Nucleon electric form factors and quark sea polarization in the Nambu-Jona-Lasinio model

Physics Letters B 278 (1992) 24-28 North-Holland

PHYSICS LETTERS B

Nucleon electric form factors and quark sea polarization in the Nambu-Jona-Lasinio model A.Z. G6rski a,b, F. Griimmer

a

and K. Goeke a

a lnstitutJ~r Theoretische Physik II, Ruhr-Universitdt Bochum, W-4630 Bochum, FRG b Institute of Nuclear Physics, Radzikowskiego 152, PL-31-342 Cracow, Poland

Received 18 June 1991; revised manuscript received 20 December 1991

The nucleon electric form factors are extracted from the semiclassically quantized solitonic solution of the Nambu-JonaLasinio model. To this end the model is solved self-consistentlyin the SU (2) sector with scalar coupling using the zero-boson and one-quark loop approximation on the chiral circle. Charge distributions, squared radii and the Coulomb energy splitting are calculated as well. For constituent masses around M= 400 MeV one obtains good agreement with the experimental data of the proton. The tail of the neutron charge distribution is made of polarized sea quarks.

In the past several years the N a m b u - J o n a - L a s i n i o ( N J L ) model [ 1 ] has become popular as an effective lowenergy Q C D theory. It is the simplest quark model to provide spontaneous breaking o f chiral symmetry and has been quite successful so far [ 2 - 6 ]. Also, there are many arguments that it can be really recovered as a long wavelength limit o f Q C D [ 7-9 ]. However, in the solitonic sector the results were usually restricted to the properties of the classical solutions [ 2-6 ]. The quantization of the chiral N a m b u - J o n a - L a s i n i o soliton was discussed in refs. [ 9,10 ] at a formal level. Just recently, two independent groups have published numerical results concerning the (iso)rotational nucleon m o m e n t o f inertia [ 11,12 ]. In this paper we compute the isoscalar and isovector nucleon electric form factors. With these fundamental objects at hand all electric properties o f the neutron and proton can be evaluated and compared with experimental data. In particular, we present neutron and proton electric form factors, charge distributions, electric mean squared radii and the Coulomb correction to the n e u t r o n - p r o t o n mass splitting. Our starting point is the two-flavor N J L lagrangian with scalar coupling: LNjL = # ( i O - - m o ) q + ~G[ (Ctq)2+ (t]i~r~,sq) z ] ,

( 1)

where G is a coupling constant and mo is the current mass o f the quarks. By standard path integral bosonisation [ 13 ] the quark theory ( 1 ) is converted into a non-linear effective meson theory with the following proper time regularized fermionic part of the action, which takes into account quark sea polarization [ 2-6,14,15 ]: Nc Serf = v -2" ~n

~ dz d4XETrrTr~ j --z q~,(xz) exp( --ZI~+I~)~,(XE),

(2)

1/A2

where our Dirac operator iI~ reads: i I ~ = i ~ - g [ a ( x ) + i 7 5 ~ t ( x ) ] , ~n(x) is any complete set of eigenfunctions. The action (2) is to be continued to the euclidean space to get a well-defined integral. The parameters including the cut-offA are determined by adjusting the model to the vacuum and meson sector as discussed in ref. [ 14 ]. The fields a and n are assumed to be in the standard self-consistent hedgehog form, i.e. a = tr(r), 7t=~z(r) and satisfying the chiral circle condition: tr 2 (r) +Tt 2 (r) = f z . The diagonalization of the Dirac operator is done following the method o f Kahana and Ripka [ l 6 ]. In this way we can compute physical observables without recourse to the derivative (gradient) expansion. Having the 24

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Volume 278, number 1,2

PHYSICS LETTERS B

19 March 1992

eigenfunctions and eigenenergies as the next step we calculate the corresponding charge densities and the Sachs electric form factors defined by

GE(q2) =

(N(p)Ij~m(0)IN(P' ) ),

(3)

where c:_ lf2T=0_l_f~2T=l "-'E-= ~'-' E " ~' E T3 and q/N is the nucleon wave function. The formula (3) is valid in the non-relativistic region, for q2 smaller than the nucleon mass ( M y ) squared, i.e. below 1 GeV that is just the case of our interest. In order to compute these matrix elements one has to quantize the soliton to obtain proton and neutron states and one has to couple the soliton to the electromagnetic field A u. All this results in the following Dirac operator: iI~ =74(i0 , +h-i£2-iA4RtQR+i747kAkR*QR).

(4)

Here, h=~,4?kO~--y4Uis the single-particle hamiltonian, R(t)-exp(-i£2t) is a general S U ( 2 ) rotation and U ~ a + i75"cA~ A. The electromagnetic current is defined as the functional derivative of the action (2), (4) with respect to A u taken at A u = 0:

j~um = ~Au Serr[Au, £2] Au=--O . F

(5)

Due to the proper time regularized form of the action we have in the exponent in (2) a term proportional to the squared hamiltonian, h 2, and several other terms containing the external electromagnetic field (Au) as well as the cranking velocity (£2). To compute (2) explicitly, first one has to make an expansion of the action (2) in terms of the £2 and A u. To this end we apply the famous F e y n m a n - S c h w i n g e r - D y s o n expansion for an operator exponent:

1 I eA+B=eA+ ~ d a e ~A Be(l-a)A+ 0 0

l-p

fob ~

dc~e "A B e ' B e ( l - a - # ) A +

(6)

....

0

where for the operator A we take A = - 042+ h 2 and B stands for the additional terms (i.e. all terms containing A u and £2). As the second step, we perform the functional derivative (5). Forj~, m the only non-zero contribution comes from the terms linear in A u (due to the condition A u = 0 in ( 5 ) ) and all other terms can be ignored. Then, we cut the expansion at the first (linear) order in £2 assuming small rotational velocity (that is, in fact, of the order of 1/Nc [ 9 ] ). Now, taking for Vn (x) in eq. (2) eigenfunctions of the unperturbed hamiltonian, h, allows us to substitute h by its corresponding eigenenergies ~n. As the next step, we quantize the collective variable £2 in the standard m a n n e r [ 17 ]. The resulting form factors are derived as

Gr=°(q2)= ~ d3xexp(-iq'x) Nc(-5- nE~val I~¢n(X)12+ G~= 1(q2) =

fd3xexp(_iq.x)-~Nc ~ d3y [ --

nepalis i g n ( + ~ . ) I~v.(x)12 ) ,

(7)

neva[Z~/nq-(X)'~A~Im(X)~II+m(y)TA~IIn(Y)En -- ('m m~all

1

-- ~

f dr ~] (~,,exp(-z,])+e,,,exp(-Z~2m)lexp(-z,2)-exp(-Z, I~A2V~ m'mn;all\ 'n'~-'m ~1_ "~ ,2__,2

× ~+ (x)zA~/~(X)q/+m(y)zA~,,(y)],

Zm)) /1

(8)

where I is the m o m e n t of inertia c o m p u t e d previously [ 10,11 ], Nc is the n u m b e r of colours (N~= 3 ), and sum25

Volume 278, number 1,2

PHYSICS LETTERS B

19 March 1992

m a t i o n is a s s u m e d o v e r r e p e a t e d i n d e x A. T h e v a l e n c e level is d e n o t e d by val. T h e i n d e x all refers to a s u m o v e r all eigenlevels i n c l u d i n g t h e v a l e n c e one. F o r q2=Othe f o r m factors G ~ =° a n d GEr = l can be s h o w n to be e q u a l to one. F o r the i s o v e c t o r o n e this is b e c a u s e the space i n t e g r a t i o n in eq. ( 7 ) basically yields the m o m e n t o f inertia, w h i c h is t h e n c a n c e l l e d by the I in the d e n o m i n a t o r . F o r the isoscalar o n e this h o l d s b e c a u s e it is n o t r e g u l a r i z e d a n d t h e n its v a l u e is i d e n t i c a l to the b a r y o n n u m b e r B = 1. Actually, as can be c h e c k e d by explicit n u m e r i c a l calculations, i m p o s i n g the p r o p e r t i m e r e g u l a r i z a t i o n changes o u r result for GEr=° by less t h a n 5%. T h e p r o t o n a n d n e u t r o n f o r m factors are s h o w n in figs. 1 a n d 2 for two v a l u e s o f the c o n s t i t u e n t m a s s M. T h e y are g i v e n in the r e g i o n q 2 < M 2 for w h i c h o u r a s s u m p t i o n t h a t recoil effects are small m i g h t be justified. A p p a r ently, the p r o t o n f o r m f a c t o r is in g o o d a g r e e m e n t w i t h the e x p e r i m e n t a l d a t a a n d also n o t v e r y sensitive to the actual c h o i c e o f M , i f M is c h o s e n a r o u n d 400 M e V . T h e n e u t r o n f o r m f a c t o r is n o t so well r e p r o d u c e d . F o r a b o u t q > 400 M e V (q2 = 0.16 G e V 2) o u r m o d e l o v e r e s t i m a t e s the e x p e r i m e n t by m o r e t h a n a f a c t o r o f two. In case o f the p r o t o n t h e sea q u a r k s c o n t r i b u t e less t h a n 5% to the electric f o r m factor. F o r the n e u t r o n , v a l e n c e q u a r k s a n d sea q u a r k s c o n t r i b u t e w i t h s i m i l a r m a g n i t u d e a n d o p p o s i t e signs. T h e s q u a r e d electric radii are

0.12

1.00

0.06

0.80

0.60

b0.00

~o-0.06

o 0.40

-0.12

0.20

-0.18

rI

0.0

I

I

I

0.1

0.2

0.3

I

04

05

0.00 0.0

0.1

02

0.3

0.4

0.5

momentum transfer [GeV~+-x-2]

momentum transfer [GeVeeee2]

Fig. 1. The neutron electric form factor as a function of the squared momentum transfer [GeV2]. The solid lines describe the total form factor (thick line) and the sea contribution (thin line) for the constituent quark mass of 418 MeV. The dashed lines display the same information for the constituent quark mass of 465 MeV. For the experimental data see ref. [ 18 ] and references therein.

Fig. 2. The proton electric form factor as a function of the squared momentum transfer [GeV 2]. The solid lines describe the total form factor (thick line) and the sea contribution (thin line) for the constituent quark mass of 418 MeV. The dashed lines display the same information for the constituent quark mass of 465 MeV. For the experimental data see ref. [ 18 ] and references therein.

Table 1 The nucleon squared radii computed for the constituent quark masses of 363 MeV and 418.5 MeV. Total, valence and sea contributions are given separately to see the influence of the vacuum polarization effects. In addition the Coulomb part of the neutron-proton splitting is listed. In the last column the experimental values are given for comparison. Quantity

Constituent quark mass 363 MeV

(r2>T=O [fm 2 ] r=l [fm2] n [fm 2]

p [fm2] AEM [MeV]

26

Experiment 418 MeV

465 MeV

total

sea

total

sea

total

sea

0.66 1.36 -0.35 1.01 --0.87

0.07 0.71 -0.32 0.39 --0.18

0.61 1.15 -0.27 0.88 -- 1.00

0.16 0.74 -0.29 0.45 --0.73

0.53 1.05 -0.26 0.79 -- 1.09

0.15 0.73 -0.29 0.44 --0.33

0.62 0.86 -0.12 0.74 --0.76+0.30

Volume 278, number 1,2 0.50

PHYSICS LETTERS B

(a)

.-.,

19 March 1992

1.5

(b) 1.0

~ 0.25

0.5 I O.O0

0.0

~

L

.

.

.

t

0.0

0.5

1.0

1.5

2.0

2.5

RADIUS[fm]

- 0 . 5

i

0.0

i

i

,

t

.

0.5

.

.

.

i

.

1.0

.

.

.

i

,

1.5

,

,

,

r

,

2.0

,

,

,

2.5

RAOIL.!S[fm]

Fig. 3. The neutron (a) and proton (b) charge distribution as a function of the radius [fm ]. The solid bold line denotes the total charge, the dashed line denotes the valence contribution (without sea polarization effects) and the solid line stands for the sea contribution only. Plotted is the quantity r2p(r). The quark mass=465 MeV, g = 5.0.

presented in table 1. Apparently the isoscalar squared radius reproduces the experiment at M = 418 MeV, the isovector one is generally too large and needs higher values of M to agree with the data. The consequence of this is that the proton squared radius is in reasonable agreement with experiment for M = 465 MeV whereas the neutron one is at least a factor of two too large in this region of M. The reason for this behaviour can perhaps be explained by the 1/Nc expansion. G E r=° is of the order O (No) whereas G T= t is of the order O (1). Actually boson loop contributions would also be of the order O( 1 ) and hence are expected to modify the result for GET=1. They are not included in the present calculations because here we concentrate on collective zero modes and the evaluation of boson loops is quite involved. The neutron form factor, being the difference of two big quantities, is generally rather sensitive and hence one expects also modifications from boson loops. Table 1 lists the Coulomb part of the proton-neutron mass splitting, as well. It is slightly lower than expected from the estimates of Gasser and Leutwyler. However, inclusion of the magnetic form factors will reduce the absolute value by about 20% [ 19 ]. Fig. 3 shows the proton and neutron charge distributions for illustration. One realizes the long negative tail in case of the neutron, which is a necessary ingredient for a negative charge radius. Actually, the sea quark contribution can be related via a gradient expansion to the contribution of a dynamic pion field. Then fig. 3 confirms the popular picture that the negative tail of the neutron charge distribution is made by the pion cloud. This is also seen at the sea quark contribution to the neutron squared charge radius. We can summarize our points. The Nambu-Jona-Lasinio model reproduces fairly well the electric form factor of the proton and the proton squared radius for constituent masses around M = 400 MeV. For these quantities the sea quarks play a minor role. In this region of M the electric part of the neutron-proton mass difference is reproduced as well. The neutron form factor is a factor of two too large and the sea quarks contribute as much as the valence quarks. The tail of the neutron charge distribution is made by sea quarks. It is a pleasure to thank Dr. Victor Petrov for many valuable discussions. This work has partially been supported by the Bundesministerium ftir Forschung und Technologie, Bonn. This work has also been supported by the Polish Comittee for Scientific Research (KBN-2-0091-91-01 ).

References [ 1 ] J. N a m b u and G. Jona-Lasinio, Phys. Rcv. 122 ( 1961 ) 345. [ 2 ] H. Reinhardt and R. W~insch, Phys. Left. B 215 ( 1988 ) 296.

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[ 3 ] Th. Meissner, F. Griimmer and K. Goeke, Phys. Lett. B 227 (1989) 296. [4] Th. Meissner and K. Goeke, Nucl. Phys. A 524 ( 1991 ) 719. [ 5 ] A. Blotz, F. D6ring, Th. Meissner and K. Goeke, Phys. Lett. B 251 (1990) 235. [6 ] Th. Meissner and K. Goeke, Z. Phys. A 229 ( 1991 ) 513. [7] R. Ball, Phys. Rep. 182 (1989) 1. [8] M. Schaden, H. Reinhardt, P. Amundsen and M. Lavelle, Nucl. Phys. B 339 (1990) 595. [9] D. Dyakonov, V. Petrov and P. Pobylitsa, Nucl. Phys. B 306 (1988) 809. [10] H. Reinhardt, Nucl. Phys. A 503 (1989) 825. [ 11 ] K. Goeke, A.Z. G6rski, F. Griimmer, Th. Meissner, H. Reinhardt and R. Wiinsch, Phys. Lett. B 256 (1991 ) 321. [ 12] M. Wakamatsu and H. Yoshiki, Nucl. Phys. A 524 ( 1991 ) 561. [13] T. Eguchi, Phys. Rev. D 14 (1976) 2755. [ 14] Th. Meissner, E. Ruiz-Arriola and K. Goeke, Z. Phys. A 336 (1990) 91. [ 15 ] J. Schwinger, Phys. Rev. 82 ( 1951 ) 664. [ 16] S. Kahana and G. Ripka, Nucl. Phys. A 419 (1984) 462. [ 17] G.S. Adkins, C.R. Nappi and E. Witten, Nucl. Phys. B 228 (1983) 552. [ 18] M.F. Gari and W. Kriimpelmann, Z. Phys. A 322 (1985) 689. [ 19 ] J. Gasser and H. Leutwyler, Phys. Rep. 87 ( 1987 ) 77.

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