cm . __ LfiB g

ELSEVIER

22 June 1995

Physics Letters B 353 (1995) 20-26

PHYSICS LETTERS B

On the strange vector form factors of the nucleon in the NJL sbliton model *

H. Weigel ’ , A. Abada, R. Alkofer, H. Reinhardt Institute for Theoretical Physics, Tiibingen University, Auf der Morgenstelle 14, D-72076 Tilbingen, Germany

Received 9 March 1995; revised manuscript received 2 May 1995 Editor: C. Mahaux

Abstract

Within the Nambu-Jona-Lasinio model strange degrees of freedom are incorporated into the soliton picture using the collective approach of Yabu and Ando. The form factors of the nucleon associated with the nonet vector current are extracted. The numerical results provide limits for the strange magnetic moment: -0.05 < ,u~ 5 0.25. For the strange magnetic form factor of the nucleon the valence quark and vacuum contributions add coherently while there are significant cancellations for the strange electric form factor.

1. Introduction

A number of experiments, which have recently been completed [ 1 ] or are up-coming [ 2,3], measure par- ity violating asymmetries in scattering processes of polarized electrons on nuclei. Although these experi- ments were initiated as precision tests of the electro- weak theory they also provide access to hadronic ‘ob- servables” like (Nlgy,slN), which e.g. enter the ma- trix elements of the neutral current between nucleon states. From a theoretical point of view it is, of course, challenging to attempt predictions on these matrix el- ements. A first estimate of the matrix element of the strange vector current between nucleon states was car- ried out in Ref. [4] performing a three-pole vector meson fit to dispersion relations [5]. Later on the matrix element (NIs~,sIN) has been studied in the

* Supported by the Deutsche Forschungsgemeinschaft (DFG) under contract number Re-856/2-Z.

1 Supported by a Habilitanden scholarship of the DFG.

Skyrme model [ 61 and the Skyrme model with vector mesons [ 71. Also the effect of 4 - o mixing in the framework of vector meson dominance has been in- vestigated [ 61. This picture has even been combined [ 81 with the kaon loop calculation of Ref. [ 31. For further details on the status of related studies we refer to the review article [9].

These investigations are based on chirally invari- ant models describing the interaction between strange mesons and the nucleon. The latter is either consid- ered as an “elementary” particle [3,8] or as a soli- ton of meson configurations [ 6,7]. Neither of these studies takes explicit account of the quark structure of the nucleon2. In order to examine effects related to the quark structure, the Nambu-Jona-Lasinio (NJL) model [ 111 represents an excellent candidate. Imitat- ing the quark flavor dynamics of QCD at low ener- gies the NJL model contains both reference to explicit

2 Bag model studies, which omit SU(3) breaking for the baryon

wave-functions, are reported in Ref. [ 101.

0370-2693/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved

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H. Weigel et al. /Physics Letters B 353 (1995) 20-26 21

quark degrees of freedom as well as the fruitful con- cepts of chiral symmetry and its spontaneous break- ing. The model provides a fair description of the pseu- doscalar and vector mesons as quark-antiquark bound states [ 12-141. In addition, it contains soliton solu- tions [ 151 in the two flavor subspace which may be identified as baryons3 . The so-called collective ap- proach [ 17,181 allows one to incorporate strange de- grees of freedom. Its application within the NJL model [ 19,201 provides an appealing possibility to explore the matrix elements of the strange vector current.

2. Baryons in the NJL model

The Lagrangian for the NJL model [ 111 with scalar and pseudoscalar degrees of freedom is defined as the sum of the free Dirac Lagrangian and a chirally invari- ant four quark interaction. By path integral bosoniza- tion this model can be converted into an effective me- son theory yielding the action [ 121

d=d,+dm

= Tr* log (i@)

1 -- 4Gtr s

d4x (Mbf - riz”(A4 + AI+) + (f?z’>‘) ,

(1)

where G denotes the effective coupling constant and Rio is the current quark mass matrix. In the case of three flavors M = S + iP is a 3 x 3 matrix containing the scalar (S) and pseudoscalar (P) meson fields. Furthermore

i@= if- (PRM+PLM+)

with PR,L =( If y5)/2 (2)

is the Dirac operator. We will assume isospin sym- metry, mz = rni =: m”; however, rnt # m”. In Eq.

( 1) we have indicated that the functional trace is UV divergent and has to be regularized. Hence one more parameter is introduced, the cut-off A. As regulariza- tion scheme we employ Schwinger’s proper time pre- scription. This requires a Wick rotation to Euclidean space (xc + -ix4 = -ir). Subsequently the real part

3 For a review see Ref. [ 161 and references therein.

of the Euclidean action is represented by a parameter integral

(3)

Here & refers to the continuation to Euclidean space. The Schwinger-Dyson equations (SDEs) for

the scalar fields yield non-vanishing vacuum ex- peCt&.iOtI values (S) = i?i = diag(m,,md,m,) = diag(m, m, m,) # ~3’. These represent the con- stituent quark masses and are a manifestation of spontaneous breaking of chiral symmetry.

In order to apply the SU(3) collective descrip- tion for baryons as chiral solitons we adopt the

parametrization

M(r,t) = R(t)S(r) R+(t)(S)R(t)C(r) R+(t)

(4)

for the (pseudo-)scalar fields [ 191. The SU(3)- matrix R(t) contains the collective coordinates. For simplicity, we have constrained the scalar fields to their vacuum configuration. The soliton configuration is characterized by the hedgehog ansutz for the chiral

field

U(r) =6”(r) =exp(i?erO(r)). (5)

The special form of the parametrization (4) guaran- tees that only the hedgehog rotates in the space of

the collective coordinates. The time dependence of the collective rotations R is measured by the angular ve- locities R+ R = (i/2) A%“.

Then A is expanded up to quadratic order in

terms of the angular velocities as well as the mass difference m - m,. From the resulting expression the collective Lagrangian, L( R, Cl") is straightfor- wardly extracted [ 191. Quantization is achieved by identifying the momenta conjugate to the angu- lar velocities with the right generators of SU(3): R, = 4L(R,W)/d&. This provides a linear re- lation between the generators and the velocities. The resulting collective Hamiltonian, H( R, R,) can be diagonalized exactly. Its eigenvalues are identi-

fied with the masses of the low-lying 4’ and 4’

22 H. We&e1 et al. / Physics Letters B 353 (1995) 20-26

baryons [ 18,6,19]. Due to SU(3) symmetry break- ing the baryon wave-functions are distorted SU(3) D-functions. As symmetry breaking increases the wave-functions of the nucleon and the A resonance approach SU(2) Wigner-functions. I.e. as symme- try breaking increases the 5s content of the nucleon decreases and vanishes for infinitely large symmetry breaking. As the effective symmetry breaking grows as NE for large Nc the OZI rule is recovered.

3. Vector currents in the SU(3) NJL model

The vector currents -(fz are obtained by introducing external sources Z$, which couple to the quark bilinear @yP ( P/2) q. Then .Zz is the derivative of the extended action with respect to these sources

J;(x) = (6)

When transforming to the flavor rotating frame (q’ =

Rt ( t) q) and performing the Wick rotation the ex-

tended Dirac operator reads

ip& (Z7;) = - & - ha - h,t - hsn

Ab Ab - ibiD,bF - a!. baD,bT. (7)

Here he = a , p + prnU% refers to the one-particle Dirac Hamiltonian in the background of the chiral soli-

ton. Furthermore

(8)

contains the analytic continuation of the angular veloc- ities, Cng = -iCF. The symmetry breaking piece is lin- ear in the difference of the constituent quark masses 4

D ab = ktr (AnRAbR+) . (9)

The static soliton enters hsB via 7 = [PL + @PR. Furthermore bt = ibt denotes the analytic continuation

In ( 10) and ( 11) el, and qP refer to the eigenvalues and eigenstates of ho. The vacuum contribution to the current is extracted from (6) by imposing anti- periodic boundary conditions for the quark fields in a Euclidean time interval T in the limit T + co [ 21,191. Again we expand (6) up to linear order in QZa and the mass difference m - m, (contained in hsn)

4 The explicit dependence on the current quark mass difference 5 The valence quark level is defined as the state with the eigen- m0 - WZ~ is completely contained in A,. value of smallest module.

of the time component of the source current. ha label the generators of the flavor group, including the singlet piece ho = mll, which enters the projector onto

strange degrees of freedom, diag(O,O, 1) = A’/& - /P/a.

To normalize the symmetry charges consistently with the linear relation between the angular velocities and the SU( 3) generators, the currents have to be ex- panded up to linear order in both h,,, and hsB. Induced kaon components have to be included in a way con- sistent with the computation of the strange moment of inertia [ 191. Leaving symmetry breaking effects apart, the time independence of the charges associated with the vector current requires to consider CP (and therefore h,,) as classical quantities constant in time.

The currents gain contributions from both, the ex- plicit occupation of the valence quark orbits and the polarized vacuum [ 211. The former is obtained by substituting the perturbation expansion for the valence quark level 5 *“,I

(10)

into the expression for the vector current ‘@‘yP (A”/

2) Q yielding

+ h.c. . (11)

H. Weigel et al./Physics Letters B 353 (1995) 20-26 23

with the regularization functions

fg= sgn(+)erfc(1~1) -w(Ep)erfc(I?/), 6 - Ep

(13)

In (12) we have chosen to regularize the W-finite imaginary part of the Euclidean action as well. Omit- ting this regularization, which essentially corresponds to a different model, amounts to assuming the limit A --+ oo in those terms which are related to an odd number of time components of Lorentz vectors. In this context (nc has to be counted as a time component. The total current is the sum

Jfl = 71valJ;Cval) + j;Cvac) (14)

with vvat = ( 1 + sgn( ~,,t) ) /2 adjusted to describe a unit baryon number configuration. From Eqs. ( 11,12) it is suggestive that the current may be cast into a sum of products of radial functions Q(r) and isospin co- variant expressions involving the SU( 3) collective co- ordinates Dab and the velocities R” [ 71. In the frame- work of the semiclassical quantization the latter are replaced by the SU(3) generators, R,. This permits to compute the spin and flavor parts of the matrix el- ements of .lE between baryon eigenstates6. We may formally write [ 71

6 3

Jf = C Vi(r) C EijkxjMakv I=1 j,k=l

J; = 2 K(r)MP. 1=9

(15)

6A typical example for these matrix elements is M$ =

(CL, dbpDarrRp), where the dRbC refer to the symmetric

structke coefficients of SU(3); see appendix A of Ref. [6] for

details.

We define Fourier transforms of the radial functions

4(1!71) = 4%7 s

drrZj0(lqlr)F(r> , Z=9 ,..., 13,

(16)

to evaluate the spatial parts of baryon matrix elements. Identifying the momentum transfer in the Breit frame ( Q2 = -q2) gives the electric GE and magnetic GM

form factors [ 221

G”,(Q2) = -mi,r-+&lql,MP3~ I=1

G:(Q2> = 2 fi<,q,>Mf. 1=9

(17)

The nucleon mass MN = 940 MeV has been intro- duced because GM is measured in nucleon magnetons. It should be noted that recoil corrections, which are subleading in ~/NC, have been ignored. Thus ( 17) is only valid in the vicinity of 141 = 0. In (17) we have kept the full flavor structure. E.g. the electro- magnetic form factor is given by the linear combi- nation Ge.*. = G3 + Gi,/& while the strange vector fo%? factor?%clude’the singlet piece Gk,M =

G’&/& - Gi,M/&. The magnetic moments of the nucleon are defined as the matrix elements pP,,, =

(=tG&(O) +G&(O)/&), andpus = (G&CO>/&- GfdO)/&),.

4. Numerical results and discussion

Expanding the action up to quadratic order in the pseudoscalar fields allows one to express the corre- sponding masses and decay constants in terms of the parameters +z, &‘, G and A. Using the pion decay con- stant fr = 93 MeV and mass mrr = 135 MeV together with the SDE in the non-strange sector determines m”, G and A for a given value of m. Then the kaon mass mK = 495 MeV and the SDE in the strange sec- tor yield m, and my for the same value of m. The kaon decay constant f~, left as a prediction, is commonly underestimated in the NJL model [ 191. In the baryon

24 H. Weigel et al. /Physics Letters B 353 (1995) 20-26

sector the mass differences 7 of the low-lying 4’ and

s’ baryons are found to be on the small side [ 19,201. This can be understood as being caused by the too small spatial extension of the self-consistent soliton. Denoting the self-consistent profile, which extremizes the static energy functional, by 0,. (r) we may intro- duce a variational parameter, A, for the extension of the soliton via

@A(r) = @,,.(Ar). (18)

Adjusting8 A to provide an optimized description of the baryon mass differences yields 0.7 5 A I 0.8. It should be remarked that the static soliton mass is rather insensitive to h; the above range gives an increase of only 10%. The dependence of the magnetic moments on h is displayed in Table 1. The isovector part of the magnetic moment of the nucleon ,UV = pP - ,u,, is sensitive to the extension of the soliton configuration. For the self-consistent soliton configuration (A = 1) ,UV is predicted to be less than half of its experimental value, 4.70. However, ,UV increases drastically with the extension of the meson profile. On the other hand the isoscalar part (,x~ + ,x~) is quite insensitive to the extension of the soliton. In view of the remarks at the end of Section 2 it is not surprising that Ifis] decreases with h: A major effect of lowering A is to increase the symmetry breaking parameters appearing in the collective Hamiltonian as e.g. is reflected by the h dependence of the baryon mass differences.

Recently rotational ~/NC corrections to /.LV have been discussed [26] ‘. As these have the same de- pendence on the collective coordinates their incorpo- ration is effectively equivalent to a modification of

4(1!71 = 0). Actually these corrections are relevant

only for the radial function already present in the two flavor model, VI in the notation of Ref. [ 7 3. Without going into detail one may therefore adjust q (141 = 0) to reproduce ,UV. Keeping all other quantities at the values associated with the self-consistent soliton

7 For the description of the absolute values of the baryon masses quantum corrections due to meson fluctuations have to be taken

into account [231. 8 Considering A as a variable is also motivated by the existence

of sizable correlations between SU( 3) symmetry breaking and the

extension of the soliton [ 241. Furthermore A < 1 is suggested by the proper inclusion of the w meson 1251. g Here we will not pursue the question of validity of the treatments

in Ref. [27].

the strange magnetic moment of the nucleon becomes

Pus = -0.03( -0.05) for m = 400(450) MeV. Al- ternatively one may consider a hybrid treatment, i.e. take A = 0.75 from the mass differences and adjust vi ( 141 = 0) to ,UV. Although the correction factor for fi is considerably smaller in that case the strange mag- netic moment PS = O.ll(O.10) is again significantly lower than the value associated with O,, and VI un- changed.

The electric radii of the nucleon, which are given by

the slopes of the associated electric form factors (Y”) =

-6WQ2G’Q21Q~_o are also shown in Table 1. Not surprisingly the radii increase with decreasing h and the experimental data are reasonably reproduced for a somewhat larger value of the scaling variable than in the case of the magnetic moments. Taking A again from optimizing the baryon mass differences gives the estimate (r’,) M -0.15...-0.25fm2.1ncontrastto the magnetic moments the radii are not subject to even- tual l/Nc rotational corrections, however, as the radii require the form factors at non-zero momentum trans- fer, 1 /NC corrections may enter when including recoil corrections. Since only an infinitesimal deviation from non-zero momentum is required these corrections are

presumably negligible [2X]. Fig. 1 exhibits that the valence quark and vacuum

contributions to Gs,s of the nucleon almost cancel each other leading to a total strange electric form fac- tor which barely deviates from zero. The decomposi- tion into ea’ and Qac refers to (14). On the other hand G$s and Qzs add up coherently to GMJ. While

Gzs drops monotonously the vacuum part develops an extremum which is also recognized in G,JQ. A non- monotonous behavior for GM,S is also known from the Skyrme model with vector meson [ 71. We have also considered the model with the imaginary part un- regularized. For the self-consistent soliton this yields a slight increase for the strange magnetic moment ps x 0.35 while (r’,) remains within the bounds given

above.

5. Summary

Using the generalized Yabu-Ando approach to the chiral soliton of the NIL model we have investigated the matrix elements of the nonet vector current be- tween nucleon states. These are of great interest be-

H. Weigel et al. /Physics Letters B 353 (1995) 20-26 25

Table 1

Baryon properties as a functions of the scaling variable ( 18)

A

m = 400 MeV

1.0 0.9 0.8 0.7

rn = 450 MeV

1.0 0.9

FXpt.

0.8 0.7

PP 1.17 1.32 1.58 2.02 1.06 1.21 1.46 1.90 2.79 Lk -0.76 -0.90 -1.17 -1.63 -0.69 -0.84 -1.09 -1.56 -1.91 i-ks 0.24 0.21 0.18 0.15 0.23 0.20 0.18 0.16 ? x(MeV) 319 219 88 199 341 255 125 144 0

;:;;I -0.18 0.66 -0.21 0.81 -0.28 1.06 -0.43 1.43 -0.16 0.61 -0.20 0.78 -0.28 1.03 -0.43 1.40 -0.12 0.74

(r2)s 0.03 -0.08 -0.22 -0.39 -0.03 -0.12 -0.26 -0.41 ?

pp, p,, and ,us are the magnetic moments of the proton, neutron and the strange magnetic moment of the nucleon, respectively. x =

Eb;uyo”S (AM,,d - A&qn)211f2 measures the deviation of the predicted mass differences from the experimental ones. The second

part contains the electric radii of the nucleon. These data are in fm’.

Strange Electric Form Factor Strange Magnetic Form Factor

0.50 I I I I I I

- total - total -_

-- -. - - - - valence - - - - valence

-. *. ----_ vacuum ----. vacuum .

0.25 - '. . . *.

'. -.

-_ --__

c - -_

r Ll 0.00 - A - -0

__-- _----

_*-- .-

-0.50 I 1 I 0.00 I I

0 250 500 750 1000 0 250 500 750 1000

Q (Mev) Q (MeV)

Pig. 1. The strange electric and magnetic form factors of the nucleon. Parameters are m = 450 MeV and h = 0.75, which provide a good

agreement for the baryon mass differences: x = 68 MeV, CJY Table 1.

26 H. Weigel et al. /Physics Letters B 353 (1995) 20-26

cause they provide information on the strangeness con- tent of the nucleon.

To some extend the model is plagued by the too small predictions for the baryon mass differences and the isovector magnetic moment of the nucleon. To im- prove on the former we have introduced a parameter for the extension of the soliton. This parameter has been fixed by optimizing the mass differences. As a consequence the effective symmetry breaking is in-

creased, lowering the (virtual) Ss pairs in the nucleon. Further modifications, as l/Nc rotational corrections, which lead to a larger PV also cause ,us to decrease. Hence we consider the result obtained from the self- consistent soliton as an upper bound. Assuming that all corrections add coherently ,us may even assume small negative values. We therefore arrive at the NJL model estimate

-0.05 5 ,us 5 0.25. (19)

The lower bound is compatible with the corresponding prediction in the Skyrme model with vector mesons [ 71. Other models give even smaller results [ 81. We have furthermore observed that the valence quark pro- vides the major contribution to ps although the vac- uum part adds coherently to the strange magnetic form factor. This is in contrast to the strange electric form factor which is characterized by a large cancellation between the valence quark and vacuum contributions and is approximately zero. The NJL, model estimate for the slope of this form factor is

-0.25 fm2 I (r”,) I -0.15 fm2. (20)

This result compares well with that of the Skyrme model [ 61. As GEJ M 0 it is easy to understand that small effects may modify this result. As an example we refer to the 4 - o mixing, which is neither included in the present model nor in the Skyrme model.

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