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Holographic estimate of the meson cloud contribution

to nucleon axial form factor

G. Ramalho

Laboratório de Física Teórica e Computacional—LFTC, Universidade Cruzeiro do Sul,

01506-000, São Paulo, SP, Brazil

and International Institute of Physics, Federal University of Rio Grande do Norte ,

Campus Universitário—Lagoa Nova CP. 1613, Natal, Rio Grande do Norte 59078-970, Brazil

(Received 25 July 2017; published 11 April 2018)

We use light-front holography to estimate the valence qua rk and the meson cloud contributions to the

nucleon axial form factor. The free couplings of the holographic model are determined by the empirical

data and by the information extracted from lattice QCD. The holographic model provides a good

description of the empirical data when we consider a meson cloud mixture of about 30% in the physical

nucleon state. The estimate of the valence quark contribution to the nucleon axial form factor compares

well with the lattice QCD data for small pion masses. Our estimate of the meson cloud contribution to the

nucleon axial form factor has a slower falloff with the square momentum transfer compared to typical

estimates from quark models with meson cloud dressing.

DOI: 10.1103/PhysRevD.97.073002

I. INTRODUCTION

In recent years, it was found that the combination of the

5D gravitational anti-de Sitter (AdS) space and conformal

field theories (CFT) can be used to study QCD in the

confining regime [1–4]. Using this formalism one can relate

the results from AdS=CFT with the results from light-front

dynamics based on a Hamiltonian that include the confin-

ing mechanism of QCD (AdS/QCD) [4]. In the limit of

massless quarks, one can relate the AdS holographic

variable z with the impact separation ζ, which measures

the distance of constituent partons inside the hadrons [4–6].

This correspondence (duality) between the two formalisms

is known as light-front holography or holographic QCD.

Over the last few years light-front holography has

been used to study several proprieties of the hadrons.

The soft-wall formulation of the light-front holography

introduces a holographic mass scale κ, which is funda-

mental for the description of the hadron spectrum (mesons

and baryons) and hadron wave functions [4,7–11]. This

scale can be estimated from the holographic expression

for the ρ mass m

≃ 2κ [4,8]. Examples of applications of

light-front holography are in the calculation of parton

distribution functions, hadron structure form factors among

others [4,5,12–22].

In the light-front formalism one can represent the wave

functions of the hadrons using an expansion of Fock states

with a well defined number of partons [4]. In the case of

baryons, the first term corresponds to the three-quark state

(qqq). The following terms are excitations associated with

a gluon, ðqqqÞg, with a quark-antiquark pair, ðqqqÞq

q, and

higher order terms. Those states can be labeled in terms of

the number of partons τ ¼ 3; 4; 5; …, respectively. The

calculation of structure form factors between baryon states

can then be performed using the light-front wave functions

and the interaction vertices associated with the respective

transition [13,16,19]. The form factors can also be expanded

in contributions from the valence quarks and in contributions

from the meson cloud [4,16,17]. Examples of calculations of

the nucleon and the nucleon to Roper electromagnetic form

factors can be found in Refs. [4,12–20].

In principle the leading twist approximation, associated

with the three-quark state, is suff icient to explain the

dominant contribution of the form factors related to the

electromagnetic transitions between baryon states, particu-

larly at large momentum transfer. In the case of the nucleon

and the Roper, the electromagnetic form factors can be

described in a good approximation by the valence quark

effects (leading twist approximation) [4,12,15,19]. There is,

however a rising interest in checking if the holography can

be used to estimate higher order corrections to the transition

form factors, particularly, in the corrections associated with

the meson cloud excitations, related in the light-front

formalism to the state ðqqqÞq

q,oforderτ ¼ 5 [16–18].

The question of whether the light-front meson cloud

contribution is important or not is pertinent, because in

Published by the American Physical Society under the terms of

the Creative Commons Attribution 4.0 International license.

Further distribution of this work must maintain attribution to

the author(s) and the published article’s title, journal citation,

and DOI . Funded by SCOAP

PHYSICAL REVIEW D 97, 073002 (2018)

2470-0010=2018=97(7)=073002(13) 073002-1 Published by the American Physical Society

principle the corrections associated with the meson cloud

should be expressed in terms of parameters related to the

microscopic structure, such as meson-baryon couplings and

the photon-meson couplings [23–25]. As discussed later, in

the case of a holographic model, the estimates of the

transition form factors depend only on the couplings

associated with quarks without explicit reference to the

substructure associated with the meson cloud.

In this work we study the axial structure of the nucleon

using a holographic model based on a soft-wall confining

potential. The weak structure of the nucleon is character-

ized by the axial form factor, G

, and the induced

pseudoscalar form factor, G

. The study of the nucleon

axial structure is important because it provides comple-

mentary information on the well known electromagnetic

structure and also because involves both strong and weak

interactions [26]. The nucleon axial form factors can be

measured in quasielastic neutrino/antineutrino scattering

with proton targets, by charged pion electroproduction on

nucleons and also in the process of muon capture by

protons [27–29]. The value of the axial form factor at

¼ 0 is determined with great accuracy by neutron β

decay [27,30].

The nucleon axial form factor has been calculated using

different frameworks [26,31–51]. Recently, also lattice

QCD simulations of the nucleon axial form factors

became available for several pion masses (m

), in the

range m

¼ 0.2–0.6 GeV [52–69].

In the present work our goal is to study the role of the

valence quarks (leading twist approximation) and the role

of the meson cloud (τ ¼ 5) in the nucleon axial form factor

. We consider in particular the holographic model from

Ref. [16], neglecting the gluon effects. We assume that the

gluon effects are included effectively in the quark structure

through the gluon dressing. In that case the next leading

order correction is associated with the quark-antiquark

excitations of the three valence quark core. In this context

the bare and the meson cloud contribution to the nucleon

axial form factor are both expressed in terms of two

independent parameters: g

and η

, associated with the

quark axial and quark induced pseudoscalar couplings [16].

To calculate the contributions associated with the

nucleon bare core and the meson cloud we use the available

experimental data and the results from lattice QCD, which

help to constrain the contributions from the pure valence

quark degrees of freedom, and therefore fix also the

contributions of the meson cloud component. In the lattice

QCD simulations with large pion masses the meson cloud

effects are very small, and the physics associated with the

valence quarks can be better calibrated.

The results from lattice QCD cannot be directly related to

the valence quark contributions to the axial form factor,

because the lattice calculations are not performed at the

physical limit (physical quark masses). The results from

lattice can, however, be extrapolated to the physical case

with the assistance of quark models that include a dynamic

dependence on the quark mass.

Once fixed the parameters of the holographic model by

the empirical and lattice QCD data, the holographic model

can be used to estimate the fraction of the meson cloud

contribution to the nucleon axial form factor. This estimate

can be compared to other estimates from quark models with

meson cloud dressing.

We conclude at the end that the holographic model

considered in the present work describes accurately the

experimental data for the nucleon axial form factor, and that

the lattice QCD data with small pion masses can be well

approximated by the estimate of the valence quark con-

tributions, in all ranges of Q

. We also conclude that the

meson cloud contribution falls off very slowly with the

square momentum transfer Q

, much slower than estimates

based on quark models.

This article is organized as follows. In Sec. II, we discuss

the formalism associated with the study of the axial

structure of the nucleon, including the axial current, para-

metrizations of the data, results from lattice QCD, as well

as theoretical models based on a valence quark core with

meson cloud dressing. In Sec. III, we present the holo-

graphic model for nucleon axial form factor considered in

the present work. The numerical results of the nucleon axial

form factor and for the estimate of the meson cloud

contributions based on the holographic model appear in

Sec. IV. The outlook and the conclusions are presented

in Sec. V.

II. BACKGROUND

We now discuss the background associated with the

study of the nucleon axial form factor. We start with the

representation of the axial current and the definition of

the axial form factors. Next, we summarize the exper-

imental status of the nucleon axial form factor G

. Later,

we explain how the experimental data can be described

within a quark model for the bare core, combined with a

meson cloud dressing of the core. Finally, we discuss the

results from lattice QCD and how those results can be

related with the function G

in the physical limit.

A. Axial current

The weak-axial transition between two nucleon states

with initial momentum p, final momentum p

, and tran-

sition momentum q ¼ p

− p, is characterized by the

weak-axial current [27,28]

ðJ

uðp



ðQ

Þγ

þG

ðQ



uðpÞ

; ð2:1Þ

where M is the nucleon mass, Q

¼ −q

, τ

(a ¼ 1,2,3)

are Pauli isospin operators and uðpÞ, uðp

Þ are the

Dirac spinors associated with the initial and final states,

G. RAMALHO PHYS. REV. D 97, 073002 (2018)

073002-2

respectively. The functions G

and G

define, respectively,

the axial-vector and the induced pseudoscalar form factors.

In the present work we restrict the analysis to the axial-

vector form factor, refereed to hereafter, simply as the axial

form factor. The leading order contribution for G

can be

estimated considering the meson pole contribution,

þQ

, derived from the partial conservation of

the axial current [26–29,35,57].

Using the spherical representation (a ¼ 0; ) we can

interpret ðJ

as the current associated with the neutral

transitions, p → p and n → n (Z

production), and the

current associated with a ¼with the W



production

(n → p and p → n transitions).

B. Experimental status

The function G

can be measured by neutrino

scattering and pion electroproduction off nucleons.

Both experiments suggest a dipole dependence G

ðQ

Þ¼

ð0Þ=ð1 þ Q

, where thevalues of M

vary between

1.03 and 1.07 GeV depending on the method [27,28].

To represent the experimental data in a general form we

consider the interval between the two functions, G

exp −

and

exp þ

, given by [26]

exp 

ðQ

Þ¼

ð1  δÞ

ð1 þ

A

; ð2:2Þ

where G

¼ 1.2723 is the experimental value of G

ð0Þ

[30], δ ¼ 0.03 is a parameter that expresses the precision

of the data, and M

A−

¼ 1.0 GeV and M

Aþ

¼ 1.1 GeV are,

respectively, the lower and upper limits from M

extracted

experimentally. The central value of the parametrization

(2.2) can be approximated by a dipole with M

≃ 1.05 GeV.

Most of the data analysis are restricted to the region

< 1 GeV

[27]. The range of the variation associated

with the parametrization of G

represented by Eq. (2.2) is

shown in Fig. 1 by the red band. The short-dashed-line

represents the central value of the parametrization.

Recently the nucleon axial form factor was determined in

the range Q

¼ 2–4 GeV

at CLAS/Jlab [70]. The new

data are consistent with the parametrization (2.2).

We discuss next, how the axial form factor can be

estimated in the context of a quark model with meson cloud

dressing of the valence quark core.

C. Theory

In a quark model with meson cloud dressing we can

represent the physical nucleon state in the form [26]

jNi¼

ﬃﬃﬃﬃﬃﬃ

½j3qiþb

jMCi; ð2:3Þ

where j3qi is the three-quark state and b

jMCi is the

meson cloud state. The coefficient b

is determined by the

normalization Z

ð1 þ b

Þ¼1, assuming that jMCi is

normalized.

In this representation Z

ﬃﬃﬃﬃﬃﬃ

p ﬃﬃﬃﬃﬃﬃ

measures the

probability of finding the qqq state in the physical nucleon

state. Consequently, 1 − Z

measures the probability of the

meson cloud component in the physical nucleon state.

In Eq. (2.3), we include only the first correction for

the meson cloud, associated with the baryon-meson states.

In principle, we should also include corrections associated

with baryon-meson-meson states. In the case of the

nucleon, however, where the meson cloud is dominated

by the pion cloud, the correction of the state jNπi provides

a good approximation to the physical nucleon state. In the

case of 1 − Z

≃ 0.3 the correction associated with the two-

pion correction is attenuated by the factor ð1 − Z

≃ 0.09.

In the calculation of the axial form factors, in order to

take into account the contribution of the meson cloud in the

form factors at the physical limit, one needs to correct

the function G

by the factor Z

, which quantifies the

contribution of the bare core to G

[26]. The effective

contribution from G

to the physical G

becomes then

. More generically, we can write

¼ Z

þð1 − Z

ÞG

; ð2:4Þ

where the second term accounts for the contribution from

the meson cloud. The function G

is the unnormalized

meson cloud contribution, estimated when we drop the

valence quark contribution.

Hereafter, we use the expression bare contribution to

refer the first term of Eq. (2.4) and meson cloud contri-

bution to refer the second term of Eq. (2.4).

An alternative representation of the meson cloud term is

ð1 − Z

ÞG

¼ Z

[26]. To convert to G

, one uses

=ð1=Z

− 1Þ. The function

can be

extracted from the data, as discussed in Ref. [26].

0.5

1.5

(GeV

)

0.4

0.8

1.2

)

Data

Bare

FIG. 1. Experimental parametrization of the data G

exp

accord-

ing to Eq. (2.2), at red, combined with the estimate of the

contribution Z

extracted from the lattice QCD data, at blue.

The short-dashed-line indicates the central value of Eq. (2.2).

HOLOGRAPHIC ESTIMATE OF THE MESON CLOUD … PHYS. REV. D 97, 073002 (2018)

073002-3

剩余12页未读，继续阅读

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