Strange vector form factors of the nucleon in the SU(3) chiral quark-soliton model with the proper kaon cloud

NUCLEAR PHYSICS A

ELs!ZVIER Nuclear Physics A 616 ( 1997) 606-624

Strange vector form factors of the nucleon in the SU(3) chiral quark-soliton model with the proper

kaon cloud Hyun-Chul Kim I, Teruaki Watabe 2, Klaus Goeke 3

Institut fir Theoretische Physik II, Posifach 102148, Ruhr-Universitiit Bochum, D-44780 Bochum, Germany

Received 25 March 1996; revised 22 October 1996

Abstract

The strange vector form factors are evaluated in the range between Q2 = 0 and Q2 = 1 GeV2 in the framework of the SU(3) chiral quark-soliton model (or semi-bosonized SU(3) Nambu- Jona-Lasinio model). The rotational l/NC and m, corrections are taken into account up to linear order. Taking care of a proper Yukawa-tail of the kaonic cloud, we get (I’)?” = -0.095 fm2 and ,uu, = -0.68 ,%?I. The results are compared with several different models. @ 1997 Elsevier Science B.V.

PACS: 12.40.-y, 14.20.Dh Keywords: Strange vector form factors, Strange magnetic moments, Strange electric charge radius, Kaonic clouds, Chiral quark-soliton model

1. Introduction

The strangeness content of the nucleon has been under a great deal of discussions for well over a decade. A few years ago, the European Muon Collaboration (EMC) [ 1 ] measured the spin structure function of the proton in deep inelastic muon scattering and showed that there is an indication of a sizable strange quark contribution. This remarkable result has been confirmed by following experiments of the Spin Muon Collaboration (SMC) [ 2,3], El42 and El43 collaborations [4,5].

’ E-mail address: [email protected]. * E-mail address: [email protected]. 3 Email address: [email protected].

03759474/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved. PIISO375-9474(96)00486-l

H.-C. Kim et aLlNuclear Physics A 616 (1997) 606-624 607

Another experiment conducted at Brookhaven [ 61 (BNL experiment 734) measuring the low-energy elastic neutrino-proton scattering came to the more or less same conclusion. Kaplan and Manohar [ 71 showed how elastic vp and ep scatterings can be used to extract not only the Gt form factors of the U( 1)~ current but also the FZ form factors of the baryon number current and furthermore how the strange quark matrix elements

blW,Y5~lP) and l I- I > p sy,s p can be evaluated. Following these suggestions, Garvey et al. [ 81 reanalyzed the above-mentioned vp elastic scattering experiment and determined proton strange form factors in particular at Q2 = 0, pointing out the shortcomings of the analysis done by Ref. [6]. The best fit of Ref. [ 81 with the smallest x2 tells Ff = 0.53 f 0.70 and Fi = -0.40 f 0.72. By comparing the different Q* dependence of &/dQ2( vp) to &/dQ2( Pp), Garvey et al. favor Ff( Q2) > 0 and F$(Q2) < 0. However, these form factors are experimentally unknown to date and have no stringent and concrete constraints on their Q2-dependence yet. There are various proposals and experiments in progress (see Refs. [ 19,201 for details). All these considerations lead to the conclusion that, in contrast to the naive quark model, it is of great importance to consider strange quarks in the nucleon seriously.

There have been several theoretical efforts to describe the strange form factors of the nucleon. The first attempt was performed by Jaffe [9]. Jaffe took advantage of Ref. [ lo], i.e. the pole fit analysis based on dispersion theory, and estimated the mean- square strange radius and magnetic moment of the nucleon:(r2)y = 0.16 f 0.06 fm2, pu, = -0.31 f 0.09 &v. More recently, Hammer et al. [ 111 updated Jaffe’s pole-fit analysis of the strange vector form factors, relying upon a new dispersion theoretic analysis of the nucleon electromagnetic form factors. In fact, Hammer et al. improved Jaffe’s prediction, giving pu, = -0.24 f 0.03 p,v and (Y”)? = 0.21 f 0.03 fm2. A noticeable point of the pole-fit analysis is that it has the different sign of the strange electric radius, compared with almost other models.

Another interesting approach is the kaon-loop calculation. The main idea of the kaon- loop calculation is that the strangeness content of the nucleon exists as a pair of K/I or a components. Koepf et al. [ 121 first evaluated ps and (r’)?, considering the possible kaon loops relevant for the strange vector form factors. However, Ref. [ 121 failed to include seagull terms which are essential to satisfy the Ward-Takahashi identity in the vector current sector. Musolf et al. [ 131 added these seagull terms and obtained pu, = -(0.31 -+ 0.40) ,!.&N and (r2)p’” = -(6.68 -+ 6.90) x 10M3 fm2. The prediction of (r”)? in the kaon-loop calculation is found to be much smaller than the pole- fit analysis. To reconcile the conflict between the pole-fit analysis and the kaon-loop calculation, Refs. [ 14,181 suggested the combination of the vector meson dominance (VMD) and w -r$ mixing in the vector-isovector channel with the kaon-loop calculation. The value of (r2)p in Ref. [ 141 appeared to be larger than that of the kaon-loop calculation but still conspicuously smaller than that of the pole-fit analysis: (r2)p =

-(2.42 + 2.45) x 10u2 fm2. Ref. [ 181 evaluated also the strange vector form factors and discussed to a great extent several different theoretical estimates.

The SU(3) Skyrme model with pseudoscalar mesons [ 151 and with vector mesons [ 161 estimated, respectively, ,uu, = -0.13, J_L~ = -0.05 and (r2)p” = -0.10 fm2,

608 H.-C. Kim et al./Nuclear Physics A 616 (1997) 606-624

(?)Ip’“” = 0.05 fm2. Most recently, Leinweber obtained pu, = -0.75 f 0.30 ,&v which appears to be much larger than other models.

In this paper, we aim at investigating the strange vector form factors and related strange observables in the SU( 3) chiral quark-soliton model (xQSM), often called semi-bosonized SU( 3) Nambu-Jona-Lasinio model (NJL) . The model is based on the interaction of quarks with Goldstone bosons and has been shown to be quite successful in reproducing static properties of the baryons such as mass splitting [ 21,221, axial constants [ 231 and magnetic moments [ 241 and their form factors [ 25,261. In a recent review [ 271, one can easily see how well the model describe the baryonic observables. In particular, since the strange vector form factors are deeply related to the electromagnetic form factors [ 9,111 being well described in xQSM, it is quite interesting to study them in the same framework.

The strangeness content of the nucleon can be interpreted in terms of the AK- and Z K-components [ 12,131. It implies that one need incorporate the kaonic cloud properly in order to calculate the strange vector form factor. In fact, the strange electric charge radius in a hedgehog model is proportional to the inverse of the kaon mass, which means that it is very sensitive to the tail of the kaonic cloud. Hence, it is of great significance to take care that the kaonic cloud has a proper Yukawa asymptotics in order to evaluate the strange vector form factors rightly. In this respect the present paper provides a clear improvement over the results of Ref. [ 271. In a recent study on the kaonic effect on the neutron electric form factor [28], it was shown that even the calculation of the neutron electric form factor requires a proper kaon tail. In the same spirit, we expect that the kaonic cloud will have a decisive effect on the strange vector form factors.

The outline of the paper is as follows: Section 2 sketches the general formalism for obtaining the strange vector form factors in the framework of xQSM. Section 3 presents the corresponding results and discuss them. Section 4 contains a summary and draws the conclusion of the present work.

2. General formalism

2.1. Basics

In this section we briefly review the formalism of xQSM. Details can be found in Ref. [ 271. We start with the low-energy partition function in Euclidean space given by the functional integral over pseudoscalar meson (ti) and quark fields(@) :

where Set is the effective action

&n[n-] = -Sp IniD.

iD represents the Dirac differential operator

(2)

H.-C. Kim et al/Nuclear Physics A 616 (1997) 606-624

iD=p(-is+r%+MUY5),

with the pseudoscalar chiral field

609

(3)

(Iy5 = exp (in-“heys) = 2 l+Y5u+ 1-B + --+.

J% is the matrix of the current quark mass given by

% = diag(m,,md, m,) = ?‘?‘@I + mg&, (5)

where A“ designate the usual Gell-Mann matrices normalized as tr( A4hb) = 2Sab. Here, we have assumed isospin symmetry (m,, = md) . A4 stands for the dynamical quark mass arising from the spontaneous chiral symmetry breaking, which is in general momentum- dependent [ 291. We regard A4 as a constant and introduce the proper-time regularization for convenience. The mo and mg in Eq. (5) are defined, respectively, by

ml1 -f md + ms ?%+md-%% mo = 3 ,m8=

2J5 . (6)

The operator iD is expressed in Euclidean space in terms of the Euclidean time derivative & and the Dirac one-particle Hamiltonian H( UR )

iD=&+H(UYS) +ph-/?priil (7)

with

CX.V H(UY5) = -

i + pMiF + pti1. (8)

fi is introduced in such a way that it produces a correct Yukawa-type asymptotic behavior of the profile function. p and LY are the well-known Dirac Hermitian matrices. The U is assumed to have a structure corresponding to the so-called trivial embedding of the SU( 2) -hedgehog into SU( 3) :

u= uo” ( > 01 ’ (9)

with

Ua=exp[in.~P(r)]. (10)

The partition function 2 of Eq. ( 1) is simplified by the stationary phase approximation, which is justified in the large NC limit of 2. One ends up with one stationary profile function P(r) which is determined numerically by solving the Euler-Lagrange equation corresponding to SS,ff/SP (r) = 0. This yields a selfconsistent classical field UO and a set of single quark energies and corresponding states E,, and P,. Note that the En and !Pfl do not constitute the nucleon IN) yet because the collective spin and and isospin quantum numbers are missing. Those are obtained by the semiclassical quantization procedure, described below in the context of the strange form factors. For this quantization, which is performed with the given profile P(r) we assume besides adiabatic rotation NC = 3.


This inconsistency with the limit of large NC is well known. It is usually accepted because due to this a diverging number of strange valence quarks, as discussed in Ref. [ 181, is avoided. Hence, our philosophy consists in assuming large NC for getting the stationary point, and NC = 3 for the collective quantization.

The information of the strange vector form factors in the nucleon is contained in the quark matrix elements as follows:

(N’(p’) IJ;IN(p)) = (N’(p’) I:~,slN(p)). (11)

The strange Dirac form factors of the nucleon are defined by the matrix elements of the J;:

(N’(p’)IJ;IN(p)) = .%v(P’> i r/2’h2) +&&‘4” w(P>,

I (12)

where q2 is the square of the four momentum transfer q2 = -Q2 with Q2 > 0. MN and UN(p) stand for the nucleon mass and its spinor, respectively. The strange quark current Ji can be expressed in terms of the baryon current and the hypercharge current:

J; = syPs = J; - J; = ijy&q, (13)

where

(14)

where NC = 3 denotes the number of colors of the quark. Qs = diag(O, 0, 1) is called strangeness operator: We employ the non-standard sign convention used by Jaffe [9] for the strange current. The baryon and hypercharge currents are equal to the singlet and octet currents, respectively.

The strange Dirac form factors Ff and Fg can be written in terms of the strange Sachs form factors, G&( Q2) and GL ( Q2) :

%(Q2> = F;(Q2> - $F;(Q2), N

G%Q2> = WQ2> + WQ2>. (15)

In the non-relativistic limit(Q2 < M$), the Sachs-type form factors G$(Q2) and G&( Q2) are related to the time and space components of the strange current, respectively:

(N’(P’) I J;(O) IN(p)) = G:(Q2> 7

(N’(P’) i J~w~N~P~~ = ~~~(~‘)ie~~~y’(~ll~~l~), (16)

where (Tk stand for Pauli spin matrices. The IA) is the corresponding spin state of the nucleon. The matrix elements of the strange quark current can be related to a correlator:

(N’(P’) I~Y,LsIWP))~_N~~ .+

(01 JNT(X, ~/2)~~;y,QsqJ~(y~ -7’P) IO). (17)

611 H.-C. Kim et al. /Nuclear Physics A 616 (1997) 606-624

The nucleon current JN can be built from NC quark fields

1

a1 . . . ffN, denote spin-flavor indices, while il . . . iN, designate color indices. The matrices IT&$?; are taken to endow the corresponding current with the quantum numbers JJ37T3Y. It is interesting to contrast the present approach with the one of the Skyrme model. In the Skyrme model one defines J; by the difference of the baryon current, J;, and the hypercharge current, Jz. Since they are of quite different nature, Ji being topological and JL being a Noether current, Forkel et al. [ 181 argue that the evaluation of strange properties via JL gives rather uncertain results. Thic argument does not hold in the present model since the J; = Zaps is obtained by the quark fields directly and Ji = J; + Ji is an outcome of the approach. Still we face also in our model the numerical problem that the strange form factor, being a small quantity, is the difference of the baryon- and hypercharge form factors being large quantities. Hence a comparison of the theory with the experiment, once to be performed, will be as delicate as e.g. in the well known case of the neutron electric form factor.

2.2. Details

In our model, Eq. ( 17) is represented by the Euclidean functional integral with regard to quark and pseudo-Goldstone fields:

(N’(P’) I@J&IN(P)) = -!j $im_ exp (W”/2 - i@/2)

X s

d3xd3yexp(-ip’.y+ip.x)SDU J2+ JD$t x JN/(Y,T/2)qt(0)py~,$,q(O)J~(x,-T/2) x exp [-/d4qbtiD$], (19)

where Z stands for the normalization factor which is expressed by the same functional integral but without the quark current operator Sy,s. Eq. ( 19) can be decomposed into valence and sea contributions:

(N’(P’) Itiy,&sqlNp))

= (N’(p’) tti&&IN(p)h + (N’(p’)I~~$,qlN(p)),ea, (20)

where

(N’(P’) IV,(O) lN(p>)v,~ = 1 PL...PNc ~rJrJ/I,&;;~;* 3 3 ;irnm exp (ip4T/2 - ipAT/2)

d3xd3y exp (-ip’ . y + ip . x)


and

(N’($) IV,(O) IN(p)),,, = _N,I+~~c ra”“aNc* * z J~J;T~T;Y~ JJ3Tr3Y $% exP (@J/2 - @@/2)

X J d3xd3yexp(-ip’.y+ip.x)

x DUcxl,(-S,~)Tr.r(O,t,l~[py,l&,lO,t,) J x fi 6 b’y T/21 &Ix, -T/2)Lyi

i=l (22)

S,, is the effective chiral action expressed by

$8 = -N,Spln [a, + H( W”) + @I - /Iid] .

In order to perform the collective quantization,

(23)

we have to integrate Eqs. (2 1) and (22) over small oscillations of the pseudo-Goldstone field around the saddle point Eq. (9). This will not be done except for the zero modes. The corresponding fluctuations of the pion fields are not small and hence cannot be neglected. The zero modes are relevant to continuous symmetries in our problem. In particular, we have to take into account the translational zero modes properly in order to evaluate form factors, since the soliton is not invariant under translation and its translational invariance is restored only after integrating over the translational zero modes. Explicitly, the zero modes are taken into account by considering a slowly rotating and translating hedgehog:

0(x, t) = A(t)U(x - Z(t))A+(t). (24)

A(t) belongs to an SU( 3) unitary matrix. The Dirac operator id in Eq. (7) can be written as

ib= (&+H(P) +A+(t)A(t) -ipi.V+PAt(t)(fFz-fil)A(t)). (25)

The corresponding collective action is expressed by

$:,ff=-NcSpIn[d,+H(UY5)+At(t)A(t)-i~i~V

-tPA+(t)(A - fil)A(t) - PA+WSpyp&SA(t)] , (26)

(27)

with the angular velocity

H.-C. Kim et al/Nuclear Physics A 616 (1997) 606-624 613

and the velocity of the translational motion

i=$Z. (28)

Hence, Eq. (21) and Eq. (22) can be written in terms of the rotated Dirac operator iD and chit-al effective action &n. The functional integral over the pseudoscalar field U is replaced by the path integral which can be calculated in terms of the eigenstates of the Hamiltonian corresponding to the collective action and these Hamiltonians can be diagonalized in an exact manner.

We take into account the rotational l/NC and m, corrections up to linear order:

1 -2’ iD

&(-p&in - tiii~)&. 7

(29)

When the mass corrections are considered, SU(3) symmetry is no more exact. Thus, the eigenfunctions of the collective Hamiltonian are neither in a pure octet nor in a pure decuplet but in mixed states with higher representations:

I8,N) = I8,N) +ciblfO,N) +c27127,N) (30)

with

Ci~=G(~-r~)12?‘l’l,, C27=~(30.fI1-4r2)I2m,. 15 75

(31)

The constant c is related to the SU(2) riV sigma term ZSU(Z) = 3/2(m,+md)a and pi designates Killi, where Ki stand for the anomalous moments of inertia defined in Ref.

[211. Having carried out a lengthy manipulation (for details, see Ref. [ 271)) we arrive at

our final expressions for the strange vector form factors. The Sachs strange electric form factor Gi is expressed as follows (see Appendix A for detail):

WQ2> = (1 - (0:;))~) B(Q2> + (D!f’JnjN2’&f2) 1

+ (D$J& 2z$f2) 2

+3 (

mu-fi+“28(D$;))N C(Q2) d3 >

- (D::‘@,N ;‘I - (WWQ2) -&(Q2>&)

- (D$‘:;))N~ (Z2K2(Q2) 7 22(Q2)Kz) . (32)

Zi and Ki are the moments of inertia and anomalous moments of inertia [ 211, respectively, B, Ii, and Ici correspond to the baryon number, moments of inertia, and the anomalous moments of inertia at Q* = 0, respectively. From Eq. (32)) we can easily see that at Q2 = 0 the strange electric form factor Gi vanishes (note that C( Q2 = 0) = 0). Making use of the relation Cf=, Dii’Ja = -&I72 and Jg = -N,/(2&), we obtain


G$( Q2 = 0) = B - Y = S. Since the net strangeness of the nucleon is zero, G; at Q2 = 0 must vanish. The final expression of the Sachs strange magnetic form factor is written (see Appendix A for detail) by

G&<Q’> = f$

4 (Q2> + (<D$ - 11J3)NT

;uz(Q2) 1

+ (d3pqD~;)~q)N~pq- da*

+ 6(mo - *)(D~:‘)NMo(Q~) + ~~~~(D~~‘D~~‘)NMo(Q~)

+ mo(Di!‘)~ 2M1(Q2> - Lr~X~(Q2) ( ds >

+ ~~(D$D$:‘)N 2M,(Q2> - $~IXI(Q~) >

+ ~rns(dgpqD~~‘D~~‘)NSpg 2

2M2(Q2> - -r2X2(Q2) . 3 )I

3. Results and discussions

In order to evaluate Eqs. (32), (33) numerically, we follow the Kahana-Ripka dis- cretized basis method [30]. However, note that it is of great importance to use a reasonably large size of the box (D M 10 fm) so as to get a numerically stable results. The present SU( 3) xQSM (equivalent to SU( 3) NIL on the chit-al circle) contains four free parameters. Two of them are fixed in the meson sector by adjusting them to the pion mass, mm = 139 MeV, the pion decay constant, fn = 93 MeV, and the kaon mass, rnK = 496 MeV. As for the fourth parameter, i.e. the constituent mass M of up and down quarks, values around M = 420 MeV have been used because they have turned out to be the most appropriate one for the description of nucleon observables and form factors (see Ref. [ 271) . In fact, M = 420 MeV is the preferred value, which is always used in this paper. For the description of the baryon sector, we choose the method of Blotz et al. [ 211 modified for a finite pion mass. The resulting strange current quark mass comes out around m, = 180 MeV. In order to illustrate the effect of the m, the calculations in the baryonic sector are performed with both m, = 0 and finite m,. One should note that a SU(3)-calculation with m, = 0 does not correspond to a SU(2) calculation, since the spaces, in which the collective quantization are performed, are different.

In the present calculation, the mass parameter %r plays a pivotal role, because it makes the solitonic profile P(r) of Eq. (10) incorporate the proper Yukawa-type tail:

1+pi” P(r) m exp C-w-) ,z,

where ,LL denotes the meson mass suppressing the tail of the profile. In fact, /.L is related to the fi in a non-linear way whose details can be extracted from the meson expansion

H.-C. Kim et al. /Nuclear Physics A 616 (1997) 606-624 61.5

0.10 ! I I

m, = 180 MeV -.---.- M = 400 MeV _

M = 420 MeV 0.08- ’ = mK ------ M=430MeV -

/__--------____ .’ -.._

__-------_____ --_ --_

1.0

Q’ [GeV’]

Fia C’ I, The strange electric form factor Gi as functions of Q* with the p = mK: The solid curve corresponds to the constituent quark mass M = 420 MeV, while dot-dashed curve draws M = 400 MeV. The dashed curve displays the case of A4 = 450 MeV. The M = 420 MeV is distinguished since all other observables of the nucleon are then basically reproduced in this model.

of Ref. [28]. In the end it turns out that when 172 = (m, + md)/2, the ,u becomes the pion mass m, = 139 MeV, while ,U corresponds to the kaon mass’mK N 490 MeV for the fi P 75 MeV. Actually, since the hedgehog formalism forces us to have just one profile function and hence all mesonic fields to have the same Yukawa mass in the tail, one has to decide if one wants the pion tail OY the kaon tail to be correct. For previous investigations of electromagnetic properties of the nucleon it was preferable to have a correct pion tail at the expense of a poor kaon tail and in this sense all calculations of Ref. [ 271 have been performed. For the present investigation of the strange vector form factors it is more desirable to have a kaon tail with a Yukawa mass, p, being equal to the kaon mass, rnK, and hence we prefer in this paper ft E 75 MeV corresponding to ,CL = rnK E 490 MeV. We know that the pion tail is now too short. However, we do not expect it to matter for the strange vector form factors. We give the results with p=m,= 139 MeV for comparison. Fig. 1 shows the strange electric form factor G$( Q2), as the constituent quark mass M is varied from 400 MeV to 450 MeV with m, = 180 MeV. The strange electric form factor Gi decreases as M increases. Fig. 2 displays the baryon and hypercharge densities weighted with r2.

In Fig. 3 the effect of the kaonic cloud is well visible. The Gi with ,U = rnK is almost three times smaller than that with ,X = m,. Apparently the present investigation with ,U = mk provides a clear progress over our present results [ 271. This remarkable outcome is in line with the recent investigation of the kaonic effects on the neutron electric form factor [ 281.

These kaonic effects can be understood more explicitly by evaluating the strange electric radii. The Sachs and Dirac mean-square strange radii are, respectively, defined

616 U.-C. Kim et al./Nuclear Physics A 616 (1997) 606-624

3

Fig. 2. The baryon and hypercharge densities as functions of r. The solid curve draws the baryon charge density with m, = 180 MeV while the dashed one is for the hypercharge density.

M = 620 MeV 0.20, 3 3 11

:

m, = 180 MeV <------- -_ I

<’ ._._

0.15 , . .

I’ . . . . _. I

/’ .._

I I ,’ I

O.lO- lI _p=mK -

I’ ____-- p”m”?r

I’

1.0

Q” [GeV’] Fig. 3. The strange electric form factor 4 as functions of Q*: The solid curve corresponds to the ,u = inK, while dashed curve draws p = m,. The constituent quark mass M and ?nn, are 420 MeV and 180 MeV, respectively.

(3)fachs = _edGj$‘) j Q&O ’

(r2)p = -(jdF;$‘) Q*=O

(3.5)

we obtain (r-2)fachslfi=nI, = -3.5 x lo-* fm2 and (r2)~IFmK = -3.2 x 10-l fm2 with

H.-C. Kim et al./Nuclear Physics A 616 (1997) 606-624 617

M = 420 MeV

0.6

Fig. 4. The strange electric density r2& as functions of r: The solid curve corresponds to the p = mu, while dashed curve draws y = mr. The constituent quark mass M and IQ are 420 MeV and 180 MeV, respectively.

the pion tail, whereas we get (r2)~hslp=nzK = -9.5 x 10e2 fm2 and (r2)FiP, =

-5.0 x 10e2 fm2 with the kaon tail. Again, we find that the case of ,U = mx is three times smaller than that of ,U = m,. The mean-square strange radii depend on the meson mass ,U which suppresses the tail of the profile, i.e. (r2)phs N l/p. From such a behavior of the (p2)yhs, we can derive the relation

(36)

Eq. (36) explains the decrease of the (r2), with p = mx. Fig. 4 illustrates the strange electric densities weighted with r2. As expected from the

above discussion, the proper treatment of the kaon cloud diminishes the strange electric density sizably.

Fig. 5 draws the strange magnetic form factor. In contrast to the G& the G& increases slowly with the increasing constituent quark mass apart from the small Q2 region (below about Q 2 = 0 2 GeV2). Fig. 6 displays the effect of the kaonic cloud on the G&. With . p increased to be ??‘LK, we find that the Gb is almost 50 % enhanced. Fig. 7 draws the corresponding magnetic densities weighted by r*.

In Table 1, the strange magnetic moments ,uu, and mean-square strange radii (r2), are displayed as a function of M and m, in the case of p = mK, while in Table 2 the same in the case of ,LL = m,. According to our philosophy the values of Table 1 with p = mx are the results of the present model. If one fixes the constituent quark mass A4 to a value of M = 420 MeV then other baryonic properties such as the octet-decuplet mass splitting and various form factors are reproduced as well. Hence, we have M = 420 MeV as canonical value. In Table 3, we have made a comparison for the pLs and (r2)yhs


0.0 I I I I

-0.2

-o’8 - ------ m, M=450MeV - = 180 MeV ~ M = 420 MeV - /1 = nlK -----.- M 400 MeV =

-1 *B!o ) I I I I I

0.2 0.4 0.6 0.8 1.0

Q” [GeV’]

Fig. 5. The strange magnetic factor Gb as functions of Q2 with the p = Ink: The solid curve corresponds to the constituent quark mass M = 420 MeV, while dot-dashed curve draws M = 400 MeV. The dashed curve displays the case of M = 450 MeV. The M = 420 MeV is distinguished since all other observables of the nucleon are then basically reproduced in this model.

M = 420 MeV

0.01 I , I

-0.8

t

-- p=m, ~ p=mK

m, = 180 MeV

I I

-%0 I I I I I

0.2 0.4 0.6 0.8

Q2 [GeV"]

1. 0

Fig. 6. The strange magnetic form factor Gt as functions of Q2: The solid curve corresponds to the p = ink, while dashed curve draws p = NZ,,.. The constituent quark mass A4 and ms are 420 MeV and 180 MeV, respectively.


M = ‘420 MeV

m, = 180 MeV -

Fig. 7. The strange magnetic density r*& as functions of r: The solid curve corresponds to the p = mu, while dashed curve draws y = ?nrr. The constituent quark mass M and m, are 420 MeV and 180 MeV, respectively.

Table I The strange magnetic moments and mean-square strange radius varying with the constituent quark mass. The kaon tail shows a Yukawa mass of p = rnK N 490 MeV. Our final values are in this table with M = 420 MeV

M 400 MeV 420 MeV 450 MeV

IQ [MeV] 0 180 0 180 0 180

PsbNl -0.66 -0.69 -0.65 -0.68 -0.63 -0.65 (r2)yrac [fm*] 0.144 -0.081 0.120 -0.051 0.090 -0.044 (r2)~hs[fin2] 0.100 -0.127 0.077 -0.095 0.049 -0.086

Table 2 The strange magnetic moments and mean-square strange radius varying with the constituent quark mass. The kaon tail shows a Yukawa mass of p = rum = 139 MeV

M 400 MeV 420 MeV 450 MeV

m, [ MeV] 0 180 0 180 0 180

/&[/&VI -0.81 -0.42 -0.78 -0.44 -0.74 -0.50 (rz)pw [fm*] -0.20 -0.36 -0.19 -0.32 -0.16 -0.27 (r2)yhs[fm2] -0.25 -0.39 -0.25 -0.35 -0.21 -0.31

between different models. We want to take the occasion to comment on Ref. [ 331 which provides calculations for

a correct pion tail and poor kaon tail, p = mT. Though Ref. [ 331 seems to use in this case the same model as the present work, there are significant differences between these two

620

Table 3

H.-C. Kim et al/Nuclear Physics A 616 (1997) 606-624

The theoretical comparison for the strange magnetic moment and mean-square strange radius between different models. M = 420 MeV, IQ = 180 MeV, and p = nr~ are used for the present work

Models PsbNI (r2)~hs[fm2] Refs.

Jaffe Hammer et al.

Koepf et al. Musolf & Burkhardt

Cohen et al. Forkel et al.

Park & Weigel Park et al. Leinweber

Alberico et al. Weigel et al.

S’J(3) xQSM

-0.31 It 0.09 -0.24 f 0.03 -2.6 x lo-*

-(0.31 --f 0.40) _

-(0.24 + 0.32) -

-0.05 -0.13

-0.75 i 0.30 -0.14

-0.05 - 0.25 -0.68

0.14f0.07 0.23 f 0.03

-0.97 x 10-2 -(2.71 - 3.23) x lo-* .(3.99 -+ 4.51) x 10-2

1.69 x lo-* 0.05

-0.11

0.055 -0.2 --f -0.1

-0.095

[91 [Ill r121 [131 I141 [I81 [161 [151 [171 [321 [331

Present work

papers. First, Weigel et al. [33] do not consider rotational l/NC corrections in contrast to the present paper. This has the immediate consequence that the magnetic moments of Weigel et al. are pP = 1.06 ,%N, pR = -0.69 PN for the nucleon 4 whereas the present work (including those corrections) yields pP = 2.20 PN, ,+, = -1.59 ,%N with a far better comparison with experiment ( ,uP = 2.79 PUN, p,, = -1.91 PN) . Furthermore, Weigel et al. regularize, besides the real part of the action, also the imaginary one. This meets problems in producing the anomaly structure and is hence avoided in the approach of the present work. In addition the calculation of Weigel et al. are not fully self-consistent but use some scaling approximations.

4. Summary and conclusion

In summary, we have calculated in the SU( 3) chiral quark-soliton model (xQSM) often called the semibosonized SU( 3) Nambu-Jona-Lasinio model, the strange electric and magnetic form factors of the nucleon, Gg and Gh including the strange magnetic moment pus, and the mean-square strange radius (r2)$. The theory takes into account rotational l/NC corrections and linear m, corrections. Choosing the parameters of the model in such a way that the kaon cloud falls off with a Yukawa mass equal to the kaon mass, we have obtained cc, = -0.68 PUN, (r’)? = -0.051 fm* and (?-*)yhs = -0.095 fm2. The results have been compared with different other models.

There are several points where the present calculations leave room for further studies. Apparently the dependence of the form factors on the value of m, is quite noticeable and probably one has to go to higher orders in perturbation theory in m,. Besides the strange

4 For this comparison, the constituent quark mass M = 450 MeV is chosen. In case of M = 420 MeV, we have obtained & = 2.39 ,UN and pn = - 1.76 ~LN [ 241.


vector form factors the strange axial form factors are also of great interest. Presently we are performing investigations to clarify these questions.

Acknowledgment

We would like to thank Chr.V. Christov, P.V. Pobylitsa, M.V. Polyakov and W. Bro- niowski for fruitful discussions and critical comments. This work has partly been sup- ported by the BMBF, the DFG and the COSY-Project (Jtilich) .

Appendix A

In this appendix, we present all formulae appearing in Eqs. (32)) (33).


GA(Q*> = +C J'd3xjl(qr) Jd3y ”

(-4.3)

G?2(Q2> = $C Jd3nji(qr)Jd3y Ino ly’;owYs{i X c). 7~“al(x)~~,1(Y)w;1*o(Y)

4,~ - J-h

Jfl(Q*> = NcC Jd3xjlCqr) Jd3y ,f

H.-C. Kim et al. /Nuclear Physics A 616 (1997) 606-624 623

M(Q2> =

+

&(Q2> =

NC 3 CJ'

d3Xjl (p-1 d3Y n J

The regularization functions for the G& are

WEi,) = J

-$$b(u; Ai) /El71 eC”Ez,

Rg(E,,,Em) = $ci ’ aXE,+E,,) -E,,exp(-[aE~+(l-cr)E~~l/nT)

J da

VW=3 aE;+(l-c~)k$$~ ’ 0

(A.5)

The cutoff parameter C$ (u; Ai) = xi c$(u - l/A:) is fixed by reproducing the pion decay constants and other mesonic properties [ 271.


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Strange vector form factors of the nucleon in the SU(3) chiral quark-soliton model with the proper kaon cloud