Gluon, pion and confinement exchange currents in the nucleon

NUCLEAR PHYSICS A

Nuclear Physics A569 (1994) 661-688 North-Holland

Gluon, pion and confinement exchange currents in the nucleon A. Buchmann Institute for Theoretical Physics, University of Hannover, Appelstrasse 2, W-3000 Hannover, Germany

E. Hemandez

’

Department of Physics, Tokyo Metropolitan University, Minami Osawa I-1, Hachioji-shi, Tokyo 192-03, Japan

K. Yazaki Department of Physics, Faculty of Science, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan Received

1 July 1993

Abstract We calculate the magnetic form factors of the nucleon in the constituent quark model, including gluon and pion exchange currents. Furthermore, we take into account the effect of configuration mixing. We also study the effect of the exchange current connected with the confinement potential.

1. Introduction The constituent

quark model

(CQM)

has been remarkably

successful in explaining

many properties of baryons and mesons, such as their excited spectra and their electromagnetic properties [l-15]. In this model, nonperturbative effects coming from multigluon and pion exchange between quarks are effectively described by a long-range contining potential and a constituent

quark mass. The residual one-gluon and one-pion exchange

interactions are then responsible for the d-N mass splitting. In addition, they admix excited configurations into the ground-state wave functions and are crucial in obtaining the experimental N and d excitation spectrum. Consequently, these residual gluon and pion degrees of freedom should also play an important role in the low energy electromagnetic properties of N and A baryons. To include them, we calculate the one-gluon and onepion exchange currents consistent with the one-gluon and one-pion exchange potentials. These exchange currents provide an effective description of the gluon and pion degrees of freedom in the electromagnetic properties above and beyond those that are already ’ Present address: Institute of Theoretical Physics, University of Regensburg, W-8400 Regensburg, Germany. 03759474/94/$07.00 0 1994 - Elsevier Science B.V. All rights reserved SSDI 0375-9474(93)E0461-G

662

A. Buchmann et al. /Exchange currents

included in the wave function. Most importantly, however, they are needed in order to satisfy the continuity equation for the total quark current. Recently, we studied the effect of gluon and pion exchange currents on the charge radii and form factors of the nucleon [ 71. While the proton charge radius and form factor were only slightly affected, the neutron charge radius and form factor were strongly influenced by gluon and pion exchange currents. Using pure (OS)3 ground-state wave functions for N and A, we found a relation between the neutron charge radius, the A-N mass splitting, and the quark core radius b, namely (Y~)~= - b2 (ii& - hfN )/ii&. Inserting the experimentally measured neutron charge radius and the d and N masses, we obtained b = 0.6 fm for the quark core radius. Finally, by including configuration mixing, relativistic corrections in the one-body charge density, and exchange currents, we could describe the charge radii and form factors fairly well up to momentum transfers q x mp, even for a small quark core radius of 0.6 fm. In the present paper we extend our study of gluon and pion exchange currents and calculate the magnetic form factors, moments, and radii of the nucleon in the CQM. Earlier works in the CQM differ widely in their estimation of pion and gluon cloud cont~butions to baryon magnetic moments. For example, the effect of pions is found to be small by some authors [S-lo] while others find an appreciable contribution of the pion cloud [ 1l- 15 1. The effect of the gluon contribution to baryon magnetic moments is even less well understood. On the other hand, in chiral bag models, the relative importance of pions and gluons seems to be better established by various authors. There, the pion cloud typically accounts for 30-5046 of the magnetic moment of the nucleon [ 16-181, while the gluon cloud generally contributes less than 10% [ 17- 19 1. See however ref. [ 20 ] for an exception. A better understanding of pion and gluon cloud effects in the CQM is clearly desirable. Although there are already a few calculations in the CQM that deal with pion [21] and gluon 122-241 exchange currents separately, there is no calculation in which gluon and pion exchange currents have been evaluated consistently with the gluon and pion exchange potentials as required by gauge invariance. In the present work, the gluon and pion cloud ~ont~butions to the magnetic properties of the nucleon are strictly related to the strong interaction hamiltonian by gauge invariance. Consequently, our calculation provides a consistent estimate of gluon and pion cloud effects in the excitation spectrum and various electromagnetic properties. In addition, we construct a gauge invariant current connected with the nonperturbative part of the hamiltonian and explore the consequences of a scalar- and vector-type confinement model on the electromagnetic properties. The paper is organized as follows. In sect. 2 we review the CQM and the determination of the parameters. The electromagnetic currents, including the pion, gluon and confinement exchange currents, are introduced in sect. 3. In sect. 4 we calculate the magnetic properties of the nucleon for (OS)3 ground-state wave functions, while in sect. 5 all Properties are recalculated including the effect of configuration mixing. The numerical results are discussed in sect. 6, and the main results of this work are summa~zed in sect. 7.

A.

~uc~~unn et al.

663

/Exchange currents

2. The ~nstitueRt quark model In the CQM a baryon is described as a nonrelativisti~ three-quark system in which the quarks interact via two-body potentials simulating the main features of the underlying theory of qu~tum c~omodyn~ics (QCD) [ 3,s 1. 2.1. THE ~MILT~NIAN The hamiltoni~

of the CQM in the case of three equal masses mq is given by

where m4 is the constituent quark mass, and Yi,pi are the spatial and momentum coordinates of the ith quark, respectively. The first two terms in Eq. (2.1) represent the nonrelativistic kinetic energy of the quarks. The kinetic energy of the center of mass motion of the three-quark system is removed by the third term, where P (see Eq. (2.5)) is the center of mass momentum of the baryon. The fact that the center of mass motion can be exactly removed is one of the main advantages of the nonrelativistic quark model. The fourth term is a two-body ba~oni~ oscilIator ~on~nement potential Vccnffri,rj)

=

-&iii

'Aj(J+i

-

Fj >",

(2.21

where Ai is the SUf3) color matrix, The first four terms of Eq. (2.1) define the unperturbed ham~tonian I&. We take the ei~enfunctions of NOas a convenient basis for expanding the baryon wave function. The residual interaction V” consists of two pieces: the one-muon exch~ge potenti~

(2.3)

x

exp(-A _ Y

exp(-Ar) Y

> ’

(2.4)

where Y = jr/ = /ri - Pj /. Here, Ci and ri denote the spin and isospin of the ith quark, respectively. In the present work we include both spin-orbit terms of VooEP, which were neglected in our previous work f7 ]_ The second term in Eq. (2.4) takes the finite size of the pion-quark vertex into account. The pion-quark cutoff mass /i describes the size of the pion-quark interaction region. The Iarger the cutoff mass A, the more point-like

A. Buchmann et al. I Exchange currents

664 the pion-quark one-pion

interaction.

For A --+ CQwe recover the full S-function

interaction

in the

exchange potential.

2.2. BARYON GROUND-STATE

WAVE FUNCTION

We start with the simplest assumption that three quarks with equal masses mq are in the lowest (0~)~ harmonic oscillator state, denoted by !&(A). To separate the internal motion from the center of mass motion we introduce the coordinates P= 1

R

=

=

II- ;(rl

-rz),

&,

+

$h

+

P,

r2

r2

+

-

%

=

),

r3),

$@*

$

Pn

p

-Pz),

=

=pt

$

:

+Pz

(PI

+

P2

-

2f’3),

+p3.

Using these coordinates, the kinetic and confinement pendent oscillators with frequency u:

12.5)

part of H separate into two inde-

2 HO

=

4f2m,

+ $m,ut2p2

+ f$

9

+ imgw2n2,

where the relation between w and aC is given by ac = hmqo2. constant b is defined as b = ,/w. The ground-state eigenfunction of & is then

The harmonic

= (&)“‘(--&)“‘exp(-$

@YA,P)

Since the color part of the baryon wave function

(2.6) oscillator

(2.7)

-6).

[3,301 must be totally antisymmetric

and the spatial part of the lowest lying (0~)~ state is completely symmetric, the spin and isospin part of the ground-state wave functions is given by the symmetric combinations cN

-131

=

4 (Qty

+

@“x”),

=d = - 131

$

@KS,

(2.8)

where +,,

(2.9a)

= ;)I(&

= l@ $)T = ;,,

(2.9b)

=

=

@PxP = \(Sr2 = O@ 4,s

=

@“x1 = I(&

= l@ 4,s

@sxs = I(&

= 1 (lp $9

The total wave function

4))(T*2

=

$)I(TlZ

YNtdj is then a direct product

08

1 C+f,r

i%(b)

= 3m, +

=

= 2,.

(2.9~)

of the spatial, spin-isospin

color parts. The masses of the N and d are given by the expectation (ul, } H 1YA}, respectively.

f)T

values (%

and

I ff 1 %I) and

We obtain

$9

+ 24&b2 - 2a,

- ;A(&, (2.10a)

3 21 1 Md (b) = 3% -+-2m9b2 + 24acb2 - 2cxs --g + ~~_-._3m{Gb3 J

1

- f&(b), (2.1Ob)

where the ind~yidua1 terms in Eq. (2. IO) are the kinetic, quadratic con~nement, ghron and piun energy contributions, respectively. In the case ofa general co~~nement potential we have to replace the third term in Eq. (2.10) by ( Vconf), Subtracting Eq. (2.1 Oa) from Eq. (2.10b) one gets (2.1 la) &-/MN = 8, +&, where 6, and S, are the pion and ghron cont~b~tions to the B-N mass splitting which are given by d,(b)

= 4crs/3&m~b3,

6,(b)

= -4&-4$$

(2.1 lb)

resp~tively. For brevity we write & = SXI1 - S,,, in the follo~ng. 2.3. ~~TER~INAT~ON

OF THE PARROTERS

FOR THE GROUND STATE

The parameters of the CQM are the quark-muon coupling constant as, the con~nement strength aC,the harmonic oscillator parameter b, and the pion-quark cutoff mass A. The con~nement strength aCand size parameter b are used as free parameters here. They are not related via ac = l/ 16m,b4, as in the case of a pure harmonic oscillator potential. In principle, the constituent quark mass mq could also be varied. Here, we keep it fixed to m4 = fiM~ = 313 MeV, in view of the many appIicatio~s of the CQM in nuclei [25-351 where this value is the most natural one. Hence, we use the constraint ~~(b~

= 3m,.

(2.12)

As in our previous calculation [ 71 we take the value n =L 4.2 fm-’ [ 3 11. That fixes the pion contribution, 6,(b), to the d-N mass splitting. We then determine as from Eq. (2,l la) and obtain Lys= ~~rn~b3(~~~

-

&fN

-

&{b)).

(2.13)

Next, we insert this expression for as into the formula for the nucleon mass MN of Eq. (2.1 Oa) and, using Eq. (2.12 1, obtain an expression for the confinement strength ac as a function of b. Finally, we determine the oscillator parameter b from the variational condition by partiat differentiation of Eq. (2.10a) with respect to b: ~~N(b~/~b

= 0.

(2.14)

666

A. Buchmann et al. /Exchange currents TABLET

Quark model parameters (i) with regularized one-pion exchange (A = 4.2 fm-t ), (ii) without pions (no n). (0~)~: without configuration mixing Quantity

(OS)3

(os)3

A [fm-‘1 b lfml as a, [ MeV.fmB2 ]

4.2 0.613 1.057 13.90

no K 0.603 1.543 25.13

If we exclude pions, Eq. (2.14) leads to a simple quadratic equation in b2. Using Eq. (2.14) to determine the variational parameter b takes the effect of the residual interactions in Eq. (2.1) approximately into account. Following the procedure just outlined we obtain the numerical values of the parameters shown in Table 1. In order to appreciate the importance of the individual contributions to the nucleon mass MN of Eq. (2. loa) we also show their numerical values, as well as the gluon (8,) and pion (8,) contributions to the A-N mass splitting in Table 2. The quantities S, and 6, are important. They indicate the relative importance of the gluon and pion cloud effects in various electromagnetic properties of the nucleon to be calculated in sect. 4. The size of the pion contribution S, of Eq. (2.1 lc) is mainly determined by the parameter A. The value of the quark core radius b, which also enters into Eq. (2.1 lc), remains largely constant if A is changed from very small values to A = cc in Eq. (2.14).

TABLE 2 Contribution of individual terms in Eq. (2.10a) to the nucleon mass and gluon (6s ) and pion (~5,) contributions to the d-N mass splitting for two cases: (i) with regularized one-pion exchange (A = 4.2 fm-‘), (ii) without pions (no x); for a (0~)~ ground-state wave function. ce: color electric part of VOGEP; cm: color magnetic part of VWEP. All entries in MeV no ~(0s)~

Term

n(Os)3

Tki”n

496.6 125.3 - 542.7 47.8 - 127.0 0.0

512.4 219.6 -805.3 73.3 0.0 0.0

191 102

293 0

ywnf

v&On $$On vpion

Total *s 6,

A. Buzhmann et al. /Exchange

c”A

300

667

currents

-

n/lN)e;cp

Fig. 1. Relative importance of gluon and pion exchange potentials in the d-N mass splitting as a function of the quark core radius b (see Eq. (2.1 I )). Dotted curves: pion contribution 8, (b) for two cases: (i) with a smeared-out S-function (A = 4.2 fm-i), (ii) without the S-function term in Eq. (2.4). Full curve: &on

In Fig. 1 we show the behavior

the (0~)~ parameter

contribution

6s (b) for the case (iI.

of 8, and 8, as a function

of the quark core radius b for

set in Table 1.

3. The electromagnetic currents and gauge invariance In this section we list the electromagnetic nonrelativistic one-body current of Fig. 2a J;;r@,q)

currents used in this work. Aside from the

= $

(3.la) (i[@ixpi,exP(&*ri)f + {Pi,eXP(i@*rj)}t, 9 we include two-body exchange currents consistent with the two-body potentials in (2.1). Here, and in the following, the three-momentum transfer of the photon is denoted by q. 3.1. PION AND GLUON EXCHANGE

CURRENTS

We consider the gluon and pion exchange currents depicted in Figs. 2b-d, J~~(ri,rj,q)

=

-~li.rl,{ejeXp(iq.ri)t(ai

9

+Uj)

xr

A. Buchmann et al. I Exchange currents

_---

7l

-_-,

7 i

Fig. 2. (a) Impulse current, (b) ghon pair current, (c) pion pair current, (d) pionic current.

(3.lb)

+ (i +-+A]$ ~~~~(~i,~j,~)

=

ie --___I 6

A2

f,2s

4x1.12~2

1

_ p2 GZ’

4

f zj

{exp(iq *l+i)iJX VrGj * Vr + (i +2 j)) x expf-v) - exp( -Ar) f r r ( )

X

Ji$(ri,rj,q)

fZ A2 4iTy2fs-p2 X{(TiXzj)3exp(iq.ri)aisj.V,

= ez-

x

exp(-w)

I

( J$(ri,rj,q)

(3.lc)

-

j-2 A2 = ea -(Zi 4zp2 if2 _ j.$

exp(-fir) I X Z_f)3bi

+ (i-j))

1 ’

(3.ld)

9

ViOj ’ Vj

1/2

dvexptiq-

X

(R-

rv))

J -1j2 X

(

Zr

fw(-L4~)

expt-&r) Lflr

IZn

LAr

>’

(3.le)

where in Eq. (J.le) we have used the following abbreviations: R = $ (ri +rj), tm(q, r) = *1‘j2. The decomposition of Eqs. (3.la,b) L,r+ivrqandLm(q,v) = [$q2(1-4v’)fm into isoscalar(IS) and isovector(IV) currents is given by the quark charge et = e( 8 + $T$. These coordinate space operators have been derived using a nonrelativistic expansion of the Feynman diagrams of Fig. 2 [ 2527,301. Eqs. (3.1 b-d) describe quark-antiquark pair creation processes induced by the external photon with subsequent annihilation of the quark-antiquark pair into a gluon (gluon pair current of Fig. 2b) or a pion (pion pair current of Fig. 2c) which is then absorbed at the site of another quark. Finally, Eq. (3. le)

A. Buchmann et al. /Exchange

II

669

”

Y

Fig. 3. Confinement

currents

exchange current.

represents the pionic current of Fig. 2d where the photon couples to the pion directly. The currents of Eq. (3.1) have also been used to calculate the elastic form factors and electrodisintegration of the deuteron in the quark model [ 2%27,301. 3.2. CONFINEMENT

CURRENT

In the previous section we considered currents corresponding to each term in our hamiltonian, with the exception of the confinement potential of Eq. (2.2). As it stands, Eq. (2.2) does not require an exchange current because it is neither isospin nor momentum dependent. However, a more general momentum-dependent confinement interaction will lead to an additional exchange current (Fig. 3). The motivation for considering such a confinement potential and current is as follows*. Some time ago, Isgur and Karl [2] proposed that the spin-orbit term in the onegluon exchange potential be suppressed in order to obtain a reasonable agreement with experimental

baryon spectra. They suggest that spin-dependent

relativistic

corrections

to

the scalar confinement potential, such as the Thomas term, could lead to the desired effect. It is therefore interesting to investigate the relativistic corrections to the confinement potential and the corresponding currents here in more detail. The scalar-isoscalar potential, including relativistic without retardation corrections, is given by [ 371

+Ci

=

‘Pivial ‘PI

@-&

q({(f7i’Pi)2>Vz}

++Y[~(cI l

+ cj’P

+

+a2).rx

In the bag model the confinement

moment of the nucleon [ 361.

jvZcj’P

to order 0 ( 1/rni ) but

j)

{(~j'Pj)2,VoS}

(PI-p21

corrections

+

+ i(bl--2).rx

$Cvf+vS)vo” (PI+P~)I).

mechanism plays also a crucial role in determining the magnetic

A. Buchmann et al. /Exchange currents

670

(3.2) From Vs we obtain, after minimal

substitution,

-Cin(ri,rj94) = --

1 eienp(iq-ri)(~‘~~i,q+f(f~).i~r)+(i*j~}, 2m,’ { (3.3)

where we have kept only the spin-dependent the commutator

local terms. Adding the current defined by

4~JSel(ri,rj,4) = [G,p/$kri)

+ P~~~(q,rj)l

(3.4)

of the lowest order term in Eq. (3.2), V,‘, and the relativistic correction (O( l/m:)) in the one-body charge density pt$, (see Eq. (3.7) ) cancels the spin-orbit-type term in JEi, and we get

=

-~{e,exp(iq.ri)VCO”fiai

X 4

+

(it-t

j)},

(3.5)

4 where we have replaced V$ by Yconf of Eq. (2.2) in Eqs. (3.2)-(3.4) to obtain the relativistically extended scalar confinement potential and current. This result agrees with the local part of the relativistic scalar-isoscalar pair current derived from Feynman diagrams [38]*.

3.3. GAUGE INVARIANCE A few remarks about gauge invariance are necessary at this stage. The one-body current of Eq. (3.1 a) and the kinetic energy term in Eq. (2.1) satisfy an independent continuity equation. Furthermore, it is well known [37] that the sum of the isovector pion pair and pionic current of Eq. (3.ld,e) Eq. (2.4) via 4

. (Jfrv4g =

(4,

[yOP=’

is connected

ri,

rj)

+

JEn

with the one-pion

(!I,

ri,

rj)

exchange potential

of

)

(r.l,,,mp r.) p.(O)(q ,Ir.) + p.(O) Imp(q ,Ir.)] 9

(3.6a)

where p:$ (q, ri ) is the nonrelativistic one-body charge density, i.e. the first term in Eq. (3.7). The gluon pair current was already investigated in refs. [25-27,301. There it is shown that the gluon pair current satisfies q.Jgqa(q,ri,rj)

=

[VOGEP

(ri,rj),/hp(4,ri)

+ Pimp(4,rj)l.

(3.6b)

* Another current of the form of Eq. (3.5) is the e meson exchange current. Recently, o exchange between quarks has been studied in the CQM [ 13,321. Since the u meson is the chiral partner of the pion its mass and coupling constant are related to the constituent quark mass and pion-quark coupling constant by chiral symmetry arguments [ 321.

A. Buchmann et al. /Exchange currents

The continuity

equation 4 .

for the confinement

671

current reads + /knp(4,rj)l.

Jconf(cl, ri, rj) = [Vmnf(ri,rj),Pimp(4,ri)

Here and in Eq. (3.6b) Pimp(!Z,ri) = eiexP(k.ri)

(

1 - &(fc’-

i0i.q

Xpj)

Q

(3.7) >

is the one-body charge density including the relativistic Darwin-Foldy and spin-orbit (2) Gauge invariance demands that we use Eq. (3.2) instead of Eq. (2.2) in corrections Pimp. determining the ground-state parameters b and aC*. However, from Eq. (2.12) it is clear that ( Vconf) must have the value quoted in Table 2 in order to reproduce the nucleon mass. Also, the value of b must be around 0.6 fm in order to be consistent with the neutron charge radius [ 71. We therefore continue to use the parameters quoted in Table 1 for the (OS) 3 ground-state calculation. For the case with configuration mixing we perform a consistent calculation and use Eq. (3.2) together with Eq. (3.5) from the beginning. Concerning relativistic corrections it is evident that consistency to order l/m: forthe total current is not established. In this order there are other terms in the hamiltonian and current that need to be included such as relativistic corrections to the kinetic energy and the spatial one-body current, as well as retardation corrections to the one-pion and onegluon exchange potentials and corresponding exchange currents. Since we do not consider these terms in the present work, we violate gauge invariance in this order. Nevertheless, we believe that our calculation includes the most important exchange current mechanisms and gives a good idea of the relative importance of these.

4. Magnetic form factors, moments and radii Next, we calculate the magnetic

form factors, moments

and radii of the nucleon.

This

involves taking matrix elements of the operators of Eqs. (3.1) and (3.5) between nucleon wave functions. We start with a pure (0~)~ wave function. This allows us to derive some analytic results for the pion, gluon and confinement contributions to the magnetic moments

and radii.

4.1. MAGNETIC

FORM FACTORS

The single-quark current contribution netic form factor is given by &&(q2)

= exp(-q2b2/6),

(see Fig. 2a) to the isoscalar and isovector mag-

FzP(q2)

= 5exp(-q2b2/6).

(4.la)

* Taking (0~)~ matrix elements of Eq. (3.2) with Vosreplaced by V conf gives an additive correction term (-24aJmi) that has to be added to the third term in Eq. (2.10a). In this case Eq. (2.14) does not lead to a reasonable solution for b, i.e. b = 0.4-0.8 fm. This may be remedied by including additional terms such as the orbit-orbit [4] term in VOGEP.

672

A. Buchmann et al. /Exchange currents

The two-body factors:

currents

of Figs. 2b-d and 3 contribute

as follows to the magnetic

form

&I&(4*)

= ru,4MN1exp(-~q2b2)Igq(l, 9 mt 4

&!&Z2)

= 2Z$&Z2),

F;iG(q2)

= -&&f$

FiiQ(q*)

= --$$zikiexp(-$q*b*)Z$)(q),

(4.lc)

F,&(q*)

= -~~MNexp(-hq'b')z,,~~q~,

(4.ld)

K%(q*)

=

(4.lb) exp(-$3*b*)(Z~~)(q)

+ Z:::)(q)),

-2~exp(-gq*6*)z~~~(q), 9

K%dq*)= W!z:&z*). Here, we have used the relation

(4.le)

between

the pion-quark

and pion-nucleon

coupling

constants, namely fzN/4n = ({)*fn29/4a [25]. The integrals Zgqa,1,$‘(q), Zynn(q) and Zeonr(q ) in Eq. (4.1) are defined in the appendix. The total isoscalar/isovector magnetic form factor is thus given by the sum of all isoscalar/isovector contributions of Eq. (4.1): Frs/rv = Z+rv ,mp + F;;r We also use the following definitions

+ F;;p

+ F;,ff’

+ Ff;?.

for the proton and neutron

FP = ; (F” + Fw),

F,, = i(F”

-F”).

(4.lf)

magnetic form factors: (4.W

4.2. MAGNETIC MOMENTS The magnetic moments are calculated by taking the q -+ 0 limit of the magnetic form factors of Eq. (4.1). In the following, we discuss the effect of the currents of Figs. 2 and 3 on the isoscalar and isovector magnetic (see Fig. 2a) the isoscalar/isovector PUN= e/2&&:

moments

separately.

magnetic moments

In impulse approximation

are in units of nuclear magnetons

(4.2a) (&l, = 1, (Z&P = 5. One can express the gluon pair (Fig. 2b) and pion pair (Fig. 2c) current contributions to the isoscalar/isovector magnetic moments in terms of the gluon and pion contributions to the A-N mass splitting as follows: (Z&a = +$b2M&,

(4.2b)

(&& = -;b*M&,. 9 In addition, there is the pionic current of Fig. 2d which contributes magnetic moment:

(4.2~)

(Z&

= +$b2M&,

(P)‘,s4# = +$$A,

only to the isovector

(4.2d)

673

A. 3u~h~nn ei al. 1 Exchange currents Here, &,, and &,, are given by the first and second term of Eq. (2.1 lc). The confinement

contribution

of Fig. 3 to the magnetic

moments

is (4.2e)

We see that this contribution and depends

is independent

only on the expectation

of the shape of the confinement

value (V”“‘).

This expectation

potential

value is fixed by

Eqs. (2.12)-(2.14). Using Eq. (2.12) to eliminate the quark mass dependence in the isoscalar pion pair and con~nement current, we obtain the following results for the sum of singlequark and exchange current contributions

(,&

to the magnetic moments

= -2 - fb*f&@,

of the proton and the neutron:

- &) + &

+MN(($+fbl)Sn.-(Pin))

+4q.

(4.3b)

Note that the isoscalar pion pair current (i.e. the third term in Eqs. (4.3)) is very small compared to the isovector pion contribution because of the extra factor 1/A@. Eqs. (4.2b)-(4.3) show that there is a close relation between the gluon and pion contributions to the A-N mass splitting and the role of gluon and pion degrees of freedom in the magnetic moments. In the (0~)~ model the relative importance of gluon and pion degrees of freedom is largely governed by the size of the pion-quark interaction region as discussed in sect. 2. Furthermore, the gluon and pion exchange current cont~butions to the magnetic moments depend explicitly on the size of the quark core. The magnetic moments

of the nucleon

are therefore quite sensitive

to the model parameters

.4 and b.

We can now study two extreme cases for the one-pion exchange contribution: (i) inclusion of the full d-function in VopEp, (ii) elimination of the S-function in VoPEP. The first choice corresponds to the limit A -+ 00 and is a special case of the formulae given in this work. However,

in the second case we must make corresponding changes in the

one-pion exchange current in order to preserve gauge invariance. We derive the matrix elements corresponding to case (ii) by first letting A go to infinity in Eq. (2.1 lc) and then removing the term &(b,A

J2r.4 72 1 + CO) = ----. 4%~’ 25 v%b3

(4.4)

Eq. (4.4) corresponds to the (0~)~ matrix element of the S-function term in Eq. (2.4). The remaining term, &,, is also shown in Fig. 1. For the magnetic moments we then obtain

A. Buchmann et al. /Exchange currents

674

04P =

3 + ;b2Nd$

- A@ ) +

(4.5a) M* = -2 - ~b2kfN(&

- s,, ) + 4M.

(4.5b) 4.3. MAGNETIC RADII

By again using the simple wave function of Eq. (2.7) and the definition

(r2)IS’IV =

-FIs,;(O) -$F’s’1v(Y2),

(4.6)

we can easily calculate the magnetic radii of the nucleon. Here, F’s”v (0) is the total form factor of Eq. (4.If) at 4 = 0, i.e. the total magnetic moment. For the lowest order impulse approximation we get (4.7a) where (p):l,’ is the impulse contribution to the magnetic moment and is given by Eq. (4.2a). For the gluon and pion pair current ~ont~butions to the magnetic radii we derive the following results: (4.7b) (4.7c) (4.7d) (4.7e)

The pionic current contribution to the isovector magnetic radius is (r”,;‘,, = -&$Mi+b4 x

(

&+$b&-

[

&

6,

+ b&, 1 - &

6,

+ b%, 1

,

(4.7f) where & (b) =: &S, (b)/6 b. For the magnetic radii due to the confinement exchange current we get

P2E,f=

-&$$b2 q

(4(Vwnf)(bl

+ b$(VConf)(b)),

(4.7g)

(4.7h) In contrast to the magnetic moments, the magnetic radii depend on the shape of the confinement potential, because the second term in Eqs, (4.7g,h) involves the partial derivative with respect to the size parameter b of the nucleon. This dependence may be helpful in discriminating between different confinement models recently proposed [39]. The total isoscalar/isovector magnetic radius (r2)*s’*vis given by the sum of all isoscaiar/isovector terms in Eq. (4.7 1. Finally, the magnetic radii of the proton and neutron are given by

P% =

-j$)$t{r2)~s~Is(o~ + (r2)~v~~v(o~),

(r2)n= ~~t(r2)1SF1S(0) 4.4. CO~~EME~

CO~~BUTIO~

-

(4.8af (4.8b)

(r2)1vFFIv(0))_

TO THE CHARGE RADII

Here, we note that the time component of the confinement current which was not considered in ref. [ 7 ] also affects the charge radii and form factors. From Feynman diagrams we derive the time component of the scalar confinement current that corresponds to Eq. (3.5) as Pc0nf(ri,ri,B)

=

1 -exp(iq.ri)ei($* w7z,)3

- iq*Vr + {Vf)Pd(ri,l‘j). (4.9)

The isosc~ar/isovector ~IclIs/Iv~q2~ conf

charge form factors associated with Eq. (4.9) are

=

heXp

(-$)4x

(~)3’~~d~~ze~p

(-$$)

(4.10) where Vlyf(p) = q&p* for the quadratic confinement potential of Eq, (2.2). As is evident from Eqs. (4.9)-(4.10~ there is also a ~ont~bution to the proton charge which, however, is negligible for the parameter set of Table 1. In the case with configuration mixing, we use only the terms propo~ional to q in order to avoid this complication. The charge radius resulting from Eq. (4.10) is given by

676 Furthermore,

A. Buchmann et al. /Exchange currents because it has the same spin-isospin

the confinement form factor

charge density

(for a (0~)~ wave fun&ion)

charge radius for the (OS)3 parameter finite constituent

structure as the single-quark

does not contribute

to the neutron

current,

charge radius and

but leads to a slight decrease of the proton

set of Table 1. This reduction

is compensated

by a

quark radius as discussed in sect. 6.

5. Configuration mixing So far we have only considered (0~)~ ground-state wave functions for the N and A. However, the residual interaction will admix higher excited configurations to the lowest (0~)~ configuration in the ground states; a process usually referred to as configuration mixing. These excited state admixtures in the baryon wave functions have a number of interesting consequences. For example, they lead, in impulse approximation, to a nonvanishing neutron form factor and a quadrupole moment for the A. We restrict ourselves to 2hw excitations and therefore have four excited states (!P$, !$, , Y&, !?$$) for N and three excited states (Yi, Y$, Y& ) for A. A detailed description of these wave functions is given in ref. [ 21. The N and A wave functions are then given by %a = as@

+ assY{ + asMY;

Iyd = bsY:

+ bsSYi + bD,YkS + bD,YdM.

+ aDMYZM + mAYi,

(5.1)

As previously suggested [2], we calculate the configuration mixing by introducing a spin-independent central potential U. This potential describes the deviation from a pure harmonic confinement potential. It is introduced to break the degeneracy of the harmonic oscillator states so that a reasonable excitation spectrum is obtained. In ref. [ 21 and our previous work [7] the anharmonic potential U was parametrized in terms of three parameters: Ee, B and A, in terms of which all U-potential matrix elements can be expressed. No explicit knowledge of the shape of U was needed. However, in this work we will use a specific shape for the anharmonic U(ri,rj)

= -Ai .nj(A

potential, + B/r + Cr),

(5.2)

in order to be able to calculate corresponding exchange currents as discussed in sect. 3.2. The particular form of Eq. (5.2) is motivated by the arguments in the review by Lucha et al. [ 31. The parameters A of ref. [7] via

A, B and C can be simply related to the parameters

Eo = f

3A + - ’ bfi

B+

12b c i%

Eo, Sz and

(5.3a)

Szdj

(5.3b)

A=$

(5.3c)

A. Buchmann et aI. /Exchange currents

677

TABLE 3 Quark model parameters (i) with regularized one-pion exchange (A = 4.2 fm-t ), (ii) without pions (no n). CM: with configuration mixing CM

Quantity A [fm-t] b lfml as a, [MeV.fme2] A [MeV] B [MeV.fm] C [MeV.fm-’ ]

In contrast

to our previous

CM

4.2 no 71 0.606 0.576 0.315 0.720 66.01 76.42 64.08 31.09 62.26 86.07 -251.43 -286.00

work [7] the parameter

Ee does not contain

the rest mass

and zero-point energy of the three quarks. In the following we use the parameters A, B, and C to parametrize the anharmonic potential U. We then diagonalize the new hamiltonian, which is obtained from Eq. (2.1) by adding the potential U in the 2fiw configuration space. Here, unlike in the (0~)~ calculation, we insert Eqs. (2.2) and (5.2) into Eqs. (3.2) and (3.5) and keep all spin-orbit andp2 terms when determining the parameters of the model. Furthermore, we include the effect of the U potential in the off-diagonal elements. This was not done in refs. [ 2, lo]. To find the best fit, we minimize x2 = xi (Expi - Cali)2/d&, with respect to the six parameters of the model (b, ac, as, A, B, C). As in the (0~)~ case, we use m, = 3 13 MeV. In Expi we include the experimental masses of Table 6, the charge and magnetic radii and the magnetic moments of the proton and neutron. For the charge radii we use

TABLE 4 Contribution of individual terms to the nucleon mass and gluon (6s) and pion (S, ) contributions to the d-N mass splitting including configuration mixing (CM): (i) with regularized one-pion exchange (A = 4.2 fm-‘), (ii) without pions (no n). All entries in MeV Term pi””

yconf

V&on vpion

Total 6s lMeV1 & [MeVl

n(CM)

no n(CM)

888.58 -469.76 -183.03 -235.77 0.02

971.73 -537.62 -434.03 0.0 0.08

110.85 188.53

289.92 0.0

A. Buchmann et al. /Exchange currents

678

TABLET

Admixture

VopEp

coefficients as defined in Eq. (5.1): (i) with (A = 4.2 fm-‘), (ii) without VoPEP (no n)

N A = 4.2 no n

A A = 4.2 no a

ass

as

a%4

a&d

apA

-0.0345 -0.0258

0.0328 0.0389

0.8210 0.8301

-0.5591 -0.5460

-0.1051 -0.1029

bs

bs,

b

0.8798 0.8900

-0.4692 -0.4500

-0.0591 -0.0565

bDM 0.0482 0.0466

(r’): = (0.83 f 0.03)2 fm2 and (r’): = -(0.34 f 0.01)2 fm2; for (r”)r = (0.83 f 0.03)2 fm2 and (r”): = (0.83 f 0.03)2 fm2. For the we use pP = 2.8 f 0.1 n.m. and p,, = - 1.9 f 0.1 n.m. These values errors than the best experimental values available [ 40-421. We use

the magnetic radii, magnetic moments have slightly larger these larger errors,

A,,, so that the search for the best fit will not be dominated by the small experimental uncertainties quoted in refs. [40-421. The results of the lit are summarized in Tables 3-6. For x2 we get x 2 = 26.62 (3 1.47) for the case with (without) pions. In Table 3, we show the values of the parameters after the inclusion of configuration mixing. While the oscillator parameter b is only slightly reduced, the quark-gluon coupling constant is drastically decreased with respect to the (0~)~ calculation. From Table 4 we see that one-pion exchange is the dominant residual interaction and that the confinement contribution to the nucleon mass becomes negative in this case. Note that S, and 6, alone do not add up to MA - MN. The difference between Ss + S, and MA - MN comes from contributions

due to kinetic, confinement

and U-potential

energies.

TABLE 6 Nucleon (N) and delta (A) spectrum with (A = 4.2 fm-t ) and without VoPEP(no a). All entries in MeV. The results of ref. [46] (W.W.) for a smeared-out pion-quark vertex are shown for comparison. Experimental numbers from ref. [ 401. Entries with (* ) are likely to disappear; therefore we used a bigger error than in ref. [40] A = 4.2 NA

939 1232 1545 1755 1687 1872 1743 2132 2223

no R NA 939 1352 1749 1799 2458

W.W. [lo] NA

1231 1848 1958 2314

936 1465 1753 1962 2062

Exp. [40] N

1235 1701 1908 1974

939 f 0.7 1440*40 1710 * 30 2100 f 200’

A 1232 f 2 1600 f 100 1920f60

A. ~~ch~~~

et al. /Exchange

currents

679

Proton and neutron magnetic moments, including gluon, pion and confinement exchange currents. (i) with pi(ii) without pions (no n). ons (,4 = 4.2 fm-l), i: impulse; g gluon; n: pion; c: confinement; t: total = impulse + gluon + pion + con~nemen~ (0~)~: without configuration mixing, (CM): with configuration mixing. The experimental proton and neutron magnetic moments [40] are = 2.793 n.m. and ,un = - 1.9 13 n.m., respectively. All !JP entries in km.

A = 4.2 P (Os13

P (CM) n (0~)~ n (CM) no II P U-W3 P (CM)

n (Ost3 n (CM)

3.000 2.978 -2.000 -1.978 3.000 2.979 -2.000 -1.979

0.578 0.213 -0.193 -0.073 0.856 0.511 -0.285 -0.175

0.159 -0.801 0.301 -0.527 -0.105 -0.211

0.534 -1.763 0.400 - 1.862

0.000 - 1.391 0.000 -0.670 0.000 0.000

2.936 2.965

2.465 2.820

0.927 -1.358 0.507 -1.646

Table 5 shows the values for the admixture coefficients for the two parameter sets (CM) of Table 5. We obtain somewhat nonstandard admixtures but we have to keep in mind that we include ail spin-orbit and p2-dependent terms in the gluon and confinement potentials as well as the nondiagonal elements of the anharmonic potential U. Finally, in table 6 we show our results for the excitation spectrum of N and d. 6. Discussion of the results 6.1. MAGNETIC MOMENTS

We discuss magnetic moments first. Using the parameters of Table 1, we obtain the numerical values for the (0~)~ wave function fip = 3 -t-0.578 - 0.307 + 0.027 + 0.439 = 3.737 n-m. and pn = -2 - 0.193 + 0.307 + 0.027 - 0.439 = -2.298 n.m., corresponding to the impulse, gluon pair, isovector pion pair, isoscalar pion pair and pionic current terms in Eq. (4.3). Due to the cancellation between the isovector pion pair and pionic currents, the pion cloud gives only a small positive (negative f cont~bution to tbe proton (neutron) magnetic moments (see Table 7). On the other hand, the gluon pair current gives a large positive (negative) contribution to the proton (neutron) magnetic moments. Thus, the total result, including pion and gluon exchange currents, is pp = 3.74 n.m. and ,&I = -2.30 n.m. This deviates from the experimental magnetic moments by 0.95 (0.39) n.m. in the case of the proton (neutron). We see that the gluon cont~bution is mainly

A. Buchmann et al. /Exchange currents

680

responsible moment

for this discrepancy.

due to the one-gluon

A large positive

contribution

to the proton

magnetic

exchange current was also found by Hwang [ 201 in the bag

model. However, the inclusion of the confinement current largely cancels the gluon and pion contribution, and we obtain a result much closer to the experiment data (see Table 7). In particular, for the isovector (isoscalar) parts of the magnetic moment we obtain 4.70 ( 1.17) n.m. compared to the experimental values 4.70 (0.88) n.m. Note, that the confinement contribution to the magnetic moment is completely fixed by the requirement that the nucleon mass has its experimental value as discussed in sect. 2.3. This gives the value of (V”“‘) as shown in Table 2. We emphasize that the numerical factors in Eq. (4.2e) depend on the Lorentz structure of the confinement model. For a vector-type con~nement, by going through steps analogous to those in Eqs. (3.2)-(3.5) we obtain (,f&

i.e. contributions cont~bution*.

= q,

(flu). =

-i&2, N

that have the wrong sign to cancel the one-gluon

(6.1) exchange

current

The inclusion of configuration mixing increases the one-pion exchange current contribution and decreases the one-gluon exchange current. This can be qualitatively understood by comparing the values for S, and Ss with and without configuration mixing in Tables 4 and 2. With configuration mixing, the pion contribution to the d-N mass splitting is enhanced by roughly a factor of two while the gluon contribution is decreased by the same factor. According to the approximate result of Eqs. (4.3) this leads to an increased pion and decreased gluon contribution to the magnetic moment. A similar reversal of the role of gluon and pion exchange currents was observed for the neutron charge radius [7] after configuration mixing was included. It is evident from Table 7 that in the case of configuration mixing the confinement exchange current also cancels the sum of gluon and pion exchange currents. At first this may seem surprising, because the confinement contribution to the nucleon mass is negative (see Table 4) and should therefore result in a positive contribution to (,u)~. However, Eq. (3.5) involves only the V, part of the confinement potential, which has a positive expectation value. This is due to the presence of off-diagonal

elements of U. By far the major part of the configuration

mixing (CM) values in Table 7 comes from the PsN and Y{ states in Eq. (5.1). Furthermore, we observe that the results for the magnetic moments are much better for the case with pions, independent of whether configuration mixing is included. All in all, the results with configuration mixing are slightly better than the (OS)3 results. In the following we provide a qualitative discussion of the role of the d-function in the pion-quark interaction based on simple (OS)’ wave functions. For the case of a pionquark interaction without the &function we get ,uP = 3 + 0.939 + 0.054- 0.009 + 0.9390.801 = 4.122n.m.and~” = -2-0.313-0.054-0.009-O-939+0.534 = -2.781 n.m., * A vector-type confinement current has the same sign as the gluon pair current. This is clear from Eq. (3.lb)

where we have to replace $asr-‘(d/drr-‘)

by r-‘d/dr

(--aCr2).

A. Buchmann et al. / Exchange currents

where, as before, the numbers

correspond

to the impulse, gluon pair, isovector pion pair,

isoscalar pion pair, pionic and confinement pionic exchange current

corresponding

but without the a-function

681

current contributions

to the one-pion

term leads to a contribution

in Eqs. (4.5 1. Thus, the

exchange potential

of Eq. (2.4)

more than twice the size of the

one in Eq. (4.3a). At the same time, the pion pair current, which is proportional

to &,

is drastically reduced. This is because 6, is reduced by a factor of about five if the 6function in Eq, (2.4) is discarded (see Fig. 1). As a result, the cancellation between the pionic and pion pair diagrams disappears, and we obtain a large positive (negative) onepion exchange current contribution to the magnetic moment of the proton Furthermore, S, is now almost solely responsible for the d-N mass splitting. the one-gluon

exchange current contribution,

which is proportional

(neutron). Therefore,

to S, is increased by

a factor of about $. Consequently, a weak one-pion exchange potential without a zero-range-type interaction leads to unacceptable results for the magnetic moments. This consequence has been overlooked by those authors who argue that the S-function of the pion should be discarded [ lo]. Here, we have shown that once a consistent calculation is performed, the weak one-pion

exchange hypothesis

is not supported

by the expe~mental

ma~etic

mo-

ments. However, it must be noted that the above argument depends on the assumption that the impulse contribution is given by Eq. (4.2a). If the single-quark contribution is reduced (e.g. by relativistic corrections) the weak one-pion exchange model could be an acceptable alternative. The case (i 1, which includes the full &function, leads to results similar to those quoted in Table 7 for a smeared-out S-function. 6.2. MAGNETIC RADII

Next, we discuss magnetic radii. We discuss the (0~)~ results first. Using Eqs. (4.7) and (4.X) and the parameters in Table 1, one calculates a = 0.654 fm and m = 0.701 fm which are smaller than the experimental magnetic radii. Evidently, this is because the impulse contribution, (r’):: E b2, is too small for quark core radii b around 0.5-0.6 fm. In ref. [ 71 it was suggested that the relativistic correction to the single-quark charge density (Darwin term ) brings the proton charge radius closer to its expe~ment~ value. A similar tendency is observed for the relativistic correction to the impulse current density [ 43 ], which leads to P2)~mp=&(02+&). and gives a correction same relativistic

(r2)?p=&(h2+&i),

term 1/4mi

correction

(rP=3(1-&),

to the nonrelativistic

impulse result. However,

(6.2) the

will also modify the impulse magnetic moments (/lP=-2(l-&)9

(6.3)

A. Buchmnn

682

et al. /Exchange

currents

TABLE 8 Proton and neutron magnetic radii, including gluon, pion and confinement exchange currents. (i) with pions (A = 4.2 fm-I), (ii) without pions (no II). imp: impulse; g: gluon; n: pion; c: confinement; t: total = impulse +gluon+ pion + confinement; (OS)? without configuration mixing. A finite quark size (ri) = 0.36(0.09 fm* is used for the (OS)~(CM) case, respectively. The experimental proton (neutron) magnetic radius is dm = 0.858 f 0.056 fm (J/W = 0.876 & 0.070 fm) [ 4 11. All entries in fm2, except total results in fm

A = 4.2 P P n n

(os)3 (CM) (0~)~ (CM)

0.752 0.346

0.112 0.016

0.139 -0.253 0.162 0.177

0.866 0.837

0.834 0.361

0.062 0.009

0.213 -0.280 0.247 0.178

0.918 0.891

0.881 0.338 1.165 0.380

0.195 0.039

0.000 -0.5 11 0.000 0.206

0.751 0.763

0.118 0.023

0.000 -0.618 0.000 0.22 1

0.752 0.790

no a P P n n

(os)3 (CM) (0~)~ (CM)

which results in very small magnetic moments, namely pr, = 0.885 n.m. and ,G = -0.590 n.m. for the proton and neutron, respectively. This means that most of the proton and neutron magnetic moments must come from &ton and pion degrees of freedom. In this work we will not pursue this possibility as it is somewhat against the spirit of the nonrelativistic quark model where the main contribution to the magnetic moments is expected to come from the single-quark current. Instead we will introduce a finite constituent quark size to obtain the correct charge and magnetic radii. The notion of a finite constituent quark size has been used before [44,45]. It is an effective means of including relativistic effects [ 441. We use a simple monopole form to describe the finite electromagnetic

extension

of the constituent F(q2)

=

quarks and pion,

1+

l

$12(rq2)’

(6.4)

and multiply Eq. (4. If) by this function. As in the case of the relativistic Darwin term considered in ref. [ 71, a finite quark size does not affect the neutron charge radius. We then obtain the results presented in Tables 8 and 9. Again, we notice that the confinement exchange current cancels the effect of gluon and pion exchange currents in the case of (0~)~ wave functions. This is not the case for a vector-type confinement. Furthermore, the inclusion of pions in the CQM gives better results for the magnetic radii.

A. Buchmann et al. /Exchange currents

683

TABLE 9 Proton and neutron charge radii, including gluon, pion and confinement exchange currents. Same notation as in Table 8. A finite quark size (ri) = 0.36(0.09) fm* is used for the (0~)~ (CM) case, respectively. The experimental proton (neutron) charge radius is dm = 0.862 5 0.012 fm [41] (dm

= 4.2 P (Os)3 P (CM)

= 0.345 ZIZ0.003 fm [41,42]). All entries in fm*, except total results in fm (r*)i

(r*h

0%

0.736 0.347

0.115 -0.061 0.041 -0.069

(r*)c

JFKi

A

n (0~)~ n (CM) no K P (OS)3 P (CM) n (Os)3 n (CM)

-0.120 0.265

0.818 0.764

0.000 -0.043

0.342 0.350

0.170 0.099

0.000 -0.208 0.000 0.329

0.828 0.866

-0.1136 -0.068

0.000 0.000 0.000 -0.051

0.337 0.353

0.000 -0.076 -0.006 -0.028 0.724 0.322 0.0 -0.006

-0.041 -0.046

After the inclusion of configuration mixing, we observe the aforementioned decrease of the gluon exchange current and a modest increase of the pion exchange current. More important, however, is that the confinement contribution changes sign with respect to the (0~)~ case. As is evident from Eqs. (4.7g,h) and (4.11), the radii depend on the shape of the confinement potential and are therefore sensitive to different confinement models. Switching off the B/r term in the U-potential leads to a negative contribution to the radii as in the (0~)~ calculation. This shows that the positive confinement contribution to the charge radius is mainly due to the B/r term in the confinement

current and not

specific to configuration mixing. Because of the positive confinement contribution to the magnetic radii a smaller size for the constituent quarks is needed compared to the (0~)~ calculation. As in the case of the magnetic moments, are included.

the magnetic radii are better described if pions

6.3. MAGNETIC FORM FACTORS Finally, we discuss the role of two-body exchange currents in the magnetic form factors (see Figs. 4 and 5). We compare our results with the empirical dipole fit F(q*)

= (1 + q*/18.23 fm-*)-*.

(6.5)

A. Buchmann et al. /Exchange currents

684

-.-.

Impulse --._

-.. Gluon

-Conlinement ... .- -:> _._ _i(

YC

-.._._._

-I ”

1

2

3

6

9

8

9

0

1

2

3

6

7

8

9

Fig. 4. Proton magnetic form factor for A = 4.2 fm-‘, including gluon, pion and confinement exchange currents. (a) Without configuration mixing; (b) with configuration mixing. Dotted curve: impulse current (impulse); short-dashed curve: gluon pair current (gluon); long-dashed curve: pion exchange current (pion); dashed-dotted curve: confinement exchange current (confinement); full curve: total= impulse + gluon + pion +confinement. Negative contributions are indicated by (-). Experimental curve: empirical dipole fit of Eq. (6.5).

Because of the cancellation of the gluon and confinement exchange currents and the smallness of the one-pion exchange current in the (0~)~ calculation, we do not observe a drastic change with respect to the nonrelativistic impulse result.

Fig. 5. Neutron magnetic form factor for A = 4.2 fm-t, including gluon, pion and confinement exchange currents. Same notation as in Fig. 4.

A. Buchmann et al. /Exchange currents In contrast tribution

685

to the (0~)~ result (Figs. 4a and 5a), where the confinement

falls off rapidly with increasing

the configuration

mixing calculation

term in the confinement

current,

momentum

transfers,

current con-

the opposite is true for

(Figs. 4b and 5b). The reason for this is the B/r

which becomes important

a term is not present in the (0~)~ calculation.

at smaller distances.

As a result we underestimate

Such

the experi-

mental data at higher momentum transfers. On the other hand, the B/r term improves the excitation spectrum. It would be interesting to study how other recently proposed confinement models [ 391 that fit the excitation spectrum affect various electromagnetic properties. All in all, the results with configuration mixing are better if exchange currents are included.

7. Conclusion To summarize, we have calculated two-body exchange current contributions to the magnetic form factors, moments and radii of the nucleon in the constituent quark model. We have included gauge invariant gluon and pion exchange currents, and, in addition, we have derived a consistent two-body current connected with the confinement mechanism. Also, we have taken into account

the effect of configuration

mixing due to gluon and

pion exchange interactions. We have shown that there is an intimate connection between the effect of gluon and pion degrees of freedom in the excitation spectrum and their influence on the low-energy electromagnetic properties of the nucleon. This is clearly seen in the (0~)~ formulae of Eqs. (4.2)-(4.7) which relate the effect of gluon and pion exchange currents on static electromagnetic properties to the corresponding effect of the gluon and pion exchange potentials on the N-d mass splitting. In particular, for the magnetic moments, we have found that the pion pair and pionic currents largely cancel each other. Thus the pion cloud gives only a small correction for the magnetic cancellation potential.

moments, depends

thus confirming on the inclusion

a previous

result [9]. We have shown that this

of the d-function

On the other hand, the gluon pair current

proton and a somewhat larging the discrepancy severe if the B-function hamiltonian because in

term in the one-pion

exchange

leads to a drastic increase of the

smaller decrease of the neutron magnetic moment, thereby enbetween theory and experiment. This discrepancy is even more part of the one-pion exchange potential is eliminated from the this case gluonic effects are enhanced. Thus, we conclude that

pions are necessary for a realistic description of the magnetic properties of the nucleon, although their direct effect on the magnetic moments and radii is not particularly large. We have also seen that the relative size of gluon and pion cloud contributions to masses and electromagnetic properties depends strongly on the baryon wave function. For the improved wave functions of sect. 5 which include configuration mixing, the effect of gluons is greatly diminished and pions are dominant. In addition, we have found that the current connected with the nonperturbative part of the hamiltonian gives very large contributions to the electromagnetic properties of

A. Buchmann et al. /Exchange currents

686

the nucleon.

In fact, it is generally

scalar confinement

the largest exchange current.

current considerably

to a better agreement

with the experimental

confinement

Moreover,

electromagnetic

current.

We have seen that the

reduces the gluon cloud contribution

and leads

data. The opposite is true for a vector-type

we have shown that the confinement

radii depends strongly on the particular

contribution

shape of the confining

to

potential.

In future work it will be necessary to study how the two-body exchange currents discussed here affect other nucleon properties (e.g. nucleon axial form factor, nucleon polarizabilities) and the electromagnetic properties of the A. Furthermore, it would be interesting to include the CJexchange potential and the corresponding exchange current as suggested by chiral symmetry arguments.

We would like to thank Dr. Y. Yamauchi

for useful discussions.

Appendix Here, we give the expressions for the integrals entering the gluon pair current of Eq. (4. lb) is

in Eqs. (4. la-e).

The function

ZNa

zgq~=4a(~)3’2Jmdpexp((A.11

0

The integrals 6:; defined as follows

needed in the isoscalar and isovector pion pair current

(4.1~) are

312

Ap2JmdPp2exp

(-g)

ji (3)

0

X

(

/.-!Yi(~Ai~)

-A$Yi(AJZp, ,

>

64.2)

with j = 0 and 2 for the isoscalar current and 312

-&$p2/dpp’cxp

(-g)

ji (s)

0

(A.31 for the isovector current. Ye(x)

Here, the functions =

yz(x) =

X (x) are defined as

exp;-x)) Yl(X) = exp;-xl

exy (1+;+;).

( > 1

+

$

,

64.4)

A. 3uc~mann et al. /Exchange currents

687

The integral Z,,, in Eq. (4.1 d) is given by

x Uv(q,p,fit) --Zv(q,p,~)).

(AS)

where the function IV is defined as Z?J(4,L%m) = t/2 J

dv

(

jo(qpfi~ko(q,P,v,M

+ j*(q~~~)g*(q,~,~,~))

-l/2

with the abbreviations

go = L exp(-LmJip) g2

=

(

L,exp(-LdZp)

(

ll+

&)s &A

and L, = (;q*( 1 - 4v2) + m2)? Finally, the integral ZeOdused in the confinement current contribution of Eq. (4.1 e) is given by Zconf= 48 (--&)3’2~dPp2exp

(-g)

jo (g)

Viy?‘fQ).

(A.61

0

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