

UNITU-THEP-5/1995

March 1995

On the strange vector form factors of the nucleon in the NJL soliton model

y

H. Weigel

z

, A. Abada, R. Alkofer, and H. Reinhardt

Institute for Theoretical Physics

T"ubingen University Auf der Morgenstelle 14 D-72076 T"ubingen, Germany

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: \Gamma 0:05 ^ _

s

^ 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.

------------ y

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

z

Supported by a Habilitanden scholarship of the DFG.

1

1. Introduction

A number of experiments, which have recently been completed [1] or are up-coming [2, 3], measure parity violating asymmetries in scattering processes of polarized electrons on nuclei. Although these experiments were initiated as precision tests of the electro-weak theory they also provide access to hadronic "observables" like hN j_sfl

_

sjN i, which e.g. enter the matrix

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 elements. A first estimate of the matrix element of the strange vector current between nucleon states was carried out in ref. [4] performing a three-pole vector meson fit to dispersion relations [5]. Later on the matrix element hN j_sfl

_

sjN i has been studied in the Skyrme model [6] and the Skyrme model

with vector mesons [7]. Also the effect of OE \Gamma ! mixing in the framework of vector meson dominance has been investigated [6]. This picture has even been combined [8] with the kaon loop calculation of ref. [3].

These investigations are based on chirally invariant models describing the interaction be- tween strange mesons and the nucleon. The latter is either considered as an "elementary" particle [3, 8] or as a soliton of meson configurations [6, 7]. Neither of these studies takes ex- plicit account of the quark structure of the nucleon. In order to examine effects related to the quark structure, the Nambu-Jona-Lasinio (NJL) model [9] represents an excellent candidate. Imitating the quark flavor dynamics of QCD at low energies the NJL model contains both reference to explicit quark degrees of freedom as well as the fruitful concepts of chiral symme- try and its spontaneous breaking. The model provides a fair description of the pseudoscalar and vector mesons as quark-antiquark bound states [10, 11, 12]. In addition, it contains soliton solutions [13] in the two flavor subspace which may be identified as baryons

a

. The

so-called collective approach [15, 16] allows one to incorporate strange degrees of freedom. Its application within the NJL model [17, 18] 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 with scalar and pseudoscalar degrees of freedom is defined as the sum of the free Dirac Lagrangian and a chirally invariant four quark interaction [9]

L = _q(i@= \Gamma ^m

0

)q + 2G

N

2

f

\Gamma 1

X

i=0



(_q

*

i

2

q)

2

+ (_q

*

i

2

ifl

5

q)

2

!

: (1)

where G denotes the effective coupling constant and ^m

0

is the current quark mass matrix.

Here we are interested in the case of three flavors, i.e. N

f

= 3. By path integral bosonization

the model can be converted into an effective meson theory yielding the action [10]

A = A

f

+ A

m

= Tr

\Lambda

log (iD=) \Gamma

1

4G

tr

Z

d

4

x

`

M

y

M \Gamma ^m

0

(M + M

y

) + ( ^m

0

)

2

'

; (2)

Here M = S + iP contains the scalar (S) and pseudoscalar (P ) meson fields. Furthermore

iD= = i@= \Gamma

`

P

R

M + P

L

M

y

'

P

R;L

= (1 \Sigma fl

5

)=2: (3)

is the Dirac operator. We will assume isospin symmetry, m

0

u

= m

0

d

=: m

0

; however, m

0

s

6= m

0

.

In eq (2) we have indicated that the functional trace is UV divergent and has to be regularized.

a

For a review see ref. [14] and references therein.

2

Hence one more parameter is introduced, the cut-off \Lambda . As regularization scheme we will solely employ Schwinger's proper time prescription. This requires a Wick rotation to Euclidean space (x

0

! \Gamma ix

4

= \Gamma io/ ). Subsequently the real part of the Euclidean action is represented by a

parameter integral

1 2

Tr

\Lambda

log

^

D=

E

D=

y

E

*

= \Gamma

1

2

Tr

Z

1

1=\Lambda

2

ds

s

exp

^

\Gamma sD=

E

D=

y

E

*

: (4)

Here D=

E

refers to the continuation to Euclidean space.

The Schwinger-Dyson equations (SDEs) for the scalar fields yield non-vanishing vacuum expectation values hSi = ^m = diag(m

u

; m

d

; m

s

) = diag(m; m; m

s

) 6= ^m

0

, to represent the

constituent quark masses and are a manifestation of spontaneous breaking of chiral symmetry.

In order to apply the SU(3) collective description for baryons as chiral solitons we adopt the parametrization

M (r; t) = R(t), (r) R

y

(t)hSiR(t), (r) R

y

(t); (5)

for the (pseudo-)scalar fields [17]. 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 ansatz for the chiral field

U (r) = ,

2

(r) = exp (i

^ r \Delta o/ \Theta (r)) : (6)

The special form of the parametrization (5) guarantees that only the hedgehog rotates in the space of the collective coordinates. It is advantageous to transform to the flavor rotating

frame q

0

= R

y

(t)q, i.e.

ifiR

y

D=

E

R = ifiD=

0

E

= \Gamma @

o/

\Gamma h

\Theta

\Gamma h

rot

\Gamma h

SB

(7)

Here h

\Theta

= ff \Delta p + fimU

fl

5

refers to the one-particle Dirac Hamiltonian in the background of

the chiral soliton. Furthermore

h

rot

=

i

2

*

a

\Omega

a

E

(8)

contains the analytic continuation (\Omega

a

E

= \Gamma i\Omega

a

) of the angular velocity measuring the time

dependence of the collective coordinates, R

y

.

R = (i=2)*

a

\Omega

a

. The symmetry breaking piece is

linear in the difference of the constituent quark masses

b

h

SB

=

m \Gamma m

s

p

3

T fi



8 X

a=1

D

8a

*

a

\Gamma *

8

!

T

y

: (9)

The static soliton enters h

SB

via T = ,P

L

+ ,

y

P

R

. Furthermore the adjoint representation of

the collective rotations D

ab

= (1=2)tr(*

a

R*

b

R

y

) has been introduced.

Then A is expanded up to quadratic order in terms of the angular velocity as well as the mass difference m\Gamma m

s

. From the resulting expression the collective Lagrangian L = L(R; \Omega

a

)

is straightforwardly extracted[17]. Quantization is achieved by identifying the momenta con- jugate to the angular velocities with the right generators of SU(3)

R

a

= \Gamma

@L(R; \Omega

a

)

@\Omega

a

; (10)

b

The explicit dependence on the current quark mass difference m

0

u

\Gamma m

0

s

is completely contained in A

m

.

3

which provides a linear relation between the generators and the velocities. The resulting collective Hamiltonian H = H(R; R

a

) can be diagonalized exactly. For details we refer to the

literature [16, 6, 17]. Here we only wish to mention that due to SU(3) symmetry breaking the eigenfunctions of H are distorted SU(3) D-functions.

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

The vector currents J

a

_

are most easily obtained by introducing external sources b

a

_

, which

couple to the quark bilinear _qfl

_

(*

a

=2)q. Then J

a

_

is the derivative of the extended action

with respect to these sources

J

a

_

(x) =

\Gamma iffi

ffib

_a

(x)

Tr log

h

ifiD=

i

b

a

_

ji

fi

fi fi

b

a

_

=0

: (11)

Transforming to the flavor rotating frame and performing the Wick rotation the extended Dirac operator reads

ifiD=

0

E

i

b

a

_

j

= \Gamma @

o/

\Gamma h

\Theta

\Gamma h

rot

\Gamma h

SB

\Gamma ib

a

4

D

ab

*

b

2

\Gamma ff \Delta b

a

D

ab

*

b

2

; (12)

where b

a

4

= ib

a

0

denotes the analytic continuation of the time component of the source current.

*

a

labels the generators of the flavor group, including the singlet piece *

0

=

q

2=31I . The

latter is of special importance for the computation of the strange vector current because it enters the projector onto strange degrees of freedom, diag(0; 0; 1) = *

0

=

p

6 \Gamma *

8

=

p

3.

To obtain a normalization consistent with (10) of the symmetry charges the currents have to be expanded up to linear order in both h

rot

and h

SB

. Induced kaon components have to be

included in a way consistent with the computation of the strange moment of inertia fi

2

[17].

Leaving symmetry breaking effects apart, the time independence of the charges associated with the vector current requires to consider h

rot

(8) (and therefore \Omega

a

) as a classical quantity

constant in time.

The currents gain contributions from both, the explicit occupation of the valence quark orbits and the polarized vacuum [19]. The former is obtained by a perturbation expansion for the valence quark level

c

\Psi

val

, i.e. by substituting

\Psi = R

8 !

:

\Psi

val

+

X

_6=val

\Psi

_

h_jh

rot

+ h

SB

jvali

ffl

val

\Gamma ffl

_

9 =

;

(13)

into the expression for the vector current

_ \Psi fl

_

(*

a

=2)\Psi yielding

J

a(val)

_

=

N

C

2

D

ab

_ \Psi

val

fl

_

*

b

\Psi

val

+

N

C

2

D

ab

X

*6=val

(

_ \Psi

*

fl

_

*

b

\Psi

val

hvalj\Omega

c

*

c

2

+ h

SB

j*i

ffl

val

\Gamma ffl

*

+ h: c:

)

: (14)

In (13) and (14) ffl

_

and \Psi

_

refer to the eigenvalues and eigenstates of h

\Theta

.

The functional traces are evaluated imposing anti-periodic boundary conditions for the quark fields in a Euclidean time interval T , with the vacuum contribution obtained from (11) in the limit T ! 1. The relevant techniques may be found in refs. [19, 17]. We thus only quote the final result for the expansion of (11) up to linear order in \Omega

a

and the mass difference

m \Gamma m

s

(contained in h

SB

)

J

a(vac)

_

=

N

C

4

D

ab

X

*

_ \Psi

*

fl

_

*

b

\Psi

*

sgn (ffl

*

) erfc

`

fi

fi fi fi

ffl

*

\Lambda

fi fi fi fi

'

+

N

C

4

D

ab

\Omega

c

X

*ae

_ \Psi

*

fl

_

*

b

\Psi

ae

haej*

c

\Omega

c

f

*ae

rot

+ h

SB

f

*ae

SB

j*i (15)

c

The valence quark level is defined as the state with the eigenvalue of smallest module.

4

with the regularization functions

f

*ae

SB

=

sgn (ffl

*

) erfc

i

fi

fi fi

ffl

*

\Lambda

fi fi fi

j

\Gamma sgn (ffl

ae

) erfc

i

fi

fi fi

ffl

ae

\Lambda

fi fi fi

j

ffl

*

\Gamma ffl

ae

; f

*ae

rot

=

1

2

f

*ae

SB

\Gamma

\Lambda p

ss

e

\Gamma ffl

2

ae

=\Lambda

2

\Gamma e

\Gamma ffl

2

*

=\Lambda

2

ffl

2

ae

\Gamma ffl

2

*

: (16)

In (15) we have chosen to regularize the UV-finite imaginary part of the Euclidean action as well. Omitting this regularization, which essentially corresponds to a different model, amounts to assuming the limit \Lambda ! 1 in those terms which are related to an odd number of time components of Lorentz vectors. In this context \Omega

c

has to be counted as a time component.

The total current is the sum

J

a

_

= j

val

J

a(val)

_

+ J

a(vac)

_

(17)

with j

val

= 0; 1 adjusted to describe a unit baryon number configuration. From eqs (14,15) it

is suggestive that the current may be cast into a sum of products of radial functions V

l

(r) and

isospin covariant expressions involving the SU(3) collective coordinates D

ab

and the velocities

\Omega

a

[7]. Upon (10) the latter are replaced by the SU(3) generators R

a

. This permits to

compute the spin and flavor parts of the matrix elements of J

a

_

between baryon eigenstates

d

.

We may formally write [7]

J

a

i

=

6 X

l=1

V

l

(r)

3 X

j;k=1

ffl

ijk

x

j

M

ak

l

and J

a

0

=

13 X

l=9

V

l

(r)M

a

l

: (18)

According to (17) the radial functions V

l

(r) separate into valence and vacuum parts. For

the explicit form of the matrix elements M in terms of SU(3) "Euler-angles" and their computation we refer to appendix A of ref. [6]. Furthermore we define Fourier transforms of the radial functions

~ V

l

(jqj) = 4ss

Z

drr

2

r

jqj

j

1

(jqjr)V

l

(r) ; l = 1; :::; 6

~ V

l

(jqj) = 4ss

Z

drr

2

j

0

(jqjr)V

l

(r) ; l = 9; :::; 13 (19)

to evaluate the spatial parts of baryon matrix elements. Identifying finally the momentum transfer in the Breit frame (Q

2

= \Gamma q

2

) allows us to extract the electric G

E

and magnetic G

M

form factors [20]

G

a

M

(Q

2

) = \Gamma 2M

N

6 X

l=1

~ V

l

(jqj)M

a3

l

and G

a

E

(Q

2

) =

13 X

l=9

~ V

l

(jqj)M

a

l

: (20)

The nucleon mass M

N

= 940MeV has been introduced because G

M

is measured in nucleon

magnetons. It should be noted that recoil corrections have been ignored. Thus (20) is only valid in the vicinity of jqj = 0. In (20) we have kept the full flavor structure. E.g. the electro-magnetic form factor is given by the linear combination G

e:m:

E;M

= G

3

E;M

+ G

8

E;M

=

p

3

while the strange vector form factors include the singlet piece G

s

E;M

= G

0

E;M

=

p

6 \Gamma G

8

E;M

=

p

3.

The magnetic moments of the nucleon are defined as the matrix elements _

p;n

= h\Sigma G

3

M

(0) +

G

8

M

(0)=

p

3i

p

and _

S

= hG

0

M

(0)=

p

6 \Gamma G

8

M

(0)=

p

3i

p

.

d

A typical example for these matrix elements is M

ak

3

= h

P

7

ff;fi=4

d

kfffi

D

aff

R

fi

i, where the d

abc

refer to the

symmetric structure coefficients of SU(3).

5

Table 1: Baryon properties as a functions of the scaling variable (21). _

p

and _

n

are the

magnetic moments of the proton and neutron, respectively. _

S

denotes our prediction for the

strange magnetic moment of the nucleon. O/ = [

P

baryons

(4M

pred

\Gamma 4M

expt

)

2

]

1=2

measures the

deviation of the predicted mass differences from the experimental ones.

m=400MeV m=450MeV expt. * 1.0 0.9 0.8 0.7 1.0 0.9 0.8 0.7 _

p

1.17 1.32 1.58 2.02 1.06 1.21 1.46 1.90 2.79

_

n

-0.76 -0.90 -1.17 -1.63 -0.69 -0.84 -1.09 -1.56 -1.91

_

S

0.24 0.21 0.18 0.15 0.23 0.20 0.18 0.16 ?

O/(MeV) 319 219 88 199 341 255 125 144 0

4. Numerical Results and discussion

Expanding the action up to quadratic order in the pseudoscalar fields allows one to express the corresponding masses and decay constants in terms of the parameters ^m; ^m

0

; G and \Lambda .

Using the pion decay constant f

ss

=93MeV and mass m

ss

= 135MeV together with the SDE in

the non-strange sector determines m

0

; G and \Lambda for a given value of m. Then the kaon mass

m

K

= 495MeV and the SDE in the strange sector yield m

s

and m

0

s

for the same value of m.

The kaon decay constant f

K

, left as a prediction, is commonly underestimated in the NJL

model [17]. In the baryon sector the mass differences

e

of the low-lying

1

2

+

and

3

2

+

baryons

are found to be on the small side [17, 18]. 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 E

cl

, by \Theta

s:c:

(r) we consider

\Theta

*

(r) = \Theta

s:c:

(*r): (21)

The driving symmetry breaking term entering the collective Hamiltonian stems from A

m

and can easily be verified to be proportional to *

\Gamma 3

. It should be remarked that E

cl

is rather

insensitive to *, choosing e.g. 0:7 ^ * ^ 0:8 yields an increase of only 10%. Furthermore such values for * are suggested by the proper inclusion of the ! meson [22] and the corresponding mass differences are in reasonable agreement with the experimental data

f

. The dependence

of the magnetic moments on * is displayed in table 1. The isovector part of the magnetic moment of the nucleon _

V

= _

p

\Gamma _

n

is sensitive to the extension of the soliton configuration.

For the self-consistent soliton configuration (* = 1) _

V

is predicted to be less than half of its

experimental value, 4.70. However, _

V

increases drastically with the extension of the meson

profile. On the other hand the isoscalar part (_

p

+ _

n

) is quite insensitive to the extension

of the soliton. Then 0:7 ^ * ^ 0:8 provides reasonable descriptions of the baryon mass differences and the magnetic moments associated with the electro-magnetic current. Hence the results shown in table 1 suggest the prediction 0:15 ^ _

S

^ 0:2.

Recently the discussion of 1=N

C

corrections to _

V

[24] came into fashion

g

. As these have

the same dependence on the collective coordinates their incorporation is effectively equivalent to a modification of

~ V

l

(jqj = 0). Actually these corrections are relevant only for the radial

function already present in the two flavor model (V

1

in the notation of ref. [7]). Without

e

For the description of the absolute values of the baryon masses quantum corrections due to meson fluc-

tuations have to be taken into account [21].

f

Considering * as a variable quantity is furthermore motivated by the existence of sizable correlations

between SU(3) symmetry breaking and the extension of the soliton [23].

g

Here we will not pursue the question of validity of the treatments in ref. [25].

6

Table 2: Electric radii of the nucleon as functions of the scaling variable (21). Data are in fm

2

.

m=400MeV m=450MeV expt. * 1.0 0.9 0.8 0.7 1.0 0.9 0.8 0.7 hr

2

i

p

0.66 0.81 1.06 1.43 0.61 0.78 1.03 1.40 0.74

hr

2

i

n

-0.18 -0.21 -0.28 -0.43 -0.16 -0.20 -0.28 -0.43 -0.12

hr

2

i

S

0.03 -0.08 -0.22 -0.39 -0.03 -0.12 -0.26 -0.41 ?

going into detail one may therefore adjust

~ V

1

(jqj = 0) to reproduce _

V

. Since this represents

quite a sizable change it provides an upper bound for the 1=N

C

corrections to the magnetic

form factors. Keeping all other quantities at the values associated with the self-consistent soliton the strange magnetic moment of the nucleon becomes _

S

= \Gamma 0:03(\Gamma 0:05) for m =

400(450)MeV. Apparently modifications, which lead to a larger _

V

causes _

S

to decrease.

From this we consider the result _

S

ss 0:25 obtained from the self-consistent soliton as an

upper bound. _

S

may even assume small negative values as in the Skyrme model with vector

mesons [7].

The electric radii of the nucleon, which are given by the slopes of the associated electric form factors hr

2

i = \Gamma 6@G

E

(Q

2

)=@Q

2

j

Q

2

=0

are shown in table 2. Not surprisingly the radii

increase with decreasing * and the experimental data are reasonably reproduced for a some- what larger value of the scaling variable than in the case of the magnetic moments. Within the reliability of the approach we estimate hr

2

S

i ss \Gamma 0:1::: \Gamma 0:2fm

2

.

Figure 1 exhibits that the valence quark and vacuum contributions to G

E;S

of the nucleon

almost cancel each other leading to a total strange electric form factor which barely deviates from zero. The decomposition into G

val

S

and G

vac

S

refers to (17). On the other hand G

val

M;S

and

G

vac

M;S

add up coherently to G

M;S

. While G

val

M;S

drops monotonously the vacuum part develops

an extremum which is also recognized in G

M;S

. A non-monotonous behavior for G

M;S

is also

known from the Skyrme model with vector meson [7].

We have also considered the model with the imaginary part unregularized. For the self- consistent soliton this yields a slight increase for the strange magnetic moment _

S

ss 0:35

while hr

2

S

i remains within the bounds given above.

5. Summary

Using the generalized Yabu-Ando approach to the chiral soliton of the NJL model we have investigated the matrix elements of the nonet vector current between nucleon states. These are of great interest because they provide information on the strangeness content of the nucleon. Furthermore these matrix elements enter the analysis of experiments attempting precision tests of the standard model.

To some extend the model is plagued by the too small prediction for the isovector magnetic moment, _

V

, of the nucleon. Modifications improving on this prediction cause the strange

magnetic moment, _

S

, to decrease. As a consequence the result related to the self-consistent

soliton represents the upper bound. A lower bound has been obtained by (over) estimating possible 1=N

C

effects, yielding

\Gamma 0:05 ^ _

S

^ 0:25: (22)

The lower bound is compatible with the corresponding prediction in the Skyrme model with vector mesons [7]. Other models give even smaller results [8]. We have furthermore observed

7

Figure 1: The strange electric and magnetic form factors of the nucleon. Parameters are m = 450MeV and * = 0:75, which provide a good agreement for the baryon mass differences: O/ = 68MeV, cf. table 1.

that the valence quark contribution (i.e. short range effects) dominates _

S

although the

vacuum 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 prediction for the slope of this form factor is

\Gamma 0:2fm

2

^ hr

2

S

i ^ \Gamma 0:1fm

2

: (23)

This result compares well with that of the Skyrme model [6]. As G

E;S

ss 0 it is easy to

understand that small effects may modify this result. As an example we refer to the OE \Gamma ! mixing, which is neither included in the present model nor in the Skyrme model.

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8

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