# Neutrino-nucleon DIS from Holographic QCD:
PDFs of sea and valence quarks, form factors, and structure functions of the
proton

Kiminad A. Mamo<EMAIL_ADDRESS>Physics Division, Argonne National Laboratory,
Argonne, Illinois 60439, USA  Ismail Zahed<EMAIL_ADDRESS>Center
for Nuclear Theory, Department of Physics and Astronomy, Stony Brook
University, Stony Brook, New York 11794-3800, USA

###### Abstract

We discuss unpolarized neutrino- and anti-neutrino-nucleon deep inelastic
scattering (DIS) using a chiral doublet of baryonic sources with explicit
symmetry breaking, in a slice of AdS5 with both a hard and soft wall. We
explicitly derive the direct and transition form factors for the vector and
axial-vector currents for the holographic dual of a proton and neutron. We use
them to derive the s-channel structure functions for neutrino and anti-
neutrino scattering on a proton and neutron in bulk. The t-channel
contributions stemming from the Pomeron and Reggeon exchanges are also
evaluated explicitly. The pertinent even and odd structure functions in the
limit of large and small parton momentum fraction $x$ are given. The results
allow for the extraction of the nonperterbative parton distribution functions
carried by the sea and valence quarks both at large-x and small-x regimes. Our
holographic PDF sets compare well with LHAPDF and CTEQ PDF sets in the large-x
and small-x regimes in the intermediate range of $Q^{2}<10~{}\rm{GeV^{2}}$.

##  I Introduction

At extremely low-x, the measured nucleon structure functions on unpolarized
nucleon targets, show a rapid growth of sea quarks and gluons at low-x H1 ;
ZEUSS . Phenomenological arguments suggest that the growth saturates GW , a
point supported by perturbative QCD arguments BK . A central question is then:
what is the primary mechanism for the growth of the sea quarks at low-x?

DIS in holography at moderate-x is different from weak coupling as it involves
hadronic and not partonic constituents POL . The large gauge coupling at the
low renormalization point, causes the color charges to rapidly deplete their
energy and momentum, making them visible to hard probes only through double
trace operators. However, because the holographic limit enjoys approximate
conformal symmetry, the structure functions and form factors exhibit various
scaling laws including the parton-counting rules BF . DIS scattering at low-x
is partonic and fully saturated HATTA . Recently, we have extended the notion
of DIS scattering to nuclei as extremal RN-AdS black holes with an emerging
Fermi surface Mamo:2018eoy .

In this paper we consider neutrino and anti-neutrino DIS scattering on an
unpolarized nucleon described by a Dirac chiral doublet using holographic QCD.
The scattering is chiefly due to neutral and charge exchanges. In QCD the
charge exchange currents discriminate between partons and anti-partons. When
combined with anti-neutrino DIS scattering, it allows for the separation
between sea and valence partons. In holography, the partonic description holds
only for very large $x$ and very small $x$ in the Regge limit. The purpose of
the present study is to construct the even and odd parity leptonic structure
functions for neutrino and anti-neutrino scattering, and use them to extract
the sea parton distribution of the nucleon as a Dirac fermion in bulk.

The organization of the paper is as follows: in section II we briefly review
the setting for the model with bulk chiral gauge fields and a doublet of Dirac
fermions in a slice of AdS5, for both the soft and hard wall. The explicit
breaking of chiral symmetry is enforced by the boundary value of a scalar bulk
field in the bi-fundamental representation. In section III we derive the
direct and transition form factors for the vector and axial-vector form
factors for both the proton and neutron. In section IV, we briefly review the
essentials of neutrino and anti-neutrino DIS scattering on unpolarized nucleon
targets, and show how the valence and sea up and down parton distribution
functions are related to the structure functions of $W^{\pm}$ exchanges. The
s-channel contributions to these structure functions are derived using the
direct and transition form factors in holographic QCD. The t-channel
contributions are also derived using the Pomeron and Reggeon exchanges in
bulk. In section V, all the results are combined to extract the valence and
sea partonic distributions at large and small x, with a detailed comparison to
the LHAPDF and CTEQ PDFs. Our conclusions are in section V. A number of
Appendices are added to support some of the derivations.

##  II holographic model

AdS/CFT duality maps a conformal and strongly coupled gauge theory in 1+3
dimensions at the boundary to a weakly coupled type-II supergravity in 1+9
dimensions in bulk, with a dilaton, an antisymmetric tensor field and
additional odd (IIA) or even (IIB) forms. AdS${}_{5}\times$S5 geometry emerges
from solitonic and BPS charged D-brane supergravity solutions in bulk with the
graviton and dilaton excitations dual to glueballs. The dual of the flavor
excitations at the boundary, are obtained through probe Dp-branes and
described by DBI and Chern-Simons effective actions.

A simple way to capture AdS/CFT duality in the non-conformal limit is to model
it using a slice of AdS5 with various bulk fields with assigned anomalous
dimensions and pertinent boundary values, in the so-called bottom-up approach
which we will follow here. We consider AdS5 both with a soft and hard wall,
with a background metric $g_{MN}=(\eta_{\mu\nu},-1)R^{2}/z^{2}$ with the flat
metric $\eta_{\mu\nu}=(1,-1,-1,-1)$ at the boundary. Confinement will be
described by a background dilaton $\phi={\kappa}^{2}z^{2}$ for mesons and
$\phi={\tilde{\kappa}}^{2}z^{2}$ for nucleons in the soft wall model. In the
hard wall model, $\phi=0$ and confinement is enforced at $z=z_{0}$.

###  II.1 Bulk vector mesons

The vector mesons fields in bulk are denoted by $L_{\mu},R_{\mu}$. They are
U(2) valued and described by the effective action HIRN ; GRIGORYAN

$\displaystyle S_{M}=$ $\displaystyle-\frac{1}{4g_{5}^{2}}\int
d^{5}xe^{-\phi(z)}\sqrt{g}\,g^{MP}g^{NQ}{\rm Tr}\bigg{(}{\cal F}^{L}_{MN}{\cal
F}^{L}_{PQ}+{\cal F}^{R}_{MN}{\cal F}^{R}_{PQ}\bigg{)}+\int
d^{5}x\,\bigg{(}\omega^{L}_{5}({\cal A})-\omega^{R}_{5}({\cal A})\bigg{)}$
(II.1)

with ${\cal F}=d{\cal A}-i{\cal A}^{2}$ in form notations and ${\cal A}={\cal
A}^{a}T^{a}$ with $T^{0}=\frac{3}{2}{\bf 1_{2}}$ and
$T^{i}=\frac{1}{2}\tau^{i}$. Note that the generators $T^{a}$ are fixed in
such a way that the electromagnetic charge of the proton is 1, while that of
the neutron is 0. The bulk U(2) vector field $V=(R+L)/2$ and axial-vector
field $A=(R-L)/2$ reduce to the QCD flavor source fields at the boundary. The
Chern-Simons contributions in (II.1) are

$\displaystyle\omega_{5}({\cal A})=\frac{N_{c}}{24\pi^{2}}\int d^{5}x\,{\rm
Tr}\left({\cal AF}^{2}+\frac{1}{2}{\cal A}^{3}{\cal F}-\frac{1}{10}{\cal
A}^{5}\right)=\frac{N_{c}}{24\pi^{2}}\int d^{5}x\,{\rm Tr}\left({\cal
A}(d{\cal A})^{2}-\frac{3i}{2}{\cal A}^{3}d{\cal A}-\frac{3}{5}{\cal
A}^{5}\right)$ (II.2)

The equation of motions for the bulk gauge fields follow by variation (${\cal
D}=d-i{\cal A}$)

$\displaystyle\frac{1}{\sqrt{g}}\left[{\cal
D}_{M},\big{(}\sqrt{g}e^{-\phi}{\cal A}^{MN}\big{)}\right]=0$ (II.3)

The coupling $g_{5}$ in (II.1) is fixed by the brane embeddings in bulk, with
${1}/{g_{5}^{2}}\equiv{3N_{c}}/(12\pi^{2})$ (D7-branes), and
${1}/{g_{5}^{2}}\equiv({3\sqrt{\lambda}}/{2^{5/2}\pi}){N_{c}}/({12\pi^{2}})$
(D9-branes). When ignoring these embedding, the standard assignement is
${1}/{g_{5}^{2}}\equiv{N_{c}}/(12\pi^{2})$ to match the vector 2-point
correlation functions of QCD in the UV CHERMAN .

###  II.2 Bulk fermionic doublet

The bulk Dirac fermion action in curved AdS5 with minimal coupling to the left
U(2) gauge fields is

$\displaystyle S_{F}=\frac{1}{2g_{5}^{2}}\int
d^{5}x\,e^{-\phi(z)}\,\sqrt{g}\,\Big{(}\mathcal{L}_{F1}+\mathcal{L}_{F2}\Big{)}+\frac{1}{2g_{5}^{2}}\int
d^{4}x\sqrt{-g^{(4)}}\,\Big{(}\mathcal{L}_{UV1}+\mathcal{L}_{UV2}\Big{)}\,,$
(II.4)

with $\mathcal{L}_{UV1,2}=(\overline{\Psi}_{1,2}\Psi_{1,2})_{z=\varepsilon}$.
This extra UV contribution is needed to maintain the bulk to boundary
correspondence. The Dirac and Pauli fermionic contributions
${\mathcal{L}}_{F1,2}=\mathcal{L}_{\rm Dirac1,2}+\mathcal{L}_{\rm Pauli1,2}$
are explicitly

$\displaystyle\mathcal{L}_{\rm Dirac1,2}$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{i}{2}\overline{\Psi}_{1,2}e^{N}_{A}\Gamma^{A}\big{(}\overrightarrow{D}_{N}^{L,R}-\overleftarrow{D}_{N}^{L,R}\big{)}\Psi_{1,2}-(\pm
M+V(z))\bar{\Psi}_{1,2}\Psi_{1,2}\bigg{)}\,,$ $\displaystyle\mathcal{L}_{\rm
Pauli1,2}$ $\displaystyle=$ $\displaystyle\pm
2g_{5}^{2}\times\eta\,\bar{\Psi}_{1,2}e^{M}_{A}e^{N}_{B}\sigma^{AB}{\cal
F}^{L,R}_{MN}\Psi_{1,2}\,,$

with $\sigma^{AB}=\frac{i}{2}[\Gamma^{A},\Gamma^{B}]$, and the left and right
covariant derivatives

$\displaystyle\overrightarrow{D}_{N}^{L}=$
$\displaystyle\overrightarrow{\partial}_{N}+\frac{1}{8}\omega_{NAB}[\Gamma^{A},\Gamma^{B}]-iL_{N}^{a}T^{a}\equiv\overrightarrow{\mathcal{D}}_{N}-iL_{N}^{a}T^{a}$
$\displaystyle\overleftarrow{D}_{N}^{L}=$
$\displaystyle\overleftarrow{\partial}_{N}+\frac{1}{8}\omega_{NAB}[\Gamma^{A},\Gamma^{B}]+iL_{N}^{a}T^{a}\equiv\overleftarrow{\mathcal{D}}_{N}+iL_{N}^{a}T^{a}$
$\displaystyle\overrightarrow{D}_{N}^{R}=$
$\displaystyle\overrightarrow{\partial}_{N}+\frac{1}{8}\omega_{NAB}[\Gamma^{A},\Gamma^{B}]-iR_{N}^{a}T^{a}\equiv\overrightarrow{\mathcal{D}}_{N}-iR_{N}^{a}T^{a}$
$\displaystyle\overleftarrow{D}_{N}^{R}=$
$\displaystyle\overleftarrow{\partial}_{N}+\frac{1}{8}\omega_{NAB}[\Gamma^{A},\Gamma^{B}]+iR_{N}^{a}T^{a}\equiv\overleftarrow{\mathcal{D}}_{N}+iR_{N}^{a}T^{a}$
(II.6)

The left and right flavor field strengths are set to ${\cal
A}^{L}_{MN}=L_{MN}$, ${\cal A}^{R}_{MN}=R_{MN}$. The Dirac or minimal
fermionic coupling is standard, but the Pauli coupling is not. It can be shown
to follow from the SUGRA action by reduction from the top-down approach.
Without it, the neutron form factor vanishes. The nucleon doublet refers to

$\displaystyle\Psi_{1,2}\equiv\begin{pmatrix}\Psi_{p1,2}\\\
\Psi_{n1,2}\end{pmatrix}\,.$ (II.7)

The nucleon fields in bulk form an iso-doublet $p,n$ with $1,2$ referring to
their $\pm=R,L$ chirality at the boundary YEE . They are dual to a boundary
chiral doublet of baryonic sources with $\Psi_{p1,2}\leftrightarrow{\cal
O}_{p,\pm}$ and $\Psi_{n1,2}\leftrightarrow{\cal O}_{n,\pm}$ with anomalous
dimensions $\pm M=\pm(\Delta-2)=\pm(\tau-3/2)$. They map onto a non-
normalizable solution or source, plus a normalizable or expectation value of
the corresponding source at the boundary. This doubling is best seen in the
top-down approach using a left and a right bulk filling brane.

The fermionic potential $V(z)={\tilde{\kappa}}^{2}z^{2}$ will be used for both
the soft and hard wall. Here $e^{N}_{A}=z\delta^{N}_{A}$ is the inverse
vielbein. The components of the spin connection are $\omega_{\mu
z\nu}=-\omega_{\mu\nu z}=\frac{1}{z}\eta_{\mu\nu}$, the Dirac gamma matrices
$\Gamma^{A}=(\gamma^{\mu},-i\gamma^{5})$ are chosen in the chiral
representation, and satisfy the flat anti-commutation relation
$\\{\Gamma^{A},\Gamma^{B}\\}=2\eta^{AB}$. The equation of motions for the bulk
Dirac chiral doublet follows by variation

$\displaystyle\bigg{(}ie^{N}_{A}\Gamma^{A}D_{N}^{L,R}-\frac{i}{2}(\partial_{N}\phi)\,e^{N}_{A}\Gamma^{A}-(\pm
M+V(z))\bigg{)}\Psi_{1,2}=0\,,$ (II.8)

with $1,2=R,L=\pm$. The inclusion of a fermionic potential which is set to the
dilaton profile for simplicity, breaks conformal symmetry and garentees the
Reggeization of the nucleon spectrum in bulk. (II.8) is supplemented with
confining boundary conditions for the hard wall, and vanishing fields
asymptotically for the soft wall. The solutions to (II.8) are briefly
discussed in Appendix B.2. In short, for the soft wall and in the absence of a
tachyon coupling in bulk, the spectrum Reggeizes with
$M_{n}^{2}=4\tilde{\kappa}^{2}(n+\tau-1)$ and the ground state proton, neutron
states with $n=0$ are degenerate. They follow from the mixed representation

$\displaystyle\Psi_{1}(p,z)$ $\displaystyle=$
$\displaystyle\psi_{R}(z)\Psi^{0}_{R}(p)+\psi_{L}(z)\Psi^{0}_{L}(p)$ (II.9)

for $n=0$, with $\psi_{R}(z\approx 0)\approx z^{\tau+1/2}$ and
$\psi_{L}(z\approx 0)\approx z^{\tau+3/2}$ for the positive parity states
$1\equiv+$ at the boundary. Similar relations follow for the negative parity
states $2\equiv-$ at the boundary through the substitution
$\psi_{R,L}\leftrightarrow\mp\psi_{L,R}$ by parity.

Note that the canonical dimension for the QCD baryonic sources is $\Delta=9/2$
which would suggest a twist $\tau=4$. However, we expect non-vanishing
anomalous dimensions to develop at strong coupling. We do not know of any
reliable calculational scheme to assess them. We will assume $\Delta$ and thus
$\tau$ as a parameter with a twist $\tau=3$ to recover the hard scattering
rules. The inclusion of additional twist contributions is discussed in VEGA .

###  II.3 Fermionic currents

For later use, it will be useful to define the bulk U(2) Dirac 1-form currents

$\displaystyle J^{aN}_{L}=$ $\displaystyle\frac{\partial\mathcal{L}_{\rm
Dirac1}}{\partial
L^{a}_{N}}=\overline{\Psi}_{1}e^{N}_{A}\Gamma^{A}T^{a}\Psi_{1}\,,$
$\displaystyle J^{aN}_{R}=$ $\displaystyle\frac{\partial\mathcal{L}_{\rm
Dirac2}}{\partial
R^{a}_{N}}=\overline{\Psi}_{2}e^{N}_{A}\Gamma^{A}T^{a}\Psi_{2}\,,$ (II.10)

and the bulk U(2) Pauli 2-form currents

$\displaystyle J^{aMN}_{L}=$ $\displaystyle\frac{\partial\mathcal{L}_{\rm
Pauli1}}{\partial
L^{a}_{MN}}=+2g_{5}^{2}\times\eta\overline{\Psi}_{1}e^{M}_{A}e^{N}_{B}\sigma^{AB}T^{a}\Psi_{1}\,,$
$\displaystyle J^{aMN}_{R}=$ $\displaystyle\frac{\partial\mathcal{L}_{\rm
Pauli2}}{\partial
R^{a}_{MN}}=-2g_{5}^{2}\times\eta\overline{\Psi}_{2}e^{M}_{A}e^{N}_{B}\sigma^{AB}T^{a}\Psi_{2}\,,$
(II.11)

in terms of which the pertinent 1- and 2-form charged currents read
respectively

$\displaystyle J_{L,R}^{+N}=$ $\displaystyle
J_{L,R}^{1N}-iJ_{L,R}^{2N}=\overline{\Psi}_{n1,2}e^{N}_{A}\Gamma^{A}\Psi_{p1,2}\,,$
$\displaystyle J_{L,R}^{-N}=$ $\displaystyle
J_{L,R}^{1N}+iJ_{L,R}^{2N}=\overline{\Psi}_{p1,2}e^{N}_{A}\Gamma^{A}\Psi_{n1,2}\,.$
(II.12)

and

$\displaystyle J_{L}^{+MN}=$ $\displaystyle
J_{L}^{1MN}-iJ_{L}^{2MN}=+2g_{5}^{2}\times\eta\overline{\Psi}_{n1}e^{M}_{A}e^{N}_{B}\sigma^{AB}\Psi_{p1}\,,$
$\displaystyle J_{L}^{-MN}=$ $\displaystyle
J_{L}^{1MN}+iJ_{L}^{2MN}=+2g_{5}^{2}\times\eta\overline{\Psi}_{p1}e^{M}_{A}e^{N}_{B}\sigma^{AB}\Psi_{n1}\,,$
$\displaystyle J_{R}^{+MN}=$ $\displaystyle
J_{R}^{1MN}-iJ_{R}^{2MN}=-2g_{5}^{2}\times\eta\overline{\Psi}_{n2}e^{M}_{A}e^{N}_{B}\sigma^{AB}\Psi_{p2}\,,$
$\displaystyle J_{R}^{-MN}=$ $\displaystyle
J_{R}^{1MN}+iJ_{R}^{2MN}=-2g_{5}^{2}\times\eta\overline{\Psi}_{p2}e^{M}_{A}e^{N}_{B}\sigma^{AB}\Psi_{n2}\,.$

###  II.4 Bulk effective action

In terms of the charged and left 1-form gauge fields
$L_{N}^{\pm}\equiv\frac{1}{\sqrt{2}}(L_{N}^{1}\mp iL_{N}^{2})$, the bulk meson
effective action (II.1) can be recast in the following form

$\displaystyle S_{M}\supset$ $\displaystyle\frac{1}{g_{5}^{2}}\int
d^{5}xe^{-\phi(z)}\sqrt{g}\,\bigg{(}-\partial^{M}L^{-N}\partial_{M}L_{N}^{+}+\partial^{M}L^{-N}\partial_{N}L_{M}^{+}-\frac{1}{4}L^{0MN}L^{0}_{MN}-\frac{1}{4}L^{3MN}L^{3}_{MN}+L\rightarrow
R\bigg{)}$ $\displaystyle+\frac{N_{c}}{48\pi^{2}}\int
d^{5}x\epsilon^{\mu\nu\rho\sigma}\bigg{(}\mathcal{L}^{CS}_{\mu\nu\rho\sigma}(L)-\mathcal{L}^{CS}_{\mu\nu\rho\sigma}(R)\bigg{)}$

with the Chern-Simons contribution restriced to the charged left-currents
through a neutral

$\displaystyle\mathcal{L}^{CS}_{\mu\nu\rho\sigma}(L)=\partial_{z}L_{\mu}^{+}\partial_{\nu}L_{\rho}^{0}L_{\sigma}^{-}+\partial_{z}L_{\mu}^{-}\partial_{\nu}L_{\rho}^{0}L_{\sigma}^{+}-\partial_{z}L_{\mu}^{+}L_{\rho}^{0}\partial_{\nu}L_{\sigma}^{-}-\partial_{z}L_{\mu}^{-}L_{\rho}^{0}\partial_{\nu}L_{\sigma}^{+}+\partial_{z}L_{\mu}^{0}\partial_{\nu}L_{\rho}^{-}L_{\sigma}^{+}+\partial_{z}L_{\mu}^{0}\partial_{\nu}L_{\rho}^{+}L_{\sigma}^{-}\,,$

where we made use of the Abelian field strengths

$\displaystyle L_{MN}^{\pm}=$
$\displaystyle\partial_{M}L^{\pm}_{N}-\partial_{N}L^{\pm}_{M}\,,$
$\displaystyle L_{MN}^{0,3}=$
$\displaystyle\partial_{M}L^{0,3}_{N}-\partial_{N}L^{0,3}_{M}\,.$ (II.16)

In terms of the 1- and 2-form currents, the bulk fermion effective action
(II.2) now reads

$\displaystyle\mathcal{L}_{F1}+\mathcal{L}_{F2}\supset$
$\displaystyle\frac{i}{2}\bar{\Psi}_{1}e^{N}_{A}\Gamma^{A}\big{(}\overrightarrow{\mathcal{D}}_{N}-\overleftarrow{\mathcal{D}}_{N}\big{)}\Psi_{1}-(M+V(z))\bar{\Psi}_{1}\Psi_{1}+\frac{1}{\sqrt{2}}L_{N}^{+}J_{L}^{-N}+\frac{1}{\sqrt{2}}L_{N}^{-}J_{L}^{+N}+L_{N}^{0}J_{L}^{0N}+L_{N}^{3}J_{L}^{3N}$
$\displaystyle+L^{+}_{MN}J_{L}^{-MN}+L^{-}_{MN}J_{L}^{+MN}+L^{0}_{MN}J_{L}^{0MN}+L^{3}_{MN}J_{L}^{3MN}+(1\leftrightarrow
2,L\leftrightarrow R,M\leftrightarrow-M)$ $\displaystyle=$
$\displaystyle\frac{i}{2}\bar{\Psi}_{1}e^{N}_{A}\Gamma^{A}\big{(}\overrightarrow{\mathcal{D}}_{N}-\overleftarrow{\mathcal{D}}_{N}\big{)}\Psi_{1}-(M+V(z))\bar{\Psi}_{1}\Psi_{1}+\frac{i}{2}\bar{\Psi}_{2}e^{N}_{A}\Gamma^{A}\big{(}\overrightarrow{\mathcal{D}}_{N}-\overleftarrow{\mathcal{D}}_{N}\big{)}\Psi_{2}-(-M+V(z))\bar{\Psi}_{2}\Psi_{2}$
(II.17)
$\displaystyle+\frac{1}{\sqrt{2}}V_{N}^{+}J_{V}^{-N}+\frac{1}{\sqrt{2}}A_{N}^{+}J_{A}^{-N}+\frac{1}{\sqrt{2}}V_{N}^{-}J_{V}^{+N}+\frac{1}{\sqrt{2}}A_{N}^{-}J_{A}^{+N}+V_{N}^{0}J_{V}^{0N}+A_{N}^{0}J_{A}^{0N}+V_{N}^{3}J_{V}^{3N}+A_{N}^{3}J_{A}^{3N}$
$\displaystyle+\frac{1}{\sqrt{2}}V^{+}_{MN}J_{V}^{-MN}+\frac{1}{\sqrt{2}}A^{+}_{MN}J_{A}^{-MN}+\frac{1}{\sqrt{2}}V^{-}_{MN}J_{V}^{+MN}+\frac{1}{\sqrt{2}}A^{-}_{MN}J_{A}^{+MN}$
$\displaystyle+V^{0}_{MN}J_{V}^{0MN}+A^{0}_{MN}J_{A}^{0MN}+V^{3}_{MN}J_{V}^{3MN}+A^{3}_{MN}J_{A}^{3MN}\,,$

where we defined

$\displaystyle J_{V,A}^{\bar{\alpha}N}=J_{L}^{\bar{\alpha}N}\pm
J_{R}^{\bar{\alpha}N}\,,\,\,\,\,J_{V,A}^{\bar{\alpha}MN}=J_{L}^{\bar{\alpha}MN}\mp
J_{R}^{\bar{\alpha}MN}\,,$

with $\bar{\alpha}\equiv-,+,0,3$, and $V_{N},A_{N}\equiv({L_{N}\pm
R_{N}})/{2}$.

###  II.5 Hard and soft wall models

We now detail the specifics of the hard and soft wall model with chiral
symmetry breaking through the use of a flavor scalar in the bi-fundamental
representation in a slab of AdS. These well motivated bottom-up models capture
the essentials of the top-down approaches and make more transparent the
essential aspects of holography. Their parameters are fixed by the brane
embeddings and reduction in higher dimensions.

####  II.5.1 Hard wall with bi-fundamentals

To distinguish vector and axial-vector spectra and introduce a scalar
$\tilde{X}$ in the bi-fundamental representation of $U(N_{f})_{L}\times
U(N_{f})_{R}$ with a scaling dimension $\Delta=3$ and bulk action

$\displaystyle{\cal
S}[\tilde{X}]=\frac{1}{2g_{5}^{2}}\int\,d^{5}x\,\sqrt{g}\,{\rm
Tr}\bigg{(}|D\tilde{X}|^{2}+3|\tilde{X}|^{2}\bigg{)}$ (II.19)

using the 1-form $D\tilde{X}=d\tilde{X}-i{\cal A}^{L}\tilde{X}+i\tilde{X}{\cal
A}^{R}$. Note by replacing $X\rightarrow\sqrt{2g_{5}^{2}}\times\tilde{X}$, we
can rewrite (II.19), in a form similar to KKSS , as

$\displaystyle{\cal S}[X]=\int\,d^{5}x\,\sqrt{g}\,{\rm
Tr}\bigg{(}|DX|^{2}+3|X|^{2}\bigg{)}\,.$ (II.20)

The equation of motion derived from the above action (II.20), for ${\cal
A}=0$, has a background solution for the bi-fundamental scalar near the UV
boundary ($z\rightarrow 0$) given by

$\displaystyle X_{0}(x,z\rightarrow
0)\approx\frac{1}{\sqrt{2g_{5}^{2}}}\Bigg{(}\frac{1}{2}M_{q}z+\frac{1}{2}\Sigma
z^{3}\Bigg{)}\equiv\frac{1}{\sqrt{2g_{5}^{2}}}v(z)$

Both the current mass matrix and the quark bilinear are diagonal,
$M_{q}=m_{q}\textbf{1}$ and $\Sigma=\sigma\textbf{1}$, with
$\Sigma^{ij}=2g_{5}^{2}\left<\bar{q}_{R}^{i}q_{L}^{j}\right>$ where
$i,j=1,2...N_{f}$ are the flavor indices. For the hard wall, the boundary
condition for the $U(N_{f})$ gauge fields is ${\cal F}^{R,L}(x,z_{0})=0$.

Vectors:

The bulk vector field $V_{M}=(V_{\mu},V_{5})$ splits into a $\mu$-transverse,
$\mu$-longitudinal and 5-contribution. The 5-longitudinal components of the
vector field, mix through a Higgs-type effect. They can be decoupled by a
pertinent choice of gauge. The transverse part of the vector field decouples,
and its mode decomposition in terms of $\psi_{n}(z)$ yields the bulk equation

$\displaystyle\partial_{z}\bigg{(}\frac{1}{z}\partial_{z}\psi_{n}(z)\bigg{)}+\frac{1}{z}m_{n}^{2}\,\psi_{n}(z)=0$
(II.22)

subject to the confining conditions $\psi_{n}(0)=\psi_{n}^{\prime}(z_{0})=0$.
The solutions are readily found as $\psi_{n}(z)\sim zJ_{1}(m_{n}z)$ with the
vector spectrum $m_{n}$ fixed by the zeros $\gamma_{0,n}$ of the Bessel
function $J_{0}(\gamma_{0,n})=0$, with normalization

$\displaystyle\int_{0}^{z_{0}}\,dz\,\sqrt{g_{xx}}\,\psi_{n}(z)\psi_{m}(z)=\delta_{nm}$
(II.23)

and the completeness relation

$\displaystyle\sum_{n=1}^{\infty}\psi_{n}(z)\psi_{n}(z^{\prime})=\sqrt{g^{xx}}\delta(z-z^{\prime})$
(II.24)

The first zero of $J_{0}$ or $\gamma_{0,1}=2.40483$, is identified with the
rho meson state $m_{1=\rho}=\gamma_{0,1}/z_{0}=0.775$ GeV which fixes the IR
scale $z_{0}=3.103/{\rm GeV}$. Note that asymptotically $\gamma_{0,n}\approx
n\pi$ with $m_{n}\approx n\pi/z_{0}$.

The bulk-to-boundary vector propagator follows from (II.22) through the
substitution $m_{n}^{2}\rightarrow-Q^{2}$ with the boundary conditions ${\cal
V}(Q,0)=1$ and $\partial_{z}{\cal V}(Q,z_{0})=0$ (confining). The solution can
be obtained in closed form

$\displaystyle{\cal V}(Q,z)=$
$\displaystyle\sum_{n}\frac{g_{5}F_{n}\psi_{n}(z)}{Q^{2}+m_{n}^{2}}$
$\displaystyle=$ $\displaystyle
Qz\bigg{(}K_{1}(Qz)+\frac{K_{0}(Qz_{0})}{I_{0}(Qz_{0})}\,I_{1}(Qz)\bigg{)}$

where we have also shown its mode decomposition with the decay constants
$g_{5}F_{n}=(-\partial_{z}\psi_{n}(z)/z)_{0}$.

Axials:

Similarly the bulk axial-vector field $A_{M}=(A_{\mu},A_{5})$ splits into a
$\mu$-transverse, $\mu$-longitudinal and 5-contribution. The 5-longitudinal
components of the axial-vector field, mix through a Higgs-type effect. They
can be decoupled by a pertinent choice of gauge, at the expense of more
coupling of the $A_{5}$ field with the tachyon field. For instance in the
R-gauge, the $A_{5}$ field is identified with the pion field. These
contributions will not be followed except when discussing the pion
contribution to the direct and transitional axial-vector form factors below.

The $\mu$-transverse part of $A_{\mu}$ is always decoupled. Its mode
decomposition in terms of $\tilde{\psi}_{n}(z)$ yields the bulk equation

$\displaystyle\partial_{z}\bigg{(}\frac{1}{z}\partial_{z}\tilde{\psi}_{n}(z)\bigg{)}-\frac{1}{2}\frac{v^{2}(z)}{z^{3}}\tilde{\psi}_{n}(z)+\frac{1}{z}\tilde{m}_{n}^{2}\,\tilde{\psi}_{n}(z)=0\,,$

Upon taking the chiral limit before the near-boundary limit (i.e.,
$m_{q}\rightarrow 0$ before $z\rightarrow 0$), (II.5.1) reduces to

$\displaystyle\partial_{z}\bigg{(}\frac{1}{z}\partial_{z}\tilde{\psi}_{n}(z)\bigg{)}-\frac{1}{2}\sigma^{2}z^{3}\tilde{\psi}_{n}(z)+\frac{1}{z}\tilde{m}_{n}^{2}\,\tilde{\psi}_{n}(z)=0$

with the same confining boundary conditions
$\tilde{\psi}_{n}(0)=\tilde{\psi}_{n}^{\prime}(z_{0})=0$ and a similar
normalization (II.23) and completeness relation (II.24). Note that (II.5.1)
can only be solved numerically.

However, we can find an equation that can be solved analytically, if we take
the near-boundary limit first without taking the chiral limit (i.e.,
$z\rightarrow 0$ with $m_{q}\neq 0$) of (II.5.1) which reduces to

$\displaystyle\partial_{z}\bigg{(}\frac{1}{z}\partial_{z}\tilde{\psi}_{n}(z)\bigg{)}-\frac{1}{2z}m_{q}^{2}\,\tilde{\psi}_{n}(z)+\frac{1}{z}\tilde{m}_{n}^{2}\,\tilde{\psi}_{n}(z)=0\,.$

Defining $\tilde{m}_{n}^{2}=m_{n}^{2}+\frac{1}{2}m_{q}^{2}$, we can recast
(II.5.1) in the form

$\displaystyle\partial_{z}\bigg{(}\frac{1}{z}\partial_{z}\tilde{\psi}_{n}(z)\bigg{)}+\frac{1}{z}m_{n}^{2}\,\tilde{\psi}_{n}(z)=0\,,$

which maps onto the vector equation (II.22) with the replacement of
$\tilde{\psi}_{n}(z)\leftrightarrow\psi_{n}(z)$, and
$\tilde{m}_{n}\leftrightarrow m_{n}$.

In the near boundary or UV limit, the differences between the axial-vector and
vector masses are only due to the explicit symmetry breaking effect with
$m_{q}\neq 0$, and vanishes in the chiral limit. In general however, the
difference is largely due to the spontaneous breaking of chiral symmetry
through $\Sigma=\sigma{\bf 1}$, as a numerical solution to (II.5.1) shows. For
$\sigma z_{0}^{3}\ll 1$, a simple parametric estimate can be obtained using
first order perturbation theory in (II.5.1)

$\displaystyle\tilde{m}_{n}^{2}\approx
m_{n}^{2}+\frac{1}{2}\int_{0}^{z_{0}}dz\sqrt{g_{xx}}\left|\psi_{n}(z)\right|^{2}(\sigma
z^{2}+m_{q})^{2}$

which reduces to the near boundary or UV limit result for $\sigma
z_{0}^{3}\rightarrow 0$. In the chiral limit (II.5.1) yields a chiral
splitting between the axials and vectors

$\displaystyle\tilde{m}_{n}^{2}\approx
m^{2}_{n}+0.38\left(\frac{1}{2}{\sigma^{2}z_{0}^{4}}\right)$ (II.31)

solely due to the chiral condensate. Note that asymptotically, the linear mass
splitting in the hard wall model vanishes, i.e.
$\tilde{m}_{n}-m_{n}\approx\sigma^{2}z_{0}^{5}/n\rightarrow 0$ for $n\gg 1$.
Below and for simplicity, we will carry the analytical analysis using the near
boundary limit, using the substitution

$\displaystyle\frac{1}{2}m_{q}^{2}\rightarrow\tilde{m}_{n}^{2}-m_{n}^{2}$
(II.32)

The bulk-to-boundary axial-vector propagator follows from (II.5.1) through the
substitution
$m_{n}^{2}\rightarrow-\tilde{Q}^{2}=-(Q^{2}+\frac{1}{2}m_{q}^{2})$ with the
boundary conditions ${\cal A}(Q,0)=1$ and $\partial_{z}{\cal A}(Q,z_{0})=0$
(confining). Note that ${\cal A}(Q,0)\neq{\cal A}(0,z)$ since ${\cal
A}(0,z)\neq 1$. The solution can be obtained in closed form as

$\displaystyle\tilde{\cal V}(Q,z)\approx$
$\displaystyle\sum_{n}\frac{g_{5}\tilde{F}_{n}\tilde{\psi}_{n}(z)}{\tilde{Q}^{2}+m_{n}^{2}}=\sum_{n}\frac{g_{5}\tilde{F}_{n}\tilde{\psi}_{n}(z)}{Q^{2}+\tilde{m}_{n}^{2}}$
$\displaystyle=$
$\displaystyle\tilde{Q}z\bigg{(}K_{1}(\tilde{Q}z)+\frac{K_{0}(\tilde{Q}z_{0})}{I_{0}(\tilde{Q}z_{0})}\,I_{1}(\tilde{Q}z)\bigg{)}$

where we have also shown its mode decomposition with the decay constants
$g_{5}\tilde{F}_{n}=(-\partial_{z}\tilde{\psi}_{n}(z)/z)_{z=0}$.

####  II.5.2 Hard wall without bi-fundamentals

Both the bulk-to-boundary vector and axial-vector propagators can be obtained
in closed form in a variant of the Sakai-Sugimoto construction SS  using a
hard wall model without the use of the bi-fundamental scalar field in bulk but
with modified boundary conditions.

Vectors:

The vector fields are still given with the same hard wall boundary conditions
$\psi_{n}(0)=\psi_{n}^{\prime}(z_{0})=0$, but the axial-vector fields satisfy
$\tilde{\psi}_{n}(0)=\tilde{\psi}_{n}(z_{0})=0$ and
$\tilde{\psi}_{n}^{\prime}(z_{0})\neq 0$. The solutions are again readily
found in the form $\psi_{n}(z)\sim zJ_{1}(m_{n}z)$ with
$m_{n}=\gamma_{0,n}/z_{0}$.

Axials:

Similarly, the axial vector spectrum follows with $\tilde{\psi}_{n}(z)\sim
zJ_{1}(\tilde{m}_{n}z)$ with $\tilde{m}_{n}=\gamma_{1,n}/z_{0}$. If $z_{0}$ is
fixed by the rho mass then the ratio of the axial-to-rho meson mass is
$\tilde{m}_{1}/M_{1}=\gamma_{1,1}/\gamma_{0,1}=1.593$ which is consistent with
the empirical ratio $m_{A}/m_{\rho}=1.587$ as in the Sakai-Sugimoto
construction. The higher excited modes fare less better empirically in both
formulations.

The bulk-to-boundary axial-vector propagator follows a similar reasoning as
the vector analogue, through the substitution
$\tilde{m}_{n}^{2}\rightarrow-Q^{2}$ with the boundary conditions $\tilde{\cal
V}(Q,0)=\tilde{\cal V}(0,z)=1$ and $\tilde{\cal V}(Q,z_{0})=0$ (confining).
The solution follows as

$\displaystyle\tilde{\cal V}(Q,z)\approx$
$\displaystyle\sum_{n}\frac{g_{5}\tilde{F}_{n}\tilde{\psi}_{n}(z)}{Q^{2}+\tilde{m}_{n}^{2}}$
$\displaystyle=$
$\displaystyle\tilde{Q}z\bigg{(}K_{1}(\tilde{Q}z)-\frac{K_{1}(\tilde{Q}z_{0})}{I_{1}(\tilde{Q}z_{0})}\,I_{1}(\tilde{Q}z)\bigg{)}$

####  II.5.3 Soft wall with bi-fundamentals

In the soft wall model with scalar bi-fundamentals, we replace (II.20) by KKSS

$\displaystyle{\cal S}[X]=\,\int\,d^{5}x\,e^{-\phi(z)}\sqrt{g}\,{\rm
Tr}\bigg{(}|DX|^{2}+3|X|^{2}\bigg{)}$

with again the same background solution and boundary identification for the
bi-fundamentals.

Vectors:

In this model, the bulk vector gauge field in terms of $\psi_{n}(z)$ yields
the bulk equation

$\displaystyle\partial_{z}\bigg{(}\frac{e^{-\phi(z)}}{z}\partial_{z}\psi_{n}(z)\bigg{)}+\frac{e^{-\phi(z)}}{z}m_{n}^{2}\,\psi_{n}(z)=0\,.$
(II.36)

The solutions are readily found as
$\psi_{n}(z)=c_{n}\tilde{\kappa}^{2}z^{2}L_{n}^{1}(\tilde{\kappa}^{2}z^{2})$,
and $m_{n}^{2}=\,\,4\tilde{\kappa}^{2}(n+1)$ for $n=0,1,...$ with
normalization coefficients $c_{n}=\sqrt{{2}/{n+1}}$ determined from the
normalization condition (for the soft wall model with background dilaton
$\phi=\tilde{\kappa}^{2}z^{2}$)

$\displaystyle\int
dz\,\sqrt{g}e^{-\phi}\,(g^{xx})^{2}\,\psi_{n}(z)\psi_{m}(z)=\delta_{nm}\,.$
(II.37)

For the soft wall model, the bulk-to-boundary vector propagator follows from
(II.36) through the substitution $m_{n}^{2}\rightarrow-Q^{2}$ with the
boundary conditions ${\cal V}(Q,0)=1$. The solution can be obtained in closed
form as

$\displaystyle{\cal V}(Q,z)$ $\displaystyle=$
$\displaystyle\sum_{n}\frac{g_{5}F_{n}\psi_{n}(z)}{Q^{2}+m_{n}^{2}}$
$\displaystyle=$
$\displaystyle\tilde{\kappa}^{2}z^{2}\,\,\Gamma(1+\frac{Q^{2}}{4\tilde{\kappa}^{2}})\,\,{\cal
U}(1+\frac{Q^{2}}{4\tilde{\kappa}^{2}};2;\tilde{\kappa}^{2}z^{2})\,,$
$\displaystyle=$
$\displaystyle\tilde{\kappa}^{2}z^{2}\int_{0}^{1}\frac{dx}{(1-x)^{2}}x^{a}{\rm
exp}\Big{[}-\frac{x}{1-x}\tilde{\kappa}^{2}z^{2}\Big{]}$

where we have also shown its mode decomposition with the decay constants
$g_{5}F_{n}=(-\partial_{z}\psi_{n}(z)/z)_{0}$, and defined
$a\equiv{Q^{2}}/{4\tilde{\kappa}^{2}}$.

Axials:

For the soft wall model, the mode decomposition of the axial-vector gauge
field in terms of $\tilde{\psi}_{n}(z)$ yields the bulk equation

$\displaystyle\partial_{z}\bigg{(}\frac{e^{-\phi(z)}}{z}\partial_{z}\tilde{\psi}_{n}(z)\bigg{)}-\frac{e^{-\phi(z)}\frac{1}{2}v(z)^{2}}{z^{3}}\tilde{\psi}_{n}(z)$
(II.39) $\displaystyle+$
$\displaystyle\frac{e^{-\phi(z)}}{z}\tilde{m}_{n}^{2}\,\tilde{\psi}_{n}(z)=0\,,$

which, upon taking the chiral limit before the near-the-boundary one (i.e.,
taking $m_{q}\rightarrow 0$ before $z\rightarrow 0$), reduces to

$\displaystyle\partial_{z}\bigg{(}\frac{e^{-\phi(z)}}{z}\partial_{z}\tilde{\psi}_{n}(z)\bigg{)}-e^{-\phi(z)}\frac{1}{2}\sigma^{2}z^{3}\tilde{\psi}_{n}(z)$
$\displaystyle+$
$\displaystyle\frac{e^{-\phi(z)}}{z}\tilde{m}_{n}^{2}\,\tilde{\psi}_{n}(z)=0\,,$

with a similar normalization (II.37). Again, note that (II.5.3) can only be
solved numerically.

But, similar to the hard wall case, we can find an equation that can be solved
analytically, for the soft wall model, if we take the near-boundary limit
first without taking the chiral one (i.e., $z\rightarrow 0$ with $m_{q}\neq
0$) of (II.39) which reduces to

$\displaystyle\partial_{z}\bigg{(}\frac{e^{-\phi(z)}}{z}\partial_{z}\tilde{\psi}_{n}(z)\bigg{)}-\frac{e^{-\phi(z)}}{z}\frac{1}{2}m_{q}^{2}\,\tilde{\psi}_{n}(z)$
$\displaystyle+$
$\displaystyle\frac{e^{-\phi(z)}}{z}\tilde{m}_{n}^{2}\,\tilde{\psi}_{n}(z)=0\,.$

Defining $m_{n}^{2}=\tilde{m}_{n}^{2}-\frac{1}{2}m_{q}^{2}$, we can rewrite
(II.5.3) as

$\displaystyle\partial_{z}\bigg{(}\frac{e^{-\phi(z)}}{z}\partial_{z}\tilde{\psi}_{n}(z)\bigg{)}+\frac{e^{-\phi(z)}}{z}m_{n}^{2}\,\tilde{\psi}_{n}(z)=0\,,$

which is essentially the same equation as the vector one (II.36) with the
replacement of $\tilde{\psi}_{n}(z)\leftrightarrow\psi_{n}(z)$, and
$M_{n}\leftrightarrow m_{n}$.

For the soft wall model, the bulk-to-boundary axial-vector propagator follows
from (II.5.3) through the substitution
$m_{n}^{2}\rightarrow-\tilde{Q}^{2}=-(Q^{2}+\frac{1}{2}m_{q}^{2})$ with the
boundary conditions $\tilde{\cal V}(Q,0)=1$. Note that $\tilde{\cal
V}(Q,0)\neq\tilde{\cal V}(0,z)$ since $\tilde{\cal V}(0,z)\neq 1$. The
approximate solution follows as

$\displaystyle\tilde{\cal V}(Q,z)\approx$
$\displaystyle\sum_{n}\frac{g_{5}\tilde{F}_{n}\tilde{\psi}_{n}(z)}{\tilde{Q}^{2}+m_{n}^{2}}=\sum_{n}\frac{g_{5}\tilde{F}_{n}\tilde{\psi}_{n}(z)}{Q^{2}+\tilde{m}_{n}^{2}}$
$\displaystyle=$
$\displaystyle\tilde{\kappa}^{2}z^{2}\,\,\Gamma\bigg{(}1+\frac{\tilde{Q}^{2}}{4\tilde{\kappa}^{2}}\bigg{)}\,\,{\cal
U}\bigg{(}1+\frac{\tilde{Q}^{2}}{4\tilde{\kappa}^{2}};2;\tilde{\kappa}^{2}z^{2}\bigg{)}=\tilde{\kappa}^{2}z^{2}\int_{0}^{1}\frac{dx}{(1-x)^{2}}x^{\tilde{a}}{\rm
exp}\Big{[}-\frac{x}{1-x}\tilde{\kappa}^{2}z^{2}\Big{]}$

where we have also shown its mode decomposition with the decay constants
$g_{5}\tilde{F}_{n}=(-\partial_{z}\tilde{\psi}_{n}(z)/z)_{0}$, and defined
$\tilde{a}\equiv{\tilde{Q}^{2}}/{4\tilde{\kappa}^{2}}$.

We emphasize that the exact form of (II.5.3) requires solving numerically
(II.39) for the normalizable modes and using the mode decomposition (first
line). Alternatively, one can solve (II.39) also numerically for the non-
normalizable modes, after the substitution
$\tilde{m}_{n}^{2}\rightarrow-Q^{2}$. However, for DIS scattering which is the
main thrust of this paper, this is not needed. Indeed, in the DIS regime with
$Qz_{0}\gg 1$, the near-boundary approximation giving (II.5.3) (second line)
is sufficient. This will be assumed throughout.

##  III Direct and transition form factors

The vector and axial-vector couplings to the Dirac fermion in bulk follow from
the Witten diagrams. They involve both the Dirac and Pauli form factors. We
note that neutrino scattering through the charged currents involve solely the
charged left-currents. Here we construct both the direct and transition form
factors needed for the vector and axial-vector currents and compare the direct
ones to the most current data on the proton and neutron. We will use these
form factors to construct the s-channel contributions for neutrino DIS
scattering on nucleon targets.

###  III.1 Direct vector and axial form factors

The direct vector and axial form factors for the proton and neutron
$V,A+N(p)\rightarrow N(p^{\prime})$, can be extracted from the boundary to
bulk three point functions with pertinent LSZ reduction using

$\displaystyle{\cal C}^{(0,3)}_{NN}(p,p^{\prime},q)=\lim_{p^{\prime
2},p^{2}\to m_{N}^{2}}(p^{\prime 2}-m_{N}^{2})(p^{2}-m_{N}^{2})\,\int
d^{4}xd^{4}y\,e^{i(p^{\prime}\cdot x-p\cdot y-q\cdot
z)}\,\left<0\bigg{|}T\bigg{(}{\cal
O}_{N}(x)\,\tilde{J}_{V,A}^{(0,3)\mu}(z)\,{\cal
O}_{N}(y)\bigg{)}\bigg{|}0\right>$

through the ratio

$\displaystyle{W}_{V,A}^{(0,3)\mu}(Q^{2})=\left<N(p^{\prime})\left|\tilde{J}_{V,A}^{(0,3)\mu}(0)\right|N(p)\right>=\frac{{\cal
C}^{(0,3)}_{NN}(p,p^{\prime},q)}{F_{N}(p^{\prime})F_{N}(p)}$

for the chargeless form factor, and similarly for the charged currents

$\displaystyle{W}_{V,A}^{\pm\mu}(Q^{2})=\left<N(p^{\prime})\left|\frac{1}{\sqrt{2}}\tilde{J}_{V,A}^{\pm\mu}(0)\,\right|N(p)\right>\,$

and the electomagnetic currents

$\displaystyle{W}_{EM}^{\mu}(Q^{2})=\left<N(p^{\prime})\left|\tilde{J}_{EM}^{\mu}(0)\right|N(p)\right>=\left<N(p^{\prime})\left|\frac{1}{3}\tilde{J}_{V}^{0\mu}(0)+\tilde{J}_{V}^{3\mu}(0)\right|N(p)\right>\,$
(III.47)

with $q^{2}=(p^{\prime}-p)^{2}=-Q^{2}$. Here $N(p)$ refers to the U(2) proton-
neutron doublet

$\displaystyle N(p)\equiv\begin{pmatrix}N_{p}(p)\\\ N_{n}(p)\end{pmatrix}\,,$
(III.48)

and the baryonic decay constant $F_{N}(p)$ is canonically defined as

$\displaystyle\left<0\left|{\cal
O}_{N}(x)\right|N(p)\right>=F_{N}(p)e^{-ip\cdot x}$ (III.49)

modulo the spin-flavor structure of the nucleon source ${\cal O}_{N}$. These
definitions are commensurate with the lattice definitions for the three point
functions and form factors Alexandrou:2017hac , with the baryonic decay
constants defined and evaluated in Tsutsui:2004qc  (and references therein).
In our case they are tied to the bulk wave functions and given in (B.2-B.176).

The chargeless currents $\tilde{J}_{V,A}^{(0,3)\mu}$ at the boundary are
identified with the quark (partonic) currents, with the quarks in the
fundamental representation of U$(N_{f}=2)$. They are sourced by the dual bulk
vector fields $V_{\mu}^{0}(Q,z\rightarrow 0)$ and $V_{\mu}^{3}(Q,z\rightarrow
0)$ at the boundary, respectively. Similarly, the dual bulk axial vector
fields $A_{\mu}^{(0,3)}(Q,z\rightarrow 0)$ and
$\frac{1}{2}\times\frac{1}{\sqrt{2}}\times A_{\mu}^{\pm}(Q,z\rightarrow 0)$ at
the boundary, are the dual of the quark currents $\tilde{J}_{A}^{(0,3)\mu}(0)$
and $2\tilde{J}_{A}^{\pm\mu}(0)$, respectively.

We now proceed to evaluate the Abelian part,
$U(1)_{V,A}^{\tilde{\alpha}}\subset U(2)$ of the Dirac and Pauli contributions
to the direct vector or axial form factors of the proton and neutron (III.1).
We will give a detailed account of the Dirac contribution to the direct parts
of the ectromagnetic and axial-vector currents, setting up this way the
various definitions and normalizations. The Pauli contributions will follow a
similar reasoning and will be only quoted.

####  III.1.1 Direct vector form factor: Dirac

The Dirac contribution to the direct part of the electromagnetic current can
be extracted from the bulk Dirac part of the action in (II.4) in the soft wall
model

$\displaystyle S_{Dirac}^{EM}[i,X]=\frac{1}{2g_{5}^{2}}\,\int
dzd^{4}y\sqrt{g}e^{-\tilde{\kappa}^{2}z^{2}}\Big{(}\frac{1}{3}V_{N}^{0}J_{V}^{0N}+V_{N}^{3}J_{V}^{3N}\Big{)}$
(III.50)

or more explicitly

$\displaystyle S_{Dirac}^{EM}[i,X]=$ $\displaystyle\frac{1}{2g_{5}^{2}}\,\int
dzd^{4}y\sqrt{g}e^{-\tilde{\kappa}^{2}z^{2}}\,\frac{z}{R}$
$\displaystyle\times\bigg{(}\bar{\Psi}_{1X}\gamma^{N}\Big{(}\frac{1}{3}V_{N}^{0}T^{0}+V_{N}^{3}T^{3}\Big{)}\Psi_{1i}+\bar{\Psi}_{2X}\,\gamma^{N}\Big{(}\frac{1}{3}V_{N}^{0}T^{0}+V_{N}^{3}T^{3}\Big{)}\Psi_{2i}\bigg{)}\,,$
$\displaystyle=$ $\displaystyle(2\pi)^{4}\delta^{4}(P_{X}-p-q)\times
F_{X}(P_{X})\times F_{N}(p)$ $\displaystyle\times\,\frac{1}{2g_{5}^{2}}\times
2g_{5}^{2}\times
e_{nucleon}\times\bar{u}_{s_{X}}(P_{X})\not{\epsilon}(q)u_{s_{i}}(p)\times\frac{1}{2}\Big{[}{\cal
I}_{L}(n_{X},Q^{2})+{\cal I}_{R}(n_{X},Q^{2})\Big{]}\,,$

following the conventions for the interaction action used in BRAGA . Here
$\epsilon^{\mu}(q)$ is the polarization of the EM probe. In the last equality
in (III.1.1) we substituted the bulk gauge fields by (LABEL:Gauge), and the
bulk fermionic currents in terms of the fermionic fields (B.176). The charge
assignments are $e_{nucleon}=1$ for the proton, and $e_{nucleon}=0$ for the
neutron. We have also defined

$\displaystyle{\cal I}_{L}(n_{X},Q^{2})\equiv{\cal
I}(M+5/2,n_{X},Q^{2})\qquad{\cal I}_{R}(n_{x},Q^{2})\equiv{\cal
I}(M+3/2,n_{X},Q^{2})$ (III.52)

with ($w=\tilde{\kappa}^{2}z^{2}$)

$\displaystyle{\cal I}({\bar{m}},n_{X},Q^{2})=$ $\displaystyle
C({\bar{m}},n_{X})\,\Gamma\bigg{(}1+\frac{Q^{2}}{4\tilde{\kappa}^{2}}\bigg{)}\,\int_{0}^{\infty}dww^{{\bar{m}}-1}e^{-w}{\cal
U}(1+\frac{Q^{2}}{4\tilde{\kappa}^{2}};2;w)\,L_{n_{X}}^{{\bar{m}}-2}(w)$
$\displaystyle=$
$\displaystyle\frac{\Gamma\left(\bar{m}\right)\bigg{(}{\frac{\Gamma\left(\bar{m}-1\right)\Gamma\left(n_{X}+\bar{m}-1\right)}{\Gamma(n_{X}+1)}}\bigg{)}^{\frac{1}{2}}}{\Gamma\left(\bar{m}-1\right)}\frac{\frac{Q^{2}}{4\tilde{\kappa}^{2}}{\Gamma\left(\frac{Q^{2}}{4\tilde{\kappa}^{2}}+n_{X}\right)}}{\Gamma\left(\frac{Q^{2}}{4\tilde{\kappa}^{2}}+n_{X}+\bar{m}\right)}$
(III.53)

and the normalization

$C({\bar{m}},n_{X})=\bigg{(}{\frac{\Gamma(n_{X}+1)}{\Gamma({\bar{m}}-1)\Gamma(n_{X}+{\bar{m}}-1)}}\bigg{)}^{\frac{1}{2}}\,,$
(III.54)

Here we have set $Q^{2}=-q^{2}$ (space-like), and used the final state mass
shell condition

$\displaystyle
P_{X}^{2}=(p+q)^{2}=M_{0}^{2}+Q^{2}\bigg{(}\frac{1}{x}-1\bigg{)}\equiv
M_{X}^{2}=4{\tilde{\kappa}}^{2}\bigg{(}n_{X}+M+\frac{1}{2}\bigg{)}$ (III.55)

to identify $n_{X}=Q^{2}(1/x-1)/4{\tilde{\kappa}}^{2}$, with
$p^{2}=M_{0}^{2}=4{\tilde{\kappa}}^{2}(M+\frac{1}{2})=8\tilde{\kappa}^{2}$ for
the initial nucleon state.

The Dirac part of the electromagnetic current (III.47) can be extracted from
(III.1.1) using

$\displaystyle{W}_{EM(Dirac)}^{\mu}(Q^{2})=\bar{u}_{s^{\prime}}(p^{\prime})\gamma^{\mu}u_{s}(p)\times
F_{1}^{EM(Dirac)}(Q)\equiv\frac{1}{F_{N}(p^{\prime})F_{N}(p)}\frac{\delta
S_{Dirac}^{EM}}{\delta\epsilon_{\mu}(q)}+\mathcal{O}(N_{c}^{-2})$ (III.56)

which amounts to the Dirac or minimal contribution to the electromagnetic form
factor ($Q^{2}<0$)

$\displaystyle F_{1}^{EM(Dirac)}(Q^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2g_{5}^{2}}\times 2g_{5}^{2}\times
e_{nucleon}\times\frac{1}{2}\Big{[}{\cal I}_{L}(n_{X}=0,Q^{2})+{\cal
I}_{R}(n_{X}=0,Q^{2})\Big{]}+\mathcal{O}(N_{c}^{-2}),$ (III.57)

or more explicitly in the soft wall model

$\displaystyle F_{1}^{EM(Dirac)}(Q)$ $\displaystyle=$ $\displaystyle
e_{nucleon}\bigg{(}\frac{1}{(\frac{Q^{2}}{4\tilde{\kappa}^{2}}+1)(\frac{Q^{2}}{4\tilde{\kappa}^{2}}+2)}+\frac{3}{(\frac{Q^{2}}{4\tilde{\kappa}^{2}}+1)(\frac{Q^{2}}{4\tilde{\kappa}^{2}}+2)(\frac{Q^{2}}{4\tilde{\kappa}^{2}}+3)}\bigg{)}+\mathcal{O}(N_{c}^{-2})$
(III.58) $\displaystyle=$ $\displaystyle
e_{nucleon}\bigg{(}\frac{m_{0}^{4}}{(Q^{2}+m_{0}^{2})(Q^{2}+m_{1}^{2})}+\frac{3m_{0}^{6}}{(Q^{2}+m_{0}^{2})(Q^{2}+m_{1}^{2})(Q^{2}+m_{2}^{2})}\bigg{)}+\mathcal{O}(N_{c}^{-2})\,.$

using the soft wall rho meson trajectory $m_{n}^{2}=4\tilde{\kappa}^{2}(n+1)$.
Note that normalizing $F_{1}^{EM(Dirac)}(0)=1$ for the proton, fixes
$1+\mathcal{O}(N_{c}^{-2})=1$. In other words, the $1/N_{c}$ corrections to
the EM form factors must vanish at $Q=0$ due to the conservation of the
electromagnetic current. Also note that (III.58) asymptotes a dipole form.

####  III.1.2 Direct axial form factor: Dirac

The direct axial form factors are derived using the same reasoning, with
${W}_{A}^{\tilde{\alpha}\mu}(Q^{2})$ (III.1) following from the pertinent
variation of the bulk Dirac action

$\displaystyle S_{Dirac}^{(0,3)Axial}[i,X]=$
$\displaystyle\frac{1}{2g_{5}^{2}}\,\int
dzd^{4}y\sqrt{g}e^{-\tilde{\kappa}^{2}z^{2}}A_{N}^{(0,3)}J_{A}^{(0,3)N}$
$\displaystyle S_{Dirac}^{\pm Axial}[i,X]=$
$\displaystyle\frac{1}{2g_{5}^{2}}\frac{1}{\sqrt{2}}\int
dzd^{4}y\sqrt{g}e^{-\tilde{\kappa}^{2}z^{2}}A_{N}^{\pm}J_{A}^{\mp N}$ (III.59)

or more explicitly for the chargeless component,

$\displaystyle S_{Dirac}^{(0,3)Axial}[i,X]=$
$\displaystyle\frac{1}{2g_{5}^{2}}\,\int
dzd^{4}y\sqrt{g}e^{-\tilde{\kappa}^{2}z^{2}}\,\frac{z}{R}\bigg{(}\bar{\Psi}_{1X}\,\gamma^{N}\,A_{N}^{(0,3)}T^{(0,3)}\,\Psi_{1i}-\bar{\Psi}_{2X}\gamma^{N}\,A_{N}^{(0,3)}T^{(0,3)}\,\Psi_{2i}\bigg{)}$
$\displaystyle\approx$ $\displaystyle(2\pi)^{4}\delta^{4}(P_{X}-p-q)\,\times
F_{X}(P_{X})\times F_{N}(p)$ $\displaystyle\times\,\frac{1}{2g_{5}^{2}}\times
2g_{5}^{2}\times
g_{Anucleon}^{(0,3)}\times\bar{u}_{s_{X}}(P_{X})\not{\epsilon}(q)\gamma^{5}u_{s_{i}}(p)\times\frac{1}{2}\Big{[}{\cal
I}_{R}(n_{X},\tilde{Q}^{2})-{\cal I}_{L}(n_{X},\tilde{Q}^{2})\Big{]}\,,$

where the substitution (II.32) was used in the second line, i.e.

$\displaystyle\tilde{Q}^{2}=Q^{2}+\frac{1}{2}m_{q}^{2}\rightarrow
Q^{2}+\tilde{m}_{0}^{2}-m_{0}^{2}$ (III.61)

Similarly, for the charged components

$\displaystyle S_{Dirac}^{\pm Axial}[i,X]\equiv$
$\displaystyle\frac{1}{2g_{5}^{2}}\times\frac{1}{\sqrt{2}}\int
dzd^{4}y\sqrt{g}e^{-\kappa^{2}z^{2}}\,\frac{z}{R}\bigg{(}\bar{\Psi}^{p/n}_{1X}\,\gamma^{N}\,A_{N}^{\pm}\,\Psi^{n/p}_{1i}-\bar{\Psi}^{p/n}_{2X}\gamma^{N}\,A_{N}^{\pm}\,\Psi^{n/p}_{2i}\bigg{)}\,,$
$\displaystyle=$ $\displaystyle(2\pi)^{4}\delta^{4}(P_{X}-p-q)\times
F_{X}(P_{X})\times F_{N}(p)$ $\displaystyle\times\frac{1}{2g_{5}^{2}}\times
2g_{5}^{2}\,\times\frac{1}{\sqrt{2}}\times
e_{Wnucleon}^{\pm}\times\bar{u}_{s_{X}}(P_{X})\not{\epsilon}^{\pm}(q)\gamma^{5}u_{s_{i}}(p)\times\frac{1}{2}\Big{[}{\cal
I}_{R}(n_{X},\tilde{Q}^{2})-{\cal I}_{L}(n_{X},\tilde{Q}^{2})\Big{]}\,,$

Here $\epsilon^{\pm}_{\mu}(q)$ is the polarization of the charged vectors.
Following the normalization of the flavor generators $T^{a}$ in (II.1), the
axial and electroweak charges are

$\displaystyle g_{Anucleon}^{0}=g_{Aproton}^{0}=g_{Aneutron}^{0}=\frac{3}{2}$
$\displaystyle g_{Anucleon}^{3}=g_{Aproton}^{3}=\frac{1}{2}$ $\displaystyle
g_{Anucleon}^{3}=g_{Aneutron}^{3}=-\frac{1}{2}$ $\displaystyle
e_{Wnucleon}^{\pm}=e_{Wproton}^{\pm}=e_{Wneutron}^{\pm}=1$ (III.63)

Finally, the isoscalar axial form factor can be extracted from (III.1.2) using

$\displaystyle{\cal W}_{A(Dirac)}^{\mu}(Q^{2})\equiv{\cal
W}_{A}^{0\mu}(Q^{2})=\bar{u}_{s^{\prime}}(p^{\prime})\gamma^{\mu}\gamma^{5}u_{s}(p)\times
F_{1}^{0A(Dirac)}(Q)=\frac{1}{F_{N}(p^{\prime})F_{N}(p)}\frac{\delta
S_{Dirac}^{0Axial}}{\delta\epsilon_{\mu}(q)}+\mathcal{O}(N_{c}^{-2})$ (III.64)

which yields the minimal Dirac contribution to the isoscalar axial form factor
as

$\displaystyle F_{1}^{0A(Dirac)}(Q)$ $\displaystyle\approx$
$\displaystyle\frac{1}{2g_{5}^{2}}\times 2g_{5}^{2}\times
g_{Anucleon}^{0}\times\frac{1}{2}\Big{[}{\cal
I}_{R}(n_{X}=0,\tilde{Q}^{2})-{\cal
I}_{L}(n_{X}=0,\tilde{Q}^{2})\Big{]}+\mathcal{O}(N_{c}^{-2})\,.$ (III.65)

More explicitly, in the soft wall model we have

$\displaystyle F_{1}^{0A(Dirac)}(Q)\approx
g_{Anucleon}^{0}\bigg{(}\frac{1}{(\frac{Q^{2}+\tilde{m}_{0}^{2}}{4\tilde{\kappa}^{2}})(\frac{Q^{2}+\tilde{m}_{0}^{2}}{4\tilde{\kappa}^{2}}+1)}-\frac{3}{(\frac{Q^{2}+\tilde{m}_{0}^{2}}{4\tilde{\kappa}^{2}})(\frac{Q^{2}+\tilde{m}_{0}^{2}}{4\tilde{\kappa}^{2}}+1)(\frac{Q^{2}+\tilde{m}_{0}^{2}}{4\tilde{\kappa}^{2}}+2)}\bigg{)}+\mathcal{O}(N_{c}^{-2})$

following (III.58). (III.1.2) reduces to a dipole form factor asymptotically.
Recall that the substitution (III.61) in the bulk-to-boundary axial-vector
propagator is justified for $Qz_{0}\gg 1$, as we noted in (II.5.3). For
$Qz_{0}\rightarrow 0$, (III.65-III.1.2) give

$\displaystyle
g_{A}^{3(Dirac)}=\frac{1}{3}g_{A}^{0(Dirac)}\approx\bigg{[}\frac{\tilde{m}_{0}^{2}}{m_{0}^{2}}-1\bigg{]}\bigg{[}\bigg{(}\frac{\tilde{m}_{0}^{2}}{m_{0}^{2}}\bigg{)}\bigg{(}\frac{\tilde{m}_{0}^{2}}{m_{0}^{2}}+1\bigg{)}\bigg{(}\frac{\tilde{m}_{0}^{2}}{m_{0}^{2}}+2\bigg{)}\bigg{]}^{-1}\rightarrow\frac{1}{6}\bigg{[}\frac{\tilde{m}_{0}^{2}}{m_{0}^{2}}-1\bigg{]}+{\cal
O}((\sigma z^{3}_{0})^{3})$ (III.67)

with the rightmost result corresponding to the leading perturbative correction
as in (II.5.1) for $\sigma z^{3}_{0}\ll 1$. Using the soft wall parameters in
KKSS  (model A) yields $m_{0}=0.775$ GeV and $\tilde{m}_{0}=1.363$ GeV,
resulting in a large Dirac leading perturbative contribution to the isovector
axial charge of the nucleon $g_{A}^{3(Dirac)}\approx 1/3$. The near-boundary
approximation is not justified in this limit, with the exact but numerical
bulk-to-boundary axial-vector propagator required.

####  III.1.3 Pion pole

Finally, we note that the axial-vector form factor is characterized by two
invariant form factors $F_{A}(q^{2})$ and $H_{A}(q^{2})$ in general, which are
defined as

$\displaystyle\left<p_{1}\left|\tilde{J}_{A}^{\mu
a}(0)\right|p_{2}\right>=\overline{u}(p_{1})\left(\gamma^{\mu}\gamma^{5}F_{A}(q^{2})+q^{\mu}\gamma^{5}H_{A}(q^{2})\right)T^{a}\,u(p_{1})$
(III.68)

with $q=p_{2}-p_{1}$. In the chiral limit, the two form factors are tied by
the conservation of the axial-vector current
$H_{A}(q^{2})=-2m_{N}F_{A}(q^{2})/q^{2}$ with $H_{A}(q^{2}\approx
0)\approx-2m_{N}g_{A}/q^{2}$ exhibiting the pion pole. The absence of this
contribution in (III.2.2) can be traced back to the $A_{5}$ field which we
have ignored. As we noted above, the $A_{5}$ mixes with the tachyon $X$. When
taken into proper consideration, this mixing after gauge fixing locks $A_{5}$
with the phase $\Pi$ of the tachyon, i.e. $X=X_{0}e^{i\Pi}$, which is
identified with the pion field. Careful considerations in bulk yield precisely
$H_{A}(q^{2})$ as expected from current conservation in the chiral limit
ABIDINK . Here we can just reinstate this contribution by inspection, with the
full pion pole $q^{2}\rightarrow q^{2}-m_{\pi}^{2}$.

###  III.2 Transition form factors

In DIS scattering of neutrinos off nucleons, we will need the transition form
factors of the left chargeless and charged currents. For that, we define the
left vector transition form factor for the process $V,L+p\rightarrow X$

$\displaystyle{\cal
W}_{V,L}^{\mu}(Q^{2})=\left<X\left|\tilde{J}_{V,L}^{\mu}(0)\right|p\right>$

with $Q^{2}=(P_{X}-p)^{2}$. We first evaluate the $U(1)_{L}$ Dirac and Pauli
contributions to the transition vector form factor (III.2) and then generalize
them to the corresponding $U(1)^{EM}_{V},U(1)^{\pm}_{L}\subset U(2)_{L}$
contributions.

####  III.2.1 $U(1)_{L}$ contributions

The minimal Dirac interaction term between the bulk $U(1)_{L}$ gauge field
$L_{N}$ and the bulk fermionic field $\Psi_{1}$ in the action is

$\displaystyle S_{Dirac}^{L}[i,X]=\frac{1}{2g_{5}^{2}}\,\int
dzd^{4}y\sqrt{g}e^{-\tilde{\kappa}^{2}z^{2}}\,\frac{z}{R}\,L_{N}\bar{\Psi}_{1X}\,\gamma^{N}\,\Psi_{1i}\,.$

following the canonical normalizations

$\displaystyle\Psi\rightarrow\sqrt{2g_{5}^{2}}\,\Psi\qquad L_{N}\rightarrow
g_{5}L_{N}$ (III.71)

which makes the couplings and power counting manifest in Witten diagrams. The
explicit form of (III.2.1) in terms of the bulk fermions is

$\displaystyle\tilde{S}_{Dirac}^{L}[i,X]=$
$\displaystyle(2\pi)^{4}\delta^{4}(P_{X}-p-q)\times F_{X}(P_{X})\times
F_{N}(p)$ $\displaystyle\times\frac{1}{2g^{2}_{5}}\times
2g_{5}^{2}\,\Big{[}\bar{u}_{s_{X}}(P_{X})\not{\epsilon}(q)\Big{(}\frac{1-\gamma^{5}}{2}\Big{)}u_{s_{i}}(p){\cal
I}_{L}(n_{x},Q^{2})+\,\bar{u}_{s_{X}}(P_{X})\not{\epsilon}(q)\Big{(}\frac{1+\gamma^{5}}{2}\Big{)}u_{s_{i}}(p){\cal
I}_{R}(n_{x},Q^{2})\Big{]}\,.$

The Dirac contribution to the $U(1)_{L}$ transition form factor reads
explicitly

$\displaystyle{\cal W}_{L,Dirac}^{\mu}(q^{2})=$
$\displaystyle\frac{1}{F_{X}(P_{X})F_{N}(p)}\frac{\delta
S_{Dirac}^{L}}{\delta\epsilon_{\mu}(q)}+\mathcal{O}(N_{c}^{-2})$
$\displaystyle=$
$\displaystyle\frac{1}{F_{X}(P_{X})F_{N}(p)}\times\frac{1}{2g_{5}^{2}}\times\frac{1}{g_{5}}\times\frac{\delta\tilde{S}_{Dirac}^{L}}{\delta\epsilon_{\mu}(q)}+\mathcal{O}(N_{c}^{-2})$
$\displaystyle=$ $\displaystyle\frac{1}{2g_{5}^{2}}\times
2g_{5}^{2}\times\,\Big{[}\bar{u}_{s_{X}}(P_{X})\gamma^{\mu}\Big{(}\frac{1-\gamma^{5}}{2}\Big{)}u_{s_{i}}(p){\cal
I}_{L}(n_{x},Q^{2})+\,\bar{u}_{s_{X}}(P_{X})\gamma^{\mu}\Big{(}\frac{1+\gamma^{5}}{2}\Big{)}u_{s_{i}}(p){\cal
I}_{R}(n_{x},Q^{2})\Big{]}+\mathcal{O}(N_{c}^{-2})$

####  III.2.2 $U(1)^{\pm}_{L}$ contributions

For neutrino and anti-neuutrino scattering, the pertinent left-handed hadronic
transition form factor is

$\displaystyle{\cal W}_{\pm}^{\mu}(Q^{2})=$
$\displaystyle\left<N(P_{X})\left|\tilde{J}^{\pm\mu}_{L}(0)\right|N(p)\right>$
(III.74)

which can be evaluated using the Dirac and Pauli contributions to the
fermionic action. To illustrate the normalizations, consider for simplicity
the contribution due to the Dirac part, which yields the interaction vertex

$\displaystyle S_{Dirac}^{\pm
L}[i,X]=\frac{1}{2g_{5}^{2}}\times\frac{1}{\sqrt{2}}\int
dzd^{4}y\sqrt{g}e^{-\tilde{\kappa}^{2}z^{2}}L_{N}^{\pm}J_{L}^{\mp
N}=\frac{1}{2g_{5}^{2}}\times\frac{1}{\sqrt{2}}\int
dzd^{4}y\sqrt{g}e^{-\tilde{\kappa}^{2}z^{2}}\,\frac{z}{R}\bar{\Psi}_{1X}^{p/n}\,\gamma^{N}\,L_{N}^{\pm}\,\Psi_{1i}^{n/p}$
$\displaystyle=(2\pi)^{4}\delta^{4}(P_{X}-p-q)\times\,\frac{1}{2g_{5}^{2}}\times
2g_{5}^{2}\times F_{X}(P_{X})F_{p/n}(p)$
$\displaystyle\times\frac{1}{\sqrt{2}}\times
e_{Wnucleon}^{\pm}\times\Big{[}\bar{u}_{s_{X}}(P_{X})\not{\epsilon}^{\pm}(q)\Big{(}\frac{1-\gamma^{5}}{2}\Big{)}u_{s_{i}}(p){\cal
I}_{L}(n_{x},Q_{L}^{2})+\,\bar{u}_{s_{X}}(P_{X})\not{\epsilon}^{\pm}(q)\Big{(}\frac{1+\gamma^{5}}{2}\Big{)}u_{s_{i}}(p){\cal
I}_{R}(n_{x},Q_{L}^{2})\Big{]}\,,$

with the corresponding transition form factor

$\displaystyle{\cal
W}_{L(Dirac)}^{\pm\mu}(Q^{2})=\frac{1}{F_{X}(P_{X})F_{p/n}(p)}\frac{1}{\frac{1}{2\sqrt{2}}}\frac{\delta
S_{Dirac}^{\pm L}}{\delta\epsilon^{\pm}_{\mu}(q)}+\mathcal{O}(N_{c}^{-2})$
$\displaystyle=\frac{1}{2g_{5}^{2}}\times 2g_{5}^{2}\times e_{Wnucleon}^{\pm}$
$\displaystyle\times
2\Big{[}\bar{u}_{s_{X}}(P_{X})\gamma^{\mu}\Big{(}\frac{1-\gamma^{5}}{2}\Big{)}u_{s_{i}}(p){\cal
I}_{L}(n_{x},Q_{L}^{2})+\,\bar{u}_{s_{X}}(P_{X})\gamma^{\mu}\Big{(}\frac{1+\gamma^{5}}{2}\Big{)}u_{s_{i}}(p){\cal
I}_{R}(n_{x},Q_{L}^{2})\Big{]}+\mathcal{O}(N_{c}^{-2})\,$
$\displaystyle=\frac{1}{2g_{5}^{2}}\times 2g_{5}^{2}\times e_{Wnucleon}^{\pm}$
$\displaystyle\times\bigg{(}\bar{u}_{s_{X}}(P_{X})\gamma^{\mu}u_{s_{i}}(p)\Big{[}{\cal
I}_{R}(n_{x},Q_{L}^{2})+{\cal
I}_{L}(n_{x},Q_{L}^{2})\Big{]}+\bar{u}_{s_{X}}(P_{X})\gamma^{\mu}\gamma^{5}u_{s_{i}}(p)\Big{[}{\cal
I}_{R}(n_{x},Q_{L}^{2})-{\cal
I}_{L}(n_{x},Q_{L}^{2})\Big{]}\bigg{)}+\mathcal{O}(N_{c}^{-2})\,.$

####  III.2.3 Charged transition form factor: Pauli

The minimal Dirac bulk interaction does not contribute to the neutron
transition form factor. This contribution arises from the $U(1)_{L}$ part of
the Pauli interaction in bulk CARLSON

$\displaystyle S_{Pauli}^{L,R}[i,X]=\frac{\eta}{2g_{5}^{2}}\int
dzd^{4}y\sqrt{g}e^{-\tilde{\kappa}^{2}z^{2}}\,\frac{z^{2}}{R^{2}}\bigg{(}\bar{\Psi}_{1X}\,\sigma^{MN}\,L_{MN}\,\Psi_{1i}\,-\bar{\Psi}_{2X}\,\sigma^{MN}\,R_{MN}\Psi_{2i}\bigg{)}$
(III.77)

The inclusion of this interaction is straightforward but tedious, with the
final result for the charged currents

$\displaystyle{\cal W}_{L}^{\pm\mu}(Q^{2})$ $\displaystyle=e_{Wnucleon}^{\pm}$
(III.78)
$\displaystyle\times\bigg{(}\bar{u}_{s_{X}}(P_{X})\gamma^{\mu}u_{s_{i}}(p)\Big{[}{\cal
I}_{R}(n_{x},Q_{L}^{2})+{\cal I}_{L}(n_{x},Q_{L}^{2})\Big{]}$
$\displaystyle\qquad+\bar{u}_{s_{X}}(P_{X})\gamma^{\mu}\gamma^{5}u_{s_{i}}(p)\Big{[}{\cal
I}_{R}(n_{x},Q_{L}^{2})-{\cal I}_{L}(n_{x},Q_{L}^{2}\Big{]}\bigg{)}$
$\displaystyle+\eta^{\pm}\bigg{(}\bar{u}_{s_{X}}(P_{X})\gamma^{\mu}u_{s_{i}}(p)\Big{[}{\cal
J}_{R}(n_{x},Q_{L}^{2})+{\cal J}_{L}(n_{x},Q_{L}^{2}\Big{]}$
$\displaystyle\qquad+\bar{u}_{s_{X}}(P_{X})\gamma^{\mu}\gamma^{5}u_{s_{i}}(p)\Big{[}{\cal
J}_{R}(n_{x},Q_{L}^{2})-{\cal J}_{L}(n_{x},Q_{L}^{2})\Big{]}\bigg{)}$
$\displaystyle+2\eta^{\pm}\,\bigg{(}\bar{u}_{s_{X}}(P_{X})\sigma^{\mu\nu}iq_{\nu}u_{s_{i}}(p)\Big{[}{\cal
I}_{LR}(n_{x},Q_{L}^{2})+{\cal I}_{RL}(n_{x},Q_{L}^{2})\Big{]}$
$\displaystyle\qquad+\bar{u}_{s_{X}}(P_{X})\sigma^{\mu\nu}iq_{\nu}\gamma^{5}u_{s_{i}}(p)\Big{[}{\cal
I}_{LR}(n_{x},Q_{L}^{2})-{\cal I}_{RL}(n_{x},Q_{L}^{2})\Big{]}\bigg{)}$
$\displaystyle-\eta^{\pm}\,\bigg{(}\bar{u}_{s_{X}}(P_{X})q^{\mu}\not{q}u_{s_{i}}(p)\Big{[}{\cal
K}_{R}(n_{x},Q_{L}^{2})+{\cal K}_{L}(n_{x},Q_{L}^{2})\Big{]}$
$\displaystyle\qquad+\,\bar{u}_{s_{X}}(P_{X})q^{\mu}\not{q}\gamma^{5}u_{s_{i}}(p)\Big{[}{\cal
K}_{R}(n_{x},Q_{L}^{2})-{\cal
K}_{L}(n_{x},Q_{L}^{2})\Big{]}\bigg{)}+\mathcal{O}(N_{c}^{-2})\,.$

with

$\displaystyle\sigma^{\mu\nu}=2S^{\mu\nu}=\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}]=i(\gamma^{\mu}\gamma^{\nu}-\eta^{\mu\nu})\,,$
(III.79)

and manifest current conservation, i.e. $q_{\mu}{\cal W}^{\pm\mu\nu}=0$ for
on-shell spinors. The integrals

$\displaystyle{\cal J}_{L},{\cal K}_{L}(n_{X},Q^{2})\equiv{\cal
J,K}(M+5/2,n_{X},Q^{2})\qquad{\cal J}_{R},{\cal K}_{R}(n_{X},Q^{2})\equiv{\cal
J,K}(M+3/2,n_{X},Q^{2})$ (III.80)

are related to the general integrals of the type (III.1.1), namely

$\displaystyle{\cal J}({\bar{m}},n_{X},Q^{2})$ $\displaystyle=$ $\displaystyle
C({\bar{m}},n_{X})\,\Gamma(1+a)\,\int_{0}^{\infty}dww^{{\bar{m}}-1}e^{-w}\bigg{(}w{\cal
U}(1+a;2;w)\bigg{)}^{\prime}\,L_{n_{X}}^{{\bar{m}}-2}(w)$ $\displaystyle=$
$\displaystyle
C({\bar{m}},n_{X})\,\Gamma(1+a)\,\int_{0}^{\infty}dww^{{\bar{m}}-1}e^{-w}\bigg{(}-a\,{\cal
U}(1+a;1;w)\bigg{)}\,L_{n_{X}}^{{\bar{m}}-2}(w)$ $\displaystyle=$
$\displaystyle
C({\bar{m}},n_{X})\frac{\Gamma({\bar{m}})\Gamma({\bar{m}}+n_{X}-1)(n_{X}-a({\bar{m}}-1))a\Gamma(a+n_{X})}{\Gamma(n_{X}+1)\Gamma\left(a+{\bar{m}}+n_{X}+1\right)}\,$
$\displaystyle=$
$\displaystyle\frac{-a({\bar{m}}-1)+n_{X}}{a+{\bar{m}}+n_{X}}\times{\cal
I}({\bar{m}},n_{X},Q^{2})\,,$ $\displaystyle{\cal K}({\bar{m}},n_{X},Q^{2})$
$\displaystyle=$ $\displaystyle\frac{1}{\tilde{\kappa}^{2}}\times{\cal
J}({\bar{m}},n_{X},Q^{2})\,,$

with $a={Q^{2}}/{4\tilde{\kappa}^{2}}$, and

$\displaystyle{\cal I}_{LR}(n_{X},Q^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{\tilde{\kappa}}\times
C(m+5/2,n_{X})\,\Gamma(1+a)\,\int_{0}^{\infty}dww^{m+1}e^{-w}{\cal
U}(1+\frac{Q^{2}}{4\tilde{\kappa}^{2}};2;w)\,L_{n_{X}}^{m+\frac{1}{2}}(w)$
$\displaystyle{\cal I}_{RL}(n_{X},Q^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{\tilde{\kappa}}\times
C(m+3/2,n_{X})\,\Gamma(1+a)\,\int_{0}^{\infty}dww^{m+2}e^{-w}{\cal
U}(1+\frac{Q^{2}}{4\tilde{\kappa}^{2}};2;w)\,L_{n_{X}}^{m-\frac{1}{2}}(w)\,.$

We have made use of the recursive relation between the confluent
hypergeometric functions

$w\;\partial_{w}\;{\cal U}(a,b,w)=(1-b)\;{\cal U}(a,b,w)-(1+a-b)\;{\cal
U}(a,b-1,w)\,,$ (III.83)

together with their integral representation

$\displaystyle{\cal
U}(a,b,w)=\frac{1}{\Gamma(a)}\int_{0}^{\infty}dt\,t^{a-1}(t+1)^{-a+b-1}e^{-tw}\,,$
(III.84)

and the property of the Laguerre polynomials

$\displaystyle
L_{n}^{a}(w)=\frac{\Gamma(a+1+n)}{\Gamma(a+1)\Gamma(n+1)}\,_{1}F_{1}\left(-n,a+1,w\right)\,,$
(III.85)

to evaluate the integrals.

####  III.2.4 $U(1)^{EM}_{V}\subset U(2)_{V}$ contributions

The electromagnetic transition vector form factor ${\cal W}_{EM}^{\mu}(Q^{2})$
as defined through

$\displaystyle{\cal
W}_{EM}^{\mu}(Q^{2})=\left<N(P_{X})\left|\tilde{J}_{EM}^{\mu}(0)\right|N(p)\right>=\left<N(P_{X})\left|\frac{1}{3}\tilde{J}_{V}^{0\mu}(0)+\tilde{J}_{V}^{3\mu}(0)\right|N(p)\right>$
(III.86)

can be extracted from (III.1.1) with the inclusion of the Pauli contribution
(III.77), using

$\displaystyle{\cal W}_{EM(Dirac)}^{\mu}(Q^{2})=$
$\displaystyle\frac{1}{F_{X}(P_{X})F_{N}(p)}\frac{\delta
S_{Dirac+Pauli}^{EM}}{\delta\epsilon_{\mu}(q)}+\mathcal{O}(N_{c}^{-2})$

The result is

$\displaystyle\epsilon_{\mu}{\cal W}_{EM}^{\mu}(Q^{2})$
$\displaystyle=\epsilon_{\mu}\bigg{(}e_{nucleon}\
\times\bar{u}_{s_{X}}(P_{X})\gamma^{\mu}u_{s_{i}}(p)\,\frac{1}{2}\,\Big{[}{\cal
I}_{R}(n_{x},Q^{2})+{\cal I}_{L}(n_{x},Q^{2})\Big{]}$ (III.88)
$\displaystyle+\eta^{p/n}\,\bar{u}_{s_{X}}(P_{X})\gamma^{\mu}u_{s_{i}}(p)\,\frac{1}{2}\,\Big{[}{\cal
J}_{R}(n_{x},Q^{2})-{\cal J}_{L}(n_{x},Q^{2})\Big{]}$
$\displaystyle+\eta^{p/n}\,\bar{u}_{s_{X}}(P_{X})\sigma^{\mu\nu}iq_{\nu}u_{s_{i}}(p)\Big{[}{\cal
I}_{LR}(n_{x},Q^{2})-{\cal
I}_{RL}(n_{x},Q^{2})\Big{]}\bigg{)}+\mathcal{O}(N_{c}^{-2})\,.$

####  III.2.5 $U(1)_{A}\subset U(2)_{A}$ contributions

The chargeless axial transition form factor ${\cal W}_{0,3}^{\mu}(Q^{2})$ as
defined through

$\displaystyle{\cal
W}_{A}^{(0,3)\mu}(Q^{2})=\left<N(P_{X})\left|\tilde{J}^{(0,3)\mu}_{A}(0)\right|N(p)\right>$
(III.89)

follows the same reasoning, with the full result including the Dirac and Pauli
contributions

$\displaystyle\epsilon_{\mu}^{(0,3)}{\cal W}_{A}^{(0,3)\mu}(Q^{2})$
$\displaystyle=\epsilon_{\mu}^{(0,3)}\bigg{(}g^{(0,3)}_{Anucleon}\times\bar{u}_{s_{X}}(P_{X})\gamma^{\mu}\gamma^{5}u_{s_{i}}(p)\,\frac{1}{2}\,\Big{[}{\cal
I}_{R}(n_{x},\tilde{Q}^{2})-{\cal I}_{L}(n_{x},\tilde{Q}^{2})\Big{]}$ (III.90)
$\displaystyle+\eta^{(0,3)}\,\bar{u}_{s_{X}}(P_{X})\gamma^{\mu}\gamma^{5}u_{s_{i}}(p)\,\frac{1}{2}\,\Big{[}{\cal
J}_{R}(n_{x},\tilde{Q}^{2})+{\cal J}_{L}(n_{x},\tilde{Q}^{2})\Big{]}$
$\displaystyle+\eta^{(0,3)}\,\bar{u}_{s_{X}}(P_{X})\sigma^{\mu\nu}iq_{\nu}\gamma^{5}u_{s_{i}}(p)\Big{[}{\cal
I}_{LR}(n_{x},\tilde{Q}^{2})+{\cal
I}_{RL}(n_{x},\tilde{Q}^{2})\Big{]}\bigg{)}+\mathcal{O}(N_{c}^{-2})\,.$

##  IV Neutrino and anti-neutrino DIS scattering in QCD

In QCD lepton nucleon scattering follows from the contraction of the leptonic
tensor and hadronic tensor through the exchange of neutral currents carried by
$\gamma,Z$ and charged currents carried by $W^{\pm}$. Some useful insights on
standard neutrino DIS scattering on a nucleon can be found in REVIEW . In this
section we briefly review the key definitions and characteristics of this
scattering as a prelude to the holographic analysis, which will make use of
the transition form factors established above for the s-wave contributions to
DIS.

###  IV.1 Structure functions for $\nu,\overline{\nu}$ scattering

An overall review of neutrino DIS scattering can be found in REVIEW , so we
will be brief in our presentation of the results for our ensuing analysis. For
unpolarized nucleons, the hadronic tensor for neutrino (antineutrino)
scattering can be organized under the strictures of Lorentz symmetry, parity
and current conservation in terms of three invariant structure functions

$\displaystyle W^{\nu,\overline{\nu}}_{\mu\nu}=$
$\displaystyle\frac{1}{4\pi}\int d^{4}x\,e^{iq\cdot
x}\left<P\left|\Big{[}\tilde{J}^{\mp}_{\mu}(x),\tilde{J}^{\pm}_{\nu}(0)\Big{]}\right|P\right>$
$\displaystyle=$
$\displaystyle\left(-\eta_{\mu\nu}+\frac{q_{\mu}q_{\nu}}{q^{2}}\right)F^{W^{\pm}}_{1}(x,q^{2})+\left(P_{\mu}+\frac{q_{\mu}}{2x}\right)\left(P_{\nu}+\frac{q_{\nu}}{2x}\right)\,\frac{2x}{q^{2}}F^{W^{\pm}}_{2}(x,q^{2})\mp
i\epsilon_{\mu\nu\alpha\beta}q^{\alpha}P^{\beta}\,\frac{x}{q^{2}}F^{W^{\pm}}_{3}(x,q^{2})$

$q_{\mu},P_{\mu}$ are the 4-momenta of the virtual current and nucleon
respectively. $x=-q^{2}/2P\cdot q$ is the Bjorken parameter which is
kinematically bounded $0\leq x\leq 1$. $F_{1,2}$ refer to the symmetric
structure functions, while $F_{3}$ to the antisymmetric one. The formers are
parity preserving, while the latter is not for neutrino and antineutrino
probes. In the DIS limit with $Q^{2}=-q^{2}\gg P^{2}$ and $x$ fixed, the
structure functions in QCD obey Bjorken scaling. In this limit, the parity
even structure functions satisfy the Callan-Gross relation $F_{2}=2xF_{1}$. We
note that analyticity allows to relate the hadronic DIS tensor (IV.1) to the
discontinuity of the forward Compton amplitude of a lepton on a nucleon

$\displaystyle 4\pi W^{\nu,\overline{\nu}}_{\mu\nu}=2\pi{\rm
Im}\,T^{\nu,\overline{\nu}}_{\mu\nu}\equiv 2\pi{\rm Im}\,i\int
d^{4}xe^{iq\cdot
x}\left<P\right|T^{*}\left(\tilde{J}^{\mp}_{\mu}(x)\tilde{J}^{\pm}_{\nu}(0)\right)\left|P\right>$
(IV.92)

To calculate the hadronic tensor (IV.1) we use the completeness of the
hadronic spectrum f

$\displaystyle
W^{\nu,\overline{\nu}}_{\mu\nu}=\frac{1}{2}\sum_{s,s_{X}}\sum_{M_{X}}\,\delta\Big{(}M_{X}^{2}-(P+q)^{2}\,\Big{)}\,{\cal
W}^{\nu,\overline{\nu}*}_{\mu}\,{\cal W}^{\nu,\overline{\nu}}_{\nu}\equiv
W^{\nu,\overline{\nu}S}_{\mu\nu}+iW^{\nu,\overline{\nu}A}_{\mu\nu}$

with the transition current matrix element (III.2), for excited states of
squared mass $M_{X}^{2}=Q^{2}(1/x-1)+m_{N}^{2}$, and
$P^{2}=M_{0}^{2}=m_{N}^{2}$. For neutrino scattring the explicit form of the
QCD quark (partonic) currents in (IV.92) are given by

$\displaystyle e\tilde{J}^{\mu}_{EM}=$ $\displaystyle
e\,\overline{q}Q\gamma^{\mu}q$ $\displaystyle e_{W}\times
2\tilde{J}^{\pm\mu}_{L}=$ $\displaystyle
e_{W}\,\overline{q}T^{\pm}\gamma^{\mu}\frac{1}{2}{(1-\gamma^{5})}q$
$\displaystyle e_{W}\tilde{J}^{\mu}_{Z}=$ $\displaystyle
e_{W}\,\tilde{J}^{\mu}_{L}-2e_{W}\,{\rm
sin^{2}\theta_{W}}\tilde{J}^{\mu}_{EM}=e_{W}\,\overline{q}\gamma^{\mu}\frac{1}{2}{(1-\gamma^{5})}q-2e_{W}\,{\rm
sin^{2}\theta_{W}}\,\overline{q}Q\gamma^{\mu}q$

with $e,e_{W}$ the electric and electroweak charges, and ${\rm
sin\theta_{W}}=\frac{e}{e_{W}}$ with the Weinberg angle
$\theta_{W}\approx\frac{\pi}{6}$.

$\displaystyle Q=\begin{pmatrix}\frac{2}{3}&0\\\
0&-\frac{1}{3}\end{pmatrix}\qquad
T^{+}=(T^{-})^{\dagger}=\begin{pmatrix}0&2\\\ 0&0\end{pmatrix}$ (IV.95)

The imaginary part in (IV.92) receives contribution from the neutral
$\gamma\gamma$, $ZZ$, $\gamma Z$ as they mix, and the charged $W^{\mp}W^{\pm}$
as they are conjugate.

###  IV.2 Unpolarized parton distributions through charged currents

Neutrino and anti-neutrino scattering on a hadron through the charged currents
yield very informative information on the parton content of a hadron in the
DIS limit. At weak coupling, the parton model gives a very simple descriptive
of the structure functions in terms of the partonic distribution functions of
the hadron. Assuming two flavors for simplicity and isospin symmetry, the
partonic model for $\nu p\rightarrow l^{-}X$ through $W^{+}$ exchange gives

$\displaystyle F_{1}^{W^{+}p}(x,Q)=$ $\displaystyle d(x,Q)+\overline{u}(x,Q)$
$\displaystyle F_{2}^{W^{+}p}(x,Q)=$ $\displaystyle
2x(d(x,Q)+\overline{u}(x,Q))$ $\displaystyle F_{3}^{W^{+}p}(x,Q)=$
$\displaystyle 2(d(x,Q)-\overline{u}(x,Q))$

while for $\overline{\nu}p\rightarrow l^{+}X$ through $W^{-}$ exchange it
gives

$\displaystyle F_{1}^{W^{-}p}(x,Q)=$ $\displaystyle u(x,Q)+\overline{d}(x,Q)$
$\displaystyle F_{2}^{W^{-}p}(x,Q)=$ $\displaystyle
2x(u(x,Q)+\overline{d}(x,Q))$ $\displaystyle F_{3}^{W^{-}p}(x,Q)=$
$\displaystyle 2(u(x,Q)-\overline{d}(x,Q))$

modulo $e_{W}^{2}$. The corresponding neutron structure functions follow by
isospin symmetry. (IV.2-IV.2) can be inverted to give the unpolarized valence
and sea parton distributions in the proton. We now proceed to evaluate these
unpolarized partonic distributions using the holographic dual of neutrino
scattering.

##  V Neutrino and anti-neutrino DIS in Holography

We now consider neutrino DIS scattering on a nucleon as a Dirac fermion in
bulk using holography, in the double limit of a large number of colors and
strong gauge coupling $\lambda=g^{2}N_{c}$. Anti-neutrino DIS scattering
follows from pertinent re-arrangements. DIS scattering on a nucleon as a bulk
dilatino using holography was first addressed in POL  and later by others
BRAGA ; HATTAX ; KOVENSKY ; Mamo:2019mka . At large-x, DIS scattering using
$U(1)_{V}$ probes follows from the direct and crossed Witten diagrams in bulk,
and at small-x it follows from Pomeron exchange in the form of a Reggeized and
warped close string exchange in bulk. We now review the analysis for a
$U(1)_{V}$ current and then extend it to the electromagnetic vector
$U(1)^{EM}_{V}\subset U(2)_{V}$, and the left-handed $U(1)^{\pm}_{L}\subset
U(2)_{L}$ currents.

###  V.1 Structure functions: Baryonic exchange in the s-channel

For unpolarized $U(1)_{L}$ scattering, the hadronic tensor is

$W_{L}^{\mu\nu}\,=\frac{1}{4\pi}\sum_{s}\,\int d^{4}y\,e^{iq\cdot y}\langle
P,s|\,\Big{[}\tilde{J}_{L}^{\mu}(y),\tilde{J}_{L}^{\nu}(0)\Big{]}\,|P,s\rangle\,,$
(V.98)

with the spectral decomposition for the s-channel contributions

$\displaystyle W_{L,s}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\sum_{s,s_{X}}\sum_{M_{X}}\int\frac{d^{4}P_{X}}{(2\pi)^{4}}\theta(P^{0}_{X})(2\pi)\delta\Big{(}P^{2}_{X}-M_{X}^{2}\,\Big{)}(2\pi)^{4}\delta^{4}(P+q-P_{X})\,\langle
P,s|\tilde{J}_{L}^{\mu}(0)|P_{X},s_{X}\rangle\,\langle
P_{X},\,s_{X}|\tilde{J}_{L}^{\nu}(0)|P,\,s\rangle\,$ (V.99) $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{s,s_{X}}\sum_{M_{X}}\,\delta\Big{(}M_{X}^{2}-(P+q)^{2}\,\Big{)}\langle
P,s|\tilde{J}_{L}^{\mu}(0)|P+q,s_{X}\rangle\,\langle
P+q,\,s_{X}|\tilde{J}_{L}^{\nu}(0)|P,\,s\rangle\,$ (V.100) $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{s,s_{X}}\sum_{M_{X}}\,\delta\Big{(}M_{X}^{2}-(P+q)^{2}\,\Big{)}\,{\cal
W}_{L}^{\mu}\,{\cal W}_{L}^{\nu\,*}\,.$ (V.101)

The t-channel Reggeized contributions will be addressed below.

###  V.2 Dirac contribution

Using the relations

$\displaystyle\epsilon_{\mu}\langle
P_{X}|{\tilde{J}}_{L,Dirac}^{\mu}(q)|P\rangle\,$ $\displaystyle=$
$\displaystyle(2\pi)^{4}\,\delta^{4}(P_{X}-P-q)\,\epsilon_{\mu}\,\langle
P+q|\tilde{J}_{L,Dirac}^{\mu}(0)|P\rangle\,=\,\mathcal{N}_{L}\times\frac{1}{g_{5}}\times\tilde{S}_{int,Dirac}^{L}[i,X]\,$
(V.103) $\displaystyle\epsilon_{\mu}\langle
P|{\tilde{J}}_{L,Dirac}^{\mu}(q)|P_{X}\rangle\,$ $\displaystyle=$
$\displaystyle(2\pi)^{4}\,\delta^{4}(P_{X}-P-q)\,\epsilon_{\mu}\,\langle
P|\tilde{J}_{L,Dirac}^{\mu}(0)|P+q\rangle\,=\,\mathcal{N}_{L}\times\frac{1}{g_{5}}\times\tilde{S}_{int,Dirac}^{L}[X,i]\,,$
(V.105)

we can make explicit the contracted hadronic tensor

$\displaystyle
W^{\mu\nu}_{L,s,Dirac}\,=\,\frac{1}{2}\,\sum_{M_{X}}\delta(M_{X}^{2}-(p+q)^{2})\,\mathcal{N}_{L}^{2}$
$\displaystyle\times\sum_{s_{i}}\sum_{s_{X}}\,\bigg{(}{\cal I}_{L}{\cal
I}_{R}\Big{(}\bar{u}_{s_{X}}\gamma^{\mu}P_{L}u_{s_{i}}\bar{u}_{s_{i}}\gamma^{\nu}P_{R}u_{s_{X}}+\bar{u}_{s_{X}}\gamma^{\mu}P_{R}u_{s_{i}}\bar{u}_{s_{i}}\gamma^{\nu}P_{L}u_{s_{X}}\Big{)}$
$\displaystyle\qquad\qquad\,\,\,+{\cal
I}^{2}_{L}\bar{u}_{s_{X}}\gamma^{\mu}P_{L}u_{s_{i}}\bar{u}_{s_{i}}\gamma^{\nu}P_{L}u_{s_{X}}\,+{\cal
I}^{2}_{R}\bar{u}_{s_{X}}\gamma^{\mu}P_{R}u_{s_{i}}\bar{u}_{s_{i}}\gamma^{\nu}P_{R}u_{s_{X}}\bigg{)}$
(V.107)

The additional normalization constant
$\mathcal{N}_{L}=({1}/{2g_{5}^{2}})\times
2g_{5}^{2}\times\tilde{\mathcal{N}}_{L}$ compensates for the missing higher
spin-j and $\mathcal{O}(g_{5}^{0})$ corrections in the s-channel. Since
$\sum_{s}(u_{s})(p)(\bar{u}_{s})(p)=\not\\!p+M$, then the contracted hadronic
tensor contributions from the s-channel are

$\displaystyle\epsilon_{\mu}\epsilon_{\nu}W_{L,s,Dirac}^{\mu\nu}=\,\sum_{M_{X}}\delta(M_{X}^{2}-(p+q)^{2})\,\,\mathcal{N}_{L}^{2}$
$\displaystyle\times 2\bigg{(}{\cal I}_{R}(n_{x}){\cal
I}_{L}(n_{x})M_{X}M_{0}\,\epsilon\cdot\epsilon+\Big{(}{\cal
I}^{2}_{R}(n_{x})+{\cal
I}^{2}_{L}(n_{x})\Big{)}\Big{(}(p\cdot\epsilon)^{2}-\frac{1}{2}(p^{2}+p\cdot
q)\,\epsilon\cdot\epsilon\Big{)}+\frac{1}{2}\Big{(}{\cal
I}_{R}^{2}(n_{x})-{\cal
I}_{L}^{2}(n_{x})\Big{)}\Big{(}i\epsilon^{\mu\nu\alpha\beta}P_{X\alpha}p_{\beta}\Big{)}\bigg{)}$

By approximating the sum by an integral in a continuous state, i.e.,

$\displaystyle\sum_{M_{X}}\delta(M_{X}^{2}-(p+q)^{2})\approx\frac{1}{4\tilde{\kappa}^{2}}\int
dn\,\delta\Big{(}\frac{s}{4\tilde{\kappa}^{2}}-\frac{M_{n}^{2}}{4\tilde{\kappa}^{2}}\Big{)}=\frac{1}{4\tilde{\kappa}^{2}}\,,$
(V.109)

we have

$\displaystyle\epsilon_{\mu}\epsilon_{\nu}W_{L,s,Dirac}^{\mu\nu}\approx\frac{\mathcal{N}_{L}^{2}}{4\tilde{\kappa}^{2}}$
$\displaystyle\times$ $\displaystyle\bigg{(}2{\cal I}_{L}(n_{x}){\cal
I}_{R}(n_{x})M_{X}M_{0}\,\epsilon\cdot\epsilon+\Big{(}{\cal
I}^{2}_{R}(n_{x})+{\cal
I}^{2}_{L}(n_{x})\Big{)}\Big{(}2(p\cdot\epsilon)^{2}-(p^{2}+p\cdot
q)\,\epsilon\cdot\epsilon\Big{)}+\Big{(}{\cal I}_{R}^{2}(n_{x})-{\cal
I}_{L}^{2}(n_{x})\Big{)}\Big{(}i\epsilon^{\mu\nu\alpha\beta}P_{X\alpha}p_{\beta}\Big{)}\bigg{)}$

The same contraction applied to the canonical hadronic tensor decomposition
(IV.1), for neutrino, with a transverse polarization $\epsilon\cdot q=0$
(which we have already used in deriving (V.2), yields

$\epsilon_{\mu}\epsilon_{\nu}W_{L}^{\mu\nu}\,=\,-\epsilon^{2}F_{1}^{L}\,+\,\frac{2x}{q^{2}}(\epsilon\cdot
p)^{2}\,F_{2}^{L}-i\epsilon_{\mu}\epsilon_{\nu}\epsilon^{\mu\nu\alpha\beta}q_{\alpha}p_{\beta}\,\frac{x}{q^{2}}F^{L}_{3}(x,q^{2})\,,$
(V.111)

A comparison of (V.111) to (V.2), allows for the extraction of the s-channel
baryonic contributions to the DIS structure functions

$\displaystyle F_{1s,Dirac}^{L}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{N}_{L}^{2}}{2\tilde{\kappa}^{2}}\bigg{(}\Big{(}{\cal
I}^{2}_{R}(n_{x})+{\cal
I}^{2}_{L}(n_{x})\Big{)}\Big{(}\frac{M_{0}^{2}}{2}+\frac{Q^{2}}{4x}\Big{)}-{\cal
I}_{R}(n_{x}){\cal
I}_{L}(n_{x})M_{0}\bigg{(}{M_{0}^{2}+Q^{2}\left(\frac{1}{x}-1\right)}\bigg{)}^{\frac{1}{2}}\bigg{)}$
$\displaystyle F_{2s,Dirac}^{L}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{N}_{L}^{2}}{4\tilde{\kappa}^{2}}\Big{(}{\cal
I}^{2}_{R}(n_{x})+{\cal I}^{2}_{L}(n_{x})\Big{)}\frac{Q^{2}}{x}\,,$
$\displaystyle F_{3s,Dirac}^{L}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{N}_{L}^{2}}{4\tilde{\kappa}^{2}}\Big{(}{\cal
I}^{2}_{R}(n_{x})-{\cal I}^{2}_{L}(n_{x})\Big{)}\frac{Q^{2}}{x}$ (V.112)

with the DIS kinematics (III.55) subsumed. Similarly, using the transition
form factor (III.2.4), we find

$\displaystyle F_{1s,Dirac}^{EM}$ $\displaystyle=$ $\displaystyle
e_{nucleon}^{2}\times F_{1s,Dirac}^{L}$ $\displaystyle F_{2s,Dirac}^{EM}$
$\displaystyle=$ $\displaystyle e_{nucleon}^{2}\times F_{2s,Dirac}^{L}$
$\displaystyle F_{3s,Dirac}^{EM}$ $\displaystyle=$ $\displaystyle 0\,.$

while using the transition form factor (III.2.2), we find

$\displaystyle F_{1s,Dirac}^{W^{\pm}}$ $\displaystyle=$ $\displaystyle
4\big{(}e^{\pm}_{Wnucleon}\big{)}^{2}\times
F_{1s,Dirac}^{L}=\frac{4\big{(}e^{\pm}_{Wnucleon}\big{)}^{2}}{e_{nucleon}^{2}}\times
F_{1s,Dirac}^{EM}$ $\displaystyle F_{2s,Dirac}^{W^{\pm}}$ $\displaystyle=$
$\displaystyle 4\big{(}e^{\pm}_{Wnucleon}\big{)}^{2}\times
F_{2s,Dirac}^{L}=\frac{4\big{(}e^{\pm}_{Wnucleon}\big{)}^{2}}{e_{nucleon}^{2}}\times
F_{2s,Dirac}^{EM}$ $\displaystyle F_{3s,Dirac}^{W^{\pm}}$ $\displaystyle=$
$\displaystyle 4\big{(}e^{\pm}_{Wnucleon}\big{)}^{2}\times
F_{3s,Dirac}^{L}\,.$

###  V.3 Dirac plus Pauli contributions

The full left current vertex with the Pauli contribution is given in (III.78).
A rerun of the preceding arguments yields the hadronic tensor for $\nu$
scattering in the untraced form ( $\bar{\nu}$ scattering follows from
$\nu\leftrightarrow\mu$)

$\displaystyle W_{L}^{+\mu\nu}=\frac{1}{16\pi{\tilde{\kappa}}^{2}}$
$\displaystyle\bigg{[}2M_{0}M_{X}\bigg{(}\alpha_{R}^{2}g^{\mu\nu}+(\beta_{R}^{2}+2\alpha_{R}\beta_{R})\,q^{\mu}q^{\nu}+\lambda_{R}^{2}(g^{\mu\nu}q^{2}-q^{\mu}q^{\nu})\bigg{)}$
(V.115)
$\displaystyle+2M_{0}\lambda_{R}\bigg{(}\alpha_{R}g^{\mu\nu}q\cdot(P_{X}+p)+\beta_{R}q^{\mu}q^{\nu}q\cdot(P_{X}+p)-i\alpha_{R}\epsilon^{\nu\mu\alpha\beta}q_{\alpha}(P_{X}+p)_{\beta}\bigg{)}$
$\displaystyle+2\bigg{(}\alpha_{R}g^{\mu}_{\bar{\mu}}+\beta_{R}q^{\mu}q_{\bar{\mu}}\bigg{)}\bigg{(}\alpha_{R}g^{\nu}_{\bar{\nu}}+\beta_{R}q^{\nu}q_{\bar{\nu}}\bigg{)}\,\bigg{(}p^{\bar{\mu}}P_{X}^{\bar{\nu}}+p^{\bar{\nu}}P_{X}^{\bar{\mu}}-g^{\bar{\mu}\bar{\nu}}p\cdot
P_{X}-i\epsilon^{\bar{\nu}\alpha\bar{\mu}\beta}P_{X\alpha}p_{\beta}\bigg{)}$
$\displaystyle-\frac{1}{2}\lambda_{R}^{2}\bigg{(}3g^{\mu\nu}(2q\cdot pq\cdot
P_{X}-q^{2}p\cdot P_{X})$
$\displaystyle\qquad\qquad+4\bigg{(}q^{2}(p^{\mu}P_{X}^{\nu}+p^{\nu}P_{X}^{\mu})+q^{\mu}q^{\nu}p\cdot
P_{X}-(q^{\nu}p^{\mu}+q^{\mu}p^{\nu})q\cdot
P_{X}-(q^{\nu}P_{X}^{\mu}+q^{\mu}P_{X}^{\nu})q\cdot p\bigg{)}\bigg{)}$
$\displaystyle+(R\rightarrow L\,,-i\rightarrow+i)\bigg{]}$

with $P_{X}=q+p$, $M_{X}^{2}=M_{0}^{2}+q^{2}(1-1/x)$ and

$\displaystyle\alpha_{R}=2(e^{+}_{W}{\cal I}_{R}+\eta{\cal
J}_{R})\qquad\beta_{R}=4\eta{\cal I}_{LR}\qquad\lambda_{R}=-2\eta{\cal K}_{R}$
(V.116)

$\displaystyle\epsilon_{\mu}^{+}\epsilon_{\nu}^{+*}\,W_{L,s}^{+\mu\nu}=\frac{1}{2}\sum_{s,s_{X}}\sum_{M_{X}}\,\delta\Big{(}M_{X}^{2}-(P+q)^{2}\,\Big{)}\,\epsilon_{\mu}^{+}{\cal
W}_{L}^{+\mu}\,\epsilon_{\nu}^{+*}{\cal W}_{L}^{+*\nu}\,$
$\displaystyle=\frac{\big{(}\mathcal{N}_{L}^{+}\big{)}^{2}}{8\tilde{\kappa}^{2}}\epsilon_{\mu}^{+}\epsilon_{\nu}^{+*}\times
8\Big{[}2\tilde{{\cal I}}_{LR}(n_{x})M_{0}M_{X}\eta^{\mu\nu}+\tilde{{\cal
I}}_{+}^{2}(n_{x})\Big{(}p^{\mu}P_{X}^{\nu}+p^{\nu}P_{X}^{\mu}-\eta^{\mu\nu}p\cdot
P_{X}\Big{)}+\tilde{{\cal
I}}_{-}^{2}(n_{x})\Big{(}i\epsilon^{\mu\nu\alpha\beta}P_{X\alpha}p_{\beta}\Big{)}\Big{]}$

where we have defined

$\displaystyle\tilde{{\cal I}}_{LR}(n_{x})$ $\displaystyle\equiv$
$\displaystyle{\cal I}_{\eta L}(n_{x}){\cal I}_{\eta
R}(n_{x})+4\eta^{2}q^{2}{\cal I}_{RL}(n_{x}){\cal I}_{LR}(n_{x})$
$\displaystyle\tilde{{\cal I}}_{+}^{2}(n_{x})$ $\displaystyle\equiv$
$\displaystyle{\cal I}_{\eta R}^{2}(n_{x})+{\cal I}_{\eta
L}^{2}(n_{x})-4\eta^{2}q^{2}\Big{(}{\cal I}_{LR}^{2}(n_{x})+{\cal
I}_{RL}^{2}(n_{x})\Big{)}$ $\displaystyle\tilde{{\cal I}}_{-}^{2}(n_{x})$
$\displaystyle\equiv$ $\displaystyle{\cal I}_{\eta R}^{2}(n_{x})-{\cal
I}_{\eta L}^{2}(n_{x})$ $\displaystyle{\cal I}_{\eta R}(n_{x})$
$\displaystyle\equiv$ $\displaystyle e^{+}_{Wnucleon}{\cal
I}_{R}(n_{x})+\eta{\cal J}_{R}(n_{x})$ $\displaystyle{\cal I}_{\eta L}(n_{x})$
$\displaystyle\equiv$ $\displaystyle e^{+}_{Wnucleon}{\cal
I}_{L}(n_{x})+\eta{\cal J}_{L}(n_{x})\,,$ (V.118)

and we have used $\epsilon^{+}\cdot q=0$, $\epsilon^{+*}\cdot q=0$,
$q_{\tilde{\nu}}q_{\tilde{\mu}}\epsilon^{\tilde{\nu}\beta\tilde{\mu}\nu}=0$,
and
$q_{\tilde{\nu}}q_{\tilde{\mu}}\epsilon^{\tilde{\mu}\alpha\mu\tilde{\nu}}=0$.

Comparing (V.3) with the contraction of (IV.1), for neutrino $\nu$, i.e.,

$\displaystyle\epsilon_{\mu}^{+}\epsilon_{\nu}^{+*}\,W_{L}^{+\mu\nu}=\epsilon_{\mu}^{+}\epsilon_{\nu}^{+*}\,W^{\nu\,,\mu\nu}=-\epsilon^{+}\cdot\epsilon^{+*}\,F_{1}^{W^{+}}\,+\,\frac{2x}{q^{2}}\,\epsilon^{+}\cdot
p\,\epsilon^{+*}\cdot
p\,F_{2}^{W^{+}}-i\epsilon^{+}_{\mu}\epsilon^{+*}_{\nu}\epsilon^{\mu\nu\alpha\beta}q_{\alpha}p_{\beta}\,\frac{x}{q^{2}}F^{W^{+}}_{3}(x,q^{2})\,,$

allows for the extraction of the structure functions

$\displaystyle F_{1s}^{W^{+}}$ $\displaystyle=$
$\displaystyle\frac{2\big{(}\mathcal{N}_{L}^{+}\big{)}^{2}}{\tilde{\kappa}^{2}}\bigg{(}\tilde{{\cal
I}}_{+}^{2}(n_{x})\Big{(}\frac{M_{0}^{2}}{2}+\frac{Q^{2}}{4x}\Big{)}-\tilde{{\cal
I}}_{LR}(n_{x})M_{0}\bigg{(}{M_{0}^{2}+Q^{2}\left(\frac{1}{x}-1\right)}\bigg{)}^{\frac{1}{2}}\bigg{)}\,,$
$\displaystyle F_{2s}^{W^{+}}$ $\displaystyle=$
$\displaystyle\frac{\big{(}\mathcal{N}_{L}^{+}\big{)}^{2}}{\tilde{\kappa}^{2}}\,\tilde{{\cal
I}}_{+}^{2}(n_{x})\frac{Q^{2}}{x}\,,$ $\displaystyle F_{3s}^{W^{+}}$
$\displaystyle=$
$\displaystyle\frac{\big{(}\mathcal{N}_{L}^{+}\big{)}^{2}}{\tilde{\kappa}^{2}}\,\tilde{{\cal
I}}_{-}^{2}(n_{x})\frac{Q^{2}}{x}\,,$ (V.120)

where we have used $P_{X_{\alpha}}=p_{\alpha}+q_{\alpha}$,
$p_{\alpha}p_{\beta}\epsilon^{\mu\nu\alpha\beta}=0$, $\epsilon^{+}\cdot q=0$,
$\epsilon^{+*}\cdot q=0$, and $M_{X}^{2}=M_{0}^{2}+q^{2}(1-1/x)$. Strict bulk-
to-boundary correspondence implies ${\cal N}_{L}^{+}=1$ in the double limit of
a large number of colors $N_{c}$ and strong gauge t Hooft gauge coupling. Here
we follow BRAGA  and assume proportionality between the bulk and boundary
structure functions with ${\cal N}_{L}^{+}$ an overall parameter that captures
parts of the finite corrections to the strict double limit. It will be fixed
by a point in the data. Similarly, we can find the structure functions for
anti-neutrino $\tilde{\nu}$ scattering through $W^{-}$ exchange as

$\displaystyle F_{1s}^{W^{-}}$ $\displaystyle=$ $\displaystyle
F_{1s}^{W^{+}}(e^{+}_{Wnucleon}\rightarrow e^{-}_{Wnucleon}\,;\,{\cal
N}_{L}^{+}\rightarrow{\cal N}_{L}^{-})\,,$ $\displaystyle F_{2s}^{W^{-}}$
$\displaystyle=$ $\displaystyle F_{2s}^{W^{+}}(e^{+}_{Wnucleon}\rightarrow
e^{-}_{Wnucleon}\,;\,{\cal N}_{L}^{+}\rightarrow{\cal N}_{L}^{-})\,,$
$\displaystyle F_{3s}^{W^{-}}$ $\displaystyle=$ $\displaystyle
F_{3s}^{W^{+}}(e^{+}_{Wnucleon}\rightarrow e^{-}_{Wnucleon}\,;\,{\cal
N}_{L}^{+}\rightarrow{\cal N}_{L}^{-})\,.$ (V.121)

###  V.4 Structure functions: Pomeron exchange in the t-channel

DIS scattering at small-x is dominated by Reggeon and Pomeron exchanges. In
this section, we first consider DIS scattering using $U(1)_{L}$ currents in
the Pomeron regime and then generalize our results to the electromagnetic
vector $U(1)^{EM}_{V}\subset U(2)_{V}$, and the left-handed
$U(1)^{\pm}_{L}\subset U(2)_{L}$ currents. The Pomeron is dual to a close
string exchange or graviton in the t-channel SIN ; JANIK ; POLX ; HATTAX ;
KOVENSKY . This is best obtained by recalling that the hadronic tentor ties to
the forward scattering amplitude of a $U(1)_{L}$ current through

$W_{L}^{\mu\nu}\,=\frac{1}{4\pi}\sum_{s}\,\int d^{4}y\,e^{iq\cdot y}\langle
P,s|\,\Big{[}\tilde{J}_{L}^{\mu}(y),\tilde{J}_{L}^{\nu}(0)\Big{]}\,|P,s\rangle\,=2\pi{\rm
Im}\,T_{L}^{\mu\nu}\,,$ (V.122)

where the Compton scattering amplitude $T_{L}^{\mu\nu}$ is given by

$\epsilon_{\mu}\epsilon_{\nu}T_{L,t}^{\mu\nu}\equiv{\cal A}^{h}_{Lp\rightarrow
Lp}(s,t)\,,$ (V.123)

for massive graviton or glueball $h_{\mu\nu}$ exchange in the t-channel, with
the explicit result given in (D.239). The Pomeron contribution as a graviton
exchange to the t-channel structure functions $F^{L}_{1t,2t}(x,Q)$ follow from

$\epsilon_{\mu}\epsilon_{\nu}W_{L,t}^{\mu\nu}\,=\,\epsilon^{2}F_{1t}^{L}\,+\,\frac{2x}{q^{2}}(\epsilon\cdot
p)^{2}\,F_{2t}^{L}\,=\,2\pi{\rm Im}\,{\cal A}^{tot}_{Lp\rightarrow
Lp}(s,t=-K^{2}=0)\,,$ (V.124)

for $\epsilon\cdot q=0$. Inserting (D.3) into (V.124) we obtain

$\displaystyle 2xF_{1t}^{L}$ $\displaystyle=$
$\displaystyle\frac{2\kappa^{2}}{g_{5}^{2}}\times\frac{\pi}{\sqrt{\lambda}}\times\Big{(}\frac{Q}{\tilde{\kappa}}\Big{)}^{2-2/\sqrt{\lambda}}\times\Big{(}\frac{1}{x}\Big{)}^{1-2/\sqrt{\lambda}}$
$\displaystyle\times$ $\displaystyle{\rm
exp}\bigg{[}-\frac{\tilde{\xi}^{2}}{2}\frac{\sqrt{\lambda}}{\log[Q^{2}/\tilde{\kappa}^{2}]+\log[1/x]}\bigg{]}\times\bigg{(}\frac{\sqrt{\lambda}}{\log[Q^{2}/\tilde{\kappa}^{2}]+\log[1/x]}\bigg{)}^{3/2}$
$\displaystyle\times$ $\displaystyle(2\pi)^{1/2}\tilde{\xi}\left(1+{\cal
O}\bigg{(}\frac{\sqrt{\lambda}}{\log[Q^{2}/\tilde{\kappa}^{2}]+\log[1/x]}\bigg{)}\right)\times\mathcal{F}(j_{0},K=0)\times
I_{\xi}^{T}(j_{0},Q=Q^{\prime})\,,$ $\displaystyle F_{2t}^{L}$
$\displaystyle=$
$\displaystyle\frac{2\kappa^{2}}{g_{5}^{2}}\times\frac{\pi}{\sqrt{\lambda}}\times\Big{(}\frac{Q}{\tilde{\kappa}}\Big{)}^{2-2/\sqrt{\lambda}}\times\Big{(}\frac{1}{x}\Big{)}^{1-2/\sqrt{\lambda}}$
$\displaystyle\times$ $\displaystyle{\rm
exp}\bigg{[}-\frac{\tilde{\xi}^{2}}{2}\frac{\sqrt{\lambda}}{\log[Q^{2}/\tilde{\kappa}^{2}]+\log[1/x]}\bigg{]}\times\bigg{(}\frac{\sqrt{\lambda}}{\log[Q^{2}/\tilde{\kappa}^{2}]+\log[1/x]}\bigg{)}^{3/2}$
$\displaystyle\times$ $\displaystyle(2\pi)^{1/2}\tilde{\xi}\left(1+{\cal
O}\bigg{(}\frac{\sqrt{\lambda}}{\log[Q^{2}/\tilde{\kappa}^{2}]+\log[1/x]}\bigg{)}\right)\times\mathcal{F}(j_{0},K=0)\times\Big{(}I_{\xi}^{T}(j_{0},Q=Q^{\prime})+I_{\xi}^{L}(j_{0},Q=Q^{\prime})\Big{)}\,,$

with $\tilde{\xi}=\gamma+\pi/2$. The pre-exponents in (V.4) are commensurate
with the expected Pomeron behavior $s^{\alpha_{\mathbb{P}}(0)-1}$ with the
intercept $\alpha_{\mathbb{P}}(0)-1=1-2/\sqrt{\lambda}$ after the
identification ${s}\sim Q^{2}/x$ in the DIS limit. The additional overall
factor of $Q$ is the left over (longitudinal) polarization from the overlaping
incoming-outgoing $U(1)_{L}$ wavefunctions. The exponents reflect on the
warped Gribov diffusion in $D_{\perp}=3$, for $2_{\perp}$ spatial dimensions
and $1_{\perp}$ holographic dimension. The result (V.4) extends to the SU(2)
currents by introducing a normalization factor $\mathcal{N}_{L}^{a=\pm}$. More
specifically, for the charged currents we have

$\displaystyle 2xF_{1t}^{W^{\pm}}$ $\displaystyle=$
$\displaystyle\big{(}\mathcal{N}_{Lt}^{\pm}\big{)}^{2}\frac{2\kappa^{2}}{g_{5}^{2}}\times\frac{\pi}{\sqrt{\lambda}}\times\Big{(}\frac{Q}{\tilde{\kappa}}\Big{)}^{2-2/\sqrt{\lambda}}\times\Big{(}\frac{1}{x}\Big{)}^{1-2/\sqrt{\lambda}}$
$\displaystyle\times$ $\displaystyle{\rm
exp}\bigg{[}-\frac{\tilde{\xi}^{2}}{2}\frac{\sqrt{\lambda}}{\log[Q^{2}/\tilde{\kappa}^{2}]+\log[1/x]}\bigg{]}\times\bigg{(}\frac{\sqrt{\lambda}}{\log[Q^{2}/\tilde{\kappa}^{2}]+\log[1/x]}\bigg{)}^{3/2}$
$\displaystyle\times$ $\displaystyle(2\pi)^{1/2}\tilde{\xi}\left(1+{\cal
O}\bigg{(}\frac{\sqrt{\lambda}}{\log[Q^{2}/\tilde{\kappa}^{2}]+\log[1/x]}\bigg{)}\right)\times\mathcal{F}(j_{0},K=0)\times
I_{\xi}^{T}(j_{0},Q=Q^{\prime})\,,$ $\displaystyle F_{2t}^{W^{\pm}}$
$\displaystyle=$
$\displaystyle\big{(}\mathcal{N}_{Lt}^{\pm}\big{)}^{2}\frac{2\kappa^{2}}{g_{5}^{2}}\times\frac{\pi}{\sqrt{\lambda}}\times\Big{(}\frac{Q}{\tilde{\kappa}}\Big{)}^{2-2/\sqrt{\lambda}}\times\Big{(}\frac{1}{x}\Big{)}^{1-2/\sqrt{\lambda}}$
$\displaystyle\times$ $\displaystyle{\rm
exp}\bigg{[}-\frac{\tilde{\xi}^{2}}{2}\frac{\sqrt{\lambda}}{\log[Q^{2}/\tilde{\kappa}^{2}]+\log[1/x]}\bigg{]}\times\bigg{(}\frac{\sqrt{\lambda}}{\log[Q^{2}/\tilde{\kappa}^{2}]+\log[1/x]}\bigg{)}^{3/2}$
$\displaystyle\times$ $\displaystyle(2\pi)^{1/2}\tilde{\xi}\left(1+{\cal
O}\bigg{(}\frac{\sqrt{\lambda}}{\log[Q^{2}/\tilde{\kappa}^{2}]+\log[1/x]}\bigg{)}\right)\times\mathcal{F}(j_{0},K=0)\times\Big{(}I_{\xi}^{T}(j_{0},Q=Q^{\prime})+I_{\xi}^{L}(j_{0},Q=Q^{\prime})\Big{)}\,.$

###  V.5 $F_{3t}^{W^{\pm}}$ structure functions: Reggeon exchange in the
t-channel

Reggeon exchange is also a t-channel contribution stemming from a spin-1
exchange induced by the Chern-Simons contribution in (IV.1). The latter allows
for the anomalous coupling of $W^{-}W^{+}\omega$ in bulk with $\omega_{\mu}$ a
spin-1 flavor singlet U(1) gauge field in bulk (the analogue of the omega-
meson). In principle, the Reggeized spin-1 exchange in bulk contributes to the
unpolarized and parity odd structure function $F_{3}$. A similar contribution
was observed for the spin structure function in KOVENSKY , following an
earlier analysis in HATTAX . More specifically, the U(1) exchange of
$L_{\mu}^{0}$ in bulk stems from (II.4) and (II.4), with the vertices

$\displaystyle L\overline{\Psi}\Psi:\quad$
$\displaystyle\int\frac{d^{4}p_{2}d^{4}p_{1}d^{4}k}{(2\pi)^{12}}(2\pi)^{4}\delta^{4}(p_{2}-k-p_{1})\,S^{k}_{L\bar{\Psi}\Psi}$
$\displaystyle LLL:\quad$
$\displaystyle\int\frac{d^{4}q^{\prime}d^{4}qd^{4}k}{(2\pi)^{12}}(2\pi)^{4}\delta^{4}(q-k-q^{\prime})\,S^{k}_{LLL}$

where we have defined

$\displaystyle S^{k}_{L\bar{\Psi}\Psi}$ $\displaystyle=$ $\displaystyle
g_{5}^{3}\,\kappa_{CS}\int
dz\,\bigg{[}\epsilon^{\mu\sigma\nu\rho}\epsilon_{\mu}^{+}(q)\epsilon_{\sigma}^{-}(q^{\prime})\partial_{z}L^{+}(q,z)\,(-ik_{\nu})\epsilon_{\rho}^{0}(k)L^{0}(k,z)L^{-}(q^{\prime},z)$
(V.128) $\displaystyle-$
$\displaystyle\epsilon^{\sigma\mu\nu\rho}\epsilon_{\sigma}^{+}(q)\epsilon_{\mu}^{-}(q^{\prime})\partial_{z}L^{-}(q^{\prime},z)\,(-ik_{\nu})\epsilon_{\rho}^{0}(k)L^{0}(k,z)L^{+}(q,z)$
$\displaystyle+$
$\displaystyle\epsilon^{\mu\sigma\rho\nu}\epsilon_{\mu}^{+}(q)\epsilon_{\sigma}^{-}(q^{\prime})\partial_{z}L^{+}(q,z)\,\epsilon_{\rho}^{0}(k)L^{0}(k,z)\,(-iq^{\prime}_{\nu})L^{-}(q^{\prime},z)$
$\displaystyle-$
$\displaystyle\epsilon^{\sigma\mu\rho\nu}\epsilon_{\sigma}^{+}(q)\epsilon_{\mu}^{-}(q^{\prime})\partial_{z}L^{-}(q^{\prime},z)\,\epsilon_{\rho}^{0}(k)L^{0}(k,z)\,(iq_{\nu})L^{+}(q,z)$
$\displaystyle-$
$\displaystyle\epsilon^{\sigma\rho\mu\nu}\epsilon_{\sigma}^{+}(q)\epsilon_{\rho}^{-}(q^{\prime})\epsilon_{\mu}^{0}(k)\partial_{z}L^{0}(k,z)\,(-iq^{\prime}_{\nu})L^{-}(q^{\prime},z)L^{+}(q,z)$
$\displaystyle+$
$\displaystyle\epsilon^{\rho\sigma\mu\nu}\epsilon_{\rho}^{+}(q)\epsilon_{\sigma}^{-}(q^{\prime})\epsilon_{\mu}^{0}(k)\partial_{z}L^{0}(k,z)(iq_{\nu})L^{+}(q,z)\,L^{-}(q^{\prime},z)\bigg{]}$
$\displaystyle=$ $\displaystyle
g_{5}^{3}\,\kappa_{CS}(-i)\epsilon^{\mu\sigma\nu\rho}\epsilon_{\mu}^{+}(q)\epsilon_{\sigma}^{-}(q^{\prime})(q^{\prime}_{\nu}+q_{\nu})\epsilon_{\rho}^{0}(k)$
$\displaystyle\times$ $\displaystyle\int
dz\,\big{(}\partial_{z}L^{0}(k,z)L(q^{\prime},z)L(q,z)-\partial_{z}L(q,z)L^{0}(k,z)L(q^{\prime},z)$

with

$\displaystyle L_{\mu}^{+}(x,z)=\epsilon^{+}_{\mu}(q)L^{+}(q,z)e^{iq\cdot
x}\,,\qquad
L_{\mu}^{-}(x,z)=\epsilon^{-}_{\mu}(q^{\prime})L^{-}(q^{\prime},z)e^{-iq^{\prime}\cdot
x}\,,\qquad L_{\mu}^{0}(x,z)=\epsilon^{0}_{\mu}(k)L^{+}(k,z)e^{-ik\cdot x}$

In the last line in (V.128), we have used the fact that
$L^{+}(q,z)=L^{-}(q,z)=L(q,z)$.

Using the relation

$\displaystyle{\rm Im}\,{\cal A}^{tot}_{Lp\rightarrow
Lp}(s,t=0)=\epsilon^{+\mu}\epsilon^{-\nu}W^{\nu}_{\mu\nu}$ (V.130)

with $W^{\nu}_{\mu\nu}$ given by (IV.1), and ${\cal A}^{tot}_{Lp\rightarrow
Lp}(s,t)$ given by (C), we can extract the structure function
$F_{3}^{W^{\pm}}(x,Q^{2})$ in the t-channel as

$\displaystyle F_{3t}^{W^{\pm}}$ $\displaystyle=$ $\displaystyle 4\times
4\times g_{5}\times\frac{1}{g_{5}^{2}}\times
g_{5}^{3}\kappa_{CS}\times\Big{(}\frac{Q}{\tilde{\kappa}}\Big{)}^{2-j_{0}-\Delta(j_{0})}\times\Big{(}\frac{s}{\tilde{\kappa}^{2}}\Big{)}^{j_{0}}\times\frac{e^{-\sqrt{\lambda}\tilde{\xi}^{2}/2\log[s/\tilde{\kappa}^{2}]}}{(\log[s/\tilde{\kappa}^{2}])^{3/2}}$
$\displaystyle\times$
$\displaystyle(\sqrt{\lambda}/2\pi)^{1/2}\;\tilde{\xi}\;\left(1+{\cal
O}\bigg{(}\frac{\sqrt{\lambda}}{\log[s/\tilde{\kappa}^{2}]}\bigg{)}\right)\times
I_{\xi}(j_{0},Q,Q^{\prime}=Q)\times\mathcal{F}_{1}^{(LN)}(j_{0},K=0)$
$\displaystyle=$ $\displaystyle 16\times
g_{5}^{2}\kappa_{CS}\times\Big{(}\frac{Q}{\tilde{\kappa}}\Big{)}^{1-\frac{1}{\sqrt{\lambda}}}\times\Big{(}\frac{1}{x}\Big{)}^{1-\frac{1}{\sqrt{\lambda}}}\times\frac{e^{-\sqrt{\lambda}\tilde{\xi}^{2}/2(\log[Q^{2}/\tilde{\kappa}^{2}]+\log[1/x])}}{(\log[Q^{2}/\tilde{\kappa}^{2}]+\log[1/x])^{3/2}}$
$\displaystyle\times$
$\displaystyle(\sqrt{\lambda}/2\pi)^{1/2}\;\tilde{\xi}\;\left(1+{\cal
O}\bigg{(}\frac{\sqrt{\lambda}}{\log[Q^{2}/\tilde{\kappa}^{2}]+\log[1/x]}\bigg{)}\right)\times
I_{\xi}(j_{0},Q,Q^{\prime}=Q)\times\mathcal{F}_{1}^{(LN)}(j_{0},K=0)\,,$

Here $s\approx\frac{Q^{2}}{x}$, $j_{0}=1-1/\sqrt{\lambda}$, $\Delta(j_{0})=2$,
and $\tilde{\xi}-\pi/2=\gamma=0.55772.....$ is Euler-Mascheroni constant.

##  VI Holographic results and comparison to data

(IV.2-IV.2) can be inverted to give the unpolarized valence and sea parton
distributions in the proton, in terms of the pertinent holographic structure
functions. The ones for the neutron follow by isospin symmetry. More
specifically, we have for the unpolarized sea of the proton

$\displaystyle x\overline{u}(x,Q)=$
$\displaystyle\frac{1}{4}\left(F_{2}^{W^{+}p}(x,Q)-xF_{3}^{W^{+}p}(x,Q)\right)\,,$
$\displaystyle x\overline{d}(x,Q)=$
$\displaystyle\frac{1}{4}\left(F_{2}^{W^{-}p}(x,Q)-xF_{3}^{W^{-}p}(x,Q)\right)\,,$

and for the unpolarized valence contribution of the proton

$\displaystyle xu_{V}(x,Q)=x(u(x,Q)-\overline{u}(x,Q))$ $\displaystyle=$
$\displaystyle\frac{1}{4}x\left(F_{3}^{W^{-}p}(x,Q)+F_{3}^{W^{+}p}(x,Q)\right)$
$\displaystyle+$
$\displaystyle\frac{1}{4}\left(F_{2}^{W^{-}p}(x,Q)-F_{2}^{W^{+}p}(x,Q)\right)\,,$
$\displaystyle xd_{V}(x,Q)=x(d(x,Q)-\overline{d}(x,Q))$ $\displaystyle=$
$\displaystyle\frac{1}{4}x\left(F_{3}^{W^{-}p}(x,Q)+F_{3}^{W^{+}p}(x,Q)\right)$
$\displaystyle-$
$\displaystyle\frac{1}{4}\left(F_{2}^{W^{-}p}(x,Q)-F_{2}^{W^{+}p}(x,Q)\right)\,,$

with the structure functions receiving contributions from the s- and t-channel

$\displaystyle
F_{2}^{W^{\pm}p}(x,Q)=F_{2s}^{W^{\pm}p}(x,Q)+F_{2t}^{W^{\pm}p}(x,Q)\,,$
$\displaystyle
F_{3}^{W^{\pm}p}(x,Q)=F_{3s}^{W^{\pm}p}(x,Q)+F_{3t}^{W^{\pm}p}(x,Q)\,,$

The charged s-channel even-parity structure functions $F_{2s}^{W^{\pm}p}(x,Q)$
are given in (V.3) and (V.3), respectively, and the t-channel structure
functions $F_{2t}^{W^{\pm}p}(x,Q)$ are given in (V.4). The charged s-channel
odd-parity structure functions $F_{3s}^{W^{\pm}p}(x,Q)$ are given in (V.3) and
(V.3), respectively. The charged t-channel odd parity structure functions
$F_{3t}^{W^{\pm}p}(x,Q)$ are given in (V.5).

Figure 1: Small-x holographic PDFs as given in (VI.1) and (VI.1) for: the
normalization coefficient $\mathcal{N}_{Lt}^{\pm}=0.309$ (the green curve),
the normalization coefficient fixed between $\mathcal{N}_{Lt}^{\pm}=0.274$ and
$\mathcal{N}_{Lt}^{\pm}=0.375$ (the blue band), ’t Hooft coupling
$\lambda=g_{YM}^{2}N_{c}=9.533$, the twist of proton $\tau=3$ (fixed by the
scaling of the electromagnetic form factor of proton in CARLSON ), and soft-
wall IR scale $\tilde{\kappa}=0.350\,\rm{GeV}$ (fixed by the mass of proton
and $\rho$ meson in CARLSON ).

###  VI.1 Results in the small-x regime

In the small-x regime, the results for the sea $\bar{u}$ and $\bar{d}$
distributions are solely due to the t-channel Pomeron exchange

$\displaystyle\lim_{x\rightarrow 0}x\overline{u}(x,Q)$ $\displaystyle=$
$\displaystyle\frac{1}{4}F_{2t}^{W^{+}p}(x,Q)\,,$
$\displaystyle\lim_{x\rightarrow 0}x\overline{d}(x,Q)$ $\displaystyle=$
$\displaystyle\frac{1}{4}F_{2t}^{W^{-}p}(x,Q)\,,$ (VI.135)

with no Pomeron contribution to the tail of the valence distributions

$\displaystyle\lim_{x\rightarrow 0}xu_{V}(x,Q)$ $\displaystyle=$
$\displaystyle 0\,,$ $\displaystyle\lim_{x\rightarrow 0}xd_{V}(x,Q)$
$\displaystyle=$ $\displaystyle 0\,.$ (VI.136)

Fig. 1 show the holographic Pomeron contribution at small-x for the $xu$ and
$xd$ distributions of the proton at a resolution $Q^{2}=6.5$ GeV2, following
from (VI.1) and (VI.1). The ′t Hooft coupling is
$\lambda=g_{YM}^{2}N_{c}=9.533$, the twist of proton is set to $\tau=3$ (fixed
by the scaling of the electromagnetic form factor of proton in CARLSON ), and
the soft-wall IR scale is fixed to $\tilde{\kappa}=0.350\,\rm{GeV}$ (to
reproduce the mass of the proton and $\rho$ meson as in CARLSON ). The green
curve uses the normalization coefficient $\mathcal{N}_{Lt}^{\pm}=0.309$. The
blue-band corresponds to the normalization coefficients fixed between
$\mathcal{N}_{Lt}^{\pm}=0.274$ and $\mathcal{N}_{Lt}^{\pm}=0.375$.

(a)  (b)  (c)  (d)  (e)  (f)  (g)  (h)

Figure 2: The reduced non-charged and unpolarized deep inelastic $e^{\pm}p$
scattering cross section $\sigma^{\pm}_{r,NC}(x,Q,s)$ at low-x versus $Q^{2}$
as given in (VI.137) for increasing $x$, at $\sqrt{s}=318$ GeV. The middle-
green curve and the blue band are the holographic results for different charge
normalizations ${\cal N}_{Lt}^{\pm}$. The red points are the H1 and ZEUS data
Abramowicz:2015mha , and the orange points at low $Q^{2}$ are the data from
E665 Adams:1996gu . See text.

###  VI.2 Fixing the charged normalization ${\cal N}_{Lt}^{\pm}$ parameters

Throughout we will use the same ′t Hooft coupling
$\lambda=g_{YM}^{2}N_{c}=9.533$ which is within the standard choice in the
most holographic constructions. The charged normalizations
$\mathcal{N}_{Lt}^{\pm}$ are fixed between $0.274$ and $0.375$ and will be
shown as a blue band in all results to follow. These normalizations are chosen
for a best fit to the reduced non-charged (r,NC) and unpolarized deep
inelastic $e^{\pm}p$ scattering cross section $\sigma^{\pm}_{r,NC}$ at low-x.
We recall, that $\sigma^{\pm}_{r,NC}$ is given by a linear combination of the
generalised structure functions

$\displaystyle\mbox{$\sigma_{r,{\rm NC}}^{\pm}$}=\mbox{$\frac{\textstyle{\rm
d^{2}}\sigma^{e^{\pm}p}_{{\rm NC}}}{\textstyle{\rm d}x_{\rm Bj}{\rm
d}Q^{2}}$}\cdot\frac{Q^{4}x_{\rm
Bj}}{2\pi\alpha^{2}Y_{+}}=\tilde{F_{2}}\mp\frac{Y_{-}}{Y_{+}}x\tilde{F_{3}}-\frac{y^{2}}{Y_{+}}\tilde{F_{\rm
L}}$ (VI.137)

with $Y_{\pm}=1\pm(1-y)^{2}$. The overall structure functions,
$\tilde{F}_{2}$, $\tilde{F}_{\rm L}$ and $x\tilde{F}_{3}$, are sums of
structure functions, ${F}_{\rm X}$, ${F}_{\rm X}^{\gamma Z}$ and ${F}_{\rm
X}^{Z}$, relating to photon exchange, photon–$Z$ interference and $Z$
exchange, respectively, and depend on the electroweak parameters as

$\tilde{F}_{2}$ $\displaystyle=$ $\displaystyle F_{2}-\kappa_{Z}v_{e}\cdot
F_{2}^{\gamma Z}+\kappa_{Z}^{2}(v_{e}^{2}+a_{e}^{2})\cdot F_{2}^{Z}~{},$
$\tilde{F}_{\rm L}$ $\displaystyle=$ $\displaystyle F_{\rm
L}-\kappa_{Z}v_{e}\cdot F_{\rm L}^{\gamma
Z}+\kappa_{Z}^{2}(v_{e}^{2}+a_{e}^{2})\cdot F_{\rm L}^{Z}~{},$
$x\tilde{F}_{3}$ $\displaystyle=$ $\displaystyle-\kappa_{Z}a_{e}\cdot
xF_{3}^{\gamma Z}+\kappa_{Z}^{2}\cdot 2v_{e}a_{e}\cdot xF_{3}^{Z}~{}$

Here $v_{e}$ and $a_{e}$ are the vector and axial-vector weak couplings of the
electron to the $Z$ boson, and
$\kappa_{Z}(Q^{2})=Q^{2}/[(Q^{2}+M_{Z}^{2})(4\sin^{2}\theta_{W}\cos^{2}\theta_{W})]$.
The values of $\sin^{2}\theta_{W}=0.23127$ and $M_{Z}=91.1876$ GeV were used
for the electroweak mixing angle and the $Z$-boson mass. In the quark-parton
model (QPM) where the kinematic variable $x_{\rm Bj}$ is equal to the
fractional momentum of the struck quark, $x$, the structure functions are
given in terms of the PDFs as

$\displaystyle(F_{2}(x,Q),F_{2}^{\gamma Z}(x,Q),F_{2}^{Z}(x,Q))$
$\displaystyle\approx$
$\displaystyle[(e_{u}^{2},2e_{u}v_{u},v_{u}^{2}+a_{u}^{2})(xu(x,Q)+x\bar{u}(x,Q))$
$\displaystyle+$
$\displaystyle(e_{d}^{2},2e_{d}v_{d},v_{d}^{2}+a_{d}^{2})(xd(x,Q)+x\bar{d}(x,Q))]~{},$
$\displaystyle(xF_{3}^{\gamma Z}(x,Q),xF_{3}^{Z}(x,Q))$ $\displaystyle\approx$
$\displaystyle 2[(e_{u}a_{u},v_{u}a_{u})(xu(x,Q)-x\bar{u}(x,Q))$
$\displaystyle+$
$\displaystyle(e_{d}a_{d},v_{d}a_{d})(xd(x,Q)-x\bar{d}(x,Q))]~{},$
$\displaystyle(F_{L}(x,Q),F_{L}^{\gamma Z}(x,Q),F_{L}^{Z}(x,Q))$
$\displaystyle\approx$ $\displaystyle(0,0,0)$ (VI.138)

with $e_{u}$ and $e_{d}$ denoting the electric charge of up- and down-type
quarks, while $v_{u,d}$ and $a_{u,d}$ are the vector and axial-vector weak
couplings of the up- and down-type quarks to the $Z$ boson.

In Fig. 2 we show the measured $\sigma_{r,NC}^{\pm}(x,Q,s)$ versus $Q^{2}$ in
GeV2 at $\sqrt{s}$ =318 GeV with increasing x-resolution. Fig. 2a follows from
(VI.137) for $x=0.000016$ using the small-x holographic PDF shown in Fig. 1.
Fig. 2b shows the same for $x=0.00005$. The middle green curve corresponds to
$\mathcal{N}_{Lt}^{\pm}=0.309$. Fig. 2c follows also from (VI.137) for
$x=0.00008$, with the middle green curve referring to
$\mathcal{N}_{Lt}^{\pm}=0.314$. Fig. 2d refers to $x=0.00013$ with the middle-
green curve referring to $\mathcal{N}_{Lt}^{\pm}=0.313$. Fig. 2e follows again
from (VI.137) at $x=0.0002$ with the middle-green curve
$\mathcal{N}_{Lt}^{\pm}=0.329$. Fig. 2f refers to $x=0.00032$ with the middle-
green curve referring to $\mathcal{N}_{Lt}^{\pm}=0.313$. Fig. 2f refers to
$x=0.0008$ with the middle-green curve referring to
$\mathcal{N}_{Lt}^{\pm}=0.335$. Fig. 2f refers to $x=0.0013$ with the middle-
green curve referring to $\mathcal{N}_{Lt}^{\pm}=0.359$. The data from the H1
and ZEUS collaboration Abramowicz:2015mha  are shown in red. The data shown in
orange at very low $Q^{2}$ are from the E665 collaboration Adams:1996gu . Note
that we have reconstructed the orange data points for the cross section
$\sigma^{\pm}_{r,NC}(x,Q,s)$ at $\sqrt{s}=318\,GeV$ from the E665 data set for
the structure function $F_{2}(x,Q)$ at $0.0008\leq x\leq 0.001$ using
Whitlow:1990dr

$\displaystyle
R(x,Q)=\frac{F_{2}}{2xF_{1}}\bigg{(}1+\frac{4M_{p}^{2}x^{2}}{Q^{2}}\bigg{)}-1=R^{1990}(x.Q)$
(VI.139)

As noted above, the blue-band follows from the normalization range fixed by
$\mathcal{N}_{Lt}^{\pm}=0.274$ and $\mathcal{N}_{Lt}^{\pm}=0.375$.

(a)  (b)  (c)  (d)

Figure 3: (a): $F_{2}(x,Q)$ as given in (VI.2) using the small-x holographic
PDFs shown in Fig. 1. The red data points are from combined H1 and ZEUS
collaborations Abramowicz:2015mha  (reconstructed from their data for the
cross section $\sigma^{\pm}_{r,NC}(x,Q,s)$ using our small-x holographic PDFs
(VI.1) and (VI.1) shown in Fig. 1); (b): $\tilde{F}_{2}(x,Q)-F_{2}(x,Q)$ as
given in (VI.2) using the small-x holographic PDFs shown in Fig. 1; (c):
$F_{2}^{\gamma Z}(x,Q)$ as given in (VI.2) using the small-x holographic PDFs
shown in Fig. 1; (d): $F_{2}^{Z}(x,Q)$ as given in (VI.2) using the small-x
holographic PDFs shown in Fig. 1.

(a)  (b)

Figure 4:  a: large-x holographic PDFs of the valence quarks in the proton as
given in (VI.3); b: large-x holographic PDFs of the sea quarks in the proton
as given in (VI.3). The blue and green bands are determined by varying
$\tilde{\tilde{\mathcal{N}}}_{L}^{+}$ between 17.715 and 35.431, and
$\tilde{\tilde{\mathcal{N}}}_{L}^{-}$ between 30.667 and 61.335. See text.

###  VI.3 Results in the large-x regime

As we noted in the introduction, DIS scattering in holographic QCD at moderate
values of partonic-x involves hadronic and not partonic constituents POL .
Indeed, in the large $N_{c}$ limit, the leading single trace twist
contributions acquire large anomalous dimensions and are suppressed. The
dominant contributions stem from double-trace operators with mesonic quantum
numbers. Another way to see this is to note that the large gauge coupling at
the low renormalization point, causes the color charges to undergo a rapid
depletion into a cascade of even weaker charges, making them visible to hard
probes only through double trace operators.

At large-x, DIS scattering is almost off the entire hadron making the
holographic approach pertinent. In this regime, the holographic limit enjoys
approximate conformal symmetry, with the structure functions and form factors
exhibiting various scaling laws including the parton-counting rules BF ; POL .
More specifically, we have for the sea contribution

$\displaystyle\lim_{x\rightarrow 1}\lim_{\frac{\kappa^{2}}{Q^{2}}\rightarrow
0}x\overline{u}(x,Q)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(F_{2s}^{W^{+}p}(x,Q)-xF_{3s}^{W^{+}p}(x,Q)\right)\,,$
$\displaystyle\lim_{x\rightarrow 1}\lim_{\frac{\kappa^{2}}{Q^{2}}\rightarrow
0}x\overline{d}(x,Q)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(F_{2s}^{W^{-}p}(x,Q)-xF_{3s}^{W^{-}p}(x,Q)\right)\,,$
(VI.140)

and for the valence contribution

$\displaystyle\lim_{x\rightarrow 1}\lim_{\frac{\kappa^{2}}{Q^{2}}\rightarrow
0}xu_{V}(x,Q)$ $\displaystyle=$
$\displaystyle\frac{1}{4}x\left(F_{3s}^{W^{-}p}(x,Q)+F_{3s}^{W^{+}p}(x,Q)\right)$
$\displaystyle+$
$\displaystyle\frac{1}{4}\left(F_{2s}^{W^{-}p}(x,Q)-F_{2s}^{W^{+}p}(x,Q)\right)\,,$
$\displaystyle\lim_{x\rightarrow 1}\lim_{\frac{\kappa^{2}}{Q^{2}}\rightarrow
0}xd_{V}(x,Q)$ $\displaystyle=$
$\displaystyle\frac{1}{4}x\left(F_{3s}^{W^{-}p}(x,Q)+F_{3s}^{W^{+}p}(x,Q)\right)$
(VI.141) $\displaystyle-$
$\displaystyle\frac{1}{4}\left(F_{2s}^{W^{-}p}(x,Q)-F_{2s}^{W^{+}p}(x,Q)\right)\,,$

with the even and odd structure functions given respectively by

$\displaystyle F_{2s}^{W^{\pm}p}(x,Q)$ $\displaystyle=$ $\displaystyle
F_{2s,Dirac}^{W^{\pm}p}(x,Q)+F_{2s,Pauli}^{W^{\pm}p}(x,Q)+F_{2s,mixed}^{W^{\pm}p}(x,Q)\,,$
$\displaystyle F_{3s}^{W^{\pm}p}(x,Q)$ $\displaystyle=$ $\displaystyle
F_{3s,Dirac}^{W^{\pm}p}(x,Q)+F_{3s,Pauli}^{W^{\pm}p}(x,Q)+F_{3s,mixed}^{W^{\pm}p}(x,Q)\,,$
(VI.142)

The large-x asymptotic of the Dirac+Pauli+Mixed structure functions following
from (V.3), can be worked out in closed form. For the even-parity structure
functions we have

$\displaystyle F_{2s,Dirac}^{W^{\pm}p}(x,Q)$ $\displaystyle=$
$\displaystyle\big{(}\tilde{\mathcal{N}}_{L}^{\pm}\big{)}^{2}\times\big{(}e^{\pm}_{Wnucleon}\big{)}^{2}\times\left(\frac{\tilde{\kappa}^{2}}{Q^{2}}\right)^{\tau-1}x^{\tau+1}(1-x)^{\tau-2}\,,$
$\displaystyle F_{2s,Pauli}^{W^{\pm}p}(x,Q)\Big{)}$ $\displaystyle=$
$\displaystyle\big{(}\tilde{\mathcal{N}}_{L}^{\pm}\big{)}^{2}\times\eta^{2}\times
4(\tau-1)^{2}\times\left(\frac{\tilde{\kappa}^{2}}{Q^{2}}\right)^{\tau-1}x^{\tau+1}(1-x)^{\tau-2}\,,$
$\displaystyle F_{2s,mixed}^{W^{\pm}p}(x,Q)\Big{)}$ $\displaystyle=$
$\displaystyle\big{(}\tilde{\mathcal{N}}_{L}^{\pm}\big{)}^{2}\times
e^{\pm}_{Wnucleon}\times\eta\times
4(\tau-1)\times\left(\frac{\tilde{\kappa}^{2}}{Q^{2}}\right)^{\tau-1}x^{\tau+1}(1-x)^{\tau-2}\,,$
(VI.143)

in agreement with a recent analysis in Jorrin:2020cil  (see their Eq. 88, Eq
105 and Eq. 132). For the odd-parity structure functions we also have

$\displaystyle F_{3s,Dirac}^{W^{\pm}p}(x,Q)$ $\displaystyle=$
$\displaystyle\big{(}\tilde{\mathcal{N}}_{L}^{\pm}\big{)}^{2}\times\big{(}e^{\pm}_{Wnucleon}\big{)}^{2}\times\left(\frac{\tilde{\kappa}^{2}}{Q^{2}}\right)^{\tau-1}x^{\tau+1}(1-x)^{\tau-2}\,,$
$\displaystyle F_{3s,Pauli}^{W^{\pm}p}(x,Q)\Big{)}$ $\displaystyle=$
$\displaystyle\big{(}\tilde{\mathcal{N}}_{L}^{\pm}\big{)}^{2}\times\eta^{2}\times
4(\tau-1)^{2}\times\left(\frac{\tilde{\kappa}^{2}}{Q^{2}}\right)^{\tau-1}x^{\tau+1}(1-x)^{\tau-2}\,,$
$\displaystyle F_{3s,mixed}^{W^{\pm}p}(x,Q)\Big{)}$ $\displaystyle=$
$\displaystyle\big{(}\tilde{\mathcal{N}}_{L}^{\pm}\big{)}^{2}\times
e^{\pm}_{Wnucleon}\times\eta\times
4(\tau-1)\times\left(\frac{\tilde{\kappa}^{2}}{Q^{2}}\right)^{\tau-1}x^{\tau+1}(1-x)^{\tau-2}\,,$
(VI.144)

also in agreement with the recent results in Jorrin:2020cil  (see their Eq.
88, Eq 104 and Eq. 131). Note that eventhough the Pauli vertex contribution
involves an additional vierbein in comparison to the Dirac one, hence an a
priori extra suppression with the z-parameter and therefore $Q^{2}$ by
duality, it is balanced by the extra z-derivative in the magnetic-like
coupling $F_{\mu z}$, causing both contributions to scale identically with
large $Q^{2}$ at large-x.

(a)  (b)

Figure 5:  a: $\sigma^{\pm}_{r,NC}(x,Q,s)$ as given in (VI.137) for $x=0.93$,
using the large-x holographic PDFs shown in Fig. 4, with the orange data
points from JLAB Malace:2009kw ; b: same as in a for $x=0.85$, with the red
data points from the combined SLAC and BCDMS collaborations Whitlow:1990dr .
See text.

(a)  (b)

(c)  (d)

(e)  (f)

Figure 6:  (a)$F_{2}(x,Q)$ as given in (VI.2) using the large-x holographic
PDFs shown in Fig. 4. The red data points are from combined SLAC and BCDMS
collaborations Whitlow:1990dr  for $x=0.65$, $x=0.75$, and $x=0.85$; (b)
$\tilde{F}_{2}(x,Q)-F_{2}(x,Q)$ as given in (VI.2) using the large-x
holographic PDFs shown in Fig. 4; (c) $F_{2}^{\gamma Z}(x,Q)$ as given in
(VI.2) using the large-x holographic PDFs shown in Fig. 4; (d)
$F_{2}^{Z}(x,Q)$ as given in (VI.2) using the large-x holographic PDFs shown
in Fig. 4; (e) $F_{3}^{\gamma Z}(x,Q)$ as given in (VI.2) using the large-x
holographic PDFs shown in Fig. 4; (g) $F_{3}^{Z}(x,Q)$ as given in (VI.2)
using the large-x holographic PDFs shown in Fig. 4.

Here $\tilde{\mathcal{N}}_{L}^{\pm}\equiv c(\tau)\times\mathcal{N}_{L}^{\pm}$
with $c(\tau)$ are undetermined coefficients, that can in principle be fixed
rigorously in a specific model. As we noted earlier, strict bulk-to-boundary
correspondence implies ${\cal N}_{L}^{\pm}=1$ in the double and dual limit.
Here we are assuming proportionality between the bulk and boundary structure
functions with ${\cal N}_{L}^{\pm}$ as overall parameters BRAGA , that capture
partially the finite corrections to the double limit. In the numerical
analysis at large-x to follow, we fit only the two parameters
$\tilde{\tilde{\mathcal{N}}}_{L}^{\pm}$ which are defined as

$\displaystyle\tilde{\tilde{\mathcal{N}}}_{L}^{\pm}\,\equiv\,\big{(}\tilde{\mathcal{N}}_{L}^{\pm}\big{)}^{2}\times\big{(}e^{\pm}_{Wnucleon}\big{)}^{2}+\big{(}\tilde{\mathcal{N}}_{L}^{\pm}\big{)}^{2}\times\eta^{2}\times
4(\tau-1)^{2}+\big{(}\tilde{\mathcal{N}}_{L}^{\pm}\big{)}^{2}\times
e^{\pm}_{Wnucleon}\times\eta\times 4(\tau-1)\,.$ (VI.145)

In terms of (VI.145), the structure functions (VI.3) simplify

$\displaystyle F_{2s}^{W^{\pm}p}(x,Q)=F_{3s}^{W^{\pm}p}(x,Q)$ $\displaystyle=$
$\displaystyle\big{(}\tilde{\tilde{\mathcal{N}}}_{L}^{\pm}\big{)}^{2}\times\left(\frac{\tilde{\kappa}^{2}}{Q^{2}}\right)^{\tau-1}x^{\tau+1}(1-x)^{\tau-2}\,.$
(VI.146)

with $\tilde{\kappa}=0.350\,\rm{GeV}$ as fixed by the mass of the proton and
$\rho$ meson in CARLSON .

(VI.146) scales as $(1/Q^{2})^{{\tau-1}}$ asymptotically in agreement with the
hard scaling laws expected from strong coupling POL , but vanishes as
$(1-x)^{\tau-2}$ at large $x$ in contrast to $(1-x)^{2\tau-3}$ suggested in
DRELL . The large-x behavior at strong coupling follows from the observation
that for the virtual photon with amplitude $1/Q$ to scatter off the nucleon as
a Dirac fermion, the latter has to shrink to a size $(1/Q)^{\tau}$, with a
scattering probability$(1/Q^{2})^{\tau-1}$. As a result, the structure
function at large $Q^{2}$ but fixed $s\sim Q^{2}(1-x)$ scales as

$\displaystyle F_{2}(x,Q)\sim
Q^{2}\bigg{|}\bigg{(}\frac{1}{Q^{2}}\bigg{)}^{\tau-1}\bigg{|}^{2}\,\bigg{(}s=Q^{2}(1-x)\bigg{)}^{\alpha}$
(VI.147)

To reproduce the hard scaling law asymptotically requires
$2\alpha+6-4\tau=2-2\tau$ or $\alpha=\tau-2$, which is the large-x scaling in
(VI.146). To recover Bjorken scaling for the structure function requires
$2\alpha+6-4\tau=0$ or $\alpha=2\tau-3$ which is the large-x scaling law
suggested in DRELL . We expect the latter to set in at very large $Q^{2}$.

In Fig. 4a we show the behavior the valence distributions $xu_{V}(x,Q)$ in the
upper blue-dark-band, and $xd_{V}(x,Q)$ in the lower green-light-band at
$Q^{2}=6.5$ GeV2, following from (VI.3). In Fig. 4b we show the behavior of
the sea distributions $x\overline{u}(x,Q)$ in the lower blue-dark-band, and
$x\overline{d}(x,Q)$ in the upper green-light-band at the same $Q^{2}=6.5$
GeV2, following from (VI.3). The normalization coefficients delimiting the
blue and green bands are determined by varying
$\tilde{\tilde{\mathcal{N}}}_{L}^{+}$ between 17.715 and 35.431, and
$\tilde{\tilde{\mathcal{N}}}_{L}^{-}$ between 30.667 and 61.335.

To assess the range of validity in parton-x of the holographic results at
large-x, we re-asses the reduced neutral charge $\sigma^{\pm}_{r,NC}(x,Q,s)$
as given in (VI.137) solely in terms of the large-x holographic PDFs. The
results are shown in Fig. 4 at low $Q^{2}$ for $\sqrt{s}=5.565$ GeV. The
orange data points are from JLAB Malace:2009kw . The red data points are from
the combined SLAC and BCDMS collaborations Whitlow:1990dr  (see Figure 5.14 in
Whitlow:1990dr ). Note that the $F_{2}(x,Q)$ data of Whitlow:1990dr  to
$\sigma^{\pm}_{r,NC}(x,Q,s)$ where converted using (VI.139). These results
show that the range of validity of the holographic results at large-x is
limited to $0.75\leq x\leq 1$ (see also below). Our results for the reported
DIS $e^{\pm}p$ in Fig. 5 are consistent with the recent holographic results
reported in FolcoCapossoli:2020pks  in the large-x regime (see Figure 5 in
FolcoCapossoli:2020pks ).

###  VI.4 Structure functions at intermediate $Q^{2}$

To investigate further the range of validity of the holographic PDFs at
intermediate $Q^{2}$ and large-x, we show in Fig. 6a the holographic structure
function $F_{2}(x)$ in (VI.2) versus $x$ at $Q^{2}=6.5$ GeV2, evaluated using
the results for the valence distributions shown in Fig. 4a. The red data
points are from the combined SLAC and BCDMS collaborations in Whitlow:1990dr
for $x=0.65$, $x=0.75$, and $x=0.85$. In Fig. 6b we show the difference
$\tilde{F}_{2}(x,Q)-F_{2}(x,Q)$ versus $x$ as given in (VI.2) for $Q^{2}=6.5$
GeV2 using also the large-x holographic PDFs shown in Fig. 4. In Figs.
4c,d,e,f we show respectively, $F_{2}^{\gamma Z}(x,Q)$, $F_{2}^{Z}(x,Q)$,
$F_{3}^{\gamma Z}(x,Q)$ and $F_{3}^{Z}(x,Q)$ versus large-x for $Q^{2}=6.5$
GeV2 using also the large-x holographic PDFs in Fig. 4. Fig. 6a shows that the
holographic results are compatible with the SLAC and BCDMS data Whitlow:1990dr
in the range $0.75\leq x\leq 1$ for $Q^{2}=6.5$ GeV2.

###  VI.5 Comparison to the empirical CTEQ, LHAPDF and NNPDF data sets

We have also compared our holographic PDF sets both for the small-x and
large-x regimes for the valence and sea distributions, to 292 PDF sets from
the CTEQ and LHAPDF projects incorporated within the ManeParse Mathematica
package Clark:2016jgm . The global comparison is displayed in Figs. 7 and
Figs. 8 for $Q^{2}=6.5$ GeV2. Figs. 8a,c show the 292 PDF sets in multiple
solid lines from CTEQ and LHAPDF data set Clark:2016jgm , in comparison to our
holographic PDF sets shown in blue-light-band for the up-distributions, and in
green-light-band for the down-distributions. The holographic growth at
intermediate-x overcomes these data sets, but joins with a smaller subset of
these data at very low-x. Figs. 8b,d show that the holographic sea at large-x
is relatively small but consistent overall with all the data sets in the
expected applicability of our approach. A similar comparison for the
holographic valence distributions is shown in Figs. 7a,c at low-x and Figs.
8b,d at large-x with consistency in both limits with the CTEQ and LHAPDF data
set Clark:2016jgm .

In Figs. 9 we comparre the holographic results for the sea distributions to
the NNPDF collaboration PDF sets with error bars (shown in red band) from the
LHAPDF projects incorporated within the ManeParse Mathematica package
Clark:2016jgm . Figs. 9a,c show our results in blue-dark-band for the sea of
up quarks, and in green-light-band for the sea of down quarks. Figs. 9b,d show
also our results for the sea of up and down quarks at large-x. The growth of
the holographic sea at low-x is larger in the intermediate range, but slower
at very low-x. At large-x, our holographic results for the sea are relatively
small but consistent with the reported data set. In Figs. 10 the valence up
and down distributions are compared to the same data set with the same
notations as for the sea. At low-x, the holographic results are consistent
with zero for the valence up and down quark distributions. At larger-x, the
holographic results are consistently larger for the up quark valence
distribution in the region of validity of the holographic construction.

(a)  (b)  (c)  (d)

Figure 7:  (a,b) Our holographic PDF sets (shown in blue band) compared to 292
PDF sets (shown in multiple solid lines) from CTEQ and LHAPDF projects
incorporated within the ManeParse Mathematica package Clark:2016jgm ; (c,d)
Our holographic PDF sets (shown in green band) compared to 292 PDF sets (shown
in multiple solid lines) from CTEQ and LHAPDF projects incorporated within the
ManeParse Mathematica package Clark:2016jgm .

(a)  (b)  (c)  (d)

Figure 8:  (a,b) Our holographic PDF sets (shown in blue band) compared to 292
PDF sets (shown in multiple solid lines) from CTEQ and LHAPDF projects
incorporated within the ManeParse Mathematica package Clark:2016jgm ; (c,d)
Our holographic PDF sets (shown in green band) compared to 292 PDF sets (shown
in multiple solid lines) from CTEQ and LHAPDF projects incorporated within the
ManeParse Mathematica package Clark:2016jgm .

(a)  (b)  (c)  (d)

Figure 9:  (a,b) Our holographic PDF sets (shown in blue band) compared to
NNPDF collaboration PDF sets with error bars (shown in red band) from LHAPDF
projects incorporated within the ManeParse Mathematica package Clark:2016jgm ;
(c,d) Our holographic PDF sets (shown in green band) compared to NNPDF
collaboration PDF sets with error bars (shown in red band) from LHAPDF
projects incorporated within the ManeParse Mathematica package Clark:2016jgm .

##  VII Conclusions

We presented a comprehensive holographic derivation of DIS neutrino-nucleon
scattering process using AdS/CFT duality formulated in a slice of AdS5 with
bulk U(2) valued vector mesons to describe a tower of iso-vector and iso-axial
particles, and a fermionic isodoublet to account for the even- and odd-parity
proton and neutron excitations. To distinguish the vectors from the axials,
chiral symmetry is broken either through a tachyon in the bi-fundamental
representation both for the hard or soft wall model, or though boundary
conditions for the hard wall. The Yukawa coupling of the bi-fundamental
tachyon field between the even-odd bulk fermionic fields was ignored, since
the DIS regime is mostly sensitive to the high lying part of the nuclon
spectrum for which the effect is negligible.

Using Witten diagrams, we have derived the pertinent Dirac and Pauli fiorm
factors for both the direct or diagonal currents and the transition or off-
diagonal currents. The results for the direct vector form factors are
consistent with those in CARLSON , and for the transition form factors they
are consistent with those in BRAGA . To our knowledge, the results for the
direct and transition axial form factors are new. We use them for a
holographic estimate of the axial coupling $g_{A}$ which is directly sensitive
to the explicit and implicit breaking of chiral symmetry.

We have used the transition form factors for the left-currents, to explicitly
construct the s-channel contributions to neutrino and anti-neutrino DIS
scattering through the pertinent Witten diagram. This has led to the explicit
identification of the s-channel holographic contributions to the even- and
odd-parity structure functions for both $W^{\pm}$ charged currents. The
t-channel contributions stemming from the Reggeized Pomeron exchange is
explicitly constructed and shown to dominate at low-x the even-parity
structure functions. The Reggeon excchange through the bulk Chern-Simons
interaction is shown to dominate the low-x odd-parity structure function.

For intermediate values of Bjorken-x, DIS scattering in holography is very
different from QCD. The leading twist-2 operator in the OPE of the JJ-
correlators acquire large anomalous dimension, and is dwarfed by the double
trace operators which carry higher twists but are protected. In this regime,
DIS scattering is off a poin-like hadron which is similar to scattering for
$x\sim 1$. When $x\ll 1$, DIS scattering in QCD is dominated by Reggeon
exchange and is dual to spin-2 or spin-1 Reggeized exchange in bulk. So
lepton-nucleon scattering in holography allows to probe the partonic content
of the nucleon in the two regimes of $x\sim 1$ and $x\ll 1$.

We have carried explicit calculations and comparison to the availabel data
both for the structure functions and PDFs from LHAPDF and CTEQ, in both of
these regimes. Our results show consistency for large-x for the intermediate
range of $Q^{2}<10$ GeV2 where scaling violations are still substantial. At
low-x our results show a somehow larger growth at intermediate but low-x. The
results at higher $Q^{2}$-resolution can be obtained through standard QCD
evolution.

##  VIII Acknowledgements

K.M. is supported by the U.S. Department of Energy, Office of Science, Office
of Nuclear Physics, contract no. DE-AC02-06CH11357, and an LDRD initiative at
Argonne National Laboratory under Project No. 2020-0020. I.Z. is supported by
the U.S. Department of Energy under Contract No. DE-FG-88ER40388.

(a)  (b)  (c)  (d)

Figure 10:  (a,b) Our holographic PDF sets (shown in green band) compared to
NNPDF collaboration PDF sets with error bars (shown in red band) from LHAPDF
projects incorporated within the ManeParse Mathematica package Clark:2016jgm ;
(c,d) Our holographic PDF sets (shown in blue band) compared to NNPDF
collaboration PDF sets with error bars (shown in red band) from LHAPDF
projects incorporated within the ManeParse Mathematica package Clark:2016jgm .

##  Appendix A.1 UV(1) gauge fields

The UV(1) gauge fields solves the equations of motion in bulk

$\displaystyle\Box V^{\mu}$ $\displaystyle+$ $\displaystyle
ze^{\tilde{\kappa}^{2}z^{2}}\partial_{z}\Big{(}e^{-\tilde{\kappa}^{2}z^{2}}\frac{1}{z}\partial_{z}V^{\mu}\Big{)}\,=\,0$
$\displaystyle\Box V_{z}$ $\displaystyle-$
$\displaystyle\partial_{z}\Big{(}\partial_{\mu}V^{\mu}\Big{)}\,=\,0\,.$
(A.1.148)

subject to the gauge condition

$\partial_{\mu}V^{\mu}\,+\,ze^{\kappa^{2}z^{2}}\partial_{z}\Big{(}e^{-\kappa^{2}z^{2}}\frac{1}{z}V_{z}\Big{)}\,=\,0\,,$
(A.1.149)

and the boundary condition

$V_{\mu}(z,y)|_{z\to 0}\,=\,\epsilon_{\mu}(q)\,e^{-iq\cdot y}\,,$ (A.1.150)

with polarization $\epsilon_{\mu}(q)$. The normalizable solutions in KK modes
are

$\displaystyle V_{\mu}(z,y)$ $\displaystyle=$
$\displaystyle\epsilon_{\mu}(q)\,e^{-iq\cdot
y}\,\Gamma\bigg{(}1+\frac{Q^{2}}{4\tilde{\kappa}^{2}}\bigg{)}\,\,\tilde{\kappa}^{2}z^{2}\,\,{\cal
U}\bigg{(}1+\frac{Q^{2}}{4\tilde{\kappa}^{2}};2;\tilde{\kappa}^{2}z^{2}\bigg{)}$
$\displaystyle V_{z}(z,y)$ $\displaystyle=$
$\displaystyle\frac{i}{2}\,\epsilon(q)\cdot q\,e^{-iq\cdot
y}\,\,\Gamma\bigg{(}1+\frac{Q^{2}}{4\tilde{\kappa}^{2}}\bigg{)}\,\,z\,\,{\cal
U}\bigg{(}1+\frac{Q^{2}}{4\tilde{\kappa}^{2}};1;\tilde{\kappa}^{2}z^{2}\bigg{)}\,,$

for $q^{2}=Q^{2}>0$, where $\,{\cal U}(a;b;w)\,$ are the confluent
hypergeometric functions of the second kind.

###  A.1.1 Bulk to boundary propagator

The non-normalizable wave function for the virtual photon is of the form
$V_{\mu}(z,y)=\mathcal{V}(Q,z)\,\epsilon_{\mu}e^{-iq\cdot y}$, with
$q^{2}=-Q^{2}<0$ and

$\displaystyle\mathcal{V}(Q,z)=g_{5}\sum_{n}\frac{F_{n}\phi_{n}(z)}{Q^{2}+m_{n}^{2}}\,.$
(A.1.152)

For the soft wall

$\displaystyle\phi_{n}(z)=c_{n}\tilde{\kappa}^{2}z^{2}L_{n}^{1}(\tilde{\kappa}^{2}z^{2})\equiv
J_{A}(m_{n},z)\,,$ (A.1.153)

with $c_{n}=\sqrt{{2}/{n+1}}$ fixed by the normalization condition

$\displaystyle\int
dz\,\sqrt{g}e^{-\phi}\,(g^{xx})^{2}\,\phi_{n}(z)\phi_{m}(z)=\delta_{nm}\,,$

Also recall that ${\cal
V}(Q,z)=\frac{1}{z^{\prime}}\partial_{z^{\prime}}G(z,z^{\prime})|_{z^{\prime}=\epsilon}$,
ties to the bulk to boundary propagator with

$\displaystyle\mathcal{V}(Q,z)=\tilde{\kappa}^{2}z^{2}\,\,\Gamma\bigg{(}1+\frac{Q^{2}}{4\tilde{\kappa}^{2}}\bigg{)}\,\,{\cal
U}\bigg{(}1+\frac{Q^{2}}{4\tilde{\kappa}^{2}};2;\tilde{\kappa}^{2}z^{2}\bigg{)}=\tilde{\kappa}^{2}z^{2}\int_{0}^{1}\frac{dx}{(1-x)^{2}}x^{a}{\rm
exp}\Big{(}-\frac{x}{1-x}\tilde{\kappa}^{2}z^{2}\Big{)}\,,$ (A.1.155)

and satisfies ${\cal V}(0,z)={\cal V}(Q,0)=1$.

###  A.1.2 Bulk to bulk propagator

The bulk-to-bulk propagator for the massive mesons, for space-like momenta
($q^{2}=-Q^{2}$), can be written as

$\displaystyle G_{\mu\nu}(z,z^{\prime})=i{\cal
T}_{\mu\nu}G(z,z^{\prime})=i\bigg{(}-\eta_{\mu\nu}+\frac{q_{\mu}q_{\nu}}{m_{n}^{2}}\bigg{)}G(z,z^{\prime})=\sum_{n}J_{V}(m_{n},z)\,\tilde{G}_{\mu\nu}(Q,m_{n})\,J_{V}(m_{n},z^{\prime})\,,$

with

$\displaystyle
G(z,z^{\prime})=-\sum_{n}\frac{\phi_{n}(z)\phi_{n}(z^{\prime})}{Q^{2}+m_{n}^{2}}\,,$
(A.1.157)

and

$\displaystyle
J_{V}(m_{n},z)=\phi_{n}(z)\,,\,\,\,\,\,\,\,\,\,\,\,\tilde{G}_{\mu\nu}(Q,m_{n})=\bigg{(}-\eta_{\mu\nu}+\frac{q_{\mu}q_{\nu}}{m_{n}^{2}}\bigg{)}\frac{-i}{Q^{2}+m_{n}^{2}}\,.$
(A.1.158)

Note that the bulk to boundary propagator follows by taking $z\rightarrow 0$,
which simplifies (A.1.157) as

$\displaystyle G(z\rightarrow
0,z^{\prime})\approx\bigg{[}-\frac{\phi_{n}(z\rightarrow
0)}{-g_{5}F_{n}}\bigg{]}\sum_{n}\frac{-g_{5}F_{n}\phi_{n}(z^{\prime})}{Q^{2}+m_{n}^{2}}=\frac{z^{2}}{2}\sum_{n}\frac{g_{5}F_{n}\phi_{n}(z^{\prime})}{Q^{2}+m_{n}^{2}}=\frac{z^{2}}{2}\mathcal{V}(Q,z^{\prime}),$
(A.1.159)

where we used the fact that

$\displaystyle
F_{n}=\frac{1}{g_{5}}\bigg{(}-e^{-\phi}\frac{1}{z^{\prime}}\partial_{z^{\prime}}\phi_{n}(z^{\prime})\bigg{)}_{z^{\prime}=\epsilon}=-\frac{2}{g_{5}}c_{n}(n+1)\tilde{\kappa}^{2}\,,$

with $\phi_{n}(z\rightarrow 0)\approx c_{n}\tilde{\kappa}^{2}z^{2}(n+1)$. Note
that the ratio in the bracket in (A.1.159) is n-independent! If we define the
meson decay constant as $f_{n}=-{F_{n}}/{m_{n}}$, we have

$\displaystyle\phi_{n}(z)=\frac{f_{n}}{m_{n}}\times
2g_{5}\tilde{\kappa}^{2}z^{2}L_{n}^{1}(\tilde{\kappa}^{2}z^{2})\,,$ (A.1.161)

in line with vector meson dominance (VMD).

##  Appendix B.2 Fermionic fields

We start by considering a general fermionic field in bulk AdS that solves the
free Dirac equation in the presence of a soft wall. For that, we remove the
dilaton field by rescaling

$\Psi(y,z)=e^{+\tilde{\kappa}^{2}z^{2}/2}\tilde{\psi}(y,z),$ (B.2.162)

with the reduced field solving

$\biggl{[}i\not\\!\partial+\gamma^{5}\partial_{z}-\frac{2}{z}\gamma^{5}-\frac{1}{z}\Big{(}M+\tilde{\kappa}^{2}z^{2}\Big{)}\biggr{]}\tilde{\psi}(y,z)=0\,,$
(B.2.163)

where $\not\\!\partial=\gamma^{\mu}\,\partial_{\mu}$. We now decompose the
reduced field into two chiral copies

$\tilde{\psi}(y,z)=\tilde{\psi}_{L}(y,z)+\tilde{\psi}_{R}(y,z)\,,\quad\tilde{\psi}_{L/R}=P_{L/R}\tilde{\psi}\,.$
(B.2.164)

with $(\gamma^{5})^{2}=1$, and $P_{L/R}=P_{\mp}=\frac{1\mp\gamma^{5}}{2}$. The
solution to (B.2.163) follows by further reduction into $L/R$ KK modes using

$\tilde{\psi}_{L/R}(y,z)=e^{-iP\cdot
y}\frac{1}{2}\Big{(}1\mp\gamma^{5}\Big{)}u_{s}(P)\,\frac{z^{2}}{R^{2}}\tilde{f}_{L/R}(z)\,,$
(B.2.165)

with $u_{s}(P)$ a free Dirac spinor in 4-dimensions, and the $L/R$ KK modes
$\tilde{f}_{L/R}(z)$ now satisfying

$\biggl{[}-\partial_{z}^{2}+\tilde{\kappa}^{4}z^{2}+2\tilde{\kappa}^{2}\Big{(}M\mp\frac{1}{2}\Big{)}+\frac{M(M\pm
1)}{z^{2}}\biggr{]}\tilde{f}_{L/R}(z)=P^{2}\,\tilde{f}_{L/R}(z)\,.$ (B.2.166)

###  B.1 Spectrum and modes

The equation of motion (B.2.166) has normalizable solutions only when $P^{2}$
has

$P^{2}_{n}=M_{n}^{2}=4\tilde{\kappa}^{2}\Big{(}n+M+\frac{1}{2}\Big{)}\,,$
(B.167)

with $n=0,1,2,...,$, which are

$\displaystyle\tilde{f}^{n}_{L}(z)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2\Gamma(n+1)}{\Gamma(n+M+3/2)}}\
\tilde{\kappa}^{M+3/2}\ z^{M+1}\ e^{-\tilde{\kappa}^{2}z^{2}/2}\
L_{n}^{M+1/2}(\tilde{\kappa}^{2}z^{2})\,,$ $\displaystyle\tilde{f}^{n}_{R}(z)$
$\displaystyle=$ $\displaystyle\sqrt{\frac{2\Gamma(n+1)}{\Gamma(n+M+1/2)}}\
\tilde{\kappa}^{M+1/2}\ z^{M}\ e^{-\tilde{\kappa}^{2}z^{2}/2}\
L_{n}^{M-1/2}(\tilde{\kappa}^{2}z^{2})$

with the normalization condition

$\displaystyle\int\limits_{0}^{\infty}dz\,\tilde{f}^{n^{\prime}}_{L/R}(z)\tilde{f}^{n}_{L/R}(z)=\delta_{n^{\prime}n}$
(B.169)

We can rewrite the normalized wavefunctions for the bulk Dirac fermion (or
proton and neutron which correspond to the $n=0$ states) as

$\displaystyle\Psi(p,z)$ $\displaystyle=$
$\displaystyle\psi_{R}(z)\Psi^{0}_{R}(p)+\psi_{L}(z)\Psi^{0}_{L}(p)\,,$
$\displaystyle\bar{\Psi}(p,z)$ $\displaystyle=$
$\displaystyle\psi_{R}(z)\bar{\Psi}^{0}_{R}(p)+\psi_{L}(z)\bar{\Psi}^{0}_{L}(p)\,,$

where for the soft-wall

$\displaystyle\psi_{R}(z)=\frac{\tilde{n}_{R}}{\tilde{\kappa}^{\tau-2}}z^{\frac{5}{2}}\xi^{\frac{\tau-2}{2}}L_{0}^{(\tau-2)}(\xi)\,,$
$\displaystyle\psi_{L}(z)=\frac{\tilde{n}_{L}}{\tilde{\kappa}^{\tau-1}}z^{\frac{5}{2}}\xi^{\frac{\tau-1}{2}}L_{0}^{(\tau-1)}(\xi)\,,$

with $\xi=\tilde{\kappa}^{2}z^{2}$ and the generalized Laguerre polynomials
$L_{n}^{(\alpha)}(\xi)$,
$\tilde{n}_{R}=\tilde{n}_{L}\tilde{\kappa}^{-1}\sqrt{\tau-1}$, and
$\tilde{n}_{L}=\tilde{\kappa}^{\tau}\sqrt{{2}/{\Gamma(\tau)}}$. The
normalization of the bulk wave functions can also be rewritten as

$\displaystyle\int_{0}^{\infty}dz\,\sqrt{g}\,e^{-\phi}\,e^{\mu}_{a}\,\psi_{R/L}^{2}(z)=\delta^{\mu}_{a}\,,$

with $\phi=\tilde{\kappa}^{2}z^{2}$, and the inverse vielbein
$e^{\mu}_{a}=\sqrt{{\left|g^{\mu\mu}\right|}}\delta^{\mu}_{a}$ (no summation
intended in $\mu$). The leading twist parameter is $\tau=3$, and the free Weyl
spinors $\Psi^{0}_{R/L}(p)=P_{\pm}u(p)$ and
$\bar{\Psi}^{0}_{R/L}(p)=\bar{u}(p)P_{\mp}$ are tied to the free spinor
normalization at the boundary

$\displaystyle\bar{u}(p)u(p)=2m_{N}\qquad\qquad
2m_{N}\times\bar{u}(p^{\prime})\gamma^{\mu}u(p)=\bar{u}(p^{\prime})(p^{\prime}+p)^{\mu}u(p)\,.$
(B.173)

###  B.2 Bulk to boundary propagator

The bulk to boundary propagator (or the nonrenormalizable mode) of the bulk
Dirac fermion is given by

$\displaystyle\Psi(p,z)=\sum_{n}\frac{\sqrt{2g_{5}^{2}}F_{n}^{R}(p)\psi^{R}_{n}(z)}{p^{2}-M_{n}^{2}}\times\Psi^{0}_{R}(p)+\sum_{n}\frac{\sqrt{2g_{5}^{2}}F_{n}^{L}(p)\psi^{L}_{n}(z)}{p^{2}-M_{n}^{2}}\times\Psi^{0}_{L}(p)\,,$
(B.174)

where the fermionic decay functions are defined as

$\displaystyle F_{n}^{R}(p)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2g_{5}^{2}}}\times
p\,\tilde{\kappa}\sqrt{\frac{2\Gamma(\tau-1+n)}{\Gamma(\tau-1)\Gamma(n+1)}}\,,$
$\displaystyle F_{n}^{L}(p)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2g_{5}^{2}}}\times
M_{n}\tilde{\kappa}\sqrt{\frac{2\Gamma(\tau-1+n)}{\Gamma(\tau-1)\Gamma(n+1)}}\,.$

Note that the on-shell bulk Dirac fermions are just the residues at the poles
$p^{2}-m_{n}^{2}$ in (B.174). The in-out baryonic states used in our DIS
analysis throughout are

$\displaystyle\Psi_{i}$ $\displaystyle=$ $\displaystyle e^{-ip\cdot
y}e^{+\tilde{\kappa}^{2}z^{2}/2}\,\frac{z^{2}}{R^{2}}\Big{[}\Big{(}\frac{1-\gamma^{5}}{2}\Big{)}u_{s_{i}}(p)\,\tilde{f}^{0}_{L}(z)\,+\,\Big{(}\frac{1+\gamma^{5}}{2}\Big{)}u_{s_{i}}(p)\,\tilde{f}^{0}_{R}(z)\Big{]}\times\sqrt{2g_{5}^{2}}\times
F_{N}(p)$ (B.176) $\displaystyle\Psi_{X}$ $\displaystyle=$ $\displaystyle
e^{-iP_{X}\cdot
y}e^{+\tilde{\kappa}^{2}z^{2}/2}\,\frac{z^{2}}{R^{2}}\Big{[}\Big{(}\frac{1-\gamma^{5}}{2}\Big{)}u_{s_{X}}(P_{X})\,\tilde{f}^{n_{X}}_{L}(z)\,+\,\Big{(}\frac{1+\gamma^{5}}{2}\Big{)}u_{s_{X}}(P_{X})\,\tilde{f}^{n_{X}}_{R}(z)\Big{]}\times\sqrt{2g_{5}^{2}}\times
F_{X}(P_{X})\,\,,$ (B.178)

where $s_{i}$ and $s_{X}$ label they in-out spin, and identified
$F_{0}^{R}=F_{0}^{L}=F_{N}(p)$ (ground state) and
$F_{n}^{R}=F_{n}^{L}=F_{X}(P_{X})$ (excited state).

###  C Yukawa coupling through the tachyon

The inclusion of a bi-fundamental tachyon field $X(x,z)$ in bulk to lift the
degeneracy between the vector and axial-vector mesons, would also imply a
Yukawa coupling between the even-odd bulk fermionic fields $\Psi_{1,2}$ which
we have not considered here. More specifically YEE

$\displaystyle{\cal S}_{12,X}=\frac{1}{2g_{5}^{2}}\frac{g_{X}}{2}\int
d^{5}x\sqrt{g}\,\left(\overline{\Psi}_{1}(x,z)X(x,z)\Psi_{2}(x,z)+\overline{\Psi}_{2}(x,z)X^{\dagger}(x,z)\Psi_{1}(x,z)\right)$
(B.180)

which would mix $1,2$ and lifts the degeneracy of the low-lying even and odd
parity states in the nucleon sector, $(\Psi_{1}\pm\Psi_{2})/\sqrt{2}$. A first