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Nucleon form factors and vNN and fNN coupling constants

S. FURUICHI(1_{) and K. WATANABE(}2₎

(1_{) Department of Physics, Rikkyo University - Toshima, Tokyo 171, Japan}

(2_{) Department of Physics, Meisei University - Hino, Tokyo 191, Japan}

(ricevuto il 9 Dicembre 1996; approvato il 3 Aprile 1997)

Summary. — By using the superconvergent dispersion relation we examine the electromagnetic form factors of nucleon taking into account the consistency with the other process, such as nucleon-nucleon scattering. The experimental data of neutron electric form factor leads to a stringent constraint on the coupling constants. It is shown that there exists a solution for which the experimental data of the form factors are reproduced very well and that for the solution the vNN coupling constant agrees with that determined by the analysis of nucleon-nucleon scattering. The vector and tensor coupling constants of fNN then become fairly large. If the fNN coupling constants are set to 0, the calculated neutron electric form factor becomes much larger than the experimental data.

PACS 13.40.Gp – Electromagnetic form factors.

1. – Introduction

In recent years experiments on the nucleon electromagnetic form factors have become very accurate and the following results have been established: although the electric and magnetic form factors of proton and the magnetic form factor of neutron are approximately given by the dipole formula GD4 1 /( 1 1 t/0 .71 )2, t being the squared

momentum transfer expressed in the unit GeV2_{, the experimental data fall off more}

rapidly than GD for large t and in the low-t region there is a small deviation from the

dipole formula. The neutron electric form factor is very small for t G4 GeV2. Theoretically, we have two typical models: the vector dominance model (VDM) and the quark model based on the perturbative QCD (PQCD). With recourse to VDM we are able to reproduce the experimental data fairly well by taking account of vector bosons given in the table of particle properties provided that the r-meson mass is taken to be 20% smaller than the experimental value. According to PQCD [1] the charge and magnetic moment form factors decrease more rapidly than the dipole formula for large t. The fall-off of the nucleon form factors agree with the prediction of PQCD, however, PQCD does not work in the low-t region. Considering that the experimental

data on the nucleon form factors have become very accurate, it is a problem of vital importance to establish a theory by which the result of PQCD is realized for large-t and the experimental data of form factors in the low-t region are explained as well.

We have assumed the dispersion relation for the form factors with respect to t. The asymptotic value of the form factors necessarily leads to the unsubtracted dispersion relation. Moreover, superconvergence constraints are required to meet the prediction of PQCD. To incorporate VDM, summation of delta functions corresponding to vector bosons is necessary for the absorptive parts, but the r-meson should be treated differently from the other vector bosons, considering that in VDM its mass must be taken much smaller that the experimental value. We have regarded the r-meson as a dynamical resonance of pions with the incorporation of uncorrelated effect for the two-pion system, that is, the contribution from the pion-nucleon scattering. Due to the uncorrelated two pions the mass of the r-meson is reduced effectively. With recourse to the superconvergent dispersion relation we analyzed the experimental data and were able to realize the data of form factors from low- and high-t region very well with the

r-meson mass kept at the experimental value.

We write the dispersion relations for the charge and magnetic moment form factors, F1I and F2I (I denoting the isospin) respectively, instead of the electric and

magnetic form factors, GI

E4 F1I2 (t/4 m2) F2I and GMI4 F1I1 F2I. The absorptive parts

of FiIare given as follows: for the high-t region, t FL2QCDwith LQCDA 2 GeV , Im FiIare taken so as to realize the prediction of PQCD. For t GL2

QCD, the absorptive parts Im FiI (i 41, 2) are the same as that of VDM except of the r-meson. For the region tGL2

QCD

with Lhadron1 .5 GeV , where the r-meson dominates, we express the isovector part

Im Fi1(i 41, 2) in terms of the helicity amplitudes for ppDNN, which are given by the pion-nucleon scattering amplitudes parametrized through the zero width approximation for the nucleon and the D( 1232 ) resonance contributions [2-4] with the parameters evaluated from the low-energy pN scattering data. For the f-meson we also have the uncorrelated KK contribution as in the case of r-meson, but we leave it out in this paper because the effect was estimated to be negligibly small [4].

In our previous analysis [4] the coupling constant of v to the nucleon became much larger than that obtained by the nucleon-nucleon scattering [5] and the fNN coupling constant also turned out to be fairly large. We obtained the following coupling constants of v and f to the nucleon: g2

v/4 p 449.6 and gf2/4 p 423.0. Similar result was obtained by Höhler et al. [6]. It seems that there is some inconsistency between the result of electromagnetic form factor and the nucleon-nucleon scattering with respect to the vNN coupling constant. It is the purpose of this paper to investigate the possibility of taking the vNN coupling constant compatible with the result of NN scattering. We also discuss on the magnitude of f coupling constant to the nucleon in connection with the fraction of the strange quark in nucleon.

The organization of this paper is given as follows: in sect. 2 we summarize the formulation of our calculations. In sect. 3 we give numerical results obtained by analyzing the experimental data of proton electric and magnetic form factors and neutron magnetic form factor. The following three cases are investigated: case I) the solution for which the v nucleon coupling constant is taken compatible with the result of NN scattering and with the experimental data of neutron electric form factor; case II) the best fit obtained without any constraint for the parameters; case III) the vector and the tensor coupling constants of f-meson to the nucleon are set to 0. Cases I) and II) reproduce the experimental data very well but for case III) the calculated neutron electric form factor does not agree with the experimental result. Section 4 is devoted to

discussions on our results. We give in the appendix functional approximation for our calculated result for the case I via the Padé formula for the sake of convenience in reproducing the theoretical form factors numerically.

2. – Superconvergent dispersion relation

According to PQCD the charge and magnetic moment form factors of nucleon are given asymptotically as follows [1]:

F1IA t22[ln (t/Q02) ]2g, F2IA t23[ln (t/Q02) ]2g.

(1)

Here Q0 is the momentum that characterizes QCD and g 4214/(3b) where b4112

2 nf/3 , with nfthe number of flavors. Taking account of (1), we assume the unsubtracted

dispersion relations for FiI

FI i(t) 4 1 p



with mpbeing the pion mass, and impose the conditions on the form factors

tn_FI

1(t) K0 (n41, 2) , tnF2I(t) K0 (n41, 2, 3) , (I 40, 1) , NtNKQ ,

(3)

which lead to the superconvergence constraints on Im FiI(t) 1 p



We write the dispersion integral (2) as

FI j(t) 4 1 p



!

where Lhadronis the hadronic cut-off. For the first part of the integral, t GL2hadron, use is

made of the same formulae and parameters for Im FjI ( hadron ) that are given in ref. [3]. The asymptotic parts Fjas (L2QCDG t), being added in order to realize the prediction of

PQCD, are taken as follows:

Fas j (t) 4



We impose the normalization conditions on FiI at t40; F1I( 0 )41/2 and F2I( 0 )4kI,

where kI _{denotes the anomalous magnetic moment of nucleon with the isospin state I.} The superconvergence conditions (4) lead to relations among the parameters appearing in (5). The hadronic and QCD quantities are thus interrelated through the superconvergence conditions.

We determine parameters so as to reproduce the experimental data of the nucleon electric and magnetic form factors. In our previous analyses we looked for the solutions

that minimize chi-square, but the dependence of the value of x2_{on the residues at the}

vector boson poles is rather weak so that a fairly large region is allowed for the residues in case we weaken the condition on the value of x2 _{a little. We analyze the}

experimental data of electromagnetic form factors by calculating the x2_{as a function of}

two residues at v andOor f poles in (5). We take account of recent data on the form factors and the vector boson resonances.

3. – Numerical results

Let us now give numerical results. The following values are taken for the cut-offs and QCD parameter: Lhadron4 1 .2 GeV , LQCD4 2 .5 GeV , Q04 0 .2 GeV and the number

of flavors is nf4 6 . Zero-width approximation is done for D( 1231 ) and for pNN and

p DN coupling constants and use the same values of parameters as in refs. [ 3 , 4 ]. For

the vector bosons we take account of the resonances in the Review of Particle Properties 1966 [7], five isoscalar and four isovectors. Among the residues at vector boson poles we separate two of them at the v andOor f-meson poles and determine the remaining ones from the condition that the chi-square for the experimental data of electric and magnetic form factors of proton and the magnetic form factor of neutron,

x024 x2(G p

E) 1x2(G p

M) 1x2(GMn), is minimized. The number of data used in the analysis

is 172 with 11 independent parameters. x02 is given as a function of the separate

parameters. To examine the dependence on v and f coupling constants we investigate the following two cases for the chosen residues:

Case a) a1vand a1 f . Case b) a1 f and a2 f .

We illustrate the contour lines of x20vs. the chosen parameters for the cases a) and

b) in fig. 1, in which we enter the constraint obtained from the chi-square value for

Fig. 1. – a) x2 0vs. a1v/Mv2and a1 f_/M2 f. b) x20vs. a1 f_/M2 fand a2 f_/M2 f.

Fig. 2. – Ratio of form factors to GD(t) 4 (11t/0.71)22 with t in GeV2. The solid curve

corresponds to the case I), the vNN coupling constant is taken consistent with analysis of N-N scattering, and the dotted one corresponds to case II), the solution that makes the chi-square value minimum. The dashed curve stands for the case III), a₁f

4 a2

f

4 0 . In a) the cases II) and III) are left out because these curves almost coincide with case I). a) GMp/mpGD. b) GEp/GD.

c) Gn

M/mnGD. d) NGEn/GDN2. %Stein et al. [8], {Price et al. [9], !Berger et al. [10], pBartel et al. [11],m Hanson et al. [12],jBorkowskj et al. [13],!Rock et al. [14],3Walker et al. [15],

sLung et al. [16],{Sill et al. [17],%Andivahis et al. [18].

squared neutron electric form factor

x2(GEn) 4

!

(

)

The shaded regions represent the constraint x2_(Gn

E) G30 for the 16 data points of

(GEn)2. In fig. 1a) we show the allowed region on the residue a1v/Mv2 evaluated via the formula in VDM, gV4 fVaV/MV2with gV and fVdenoting the coupling constants of vector boson V to the nucleon and to g, respectively. The dashed region is obtained from the v nucleon coupling constant, g2

Fig. 3. – a) F₁p/GD. b) F2p/kpGD. c) F1n/GD. d) F2n/mnGD. Here kp4 mp2 1 . The data in a)-b) are

taken from Walker et al. [19].

Fig. 4. – Q2_F 2

p_/k pF1

p_.

TABLEI. – Residue and parameters determined by the analyses.

MV(GeV/c2) Case I) x204 197.7 Case II) x204 193.7 Case III) x204 203.7

aV 1 /MV2 a2V/MV2 a1V/MV2 a2V/MV2 a1V/MV2 a2V/MV2 I 40 0.78194 1.019413 1.4190 1.649 1.680 0.62 1.2 2 5.636 24.05 219.74 0.630 2 1.916 4.136 214.47 11.56 1.299 20.507 21.914 9.758 28.138 0.510 2 1.886 4.992 219.18 15.50 0.576 0.0 0.995 26.240 5.165 0.458 0.0 2 4.77 26.091 221.841 I 41 0.7699 1.465 1.70 2.15 0.1561 3.512 2 3.077 0.4587 3.530 2 3.189 2.048 2 0.1481 0.1673 2.885 22.594 0.3767 3.618 2 3.584 2.436 2 0.2193 22.727 1.785 21.425 0.1756 215.397 2 1.318 0.2132 0.1894 g₁ g₂ C0 CV 1 /L4QCD CV 2 2 /L6QCD CS 1 /L4QCD CS 2 /L6QCD 2.304 17.07 216.58 0.6829 2 0.5908 0.7049 2 0.7221 2.384 17.53 210.21 0.9182 2 0.3542 0.4879 2 1.636 3.302 16.96 216.29 0.7118 2 1.784 0.7225 5.170

TABLEII. – Vector coupling constants of vector boson to the nucleon. rNN coupling constant is the effective one calculated via the zero-width approximation for Im Fi1.

Vector boson g2

V/4 p

case I) case II) case III)

r v f 0.0491 8.90 19.01 0.0546 39.08 3.39 14.97 7.69 0.0

through the analysis of nucleon-nucleon scattering [5]. We give our result for

GMp/mpGD, G p

E/GD, GMn/mnGD and (GEn/GD)2 in fig. 2a)-d) for the following cases: I) a1v

and a1f satisfy the condition of Grein-Kroll constraint and x2(GEn) G30, II) the best fit

solution that minimizes x20 and III) a1

f 4 a2

f

4 0 . The parameters determined by the analysis are summarized in table I.

The ratio of the charge and the magnetic moment form factors to the dipole formula, FiN/GD (i 41, 2), are given in fig. 3a)-d) for the proton and the neutron. We

illustrate Q2_F 2

p

/kpF1 p

in fig. 4, where kpis the proton anomalous magnetic moment.

Our results agree with the experimental data very well for Fipand for Q2F2 p

/kpF1 p

as well as GEp, G p M and G

n

M for the cases I)-III) but for the case III) the calculated

neutron electric form factor becomes too large.

We calculate the coupling constants of the v, f and nucleon by using the formula in VDM with fr4 5 .031 , fv4 17 .06 and ff4 12 .88 and summarize them in table II. The

rNN coupling constant is the effective one which is calculated via the zero width

approximation for the absorptive parts Im FjI ( hadron )[4].

We obtain the electromagnetic form factors that realize the experimental data by using the vNN coupling constant compatible with that determined from the analysis of nucleon-nucleon scattering. It must be noticed that for this solution the value of x2

becomes larger than the best fit by almost 4 for 172 data points. The vNN coupling constant is also consistent with our previous result determined through the systematic analysis of the pion-pion, pion-nucleon and nucleon-nucleon scattering [20] where it was determined as g2

v/4 p 45.63–9.39.

4. – Discussions

Let us discuss on the fNN coupling constant. The contour lines in x2 _{depicted in}

fig. 1b) show that the tensor coupling of the f-meson cannot be taken small if the data of neutron electric form factor is taken into account, although the vector coupling may possibly be small. The vector coupling constant is g2

f4 4 p 4 19 .01 for the case I) compatible with the result of Grein-Kroll, and for the best fit, the case II), g2

f/4 p 4 3 .39 . For the case III), in which the residue at the f-meson pole vanishes, a₁f

4 a2

f 4 0 , we get x204 204 so that the proton electric and magnetic and the neutron magnetic

form factors are reproduced fairly well, but the calculated result for the neutron charge form factor, illustrated by the dashed curve in fig. 2d), becomes much larger than the experimental value; x2_{becomes as large as x}2_(Gn

E) 40.93103. As is seen from

fig. 1b), the constraint from the neutron electric form factor is stringent so that we must take a₂f/Mf2A 21 .5 if we impose the condition x2(GEn) G30. The contribution

from f-meson cannot be neglected to explain the experimental data of electromagnetic form factors. The fractions of strange quark in nucleon results to be fairly large so that OZI rule is badly violated.

To conclude the paper we give the estimate of the ratio of effective coupling constant of the strange quark to the u- or d-quarks, ri4 g×i

(

)

(

(

dd) /k2

)

)

f_{by the formulae r} i4 2k2

(

)

(

f tan u 2 (aiv/Mv2)( 1 /tan u)

)

u being the v-f mixing angle. We obtain the following results for the f-v mixing angle u 437.087: for the case I) r14 21 .57 and r24 2 .12 and for the best-fit solution II) r14

2 .26 , r24 2 .54 , and for the case III) r14 r24 2 0 .032 . It is possible to take the value of

Nr1N small by weakening the condition for the neutron electric form factor and the

Grein-Kroll constraint a little, but the value of r2 is hardly reduced. For instance, for

a1v/Mv24 0 .6 and a1f/M 2

f4 0 .2 we have r14 2 0 .29 with x204 198 .5 and x2(GEn) 431.7.

The tensor part of the residues are then obtained to be av

2 /Mv24 0 .708 and a2f/M 2

f4 2 1 .08 , and consequently we have r24 1 .08 . The constraint from the neutron electric

form factor leads to a severe condition on r2. We note that the recent experiment on the

f-meson production in pp annihilation at rest [22] implies a fairly large fraction of

* * *

The authors thank Prof. M. NAKAGAWAfor valuable discussions and comments.

AP P E N D I X

We represent our calculated results by a functional approximation of the Padé formula [23] for reproducing the theoretical form factors. The ratio of the form factors to the dipole formula is expressed as follows:

G/GD4 a01

!

!

where x 4 ln(11t) with t given in terms of GeV2and a04 1 for G p E, G

p

M, GMn and a04 0

for GEn. Here we give the formula for the case I), represented by the solid curves in figs.

2-4, in which the v vector coupling to the proton is taken compatible with that determined by the nucleon-nucleon scattering. We find that by taking N 43 in (A.1) we are able to realize our theoretical results within the error of 0.1%. We have the following results: GMp/mpGD4 1 14.4359x12.2337x 2 1 0.25568 x3 1 14.8528x11.3947x2 2 0.033834 x31 0.33205 x4 , GEp/GD4 1 13.5104x21.6773x 2 1 0.16450x3 1 13.9881x22.7336x2 1 0.96081 x32 0.15320 x4 , Gn M/mnGD4 1 13.2464x10.31357x2 2 0.19954 x3 1 13.8546x21.4406x2 1 0.71061 x31 0.0037074 x4 , Gn E/GD4 0.81397 x 20.86083x2 1 0.34643x3 1 11.4510x21.2002x2 1 0.60461 x32 0.051526 x4 ,

where mp and mn stand for the magnetic moment of proton and neutron, respectively.

The formulae are valid for GMp/mpGDup to t 450 GeV2, for G p

E/GD and GMn/mnGD up to

t 420 GeV2_{and for G}n

E/GD up to t 410 GeV2.

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