be

(18)Λ= [a0 − 4kd2(p− 1)φ

]e− 4

p φ − 2dp2 e−4φ

[Q2(1+ g2)e

4g2p φ

p − 1− g2+ P 2(1− gp)e

− 4gpp φ

p − 1+ gp

],

correspondingA, φ are given by Eqs. (11), (15) withd1 being as above, whereasU turns out to be

eU = e4p φ

(p/2)2

[ce−2φ + a0

d2+ k

(p2 +p − 2

) − 4k(p− 1)φ

]

+ 8Q2dp−12 e

− 4(p−2−g2)p

φ

(p − 2− 2g2)(p− 1− g2)+ 8P 2d

p−12 e

− 4(p−2+gp)

pφ

(p− 2+ 2gp)(p − 1+ gp).

(iii) π1 = 0,π2 = 0,π3 = 0.To avoid root branches we will work in terms ofd1 assuming that it is related tog2 via the relationg2 = pd1−1/d1,


thenΛ is

(19)Λ= (a0 + 8Q2d1d

p2 φ

)e− 4

pd1φ + kd2p

π1(p − 1)e− 4d1

p φ − 2P 2dp

2

π3(1− gpd1)e

− 4(pd1+gp)

p φ,


eU = ce2

1−(p−1)d21

pd1φ + 4kd4

1(p− 1)e4φpd1

π1(pd21 − π1)

+ 8Q2d21d

p−12

pd21 + π1

e4d1p φ

p

[4d1φ + a0

2Q2dp

2

− 3pd21 + π1

pd21 + π1

]

(20)+ 8P 2d41d

p−12 e

−4π3−d2

1pd1

φ

π3(π3 − d1(d1 − gp)).

(iv) π1 = 0,π2 = 0,π3 = 0.This case is identical to the previous one if one replaces everywhereg2 with −gp and interchangesQ andP .

(v) π1 = 0,π2 = 0,π3 = 0.Therefore, we have the following two setsg2, d1 = ±p− 1,1. We choose the plus branch thenΛ is

(21)Λ= [a0 + 4d2

(2Q2d

p−12 − k(p − 1)

)φ]e− 4

p φ − 2P 2dp

2 (1− gp)

p − 1+ gpe− 4(p+gp)

p φ,


eU = e4p φ

p2

[ce−2φ + 4

(a0

d2+ k

(p2 + p− 2

) − 6Q2

d1−p2

)+ 16φ

(2Q2

d1−p2

− k(p − 1)

)]

+ 8P 2dp−12 e

42−p−gp

p φ

(p − 1+ gp)(p − 2+ 2gp).

(vi) π1 = 0,π2 = 0,π3 = 0.Similarly, we have the following two setsgp, d1 = ±1− p,1. One can use the expressions from the previouscase but has to replace in themgp with −g2 and interchangeQ andP .

(vii) π1 = 0,π2 = 0,π3 = 0.It contains the conditiong2 + gp = 0, so we can excludegp . Besides, to avoid root branches we will work again interms ofd1 assuming that it is related tog2 via the relationg2 = pd1 − 1/d1. We have

(22)Λ= [a0 + 8d1d

p

2

(Q2 + P 2)φ]

e− 4φ

pd1 + kd2p(p − 1)

π1e4d1pφ

,

correspondingA, φ are given by Eqs. (11), (15) withd1 being as above, whereas forU one can formally useEq. (20) without the last term (∼ P 2) and withQ2 being replaced withQ2 + P 2.

(viii) π1 = 0,π2 = 0,π3 = 0.Therefore,g2 + gp = 0 and we have the following two setsg2, gp, d1 = ±p− 1,1− p,1. We choose the plusbranch thenΛ is

(23)Λ= [a0 + 4d2

(2dp−1

2

(Q2 + P 2) − k(p− 1)

)φ]e− 4

pφ,

correspondingA, φ are given by Eqs. (11), (15) withd1 being as above, whereas forU one can formally use thecorresponding expression from (v) but without the last term (∼ P 2) and withQ2 being replaced withQ2 + P 2.

Thus, we have enumerated the basic solutions, which correspond to the model (14) within frameworks of thelinear class. Of course, we have mentioned just a few examples.


3.2. String-inspired model: discussion of solutions

Analyzing the solutions above, one can see that the set (i) is the largest set of solutions due to the parameterd1being non-fixed there. In this section we will study the solutions (i) in details. In view of future considerations, letus first redefine the constants

(24)c= −8µdp1+d−2

12

2 , a0 =Λ0dd−2

12 ,

wherep1 ≡ p − 1. The next step is to switch coordinates to the infinite-observer frame of reference

(25)eA(r) =[

1+ d21

2d21d

12d

−21

2

r

] 2d21

1+d21 −→ r2, 2d

12d

−21

2 t −→ t,

then in new coordinates we obtain

ds2 = −dt2[kp1d

41r

2d−21

π1(p1d21 + 1)

− 2µ

rp1−d−21

+ Λ0d21r

2

p((p + 1)d21 − 1)

+ 2d41d

g2d−11

2 Q2r2(1−π2d−21 )

π2(p1d21 − 2g2d1 − 1)

+ 2d41d

−gpd−11

2 P 2r2(1−π3d−21 )

π3(p1d21 + 2gpd1 − 1)

]

+ dr2[

kp1d41

π1(p1d21 + 1)

− 2µ

rp1+d−21

+ Λ0d21r

2π1d−21

p((p + 1)d21 − 1)

+ 2d41d

g2d−11

2 Q2r2(g2d−11 −p1)

π2(p1d21 − 2g2d1 − 1)

+ 2d41d

−gpd−11

2 P 2r2(p1+gpd−11 )

π3(p1d21 + 2gpd1 − 1)

]

+ r2ds2(p,k), eφ = (√

d2 r) p

2 d−11 ,

(26)F = dg2d

−11

2 Q

rp−2g2d−11 −d−2

1

dr ∧ dt, F(p)M...N = P

rpεM...N .

The remainder of this section will be devoted to the studies of this solution at non-fixed large values of|d1|.Assuming|d1| max1, |g2|, |gp|, we obtain that up to the orderO[1/d2

1] (here and below it is supposed to bethe default precision of calculations) the metric above takes the habitual form

ds2 = −eU(r) dt2 + e−U(r) dr2 + r2 ds2(p,k),

with

(27)eU(r) = k − 2µ

rp1+ Λ0r

2

p(p + 1)+ ∆−Θ ln(rp1/η)

r2p1,

where we have defined the following constants

∆= 2

pp1

(Z2 + 3p− 1

pp1d1W

), Θ = − 4W

pp21d1

,

(28)η = d−p1/22 , W = g2Q

2 − gpP2, Z2 =Q2 + P 2,


and it is implied thatp > 1 (lower-dimensional cases will be separately considered after). Also, theO[d−21 ]-

asymptotical form of the dilaton potential (16) is

(29)Λ=Λ0 − 2dp2W

pd1e−4d1φ.

The first, second and third terms in the metric above is theD-dimensional Schwarzschild–de-Sitter. The termproportional to∆ is nothing but theD-dimensional Reissner–Nordström with the only difference that the effectivedyonic charge is the standard one plus a small correction of orderd−1

1 . The last term, proportionalΘ, is definitelysomething new, and below we will study its influence in details.

From now we will work with the spherical casek = 1, besides we will neglect the cosmological constant forsimplicity. Then the information about the global structure of the metric can be read off from the intersection oftwo curves described by the following algebraic equation

(30)x2 − 2µx +∆≡ (x − δ+)(x − δ−)=Θ ln(x/η),

wherex = rp1 andδ± = µ(1 ± √1−∆/µ2 ). It is useful to keep in mind thatΘ is small (∼ d−1

1 ) that simplifiessubject matter. This smallness in fact means that for the whole region except perhapsx → 0 andx → +∞ thevalue of the logarithm in the equation above should be assumed small in comparison with the parametersµ, ∆ andη.

Caseµ2 > ∆. If Θ ≡ 0 (that may happen not only whend1 = ∞ but also wheng2Q2 = gpP

2) this casecorresponds to theD-dimensional Reissner–Nordström black hole. Otherwise we have to solve the transcendentalEq. (30) with realδ’s. Fortunately, it can be done analytically with the use ofΘ ’s smallness. Solving it, we obtainthat we still have two horizons but their radii acquire a correction:

(31)rH± =[δ± + Θ ln(δ±/η)

2(δ± −µ)

]1/p1

,

and the corresponding Hawking temperatures are calculated to be

(32)TH± = p1δ−p/p1

2π

[δ± −µ− Θ

2δ±

(1+ (δ± − pµ) ln(δ±/η)

p1(δ± −µ)

)],

an absolute value is implied.

Caseµ2 = ∆. Without theΘ-perturbation this case corresponds to theD-dimensional extremal Reissner–Nordström black hole. It turns out that the series expansion used in the previous case fails (diverges) so we haveto invent another one. The non-perturbed horizon appears atx = µ. We are interested in small deviations from thenon-perturbed case so it is natural to expand Eq. (30) with respect tox up to the third order near this point. Weobtain that Eq. (30) becomes the quadratic equation,(

1+ Θ

2µ2

)x2 − 2µ

(1+ Θ

µ2

)x +µ2 =Θ

[ln(µ/η)− 3

2

],

from which one concludes that extremality is broken and the extreme horizon is shifted and split into two ones,with the radii

(33)rH± =[µ+ Θ

2µ± √

Θ ln(µ/η)

]1/p1

.

Here, the term proportional toΘ shifts the horizon outward or inward (depending on the sign ofW/d1) whereasthe term proportional to

√Θ describes the split. It is curious that in the particular caseη = µ the extremality is


again restored up toO[d−21 ]. The corresponding Hawking temperatures are

(34)TH± =√Θ ln(µ/η)

2πµ(p+p1)/p1

[p1µ± p

√Θ ln(µ/η)

],

and they do vanish not only whend1 = ∞ but also atη = µ.

Caseµ2 <∆. If Θ ≡ 0 then the solution describes the naked Reissner–Nordström singularity. There is a stronghope that theΘ-perturbation “dresses” the singularity, i.e., creates a horizon around it. To prove it, one has to showthe conditions at which the parabola and logarithmic curve have the intersection point(s) even if the former doesnot cross anx-axis. The intuitive solution for this is to require the minimum point of the parabola to be as closely aspossible to thex-axis, hence, to the logarithmic curve, because the latter is small. The distance from the minimumpoint of the parabola to thex-axis equals to∆−µ2, so∆ must be equal toµ2 plus a small positive correction, say

(35)∆= µ2 + ∣∣const·d−11

∣∣.Again, we expand Eq. (30) near the minimum point of parabola and obtain the quadratic equation(

1+ Θ

2µ2

)x2 − 2µ

(1+ Θ

µ2

)x +∆=Θ

[ln(µ/η)− 3

2

].

If it has complex roots then the singularity is naked otherwise it is hidden under at least one horizon. One can checkthat this equation in general case does not have real roots but if∆ is

∆= µ2 − 2Θµ2 ln(µ/η)

Θ − 4µ2= µ2 + Θ

2ln(µ/η),

i.e., of the form (35), providedd−11 W ln (µ/η) is non-positive, then the imaginary part vanishes, so one does have

the purely real double root. It means that we have found an example when a singularity is dressed by the singlehorizon. Its radius is

(36)rH± =[µ+ Θ

2µ

]1/p1

,

but with the Hawking temperature,

(37)TH = p1Θ ln(µ/η)

4πµ(p+p1)/p1,

being of orderO[d−11 ], rather thanO[d−1/2

1 ] as in previous case.

As a final part of this section, we have to study the low-dimensional case. Indeed, the majority of Eqs. (27)–(37)are not applicable whenD = 3 or 2, i.e., when the number of spatial dimensions is, respectively, two and one.The two-dimensional case is of no interest here because all the solutions were derived assumingp = 0 for obviousreasons. In the 3D case when|d1| is large, instead of Eq. (27) we obtain

(38)eU(r) = ζ − 2Z2 ln(r√d2

) + Λ0r2

2,

whereZ2 is as above and it is denoted

ζ = g2pQ

2 + g22P

2

2g22g

2p

− d1(gpQ2 − g2P

2)

g2gp−Z2 − 2µ,

d2 is assumed positive for definiteness. The scalar, Maxwell andp-form fields (26) do not undergo principalchanges in the sense that they are not singular whenp → 1. However, it is worth to note thatp-form becomes the


plain vertex-type vector field with the only non-zero componentF(1) = f (r) dϕ, whereϕ is an angular coordinate.It cannot be represented as an external derivative of some potential, therefore, it is not possible to derive it from the3D variational principle—in the action (1) it may appear only as a non-dynamical source term

(39)ΨF(1)αFα(1) = Ψf (r)2 = Λ(φ),

because dilaton is an invertible function ofr. Nevertheless, one can consider the 1-form contribution formally, sobelow we will not imposeP ≡ 0.

The solution is essentially cosmological—the metric (38) tends to de-Sitter one (providedζ is positive), and hasthe only singularity atr = 0 providedZ = 0. Its global structure is, however, non-trivial and crucially depends onvalues of the parameters. Simple analysis shows that: (a) whenΛ0 is negative, we have a single horizon regardlessof what other parameters are; (b) whenΛ0 is positive, we have the naked singularity, one extreme horizon, twohorizons, depending on whether the value

ζ +Z2 −Z2 ln(2d2Z

2/Λ0)

is positive, zero or negative, respectively; (c) whenΛ0 vanishes, we have single horizon with

(40)rH =√d2eζ/Z

2,

but the solution is not asymptotically flat (despite the curvature invariants do vanish asymptotically) so this is stillcosmological, rather than black hole, horizon.

To summarize this section: we have demonstrated that the two-parametric family of exact solutions (i) at largevalues of one of the parameters describes the solution which is theD-dimensional Reissner–Nordström–de-Sittersolution but with “renormalized” dyonic charge plus a perturbative non-constant (logarithmic) correction (27).It is also shown that this correction despite its smallness has significant influence on the global structure of thenon-perturbed solution—it may shift and split horizons, break down extremality, and dress the naked singularity.

4. Integrability classes and dilaton potential problem

In the previous section we studied some particular class as the fruitful example of the proposed approach’spower. Other integrability classes (not talking about models) are so diverse and numerous that in principle nevercan be covered all. OtherΞ , Ψ , Λ that may appear from a concrete problem can be paired up within our class in asimilar manner. Despite this pairing is an artificial procedure the generated exact solutions are better than numericalstudies from scratch, besides ones can verify or falsify qualitative approaches and results.

Now, it is a good time to coin the advantages that come after the proven separability of static EMATD theory.This section will be devoted to the generic physical significance of the integrability classes given by relationsbetweenA and dilaton alike (10). Here we are going to justify the point that the integrability classes of such akind is not only a mathematical object but also can play key role in some fundamental aspects of field theory andtheoretical cosmology.

For instance, the integrability classes may help with the problem of unknown dilaton potentialΛ (but, of course,by themselves they cannot provide a complete answer). When supersymmetry is unbroken then the dilaton is inthe same supermultiplet as the graviton and hence cannot acquire a mass. However, in low-energy region thesupersymmetry is broken so the assumptionΛ≡ 0 is inconsistent with observations [7]. Nowadays it is the strongproblem that the dependence ofΛ on scalar field is precisely known neither from string theory nor from cosmology-related experiments. The one of the ideas of gettingΛ(φ) comes from supersymmetry and supergravity—despitethe theories with the dilaton potential of general type are non-supersymmetric as a rule, some non-trivial potentialscan be justified by supergravity models. The flaw, however, is that supergravity cannot predict dilaton potentialuniquely—it has been already proposed an enormous amount of them [8].


The integrability classes, which are incidentally based on dependenceA(φ), bring the view on theΛ-problemfrom the viewpoint different from the above-mentioned ones. First, one should make the important observationthat the dependenceA(φ) is more universal than, e.g.,U(φ) orA(U). Indeed, if the metric is in the gauge (2) thenA∼ lndetg is related to the radius of (compact) factor space and thus determines the geometrical scale of extrap

dimensions. On the other hand, the dilaton field was introduced historically to consider the gravitational constantand speed of light as variable values, and it describes thus the rigidity of spacetime. Therefore,A(φ) symbolizesthe dependence

(41)A(φ)∼ graviton-scale(dilaton-scale),

or, the “size (rigidity)” one. In view of this, the existence proof of above-mentioned integrability classes claims thatwith each such a dependence it is associated a unique Ξ–Ψ –Λ triplet. Let us for clarity disregard thep-form, asin Ref. [2]. Then, if one knows both the dependenceA(φ) and the explicit form of the dilaton-Maxwell couplingΞ thenΛ is uniquely determined by the class equation. Occasionally, in our caseΞ is known from (perturbative)string theory so the problem now is what is the explicit relation ofA to dilaton.

So far, we do not have any clear idea on the latter. But we sure that the above-studied linear class (and, therefore,models therein) is at least a first-order approximation if one expands the yet unknown for sure True FunctionA(φ)

in Taylor series with respect to dilaton. It should be also noted that the linear class is distinct from others not onlybecause it is given by a first- rather than second-order ODE with respect toΞ andΛ but also because it possessesa certain discrete symmetry that alike the electric-magnetic duality in pure EMD is broken atD = 4 and thus itforcesD = 4 to be a magic number again, see the paragraph after Eq. (13) and Ref. [2].

Acknowledgements

I am grateful to Edward Teo (DAMTP, University of Cambridge and National University of Singapore) andCristian Stelea (University “Alexandru Ioan Cuza”, Iasi, Romania and National University of Singapore) forsuggesting the theme and helpful discussions.

References

[1] M. Cvetic, A.A. Tseytlin, Nucl. Phys. B 416 (1994) 137;K.-L. Chan, Mod. Phys. Lett. A 12 (1997) 1597;J.X. Lu, Nucl. Phys. B 409 (1993) 290;J.H. Horne, G.T. Horowitz, Nucl. Phys. B 399 (1993) 169;R. Gregory, J.A. Harvey, Phys. Rev. D 47 (1993) 2411.

[2] K.G. Zloshchastiev, Phys. Rev. D 64 (2001) 084026.[3] S.G. Matinyan, E.B. Prokhorenko, G.K. Savvidy, Nucl. Phys. B 298 (1988) 414.[4] G.W. Gibbons, K. Maeda, Nucl. Phys. B 298 (1988) 741.[5] B. Zhou, C. Zhu, Commun. Theor. Phys. 32 (1999) 173;

K.S. Stelle, hep-th/9803116;T. Maki, K. Shiraishi, Class. Quantum Grav. 11 (1994) 2781;M.J. Duff, J.X. Lu, Nucl. Phys. B 416 (1994) 301;G.T. Horowitz, A. Strominger, Nucl. Phys. B 360 (1991) 197;A. Dabholkar, G. Gibbons, J.A. Harvey, F. Ruiz Ruiz, Nucl. Phys. B 340 (1990) 33;M.J. Duff, K.S. Stelle, Phys. Lett. B 253 (1991) 113.

[6] S.J. Poletti, J. Twamley, D.L. Wiltshire, Phys. Rev. D 51 (1995) 5720;S.J. Poletti, J. Twamley, D.L. Wiltshire, Class. Quantum Grav. 12 (1995) 1753;A. Lukas, B.A. Ovrut, D. Waldram, Nucl. Phys. B 495 (1997) 365.

[7] C. Will, Theory and Experiment in Gravitational Physics, Cambridge Univ. Press, Cambridge, 1981.[8] S.J. Gates, B. Zwiebach, Phys. Lett. B 123 (1983) 200;


N.P. Warner, Nucl. Phys. B 231 (1984) 250;C.M. Hull, N.P. Warner, Nucl. Phys. B 253 (1985) 675;L. Castellani, A. Ceresole, R. D’Auria, S. Ferrara, P. Fre, E. Maina, Phys. Lett. B 161 (1985) 91;K.T. Mahanthappa, G.M. Staebler, Z. Phys. C 33 (1987) 537;R. D’Auria, S. Ferrara, P. Fre, Nucl. Phys. B 359 (1991) 705.



Cumulative author index to volumes 521–527

Abaev, V.,521, 158Abazov, V.M.,525, 211Abbaneo, D.,526, 34, 191, 206Abbiendi, G.,521, 181;523, 35;526, 221, 233Abbott, B.,525, 211Abdallah, J.,525, 17Abdel-Bary, M.,522, 16Abd El-Samad, S.,522, 16Abdesselam, A.,525, 211Abe, K.,524, 33, 33;526, 247, 247, 258, 258Abe, R.,524, 33;526, 247, 258Abe, T.,524, 33;526, 247, 258Abolins, M.,525, 211Abramov, V.,525, 211Abreu, M.C.,521, 195Abreu, P.,525, 17Achard, P.,524, 44, 55, 65;526, 269;527, 29Acharya, B.S.,525, 211Achiman, Y.,523, 304Adachi, I.,524, 33;526, 247, 258Adam, W.,525, 17Adamian, G.G.,526, 322Adams, D.L.,525, 211Adams, M.,525, 211Added, N.,527, 187Adloff, C., 523, 234;525, 9Adriani, O.,524, 44, 55, 65;526, 269;527, 29Adzic, P.,525, 17Affholderbach, K.,526, 34, 191, 206Aglietti, U., 522, 83Agnello, M.,527, 39Agostino, L.,523, 53;525, 205Aguilar-Benitez, M.,524, 44, 55, 65;526, 269;527, 29Ahmed, S.N.,525, 211Ahn, B.S.,524, 33;526, 247, 258Ahrens, J.,526, 287Aiche, M.,521, 165Aihara, H.,524, 33;526, 247, 258Ainsley, C.,521, 181;523, 35;526, 221, 233Akatsu, M.,524, 33;526, 247, 258Akemann, G.,524, 400Akeroyd, A.G.,525, 81, 315Åkesson, P.F.,521, 181;523, 35;526, 221, 233

Akhmedov, E.Kh.,521, 79Akhmetshin, R.R.,527, 161Akimov, D., 524, 245Alberto, P.,523, 273Albrecht, T.,525, 17Alcaraz, J.,524, 44, 55, 65;526, 269;527, 29Alderweireld, T.,525, 17Alemanni, G.,524, 44, 55, 65;526, 269;527, 29Alemany-Fernandez, R.,525, 17Aleonard, M.M.,521, 165ALEPH Collaboration,526, 34, 191, 206Alessandro, B.,521, 195Alexa, C.,521, 195Alexander, G.,521, 181;523, 35;526, 221, 233Alexeev, G.D.,525, 211Aliev, T.M., 527, 193Alimonti, G., 523, 53;525, 205Alkhazov, G.,521, 171;522, 233;523, 22Allaby, J.,524, 44, 55, 65;526, 269;527, 29Allison, J.,521, 181;523, 35;526, 221, 233Allmendinger, T.,525, 17Allport, P.P.,525, 17Almehed, S.,525, 17Aloisio, A., 524, 44, 55, 65;526, 269;527, 29Alton, A., 525, 211Alves, G.A.,525, 211Alviggi, M.G., 524, 44, 55, 65;526, 269;527, 29Amaldi, U.,525, 17Amapane, N.,525, 17Amato, S.,525, 17Amelino-Camelia, G.,522, 133Amos, N.,525, 211Anagnostou, G.,521, 181;523, 35;526, 221, 233Anashkin, E.,525, 17Anashkin, E.V.,527, 161Anderhub, H.,524, 44, 55, 65;526, 269;527, 29Anderson, E.W.,525, 211Anderson, K.J.,521, 181;523, 35;526, 221, 233Andreazza, A.,525, 17Andreev, V.,523, 234;525, 9Andreev, V.P.,524, 44, 55, 65;526, 269;527, 29Andrieu, B.,523, 234;525, 9Andringa, S.,525, 17Anjos, J.C.,523, 29, 53;525, 205

0370-2693/2002 Published by Elsevier Science B.V.PII: S0370-2693(02)01225-X


Cumulative author index to volumes 521–527 (2002) 226–261 227

Anjos, N.,525, 17Anklin, H., 524, 26Ansari, A.,525, 255Anselmo, F.,524, 44, 55, 65;526, 269;527, 29Anthonis, T.,523, 234;525, 9Antilogus, P.,525, 17Antonelli, A., 526, 34, 191, 206Antonelli, M., 526, 34, 191, 206Antonenko, N.V.,526, 322Antusch, S.,525, 130Aoi, N., 522, 227Aoki, S.,527, 173Aoyama, H.,521, 400Aoyama, S.,521, 376Apel, W.-D.,525, 17Arbuzov, A.B.,524, 99;527, 161Arcelli, S.,521, 181;523, 35;526, 221, 233Arefiev, A.,524, 44, 55, 65;526, 269;527, 29Arena, V.,523, 53;525, 205Arik, E., 527, 173Arkadov, V.,523, 234;525, 9Armenise, N.,527, 173Armesto, N.,527, 92Armstrong, S.R.,526, 34, 191, 206Arnaldi, R.,521, 195Arnoud, Y.,525, 17, 211Artamonov, A.,527, 173Artru, X., 525, 41Asai, S.,521, 181;523, 35;526, 221, 233Asaka, T.,521, 329;523, 199Asano, Y.,524, 33;526, 247, 258Ask, S.,525, 17Asman, B.,525, 17Aso, T.,524, 33;526, 247, 258Astier, P.,526, 278;527, 23Astrua, M.,527, 39Astvatsatourov, A.,523, 234;525, 9Atag, S.,522, 76Atamantchouk, A.G.,521, 171;522, 233;523, 22Atayan, M.,521, 195Augustin, J.E.,525, 17Augustinus, A.,525, 17Aulchenko, V.,524, 33;526, 247, 258Aulchenko, V.M.,527, 161Aushev, T.,524, 33;526, 247, 258Autiero, D.,526, 278;527, 23Avila, C., 525, 211Awunor, O.,526, 191, 206Axen, D.,521, 181;523, 35;526, 221, 233Azemoon, T.,524, 44, 55, 65;526, 269;527, 29Aziz, T., 524, 44, 55, 65;526, 269;527, 29Azuelos, G.,521, 181;523, 35;526, 221, 233Azzurri, P.,526, 34, 191, 206

Baarmand, M.,524, 44, 55Baarmand, M.M.,525, 211Babaev, A.,523, 234;525, 9Babintsev, V.V.,525, 211

Babu, K.S.,522, 287;525, 289Babukhadia, L.,525, 211Back, H.O.,525, 29Bacon, T.C.,525, 211Badaud, F.,526, 34, 191, 206Baden, A.,525, 211Baek, S.,525, 315Bagliesi, G.,526, 34, 191, 206Baglin, C.,521, 195Bagnaia, P.,524, 44, 55, 65;526, 269;527, 29Bähr, J.,523, 234;525, 9Bai, X., 527, 50Baier, V.,525, 41Bailey, I.,521, 181;523, 35;526, 221, 233Baillon, P.,525, 17Bajc, B.,525, 189Bajo, A.,524, 44, 55, 65;526, 269;527, 29Bak, D.,527, 131Bakich, A.M.,524, 33;526, 247, 258Baksay, G.,524, 44, 55, 65;526, 269;527, 29Baksay, L.,524, 44, 55, 65;526, 269;527, 29Balaji, K.R.S.,524, 153Balata, M.,525, 29Balatz, M.Y.,521, 171;522, 233;523, 22Balázs, C.,525, 219Baldew, S.V.,524, 44, 55, 65;526, 269;527, 29Baldin, B.,525, 211Baldisseri, A.,526, 278;527, 23Baldit, A., 521, 195Baldo, M.,525, 261;526, 19Baldo-Ceolin, M.,526, 278;527, 23Ballestrero, A.,525, 17Balm, P.W.,525, 211Balog, J.,523, 211Bambade, P.,525, 17Ban, Y.,524, 33;526, 247, 258Banas, E.,524, 33;526, 247, 258Banerjee, S.,524, 44, 55, 65;525, 211;526, 269;527, 29Banerjee, Sw.,524, 44, 55, 65;526, 269;527, 29Banner, M.,526, 278;527, 23Banzarov, V.Sh.,527, 161Baran, A.,526, 329Baranov, P.,523, 234;525, 9Barate, R.,526, 34, 191, 206Barberio, E.,521, 181;523, 35;526, 221, 233Barberis, E.,525, 211Barberis, S.,525, 205Barbier, R.,525, 17Barbuto, E.,527, 173Barczyk, A.,524, 44, 55, 65;526, 269;527, 29Bardin, D.,525, 17Barillère, R.,524, 44, 55, 65;526, 269;527, 29Baringer, P.,525, 211Barker, G.,525, 17Barklow, T.,526, 191, 206Barkov, L.M.,527, 161Barlow, R.J.,521, 181;523, 35;526, 221, 233Baroncelli, A.,525, 17

228 Cumulative author index to volumes 521–527 (2002) 226–261

Barr, S.M.,525, 289Barreau, G.,521, 165Barrelet, E.,523, 234;525, 9Barreto, J.,525, 211Bartalini, P.,524, 44, 55, 65;526, 269;527, 29Bartel, W.,523, 234;525, 9Bartlett, J.F.,525, 211Bartsch, P.,524, 26Baru, S.E.,527, 161Basak, S.,522, 350Bashtovoy, N.S.,527, 161Basile, M.,524, 44, 55, 65;526, 269;527, 29Bassler, U.,525, 211Bassompierre, G.,526, 278;527, 23Batalova, N.,524, 44, 55, 65;526, 269;527, 29Bate, P.,523, 234;525, 9Batley, R.J.,521, 181;523, 35;526, 221, 233Battaglia, M.,525, 17Battiston, R.,524, 44, 55, 65;526, 269;527, 29Baubillier, M.,525, 17Bauer, D.,525, 211Baumann, D.,524, 26Bay, A.,524, 44, 55, 65;526, 269;527, 29Bazeia, D.,521, 418Bean, A.,525, 211Beane, S.R.,521, 47Bearden, I.G.,523, 227Beau, T.,525, 29Beaudette, F.,525, 211Beavis, D.,523, 227Becattini, F.,524, 44, 55, 65;526, 269;527, 29Bechtle, P.,526, 221, 233Becirevic, D.,524, 115Beck, R.,526, 287Becker, J.,523, 234;525, 9Becker, U.,524, 44, 55, 65;526, 269;527, 29Becks, K.-H.,525, 17Bedaque, P.F.,524, 137Bediaga, I.,523, 53;525, 205Bedjidian, M.,521, 195Begalli, M.,525, 17Begel, M.,525, 211Beglarian, A.,523, 234;525, 9Behari, S.,524, 33;526, 247, 258Behera, P.K.,524, 33;526, 247, 258Behner, F.,524, 44, 55, 65;526, 269;527, 29Behnke, O.,523, 234;525, 9Behnke, T.,521, 181;523, 35;526, 221, 233Behrmann, A.,525, 17Beier, C.,523, 234;525, 9Belhaj, A.,523, 191Bell, K.W., 521, 181;523, 35;526, 221, 233Bell, P.J.,521, 181;523, 35;526, 221, 233Bella, G.,521, 181;523, 35;526, 221, 233Belle Collaboration,524, 33;526, 247, 258Bellerive, A.,521, 181;523, 35;526, 221, 233Belli, P.,527, 182Bellini, G., 525, 29

Bellucci, L.,524, 44, 55, 65;526, 269;527, 29Bellucci, S.,522, 345Bellunato, T.,525, 17Beloborodov, K.,525, 41Belousov, A.,523, 234;525, 9Belyaev, A.,525, 211Belyaev, V.B.,522, 222Bencivenni, G.,526, 34, 191, 206Benekos, N.,525, 17Benelli, G.,521, 181;523, 35;526, 221, 233Benhar, O.,527, 73Benisch, T.,523, 234;525, 9Bennhold, C.,527, 99Benslama, K.,526, 278;527, 23Bentley, M.A.,525, 49Benvenuti, A.,525, 17Benziger, J.,525, 29Beolè, S.,521, 195Berat, C.,525, 17Berbeco, R.,524, 44, 55, 65;526, 269;527, 29Berdugo, J.,524, 44, 55, 65;526, 269;527, 29Berezhiani, Z.,522, 107Berezinsky, V.,521, 287Berger, Ch.,523, 234;525, 9Berger, E.R.,523, 265Berges, P.,524, 44, 55, 65;526, 269;527, 29Berggren, M.,525, 17Beri, S.B.,525, 211Berkelman, K.,526, 34, 191, 206Bernabei, R.,527, 182Bernardi, G.,525, 211Berndt, T.,523, 234;525, 9Berntzon, L.,525, 17Bertram, I.,525, 211Bertrand, D.,525, 17Bertucci, B.,524, 44, 55, 65;526, 269;527, 29Besancon, M.,525, 17Besliu, C.,523, 227Besson, A.,525, 211Besson, N.,525, 17;526, 278;527, 23Betev, B.L.,524, 44, 55, 65;526, 269;527, 29Bethke, S.,521, 181;523, 35;526, 221, 233Beuselinck, R.,525, 211;526, 34, 191, 206Bewick, A.,524, 245Beyer, M.,521, 33Bezzubov, V.A.,525, 211Bhat, P.C.,525, 211Bhatnagar, V.,525, 211Bhattacharjee, M.,525, 211Bianco, S.,523, 53;525, 205Biasini, M.,524, 44, 55, 65;526, 269;527, 29Biebel, O.,521, 181;523, 35;526, 221, 233Bieniek, A.,526, 329Biglietti, M., 524, 44, 55, 65;526, 269;527, 29Biland, A.,524, 44, 55, 65;526, 269;527, 29Binnie, D.M.,526, 34, 191, 206Bird, I., 526, 278;527, 23Bizot, J.C.,523, 234;525, 9


Blair, G.A.,526, 34, 191, 206Blaising, J.J.,524, 44, 55, 65;526, 269;527, 29Blaizot, J.-P.,523, 143Błaut, A.,521, 364Blazey, G.,525, 211Bleicher, M.,526, 309Blekman, F.,525, 211Blessing, S.,525, 211Bloch, D.,525, 17Bloch-Devaux, B.,526, 34, 191, 206Blom, M., 525, 17Blondel, A.,526, 34, 191, 206Bloodworth, I.J.,521, 181;523, 35;526, 221, 233Blumenfeld, B.,526, 278;527, 23Blumenschein, U.,526, 191, 206Blyakhman, Y.,523, 227Blyth, S.C.,524, 44, 55, 65;526, 269;527, 29Bobbink, G.J.,524, 44, 55, 65;526, 269;527, 29Bobisut, F.,526, 278;527, 23Boca, G.,523, 53;525, 205Boccali, T.,526, 34, 191, 206Bochek, G.,525, 41Bock, P.,521, 181;523, 35;526, 221, 233Boeglin, W.U.,524, 26Boehme, J.,523, 234;525, 9Boehnlein, A.,525, 211Boeriu, O.,521, 181;523, 35;526, 221, 233Bogdanov, A.,525, 41Bøggild, H.,523, 227Bohinc, K.,524, 26Böhm, A.,524, 44, 55, 65;526, 269;527, 29Böhm, R.,524, 26Böhme, J.,521, 181;523, 35;526, 221, 233Böhrer, A.,526, 34, 191, 206Boivin, D., 521, 165Boix, G.,526, 34, 191, 206Bojko, N.I., 525, 211Boldea, V.,521, 195Boldizsar, L.,524, 44, 55, 65;526, 269;527, 29Bologna, G.,526, 34, 191, 206Bonacorsi, D.,521, 181;523, 35;526, 221, 233Bonanno, A.,527, 9Bondar, A.,524, 33;526, 247, 258Bondar, A.E.,527, 161Bondar, N.F.,521, 171;522, 233;523, 22Bondarev, D.V.,527, 161Bonelli, G.,521, 383Bonesini, M.,525, 17Bonetti, S.,525, 29Bonissent, A.,526, 34, 191, 206Bonneaud, G.,526, 34Bonomi, G.,523, 53;525, 205Bonora, L.,521, 421Boonekamp, M.,525, 17Booth, C.N.,526, 34, 191, 206Booth, P.S.L.,525, 17Borcherding, F.,525, 211Bordalo, P.,521, 195

Borean, C.,526, 191, 206Borges, G.,521, 195Borgia, B.,524, 44, 55, 65;526, 269;527, 29Borisov, G.,525, 17Bos, K.,525, 211Boschi-Filho, H.,525, 164Boschini, M.,523, 53;525, 205Bose, T.,525, 211Bossi, F.,526, 34, 191, 206Boston, A.J.,523, 13Botner, O.,525, 17Botta, E.,527, 39Bottai, S.,524, 44, 55, 65;526, 269;527, 29Boubekeur, L.,524, 342Bouchez, J.,526, 278;527, 23Boucrot, J.,526, 34, 191, 206Boudrie, R.L.,523, 1Boudry, V.,523, 234;525, 9Bouhova-Thacker, E.,526, 34, 191, 206Boumediene, D.,526, 191, 206Bouquet, B.,525, 17Bourilkov, D.,524, 44, 55, 65;526, 269;527, 29Bourquin, M.,524, 44, 55, 65;526, 269;527, 29Boutemeur, M.,521, 181;523, 35;526, 221, 233Bowcock, T.J.V.,525, 17Bowdery, C.K.,526, 34, 191, 206Boyd, S.,526, 278;527, 23Boyko, I.,525, 17Bozek, A.,524, 33;526, 247, 258Bozza, C.,527, 173Braccini, S.,524, 44, 55, 65;526, 269;527, 29Bracco, M.E.,521, 1Bracko, M.,525, 17Braga, N.R.F.,525, 164Bragin, A.V.,527, 161BRAHMS Collaboration,523, 227Braibant, S.,521, 181;523, 35;526, 221, 233Branco, G.C.,526, 104Brandt, A.,525, 211Brandt, S.,526, 34, 191, 206Branson, J.G.,524, 44, 55, 65;526, 269;527, 29Braunschweig, W.,523, 234;525, 9Bravo, S.,526, 34, 191, 206Brax, P.,521, 105Breedon, R.,525, 211Brenner, R.,525, 17Bressani, T.,527, 39Brient, J.-C.,526, 34, 191, 206Brigliadori, L., 521, 181;523, 35;526, 221, 233Brihaye, Y.,524, 227Brinkmann, K.-Th.,522, 16Briskin, G.,525, 211Brisson, V.,523, 234;525, 9Brochu, F.,524, 44, 55, 65;526, 269;527, 29Brock, R.,525, 211Brodet, E.,525, 17Brodzicka, J.,525, 17Bröker, H.-B.,523, 234;525, 9


Brondi, A.,521, 165Broniowski, W.,526, 329Brooijmans, G.,525, 211Bross, A.,525, 211Browder, T.E.,524, 33;526, 247, 258Brown, D.P.,523, 234;525, 9Brown, R.M.,521, 181;523, 35;526, 221, 233Bruce, A.M.,525, 49Bruckman, P.,525, 17Brückner, W.,523, 234;525, 9Bruncko, D.,523, 234;525, 9Brunelière, R.,526, 191, 206Brunet, J.M.,525, 17Bruno, N.R.,522, 133Bruski, N.,527, 173Brzychczyk, J.,523, 227Buccella, F.,524, 241Buchbinder, I.L.,523, 338;524, 208Buchholz, D.,525, 211Büchler, M.,521, 22, 29Buchmüller, O.,526, 34, 191, 206Buchmüller, W.,521, 291;523, 199Buck, C.,525, 29Budick, B.,523, 227Buehler, M.,525, 211Buénerd, M.,521, 139Bueno, A.,526, 278;527, 23Buescher, V.,525, 211Bugaev, K.A.,523, 255Bugge, L.,525, 17Buijs, A., 524, 44, 55, 65;526, 269;527, 29Bukin, A., 525, 41Bülte, A.,527, 173Bunyatov, S.,526, 278;527, 23Bunyatyan, A.,523, 234;525, 9Buontempo, S.,527, 173Burckhart, H.J.,521, 181;523, 35;526, 221, 233Burdin, S.,525, 41Bürger, J.,523, 234;525, 9Burger, J.D.,524, 44, 55, 65;526, 269;527, 29Burger, W.J.,524, 44, 55, 65;526, 269;527, 29Burgio, G.F.,526, 19Burrage, A.,523, 234;525, 9Burtovoi, V.S.,525, 211Büscher, M.,521, 158, 217Buschhorn, G.,523, 234;525, 9Buschmann, P.,525, 17Büsser, F.W.,523, 234;525, 9Bussière, A.,521, 195Butler, J.M.,525, 211Butler, J.N.,523, 53;525, 205Butt, R.D.,526, 295Bystritskaya, L.,523, 234;525, 9

Caccianiga, B.,525, 29Cadonati, L.,525, 29Cadoni, M.,522, 126Cai, R.-G.,525, 331

Cai, X.D.,524, 44, 55, 65;526, 269;527, 29Çakır, O.,522, 76Calaprice, F.,525, 29Caldas, H.C.G.,523, 293Callot, O.,526, 34, 191, 206Calmet, X.,525, 297;526, 90Calvi, M., 525, 17Calvo, D.,527, 39Cameron, J.A.,525, 49Cameron, W.,526, 34, 191, 206Camilleri, L.,526, 278;527, 23Camino, J.M.,525, 337Cammin, J.,521, 181;523, 35;526, 221, 233Campana, P.,526, 34, 191, 206Campana, S.,526, 221, 233Campbell, A.J.,523, 234;525, 9Camporesi, T.,525, 17Canale, V.,525, 17Canelli, F.,525, 211Cano, F.,521, 225Cao, J.,523, 234;525, 9Capell, M.,524, 44, 55, 65;526, 269;527, 29Capelli, L.,521, 195Capon, G.,526, 34, 191, 206Cappella, F.,527, 182Cappiello, L.,522, 139Cara Romeo, G.,524, 44, 55, 65;526, 269;527, 29Cardini, A.,526, 278;527, 23Carena, F.,525, 17Carimalo, C.,525, 17Carlino, G.,524, 44, 55, 65;526, 269;527, 29Carnegie, R.K.,521, 181;523, 35;526, 221, 233Caron, B.,521, 181;523, 35;526, 221, 233Caron, S.,523, 234;525, 9Carpenter, M.P.,525, 49Carr, J.,526, 34, 191, 206Carrillo, S.,523, 53;525, 205Carrington, M.E.,523, 221Carta, P.,522, 126Cartacci, A.,524, 44, 55, 65;526, 269;527, 29Carter, A.A.,521, 181;523, 35;526, 221, 233Carter, G.W.,524, 297;525, 249Carter, J.R.,521, 181;523, 35;526, 221, 233Cartwright, S.,526, 34, 191, 206Carvalho, W.,525, 211Casado, M.P.,526, 34, 191, 206Casalbuoni, R.,524, 144Casaus, J.,524, 44, 55, 65;526, 269;527, 29Casey, B.C.K.,524, 33;526, 247, 258Casey, D.,521, 171;525, 211Casilum, Z.,525, 211Casimiro, E.,523, 53;525, 205Cassing, W.,521, 217Cassol-Brunner, F.,523, 234;525, 9Castagneto, L.,527, 173Castanier, C.,521, 195Casten, R.F.,527, 55Castilla-Valdez, H.,525, 211


Castor, J.,521, 195Castro, N.,525, 17Catanesi, M.G.,527, 173Cattaneo, M.,526, 34, 191, 206Cattaneo, P.W.,526, 278;527, 23Caurier, E.,522, 240Cavallari, F.,524, 44, 55, 65;526, 269;527, 29Cavallo, F.,525, 17Cavallo, N.,524, 44, 55, 65;526, 269;527, 29Cavanaugh, R.,526, 34, 191, 206Cavasinni, V.,526, 278;527, 23Cawlfield, C.,523, 53;525, 205Cecchet, G.,525, 29Cecchi, C.,524, 44, 55, 65;526, 269;527, 29Centelles, M.,523, 67Cerrada, M.,524, 44, 55, 65;526, 269;527, 29Cerulli, R.,527, 182Cerutti, F.,526, 34, 191, 206Cervera-Villanueva, A.,526, 278;527, 23Chaichian, M.,524, 161;526, 132;527, 149Chakraborty, D.,525, 211Challis, R.,526, 278Chamizo, M.,524, 44, 55, 65;526, 269;527, 29Chan, K.M.,525, 211Chang, C.Y.,521, 181;523, 35;526, 221, 233Chang, P.,524, 33;526, 247, 258Chang, Y.H.,524, 44, 55, 65;526, 269;527, 29Chao, Y.,524, 33;526, 247, 258Chapkin, M.,525, 17Charlton, D.G.,521, 181;523, 35;526, 221, 233Charpentier, Ph.,525, 17Chasman, C.,523, 227Chasman, R.R.,524, 81Chaudhuri, A.K.,527, 80Chaurand, B.,521, 195Checchia, P.,525, 17Chehab, R.,525, 41Chekulaev, S.V.,525, 211Chemarin, M.,524, 44, 55, 65;526, 269;527, 29Chemin, J.F.,521, 165Chen, A.,524, 44, 55, 65;526, 269;527, 29Chen, C.-H.,521, 315;525, 56Chen, G.,524, 44, 55, 65;526, 269;527, 29Chen, G.-Y.,522, 27Chen, G.M.,524, 44, 55, 65;526, 269;527, 29Chen, H.F.,524, 44, 55, 65;526, 269;527, 29Chen, H.S.,524, 44, 55, 65;526, 269;527, 29Chen, J.-W.,523, 73, 107Chen, M.,525, 29Cheng, H.-C.,521, 308Cheon, B.G.,524, 33;526, 247, 258Chernyak, D.V.,527, 161Cheung, H.W.K.,523, 53;525, 205Chevallier, M.,525, 41Chevrot, I.,521, 195Cheynis, B.,521, 195Chiapparini, M.,521, 1Chiara, C.J.,523, 13

Chiarella, V.,526, 34, 191, 206Chiavassa, E.,521, 195Chiefari, G.,524, 44, 55, 65;526, 269;527, 29Chierici, R.,525, 17Chikawa, M.,527, 173Chimento, L.P.,521, 133Chiodini, G.,523, 53;525, 205Chishtie, F.A.,521, 434Chistov, R.,524, 33;526, 247, 258Chiu, T.-W.,521, 429Chivukula, R.S.,521, 239;525, 175Chizhov, M.,527, 173Chliapnikov, P.,525, 17Chliapnikov, P.V.,525, 1Chmeissani, M.,526, 34, 191, 206Cho, D.K.,525, 211Cho, K.,523, 53;525, 205Cho, Y.M.,525, 347Choi, S.,525, 211Choi, S.-K.,524, 33;526, 247, 258Choi, Y.,524, 33;526, 247, 258Chopra, S.,525, 211CHORUS Collaboration,527, 173Christensen, C.H.,523, 227Christenson, J.H.,525, 211Christiansen, P.,523, 227Chu, C.-S.,524, 389Chukanov, A.,526, 278Chun, E.J.,525, 114Chung, M.,525, 211Chung, S.U.,525, 17Chung, Y.S.,523, 53;525, 205Cibor, J.,523, 227Cicalò, C.,521, 195Cieslik, K.,525, 17Cifarelli, L., 524, 44, 55, 65;526, 269;527, 29Cinausero, M.,521, 165Cindolo, F.,524, 44, 55, 65;526, 269;527, 29Cinquini, L.,523, 53;525, 205Cirigliano, V.,522, 245Ciulli, V., 526, 34, 191, 206Cizeron, R.,525, 41Claes, D.,525, 211Clare, I.,524, 44, 55, 65;526, 269;527, 29Clare, R.,524, 44, 55, 65;526, 269;527, 29Clark, A.R.,525, 211Clarke, D.,523, 234;525, 9Clarke, D.P.,526, 191, 206Clarke, P.E.L.,521, 181;523, 35;526, 221, 233Claudino, T.,521, 195Clavelli, L., 523, 249;526, 360Clay, E.,521, 181;523, 35;526, 221, 233Clement, H.,522, 16Clerbaux, B.,523, 234;525, 9; 526, 34, 191, 206Clifft, R.W., 526, 34, 191, 206Cocco, A.G.,527, 173Cohen, I.,521, 181;523, 35;526, 221, 233Coignet, G.,524, 44, 55, 65;526, 269;527, 29


Colaleo, A.,526, 34, 191, 206Colangelo, G.,521, 22, 29Colas, P.,526, 34, 191, 206Coles, J.,526, 191, 206Colino, N.,524, 44, 55, 65;526, 269;527, 29Collard, C.,523, 234;525, 9Collazuol, G.,526, 278;527, 23Collins, P.,525, 17Combley, F.,526, 34, 191, 206Comets, M.P.,521, 195Coney, L.,525, 211Conforto, G.,526, 278;527, 23Connolly, B.,525, 211Constans, N.,521, 195Constantinescu, S.,521, 195Conta, C.,526, 278;527, 23Contalbrigo, M.,526, 278;527, 23Contino, R.,523, 347Contreras, J.G.,523, 234;525, 9Contri, R.,525, 17Cooper, P.S.,521, 171;522, 233;523, 22Cooper, W.E.,525, 211Coppage, D.,525, 211Coppens, Y.R.,523, 234;525, 9Corradini, O.,521, 96Cortese, P.,521, 195Cosme, G.,525, 17Cossutti, F.,525, 17Costa, M.J.,525, 17Costantini, S.,524, 44, 55, 65;526, 269;527, 29COSY-TOF Collaboration,522, 16Couchman, J.,521, 181;523, 35;526, 221, 233Coughlan, J.A.,523, 234;525, 9Coulter, P.,526, 360Courtin, S.,521, 165Cousinou, M.-C.,523, 234;525, 9Cousins, R.,526, 278;527, 23Covi, L., 523, 199Cowan, G.,526, 34, 191, 206Cox, B.E.,523, 234;525, 9Coyle, P.,526, 34, 191, 206Cozzika, G.,523, 234;525, 9Cranmer, K.,526, 34, 191, 206Crawley, B.,525, 17Creanza, D.,526, 34, 191, 206Crema, E.,527, 187Crennell, D.,525, 17Crépé-Renaudin, S.,525, 211Crespo, J.M.,526, 34, 191, 206Cruz, N.,521, 343Csilling, A., 521, 181;523, 35;526, 221, 233Cuautle, E.,525, 205Cucchieri, A.,524, 123Cucciarelli, S.,524, 44, 55, 65;526, 269;527, 29Cudell, J.R.,526, 413Cuevas, J.,525, 17Cuffiani, M.,521, 181;523, 35;526, 221, 233Cumalat, J.P.,523, 53;525, 205

Cummings, M.A.C.,525, 211Curtil, C.,526, 191, 206Cutts, D.,525, 211Cvach, J.,523, 234;525, 9

Dado, S.,521, 181;523, 35;526, 221, 233Dadoun, O.,525, 29Dai, C.J.,527, 182Dai, T.S.,524, 55Dai, Y., 525, 301Dainton, J.B.,523, 234;525, 9Dallavalle, G.M.,521, 181;523, 35;526, 221, 233Dallison, S.,521, 181;523, 35;526, 221, 233Dalmau, J.,525, 17D’Ambrosio, N.,527, 173Damgaard, P.H.,524, 400Da Motta, H.,525, 211D’Angelo, D.,525, 29D’Angelo, P.,523, 53;525, 205Daniels, D.,526, 278;527, 23Darabi, F.,527, 1Dasgupta, M.,526, 295Da Silva, A.J.,521, 119Da Silva, T.,525, 17Da Silva, W.,525, 17Dass, G.V.,521, 267Datta, A.,524, 161;526, 111Dau, W.D.,523, 234;525, 9Daum, K.,523, 234;525, 9Dauvergne, D.,525, 41Dauwe, L.J.,521, 171;522, 233;523, 22Davenport, T.F.,523, 53;525, 205David, A.,526, 34, 191, 206Davidenko, G.V.,521, 171;522, 233;523, 22Davidge, D.,524, 245Davids, C.N.,525, 49Davidsson, M.,523, 234;525, 9Davier, M.,526, 34, 191, 206Davies, G.,526, 191, 206Davis, G.A.,525, 211Davis, K.,525, 211Dawson, J.,524, 245De, A.K.,522, 350De, K.,525, 211De A. Bicudo, P.J.,521, 441Deandrea, A.,526, 79De Angelis, A.,525, 17De Asmundis, R.,524, 44, 55, 65;526, 269;527, 29De Bari, A.,525, 29Debbe, R.,523, 227De Bellefon, A.,525, 29De Boer, W.,525, 17De Bonis, I.,526, 34, 191, 206Debreczeni, J.,524, 44, 55, 65;526, 269;527, 29Debruyne, D.,527, 62Decamp, D.,526, 34, 191, 206De Clercq, C.,525, 17De Falco, A.,521, 195


De Fazio, F.,521, 15De Filippis, N.,526, 191, 206Defu, H.,523, 221Degaudenzi, H.,526, 278;527, 23Déglon, P.,524, 44, 55, 65;526, 269;527, 29Degré, A.,524, 44, 55, 65;526, 269;527, 29Deiters, K.,524, 44, 55, 65;526, 269;527, 29De Jong, M.,527, 173De Jong, P.,524, 44, 55, 65;527, 29De Jong, S.J.,525, 211De Kerret, H.,525, 29De la Cruz, B.,524, 44, 55, 65;526, 269;527, 29Delbar, T.,527, 173Delcourt, B.,523, 234;525, 9De Lellis, G.,527, 173De Leon, J.P.,523, 311Delépine, D.,522, 95Delerue, N.,523, 234;525, 9Del Estal, M.,523, 67Delius, G.W.,522, 335;524, 401Dellacasa, G.,521, 195Della Ricca, G.,525, 17Della Volpe, D.,524, 44, 55, 65;526, 269;527, 29Delmeire, E.,524, 44, 55, 65;526, 269;527, 29De Lotto, B.,525, 17DELPHI Collaboration,525, 17Del Prete, T.,526, 278;527, 23Del Signore, K.,525, 211De Marco, N.,521, 195De Maria, N.,525, 17Demarteau, M.,525, 211De Min, A., 525, 17Demina, R.,525, 211Demine, P.,525, 211De Miranda, J.M.,523, 53;525, 205Demirchyan, R.,523, 234;525, 9Denes, P.,524, 44, 55, 65;526, 269;527, 29Denisov, D.,525, 211Denisov, S.P.,525, 211Denisov, V.Yu.,526, 315DeNotaristefani, F.,524, 44, 55, 65;526, 269;527, 29De Oliveira, H.P.,526, 1De Palma, M.,526, 34, 191, 206De Paula, L.,525, 17Derbin, A.,525, 29De Risi, G.,521, 335De Roeck, A.,521, 181;523, 35, 234;525, 9; 526, 221, 233Derome, L.,521, 139De Rosa, G.,527, 173Dersch, U.,521, 171;522, 233;523, 22Dervan, P.,521, 181;523, 35;526, 221, 233De S. Pires, C.A.,522, 297Desai, S.,525, 211De Salvo, A.,524, 44, 55, 65;526, 269;527, 29De Santo, A.,526, 278;527, 23Desch, K.,521, 181;523, 35;526, 221, 233De Serio, M.,527, 173Desplanques, B.,521, 225

Dessagne, S.,526, 191, 206Deutsch, M.,525, 29Devaux, A.,521, 195De Vivie de Régie, J.-B.,526, 191, 206Devlin, M., 523, 13Dewald, A.,524, 252De Wolf, E.A.,521, 181;523, 35, 234;525, 9; 526, 221, 233Dhamotharan, S.,526, 34, 191, 206Dhawan, S.,527, 161D’Hondt, J.,525, 17Diaconu, C.,523, 234;525, 9Di Capua, E.,527, 173Di Capua, F.,527, 173Di Ciaccio, L.,525, 17DiCorato, M.,523, 53;525, 205Di Credico, A.,525, 29Dicus, D.A.,525, 175Diehl, H.T.,525, 211Diehl, M., 523, 265Diemoz, M.,524, 44, 55, 65;526, 269;527, 29Dienes, B.,521, 181;523, 35;526, 221, 233Dierckxsens, M.,524, 44, 55, 65;526, 269;527, 29Diesburg, M.,525, 211Dietl, H., 526, 34, 191, 206Dignan, T.,526, 278;527, 23Dileç, B.,522, 76Di Lella, L., 526, 278;527, 23Dimopoulos, S.,526, 393Dimova, T.,525, 41Dinçer, Y.,521, 7Ding, X.-M., 523, 367Dingfelder, J.,523, 234;525, 9Dini, P.,523, 53;525, 205Dionisi, C.,524, 44, 55, 65;526, 269;527, 29Di Pierro, M.,525, 360Dirkes, G.,522, 233Di Simone, A.,525, 17Dissertori, G.,526, 34, 191, 206Distler, M.O.,524, 26Dita, S.,521, 195Dittmar, M.,524, 44, 55, 65;526, 269;527, 29Dixit, M.S., 521, 181;523, 35Dixon, P.,523, 234;525, 9DØ Collaboration,525, 211Do Couto e Silva, E.,526, 278;527, 23Dodonov, V.,523, 234;525, 9Dolgolenko, A.G.,521, 171;522, 233;523, 22Dong, L.Y.,524, 33;526, 247, 258Donkers, M.,521, 181;523, 35;526, 221, 233Donnachie, A.,526, 413Donoghue, J.F.,522, 245Dore, U.,527, 173Doria, A.,524, 44, 55, 65;526, 269;527, 29Dorn, H.,524, 389Dornan, P.J.,526, 34, 191, 206Doroba, K.,525, 17Dos Reis, A.C.,523, 53;525, 205Doucet, M.,527, 173


Doulas, S.,525, 211Dova, M.T.,524, 44, 55, 65;526, 269;527, 29Dowell, J.D.,523, 234;525, 9Drapier, O.,521, 195Drechsel, D.,522, 27Drees, J.,525, 17Drees, M.,525, 130Drevermann, H.,526, 34, 191, 206Dris, M., 525, 17Droutskoi, A.,523, 234;525, 9Drozdetsky, A.,525, 41Drutskoy, A.,524, 33;526, 247, 258Druzhinin, V.,525, 41Dshemuchadse, S.,522, 16Dubak, A.,523, 234;525, 9Dubbert, J.,521, 181;523, 35;526, 221, 233Dubrovin, M.,525, 41Duchesneau, D.,524, 44, 55, 65;526, 269;527, 29Duchovni, E.,521, 181;523, 35;526, 221, 233Duckeck, G.,521, 181;523, 35;526, 221, 233Ducros, Y.,525, 211Ducroux, L.,521, 195Dudko, L.V.,525, 211Duensing, S.,525, 211Duerdoth, I.P.,521, 181;523, 35;526, 221, 233Duflot, L., 525, 211;526, 34, 191, 206Dugad, S.R.,525, 211Duinker, P.,524, 44, 55, 65;526, 269;527, 29Dumarchez, J.,526, 278;527, 23Dumitru, A.,525, 95Dunne, G.V.,526, 55Duperrin, A.,525, 211Duprel, C.,523, 234;525, 9Durell, J.L.,523, 13Dutz, H.,522, 16Dymarsky, A.Ya.,527, 125Dymov, S.,521, 158Dyshkant, A.,525, 211Dzyubenko, G.B.,521, 171;522, 233;523, 22

Ealet, A.,526, 191, 206Eaton, G.H.,527, 43Echenard, B.,524, 44, 55, 65;526, 269;527, 29Eckerlin, G.,523, 234;525, 9Eckstein, D.,523, 234;525, 9Edelstein, R.,521, 171;522, 233;523, 22Edera, L.,525, 205Edgecock, T.R.,526, 34, 191, 206Edmunds, D.,525, 211Efremenko, V.,523, 234;525, 9Efremov, A.V.,522, 37Egli, S.,523, 234;525, 9Eichler, R.,523, 234;525, 9Eidelman, S.,524, 33;526, 247, 258Eidelman, S.I.,527, 161Eigen, G.,525, 17Eiges, V.,524, 33;526, 247, 258Eisele, F.,523, 234;525, 9

Eisenhandler, E.,523, 234;525, 9Ekelof, T.,525, 17El-Aidi, R., 527, 173Elias, V.,521, 434Eline, A.,524, 44, 55, 65;526, 269;527, 29Elisei, F.,525, 29Ellerbrock, M.,523, 234;525, 9Ellert, M., 525, 17Ellis, G.,526, 191, 206Ellis, J.,525, 308Ellis, M., 526, 278;527, 23Ellison, J.,525, 211El Mamouni, H.,524, 44, 55, 65;526, 269;527, 29Elsen, E.,523, 234;525, 9Elsing, M.,525, 17Eltzroth, J.T.,525, 211Elvira, V.D., 525, 211Emediato, L.,521, 171;522, 233;523, 22Emery, S.,526, 34Emparan, R.,526, 393Enari, Y.,524, 33Enberg, R.,524, 273Endler, A.M.F.,521, 171;522, 233;523, 22Endo, M.,525, 121Engelfried, J.,521, 171;522, 233;523, 22Engelmann, R.,525, 211Engh, D.,523, 53;525, 205Engler, A.,524, 44, 55, 65;526, 269;527, 29Eno, S.,525, 211Enqvist, K.,526, 9Entem, D.R.,524, 93Epele, L.N.,523, 102Eppley, G.,525, 211Eppling, F.J.,524, 44, 55, 65;526, 269;527, 29Erba, S.,525, 205Erdmann, M.,523, 234;525, 9Erdmann, W.,523, 234;525, 9Ereditato, A.,527, 173Erhardt, A.,522, 16Erler, J.,521, 114Ermolaev, B.I.,522, 57Ermolov, P.,525, 211Eroshin, O.V.,525, 211Eschrich, I.,521, 171;522, 233;523, 22Escobar, C.O.,521, 171;522, 233;523, 22Espagnon, B.,521, 195Espirito Santo, M.C.,525, 17Estrada, J.,525, 211Etenko, A.,525, 29Etesi, G.,521, 391Etzion, E.,521, 181;523, 35;526, 221, 233Evans, H.,525, 211Evdokimov, A.V.,521, 171;522, 233;523, 22Evdokimov, V.N.,525, 211Everton, C.W.,526, 247, 258Ewald, I.,524, 26Ewers, A.,524, 44, 55, 65;526, 269;527, 29Extermann, P.,524, 44, 55, 65;526, 269;527, 29


Eyrich, W.,522, 16

Fabbri, F.,521, 181;523, 35;526, 221, 233Fabbri, F.L.,523, 53;525, 205Fabbro, B.,526, 191, 206Fabris, D.,521, 165Faddeev, L.,525, 195Fahland, T.,525, 211Falagan, M.A.,524, 44, 55, 65;526, 269;527, 29Falciano, S.,524, 44, 55, 65;526, 269;527, 29Falcone, D.,524, 241Falkowski, A.,521, 105Fallon, P.,525, 49Falvard, A.,526, 34, 191, 206Fanchiotti, H.,523, 102Fang, F.,524, 33;526, 247, 258Fanourakis, G.,525, 17Fantoni, S.,527, 73Fargeix, J.,521, 195Fassouliotis, D.,525, 17Faulkner, P.J.W.,523, 234;525, 9Favara, A.,524, 44, 55, 65;526, 269;527, 29Favart, D.,527, 173Favart, L.,523, 234;525, 9Fay, J.,524, 44, 55, 65;526, 269;527, 29Fayolle, D.,526, 191, 206Fazio, T.,526, 278;527, 23Fearing, H.W.,521, 204Fedin, O.,524, 44, 55, 65;526, 269;527, 29Fedotov, A.,523, 234;525, 9Fedotovich, G.V.,527, 161Feher, S.,525, 211Fein, D.,525, 211Feindt, M.,525, 17Felcini, M.,524, 44, 55, 65;526, 269;527, 29Feld, L.,521, 181;523, 35;526, 221, 233Feldman, G.J.,526, 278;527, 23Feliciello, A.,527, 39Felst, R.,523, 234;525, 9Ferbel, T.,521, 171;525, 211Ferencei, J.,523, 234;525, 9Ferguson, D.P.S.,526, 34, 191, 206Ferguson, T.,524, 44, 55, 65;526, 269;527, 29Fernandez, E.,526, 34, 191, 206Fernandez, J.,525, 17Fernandez-Bosman, M.,526, 34, 191, 206Fernholz, R.,525, 29Ferrari, P.,521, 181;523, 35;526, 221, 233Ferrari, R.,526, 278;527, 23Ferrer, A.,525, 17Ferrère, D.,526, 278;527, 23Ferro, F.,525, 17Ferron, S.,523, 234;525, 9Fesefeldt, H.,524, 44, 55, 65;526, 269;527, 29Fiandrini, E.,524, 44, 55, 65;526, 269;527, 29Fiedler, F.,521, 181;523, 35;526, 221, 233Field, J.H.,524, 44, 55, 65;526, 269;527, 29Filges, D.,522, 16

Filimonov, I.S.,521, 171;522, 233;523, 22Filippi, A., 522, 16;527, 39Filthaut, F.,524, 44, 55, 65;525, 211;526, 269;527, 29Finch, A.J.,526, 34, 191, 206Fiolhais, M.,523, 273Fiorentini, G.,526, 186Fioretto, E.,521, 165Fiorillo, G., 527, 173Fischbach, E.,526, 355Fischer, P.,521, 357Fisher, P.H.,524, 44, 55, 65;526, 269;527, 29Fisher, W.,524, 44, 55, 65;526, 269;527, 29Fisk, H.E.,525, 211Fisk, I.,524, 44, 55, 65;526, 269;527, 29Fisyak, Y.,525, 211Flagmeyer, U.,525, 17Flaminio, V.,526, 278;527, 23Flanagan, É.É.,522, 155Flattum, E.,525, 211Fleck, I.,521, 181;523, 35;526, 221, 233Fleischer, M.,523, 234;525, 9Fleming, Y.H.,523, 234;525, 9Fleuret, F.,525, 211Flügge, G.,523, 234;525, 9Foà, L.,526, 34, 191, 206Focardi, E.,526, 34, 191, 206FOCUS Collaboration,523, 53;525, 205Foeth, H.,525, 17Fokitis, E.,525, 17Fomenko, A.,523, 234;525, 9Force, P.,521, 195Forconi, G.,524, 44, 55, 65;526, 269;527, 29Ford, M.,521, 181;523, 35;526, 221, 233Ford, R.,525, 29Foresti, I.,523, 234;525, 9Formánek, J.,523, 234;525, 9Fortner, M.,525, 211Forty, R.W.,526, 34, 191, 206Fossan, D.B.,523, 13Foster, F.,526, 34, 191, 206Fouchez, D.,526, 191, 206Fox, H.,525, 211Frame, K.C.,525, 211Franco, D.,525, 29Frank, M.,526, 34, 191, 206Franke, F.,526, 370Franke, G.,523, 234;525, 9Frankland, L.,525, 49Fraternali, M.,526, 278;527, 23Frederico, T.,521, 33Freeman, S.J.,523, 13Freiesleben, H.,522, 16Freire, F.,526, 405Frekers, D.,527, 173Freudenreich, K.,524, 44, 55, 65;526, 269;527, 29Freudiger, B.,525, 29Frey, A.,521, 181;523, 35;526, 221, 233Fried, H.M.,524, 233


Friedman, E.,524, 87Friedrich, J.,524, 26Friedrich, J.M.,524, 26Fritsch, M.,522, 16Fritzsch, H.,525, 297;526, 90Fu, S.,525, 211Fuess, S.,525, 211Fujii, H., 524, 33;526, 247, 258Fujii, M., 525, 143;527, 106Fukuda, N.,522, 227Fukunaga, C.,524, 33;526, 247, 258Fukushima, M.,524, 33;526, 247, 258Fulda-Quenzer, F.,525, 17Fülöp, Zs.,522, 227Furetta, C.,524, 44, 55, 65;526, 269;527, 29Fürtjes, A.,521, 181;523, 35;526, 221, 233Fuster, J.,525, 17Futakami, U.,522, 227Futyan, D.I.,521, 181;523, 35;526, 221, 233

Gaardhøje, J.J.,523, 227Gabathuler, E.,523, 234;525, 9Gabathuler, K.,523, 234;525, 9Gabyshev, N.,524, 33;526, 247, 258Gabyshev, N.I.,527, 161Gacsi, Z.,522, 227Gagnon, P.,521, 181;523, 35;526, 221, 233Gaillard, J.-M.,526, 278;527, 23Gaines, I.,523, 53;525, 205Gajdosik, T.,523, 249;526, 360Gál, J.,521, 165Galaktionov, Yu.,524, 44, 55, 65;526, 269;527, 29Galbiati, C.,525, 29Gallas, E.,525, 211Gallio, M., 521, 195Galyaev, A.N.,525, 211Gandelman, M.,525, 17Gangler, E.,526, 278;527, 23Ganguli, S.N.,524, 44, 55, 65;526, 269;527, 29Ganis, G.,526, 34, 191, 206Gao, M.,525, 211Gao, Y.,526, 34, 191, 206Garbincius, P.H.,523, 53;525, 205Garcia, C.,525, 17Garcia, F.G.,521, 171;522, 233;523, 22Garcia-Abia, P.,524, 44, 55, 65;526, 269;527, 29Garcia-Bellido, A.,526, 191, 206García Canal, C.A.,523, 102Gardner, C.L.,524, 21Gardner, R.,523, 53;525, 205Garmash, A.,524, 33;526, 247, 258Garren, L.A.,523, 53;525, 205Garrido, Ll.,526, 34, 191, 206Garvey, J.,523, 234;525, 9Gary, J.W.,521, 181;523, 35;526, 221, 233Gasperini, M.,521, 335Gaspero, M.,521, 171;522, 233;523, 22Gassner, J.,523, 234;525, 9

Gataullin, M.,524, 44, 55, 65;526, 269;527, 29Gatignon, L.,525, 41Gatti, F.,525, 29Gatto, R.,524, 144Gattringer, C.,522, 194Gavillet, Ph.,525, 17Gavrilov, V.,525, 211Gavrilov, Y.K.,521, 195Gay, P.,526, 34, 191, 206Gaycken, G.,521, 181;523, 35;526, 221, 233Gay Ducati, M.B.,521, 259Gayler, J.,523, 234;525, 9Gazdzicki, M.,523, 255Gazis, E.,525, 17Gazzana, S.,525, 29Geich-Gimbel, C.,521, 181;523, 35;526, 221, 233Geiser, A.,526, 278;527, 23Gele, D.,525, 17Gelletly, W.,525, 49Genik, R.J.,525, 211Genser, K.,525, 211Gentile, S.,524, 44, 55, 65;526, 269;527, 29Georgi, J.,522, 16Geppert, D.,526, 278;527, 23Geralis, T.,525, 17Gerber, C.E.,525, 211Gerhards, R.,523, 234;525, 9Gerlich, C.,523, 234;525, 9Gerschel, C.,521, 195Gershon, T.,524, 33;526, 247, 258Gershtein, Y.,525, 211Geweniger, C.,526, 34, 191, 206Ghazaryan, S.,523, 234;525, 9Gheerardyn, J.,525, 322Ghete, V.M.,526, 34, 191, 206Ghosh, S.,522, 189Giacomelli, G.,521, 181;523, 35;526, 221, 233Giacomelli, P.,521, 181;523, 35;526, 221, 233Giagu, S.,524, 44, 55, 65;526, 269;527, 29Giammanco, A.,526, 34, 191, 206Giammarchi, M.,523, 53;525, 205Giammarchi, M.G.,525, 29Gianini, G.,523, 53;525, 205Giannini, G.,526, 34, 191, 206Gianotti, F.,526, 34, 191, 206Giassi, A.,526, 34, 191, 206Gibin, D.,526, 278;527, 23Giller, I., 521, 171;522, 233;523, 22Gillitzer, A., 522, 16Gilmartin, R.,525, 211Ginther, G.,525, 211Girone, M.,526, 34, 191, 206Girotti, H.O.,521, 119Girtler, P.,526, 34, 191, 206Giubellino, P.,521, 195Giugni, D.,525, 29Giunta, M.,526, 221, 233Glass, G.,523, 1


Glenzinski, D.,521, 181;523, 35Gninenko, S.,526, 278Gninenko, S.N.,527, 23Göbel, C.,523, 53;525, 205Göckeler, M.,522, 194Godley, A.,526, 278;527, 23Goeger-Neff, M.,525, 29Goeke, K.,522, 37Goerlich, L.,523, 234;525, 9Gogitidze, N.,523, 234;525, 9Gogoladze, I.,522, 107;525, 189Gokalp, A.,525, 273Gokieli, R.,525, 17Goldberg, J.,521, 181;523, 35;526, 221, 233;527, 173Goldberg, M.,523, 234;525, 9Golli, B., 523, 273Golob, B.,525, 17Golovtsov, V.L.,521, 171;522, 233;523, 22Golowich, E.,522, 245Golubchikov, A.,525, 29Golubev, V.,525, 41Golubeva, M.B.,521, 195Gomes, M.,521, 119Gomes, P.R.S.,526, 295Gómez, B.,525, 211Gómez, G.,525, 211Gomez-Cadenas, J.-J.,526, 278;527, 23Gomez-Ceballos, G.,525, 17Gomi, T.,522, 227Gomis, J.,524, 170Goncalves, P.,525, 17Goncharov, P.I.,525, 211Gong, Z.F.,524, 44, 55, 65;526, 269;527, 29Gonin, M.,521, 195González, P.,521, 225González, S.,526, 34, 191, 206González-Díaz, P.F.,522, 211Gonzalez-Garcia, M.C.,527, 23, 199González Solís, J.L.,525, 211Gorbachev, D.A.,527, 161Gorbunov, P.,527, 173Gordon, A.,524, 33;526, 247, 258Gordon, H.,525, 211Gorenstein, M.I.,523, 255;524, 265Goretti, A.,525, 29Goss, L.T.,525, 211Gosset, J.,526, 278;527, 23Gößling, C.,526, 278;527, 23Goswami, S.,526, 27Gottschalk, E.,523, 53;525, 205Gouanère, M.,526, 278;527, 23Gouffon, P.,521, 171;522, 233;523, 22Gould, M.D.,523, 367Gounaris, G.J.,525, 63Gounder, K.,525, 211Goussiou, A.,525, 211Goy, C.,526, 34, 191, 206Grab, C.,523, 234;525, 9

Gracey, J.A.,525, 89Graf, N.,525, 211Graham, G.,525, 211Graham, K.,521, 181;523, 35;526, 221, 233Gram, P.A.M.,523, 1Grandchamp, L.,523, 60Grannis, P.D.,525, 211Grant, A.,521, 239;526, 278;527, 23Grässler, H.,523, 234;525, 9Graugés, E.,526, 34, 191, 206Graziani, E.,525, 17Graziani, G.,526, 278;527, 23Grebenuk, A.A.,527, 161Greco, M.,522, 57Green, J.A.,525, 211Green, M.G.,526, 34, 191, 206Greening, T.C.,526, 34, 191, 206Greenlee, H.,525, 211Greenshaw, T.,523, 234;525, 9Greenwood, Z.D.,525, 211Grégoire, G.,527, 173Greiner, W.,524, 265Grella, G.,527, 173Grenier, G.,524, 44, 55, 65;526, 269;527, 29Grichine, V.M.,525, 225Grieb, C.,525, 29Grigorian, A.A.,521, 195Grigorian, S.,521, 195Grigoriev, D.N.,527, 161Grimm, O.,524, 44, 55, 65;526, 269;527, 29Grimus, W.,521, 267Grindhammer, G.,523, 234;525, 9Grinstein, B.,526, 345Grinstein, S.,525, 211Grishina, V.Yu.,521, 217Grivaz, J.-F.,526, 34, 191, 206Groer, L.,525, 211Grosdidier, G.,525, 17Gross, E.,521, 181;523, 35;526, 221, 233Grossiord, J.Y.,521, 195Grotowski, K.,523, 227Gruenewald, M.W.,524, 44, 55, 65;526, 269;527, 29Grumiller, D.,521, 357Grünendahl, S.,525, 211Grunhaus, J.,521, 181;523, 35;526, 221, 233Grupen, C.,526, 34, 191, 206Gruwé, M.,521, 181;523, 35;526, 221, 233Grzelak, K.,525, 17Guber, F.F.,521, 195Guevara, R.,525, 189Guglielmi, A.,526, 278;527, 23Guichard, A.,521, 195Guida, M.,524, 44, 55, 65;526, 269;527, 29Güler, M.,527, 173Gulkanyan, H.,521, 195Gülmez, E.,522, 233;523, 22Gülmez, E.E.,521, 171Günther, P.O.,521, 181;523, 35;526, 221, 233


Guo, R.,524, 33;526, 247, 258Guo, X.,523, 88Gupta, A.,521, 181;523, 35;525, 211;526, 221, 233Gupta, K.S.,526, 121Gupta, V.K.,524, 44, 55, 65;526, 269;527, 29Gurtu, A.,524, 44, 55, 65;526, 269;527, 29Gurzhiev, S.N.,525, 211Gutay, L.J.,524, 44, 55, 65;526, 269;527, 29Gutierrez, G.,525, 211Gutierrez, P.,525, 211Gutowski, J.,525, 150Guy, J.,525, 17Gyulassy, M.,526, 301

H1 Collaboration,523, 234;525, 9Haag, C.,525, 17Haas, D.,524, 44, 55, 65;526, 269;527, 29Haas, F.,521, 165Haba, J.,524, 33;526, 247, 258Hadig, T.,523, 234;525, 9Hadley, N.J.,525, 211Hagel, K.,521, 165;523, 227Haggerty, H.,525, 211Hagner, C.,525, 29;526, 278;527, 23Hagner, T.,525, 29Hagopian, S.,525, 211Hagopian, V.,525, 211Hahn, F.,525, 17Hahn, S.,525, 17Haidt, D.,523, 234;525, 9Hajdu, C.,521, 181;523, 35;526, 221, 233Hajduk, L.,523, 234;525, 9Hakobyan, R.,521, 195Hall, R.E.,525, 211Haller, J.,523, 234;525, 9Halley, A.,526, 34Hallgren, A.,525, 17Hamacher, K.,525, 17Hamaguchi, K.,525, 143Hamann, M.,521, 181;523, 35;526, 221, 233Hamasaki, H.,524, 33;526, 247, 258Hamilton, K.,525, 17Hammer, C.,521, 171Hampel, W.,525, 29Hanagaki, K.,526, 247, 258Handa, F.,524, 33;526, 247, 258Handler, T.,523, 53;525, 205Hanke, P.,526, 34, 191, 206Hanlet, P.,525, 211Hansen, J.,525, 17Hansen, J.B.,526, 34, 191, 206Hansen, J.D.,526, 34, 191, 206Hansen, J.R.,526, 34, 191, 206Hansen, O.,523, 227Hansen, P.H.,526, 34, 191, 206Hansen, S.,525, 211Hanson, G.G.,521, 181;523, 35;526, 221, 233Hansper, G.,526, 34

Hara, K.,524, 33;526, 247, 258Hara, T.,524, 33;526, 247, 258;527, 173Harder, K.,521, 181;523, 35;526, 221, 233Harding, B.,525, 29Harel, A.,521, 181;523, 35;526, 221, 233Harin-Dirac, M.,521, 181;523, 35;526, 221, 233Haroutunian, R.,521, 195Hart, A.,523, 280Hartmann, B.,524, 227Hartmann, F.X.,525, 29Hartmann, M.,521, 158Harvey, J.,526, 34, 191, 206Hastings, N.C.,524, 33;526, 247, 258Hatzifotiadou, D.,524, 44, 55, 65;526, 269;527, 29Haug, S.,525, 17Hauger, M.,524, 26Hauler, F.,525, 17Hauptman, J.M.,525, 211Hauschild, M.,521, 181;523, 35;526, 221, 233Hauschildt, J.,521, 181;523, 35;526, 221, 233Hawkes, C.M.,521, 181;523, 35;526, 221, 233Hawkings, R.,521, 181;523, 35;526, 221, 233Hayashii, H.,524, 33;526, 247, 258Hayes, O.J.,526, 34, 191, 206Haynes, W.J.,523, 234;525, 9Hays, C.,525, 211Hazumi, M.,524, 33;526, 247, 258He, H.,526, 191, 206He, H.-J.,525, 175He, K.,521, 171;522, 233;523, 22Hebbeker, T.,524, 44, 55, 65;526, 269;527, 29Hebert, C.,525, 211Hedberg, V.,525, 17Hedin, D.,525, 211Heenan, E.M.,524, 33;526, 247, 258Heenen, P.H.,523, 13Hegedus, Á.,523, 211Heinemann, B.,523, 234;525, 9Heinmiller, J.M.,525, 211Heinson, A.P.,525, 211Heintz, U.,525, 211Heinzelmann, G.,523, 234;525, 9Heister, A.,526, 34, 191, 206Hemingway, R.J.,521, 181;523, 35;526, 221, 233Henderson, R.C.W.,523, 234;525, 9Hengstmann, S.,523, 234;525, 9Hennecke, M.,525, 17Henrard, P.,526, 34Henschel, H.,523, 234;525, 9Hensel, C.,521, 181;523, 35;526, 221, 233Hepp, V.,526, 34, 191, 206Heremans, R.,523, 234;525, 9Hernandez, H.,523, 53;525, 205Hernández, R.,521, 371Hernando, J.,526, 278;527, 23Herr, H.,525, 17Herrera, G.,523, 29, 234;525, 9Herten, G.,521, 181;523, 35;526, 221, 233


Hervé, A.,524, 44, 55, 65;526, 269;527, 29Herynek, I.,523, 234;525, 9Herzog, C.P.,526, 388Hess, J.,526, 191, 206Hesselbach, S.,526, 370Hesselbarth, D.,522, 16Hesselink, W.H.A.,523, 6Heuer, R.D.,521, 181;523, 35;526, 221, 233Heusse, Ph.,526, 34, 191, 206Heusser, G.,525, 29Higuchi, I.,524, 33;526, 247, 258Higuchi, T.,524, 33;526, 247, 258Higurashi, Y.,522, 227Hildebrandt, M.,523, 234;525, 9Hildreth, M.D.,525, 211Hilgers, M.,523, 234;525, 9Hill, J.C., 521, 181;523, 35;526, 221, 233Hill, R.D., 526, 191, 206Hiller, K.H., 523, 234;525, 9Hinde, D.J.,526, 295Hirenzaki, S.,527, 69Hirosky, R.,525, 211Hirschfelder, J.,524, 44, 55, 65;526, 269;527, 29Hladký, J.,523, 234;525, 9Hobbs, J.D.,525, 211Hodgson, P.N.,526, 191, 206Hoeneisen, B.,525, 211Hofer, H.,524, 44, 55, 65;526, 269;527, 29Hoffman, K.,521, 181;523, 35;526, 221, 233Hoffmann, D.,523, 234;525, 9Hohlmann, M.,524, 65;526, 269;527, 29Hojo, T.,524, 33;526, 247, 258Hokuue, T.,524, 33;526, 247, 258Hölldorfer, F.,526, 191, 206Holm, A., 523, 227Holme, A.K.,523, 227Holmgren, S.-O.,525, 17Holt, P.J.,525, 17Holzner, G.,524, 44, 55, 65;526, 269;527, 29Homer, R.J.,521, 181;523, 35;526, 221, 233Honegger, A.,524, 26Horisberger, R.,523, 234;525, 9Horváth, D.,521, 181;523, 35;526, 221, 233Hosack, M.,523, 53;525, 205Hoshi, Y.,524, 33;526, 247, 258Hoshina, K.,524, 33;526, 247, 258Hoshino, K.,527, 173Hossain, K.R.,521, 181;523, 35;526, 221, 233Höting, P.,523, 234;525, 9Hou, S.R.,524, 33, 44, 55, 65;526, 247, 258, 269;527, 29Hou, W.-S.,524, 33;526, 247, 258Houlden, M.A.,525, 17Howard, A.S.,524, 245Howard, R.,521, 181;523, 35;526, 221, 233Hsu, S.-C.,524, 33;526, 247, 258Hu, H.,526, 34, 191, 206Hu, Y., 524, 44, 55, 65;526, 269;527, 29Huang, C.-S.,525, 107

Huang, H.-C.,524, 33;526, 247, 258Huang, X.,526, 34, 191, 206Huang, Y.,525, 211Hubbard, D.,526, 278;527, 23Huber, P.,523, 151Hughes, G.,526, 34, 191, 206Hughes, V.W.,527, 161Huitu, K., 524, 161Hultqvist, K.,525, 17Hüntemeyer, P.,521, 181;523, 35;526, 221, 233Huovinen, P.,526, 301Hurling, S.,523, 234;525, 9Hurst, P.,526, 278;527, 23Hutchcroft, D.E.,526, 34, 191, 206Hüttmann, K.,526, 34, 191, 206Hyett, N.,526, 278;527, 23

Iacopini, E.,526, 278;527, 23Iancu, E.,523, 143Ianni, A.,525, 29Ianni, A.M.,525, 29Iaselli, G.,526, 34, 191, 206Iashvili, I., 525, 211Iazzi, F.,527, 39Ibbotson, M.,523, 234;525, 9Ichihara, T.,521, 153Idzik, M., 521, 195Ieva, M.,527, 173Igarashi, Y.,524, 33;526, 164, 247, 258Iglesias, A.,521, 96Ignatov, F.V.,527, 161Igo-Kemenes, P.,521, 181;523, 35;526, 221, 233Iijima, T., 524, 33;526, 247, 258Ikeda, H.,524, 33;526, 247, 258Illingworth, R.,525, 211Imai, N.,522, 227Inami, K.,524, 33;526, 247Incicchitti, A., 527, 182Ingelman, G.,524, 273Inoue, S.,524, 15Inzani, P.,523, 53;525, 205Iori, M., 521, 171;522, 233;523, 22Ishida, K.,527, 43Ishida, S.,521, 153Ishihara, M.,522, 227Ishii, K., 521, 181;523, 35;526, 221, 233Ishikawa, A.,524, 33;526, 247, 258Ishino, H.,524, 33;526, 247, 258Issever, Ç.,523, 234;525, 9Ito, A.S.,525, 211Ito, H., 523, 227Ito, M., 524, 357Itoh, K., 526, 164Itoh, K.S.,521, 153Itoh, R.,524, 33;526, 247, 258Ivaniouchenkov, I.,524, 245Ivanov, E.A.,524, 208Ivanov, V.G.,524, 259


Iwasa, N.,522, 227Iwasaki, H.,522, 227;524, 33;526, 247, 258Iwasaki, Y.,524, 33;526, 247, 258

Jacholkowska, A.,526, 34, 191, 206Jackson, D.J.,524, 33;526, 247, 258Jackson, J.N.,525, 17Jacob, A.M.,527, 187Jacquet, M.,523, 234;525, 9Jadach, S.,523, 117Jaffre, M.,523, 234;525, 9Jaffré, M.,525, 211Jain, S.,525, 211Jakob, B.,522, 16Jakobs, K.,526, 34, 191, 206Jakobsen, E.,523, 227Jalocha, P.,524, 33;525, 17;526, 247, 258Janauschek, L.,523, 234;525, 9Jang, H.K.,524, 33;526, 247, 258Janot, P.,526, 34, 191, 206Jans, E.,523, 6Janssen, B.,526, 144Janssen, S.,527, 62Janssen, X.,523, 234;525, 9Janssens, R.V.F.,525, 49Jarlskog, Ch.,525, 17Jarlskog, G.,525, 17Jarry, P.,525, 17Jawahery, A.,521, 181;523, 35;526, 221, 233Jeans, D.,525, 17Jejcic, A.,525, 41Jemanov, V.,523, 234;525, 9Jennewein, P.,524, 26Jeremie, H.,521, 181;523, 35;526, 221, 233Jesik, R.,525, 211Jezequel, S.,526, 191, 206Ji, X., 523, 73, 107Jin, B.N.,524, 44, 55, 65;526, 269;527, 29Jin, S.,526, 34, 191, 206Jipa, A.,523, 227Johansson, E.K.,525, 17Johansson, P.D.,525, 17Johns, K.,525, 211Johns, W.E.,523, 53;525, 205Johnson, C.,523, 234;525, 9Johnson, D.P.,523, 234;525, 9Johnson, M.,525, 211Johnson, M.B.,523, 127Jokinen, A.,526, 9Jolie, J.,524, 252;527, 55Jolos, R.V.,521, 146;524, 252;526, 322Jonckheere, A.,525, 211Jones, C.R.,521, 181;523, 35;526, 221, 233Jones, L.T.,526, 191, 206Jones, L.W.,524, 44, 55, 65;526, 269;527, 29Jones, M.A.S.,523, 234;525, 9Jones, R.W.L.,526, 34, 191, 206Jones, W.G.,524, 245

Jonke, L.,526, 149Jönsson, L.,523, 234;525, 9Jonsson, P.,525, 17Joram, C.,525, 17Jørdre, J.I.,523, 227Jørgensen, C.E.,523, 227Josa-Mutuberría, I.,524, 44, 55, 65;526, 269;527, 29Joseph, C.,526, 278;527, 23Joshi, J.,524, 245Joss, D.T.,525, 49Jost, B.,526, 34, 191, 206Jöstlein, H.,525, 211Jouan, D.,521, 195Jourdan, J.,524, 26Jousset, J.,526, 34, 191, 206Jovanovic, P.,521, 181;523, 35;526, 221, 233Juget, F.,526, 278;527, 23Jun, S.Y.,521, 171;522, 233;523, 22Jundt, F.,523, 227Jung, H.,523, 234;525, 9Jungermann, L.,525, 17Junk, T.R.,521, 181;523, 35;526, 221, 233Juste, A.,525, 211

Kacharava, A.,521, 158Kachelhoffer, T.,526, 191, 206Kado, M.,526, 34, 191, 206Käfer, D.,524, 44, 55, 65;526, 269;527, 29Kagan, R.,524, 33Kageyama, A.,527, 206Kahl, W.,525, 211Kahn, S.,525, 211Kahrau, M.,524, 26Kajfasz, E.,525, 211Kakushadze, Z.,521, 96Kalinin, A.M., 525, 211Kalinin, S.,527, 173Kalliomäki, A., 524, 153Kamalov, S.,526, 287Kamalov, S.S.,522, 27Kambor, J.,521, 22, 29Kanaya, N.,521, 181;523, 35;526, 221, 233Kaneko, S.,527, 206Kang, J.H.,524, 33;526, 247, 258Kang, J.S.,523, 53;524, 33;525, 205;526, 247, 258Kang, S.K.,521, 61Kant, D.,523, 234;525, 9Kanzaki, J.,521, 181;523, 35;526, 221, 233Kao, W.-F.,522, 257Kapichine, M.,523, 234;525, 9Kapusta, F.,525, 17Kapusta, P.,524, 33;526, 247, 258Karapetian, G.,521, 181;523, 35;526, 221, 233Karavitcheva, T.L.,521, 195Karlen, D.,521, 181;523, 35;526, 221, 233Karlsson, M.,523, 234;525, 9Karmanov, D.,525, 211Karmgard, D.,525, 211


Karpov, S.V.,527, 161Karsch, L.,522, 16Karschnick, O.,523, 234;525, 9Karshenboim, S.G.,524, 259Kartvelishvili, V., 521, 181;523, 35;526, 221, 233Kasdorp, W.-J.,523, 6Kasper, P.H.,523, 53;525, 205Katayama, N.,524, 33;526, 247, 258Kato, M.,527, 43Katsanevas, S.,525, 17Katsoufis, E.,525, 17Kaur, M.,524, 44, 55, 65;526, 269;527, 29Kawagoe, K.,521, 181;523, 35;526, 221, 233Kawai, H.,524, 33, 33;526, 247, 247, 258, 258Kawamoto, T.,521, 181;523, 35;526, 221, 233Kawamura, H.,523, 111Kawamura, N.,524, 33;526, 247, 258;527, 43Kawamura, T.,527, 173Kawasaki, T.,524, 33;526, 247, 258Kaya, A.,524, 348Kaya, M.,521, 171;522, 233;523, 22Kayis-Topaksu, A.,527, 173Kayser, F.,526, 191, 206Kazanin, V.F.,527, 161Kechkin, O.V.,522, 166;523, 323Keeler, R.K.,521, 181;523, 35;526, 221, 233Kehoe, R.,525, 211Keil, F., 523, 234;525, 9Keller, N.,523, 234;525, 9Kellogg, R.G.,521, 181;523, 35;526, 221, 233Kennedy, B.W.,521, 181;523, 35;526, 221, 233Kennedy, J.,523, 234;525, 9; 526, 191, 206Kent, N.,526, 278Kenyon, I.R.,523, 234;525, 9Kephart, T.W.,522, 315Keppler, P.,525, 41Keranen, R.,525, 17Kermiche, S.,523, 234;525, 9Kernel, G.,525, 17Kersevan, B.P.,525, 17Kersten, J.,525, 130Kersting, N.,527, 115Keutgen, T.,523, 227Khanov, A.,525, 211Kharchilava, A.,525, 211Kharzeev, D.,523, 79Khavaev, A.,522, 181Khazin, B.I.,527, 161Khovansky, V.,527, 173Khruschev, V.V.,525, 283Kichimi, H., 524, 33;526, 247, 258Kienzle-Focacci, M.N.,524, 44, 55, 65;526, 269;527, 29Kiesling, C.,523, 234;525, 9Kiiskinen, A.,525, 17Kiko, J.,525, 29Kile, J.,526, 34, 191, 206Kilian, K., 522, 16Kilmer, J.,521, 171;522, 233;523, 22

Kim, C., 523, 205Kim, D.H., 521, 181;523, 35;526, 221, 233Kim, D.W., 524, 33;526, 247, 258Kim, D.Y., 523, 53;525, 205Kim, E.J.,523, 227Kim, H., 524, 33, 33;526, 247, 247, 258, 258Kim, H.B., 527, 18Kim, H.J.,524, 33;526, 247, 258Kim, H.O., 524, 33;526, 247, 258Kim, J.E.,527, 18Kim, J.K.,524, 44, 55, 65;526, 269;527, 29Kim, K.-Y., 523, 357Kim, S.K., 524, 33;525, 211;526, 247, 258Kim, T.H., 524, 33;526, 247, 258Kim, V.T., 521, 171;522, 233;523, 22King, B.T.,525, 17Kinoshita, K.,524, 33;526, 247, 258Kirkby, J.,524, 44, 55, 65;526, 269;527, 29Kirsanov, M.,526, 278Kirsanov, M.M.,527, 23Kirsch, R.,525, 41Kirsten, T.,525, 29Kisslinger, L.S.,523, 127Kitabayashi, T.,524, 308Kittel, W., 524, 44, 55, 65;526, 269;527, 29Kjaer, N.J.,525, 17Kjellberg, P.,523, 234;525, 9Klebanov, I.R.,526, 388Klein, K., 521, 181;523, 35;526, 221, 233Klein, M., 523, 234;525, 9Kleinknecht, K.,526, 34, 191, 206Kleinwort, C.,523, 234;525, 9Klier, A., 521, 181;523, 35;526, 221, 233Klima, B., 525, 211Klimenko, A.A.,522, 222Klimentov, A.,524, 44, 55, 65;526, 269;527, 29Klimov, O., 526, 278;527, 23Kluberg, L.,521, 195Klug, T., 524, 252Kluge, E.E.,526, 34, 191, 206Kluge, T.,523, 234;525, 9Kluit, P., 525, 17Kluth, S.,521, 181;523, 35;526, 221, 233Kneringer, E.,526, 34, 191, 206Knies, G.,523, 234;525, 9Knuteson, B.,525, 211Ko, B.R.,523, 53;525, 205Ko, W., 525, 211Kobakhidze, A.,522, 107Kobayashi, T.,521, 181;523, 35;526, 221, 233Kobel, M.,521, 181;523, 35;526, 221, 233Kobes, R.,523, 221Koblitz, B., 523, 234;525, 9Koch, H.,522, 16Kochenda, L.M.,521, 171;522, 233;523, 22Kodaira, J.,523, 111Kodama, K.,527, 173Kohli, J.M.,525, 211


Kokkinias, P.,525, 17Kokkonen, J.,526, 278;527, 23Kokott, T.P.,521, 181;523, 35;526, 221, 233Kolev, D.,527, 173Kolya, S.D.,523, 234;525, 9Komamiya, S.,521, 181;523, 35;526, 221, 233Komarov, V.I.,521, 158;524, 303Komatsu, M.,527, 173Kondratyuk, L.A.,521, 217Kondratyuk, S.,521, 204König, A.C.,524, 44, 55, 65;526, 269;527, 29Königsmann, K.,522, 233Konishi, H.,524, 33;526, 247, 258Konorov, I.,521, 171;522, 233;523, 22Konstantinidis, N.,526, 34, 191, 206Koop, I.A.,527, 161Kopal, M.,524, 44, 55, 65;526, 269;527, 29Koptev, V.,521, 158Korbel, V.,523, 234;525, 9Korga, G.,525, 29Kormos, L.,526, 233Korpar, S.,524, 33;526, 247, 258Korschinek, G.,525, 29Kostka, P.,523, 234;525, 9Kostritskiy, A.V., 525, 211Kostyuk, A.P.,524, 265Kotcher, J.,525, 211Kotelnikov, S.K.,523, 234;525, 9Kothari, B.,525, 211Kotwal, A.V., 525, 211Kou, E.,525, 240Kourkoumelis, C.,525, 17Koutouev, R.,523, 234;525, 9Koutov, A.,523, 234;525, 9Koutsenko, V.,524, 44, 55, 65;526, 269;527, 29Kouznetsov, O.,525, 17Kovzelev, A.,526, 278Kovzelev, A.V.,527, 23Kowalewski, R.V.,521, 181;523, 35;526, 221, 233Kowalski-Glikman, J.,521, 364;522, 133Kozanecki, W.,526, 34Kozelov, A.V.,525, 211Kozhevnikov, A.P.,521, 171;522, 233;523, 22Kozik, T., 523, 227Kozlov, Y., 525, 29Kozlovsky, E.A.,525, 211Kräber, M.,524, 44, 55, 65;526, 269;527, 29Kraemer, R.W.,524, 44, 55, 65;526, 269;527, 29Krämer, T.,521, 181;523, 35;526, 221, 233Krane, J.,525, 211Krasnikov, N.V.,527, 23Krasnoperov, A.,526, 278;527, 23Krehbiel, H.,523, 234;525, 9Krenz, W.,524, 44, 55, 65;526, 269;527, 29Kreß, J.,522, 16Kress, T.,521, 181;523, 35;526, 221, 233Kreymer, A.E.,523, 53;525, 205Krieger, P.,521, 181;523, 35;526, 221, 233

Krishnaswamy, M.R.,525, 211Kriss, B.J.,523, 1Krivkova, P.,525, 211Krivshich, A.G.,521, 171;522, 233;523, 22Križan, P.,524, 33;526, 247, 258Krokovny, P.,524, 33;526, 247, 258Krokovny, P.P.,527, 161Krop, D.,521, 181;523, 35;526, 221, 233Kroseberg, J.,523, 234;525, 9Krüger, A.,524, 44, 55, 65;526, 269;527, 29Krüger, H.,521, 171;522, 233;523, 22Krüger, K.,523, 234;525, 9Krumstein, Z.,525, 17Krusche, B.,526, 287Kryemadhi, A.,523, 53;525, 205Krygier, K.W.,524, 26Kryn, D., 525, 29Krzywdzinski, S.,525, 211Kubantsev, M.,525, 211Kubantsev, M.A.,521, 171;522, 233;523, 22Kubarovsky, V.P.,521, 171;522, 233;523, 22Kubo, T.,522, 227Kubon, G.,524, 26Kucharczyk, M.,525, 17Kudo, K.,527, 43Kudryavtsev, V.A.,524, 245Kuhl, T., 521, 181;523, 35;526, 221, 233Kuhlmann, E.,522, 16Kuhn, D.,526, 34, 191, 206Kuhr, T.,523, 234;525, 9Kulasiri, R.,524, 33;526, 247, 258Kuleshov, S.,525, 211Kulibaba, V.,525, 41Kulik, Y., 525, 211Kulyavtsev, A.I.,521, 171;522, 233;523, 22Kumar, S.,524, 33;526, 247, 258Kummer, W.,521, 357Kunibu, M.,522, 227Kunin, A., 524, 44, 55, 65;526, 269;527, 29Kunori, S.,525, 211Kupco, A.,525, 211Küpper, A.,523, 234;525, 9Kupper, M.,521, 181;523, 35;526, 221, 233Kuraev, E.A.,527, 161Kurbatov, V.,521, 158Kurca, T.,523, 234;525, 9Kurdadze, L.M.,527, 161Kurepin, A.B.,521, 195Kurokawa, M.,522, 227Kuropatkin, N.P.,521, 171;522, 233;523, 22Kurosawa, K.,527, 43Kurowska, J.,525, 17Kurshetsov, V.F.,521, 171;522, 233;523, 22Kushnirenko, A.,521, 171;522, 233;523, 22Kustov, D.,526, 278Kutschke, R.,523, 53;525, 205Kuzenko, S.M.,522, 320Kuzmin, A.,524, 33;526, 247, 258


Kuzmin, A.S.,527, 161Kuznetsov, V.,526, 278Kuznetsov, V.E.,525, 211Kuznetzov, I.V.,522, 222Kwak, J.W.,523, 53;525, 205Kwan, S.,521, 171;522, 233;523, 22Kwon, O.-K.,523, 205Kwon, Y.-J.,524, 33;526, 247, 258Kyae, B.,526, 379Kyberd, P.,521, 181;523, 35;526, 221, 233Kyriakis, A., 526, 34, 191, 206

L3 Collaboration,524, 44, 55, 65;526, 269;527, 29Lacaprara, S.,526, 278;527, 23Lach, J.,521, 171;522, 233;523, 22Lachaud, C.,526, 278;527, 23Ladron de Guevara, P.,524, 44, 55, 65;526, 269;527, 29Lafferty, G.D.,521, 181;523, 35;526, 221, 233Laforge, B.,525, 17, 219LaFosse, D.R.,523, 13Laget, J.M.,523, 6Lahiff, A.D., 521, 204Lahmann, R.,523, 234;525, 9Lakic, B.,526, 278;527, 23Laktineh, I.,524, 44, 55, 65;526, 269;527, 29Lalak, Z.,521, 105Lamb, D.,523, 234;525, 9Lamberto, A.,521, 171;522, 233;523, 22Lamsa, J.,525, 17Lançon, E.,526, 34, 191, 206Landi, G.,524, 44, 55, 65;526, 269;527, 29Landon, M.P.J.,523, 234;525, 9Landsberg, G.,525, 211Landsberg, L.G.,521, 171;522, 233;523, 22Landshoff, P.V.,526, 413Landsman, H.,521, 181;523, 35;526, 221, 233Lane, G.J.,523, 13Lang, C.B.,522, 194Langanke, K.,522, 240Lange, J.S.,524, 33;526, 247, 258Lange, W.,523, 234;525, 9Langfelder, P.,521, 96Lanske, D.,521, 181;523, 35;526, 221, 233Lanza, A.,526, 278;527, 23Laporta, S.,523, 95La Rana, G.,521, 165Larin, I., 521, 171;522, 233;523, 22La Rotonda, L.,526, 278;527, 23Larsen, T.M.,523, 227Laštovicka, T.,523, 234;525, 9Laubenstein, M.,525, 29Laurelli, P.,526, 34, 191, 206Lautesse, Ph.,525, 41Laveder, M.,526, 278;527, 23Lawson, I.,521, 181;523, 35;526, 221, 233Lawson, T.B.,524, 245Laycock, P.,523, 234;525, 9Layter, J.G.,521, 181;523, 35;526, 221, 233

Lebailly, E.,523, 234;525, 9Lebeau, M.,524, 44, 55, 65;526, 269;527, 29Lebedenko, V.,524, 245Lebedev, A.,523, 234;524, 44, 55, 65;525, 9; 526, 269;527, 29Lebedev, O.,521, 71Le Bornec, Y.,521, 195Lebrun, P.,524, 44, 55, 65;526, 269;527, 29Lechner, K.,524, 199Lechtenfeld, O.,523, 178Lecomte, P.,524, 44, 55, 65;526, 269;527, 29Lecoq, P.,524, 44, 55, 65;526, 269;527, 29Le Coultre, P.,524, 44, 55, 65;526, 269;527, 29Leddy, M.J.,523, 13Leder, G.,524, 33;525, 17;526, 247, 258Ledroit, F.,525, 17Lee, B.-H.,523, 357Lee, C.,523, 205Lee, H.J.,524, 44, 55Lee, H.W.,525, 347Lee, I.Y.,523, 13Lee, J.-P.,526, 61Lee, J.H.,523, 227Lee, K.B.,523, 53;525, 205Lee, K.Y.,521, 61Lee, S.H.,524, 33;526, 247, 258Lee, W.M.,525, 211Lee, Y.K.,523, 227Lees, J.-P.,526, 34, 191, 206Leflat, A.,525, 211Lefrançois, J.,526, 34Leggett, C.,525, 211Le Goff, J.M.,524, 44, 55, 65;526, 269;527, 29Lehmann-Dronke, B.,521, 55Lehner, F.,525, 211Lehner, M.J.,524, 245Lehto, M.,526, 34, 191, 206Leibenguth, G.,526, 191, 206Leikin, E.M.,521, 171;522, 233;523, 22Leinonen, L.,525, 17Leins, A.,521, 181;523, 35;526, 221, 233Leinson, L.B.,522, 358Leißner, B.,523, 234;525, 9Leiste, R.,524, 44, 55, 65;526, 269;527, 29Leitner, R.,525, 17Lellouch, D.,521, 181;523, 35;526, 221, 233Lemaire, M.-C.,526, 34, 191, 206Lemaitre, V.,526, 191, 206Lemonne, J.,525, 17Lemrani, R.,523, 234;525, 9Lendermann, V.,523, 234;525, 9Lendvai, C.,525, 29Lenzen, G.,525, 17Lenzi, S.M.,525, 49Leonidopoulos, C.,525, 211Lepe, S.,521, 343Lepeltier, V.,525, 17Lepora, N.F.,524, 383Lerma, F.,523, 13


Leroux, K.,526, 97Leroy, O.,526, 34, 191, 206Lesiak, T.,525, 17Letessier-Selvon, A.,526, 278;527, 23Letts, J.,521, 181;523, 35;526, 221, 233Leveraro, F.,523, 53;525, 205Levin, E.,521, 233;523, 79Levinson, L.,521, 181;523, 35;526, 221, 233Levonian, S.,523, 234;525, 9Levtchenko, P.,524, 44, 55, 65;526, 269;527, 29Levy, J.-M.,526, 278;527, 23Li, C., 524, 44, 55, 65;526, 269;527, 29Li, J., 525, 211Li, Q.Z., 525, 211Li, S.-Y., 527, 85Li, X., 525, 211Li, Y., 521, 171;522, 233;523, 22Li, Z., 527, 50, 50Lian, G.,527, 50Liebig, W.,525, 17Liesenfeld, A.,524, 26Ligabue, F.,526, 34, 191, 206Ligeti, Z., 526, 345Lightfoot, P.K.,524, 245Liguori, G.,523, 53;525, 205Likhoded, S.,524, 44, 55, 65;526, 269;527, 29Liko, D., 525, 17Lillich, J., 521, 181;523, 35;526, 221, 233Lima, J.G.R.,525, 211Lin, C.H., 524, 44, 55, 65;526, 269;527, 29Lin, G.-L., 522, 257Lin, J.,526, 34, 191, 206Lin, W.T., 524, 44, 55, 65;526, 269;527, 29Lincoln, D.,525, 211Linde, F.L.,524, 44, 55, 65;526, 269;527, 29Lindner, M.,525, 130Lindstroem, M.,523, 234;525, 9Link, J.M.,523, 53;525, 205Linn, S.L.,525, 211Linnemann, J.,525, 211Linssen, L.,526, 278;527, 23Lipniacka, A.,525, 17Lipton, R.,525, 211Lissia, M.,521, 287List, B., 523, 234;525, 9Lista, L.,524, 44, 55, 65;526, 269;527, 29Lister, C.J.,525, 49Litke, A.M., 526, 34, 191, 206Littlewood, C.,521, 181;523, 35;526, 221, 233Liu, G., 525, 301Liu, J.T.,525, 157Liu, W., 527, 50Liu, Z., 522, 227Liu, Z.A., 524, 44, 55, 65;526, 269;527, 29Liubarsky, I.,524, 245Liventsev, D.,524, 33;526, 247, 258Ljubicic, A., 526, 278;527, 23Lloyd, S.L.,521, 181;523, 35;526, 221, 233

Lobodzinska, E.,523, 234;525, 9Lobodzinski, B.,523, 234;525, 9Locci, E.,526, 34, 191, 206Loebinger, F.K.,521, 181;523, 35;526, 221, 233Logashenko, I.B.,527, 161Loginov, A.,523, 234;525, 9Lohmann, W.,524, 44, 55, 65;526, 269;527, 29Loktionova, N.,523, 234;525, 9Lombardi, P.,525, 29Lombardo, F.C.,523, 317Lo Monaco, L.,525, 261London, D.,526, 97Long, G.D.,521, 181;523, 35Long, J.,526, 278;527, 23Longo, E.,524, 44, 55, 65;526, 269;527, 29Loomis, C.,526, 191, 206Lopes, J.H.,525, 17Lopez, A.M.,523, 53;525, 205Lopez, J.,526, 191, 206Lopez, J.M.,525, 17Losev, A.,522, 327Losty, M.J.,521, 181;523, 35Loukas, D.,525, 17Lourenço, C.,521, 195Loverre, P.F.,527, 173Løvhøiden, G.,523, 227Lozéa, A.,521, 1Lu, J.,521, 181;523, 35;526, 221, 233Lu, R.-S.,524, 33;526, 247, 258Lu, Y.S.,524, 44, 55, 65;526, 269;527, 29Lübelsmeyer, K.,524, 44, 55, 65;526, 269;527, 29Lubicz, V.,524, 115Lubimov, V.,523, 234;525, 9Lublinsky, M.,521, 233Luci, C.,524, 44, 55, 65;526, 269;527, 29Luckey, D.,524, 55Lucotte, A.,525, 211Lüders, S.,523, 234;525, 9Ludovici, L., 527, 173Ludwig, J.,521, 181;523, 35;526, 221, 233Lueking, L.,525, 211Lüke, D.,523, 234;525, 9Lukin, P.A.,527, 161Luksys, M.,521, 171;522, 233;523, 22Luminari, L.,524, 44, 55, 65;526, 269;527, 29Lunardon, M.,521, 165Lundstedt, C.,525, 211Lunghi, E.,521, 320Lungov, T.,521, 171;522, 233;523, 22Luo, C.,525, 211Lupi, A., 526, 278;527, 23Lüscher, R.,524, 245Lustermann, W.,524, 44, 55, 65;526, 269;527, 29Lütjens, G.,526, 34, 191, 206Lutz, P.,525, 17Lykasov, G.I.,527, 73Lynch, J.G.,526, 34, 191, 206Lyons, L.,525, 17


Lysenko, A.P.,527, 161Lyth, D.H., 524, 1, 5;526, 173Lytkin, L., 523, 234;525, 9

Ma, B.-Q.,523, 260Ma, E.,525, 101Ma, W.G.,524, 44, 55, 65;526, 269;527, 29Maalampi, J.,524, 153Macchiavelli, A.O.,523, 13Macchiolo, A.,521, 181;523, 35;526, 221, 233Macciotta, P.,521, 195Mac Cormick, M.,521, 195Machefert, F.,526, 191, 206Machleidt, R.,524, 93Machulin, I.,525, 29Maciel, A.K.A., 525, 211MacKay, N.J.,522, 335;524, 401MacNaughton, J.,524, 33;525, 17;526, 247, 258Macpherson, A.,521, 181;523, 35;526, 221, 233Madaras, R.J.,525, 211Mader, W.,521, 181;523, 35;526, 221, 233Maekawa, N.,521, 42Maggi, G.,526, 34, 191, 206Maggi, M.,526, 34, 191, 206Magnin, J.,523, 29, 53;525, 205Mahlke-Krüger, H.,523, 234;525, 9Majka, Z.,523, 227Major, J.,525, 41Makeev, A.,523, 227Maklioueva, I.,527, 173Malden, N.,523, 234;525, 9Maleev, V.P.,521, 171;522, 233;523, 22Malek, A.,525, 17Maley, P.,526, 191, 206Malgeri, L.,524, 44, 55, 65;526, 269;527, 29Malik, R.P.,521, 409Malinin, A., 524, 44, 55, 65;526, 269;527, 29Malinovski, E.,523, 234;525, 9Malinovski, I.,523, 234;525, 9Maltezos, S.,525, 17Maltman, K.,522, 245Malvezzi, S.,523, 53;525, 29, 205Malyshev, V.L.,525, 211Maña, C.,524, 44, 55, 65;526, 269;527, 29Manankov, V.,525, 211Mandl, F.,525, 17Maneira, J.,525, 29Mangeol, D.,524, 44, 55, 65;526, 269;527, 29Mannarelli, M.,524, 144Männer, W.,526, 34, 191, 206Mannert, C.,526, 34, 191, 206Manno, I.,525, 29Mannocchi, G.,526, 34, 191, 206Mans, J.,524, 44, 55, 65;526, 269;527, 29Manuzio, G.,525, 29Mao, C.,521, 171;522, 233;523, 22Mao, D.,521, 171;522, 233;523, 22Mao, H.S.,525, 211

Mao, Z.,521, 171;522, 233;523, 22Maracek, R.,523, 234;525, 9Marage, P.,523, 234;525, 9Marcellini, S.,521, 181;523, 35;526, 221, 233Marcello, S.,522, 16;527, 39Marchant, T.E.,521, 181;523, 35;526, 221, 233Marchetti, P.A.,524, 199Marchionni, A.,526, 278;527, 23Marco, J.,525, 17Marco, R.,525, 17Marechal, B.,525, 17Margoni, M.,525, 17Marin, J.-C.,525, 17Marinelli, N., 526, 34, 191, 206Mariotti, C.,525, 17Markou, A.,525, 17Markou, C.,526, 34, 191, 206Marks, J.,523, 234;525, 9Marques, J.,523, 273Marshall, R.,523, 234;525, 9Marshall, T.,525, 211Martelli, F.,526, 278;527, 23Martemianov, A.,525, 29Martin, A.D.,524, 107Martin, A.J.,521, 181;523, 35;526, 221, 233Martin, F.,526, 191, 206Martin, J.P.,521, 181;523, 35;524, 44, 55, 65;526, 221, 233, 269;

527, 29Martin, M.I., 525, 211Martinelli, G.,524, 115Martinez, G.,521, 181;523, 35;526, 221, 233Martinez, M.,526, 34, 191, 206Martínez-Pinedo, G.,522, 240;525, 49Martinez-Rivero, C.,525, 17Martyn, H.-U.,523, 234;525, 9Martyniak, J.,523, 234;525, 9Maru, N.,522, 117Marucho, M.,523, 102Marwinski, S.,522, 16Marzano, F.,524, 44, 55, 65;526, 269;527, 29Marzari-Chiesa, A.,521, 195Masciocchi, S.,522, 233Masera, M.,521, 195Masetti, F.,525, 29Masetti, G.,521, 181;523, 35;526, 221, 233Mashimo, T.,521, 181;523, 35;526, 221, 233Masik, J.,525, 17Maslov, N.,525, 41Masoni, A.,521, 195Massafferri, A.,523, 53;525, 205Mastroyiannopoulos, N.,525, 17Masuda, T.,521, 376Matchev, K.T.,521, 308Mateos, T.,524, 170Mathew, P.,521, 171;522, 233;523, 22Mato, P.,526, 34, 191, 206Matorras, F.,525, 17Matsubara, T.,524, 33;526, 247, 258


Matsuda, Y.,527, 43Matsumoto, S.,524, 33;526, 247, 258Matsumoto, T.,524, 33;526, 247, 258Matsuzaki, T.,527, 43Matteuzzi, C.,525, 17Matthews, J.L.,523, 1Mattiello, S.,521, 33Mättig, P.,521, 181;523, 35;526, 221, 233Mattson, M.,521, 171;522, 233;523, 22Matveev, V.,521, 171;522, 233;523, 22Mauritz, K.M.,525, 211Mauro, S.,522, 16Maxfield, S.J.,523, 234;525, 9Mayorov, A.A.,525, 211Mazumdar, K.,524, 44, 55, 65;526, 269;527, 29Mazzitelli, F.D.,523, 317Mazzucato, F.,525, 17Mazzucato, M.,525, 17Mazzucato, U.,525, 29McArthur, I.N.,522, 320McBreen, E.,523, 227McCarthy, R.,525, 211McCarty, K.,525, 29McCliment, E.,521, 171;522, 233;523, 22McDonald, W.J.,521, 181;523, 35;526, 221, 233McKenna, J.,521, 181;523, 35;526, 221, 233McMahon, T.,525, 211McMahon, T.J.,521, 181;523, 35;526, 221, 233McMillan, J.E.,524, 245McNamara, P.A.,526, 34, 191, 206McNeil, R.R.,524, 44, 55, 65;526, 269;527, 29Mc Nulty, R.,525, 17McPherson, R.A.,521, 181;523, 35;526, 221, 233Méchain, X.,526, 278;527, 23Medcalf, T.,526, 34, 191, 206Meer, D.,523, 234;525, 9Meessen, P.,525, 322;526, 144Mehta, A.,523, 234;525, 9Meier, K.,523, 234;525, 9Meijers, F.,521, 181;523, 35;526, 221, 233Meinhard, H.,527, 173Melanson, H.L.,525, 211Mele, S.,524, 44, 55, 65;526, 269;527, 29Meljanac, S.,526, 149Melzer, O.,527, 173Menasce, D.,523, 53;525, 205Mendes, S.O.,522, 1Mendez, H.,523, 53;525, 205Mendez, L.,523, 53;525, 205Mendez-Lorenzo, P.,521, 181;523, 35;526, 221, 233Mendiburu, J.-P.,526, 278;527, 23Menezes, J.,521, 418Menges, W.,521, 181;523, 35;526, 221, 233Merino, G.,526, 34, 191, 206Merkel, H.,524, 26Merkin, M., 525, 211Merle, E.,526, 34, 191, 206Merlo, M.M., 523, 53;525, 205

Merola, L.,524, 44, 55, 65;526, 269;527, 29Meroni, C.,525, 17Meroni, E.,525, 29Merritt, F.S.,521, 181;523, 35;526, 221, 233Merritt, K.W., 525, 211Mes, H.,521, 181;523, 35;526, 221, 233Meschini, M.,524, 44, 55, 65;526, 269;527, 29Messina, M.,527, 173Messineo, A.,526, 34, 191, 206Metag, V.,526, 287Metzger, W.J.,524, 44, 55, 65;526, 269;527, 29Meyer, A.B.,523, 234;525, 9Meyer, H.,523, 234;525, 9Meyer, J.,523, 234;525, 9; 526, 79Meyer, J.-P.,526, 278;527, 23Meyer, P.-O.,523, 234;525, 9Meyer, W.,522, 16Meyer, W.T.,525, 17Mezzadri, M.,523, 53;525, 205Mezzetto, M.,526, 278;527, 23Miao, C.,525, 211Michael, C.,525, 360Michel, B.,526, 34, 191, 206Michel, P.,522, 16Michelini, A., 521, 181;523, 35;526, 221, 233Miettinen, H.,525, 211Migliore, E.,525, 17Migliozzi, P.,527, 173Mihalcea, D.,525, 211Mihara, S.,521, 181;523, 35;526, 221, 233Mihul, A., 524, 44, 55, 65;526, 269;527, 29Mikami, Y., 524, 33;526, 247, 258Mikenberg, G.,521, 181;523, 35;526, 221, 233Mikhailov, K.Yu., 527, 161Mikirtychiants, S.,521, 158Mikocki, S.,523, 234;525, 9Milazzo, L.,523, 53;525, 205Milcent, H.,524, 44, 55, 65;526, 269;527, 29Miller, D.J.,521, 181;523, 35;526, 221, 233Miller, J.P.,527, 161Miller, M.B., 522, 222Milstead, D.,523, 234;525, 9Milstein, A.I., 527, 161Minakata, H.,526, 335Minard, M.-N.,526, 34, 191, 206Minemura, T.,522, 227Mintchev, M.,524, 363Miquel, R.,526, 34, 191, 206Mir, Ll.M., 526, 34, 191, 206Mirabelli, G.,524, 44, 55, 65;526, 269;527, 29Miramonti, L.,525, 29Miranda, O.G.,521, 299Mirles, A., 523, 53Mironov, A., 524, 217Mishima, S.,521, 252Mishra, C.S.,525, 211Mishra, S.R.,526, 278;527, 23Misiejuk, A., 526, 34, 191, 206


Mitaroff, W., 525, 17Mitchell, R.,523, 53;525, 205Miyabayashi, K.,524, 33;526, 247, 258Miyake, H.,524, 33;526, 247, 258Miyanishi, M.,527, 173Miyata, H.,524, 33;526, 247, 258Mizoguchi, S.,523, 351Mjoernmark, U.,525, 17Mkrtchyan, T.,523, 234;525, 9Mnich, J.,524, 44, 55, 65;526, 269;527, 29Moa, T.,525, 17Moch, M.,525, 17Moed, S.,521, 181;523, 35;526, 221, 233Moenig, K.,525, 17Mohanty, G.B.,524, 44, 55, 65;526, 269;527, 29Mohapatra, R.N.,522, 287Mohr, R.,523, 234;525, 9Mohr, W.,521, 181;523, 35;526, 221, 233Mohrdieck, S.,523, 234;525, 9Moinester, M.A.,521, 171;522, 233;523, 22Mokhov, N.,525, 211Molchanov, V.V.,521, 171;522, 233;523, 22Möller, K., 522, 16Molnár, J.,521, 165Moloney, G.R.,524, 33;526, 247, 258Mondal, N.K.,525, 211Mondragon, M.N.,523, 234;525, 9Monge, R.,525, 17Montecchia, F.,527, 182Monteil, S.,526, 34, 191, 206Monteiro, D.S.,527, 187Montenegro, J.,525, 17Monteno, M.,521, 195Montgomery, H.E.,525, 211Montiel, E.,523, 53;525, 205Montret, J.-C.,526, 34Montvay, I.,527, 155Monzani, M.E.,525, 29Moore, R.W.,525, 211Moorhead, G.F.,526, 278;527, 23Moraes, D.,525, 17Moreau, F.,523, 234;525, 9Morelos, A.,521, 171;522, 233;523, 22Moreno, S.,525, 17Morettini, P.,525, 17Mori, S.,524, 33;526, 247, 258Mori, T., 521, 181;523, 35;524, 33;526, 221, 233, 247, 258Moriyama, S.,522, 177Moro, R.,521, 165Moroi, T., 522, 215;525, 121Moroni, L., 523, 53;525, 205Morozov, A.,523, 234;524, 217;525, 9Morris, C.L.,523, 1Morris, J.V.,523, 234;525, 9Morsch, H.P.,522, 16Mörtel, H.,522, 16Moser, H.-G.,526, 34, 191, 206Mostafa, M.,525, 211

Motobayashi, T.,522, 227Motovilov, A.K., 522, 222Motyka, L.,524, 107, 273Moutoussi, A.,526, 34, 191, 206Muanza, G.S.,524, 44, 55, 65;526, 269;527, 29Muciaccia, M.T.,527, 173Mück, W.,522, 139Mueller, A.H.,523, 243Mueller, U.,525, 17Muenich, K.,525, 17Muijs, A.J.M.,524, 44, 55, 65;526, 269;527, 29Mukherjee, A.,526, 295Mukhin, V.A., 521, 171Mulders, M.,525, 17Müller, A.-S.,526, 191, 206Müller, K., 523, 234;525, 9Müller, U., 524, 26Multamäki, T.,526, 9Mundim, L.,525, 17Murakami, A.,526, 247, 258Murakami, B.,526, 157Murín, P.,523, 234;525, 9Murray, M.,523, 227Murray, W.,525, 17Murtas, F.,526, 34, 191, 206Murtas, G.P.,526, 34, 191, 206Muryn, B.,525, 17Musicar, B.,524, 44, 55, 65;526, 269;527, 29Musico, P.,525, 29Musiri, S.,524, 192Musso, A.,521, 195Musy, M.,524, 44, 55, 65;526, 269;527, 29Mutter, A.,521, 181;523, 35;526, 221, 233Myatt, G.,525, 17Myklebust, T.,525, 17

NA50 Collaboration,521, 195Nagai, K.,521, 181;523, 35;526, 221, 233Nagamine, K.,527, 43Nagamine, T.,524, 33;526, 247, 258Nagasaka, Y.,524, 33;526, 247, 258Nagashima, Y.,524, 33;526, 247, 258Nagovizin, V.,523, 234;525, 9Nagy, E.,525, 211Nagy, S.,524, 44, 55, 65;526, 269;527, 29Nakadaira, T.,524, 33;526, 247, 258Nakamura, I.,521, 181;523, 35;526, 221, 233Nakamura, K.,527, 173Nakamura, M.,527, 173Nakamura, S.N.,527, 43Nakano, E.,524, 33;526, 247, 258Nakano, T.,527, 173Nakao, M.,524, 33;526, 247, 258Nakayama, N.,521, 400Nam, J.W.,524, 33;526, 247, 258Nang, F.,525, 211Nanopoulos, D.V.,525, 308Napolitano, M.,524, 44, 55, 65;526, 269;527, 29


Narain, M.,525, 211Narasimham, V.S.,525, 211Nardulli, G.,524, 144Narison, S.,522, 266;526, 414Narita, K.,527, 173Naroska, B.,523, 234;525, 9Nassiakou, M.,525, 17Natale, S.,524, 65;526, 269;527, 29Natkaniec, Z.,524, 33;526, 247, 258Natowitz, J.,523, 227Natowitz, J.B.,521, 165Naumann, J.,523, 234;525, 9Naumann, L.,522, 16Naumann, N.A.,525, 211Naumann, Th.,523, 234;525, 9Naumov, D.,526, 278;527, 23Navarra, F.S.,521, 1Navarria, F.,525, 17Nawrocki, K.,525, 17Neal, H.A.,521, 181;523, 35;525, 211;526, 221, 233Nebbia, G.,521, 165Necco, S.,523, 135Nédélec, P.,526, 278;527, 23Neder, H.,525, 29Nefedov, Yu.,526, 278;527, 23Negret, J.P.,525, 211Negroni, S.,525, 211Negus, P.,526, 34, 191, 206Nehring, M.,523, 53;525, 205Neichi, K.,524, 33;526, 247, 258Nekipelov, M.,521, 158Nellen, G.,523, 234;525, 9Nelson, K.D.,521, 171;522, 233;523, 22Nemecek, S.,525, 17Nemes, M.C.,523, 293Nemitkin, A.V.,521, 171;522, 233;523, 22Neoustroev, P.V.,521, 171;522, 233;523, 22Nersessian, A.,522, 345Nessi-Tedaldi, F.,524, 44, 55, 65;526, 269;527, 29Nesterenko, I.N.,527, 161Neuhausen, R.,524, 26Newman, H.,524, 44, 55, 65;526, 269;527, 29Newman, P.R.,523, 234;525, 9Newsom, C.,521, 171;522, 233;523, 22Ngac, A.,526, 34, 191, 206Nguyen-Mau, C.,526, 278;527, 23Nicholls, T.C.,523, 234;525, 9Nicolaidou, R.,525, 17Niebergall, F.,523, 234;525, 9Niebuhr, C.,523, 234;525, 9Niedermeier, L.,525, 29Nielsen, B.S.,523, 227Nielsen, J.,526, 34, 191, 206Nielsen, M.,521, 1Niemi, A.J.,525, 195Niessen, T.,524, 44, 55, 65;526, 269;527, 29Niezurawski, P.,525, 17Niizeki, T., 521, 153

Nikolenko, M.,525, 17Nikoli c, H.,527, 119Nilles, H.P.,522, 304Nilov, A.P.,521, 171;522, 233;523, 22Nilsson, B.S.,526, 34, 191, 206Nisati, A.,524, 44, 55, 65;526, 269;527, 29Nishida, S.,524, 33;526, 247, 258Nishimori, N.,521, 153Nisius, R.,521, 181;523, 35;526, 221, 233Nitoh, O.,524, 33;526, 247, 258Niu, K., 527, 173Niwa, K., 527, 173Nix, O., 523, 234;525, 9Noguchi, S.,524, 33;526, 247, 258Noguera, S.,521, 225Nojiri, S., 521, 87;523, 165NOMAD Collaboration,526, 278;527, 23Nonaka, N.,527, 173Normand, Ch.,524, 26Norton, P.R.,526, 34, 191, 206Novozhilov, V.,522, 49Novozhilov, Yu.,522, 49Nowacki, F.,522, 240Nowak, G.,523, 234;525, 9Nowak, H.,524, 44, 55, 65;526, 269;527, 29Nowak-Szczepaniak, D.,521, 364Nowell, J.,526, 191, 206Nozaki, T.,526, 247, 258Nunnemann, T.,525, 211Nurmagambetov, A.J.,524, 185Nurushev, S.B.,521, 171;522, 233;523, 22Nussinov, S.,526, 137Nuzzo, S.,526, 34, 191, 206Nyakó, B.M.,521, 165Nygren, A.,525, 17

Oberauer, L.,525, 29Oblakowska-Mucha, A.,525, 17Obolensky, M.,525, 29Obraztsov, V.,525, 17Ocherashvili, A.,521, 171;522, 233;523, 22Odintsov, S.D.,521, 87;523, 165Ofierzynski, R.,524, 44, 55, 65;526, 269;527, 29Ogawa, S.,524, 33;526, 247, 258;527, 173Oguri, V.,525, 211Oh, A.,521, 181;523, 35;526, 221, 233Ohlsson, T.,522, 280Ohnishi, T.,521, 153Ohshima, T.,524, 33;526, 247, 258Ohta, N.,527, 131Oi, M., 525, 255Okabe, T.,524, 33;526, 247, 258Okamura, H.,521, 153Okhapkin, V.S.,527, 161Okpara, A.,521, 181;523, 35;526, 221, 233Okuno, S.,524, 33;526, 247, 258Okusawa, T.,527, 173Olaya, D.,523, 53;525, 205


Olchanski, K.,523, 227Oldeman, R.G.C.,527, 173O’Leary, C.D.,525, 49Olive, K.A., 522, 304;525, 308Olivier, B., 525, 211Olness, J.,523, 227Olsen, S.L.,524, 33;526, 247, 258Olshevski, A.,525, 17Olsson, J.E.,523, 234;525, 9Onderwater, C.J.G.,523, 6O’Neale, S.W.,521, 181;523, 35;526, 221, 233O’Neil, D., 525, 211Onel, Y.,521, 171;522, 233;523, 22Onengüt, G.,527, 173Ono, L.K.,527, 187Onofre, A.,525, 17OPAL Collaboration,521, 181;523, 35;526, 221, 233Opher, R.,522, 1Orava, R.,525, 17Oreglia, M.J.,521, 181;523, 35;526, 221, 233O’Reilly, B., 523, 53;525, 205Orellana, F.,521, 22, 29Orestano, D.,526, 278;527, 23Organtini, G.,524, 44, 55, 65;526, 269;527, 29Orito, S.,521, 181;523, 35;526, 221, 233Ortica, F.,525, 29Oset, E.,527, 69, 99Osetrov, S.B.,522, 222O’Shea, V.,526, 34, 191, 206Oshima, N.,525, 211Osterberg, K.,525, 17Ostrowicz, W.,524, 33;526, 247, 258Otsu, H.,521, 153Ouerdane, D.,523, 227Ouraou, A.,525, 17Ouyang, Q.,526, 34, 206Overhauser, A.W.,526, 355Owens, R.O.,526, 287Oyanguren, A.,525, 17Ozaki, H.,524, 33;526, 247, 258Ozel, E.,521, 171;522, 233;523, 22Ozerov, D.,523, 234;525, 9Ozkorucuklu, S.,521, 171;522, 233;523, 22Özpineci, A.,527, 193

Pacheco, A.,526, 34, 191, 206Padley, P.,525, 211Paganoni, M.,525, 17Pahl, C.,521, 181;523, 35;526, 221, 233Paiano, S.,525, 17Pajares, C.,527, 92Pak, D.G.,525, 347Pakhlov, P.,524, 33;526, 247, 258Palacios, J.P.,525, 17Palka, H.,524, 33;525, 17;526, 247, 258Palla, F.,526, 34, 191, 206Palladino, V.,527, 173Pallavicini, M.,525, 29

Pallin, D.,526, 34, 191, 206Palomares, C.,524, 44, 55, 65;526, 269;527, 29Pan, L.J.,525, 211Pan, Y.B.,526, 34, 191, 206Panassik, V.,523, 234;525, 9Pandoulas, D.,524, 44, 55, 65;526, 269;527, 29Paneque, D.,526, 191, 206Panman, J.,527, 173Pantea, D.,523, 53;525, 205Paolucci, P.,524, 44, 55, 65;526, 269;527, 29Papadopoulos, I.M.,527, 173Papadopoulou, Th.D.,525, 17Papageorgiou, K.,525, 211Pape, L.,525, 17Papp, L.,525, 29Para, A.,525, 211Paramatti, R.,524, 44, 55, 65;526, 269;527, 29Parashar, N.,525, 211Paris, A.,523, 53;525, 205Park, C.S.,524, 33;526, 247, 258Park, C.W.,524, 33;526, 247, 258Park, H.,523, 53;524, 33;525, 205;526, 247, 258Park, I.Y.,524, 185Park, K.S.,524, 33;526, 247, 258Park, S.C.,523, 161Parkes, C.,525, 17Parodi, F.,525, 17Parrini, G.,526, 34, 191, 206Partridge, R.,525, 211Parua, N.,525, 211Parzefall, U.,525, 17Päs, H.,522, 315Pascaud, C.,523, 234;525, 9Paschos, E.A.,525, 63Pascoli, S.,524, 319Pascolo, J.M.,526, 191, 206Pashnev, A.,523, 338Passalacqua, L.,526, 34, 191, 206Passaleva, G.,524, 44, 55, 65;526, 269;527, 29Passeri, A.,525, 17Passon, O.,525, 17Pastore, F.,526, 278;527, 23Pasyuk, E.A.,523, 1Pásztor, G.,521, 181;523, 35;526, 221, 233Patel, G.D.,523, 234;525, 9Pater, J.R.,521, 181;523, 35;526, 221, 233Paterno, M.,525, 211Patra, S.K.,523, 67Patricelli, S.,524, 44, 55, 65;526, 269;527, 29Patrick, G.N.,521, 181;523, 35;526, 221, 233Patwa, A.,525, 211Paul, E.S.,523, 13Paul, T.,524, 44, 55, 65;526, 269;527, 29Pauluzzi, M.,524, 44, 55, 65;526, 269;527, 29Paus, C.,524, 44, 55, 65;526, 269;527, 29Pauss, F.,524, 44, 55, 65;526, 269;527, 29Pavón, D.,521, 133Pawlik, B.,525, 211


Payre, P.,526, 34, 191, 206Peak, C.D.,524, 245Peak, L.S.,524, 33;526, 247, 258, 278;527, 23Pearson, M.R.,526, 34, 191, 206Pedace, M.,524, 44, 55, 65;526, 269;527, 29Pedrini, D.,523, 53;525, 205Peez, M.,523, 234;525, 9Pellegrino, A.R.,523, 6Peloso, M.,522, 304Peña-Garay, C.,521, 299;527, 199Pennacchio, E.,526, 278;527, 23Pennington, M.R.,521, 15Pensotti, S.,524, 44, 55, 65;526, 269;527, 29Penzo, A.,521, 171;522, 233;523, 22Pepe, I.M.,523, 53;525, 205Pepe-Altarelli, M.,526, 34, 191, 206Peralta, L.,525, 17Perasso, L.,525, 29Perepelitsa, V.,525, 17Perevalova, I.,523, 249Pérez, A.,522, 358Perez, E.,523, 234;525, 9Perez, P.,526, 34, 191, 206Pérez-Lorenzana, A.,522, 297Perkins, J.,525, 211Perret, P.,526, 34, 191, 206Perret-Gallix, D.,524, 44, 55, 65;526, 269;527, 29Perrotta, A.,525, 17Perroud, J.-P.,524, 33;526, 247, 258Pesen, E.,527, 173Pessard, H.,526, 278;527, 23Pestieau, J.,524, 395Petcov, S.T.,524, 319Peters, M.,524, 33;526, 247, 258Peters, O.,525, 211Petersen, B.,524, 44, 55, 65;526, 269;527, 29Petiau, P.,521, 195Petitjean, Th.,524, 26Petrenko, S.V.,521, 171;522, 233;523, 22Pétroff, P.,525, 211Petrolini, A.,525, 17Petrov, A.Yu.,521, 119Petrus, A.,521, 158Petti, R.,526, 278;527, 23Philipsen, O.,521, 273Phillips, J.P.,523, 234;525, 9Piccolo, D.,524, 44, 55, 65;526, 269;527, 29Piccotti, A.,521, 195Piechocki, W.,526, 127Piedra, J.,525, 17Piegaia, R.,525, 211Pierella, F.,524, 44, 55, 65;526, 269;527, 29Pieri, L.,525, 17Pierre, F.,525, 17Pietrzyk, B.,526, 34, 191, 206Piilonen, L.E.,524, 33;526, 247, 258Pilcher, J.E.,521, 181;523, 35;526, 221, 233Pilo, L., 523, 347

Pimenta, M.,525, 17Pinfold, J.,521, 181;523, 35;526, 221, 233Pioppi, M.,524, 44, 55, 65;526, 269;527, 29Piotto, E.,525, 17Pire, B.,523, 265Pirner, H.-J.,526, 79Pirner, H.J.,521, 279Piroué, P.A.,524, 44, 55, 65;526, 269;527, 29Pisarski, R.D.,525, 95Pistolesi, E.,524, 44, 55, 65;526, 269;527, 29Pitzl, D.,523, 234;525, 9Pizzi, J.R.,521, 195Placci, A.,526, 278;527, 23Płaczek, W.,523, 117Plane, D.E.,521, 181;523, 35;526, 221, 233Płaneta, R.,523, 227Plettner, Ch.,522, 16Plyaskin, V.,524, 44, 55, 65;526, 269;527, 29Pocar, A.,525, 29Podobnik, T.,525, 17Pogodin, P.,521, 171;522, 233;523, 22Pohl, M.,524, 44, 55, 65;526, 269;527, 29Poireau, V.,525, 17Poizat, J.-C.,525, 41Pojidaev, V.,524, 44, 55, 65;526, 269;527, 29Pol, M.E.,525, 17Polesello, G.,526, 278;527, 23Poli, B.,521, 181;523, 35;526, 221, 233Pollmann, D.,526, 278;527, 23Polok, G.,525, 17Polok, J.,521, 181;523, 35;526, 221, 233Polunin, A.A.,527, 161Polyarush, A.,526, 278;527, 23Pontoglio, C.,523, 53;525, 205Pooth, O.,521, 181;523, 35;526, 221, 233Pope, B.G.,525, 211Popkov, E.,525, 211Popov, A.D.,523, 178Popov, A.S.,527, 161Popov, B.,526, 278;527, 23Porfyriadis, P.I.,525, 63Poropat, P.,525, 17Pöschl, R.,523, 234;525, 9Pospischil, Th.,524, 26Postema, H.,524, 55Potachnikova, I.,523, 234;525, 9Pothier, J.,524, 44, 55, 65;526, 269;527, 29Potokar, M.,524, 26Potylitsin, A.,525, 41Poulose, P.,525, 71Poulsen, C.,526, 278;527, 23Povh, B.,522, 233;523, 234;525, 9Pozdniakov, V.,525, 17Prakash, M.,525, 249Prange, G.,526, 34, 191, 206Prasanna, A.R.,526, 27Prebys, E.,526, 247Prelz, F.,523, 53;525, 205


Prešnajder, P.,526, 132;527, 149Prete, G.,521, 165Prino, F.,521, 195Privitera, P.,525, 17Procario, M.,521, 171;522, 233;523, 22Prokofiev, D.,524, 44, 55, 65;526, 269;527, 29Prokofiev, D.O.,524, 44, 55, 65;526, 269;527, 29Prosper, H.B.,525, 211Prosperi, D.,527, 182Protopopescu, S.,525, 211Prutskoi, V.A.,521, 171;522, 233;523, 22Przybycien, M.B.,525, 211Przysiezniak, H.,526, 191, 206Puddu, G.,521, 195Pukhaeva, N.,525, 17Pullia, A.,525, 17Purlatz, T.A.,527, 161Putz, J.,526, 191, 206Putzer, A.,526, 34, 191, 206

Qian, J.,525, 211Qiao, C.-F.,523, 111Qiu, J.,523, 88;525, 265Quadt, A.,521, 181;523, 35;526, 221, 233Quartieri, J.,524, 44, 55, 65;526, 269;527, 29Quast, G.,526, 34, 191, 206Quenby, J.J.,524, 245Quinones, J.,523, 53;525, 205Quintans, C.,521, 195Quyang, Q.,526, 191

Rabbertz, K.,521, 181;523, 35, 234;525, 9; 526, 221, 233Rädel, G.,523, 234;525, 9Radicioni, E.,527, 173Raghavan, R.S.,525, 29Ragusa, F.,526, 34, 191, 206Rahal-Callot, G.,524, 44, 55, 65;526, 269;527, 29Rahaman, M.A.,524, 44, 55, 65;526, 269;527, 29Rahimi, A.,523, 53;525, 205Raics, P.,524, 44, 55, 65;526, 269;527, 29Raine, C.,526, 34Raja, N.,524, 44, 55, 65;526, 269;527, 29Raja, R.,525, 211Rajagopalan, S.,525, 211Rakow, P.E.L.,522, 194Ramallo, A.V.,525, 337Ramberg, E.,521, 171;522, 233;523, 22;525, 211Ramelli, R.,524, 44, 55, 65;526, 269;527, 29Ramello, L.,521, 195Rames, J.,525, 17Rami, F.,523, 227Ramirez, J.E.,523, 53;525, 205Ramler, L.,525, 17Ramos, A.,527, 99Ramos, S.,521, 195Rancoita, P.G.,524, 44, 55, 65;526, 269;527, 29Rander, J.,526, 34, 191, 206

Ranieri, A.,526, 34, 191, 206Ranieri, R.,524, 44, 55, 65;526, 269;527, 29Ranjard, F.,526, 34, 191, 206Ranucci, G.,525, 29Rapidis, P.A.,525, 211Rapp, R.,523, 60Rappazzo, G.F.,521, 171;522, 233;523, 22Rashba, T.I.,521, 299Raso, G.,526, 34, 191, 206Raspereza, A.,524, 44, 55, 65;526, 269;527, 29Rathmann, F.,524, 303Rathouit, P.,527, 23Rato Mendes, P.,521, 195Ratti, S.P.,523, 53;525, 205Ratz, M.,525, 130Rau, W.,525, 29Rauschenberger, J.,523, 234;525, 9Razeto, A.,525, 29Razis, P.,524, 44, 55, 65;526, 269;527, 29Razmyslovich, B.V.,521, 171;522, 233;523, 22Read, A.,525, 17Reay, N.W.,525, 211Rebecchi, P.,525, 17Rebhan, A.,523, 143Recksiegel, S.,525, 81Regan, P.H.,525, 49Rehn, J.,525, 17Reid, D.,525, 17Reimer, P.,523, 234;525, 9Reinhardt, R.,525, 17Reisert, B.,523, 234;525, 9Reiter, P.,525, 49Rembser, C.,521, 181;523, 35;526, 221, 233Remillieux, J.,525, 41Ren, D.,524, 44, 55, 65;526, 269;527, 29Renardy, J.-F.,526, 34, 191, 206Renk, B.,526, 34, 191, 206Renkel, P.,521, 181;523, 35;526, 221, 233Renton, P.,525, 17Rescigno, M.,524, 44, 55, 65;526, 269;527, 29Resconi, E.,525, 29Reucroft, S.,524, 44, 55, 65;525, 211;526, 269;527, 29Reuter, M.,527, 9Reyes, M.,523, 53;525, 205Reyna, D.,523, 234;525, 9Riccardi, C.,523, 53;525, 205Riccati, L.,521, 195Ricci, B.,526, 186Ricciardi, S.,527, 173Richard, F.,525, 17Richter, M.,522, 16;523, 304Rick, H.,521, 181;523, 35;526, 221, 233Rico, J.,526, 278;527, 23Ridel, M.,525, 211Ridky, J.,525, 17Riemann, P.,526, 278Riemann, S.,524, 44, 55, 65;526, 269;527, 29Righini, P.,527, 173


Rijssenbeek, M.,525, 211Riles, K.,524, 44, 55, 65;526, 269;527, 29Rindani, S.D.,525, 71Ringwald, A.,525, 135Ripp-Baudot, I.,525, 17Risler, C.,523, 234;525, 9Rivelles, V.O.,521, 119Rivera, C.,523, 53;525, 205Rivers, R.J.,523, 317Rizatdinova, F.,525, 211Rizvi, E.,523, 234;525, 9Rizzi, V., 521, 165Roberts, B.L.,527, 161Robertson, N.A.,526, 34, 191, 206Robmann, P.,523, 234;525, 9Rockwell, T.,525, 211Roco, M.,525, 211Roda, C.,526, 278;527, 23Roderburg, E.,522, 16Rodning, N.,521, 181;523, 35;526, 221, 233Rodriguez, D.,525, 17Rodriguez, J.L.,526, 247, 258Roe, B.P.,524, 44, 55, 65;526, 269;527, 29Rogalyov, R.N.,521, 243Rohe, D.,524, 26Röhrich, D.,523, 227Rolandi, L.,526, 34, 191, 206Romana, A.,521, 195Romano, G.,527, 173Romero, A.,525, 17Romero, L.,524, 44, 55, 65;526, 269;527, 29Ronchese, P.,525, 17Roney, J.M.,521, 181;523, 35;526, 221, 233Roosen, R.,523, 234;525, 9Root, N.,524, 33;526, 247, 258Root, N.I.,527, 161Rosa, G.,527, 173Rosati, S.,521, 181;523, 35;526, 221, 233Rosca, A.,524, 44, 55, 65;526, 269;527, 29Roscoe, K.,521, 181;523, 35;526, 221, 233Rosenberg, E.,525, 17Rosenbleck, C.,524, 44, 55, 65;526, 269;527, 29Rosier-Lees, S.,524, 44, 55, 65;526, 269;527, 29Rosner, G.,524, 26Rosowsky, A.,526, 191, 206Rostovtsev, A.,523, 234;525, 9Roth, S.,524, 44, 55, 65;526, 269;527, 29Rothberg, J.,526, 34, 191, 206Roudeau, P.,525, 17Rougé, A.,526, 34, 191, 206Roussarie, A.,526, 34Rousseau, D.,526, 34, 191, 206Roux, B.,524, 44, 55, 65;526, 269;527, 29Rovelli, T.,525, 17Rovere, M.,523, 53;525, 205Rowley, N.,521, 165Roy, D.P.,525, 101Roy, S.,525, 101

Royon, C.,525, 211Rozanov, A.,527, 173Rozanska, M.,524, 33;526, 247, 258Rozen, Y.,521, 181;523, 35;526, 221, 233Ruban, A.A.,527, 161Rubbia, A.,526, 278;527, 23Rubinov, P.,525, 211Rubio, B.,525, 49Rubio, J.A.,524, 44, 55, 65;526, 269;527, 29Ruchti, R.,525, 211Rud, V.I.,521, 171;522, 233;523, 22Rudolph, G.,526, 34, 191, 206Ruggieri, F.,526, 34, 191, 206Ruggiero, G.,524, 44, 55, 65;526, 269;527, 29Ruhlmann-Kleider, V.,525, 17Ruiz, H.,526, 34, 191, 206Rumpf, M.,526, 34Runge, K.,521, 181;523, 35;526, 221, 233Rusakov, S.,523, 234;525, 9Russ, J.,521, 171;522, 233;523, 22Rust, D.R.,521, 181;523, 35;526, 221, 233Rutherfoord, J.,525, 211Rutherford, S.A.,526, 191, 206Ryabtchikov, D.,525, 17Rybicki, K., 523, 234;524, 33;525, 9; 526, 258Ryckebusch, J.,527, 62Rykaczewski, H.,524, 44, 55, 65;526, 269;527, 29Ryskin, M.G.,524, 107Ryskulov, N.M.,527, 161Ryuko, J.,524, 33;526, 247, 258

Sabelnikov, A.,525, 29Sabirov, B.M.,525, 211Sachrajda, C.T.,525, 360Sachs, K.,521, 181;523, 35;526, 221, 233Sadovsky, A.,525, 17Saeki, T.,521, 181;523, 35;526, 221, 233Sagawa, H.,524, 33;526, 247, 258Saha, B.,524, 252Sahr, O.,521, 181;523, 35;526, 221, 233Sahu, P.K.,526, 19Saidi, E.H.,523, 191Saito, A.,522, 227Saitta, B.,527, 173Sajot, G.,525, 211Sakai, H.,521, 153Sakai, Y.,524, 33;526, 247, 258Sakamoto, H.,524, 33;526, 247, 258Sakamoto, N.,521, 153Sakharov, A.,524, 44, 55, 65;526, 269;527, 29Sakurai, H.,522, 227Sala, S.,523, 53;525, 205Salgado, C.A.,521, 211Salicio, J.,524, 44, 55, 65;526, 269;527, 29Salmi, L.,525, 17Salt, J.,525, 17Salvatore, F.,526, 278;527, 23Salvo, C.,525, 29


Samanta, A.,526, 111Samset, B.H.,523, 227Sanchez, E.,524, 44, 55, 65;526, 269;527, 29Sánchez-Hernández, A.,523, 53;525, 205Sanda, A.I.,525, 240Sander, H.-G.,526, 34, 191, 206Sanders, M.P.,524, 44, 55, 65;526, 269;527, 29Sanders, S.J.,523, 227Sanguinetti, G.,526, 34, 191, 206Sankey, D.P.C.,523, 234;525, 9Sannino, F.,527, 142Santacesaria, R.,527, 173Santoro, A.,525, 211Santos, H.,521, 195Santos, M.M.,521, 418Santra, A.B.,526, 19Sanyal, A.K.,524, 177Sarantites, D.G.,523, 13Saremi, S.,524, 44, 55, 65;526, 269;527, 29Sarkar, S.,524, 44, 55, 65;526, 269;527, 29Sarkisyan, E.K.G.,521, 181;523, 35;526, 221, 233Sarma, D.,522, 189Sasaki, K.,522, 22Satapathy, M.,524, 33;526, 247, 258Sato, M.,521, 400Sato, O.,527, 173Sato, Y.,527, 173Satou, Y.,521, 153Satpathy, A.,526, 247, 258Satta, A.,527, 173Saturnini, P.,521, 195Sauter, W.K.,521, 259Savcı, M.,527, 193Savoy-Navarro, A.,525, 17Savrin, V.I.,525, 283Sawado, N.,524, 289Sawyer, L.,525, 211Sbarra, C.,521, 181;523, 35Scalas, E.,521, 195Scardaoni, R.,525, 29Schael, S.,526, 34, 191, 206Schäfer, A.,521, 55;522, 194Schäfer, C.,524, 44, 55, 65;526, 269;527, 29Schäfer, T.,522, 67Schäfer, W.,527, 142Schahmaneche, K.,526, 278;527, 23Schaile, A.D.,521, 181;523, 35;526, 221, 233Schaile, O.,521, 181;523, 35;526, 221, 233Schamberger, R.D.,525, 211Schamlott, A.,522, 16Scharff-Hansen, P.,521, 181;523, 35;526, 221, 233Schegelsky, V.,524, 44, 55, 65;526, 269;527, 29Scheglov, Y.,522, 233Scheid, W.,526, 322Scheins, J.,523, 234;525, 9Schellman, H.,525, 211Scheurer, J.N.,521, 165Schiavon, P.,521, 171;522, 233;523, 22

Schilling, F.-P.,523, 234;525, 9Schlatter, D.,526, 34, 191, 206Schleper, P.,523, 234;525, 9Schmeling, S.,526, 191, 206Schmidt, B.,526, 278;527, 23Schmidt, D.,523, 234, 234;525, 9, 9Schmidt, S.,523, 234;525, 9Schmidt, T.,526, 278Schmidt-Kaerst, S.,524, 44, 55, 65;526, 269;527, 29Schmieden, H.,524, 26Schmitt, S.,523, 234;525, 9Schmitz, D.,524, 44, 55, 65;526, 269;527, 29Schneider, M.,523, 234;525, 9Schneider, O.,524, 33;526, 34Schoeffel, L.,523, 234;525, 9Schoenert, S.,525, 29Schöning, A.,523, 234;525, 9Schönmeier, P.,522, 16Schopper, H.,524, 44, 55, 65;526, 269;527, 29Schörner, T.,523, 234;525, 9Schotanus, D.J.,524, 44, 55, 65;526, 269;527, 29Schrenk, S.,524, 33;526, 247, 258Schröder, M.,521, 181;523, 35;526, 221, 233Schröder, V.,523, 234;525, 9Schroeder, W.,522, 16Schubert, C.,526, 55Schuller, J.-P.,526, 34Schulte-Wissermann, M.,522, 16Schultz-Coulon, H.-C.,523, 234;525, 9Schulze, H.-J.,526, 19Schumacher, M.,521, 181;523, 35;526, 221, 233Schwanda, C.,525, 17Schwanenberger, C.,523, 234;525, 9Schwartzman, A.,525, 211Schweitzer, P.,522, 37Schwenzer, K.,526, 79Schwering, B.,525, 17Schwering, G.,524, 44, 55, 65;526, 269;527, 29Schwick, C.,521, 181;523, 35;526, 221, 233Schwickerath, U.,525, 17Schwindling, J.,526, 34Sciabà, A.,526, 34, 191, 206Sciacca, C.,524, 44, 55, 65;526, 269;527, 29Scomparin, E.,521, 195Scott, W.G.,521, 181;523, 35;526, 221, 233Scrucca, C.A.,525, 169Seager, P.,526, 191, 206Sears, J.M.,523, 13Sedgbeer, J.K.,526, 34, 191, 206Sedlák, K.,523, 234;525, 9Sefkow, F.,523, 234;525, 9Segar, A.,525, 17Segoni, I.,523, 53;525, 205Sehgal, L.M.,521, 7; 525, 71Seidel, H.,525, 29Seiler, E.,525, 355Sekiguchi, K.,521, 153Sekulin, R.,525, 17


SELEX Collaboration,521, 171;522, 233;523, 22Selvaggi, G.,526, 34, 191, 206Semenov, S.,524, 33;526, 247, 258Semenov, S.V.,525, 283Semikoz, V.B.,521, 299Semple, A.T.,523, 13Sen, N.,525, 211Sen, S.,526, 121Senami, M.,524, 332SenGupta, S.,521, 350Senjanovic, G.,525, 189Senyo, K.,524, 33;526, 247, 258Serata, M.,522, 227Serci, S.,521, 195Serednyakov, S.,525, 41Serin, L.,526, 191, 206Serin-Zeyrek, M.,527, 173Sermyagin, A.V.,522, 222Serone, M.,525, 169Servoli, L.,524, 44, 55, 65;526, 269;527, 29Settles, R.,526, 34, 191, 206Seuster, R.,521, 181;523, 35;526, 221, 233Sever, R.,527, 173Sevior, M.,526, 278;527, 23Sevior, M.E.,524, 33;526, 247, 258Seweryniak, D.,525, 49Seyfarth, H.,524, 303Sghedoni, R.,522, 83Sguazzoni, G.,526, 34, 191, 206Shabalina, E.,525, 211Shabanov, S.V.,522, 201Shafi, Q.,522, 102;526, 379Shahoyan, R.,521, 195Shamanov, V.,527, 173Shamov, A.G.,527, 161Shary, V.,525, 41Shatunov, Yu.M.,527, 161Sheaff, M.,523, 53;525, 205Shears, T.G.,521, 181;523, 35;526, 221, 233Sheetz, R.A.,523, 227Sheikh-Jabbari, M.M.,526, 132;527, 149Shekelyan, V.,523, 234;525, 9Sheldon, P.D.,523, 53;525, 205Shen, B.C.,521, 181;523, 35;526, 221, 233Shepherd-Themistocleous, C.H.,521, 181;523, 35;526, 221, 233Sherwood, P.,521, 181;523, 35;526, 221, 233Shevchenko, S.,524, 44, 55, 65;526, 269;527, 29Sheviakov, I.,523, 234;525, 9Shibuya, H.,524, 33;526, 247, 258;527, 173Shifman, M.,522, 327Shima, K.,521, 67;525, 183Shimoura, S.,522, 227Shimoyama, N.,527, 206Shiu, G.,521, 114Shivarov, N.,524, 44, 55, 65;526, 269;527, 29Shivpuri, R.K.,525, 211Shneidman, T.M.,526, 322Short, B.J.,522, 335;524, 401

Shorto, J.M.B.,527, 187Shoutko, V.,524, 44, 55, 65;526, 269;527, 29Shpakov, D.,525, 211Shrock, R.,526, 137Shtarkov, L.N.,523, 234;525, 9Shumilov, E.,524, 44, 55, 65;526, 269;527, 29Shupe, M.,525, 211Shuryak, E.V.,524, 297Shutt, T.,525, 29Shvorob, A.,524, 44, 55, 65;526, 269;527, 29Shwartz, B.,524, 33;526, 247, 258Shwartz, B.A.,527, 161Sibidanov, A.L.,527, 161Sick, I.,524, 26Sidorov, V.A.,527, 161Sidwell, R.A.,525, 211Siebel, M.,525, 17Siedenburg, T.,524, 44, 55, 65;526, 269;527, 29Sieler, U.,526, 34, 191, 206Sigaudo, F.,521, 195Sillou, D.,526, 278;527, 23Silva, S.,521, 195Silva-Marcos, J.I.,526, 104Silvestrini, L.,525, 169Silvestris, L.,526, 34, 191, 206Simak, V.,525, 211Simgen, H.,525, 29Simmons, E.H.,521, 239;526, 365Simões, R.F.,527, 187Simon, J.,521, 171;522, 233;523, 22Simone, S.,527, 173Simopoulou, E.,526, 34, 191, 206Singh, H.,525, 211Singh, J.B.,524, 33;525, 211;526, 247, 258Siopsis, G.,524, 192Širca, S.,524, 26Sirignano, C.,527, 173Sirois, Y.,523, 234;525, 9Sirotenko, V.,525, 211Sisakian, A.,525, 17Sitnikov, A.I., 521, 171;522, 233;523, 22Sitta, M.,521, 195Skorokhvatov, M.,525, 29Skow, D.,521, 171;522, 233;523, 22Skrinsky, A.N.,527, 161Skrzypek, M.,523, 117Skuja, A.,521, 181;523, 35;526, 221, 233Slattery, P.,521, 171;525, 211Sloan, T.,523, 234;525, 9Smadja, G.,525, 17Smakhtin, V.P.,527, 161Smirnov, O.,525, 29Smirnov, P.,523, 234;525, 9Smirnov, V.A.,524, 129Smirnova, O.,525, 17Smith, A.M.,521, 181;523, 35;526, 221, 233Smith, C.,524, 395Smith, D.,526, 191, 206


Smith, E.,525, 211Smith, J.F.,523, 13Smith, R.P.,525, 211Smith, V.J.,521, 171;522, 233;523, 22Smizanska, M.,526, 191, 206Smolnikov, A.A.,522, 222Snihur, R.,525, 211Snopkov, I.G.,527, 161Snow, G.A.,521, 181;523, 35;526, 221, 233Snow, G.R.,525, 211Snow, J.,525, 211Snyder, S.,525, 211So, H.,526, 164Sobie, R.,521, 181;523, 35;526, 221, 233Sobolev, Yu.G.,522, 222Sochichiu, C.,522, 345Soffer, J.,522, 22Sokolov, A.,525, 17Söldner-Rembold, S.,521, 181;523, 35;526, 221, 233Soler, F.J.P.,526, 278;527, 23Solodov, E.P.,527, 161Solomon, J.,525, 211Soloviev, Y.,523, 234;525, 9Sommer, J.,526, 34Sommer, R.,523, 135Son, D.,524, 44, 55, 65;526, 269;527, 29Son, D.T.,522, 67Sonderegger, P.,521, 195Song, H.S.,523, 161Song, J.S.,527, 173Song, Y.,525, 211Sonnenschein, A.,525, 29Sopczak, A.,525, 17Sorin, A.,521, 421Sorín, V.,525, 211Sorrentino, S.,527, 173Sosebee, M.,525, 211Sosin, Z.,523, 227Sosnowski, R.,525, 17Sotnikov, A.,525, 29Sotnikova, N.,525, 211Sousa, D.,527, 92Soustruznik, K.,525, 211South, D.,523, 234;525, 9Souza, M.,525, 211Sozzi, G.,526, 278;527, 23Spada, F.R.,527, 173Spagnolo, P.,526, 34, 191, 206Spagnolo, S.,521, 181;523, 35;526, 221, 233Spano, F.,521, 181;523, 35;526, 221, 233Spaskov, V.,523, 234;525, 9Spassov, T.,525, 17Specka, A.,523, 234;525, 9Spillantini, P.,524, 44, 55, 65;526, 269;527, 29Spitzer, H.,523, 234;525, 9Spooner, N.J.C.,524, 245Sproston, M.,521, 181;523, 35;526, 221, 233Srivastava, M.,521, 171;522, 233;523, 22

Stahl, A.,521, 181;523, 35;526, 221, 233Stamen, R.,523, 234;525, 9Stanic, S.,524, 33;526, 247, 258Stanitzki, M.,525, 17Stanton, N.R.,525, 211Starosta, K.,523, 13Staszel, P.,523, 227Steele, D.,526, 278;527, 23Steinbrück, G.,525, 211Steiner, V.,521, 171;522, 233;523, 22Steinke, M.,522, 16Stella, B.,523, 234;525, 9Stenson, K.,523, 53;525, 205Stenzel, H.,526, 34, 191, 206Stepanov, P.Yu.,527, 161Stepanov, V.,521, 171;522, 233;523, 22Stephanov, M.A.,522, 67Stephens, K.,521, 181;523, 35;526, 221, 233Stephens, R.W.,525, 211Steuer, M.,524, 44, 55, 65;526, 269;527, 29Stichel, P.C.,526, 399Stichelbaut, F.,525, 211Stickland, D.P.,524, 44, 55, 65;526, 269;527, 29Stiegler, U.,526, 278;527, 23Stiewe, J.,523, 234;525, 9Stinzing, F.,522, 16Stipcevic, M., 526, 278;527, 23Stocchi, A.,525, 17Stöcker, H.,524, 265;526, 309Stoker, D.,525, 211Stolarczyk, Th.,526, 278;527, 23Stolin, V.,525, 211Stone, A.,525, 211Stoyanov, B.,524, 44, 55, 65;526, 269;527, 29Stoyanova, D.A.,525, 211Straessner, A.,524, 44, 55, 65;526, 269;527, 29Strakhovenko, V.,525, 41Strang, M.A.,525, 211Straumann, U.,523, 234;525, 9Strauss, J.,525, 17Strauss, M.,525, 211Ströher, H.,521, 158, 217;526, 287Ströhmer, R.,521, 181;523, 35;526, 221, 233Strolin, P.,527, 173Strom, D.,521, 181;523, 35;526, 221, 233Strong, J.A.,526, 34, 191, 206Strovink, M.,525, 211Stugu, B.,525, 17Stumpf, L.,521, 181;523, 35;526, 221Stutte, L.,521, 171;522, 233;523, 22;525, 211Su, W.-C.,525, 201Sudhakar, K.,524, 44, 55, 65;526, 269;527, 29Sugai, H.,527, 43Sugi, A.,524, 33Sugino, F.,522, 145Sugiyama, A.,524, 33;526, 247, 258Sugiyama, H.,526, 335Sukhanov, A.I.,527, 161


Sukhatme, U.,527, 73Sukhotin, S.,525, 29Sultanov, G.,524, 44, 55, 65;526, 269;527, 29Sumisawa, K.,524, 33;526, 247, 258Sumiyoshi, T.,524, 33;526, 247, 258Sumner, T.J.,524, 245Sun, G.Y.,522, 16Sun, L.Z.,524, 44, 55, 65;526, 269;527, 29Sur, S.,521, 350Surrow, B.,521, 181;523, 35;526, 221, 233Sushkov, S.,524, 44, 55, 65;526, 269;527, 29Suter, H.,524, 44, 55, 65;526, 269;527, 29Suzuki, K.,524, 33Suzuki, S.,524, 33;526, 247, 258Suzuki, S.Y.,524, 33;526, 247, 258Svensson, C.E.,525, 49Svoiski, M.,521, 171;522, 233;523, 22Swain, J.D.,524, 44, 55, 65;526, 269;527, 29Swain, S.K.,524, 33;526, 247, 258Swart, M.,523, 234;525, 9Swynghedauw, M.,526, 34, 191, 206Sylvia, C.,525, 41Szczekowski, M.,525, 17Szczepaniak, A.P.,526, 72Szeptycka, M.,525, 17Szillasi, Z.,524, 44, 55, 65;526, 269;527, 29Sznajder, A.,525, 211Szumlak, T.,525, 17

Tabarelli, T.,525, 17Taffard, A.C.,525, 17Takahashi, T.,522, 215;524, 33;526, 247, 258Takasaki, F.,524, 33;526, 247, 258Takeda, N.,527, 43Takeuchi, S.,522, 227Takita, M.,524, 33;526, 247, 258Talby, M.,525, 211;526, 34Tamai, K.,524, 33;526, 247, 258Tamura, N.,524, 33;526, 247, 258Tan, Y.,523, 1Tanaka, J.,524, 33;526, 247, 258Tanaka, K.,523, 111Tanaka, M.,524, 33;526, 247, 258Tanaka, R.,526, 191, 206Tanaka, T.,521, 400Tanaka, Y.,524, 33;526, 247, 258Tanase, M.,527, 43Tang, X.W.,524, 44, 55, 65;526, 269;527, 29Tanii, Y., 525, 183Tanimoto, M.,527, 206Tarasenkov, V.,525, 29Tareb-Reyes, M.,526, 278;527, 23Tarem, S.,521, 181;523, 35;526, 221, 233Tarjan, P.,524, 44, 55, 65;526, 269;527, 29Tarrago, X.,521, 195Tartaglia, R.,525, 29Tasevsky, M.,521, 181;523, 35;526, 221, 233Taševský, M.,523, 234;525, 9

Tasinato, G.,524, 342Tatar, R.,523, 185Tauscher, L.,524, 44, 55, 65;526, 269;527, 29Tavartkiladze, Z.,522, 102Taxil, P.,522, 89Taylor, G.,526, 34, 191, 206Taylor, G.N.,524, 33;526, 247, 258, 278;527, 23Taylor, L.,524, 44, 55, 65;526, 269;527, 29Taylor, R.J.,521, 181;523, 35;526, 221, 233Taylor, W.,525, 211Tchernyshov, V.,523, 234;525, 9Tchetchelnitski, S.,523, 234;525, 9Tegenfeldt, F.,525, 17Teixeira-Dias, P.,526, 34, 191, 206Tejessy, W.,526, 34, 191, 206Tellili, B., 524, 44, 55, 65;526, 269;527, 29Tempesta, P.,526, 34, 191, 206Templon, J.A.,523, 6Tenchini, R.,526, 34, 191, 206Tentindo-Repond, S.,525, 211Teper, M.,523, 280Teramoto, Y.,524, 33;526, 247, 258Terentyev, N.K.,521, 171;522, 233;523, 22Tereshchenko, V.,526, 278;527, 23Terranova, F.,525, 17Teschner, J.,521, 127Testera, G.,525, 29Teubert, F.,526, 34, 191, 206Teuscher, R.,521, 181;523, 35;526, 221, 233Teyssier, D.,524, 44, 55, 65;526, 269;527, 29Tezuka, I.,527, 173Thomas, A.W.,526, 72Thomas, G.P.,521, 171;522, 233;523, 22Thomas, J.,521, 181;523, 35;526, 221, 233Thompson, A.S.,526, 34, 191, 206Thompson, G.,523, 234;525, 9Thompson, J.A.,527, 161Thompson, J.C.,526, 34, 191, 206Thompson, L.F.,526, 34, 191, 206Thompson, P.D.,523, 234;525, 9Thomson, M.A.,521, 181;523, 35;526, 221, 233Thorsteinsen, T.F.,523, 227Tiator, L.,522, 27Tilquin, A., 526, 191, 206Timmermans, C.,524, 44, 55, 65;526, 269;527, 29Timmermans, J.,525, 17Timmers, H.,526, 295Ting, S.C.C.,524, 44, 55, 65;526, 269;527, 29Ting, S.M.,524, 44, 55, 65;526, 269;527, 29Tinti, N., 525, 17Tioukov, V.,527, 173Titov, V.M., 527, 161Tittel, K., 526, 34, 191, 206Tkatchev, L.,525, 17Tobe, K.,526, 157Tobien, N.,523, 234;525, 9Tobin, M.,525, 17Todorovova, S.,525, 17


Tolun, P.,527, 173Tomalin, I.R.,526, 34, 191, 206Tome, B.,525, 17Tomoto, M.,524, 33;526, 247, 258Tomura, T.,524, 33;526, 247, 258Tonazzo, A.,525, 17Tonin, M.,524, 199Tonwar, S.C.,524, 44, 55, 65;526, 269;527, 29Topilskaya, N.S.,521, 195Toropin, A.,526, 278Toropin, A.N.,527, 23Torrence, E.,521, 181;523, 35;526, 221, 233Tortosa, P.,525, 17Toshito, T.,527, 173Tóth, J.,524, 44, 55, 65;526, 269;527, 29Toublan, D.,522, 67Touchard, A.-M.,526, 278;527, 23Tovey, D.R.,524, 245Tovey, S.N.,524, 33;526, 247, 258, 278;527, 23Toya, D.,521, 181;523, 35;526, 221, 233Trabelsi, A.,526, 34, 191, 206Trabelsi, K.,526, 247, 258Tramontano, F.,524, 241Tran, M.-T.,526, 278;527, 23Travnicek, P.,525, 17Traynor, D.,523, 234;525, 9Trefzger, T.,521, 181;523, 35;526, 221, 233Treille, D.,525, 17Trentadue, L.,522, 83Tricoli, A., 521, 181;523, 35;526, 221, 233Tricomi, A., 526, 34, 191, 206Trigger, I.,521, 181;523, 35;526, 221, 233Tripathi, S.M.,525, 211Trippe, T.G.,525, 211Tristram, G.,525, 17Trochimczuk, M.,525, 17Trocmé, B.,526, 191, 206Trócsányi, Z.,521, 181;523, 35;526, 221, 233Troncon, C.,525, 17Troyan, S.I.,522, 57Trueb, Ph.,524, 26Truöl, P.,523, 234;525, 9Tseng, J.-J.,522, 257Tsenov, R.,527, 173Tsesmelis, E.,526, 278;527, 23Tsipolitis, G.,523, 234;525, 9Tsuboyama, T.,524, 33;526, 247, 258Tsuda, M.,521, 67;525, 183Tsujikawa, S.,526, 179Tsukamoto, T.,524, 33;526, 247, 258Tsukerman, I.,527, 173Tsulaia, M.,523, 338Tsur, E.,521, 181;523, 35;526, 221, 233Tsurin, I.,523, 234;525, 9Tu, H.,525, 135Tuchming, B.,526, 191, 206Tully, C., 524, 44, 55, 65;526, 269;527, 29Tung, K.L.,524, 44, 55, 65;526, 269;527, 29

Turcot, A.S.,525, 211Tureanu, A.,526, 132;527, 149Turluer, M.-L.,525, 17Turnau, J.,523, 234;525, 9Turner-Watson, M.F.,521, 181;523, 35;526, 221, 233Turney, J.E.,523, 234;525, 9Tuts, P.M.,525, 211Tveter, T.S.,523, 227Tyapkin, I.A.,525, 17Tyapkin, P.,525, 17Tye, S.-H.H.,522, 155Tzamarias, S.,525, 17Tzamariudaki, E.,523, 234;525, 9

Uchida, Y.,524, 55Udluft, S.,523, 234;525, 9Ueda, I.,521, 181;523, 35;526, 221, 233Uehara, S.,524, 33;526, 247, 258Uematsu, T.,522, 22Ueno, K.,524, 33;526, 247, 258Uesaka, T.,521, 153Uiterwijk, J.W.E.,527, 173Ujvári, B., 521, 181;523, 35;526, 221, 233UKQCD Collaboration,523, 280;525, 360Ulbricht, J.,524, 44, 55, 65;526, 269;527, 29Ullaland, O.,525, 17Ulrichs, J.,526, 278;527, 23Unno, Y.,524, 33;526, 247, 258Uno, S.,524, 33;526, 247, 258Urban, M.,523, 234;525, 9Uretsky, J.L.,523, 299Uribe, C.,523, 53;525, 205Usai, G.L.,521, 195Ushida, N.,527, 173Ushiroda, Y.,524, 33;526, 247, 258Usik, A., 523, 234;525, 9Utyuzh, O.V.,522, 273Uvarov, L.N.,521, 171;522, 233;523, 22Uvarov, V.,525, 17Uzikov, Yu.N.,521, 158;524, 303

Vaandering, E.W.,523, 53;525, 205Vacavant, L.,526, 278;527, 23Vachon, B.,521, 181;523, 35;526, 221, 233Vahsen, S.E.,524, 33;526, 247Vainshtein, A.,522, 327Valassi, A.,526, 34, 191, 206Valdata-Nappi, M.,526, 278;527, 23Valente, E.,524, 44, 55, 65;526, 269;527, 29Valenti, G.,525, 17Valkár, S.,523, 234;525, 9Valkárová, A.,523, 234;525, 9Vallage, B.,526, 34, 191, 206Valle, J.W.F.,521, 299;523, 151Vallée, C.,523, 234;525, 9Valuev, V.,526, 278;527, 23Van Cauteren, T.,527, 62


Van Dalen, J.A.,524, 44, 55, 65;526, 269;527, 29Van Dam, P.,525, 17Van Dantzig, R.,527, 173Van de Vyver, B.,527, 173Van de Walle, R.T.,524, 44, 55, 65;526, 269;527, 29Van Dierendonck, D.,524, 44, 55, 65;526, 269;527, 29Van Eldik, J.,525, 17Van Gulik, R.,524, 44, 55, 65;526, 269;527, 29Vanhove, P.,522, 145Vaniev, V.,525, 211Van Kooten, R.,525, 211Van Leeuwe, J.J.,523, 6Van Lysebetten, A.,525, 17Van Mechelen, P.,523, 234;525, 9Vannerem, P.,521, 181;523, 35;526, 221, 233Vannucci, F.,526, 278;527, 23Van Remortel, N.,525, 17Van Vulpen, I.,525, 17Vardaci, E.,521, 165Varelas, N.,525, 211Varvell, K.E.,524, 33;526, 247, 258, 278;527, 23Vasiliev, A.N.,521, 171;522, 233;523, 22Vasiliev, S.I.,522, 222Vassilevich, D.V.,521, 357Vassiliev, S.,523, 234;525, 9Vavilov, D.V., 521, 171;522, 233;523, 22Vayaki, A.,526, 34, 191, 206Vazdik, Y.,523, 234;525, 9Vázquez, F.,523, 53;525, 205Vegni, G.,525, 17Veillet, J.-J.,526, 34, 191, 206Veloso, F.,525, 17Veltri, M., 526, 278;527, 23Venturi, A.,526, 34, 191, 206Venus, W.,525, 17Verbaarschot, J.J.M.,522, 67Verbeure, F.,525, 17Vercellin, E.,521, 195Vercesi, V.,526, 278;527, 23Verderi, M.,526, 34Verdier, P.,525, 17Verdini, P.G.,526, 34, 191, 206Verebryusov, V.S.,521, 171;522, 233;523, 22Vertogradov, L.S.,525, 211Verzi, V., 525, 17Verzocchi, M.,521, 181;523, 35;526, 221, 233Veszpremi, V.,524, 44, 55, 65;526, 269;527, 29Vesztergombi, G.,524, 44, 55, 65;526, 269;527, 29Vetlitsky, I., 524, 44, 55, 65;526, 269;527, 29Vichnevski, A.,523, 234;525, 9Vicinanza, D.,524, 44, 55, 65;526, 269;527, 29Victorov, V.A., 521, 171;522, 233;523, 22Vidal-Sitjes, G.,526, 278;527, 23Videau, H.,526, 34, 191, 206Videau, I.,526, 34, 191, 206Videbæk, F.,523, 227Vieira, J.-M.,526, 278;527, 23Viertel, G.,524, 44, 55, 65;526, 269;527, 29

Viesti, G.,521, 165Vignaud, D.,525, 29Vilain, P.,527, 173Vilanova, D.,525, 17Vilja, I., 526, 9Villa, S., 524, 44, 55, 65;526, 269;527, 29Villatte, L., 521, 195Villeneuve-Seguier, F.,525, 211Viñas, X.,523, 67Vincent, S.M.,525, 49Vinogradova, T.,526, 278;527, 23Violini, P., 527, 29Virey, J.-M.,522, 89Vishnyakov, V.E.,521, 171;522, 233;523, 22Vissani, F.,522, 95Visschers, J.L.,527, 173Vitale, L., 525, 17Vitale, S.,525, 29Vitev, I., 526, 301Vitulo, P.,523, 53;525, 205Vivargent, M.,524, 44, 55, 65;526, 269;527, 29Vlachos, S.,524, 44, 55, 65;526, 269;527, 29Vnukov, I.,525, 41Vodopianov, I.,524, 44, 55, 65;526, 269;527, 29Vogel, H.,524, 44, 55, 65;526, 269;527, 29Vogel, P.,522, 240Vogelaar, R.B.,525, 29Vogt, H.,524, 44, 55, 65;526, 269;527, 29Volkov, A.A., 525, 211Volkov, M.S.,524, 369Vollmer, C.F.,521, 181;523, 35;526, 221, 233Voloshin, M.B.,524, 376Von Brentano, P.,521, 146;524, 252;527, 55Von Feilitzsch, F.,525, 29Von Krogh, J.,521, 181;523, 35;526, 221, 233Von Wimmersperg-Toeller, J.H.,526, 34, 34, 191, 206Vorobiev, A.P.,525, 211Vorobiev, I.,524, 44, 55, 65;526, 269;527, 29Vorobyov, A.A.,521, 171;522, 233;523, 22;524, 44, 55, 65;

526, 269;527, 29Vorwalter, K.,521, 171;522, 233;523, 22Voss, H.,521, 181;523, 35;526, 221, 233Vossebeld, J.,521, 181;523, 35;526, 221, 233Vrba, V.,525, 17Vyrodov, V.,525, 29

Wäänänen, A.,526, 34, 191, 206Wachsmuth, H.,526, 34, 191, 206Wacker, K.,523, 234;525, 9Wada, R.,521, 165;523, 227Wadhwa, M.,524, 44, 55, 65;526, 269;527, 29Wagner, A.,524, 26Wagner, G.J.,522, 16Wagner, M.,522, 16Wahl, H.D.,525, 211Wahlen, H.,525, 17Wakasa, T.,521, 153Walcher, T.,522, 10


Walcher, Th.,524, 26Walker, P.M.,525, 255Waller, D.,521, 181;523, 35;526, 221, 233Wallny, R.,523, 234;525, 9Wallraff, W., 524, 44, 55, 65;526, 269;527, 29Wands, D.,524, 5Wang, C.C.,524, 33;526, 247, 258Wang, C.H.,524, 33;526, 247, 258Wang, H.,525, 211Wang, J.,521, 308Wang, J.G.,524, 33;526, 247, 258Wang, M.,524, 55Wang, M.-Z.,524, 33;526, 247, 258Wang, T.,526, 34, 191, 206Wang, X.-N.,526, 301;527, 85Wang, X.L.,524, 44, 55, 65;526, 269;527, 29Wang, Y.,527, 50Wang, Z.-M.,525, 211Wang, Z.M.,524, 44, 55, 65;526, 269;527, 29Warchol, J.,525, 211Ward, B.F.L.,523, 117Ward, C.K.,524, 245Ward, C.P.,521, 181;523, 35;526, 221, 233Ward, D.R.,521, 181;523, 35;526, 221, 233Ward, J.,526, 34Ward, J.J.,526, 191, 206Warner, D.D.,525, 49Warner, N.P.,522, 181Warren, G.,524, 26Was, Z.,523, 117Washbrook, A.J.,525, 17Wasserbaech, S.,526, 34, 191, 206Wasserman, I.,522, 155Watanabe, Y.,524, 33;526, 247, 258Watanabe, Y.X.,522, 227Watari, T.,527, 106Watkins, P.M.,521, 181;523, 35;526, 221, 233Watson, A.T.,521, 181;523, 35;526, 221, 233Watson, N.K.,521, 181;523, 35;526, 221, 233Watts, G.,525, 211Waugh, B.,523, 234;525, 9Wayne, M.,525, 211Weber, F.V.,526, 278;527, 23Weber, G.,523, 234;525, 9Weber, H.J.,521, 33Weber, M.,523, 234;524, 44, 55, 65;525, 9; 526, 269;527, 29Webster, M.,523, 53;525, 205Weerts, H.,525, 211Wegener, D.,523, 234;525, 9Wei, L., 525, 107Weis, M.,524, 26Weiser, C.,525, 17Weisse, T.,526, 278;527, 23Well, J.D.,526, 157Wells, P.S.,521, 181;523, 35;526, 221, 233Wen, W.Y.,525, 157Wengler, T.,521, 181;523, 35;526, 221, 233Wermes, N.,521, 181;523, 35;526, 221, 233

Werner, C.,523, 234;525, 9Werner, M.,523, 234;525, 9Werner, N.,523, 234;525, 9Werner, S.,526, 34, 191, 206Werner, V.,521, 146;527, 55Wessling, M.E.,523, 331Wesson, P.S.,527, 1Wetterich, C.,522, 5; 525, 277Wetterling, D.,521, 181;523, 35;526, 221, 233White, A.,525, 211White, G.,523, 234;525, 9White, J.T.,525, 211White, R.,526, 191, 206Whiteson, D.,525, 211Wicke, D.,525, 17Wickens, J.,525, 17Wiedenmann, W.,526, 34, 34, 191, 206Wieloch, A.,523, 227Wienemann, P.,524, 44, 55, 65;526, 269;527, 29Wiesand, S.,523, 234;525, 9Wijngaarden, D.A.,525, 211Wilk, G., 522, 273Wilkens, H.,524, 44, 55, 65;526, 269;527, 29Wilkin, C., 521, 158Wilkinson, G.,525, 17Wilksen, T.,523, 234;525, 9Williams, S.J.,525, 49Willis, N., 521, 195Willis, S., 525, 211Wilms, A., 522, 16Wilquet, G.,527, 173Wilson, F.F.,526, 278;527, 23Wilson, G.W.,521, 181;523, 35;526, 221, 233Wilson, J.A.,521, 181;523, 35;526, 221, 233Wilson, J.R.,523, 53;525, 205Wimpenny, S.J.,525, 211Winde, M.,523, 234;525, 9Winter, G.-G.,523, 234;525, 9Winter, K.,527, 173Winter, M.,525, 17Winton, L.J.,526, 278;527, 23Wirth, S.,522, 16Wise, B.B.,523, 331Wiss, J.,523, 53;525, 205Wissing, Ch.,523, 234;525, 9Witek, M., 525, 17Włodarczyk, Z.,522, 273Wobisch, M.,523, 234;525, 9Woehrling, E.-E.,523, 234;525, 9Wöhrle, H.,524, 26Wojcik, M., 525, 29Wolf, G., 526, 34, 191, 206Wolfenstein, L.,524, 319Womersley, J.,525, 211Won, E.,524, 33;526, 247, 258Wood, D.R.,525, 211Woodahl, B.,526, 355Woodard, R.P.,524, 233


Wu, J.,526, 34, 191, 206Wu, S.L.,526, 34, 191, 206Wu, S.X.,524, 55Wu, X., 526, 34, 191, 206Wünsch, E.,523, 234;525, 9Wunsch, M.,526, 34, 191, 206Wyatt, A.C.,523, 234;525, 9Wyatt, T.R.,521, 181;523, 35;526, 221, 233Wyler, D.,521, 291, 320Wynhoff, S.,524, 44, 55, 65;526, 269;527, 29

Xia, L., 524, 44, 55, 65;526, 269;527, 29Xiang, L.,523, 171Xie, Y., 526, 34, 191, 206Xiong, W.,523, 53;525, 205Xu, Q.,525, 211, 301Xu, R.,526, 34, 191, 206Xu, Z.Z.,524, 44, 55, 65;526, 269;527, 29Xue, S.,526, 34, 191, 206

Yabsley, B.D.,524, 33;526, 247, 258, 278;527, 23Yager, P.M.,523, 53;525, 205Yako, K.,521, 153Yamada, K.,522, 227Yamada, R.,525, 211Yamada, Y.,524, 33;526, 247, 258Yamaga, M.,524, 33;526, 247, 258Yamaguchi, A.,524, 33;526, 247, 258Yamamoto, H.,526, 247, 258Yamamoto, J.,524, 44, 55, 65;526, 269;527, 29Yamamoto, K.,524, 332Yamashita, S.,521, 181;523, 35;526, 221, 233Yamashita, Y.,524, 33;526, 247, 258Yamauchi, M.,524, 33;526, 247Yamin, P.,525, 211Yanagisawa, Y.,522, 227Yanaka, S.,524, 33;526, 247, 258Yang, B.Z.,524, 44, 55, 65;526, 269;527, 29Yang, C.G.,524, 44, 55, 65;526, 269;527, 29Yang, H.J.,524, 44, 55, 65;526, 269;527, 29Yang, H.S.,523, 357Yang, J.-J.,526, 50Yang, M.,524, 44, 55, 65;526, 269;527, 29Yang, S.N.,522, 27Yaschenko, S.,521, 158Yashima, J.,524, 33;526, 247, 258Yasuè, M.,524, 308Yasuda, T.,525, 211Yatsunenko, Y.A.,525, 211Yeh, S.C.,524, 44, 55, 65;526, 269;527, 29Yilmaz, O.,525, 273Yip, K., 525, 211Yokoyama, J.,524, 15Yokoyama, M.,524, 33;526, 247, 258Yoneda, K.,522, 227Yoon, C.S.,527, 173Yoshida, K.,524, 33;526, 247, 258

You, J.,521, 171;522, 233;523, 22Youssef, S.,525, 211Yu, J.,525, 211Yu, Z., 524, 161;525, 211Yuan, C.,526, 34, 191, 206Yuan, Y.,524, 33;526, 247, 258Yudin, Yu.Y.,527, 161Yue, C.,525, 301Yusa, Y.,524, 33;526, 247, 258Yushchenko, O.,525, 17

Zaccone, H.,526, 278;527, 23Žácek, J.,523, 234;525, 9Zacek, V.,521, 181;523, 35;526, 221, 233Zachariadou, K.,526, 34, 191, 206Zaimidoroga, O.,525, 29Zakharov, V.I.,522, 266Zálešák, J.,523, 234;525, 9Zalewska, A.,525, 17Zalewski, P.,525, 17Zalite, An.,524, 44, 55, 65;526, 269;527, 29Zalite, Yu.,524, 44, 55, 65;526, 269;527, 29Zallo, A., 523, 53;525, 205Zanabria, M.,525, 211Zavrtanik, D.,525, 17Zeier, M.,524, 26Zeitnitz, C.,526, 34, 191, 206Zelevinsky, V.V.,523, 1Zeng, S.,527, 50Zer-Zion, D.,521, 181;523, 35;526, 221, 233Zeyrek, M.T.,527, 173Zgura, I.S.,523, 227Zhang, B.,523, 260Zhang, C.C.,524, 33;526, 247, 258Zhang, J.,524, 33;526, 34, 191, 206, 247, 258Zhang, L.,526, 34, 191, 206Zhang, X.,525, 211, 265Zhang, Y.,523, 53;525, 205Zhang, Y.-J.,523, 260Zhang, Y.-Z.,523, 367Zhang, Z.,523, 234;525, 9Zhang, Z.P.,524, 44, 55, 65;526, 269;527, 29Zhao, H.W.,524, 33;526, 247, 258Zhao, J.,524, 26, 44, 55, 65;526, 269;527, 29Zhao, W.,521, 171;522, 233;526, 34, 191, 206Zhaog, W.,523, 22Zheng, H.,525, 211Zheng, S.,521, 171;522, 233;523, 22Zheng, Y.,524, 33;526, 247, 258Zhilich, V., 524, 33;526, 247, 258Zhokin, A.,523, 234;525, 9Zhou, B.,525, 211Zhou, Z.,525, 211Zhu, G.Y.,524, 44, 55, 65;526, 269;527, 29Zhu, R.Y.,524, 44, 55, 65;526, 269;527, 29Zhu, S.,524, 283Zhu, W.,523, 88Zhu, Z.H.,521, 171


Zhuang, H.L.,524, 44, 55, 65;526, 269;527, 29Zichichi, A., 524, 44, 55, 65;526, 269;527, 29Ziegler, T.,526, 191, 206Zielinski, M., 521, 171;525, 211Zieminska, D.,525, 211Zieminski, A.,525, 211Zihlmann, B.,524, 26Zilizi, G., 524, 44, 55, 65;526, 269;527, 29Zimdahl, W.,521, 133Zimin, N.I., 525, 17Zimmermann, B.,524, 44, 55, 65;526, 269;527, 29Zintchenko, A.,525, 17Zito, G.,526, 34, 191, 206Zloshchastiev, K.G.,527, 215Zobernig, G.,526, 34, 191, 206Zöller, M., 524, 44, 55, 65Zöller, M.Z.,526, 269;527, 29Zoller, Ph.,525, 17Zomer, F.,523, 234;525, 9

Žontar, D.,524, 33;526, 247, 258Zsembery, J.,523, 234;525, 9Zuber, K.,526, 278;527, 23Zucchelli, P.,527, 173Zuccon, P.,526, 278;527, 23Zukanovich, R.,521, 171Zukanovich-Funchal, R.,522, 233;523, 22Zumbro, J.D.,523, 1Zupan, M.,525, 17Zur Nedden, M.,523, 234;525, 9Zutshi, V.,525, 211Zuzel, G.,525, 29Zverev, E.G.,525, 211Zverev, S.G.,527, 161Zwanziger, D.,524, 123Zwirner, F.,525, 169Zylberstejn, A.,525, 211

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Gravitational conformal invariance and coupling constants ... v.527.pdf · Gravitational conformal invariance and coupling constants in Kaluza–Klein theory F. Darabia,b,c,P.S.Wessona