Available via license: CC BY 4.0
Content may be subject to copyright.
arXiv:1703.03819v1 [hep-ph] 10 Mar 2017
ITEP-TH-08/17
Valence quark contributions for the γ∗N→N(1440) form factors
from Light-Front holography
G. Ramalho aand D. Melnikov a,b
aInternational Institute of Physics, Federal University of Rio Grande do Norte,
Campus Universit´ario - Lagoa Nova, CP. 1613 Natal Rio Grande do Norte 59078-970, Brazil and
bInstitute for Theoretical and Experimental Physics,
B. Cheremushkinskaya 25, 117218 Moscow, Russia
(Dated: March 14, 2017)
The structure of the nucleon and the first radial excitation of the nucleon, the Roper, N(1440),
is studied within the formalism of Light-Front holography. The nucleon elastic form factors and
γ∗N→N(1440) transition form factors are calculated under the assumption of the dominance of
the valence quark degrees of freedom. Contrary to the previous studies, the bare parameters of
the model associated with the valence quark are fixed by the empirical data for large momentum
transfer (Q2) assuming that the corrections to the three-quark picture (meson cloud contributions)
are suppressed. The γ∗N→N(1440) transition form factors are then calculated without any
adjustable parameters. Our estimates are compared with results from models based on valence
quarks and others. The model compares well with the γ∗N→N(1440) transition form factor data,
suggesting that meson cloud effects are not large, except in the region Q2<1.5 GeV2. In particular,
the meson cloud contributions for the Pauli form factor are small.
I. INTRODUCTION
Within the nucleon excited states (N∗) the N(1440)
resonance, also known as Roper, plays a special role.
Contrarily to the ∆(1232) and other nucleon excitations,
the Roper was not identified as a bump in a reaction
cross-section but was instead found in the analysis of
phase-shifts [1]. Nowadays there is evidence that the
Roper should be identified as the first radial excitation of
the nucleon quark core, although meson excitations are
also important for the internal structure.
Calculations based on valence quark degrees of free-
dom are consistent with the γ∗N→N(1440) transition
form factors for large Q2(Q2>2 GeV2) [2–7]. However,
estimates based exclusively on quark degrees of freedom
fail to describe the small Q2data (Q2<2 GeV2) [2, 8, 9].
The gap between valence quark models and the data at
low Q2has been interpreted as the manifestation of the
meson cloud effects [3, 4, 7, 10]. When the meson cloud
contributions are included in quark models the estimates
approach the data [11–16].
Besides, the calculations based on dynamical coupled-
channel reaction models, where the baryon excitations
are described as baryon-meson states with extended
baryon cores [17, 18], corroborates the importance of the
meson cloud effects. In those models the mass associated
with the Roper bare core is about 1.7 GeV. Only when
the meson cloud dressing is considered the Roper mass
is reduced to the experimental value [19]. The interpre-
tation of the Roper as a radial excitation of the nucleon
combined with a dynamical meson cloud dressing solves
the long standing problem of the Roper mass in the con-
text of a quark model [21].
The Roper decay widths into γN ,πN and ππN are
large comparative to other N∗decays. Those decay
widths are also difficult to explain in the context of a
quark model. The meson cloud dressing helps to explain
the N(1440) →γN width, where the contributions as-
sociated with baryon-meson-meson states play an impor-
tant role [14, 15, 19, 20].
Overall the recent developments in the study of the
Roper electromagnetic structure point to the picture of
a radial excitation of the nucleon surrounded by a cloud
of mesons [10, 15, 19, 22–24]. There is therefore a strong
motivation to study the Roper internal structure and to
disentangle the effects of the valence quark component
from the meson cloud component. In particular, one can
use the knowledge of the baryon core structure to infer
the contribution due to the meson cloud. This procedure
was used in previous works based on different frameworks
for the baryon core [3, 4, 7, 10].
In the case of the nucleon the valence quark degrees of
freedom produce the dominant effect in the elastic form
factors. The meson cloud contribution to the nucleon
wave function is estimated to be of the order of a few
percent [25–28]. As for the Roper, the meson cloud seems
to play a more prominent role. We conclude at the end
that the valence quark degrees of freedom provide a very
good description of the data, but the meson cloud con-
tribution are important below 1.5 GeV2, particularly for
the Dirac form factor.
In the present work we propose a new framework to an-
alyze the role of the valence quarks and the meson cloud
in the Roper. We use lLight-Front holography to esti-
mate the leading order (lowest Fock state) contribution
2
to the γ∗N→N(1440) transition form factors. Since
the leading order calculation includes only pure valence
quark effects in the baryon wave functions (qqq contri-
butions), there is no contribution from the meson cloud.
In those conditions the gap between the calculations and
the data must be essentially the consequence of the me-
son cloud effects.
The Light-Front (LF) formalism is particularly ap-
propriate to study hadron systems, ruled by QCD, and
to describe the hadronic structure in terms of the con-
stituents [29–31]. The LF wave functions (LFWF) are
relativistic and frame independent [30, 32]. The con-
nection between LF quantization of QCD and anti-de-
Sitter conformal field theory (AdS/CFT) leads to Light-
Front holography [30, 31, 33]. LF holography have been
used to study the structure of hadron properties, such
as the hadron mass spectrum, parton distribution func-
tions, meson and baryon form factors etc. [8, 30, 31, 33–
47]. In particular, the formalism was recently applied
to the study of the nucleon [48–54] and Roper [31, 55]
electromagnetic structure.
An important advantage of the LF formalism applied
to hadronic physics is the systematic expansion of the
wave functions into Fock states with different number
of constituents [30, 32, 56]. In the case of the baryons
the leading order contributions is restricted to the three-
valence quark configuration. Although the restriction to
the lowest order Fock state (three valence-quark system)
may look as a rough simulation of the real world, it
may provide an excellent first approximation when the
confinement is included in the LFWF, defined at the
Light-Front time [25, 33, 36]. In those conditions non-
perturbative physics is effectively taken into account by
LF holography [30, 36].
Under the assumption that LF holography can de-
scribe accurately the large-Q2region dominated by va-
lence quark degrees of freedom, we calibrate the free pa-
rameters of the model by the available data above a given
threshold Q2≥Q2
m(Q2
m= 1.5–2.5 GeV2). This pro-
cedure differs from the previous studies where the free
parameters, associated with the nucleon anomalous mag-
netic moments were fixed at Q2= 0, where the meson
cloud contamination is expected to be stronger. Once
the free parameters are fixed by the nucleon data, one
uses the model to calculate the γ∗N→N(1440) transi-
tion form factors. Since no parameter is adjusted by the
Roper data, our calculations are true predictions for the
γ∗N→N(1440) transition form factors.
This article is organized as follows: In Sec. II we dis-
cuss LF holography. In the following section (Sec. III),
we review the results for the nucleon form factors and
fix the parameters by the nucleon data. Our results for
the valence quark contribution to the γ∗N→N(1440)
transition form factors and their discussion are presented
in Sec. IV. The outlook and conclusions are presented in
Sec. V.
II. LIGHT-FRONT HOLOGRAPHY
It is by now well known that string theory or grav-
ity in anti-de-Sitter (AdS) space can provide a descrip-
tion of some lower-dimensional (conformal) gauge theo-
ries (CFT) at strong coupling [57–59]. This AdS/CFT
correspondence, or holography, can, with certain im-
portant restrictions, be applied to QCD-like theories,
e.g. [60–62]. It remains an open question, however,
whether the top-down string theory methods are prac-
tical enough to model QCD itself. Instead it proved effi-
cient to apply bottom-up approaches to QCD [41, 63, 64].
Such approaches are motivated phenomenologically by
the string models, but lack a first-principle justifica-
tion. Among those, the approach of LF holography is
based on a comparison of the results of the LF quantiza-
tion of QCD [65, 66] with the predictions of holographic
(AdS/CFT) models [30, 31, 50, 51, 53].
A. Light-Front QCD vs AdS/CFT
Light-Front quantization provides a powerful tool
to study systems ruled by microscopic QCD dynam-
ics. It starts with introducing a special (Light-Front)
parametrization of the Hamiltonian, which consequently
acts on a Hilbert space of the LFWF spanned by partonic
Fock states associated with quarks and gluons [30, 31].
Using LF parametrization one can separate the rela-
tive and the overall kinematics of partons obtaining wave
functions ψn(κi,k⊥i) where κiis the fraction of momen-
tum of the parton i,k⊥iis the transverse momentum
and nis the number of components of the Fock state.
Functions ψn(κi,k⊥i) define the probability of the given
n-parton Fock state. In the massless quark limit, consid-
ering the 2D Fourier transform from k⊥to the transverse
impact variable b⊥, one passes to a wave function defined
in the transverse impact space, ψn(κ,b⊥) [30].
In the semiclassical approximation, when the quark
loops are neglected and the quarks are massless, one can
replace the dependence on the variables κand b⊥in ψn
by dependence on a single parameter ζ=κ(1 −κ)|b⊥|,
except for an overall factor f(κ) = pκ(1 −κ) [30, 34,
53]. The variable ζmeasures the separation between the
active quark and the remaining spectator partons.
The key observation of LF holography is the equiva-
lence of characteristic LFWFs ψn(κ, ζ ) and wave func-
tions of matter fields in five-dimensional anti-de-Sitter
(AdS5) space, assuming that one identifies ζwith the
radial coordinate of AdS5[30, 34, 67]. The residual κ
dependent factor f(κ) is not relevant in the holographic
correspondence, since its effect can be absorbed in the
normalization factors in the 5D theory calculation of ma-
trix elements.
The full LFWF is an expansion over n-parton Fock
states |niwith coefficients ψn(κ, ζ). In the standard ter-
minology those are the states of a given twist τ≡n. The
purpose of this work is to understand the leading twist
3
contribution (τ= 3), which corresponds to the valence
quark approximation to certain QCD processes.
To study the electromagnetic properties of the nucleon
excitations we note that a three-quark system (τ= 3),
as the nucleon and the nucleon radial excitations can be
regarded as two-body systems with an active quark and
a spectator cluster with masses (eigenvalues) and wave
functions determined by the LF wave equation [30].
B. 5D fermion in AdS space
In the bottom-up holographic approach to QCD the
nucleon and the nucleon excitations can be introduced
as fermion fields in five-dimensional AdS space [68]. To
define this curved space one requires the metric tensor
g, which can be conveniently introduced via line element
ds2in the Poincar´e coordinates:
ds2=R2
z2ηµν dxµdxν−dz2,(2.1)
where ηµν = diag(+,−,−,−) is the 4D Minkowski metric
tensor and Ris a parameter called AdS radius. The
latter sets the scale for the space’s curvature. AdS5radial
coordinate zis to be identified with the LF parameter ζ.
From the metric one can define the Dirac operator ˆ
D,
which we write as
ˆ
D=i
2eM
AΓA←→
∂M+1
8ωAB
M[ΓA,ΓB].(2.2)
Here ΓA,A, B = 0,1,2,3, z is the standard set of five
Dirac gamma matrices and we choose to represent in the
chiral basis. Frame eM
Aand spin-connection ωAB
Mten-
sor fields can be computed from the metric tensor. For
completeness we summarize the explicit formulas here
eM
AΓA=z
R(γµ,−iγ5),(2.3)
ωAB
M=1
z(ηAz δB
M−ηBz δA
M) (2.4)
It is also understood that A←→
∂MB=A(∂MB)−(∂MA)B.
Substantiating the above claim, the kinematic proper-
ties of nucleons, e.g. the spectrum, can be described by
a theory of a massive fermion in AdS5:
SN=Zd4x dz p−det gΨˆ
D − µ−ΦΨ,(2.5)
where µis the 5D fermion mass to be fixed below.
The function Φ(z) is an effective scalar potential,
whose role is to introduce the complex dynamics of QCD,
essentially the confinement. When Φ = 0, the equations
of motion of the 5D theory can be cast in the form corre-
sponding to the LF Schr¨odinger equation in the confor-
mal limit [30]. A phenomenologically compelling choice
of the potential is [49, 50]
Φ = κ2z2.(2.6)
The dimensionful parameter κbreaks conformal symme-
try and introduces a mass scale, which determines the
meson and baryon spectrum [30, 31, 36]. It can be thus
related to ΛQCD. A conventional way to introduce this
potential is through the dilaton field background [49, 69].
This procedure makes the connection with top-down
AdS/CFT constructions. The corresponding bottom-up
case is dubbed the soft-wall model.
In four dimensions the four-component Dirac spinor
describes two independent chirality modes. Although
five-dimensional Dirac spinor Ψ equally has four com-
ponents, it will correspond to a nucleon with a spe-
cific single choice of chirality. This is a consequence
of boundary conditions necessary in the holographic ap-
proach [49, 50, 70–72]. In order to preserve CP invari-
ance, the second chirality mode is introduced similarly as
Eq. (2.5), but with an opposite sign in front of µand Φ,
e.g. [50]. For simplicity of presentation we only review
one chirality mode here.
C. Wave equations and wave functions
In order to solve the Dirac equations derived from ac-
tion (2.5) the fermion field (spin 1/2 and positive parity)
Ψ is decomposed into its left- and right- chirality compo-
nents:
Ψ(x, z) = ΨL(x, z ) + ΨR(x, z),(2.7)
where ΨL/R =1
2(1 ∓γ5)Ψ. The solution can then be
found via separation of variables, assuming a plane wave
dependence on the 4D spacetime coordinates
ΨL/R(x, z) = ψL/R(x)z2FL/R(z),(2.8)
where left and right Weyl spinors ψL/R(x) =
ψL/R,P eiP ·xsatisfy the 4D Dirac equation with mass
M2=P2. We also pull out the z2factor for convenience.
In the end, one is left with a pair of coupled equations
for scalar profile functions FL/R(z):
±∂z+µR + Φ
zFL/R(z) = M FR/L (z).(2.9)
The equations must be solved with an appropriate
choice of boundary conditions. First, holography re-
quires regularity of the wave function Ψ(x, z ) as z→ ∞.
For z→0 there are two linearly independent solutions
FL/R ∼z∓µR. In holography one usually chooses to call
one of those solutions the source and another one the
vev of the operator dual to the bulk (5D) field Ψ – in
this case baryon interpolating operator. For the problem
in question it only makes sense to choose the vev solu-
tion as FR∼zµR and the source FL= 0. The opposite
choice, source FR= 0 and vev FL∼z−µR leads to a
non-physical spectrum. Note that for the opposite chi-
rality case one changes µ→ −µ, and the two choices of
the z→0 boundary conditions are interchanged, with
the physical one being FR= 0 and FL∼zµR.
4
After fixing the boundary conditions it is standard to
rewrite system (2.9) as a second order Schr¨odinger-like
differential equation for either FR, or FL. Since we are
solving the boundary problem, any solution can be ex-
panded over an eigenfunction basis, labeled by an integer
n. Equation
−d2
dz2FR+m(m−1)
z2FR+ 2κ2m+1
2FR
+κ4z2FR=M2
nFR,(2.10)
has a set of eigenfunctions FL/R,n (z) expressed in terms
of the generalized Laguerre polynomials Lα
n:
FR,n(z)∝zme−κ2z2/2Lm−1/2
n(κ2z2).(2.11)
The second component FLcan now be calculated using
Eqs. (2.9):
FL,n(z)∝zm+1 e−κ2z2/2Lm+1/2
n(κ2z2).(2.12)
These eigenfunctions correspond to eigenvalues given by
M2
n= 4κ2n+m+1
2.(2.13)
We recall that in the previous expressions m≡µR.
The eigenvalues provide the result for the 4D spec-
trum of nucleon excitations with radial excitation quan-
tum number nand angular momentum quantum number
linear in the parameter m, which we will fix below. The
above choice of the potential (2.6) leads to the spectrum
consistent with Regge trajectories [30, 31, 41].
Summing up, the full solution for the 4D positive chi-
rality mode reads
Ψ+
n(x, z) = z2FL,n (z)χn(x)
FR,n(z)χn(x),(2.14)
where χn(x) are two component spinors related with
the Weyl spinors: ψ†
L(x) = (χ†
n(x),0) and ψ†
R(x) =
(0, χ†
n(x)). The negative chirality modes is obtained via
an appropriate change of signs and exchange of the solu-
tions for FL/R [50, 55]:
Ψ−
n(x, z) = z2FR,n (z)χn(x)
−FL,n(z)χn(x).(2.15)
Using the component associated with the index nwe
can write the fermion fields with positive and negative
parities as
Ψ±(x, z) = 1
√2X
n
Ψ±
n(x, z).(2.16)
D. Interactions
In hadron physics, information on the structure of
nucleon resonances is contained in the matrix elements
of interaction currents between the initial and final
hadrons, including the nucleon and the nucleon excita-
tions (baryons). In holography, 4D operators of con-
served currents Jµare described by massless gauge fields
VMin AdS5space. Interacting bulk baryon fields Ψ cou-
ple to the gauge fields, which is a dual holographic de-
scription of the coupling of baryons to conserved currents.
Matrix elements of the currents, can be computed from
the interaction terms, which are, schematically, overlaps
of the bulk fields with the source gauge field [73]:
Sint =Zd4x dz p−det g¯
Ψ(x, z)ˆ
V(x, z)Ψ(x, z ).(2.17)
Here ˆ
Vencodes a coupling of field VMto fermions. We
consider in particular the decomposition
ˆ
V(x, z) = ˆ
V0(x, z) + ˆ
V1(x, z) + ˆ
V2(x, z),(2.18)
as discussed next. For simplicity, we only present the in-
teraction for positive chirality. For the opposite chirality
case one has to adjust the signs in the above expression
to preserve the CP invariance [50].
The term ˆ
V0(x, z) is representing the minimal Dirac
coupling given by [49]
ˆ
V0(x, z) = ˆ
QΓMVM(x, z),(2.19)
where VMcould be either the abelian dual of the electro-
magnetic current, or the non-abelian isovector current.
Following the parametrization of [50, 55] ˆ
Q=eN=
1
2(1 + τ3), the nucleon charge (N=p, n), for the elas-
tic electromagnetic transitions, and ˆ
Q=τ3for the tran-
sition between the nucleon and higher mass JP=1
2
+
resonances (τ3is the Pauli isospin operator). This mini-
mal coupling only yields the Dirac form factor.
To produce the Pauli form factor more input from the
holographic side is necessary. Abidin and Carlson [49]
proposed to consider a non-minimal extension of the cou-
pling ˆ
V0, adding a term
ˆ
V1(x, z) = i
4ηN[ΓM,ΓN]VMN (x, z ),(2.20)
where VMN =∂MVN−∂NVM+ [VM, VN] and one can
consider different couplings ηSand ηVfor the isoscalar
and isovector parts respectively [ηN=1
2(ηS+ηVτ3)].
As one may expect from its Lorentz structure, this term
yields the Pauli form factors, as well as a correction to
the Dirac form factors. Moreover, since the non-minimal
term contains derivatives of the field VMand extra pow-
ers of z, the Pauli form factor turns out to be subleading
with respect to the Dirac form factor at large momentum
transfer, as expected, while the correction to the Dirac
form factor is of the same order. The effect of the term
V1on the nucleon form factors is studied in Refs. [31, 49–
52, 55]. We provide additional details in what follows.
More non-minimal couplings can be considered. In
Refs. [50, 55] the following non-minimal (isovector) term
was suggested
ˆ
V2(x, z) = gVτ3ΓMγ5VM(x, z).(2.21)
5
where gVis a coupling constant. This coupling is consis-
tent with gauge invariance and discrete symmetries, that
can be added to the 5D action to improve the fitting of
experimental data [50].
One first observation about this term is that it naively
breaks CP in 4D. However, this is not the case, since
5D fermion Ψ describes only one 4D chirality. Adding a
similar interaction term for the opposite chirality, with
the opposite sign of coupling, ensures 4D CP invariance.
A more serious issue is that Eq. (2.21) breaks 5D co-
variance. Adding such term requires an implicit back-
ground 5D vector field. Such background fields are more
common in the higher-dimensional top-down holographic
models with flux compactifications. Here we assume that
such a flux compactification exists and we assess its effect
on the observable form factors. We further comment on
the general structure of the term (2.17) in Ref. [74].
In order to calculate transition matrix elements of the
currents using Eq. (2.17) one needs to specify the 5D
gauge field VM(x, z). Similarly to fermions, a bosonic
gauge field dual to the isovector or isoscalar current is
introduced by a 5D Lagrangian. Specifically, one needs
to solve equations following from the action
SJ=−Zd4x dz p−det ge−Φ
4Tr VM N VMN ,(2.22)
subject to appropriate boundary conditions. These
boundary conditions differ from the boundary conditions
for the fields associated with the nucleon and the nu-
cleon excitations characterized by the quantum numbers
nand m, since they have to be of the source type at
z= 0 (the leading solution as opposed to the sublead-
ing vev-type). The regularity condition requires VMto
vanish for z→ ∞.
In what follows we will be interested in the abelian
electromagnetic current, so the non-abelian details of this
discussion can be omitted. Altogether, the solution can
be cast in the form [50, 55, 75]
Vµ(x, z) = Zd4q
(2π)4e−iq·xǫµ(q)V(−q2, z),(2.23)
where ǫµis the photon polarization vector, and
V(Q2, z) = κ2z2Z1
0
dκ
(1 −κ)2xQ2
4κ2e−κ2z2κ
1−κ,(2.24)
with Q2=−q2. In the limit z→0, one has V(Q2,0) = 1.
Substituting Eqs. (2.23) and (2.24) together with
the modes (2.14) and (2.15) into the interaction La-
grangian (2.17) yields matrix elements of the current up
to some normalization terms, from which form factors
are read off directly. We present the results in Sec. III
for the nucleon and in Sec. IV for the Roper.
E. Mass spectrum and more
In this section we specify the details of the holographic
model and relating it to the observable spectrum of the
nucleon radial excitations and the ρmesons.
First, the nucleon (ΨN0) and the Roper (ΨN1) are the
states with radial quantum numbers n= 0,1 and angular
momentum L= 0. Any of these states must be regarded
as a superposition of the given twist (number of partons)
Fock states, so we assume that the 5D mass µis encoding
both the angular momentum Land the twist τ,µR =
m(L, τ ).
A way to fix the relation for L= 0 is to analyze the
large Q2scaling of FL/R,n (z) and consequently the form
factors. As discussed later, the choice m=τ−3/2
yields the correct falloff estimated by perturbative QCD
(pQCD) [76]. Consequently, the spectrum (2.13) takes
the following form [30, 31, 55],
MNn = 2κ√n+τ−1.(2.25)
Thus, holography provides an estimate for the spectrum
of the nucleon radial excitations in terms of a single scale
parameter κ. Similarly, one obtains the spectrum of
mesons and baryons with different angular momenta and
parity [30, 31, 36].
The ρmeson (mass mρ) is a traditional reference
for the hadron states. In LF holography we can write
mρ= 2κ. The nucleon mass, for example, approximately
satisfies MN=√2mρ. If we assume that the nucleon
mass is primarily composed of the leading twist τ= 3
contribution, MN= 2√2κ, then the phenomenological
value of κis fixed at
κ=mρ
2∼ΛQCD .(2.26)
The leading twist estimate of the masses of nucleon ra-
dial radial excitations is now given by MNn = 2κ√n+ 2,
n= 0,1,2.... This yields MR=MN1= 2√3κfor the
Roper mass in leading twist approximation (τ= 3).
It is worth mentioning that the LF estimate of the
Roper mass, MR≃√3mρ, is not so accurate as for the
nucleon and the ρmeson. This is consistent with a gen-
eral expectation that leading twist approximation is not
so accurate for baryons as it is for mesons. In particu-
lar for the Roper, there are indications that the baryon-
meson-meson corrections of the twist order τ= 7 are
important [19, 20]. In the next subsection we collect fur-
ther comments on the spectrum and other issues of LF
holography.
The spectrum of the hadrons, as predicted by LF holo-
graphic approach has further issues. A similar approach
to mesons yields similar Regge-behaved spectra as the
one given by Eq. (2.25). For the ρmeson family one
gets mρn = 2κ√n+ 1 for n= 0,1,2,... [41, 44, 75].
Brodsky and Teramond argue, however, that those twist-
2 mass poles should be shifted to their physical values,
suggesting that mρn = 2κ√2n+ 1 [30, 36, 54]. These
formula gives a very good agreement with the vector me-
son masses for κ≃0.385 GeV [30, 36], as in Eq. (2.26),
provided that mρ≃770 MeV. Similar values were used
in Refs. [50, 55].
6
F. Comments on interpolating operators in LF
holographic approach
The scaling dimension ∆ is defined by the behavior
of the wave function in the limit z→0: Ψ(x, z)∼
z∆[33, 49]. The holographic approach relates the 5D
mass parameter mwith the scaling dimension ∆ of the
hadron interpolating operator. For example, for the vec-
tor field this relation is
∆ = 2 + pm2+ 1 .(2.27)
For the massless vector field VMabove, this infers ∆ = 3,
the correct scaling dimension of a current operator [33].
For fermions however the relation is
∆ = 2 + |m|.(2.28)
In the leading twist this yields ∆ = 7/2 for the nu-
cleon/Roper interpolating operator, in contrast to the
scaling dimension ∆O= 9/2 of the operator O=qqq,
composed of three quark fields. This discrepancy is
even more pronounced if one applies the identification
m(τ, L) used in Ref. [30]. Indeed, comparing Eqs. (2.11)
and (2.13) with Eqs. (5.34) and (5.38) of Ref. [30] one
ends up with the relation m=ν+ 1/2, which implies
∆ = 5/2.
In QCD the (anomalous) scaling dimensions of the in-
terpolating operators are not well-defined, because of the
logarithmic running of the gauge coupling. In a putative
conformal theory, which would approximate the slow run-
ning of the coupling, the anomalous scaling dimensions
would exist and correspond to a dual gravity theory of
the type studied here. In particular, the massless vector
field of a gravity dual would correspond to ∆ = 3 con-
served symmetry current. In the meantime we have to
accept that the correct choice of the operator dimension
leads to unpleasant effects like incorrect scaling of the
form factors for very large Q2.
One can argue, in principle, that predictions of holo-
graphic models are generally valid in the strongly coupled
regime of a theory, thus the pQCD prediction should not
be compared with the outcome of the holographic anal-
ysis. We believe, however, that this argument does not
quite apply to the scaling of the form factors, since the
pQCD prediction is rather based on the assumption of a
particular partonic structure of the hadron, which is not
exactly the full perturbative quark picture [76].
We hope that some more advanced holographic model
can resolve the issues mentioned above. In particu-
lar, top-down models, as more complex ones, naturally
share more intricate details with QCD. A prototypical
example would be (the glueball sector of) the Klebanov-
Strassler theory [61, 77]. Albeit supersymmetric, this
theory encodes a logarithmically running gauge cou-
pling, thus, extracting dimensions of the operators in
this theory requires removing the logarithmic scale de-
pendence (e.g. [78–80]). Moreover, in this theory, the
states with the same quantum numbers tend to mix with
each other. Therefore the mass eigenstates are superpo-
sitions of states with different twist. Mixing has effect of
changing the resulting spectrum as well as shifting the
values of the dimensions [79–83].
III. NUCLEON ELECTROMAGNETIC FORM
FACTORS
The transition current Jµbetween two nucleon states
(elastic transition) can be expressed, omitting the asymp-
totic spin states (spinors) and the electric charge e,
as [8, 26]
Jµ=F1N(Q2)γµ+F2N(Q2)iσµν qν
2M,(3.1)
where FiN (i= 1,2) define, respectively, the Dirac and
Pauli form factors of the proton (N=p) and neutron
(N=n). In the LF formalism, the two form factors ap-
pear in a spin-nonflip (F1N) and a spin-flip (F2N) tran-
sitions [56, 84].
The calculation of the overlaps (2.17) in the holo-
graphic model with the interaction (2.18), where one re-
places Ψ and ¯
Ψ by the appropriate initial and final nu-
cleon state modes, in this case the nucleon modes ΨN0.
The leading twist case (τ= 3), the Dirac form factor is
determined by the functions F2
L,0and F2
R,0, and the Pauli
form factor is determined by overlap of the FL,0and FR,0
components [50, 55]. The final expressions for the form
factors are then [50]:
F1N=eN
a+ 6
(a+ 1)(a+ 2)(a+ 3) +
gVδN
a
(a+ 1)(a+ 2)(a+ 3) +
ηN
2a(2a−1)
(a+ 1)(a+ 2)(a+ 3)(a+ 4),(3.2)
F2N=ηN
MN
2√2κ
48
(a+ 1)(a+ 2)(a+ 3),(3.3)
where a=Q2
4κ2and κis the holographic scale discussed
in the previous section. In Eqs. (3.2)-(3.3), δN=±1
(δp= 1, δn=−1) and ηNtake different values for proton
and neutron. In Eq. (3.3) we can replace MN
2√2κby unity,
if 2√2κis a good approximation for the nucleon mass.
In an exact SU (2)-flavor model one has ηn=−ηp.
These parameters can be determined by the proton and
neutron bare anomalous magnetic moment κb
pand κb
n(we
use the upper index bto indicate the bare values). In a
model with no meson cloud one can write F2p(0) ≡κp=
8ηpand F2n(0) ≡κn= 8ηn, assuming MN= 2√2κ.
The holographic results (3.2)-(3.3) yield the correct
perturbative QCD behavior for the nucleon electromag-
netic form factors: F1N∝1/Q4and F2N∝1/Q6, as
consequence of m= 3/2. For a more detailed discussion
see Refs. [49, 50, 55].
7
0 1 2 3 4 5
0
1
2
3
4
GMp(Q2)
0 1 2 3 4 5
0
0.5
1
1.5
2
2.5
3
-GMn(Q2)
0 1 2 3 4 5
Q2 (GeV2)
0
0.2
0.4
0.6
0.8
1
GEp(Q2)
0 1 2 3 4 5
Q2 (GeV2)
0
0.02
0.04
0.06
0.08
GEn(Q2)
FIG. 1: Proton and neutron electric and magnetic form factors for the parametrization from Table I. In the case of GEn the dashed-line
corresponds to the parametrization with Q2≥2.0 GeV2. Data from Refs. [85–87].
As previously discussed the non-minimal couplings ηN
and gVare included phenomenologically. As can be seen
from Eqs. (3.2)-(3.3), the minimal coupling only produces
the Dirac form factor. The non-minimal couplings gen-
erate a non-zero F2Nand adds an extra contribution to
the Dirac form factor (proportional to ηN) [37]. Relations
of the type (3.2)-(3.3) were derived for the first time in
Ref. [49] for the case gV= 0.
To determine the values of gVand ηNwe performed a
fit to the data Q2≥Q2
mwhere Q2
mis the threshold of the
data included in the fit. We considered the cases Q2
m=
1.5, 2.0 and 2.5 GeV2, since the meson cloud effects are in
principle suppressed for large Q2, but we do not know a
priory the threshold of the suppression. We use the data
from Refs. [85–87]. Check Refs. [27, 28] for a detailed
description of the database.
From the fits, we conclude that the results for the neu-
tron electric form factor GE n are very sensitive to the
value of Q2
m. If the low Q2is included in the fit (Q2
m<
1.0 GeV2) the best fit favors a negative GEn for low Q2,
in conflict with the measured data. This result is very
pertinent, because it is known that meson cloud effects,
in particular the pion cloud, is very important for the
description of the GEn at low Q2. Since the pion cloud
contributions are not included in the parametrization of
Eqs. (3.2)-(3.3), it is not surprising to see that the fit
with Q2
m= 1.5 GeV2fails to describe the GE n data. It
can be surprising, however, to note that the best descrip-
tion of the nucleon data occurs for Q2>1.5 GeV2in the
fits with Q2
m= 2.0 and 2.5 GeV2, when only a few data
points for GEn are considered. One then concludes that
the shape of GE n is determined by the form factors GEp,
GMp and GM n . This result shows the consistence of the
fitting procedure.
The parameters obtained for the fits with Q2
m= 1.5,
2.0 and 2.5 GeV2are presented in Table I. We can con-
clude that the parameters are sensitive to the data in-
cluded in the fit. In particular, gVdepends strongly of
the threshold Q2
m.
The results obtained for the nucleon electric (GEN =
F1N−Q2
4M2
NF2N) and magnetic (GMN =F1N+F2N) form
factors using the parameters from Table I are presented
in Fig. 1. In the figure, we include a band to represent
the interval between the model with Q2
m= 1.5 GeV2and
2.5 GeV2. In the calculations we used κ= 0.385 GeV,
in order to have a good description of the ρmesons and
nucleon masses.
8
Q2
m(GeV2)gVηpηn
1.5 1.571 0.378 −0.326
2.0 1.370 0.410 −0.345
2.5 1.275 0.424 −0.361
TABLE I: Parameters gV,ηpand ηnobtained by the fit of
the nucleon form factor data with Q2≥Q2
m.
In general the estimates from different parametriza-
tions are very close, except, for the neutron electric form
factor. The function GEn is in fact very sensitive to the
parameters gV,ηpand ηn. In the graph of GEn the
lower limit corresponds to Q2
m= 1.5 GeV2and the up-
per limit to the case Q2
m= 2.5 GeV2. The intermediate
case (Q2
m= 2.0 GeV2) is included in order to emphasize
the strong dependence of GE n with the threshold Q2
m
used in the fit, and it is represented by the dashed-line.
Concerning the remaining form factors, one can notice
an overestimation of the low Q2data (Q2<1 GeV2) for
GMp and −GM n . These results may be interpreted as
the manifestation of the pion cloud effects, not included
in the formalism associated with Eqs. (3.2)-(3.3).
In the past [37, 50, 55] the parameters ηNand gVhave
been fixed by the static properties of the nucleon, such as
the anomalous magnetic moments and the nucleon elec-
tric charge radius. Since those observables depend on the
meson cloud, we choose in this work to fix the parame-
ters in a region where the meson cloud is significantly
reduced, in order to obtain a more accurate estimate of
the bare parameters.
In the recent work the Brodsky-Teramond model for
the nucleon [54] was improved with the inclusion of the
explicit pion cloud contributions (q¯qstates) with a few
adjustable parameters. It was shown that, indeed, the
inclusion of the pion cloud contribution (τ= 5) is funda-
mental for an accurate description of GEn .
In order to understand the role of the valence quark
degrees of freedom we restrict the present study to the
the leading twist contribution (τ= 3). Once fixed the
parameters of the model (bare parameters) one can use
those parameters to calculate the valence quark contri-
butions for the γ∗N→N(1440) transition form factors.
IV. γ∗N→N(1440) FORM FACTORS
We consider now the γ∗N→N(1440) transition form
factors. Again omitting the spinors of the nucleon and
the Roper (and the electric charge e), the transition cur-
rent Jµcan be expressed as [3]
Jµ=F∗
1N(Q2)γµ−6qqµ
q2+F∗
2N(Q2)iσµν qν
M+MR
,
(4.1)
where F∗
1Nand F∗
2N(N=p, n) are the Dirac and Pauli
transition form factors, respectively.
Using the holographic wave functions for the nucleon
and the Roper combined with the interaction (2.18), one
concludes that the Dirac and Pauli form factors associ-
ated with the γ∗N→N(1440) in leading twist [55] are
F∗
1N=δN
a(√2a+c1)
(a+ 1)(a+ 2)(a+ 3)(a+ 4) +
gVδN
a(√2a+c2)
(a+ 1)(a+ 2)(a+ 3)(a+ 4) +
ηN
2a(2√2a2−c3a+c4)
(a+ 1)(a+ 2)(a+ 3)(a+ 4)(a+ 5),(4.2)
F∗
2N=ηNMR+MN
MR2MR
2√3κ
×6√3(c5a−4)
(a+ 1)(a+ 2)(a+ 3)(a+ 4),(4.3)
where c1= 4√2+3√3, c2= 4√2−3√3, c3= 9(√3−√2),
c4= 3√3−5√2 and c5= 2 + √6.
In Eq. (4.3) we can replace MR
2√3κby unity if 2√3κis a
good approximation to the Roper mass. In comparison
with Ref. [55] we include an extra factor MR+MN
MRin order
to be consistent with the more usual definition of the
transition form factors (4.1).
The results for the nucleon to Roper transition form
factors associated with the parameters gV,ηpand ηn
discussed previously (Table I), are presented in Fig. 2,
for the proton target (N=p). As for the nucleon we
use κ= 0.385 GeV. The data presented here are those
from CLAS/Jefferson Lab for single pion production [88]
and double pion production [89], and it is collected in
the database [90]. We do not discuss the results for the
neutron target (N=n) because they are restricted to
the photon point.
In Fig. 2, we can see that, contrary to the nucleon
case, the results are almost insensitive to the variation
of parameters, therefore we consider only the upper and
lower cases (Q2
m= 1.5 and 2.5 GeV2), delimited by the
band. In the figure one can see that the holographic
approach based on Eqs. (4.2)-(4.3) in the leading twist
approximation gives a very good description of the large
Q2data (Q2>1.8 GeV2) in the range gV= 1.28,...,1.57,
ηp= 0.38,...,0.42 and ηn=−(0.36,...,0.32).
In the above holographic approach one obtains a clear
estimate of the valence quark contributions based on
framework that includes only one pre-defined parame-
ter: the mass scale κ. Since the coefficients gV,ηp,ηn
are determined by the nucleon form factors, one can con-
sider this estimate of the valence quark contributions for
the nucleon to Roper form factors as a parameter-free
prediction.
The present results for F∗
1pand F∗
2pare consistent with
estimates of the form factors based on different frame-
works, such as the results of Refs. [2–4, 7, 10]. In partic-
ular, the present model and the results of the covariant
9
0 2 4 68 10
Q2 (GeV2)
0
0.05
0.1
0.15
0.2
F1p* (Q2)
02468 10
Q2 (GeV2)
-0.4
-0.2
0
0.2
F2p* (Q2)
FIG. 2: γ∗N→N(1440) transition form factors F∗
1pand F∗
2p. The dashed-line is the result from the Light-Front holographic model
from Ref. [31]. Black dots from Ref. [88]; red dots from Ref. [89].
quark model from Refs. [3, 4, 26] are very close for Q2>5
GeV2.
The original holographic calculation of the nucleon to
Roper transition form factors was done in Ref. [55]. In
that work higher Fock states (twist 4 and 5 contributions)
were also taken into account. However the effective con-
tribution of the higher Fock states was determined by
the equation for the Roper mass, and not by the physics
associated with the form factors (dominance of the lead-
ing twist contributions for large Q2). In addition, the
overlap between the nucleon and Roper states was deter-
mined in Ref. [55] using coefficients adjusted to the form
factor data.
In the present work we rather prefer to concentrate on
deriving a clean estimate of the valence quark contribu-
tion using an alternative method of fixing the parameters
of the model, instead of trying to estimate the quark-
antiquark and gluon contributions using LF holography.
Since we do not consider higher twist contributions, the
overlap between the states is just the result of the overlap
between the valence quark states of both baryons with no
unknown coefficients.
Although we conclude that the leading twist calcula-
tion is a good approximation for the form factors, one
still notes that it may not be sufficient for a satisfactory
estimate for the mass of the Roper. Different estimates
of the mass suggest, in fact, that higher Fock states are
crucial for the explanation of the experimental value [19–
21].
In Fig. 2, we also present the estimate of the LF holo-
graphic model from Teramond and Brodsky for F∗
1p[31].
In their formulation, the Dirac form factor is expressed
by an analytic function dependent on the ρmeson
masses [31, 36]. The closeness between results is a conse-
quence of the use of the holographic masses instead of the
physical masses. We note at this point, that, the relations
(4.2)-(4.3) are also analytic parametrization of the Roper
form factors, since the form factors are expressed in terms
of functions of a=Q2
m2
ρ, using m2
ρ= 4κ2. We leave how-
ever the comparison between analytic parametrization of
the Roper form factors to a separated work [91].
In the graph for F∗
1pone can notice a deviation between
our estimate and the data for Q2<1 GeV2. This result
can be interpreted as the consequence of the meson cloud
contributions not included in our leading twist analysis.
Similar results were obtained in independent works [2, 3,
7]. Sizable contributions of the meson cloud effects were
found in Refs. [15, 16].
As for the results for F∗
2p, one can note that the
parametrization based on the LF holography gives nega-
tive values for small Q2, and are close to the data. In par-
ticular we estimate the change of sign in F∗
2pfor Q2≈0.5
GeV2. The results for F∗
2psuggest that contrarily to the
form factor F∗
1p, where the meson cloud are sizable, for
F∗
2pthe meson cloud contribution are very small at low
Q2. To the best of our knowledge, it is the first time that
this fact was observed in the context of a quark model.
There is, nevertheless, a discrepancy between the model
and the data in the region between 0.9 and 1.5 GeV2.
Concerning the F∗
2pdata, one can note that in the re-
gion 0.9–1.5 GeV2there are only two datapoints associ-
ated the two pion electroduction CLAS data [89]. It is
important to check if the new data associated with the
one pion production confirms the trend from Ref. [89]. At
the moment one can conclude that the holographic model
gives a very good description of the F∗
2pdata except for
three data points in the range 0.9–1.5 GeV2(underesti-
mation of the data in about 60%).
Discussion
Summarizing the results presented here for the nucleon
and Roper form factors, one can conclude that the lead-
ing twist approximation provides a very good estimate for
10
both nucleon and Roper form factors. This result is con-
sistent with the LF formalism, because in the Drell-Yan-
West frame the contributions from higher Fock states are
small [25, 30, 33]. It is expected however that the me-
son cloud effects provide significant contributions to the
γ∗N→N∗transition form factors for some nucleon ex-
citations N∗for small Q2[8, 92–100].
In the present work, the meson cloud effect is observed
in particular for the form factors of the Roper: F∗
1p, and
in the region 0.9–1.5 GeV2for F∗
2p. Those meson cloud
corrections come from higher twist contributions to the
Light-Front wave functions. For higher mass resonances
it is expected that higher twist corrections also give im-
portant contributions to the transition form factors at
low Q2. In that case LF holography can be used to es-
timate mainly the contribution of the quark core. Al-
though the estimate for low Q2may appear rough, since
the calculation is based on massless quarks (and the phys-
ical quarks have mass), for large Q2the estimate is ex-
pected to be accurate due to the dominance of the valence
quark degrees of freedom.
In the future one can use holography to estimate tran-
sition form factors in the leading twist approximation for
other N∗states. Those estimates are expected to be ac-
curate at large Q2, since they are based on valence quark
degrees of freedom. At low Q2LF holography may fail
in leading twist, since the framework provides only the
valence quark contributions to the form factors. That
information can be, however, very useful to understand
the role of the meson cloud contributions. On one hand,
we can use the comparison with the data to estimate
the effect of the meson cloud contribution. On the other
hand, estimates of the bare core can also be used as in-
put to dynamical coupled-channel reaction models in the
parametrization of baryon bare core [17–19].
It is worth mentioning that some authors interpret the
Roper as a dynamically-generated baryon-meson reso-
nance, without an explicit reference to three-quark sys-
tems, except for the nucleon and the ∆(1232) [101–
104]. As far as we know, there are no calculations of
γ∗N→N(1440) transition form factors for intermedi-
ate and large Q2, based on dynamically-generated reso-
nance models for the Roper. Future lattice QCD sim-
ulations can help to understand the role of the baryon-
meson contribution for the transition form factors at low
Q2[103, 104].
V. OUTLOOK AND CONCLUSIONS
In the present work we apply the LF holography in
the soft-wall approximation to the study of the elec-
tromagnetic structure of the nucleon and nucleon ex-
citations. More specifically we study the nucleon and
γ∗N→N(1440) transition form factors in the leading
twist approximation.
Since in the leading twist approximation the transition
form factors are determined by the valence quark degrees
of freedom, in the present approach we estimate the con-
tribution of valence quarks to the electromagnetic form
factors. The Light-Front wave functions are determined
by an appropriate choice of boundary conditions in the
5D space and by the expected pQCD falloff for the Dirac
and Pauli form factors (bottom-up approach to QCD).
The Light-Front holography in the soft-wall version
was used previously in the study of the nucleon elastic
form factors, in leading twist and higher orders. How-
ever, since in those works the couplings are adjusted to
the dressed couplings (empirical anomalous magnetic mo-
ments) they may describe well the low Q2data, but fail in
the description of the large-Q2data. In the present work
we fix the free parameters of the model associated with
the bare couplings using the large Q2data for the nu-
cleon, where the effect of the meson cloud contributions
is significantly reduced.
Our expressions for the electromagnetic form factors
depend only of the holographic mass scale κand of three
bare couplings: gV,ηpand ηn. Once determined the
bare couplings by the nucleon data, the model is used to
predict the γ∗N→N(1440) transition form factors.
Our results for the γ∗N→N(1440) transition form
factors compare well with the empirical data. For the
Dirac form factor, the deviation observed for Q2<1.0
GeV2is compatible with the interpretation that meson
cloud contributions are important in that region. As for
the Pauli form factor, our estimate is very close to the
data, both at low and at large Q2, except for 3 datapoints
in the range 0.9–1.5 GeV2. The result at low Q2, suggests
that the meson cloud contributions for F∗
2pare small. As
far as we know this is the first time that this effect is
observed.
The method used in the present work for the Roper can
in the future be extended for higher mass nucleon excita-
tions. The use of the Light-Front holography in leading
twist provides then a natural method to estimate the va-
lence quark contributions for the transition form factors.
The effect of the meson cloud can then be estimated from
the comparison with the experimental data.
Theoretical estimates of the bare core contributions
are very important for the study of the baryon-meson
reactions and nucleon electroproduction reactions. The
results from Light-Front holography may be used as in-
put to dynamical coupled-channel reaction models in the
theoretical study of those reactions.
Acknowledgments
G. R. thanks Valery Lyubovitskij for useful clarifica-
tions about the calculation of the transition form fac-
tors. This work was supported by the Universidade Fed-
eral do Rio Grande do Norte/Minist´erio da Educa¸c˜ao
(UFRN/MEC). The work of D. M. was also partially
supported by the RFBR grant 16-01-00291.
11
[1] L. D. Roper, Phys. Rev. Lett. 12, 340 (1964).
[2] I. G. Aznauryan, Phys. Rev. C 76, 025212 (2007)
[nucl-th/0701012].
[3] G. Ramalho and K. Tsushima, Phys. Rev. D 81, 074020
(2010) [arXiv:1002.3386 [hep-ph]].
[4] G. Ramalho and K. Tsushima, Phys. Rev. D 89, 073010
(2014) [arXiv:1402.3234 [hep-ph]].
[5] G. Ramalho and K. Tsushima, AIP Conf. Proc. 1374,
353 (2011) [arXiv:1010.2765 [hep-ph]].
[6] E. Santopinto and M. M. Giannini, Phys. Rev. C 86,
065202 (2012) [arXiv:1506.01207 [nucl-th]].
[7] J. Segovia, B. El-Bennich, E. Ro jas, I. C. Cloet,
C. D. Roberts, S. S. Xu and H. S. Zong, Phys. Rev.
Lett. 115, 171801 (2015) [arXiv:1504.04386 [nucl-th]].
[8] I. G. Aznauryan et al., Int. J. Mod. Phys. E 22, 1330015
(2013) [arXiv:1212.4891 [nucl-th]].
[9] I. G. Aznauryan and V. D. Burkert, Prog. Part. Nucl.
Phys. 67, 1 (2012) [arXiv:1109.1720 [hep-ph]].
[10] I. G. Aznauryan and V. D. Burkert, Phys. Rev. C 85,
055202 (2012) [arXiv:1201.5759 [hep-ph]].
[11] F. Cano and P. Gonzalez, Phys. Lett. B 431, 270 (1998)
[arXiv:nucl-th/9804071].
[12] L. Tiator, D. Drechsel, S. Kamalov, M. M. Giannini,
E. Santopinto and A. Vassallo, Eur. Phys. J. A 19, 55
(2004) [nucl-th/0310041].
[13] Q. B. Li and D. O. Riska, Phys. Rev. C 74, 015202
(2006) [arXiv:nucl-th/0605076].
[14] D. Y. Chen and Y. B. Dong, Commun. Theor. Phys.
50, 142 (2008).
[15] B. Golli, S. Sirca and M. Fiolhais, Eur. Phys. J. A 42,
185 (2009) [arXiv:0906.2066 [nucl-th]].
[16] I. T. Obukhovsky, A. Faessler, D. K. Fedorov,
T. Gutsche and V. E. Lyubovitskij, Phys. Rev.
D84, 014004 (2011) [arXiv:1104.0957 [hep-ph]];
I. T. Obukhovsky, A. Faessler, T. Gutsche and
V. E. Lyubovitskij, Phys. Rev. D 89, 014032 (2014)
[arXiv:1306.3864 [hep-ph]].
[17] V. D. Burkert and T. S. H. Lee, Int. J. Mod. Phys. E
13, 1035 (2004) [nucl-ex/0407020].
[18] B. Julia-Diaz, H. Kamano, T. S. H. Lee, A. Matsuyama,
T. Sato and N. Suzuki, Phys. Rev. C 80, 025207 (2009).
[arXiv:0904.1918 [nucl-th]].
[19] N. Suzuki, B. Julia-Diaz, H. Kamano, T.-S. H. Lee,
A. Matsuyama and T. Sato, Phys. Rev. Lett. 104,
042302 (2010) [arXiv:0909.1356 [nucl-th]].
[20] J. Gegelia, U. G. Meißner and D. L. Yao, Phys. Lett. B
760, 736 (2016) [arXiv:1606.04873 [hep-ph]].
[21] S. Capstick and W. Roberts, Prog. Part. Nucl. Phys.
45, S241 (2000) [nucl-th/0008028].
[22] T. Bauer, S. Scherer and L. Tiator, Phys. Rev. C 90,
015201 (2014) [arXiv:1402.0741 [nucl-th]].
[23] K. L. Wang, L. Y. Xiao and X. H. Zhong,
arXiv:1702.04604 [nucl-th].
[24] D. S. Roberts, W. Kamleh and D. B. Leinweber, Phys.
Rev. D 89, 074501 (2014) [arXiv:1311.6626 [hep-lat]].
[25] C. F. Perdrisat, V. Punjabi and M. Vander-
haeghen, Prog. Part. Nucl. Phys. 59, 694 (2007)
[hep-ph/0612014].
[26] F. Gross, G. Ramalho and M. T. Pe˜na, Phys. Rev. C
77, 015202 (2008) [nucl-th/0606029].
[27] G. Ramalho and K. Tsushima, Phys. Rev. D 84, 054014
(2011) [arXiv:1107.1791 [hep-ph]].
[28] G. Ramalho, K. Tsushima and A. W. Thomas, J. Phys.
G40, 015102 (2013) [arXiv:1206.2207 [hep-ph]].
[29] G. F. de Teramond and S. J. Brodsky, Phys. Rev. Lett.
102, 081601 (2009) [arXiv:0809.4899 [hep-ph]].
[30] S. J. Brodsky, G. F. de Teramond, H. G. Dosch and
J. Erlich, Phys. Rept. 584, 1 (2015) [arXiv:1407.8131
[hep-ph]].
[31] G. F. de Teramond and S. J. Brodsky, arXiv:1203.4025
[hep-ph]. (Hadron Physics, Ferrara, Italy 2011).
[32] S. J. Brodsky, H. C. Pauli and S. S. Pinsky, Phys. Rept.
301, 299 (1998) [hep-ph/9705477].
[33] S. J. Brodsky and G. F. de Teramond, Phys. Rev. D 77,
056007 (2008) [arXiv:0707.3859 [hep-ph]].
[34] S. J. Brodsky and G. F. de Teramond, Phys. Rev. Lett.
96, 201601 (2006) [hep-ph/0602252].
[35] G. F. de Teramond, H. G. Dosch and S. J. Brodsky,
Phys. Rev. D 87, 075005 (2013) [arXiv:1301.1651 [hep-
ph]].
[36] G. F. de Teramond, H. G. Dosch and S. J. Brodsky,
Phys. Rev. D 91, 045040 (2015) [arXiv:1411.5243 [hep-
ph]].
[37] D. Chakrabarti and C. Mondal, Eur. Phys. J. C 74,
2962 (2014) [arXiv:1402.4972 [hep-ph]].
[38] H. Forkel, M. Beyer and T. Frederico, JHEP 0707, 077
(2007) [arXiv:0705.1857 [hep-ph]].
[39] H. R. Grigoryan and A. V. Radyushkin, Phys. Lett. B
650, 421 (2007) [hep-ph/0703069].
[40] A. Ballon-Bayona, G. Krein and C. Miller, Phys. Rev.
D91, 065024 (2015) [arXiv:1412.7505 [hep-ph]].
[41] A. Karch, E. Katz, D. T. Son and M. A. Stephanov,
Phys. Rev. D 74, 015005 (2006) [hep-ph/0602229].
[42] A. Ballon-Bayona, H. Boschi-Filho, N. R. F. Braga,
M. Ihl and M. A. C. Torres, Phys. Rev. D 86, 126002
(2012) [arXiv:1209.6020 [hep-ph]].
[43] H. R. Grigoryan, T.-S. H. Lee and H. U. Yee, Phys. Rev.
D80, 055006 (2009) [arXiv:0904.3710 [hep-ph]].
[44] T. Branz, T. Gutsche, V. E. Lyubovitskij, I. Schmidt
and A. Vega, Phys. Rev. D 82, 074022 (2010)
[arXiv:1008.0268 [hep-ph]].
[45] A. Vega, I. Schmidt, T. Gutsche and V. E. Lyubovitskij,
Phys. Rev. D 83, 036001 (2011) [arXiv:1010.2815 [hep-
ph]].
[46] M. Ahmady, R. Sandapen and N. Sharma, Phys. Rev.
D94, no. 7, 074018 (2016) [arXiv:1605.07665 [hep-ph]].
[47] S. Mamedov, B. B. Sirvanli, N. Huseynova and
I. Atayev, arXiv:1609.00167 [hep-th].
[48] D. K. Hong, M. Rho, H. U. Yee and P. Yi, Phys. Rev.
D77, 014030 (2008) [arXiv:0710.4615 [hep-ph]].
[49] Z. Abidin and C. E. Carlson, Phys. Rev. D 79, 115003
(2009) [arXiv:0903.4818 [hep-ph]].
[50] T. Gutsche, V. E. Lyubovitskij, I. Schmidt and A. Vega,
Phys. Rev. D 86, 036007 (2012) [arXiv:1204.6612 [hep-
ph]].
[51] D. Chakrabarti and C. Mondal, Eur. Phys. J. C 73,
2671 (2013) [arXiv:1307.7995 [hep-ph]].
[52] T. Maji and D. Chakrabarti, Phys. Rev. D 94, 094020
(2016) [arXiv:1608.07776 [hep-ph]].
[53] T. Liu and B. Q. Ma, Phys. Rev. D 92, 096003 (2015)
[arXiv:1510.07783 [hep-ph]].
[54] R. S. Sufian, G. F. de Teramond, S. J. Brodsky, A. Deur
12
and H. G. Dosch, Phys. Rev. D 95, 014011 (2017)
[arXiv:1609.06688 [hep-ph]].
[55] T. Gutsche, V. E. Lyubovitskij, I. Schmidt and A. Vega,
Phys. Rev. D 87, 016017 (2013) [arXiv:1212.6252 [hep-
ph]].
[56] S. J. Brodsky, D. S. Hwang, B. Q. Ma and I. Schmidt,
Nucl. Phys. B 593, 311 (2001) [hep-th/0003082].
[57] J. M. Maldacena, Int. J. Theor. Phys. 38, 1113
(1999) [Adv. Theor. Math. Phys. 2, 231 (1998) ]
[hep-th/9711200].
[58] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys.
Lett. B 428, 105 (1998) [hep-th/9802109].
[59] E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998)
[hep-th/9802150].
[60] E. Witten, Adv. Theor. Math. Phys. 2, 505 (1998)
[hep-th/9803131].
[61] I. R. Klebanov and M. J. Strassler, JHEP 0008, 052
(2000) [hep-th/0007191].
[62] T. Sakai and S. Sugimoto, Prog. Theor. Phys. 113, 843
(2005) [hep-th/0412141].
[63] J. Erlich, E. Katz, D. T. Son and M. A. Stephanov,
Phys. Rev. Lett. 95, 261602 (2005) [hep-ph/0501128].
[64] L. Da Rold and A. Pomarol, Nucl. Phys. B 721, 79
(2005) [hep-ph/0501218].
[65] S. D. Drell and T. M. Yan, Phys. Rev. Lett. 24, 181
(1970).
[66] G. B. West, Phys. Rev. Lett. 24, 1206 (1970).
[67] A. V. Radyushkin, Acta Phys. Polon. B 26, 2067 (1995)
[hep-ph/9511272].
[68] D. K. Hong, M. Rho, H. U. Yee and P. Yi, Phys. Rev.
D76, 061901 (2007) [hep-th/0701276 [HEP-TH]].
[69] T. Gutsche, V. E. Lyubovitskij, I. Schmidt and A. Vega,
Phys. Rev. D 85, 076003 (2012) [arXiv:1108.0346 [hep-
ph]].
[70] M. Henningson and K. Sfetsos, Phys. Lett. B 431, 63
(1998) [hep-th/9803251].
[71] W. Mueck and K. S. Viswanathan, Phys. Rev. D 58,
106006 (1998) [hep-th/9805145].
[72] M. Henneaux, [hep-th/9902137].
[73] J. Polchinski and M. J. Strassler, Phys. Rev. Lett. 88,
031601 (2002) [hep-th/0109174].
[74] D. Melnikov and G. Ramalho, in preparation.
[75] H. R. Grigoryan and A. V. Radyushkin, Phys. Rev. D
76, 095007 (2007) [arXiv:0706.1543 [hep-ph]].
[76] C. E. Carlson and F. Gross, Phys. Rev. D 36, 2060
(1987); C. E. Carlson and N. C. Mukhopadhyay, Phys.
Rev. Lett. 81, 2646 (1998) [hep-ph/9804356].
[77] M. J. Strassler, hep-th/0505153.
[78] M. Krasnitz, JHEP 0212, 048 (2002) [hep-th/0209163].
[79] A. Dymarsky, D. Melnikov and A. Solovyov, JHEP
0905, 105 (2009) [arXiv:0810.5666 [hep-th]].
[80] I. Gordeli and D. Melnikov, arXiv:1311.6537 [hep-ph].
[81] M. Berg, M. Haack and W. Mueck, Nucl. Phys. B 736,
82 (2006) [hep-th/0507285].
[82] I. Gordeli and D. Melnikov, JHEP 1108, 082 (2011)
[arXiv:0912.5517 [hep-th]].
[83] I. Gordeli, Singlet Glueballs In Klebanov-Strassler The-
ory, PhD thesis; A. Dymarsky and D. Melnikov, unpub-
lished.
[84] S. J. Brodsky and S. D. Drell, Phys. Rev. D 22, 2236
(1980).
[85] J. Arrington, W. Melnitchouk and J. A. Tjon, Phys.
Rev. C 76, 035205 (2007) [arXiv:0707.1861 [nucl-ex]].
A. J. R. Puckett et al., Phys. Rev. Lett. 104, 242301
(2010) [arXiv:1005.3419 [nucl-ex]].
[86] M. Ostrick et al., Phys. Rev. Lett. 83, 276 (1999);
C. Herberg et al., Eur. Phys. J. A 5, 131 (1999);
D. I. Glazier et al., Eur. Phys. J. A 24, 101 (2005)
[arXiv:nucl-ex/0410026]; I. Passchier et al., Phys. Rev.
Lett. 82, 4988 (1999) [arXiv:nucl-ex/9907012]; T. Eden
et al., Phys. Rev. C 50, 1749 (1994); H. Zhu et al.
[E93026 Collaboration], Phys. Rev. Lett. 87, 081801
(2001) [arXiv:nucl-ex/0105001]; R. Madey et al. [E93-
038 Collaboration], Phys. Rev. Lett. 91, 122002 (2003)
[arXiv:nucl-ex/0308007]; G. Warren et al. [Jefferson Lab
E93-026 Collaboration], Phys. Rev. Lett. 92, 042301
(2004) [arXiv:nucl-ex/0308021]; S. Riordan et al., Phys.
Rev. Lett. 105, 262302 (2010) arXiv:1008.1738 [nucl-
ex]; R. Schiavilla and I. Sick, Phys. Rev. C 64, 041002
(2001) [arXiv:nucl-ex/0107004].
[87] P. E. Bosted, Phys. Rev. C 51, 409 (1995);
G. Kubon et al., Phys. Lett. B 524, 26 (2002)
[arXiv:nucl-ex/0107016]; H. Anklin et al., Phys. Lett. B
428, 248 (1998); H. Anklin et al., Phys. Lett. B 336, 313
(1994); J. Lachniet et al. [CLAS Collaboration], Phys.
Rev. Lett. 102, 192001 (2009) [arXiv:0811.1716 [nucl-
ex]].
[88] I. G. Aznauryan et al. [CLAS Collaboration], Phys. Rev.
C80, 055203 (2009) [arXiv:0909.2349 [nucl-ex]].
[89] V. I. Mokeev et al. [CLAS Collaboration], Phys.
Rev. C 86, 035203 (2012) [arXiv:1205.3948 [nucl-ex]];
V. I. Mokeev et al., Phys. Rev. C 93, 025206 (2016)
[arXiv:1509.05460 [nucl-ex]].
[90] V. I. Mokeev, https://userweb.jlab.org/~mokeev/
resonance_electrocouplings/
[91] G. Ramalho, in preparation.
[92] G. Ramalho and M. T. Pe˜na, Phys. Rev. D 80, 013008
(2009) [arXiv:0901.4310 [hep-ph]].
[93] G. Ramalho and K. Tsushima, Phys. Rev. D 82, 073007
(2010) [arXiv:1008.3822 [hep-ph]].
[94] G. Ramalho, D. Jido and K. Tsushima, Phys. Rev. D
85, 093014 (2012) [arXiv:1202.2299 [hep-ph]].
[95] G. Ramalho and K. Tsushima, Phys. Rev. D 88,
053002 (2013) [arXiv:1307.6840 [hep-ph]]; G. Ramalho
and K. Tsushima, Phys. Rev. D 87, 093011 (2013)
[arXiv:1302.6889 [hep-ph]]
[96] G. Ramalho and M. T. Pe˜na, Phys. Rev. D 89, 094016
(2014) [arXiv:1309.0730 [hep-ph]].
[97] G. Ramalho, Phys. Rev. D 90, 033010 (2014)
[arXiv:1407.0649 [hep-ph]].
[98] G. Ramalho, M. T. Pe˜na, J. Weil, H. van Hees
and U. Mosel, Phys. Rev. D 93, 033004 (2016)
[arXiv:1512.03764 [hep-ph]].
[99] G. Ramalho, Phys. Rev. D 94, 114001 (2016)
[arXiv:1606.03042 [hep-ph]].
[100] G. Ramalho, arXiv:1612.09555 [hep-ph].
[101] O. Krehl, C. Hanhart, S. Krewald and J. Speth, Phys.
Rev. C 62, 025207 (2000) [nucl-th/9911080].
[102] M. Doring, C. Hanhart, F. Huang, S. Krewald and
U.-G. Meissner, Nucl. Phys. A 829, 170 (2009)
[arXiv:0903.4337 [nucl-th]].
[103] Z. W. Liu, W. Kamleh, D. B. Leinweber, F. M. Stokes,
A. W. Thomas and J. J. Wu, arXiv:1607.04536 [nucl-th].
[104] C. B. Lang, L. Leskovec, M. Padmanath and
S. Prelovsek, Phys. Rev. D 95, 014510 (2017)
[arXiv:1610.01422 [hep-lat]].
