# Gluon parton densities in soft-wall AdS/QCD

Valery E. Lyubovitskij  Institut für Theoretische Physik, Universität
Tübingen, Kepler Center for Astro and Particle Physics, Auf der Morgenstelle
14, D-72076 Tübingen, Germany  Departamento de Física y Centro Científico
Tecnológico de Valparaíso-CCTVal, Universidad Técnica Federico Santa María,
Casilla 110-V, Valparaíso, Chile  Department of Physics, Tomsk State
University, 634050 Tomsk, Russia  Tomsk Polytechnic University, 634050 Tomsk,
Russia  Ivan Schmidt  Departamento de Física y Centro Científico Tecnológico
de Valparaíso-CCTVal, Universidad Técnica Federico Santa María, Casilla 110-V,
Valparaíso, Chile

###### Abstract

We study the gluon parton densities [parton distribution functions (PDFs),
transverse momentum distributions (TMDs), generalized parton distributions
(GPDs)] and form factors in soft-wall AdS/QCD. We show that the power behavior
of gluon parton distributions and form factors at large values of the light-
cone variable and large values of square momentum is consistent with quark
counting rules. We also show that the transverse momentum distributions
derived in our approach obey the model-independent Mulders-Rodrigues
inequalities without referring to specific model parameters. All gluon parton
distributions are defined in terms of the unpolarized and polarized gluon PDFs
and profile functions. The latter are related to gluon PDFs via differential
equations.

##  I Introduction

The soft-wall AdS/QCD model Karch:2006pv -Andreev:2006vy , based on breaking
of conformal symmetry due a quadratic dilaton field, has achieved important
progress in the description and understanding of hadron structure (mass
spectrum, parton distributions, form factors, thermal properties, etc.)
Brodsky:2014yha . One of the main advantages of the soft-wall AdS/QCD is the
analytical implementation of quark counting rules Brodsky:1973kr , in the
description of hadronic form factors at large $Q^{2}$ (power scaling)
Brodsky:2014yha -Lyubovitskij:2020gjz . Together with form factors, parton
distributions of quarks and gluons in hadrons play important role in the QCD
description of hadron structure and spin physics (see, e.g., Refs. Lin:2017snn
; Lin:2020rut ; Angeles-Martinez:2015sea ; Diehl:2013xca ; Boer:2011fh  for
reviews). Based on QCD factorization, one can separate effects of strong
interactions at small and long distances, characterizing respectively
perturbative and nonperturbative dynamics of quarks and gluons. In particular,
the nonperturbative part is parametrized by parton distribution functions,
which are universal functions for each hadron and independent of the specific
process. Since these universal parton distributions cannot be directly
calculated in QCD, they are either extracted from data (world data analysis)
or calculated using lattice QCD, or in QCD motivated approaches (light-front
QCD, AdS/QCD, quark and potential models, etc.), which have been applied to
extract or predict the PDFs, TMDs, and GPDs (for a recent overview see, e.g.,
Ref. Lin:2020rut ).

As happens with form factors, the partonic distributions obey model-
independent scaling rules at large $x$. The starting point for these rules was
the derivation of the Drell-Yan-West (DYW) relation Drell:1969km  between the
large-$Q^{2}$ behavior of nucleon electromagnetic form factors and the
large-$x$ behavior of the structure functions (see also Ref. Bloom:1970xb  for
the extension to inelastic scattering) and quark counting rules Brodsky:1973kr
. Based on the results of Refs. Drell:1969km ; Bloom:1970xb ; Brodsky:1973kr
the behavior of the quark PDF in nucleon $q_{v}(x)\sim(1-x)^{p}$ at $x\to 1$
was related with the scaling of the proton Dirac form factor
$F_{1}^{p}(Q^{2})\sim 1/(Q^{2})^{(p+1)/2}$ at large $Q^{2}$, where the
parameter $p$ is related to the number of constituents in the proton (or twist
$\tau$) as $p=2\tau-3$ Drell:1969km ; Blankenbecler:1974tm . The model-
independent predictions of perturbative QCD (pQCD) for the GPDs of pion ${\cal
H}_{q}^{\pi}(x,Q^{2})$ and nucleon ${\cal H}_{q}^{N}(x,Q^{2})$, ${\cal
E}_{q}^{N}(x,Q^{2})$, are given in Ref. Yuan:2003fs , at large $x$ and finite
$Q^{2}$, as:

$\displaystyle{\cal H}_{q}^{\pi}(x,Q^{2})\sim(1-x)^{2}\,,\quad{\cal
H}_{q}^{N}(x,Q^{2})\sim(1-x)^{3}\,,\quad{\cal
E}_{q}^{N}(x,Q^{2})\sim(1-x)^{5}\,.$ (1)

The prediction of pQCD for the pion PDF $q_{\pi}(x)\sim(1-x)^{2}$ at large
$x$, which follows from the prediction for GPDs Yuan:2003fs ), was supported
by the updated analysis Aicher:2010cb  of the E615 data Conway:1989fs  on the
cross section of the Drell-Yan (DY) process $\pi^{-}N\to\mu^{+}+\mu^{-}X$,
including next-to-leading logarithmic threshold resummation effects:
$q_{\pi}(x)\sim(1-x)^{2.03}$ at the initial scale $\mu_{0}=0.63$ GeV
Aicher:2010cb .

In the case of gluons the large and small $x$ behavior of parton densities was
studied in Brodsky:1989db ; Brodsky:1994kg . In particular, the QCD
constraints on unpolarized $G(x)$ and polarized $\Delta G(x)$ gluon PDFs in
nucleons have been derived in Brodsky:1989db :

$\displaystyle G(x)$ $\displaystyle=$
$\displaystyle\frac{N}{x}\,\biggl{[}5(1-x)^{4}-4(1-x)^{5}+(1-x)^{6}\biggr{]}=\frac{N}{x}\,(1-x)^{4}\,\biggl{[}2+2x+x^{2}\biggr{]}\,,$
$\displaystyle\Delta G(x)$ $\displaystyle=$
$\displaystyle\frac{N}{x}\,\biggl{[}5(1-x)^{4}-4(1-x)^{5}-(1-x)^{6}\biggr{]}=N\,(1-x)^{4}\,\biggl{[}6-x\biggr{]}\,,$
(2)

where $N$ is the normalization constant.

On the other hand, the densities $G(x)$ and $\Delta G(x)$ can be written as
combinations of the helicity-aligned $G^{+}(x)=G_{g\uparrow/N\uparrow}(x)$ and
helicity-antialigned $G^{-}(x)=G_{g\downarrow/N\uparrow}(x)$ gluon
distributions

$\displaystyle G(x)=G^{+}(x)+G^{-}(x)\,,\quad\Delta G(x)=G^{+}(x)-G^{-}(x)\,,$
(3)

where

$\displaystyle G^{+}(x)$ $\displaystyle=$
$\displaystyle\frac{N}{x}\,(1-x)^{4}\,\biggl{[}1+4x\biggr{]}\,,$
$\displaystyle G^{-}(x)$ $\displaystyle=$
$\displaystyle\frac{N}{x}\,(1-x)^{6}\,.$ (4)

Thus, at large $x$ the power scaling of gluon PDFs and their ratios read
Brodsky:1989db

$\displaystyle G^{+}(x)\sim(1-x)^{4}\,,\quad
G^{-}(x)\sim(1-x)^{6}\,,\quad\frac{G^{-}(x)}{G^{+}(x)}\sim(1-x)^{2}\,,$ (5)

which is consistent with QCD constraints Bjorken:1969mm ; Gribov:1972ri ,
dictated matching the signs of the quark and gluon helicities and the even
power scaling of gluon PDFs.

At small $x$ the gluon asymmetry ratio $\Delta G/G$ behaves as

$\displaystyle\frac{\Delta G(x)}{G(x)}\to N_{q}x\,,$ (6)

where $N_{q}$ is the number of valence quarks in a specific hadron (e.g.,
$N_{q}=3$ in case of nucleon). The scaling rule (6) is consistent with Reggeon
exchange arguments Brodsky:1988ip .

The moments of the gluon PDFs, the momentum fraction $\langle x_{g}\rangle$
and the helicity $\Delta G$ carried by intrinsic gluons in nucleon, obtained
in Ref. Brodsky:1988ip , are

$\displaystyle\langle
x_{g}\rangle=\int\limits_{0}^{1}dxxG(x)=\frac{10}{21}N\,,\quad\Delta
G=\int\limits_{0}^{1}dx\Delta G(x)=\frac{7}{6}N\,.$ (7)

The ratio of these two moments $\Delta G/\langle x_{g}\rangle=\frac{49}{20}$
is independent of the $N$.

In Ref. Brodsky:1994kg  a slightly different small $x$ constraint on the ratio
$\Delta G/G$, due to color coherence of gluon couplings, has been implemented,
leading to the following form of gluon PDFs

$\displaystyle G(x)$ $\displaystyle=$
$\displaystyle\frac{N}{x}\,(1-x)^{4}\biggl{[}1+(1-x)^{2}\biggr{]}\,,$
$\displaystyle\Delta G(x)$ $\displaystyle=$
$\displaystyle\frac{N}{x}\,(1-x)^{4}\biggl{[}1-(1-x)^{2}\biggr{]}\,.$ (8)

One can see that the two sets of gluon PDFs, presented in Eqs. (I) and (I),
differ by the presence of an extra linear term $+4x$ in the set of Ref.
Brodsky:1989db . As a consequence, the two models considered in Refs.
Brodsky:1989db ; Brodsky:1994kg  produce different results for the moments of
gluon PDFs. In particular, the model considered in Ref. Brodsky:1994kg  gives:

$\displaystyle 1=\langle x_{g}\rangle=\frac{12}{35}N\,,\quad\Delta
G=\int\limits_{0}^{1}dx\Delta G(x)=\frac{11}{30}N\,,\quad\frac{\Delta
G}{\langle x_{g}\rangle}=\frac{77}{72}\,.$ (9)

As we know, the PDFs are related to the transverse momentum dependent (TMD)
parton distributions, upon integration of the latter over the struck parton
transverse momentum ${\bf k}_{\perp}$. TMDs provide a three-dimensional
picture of hadrons and their knowledge is important for the description of QCD
processes, using TMD factorization at small values of the transverse momentum
of particles produced in hadronic collisions. At present the TMDs are under
intensive study both experimentally and theoretically (for recent progress
see, e.g., Refs. Lin:2020rut ; Angeles-Martinez:2015sea ). Leading twist quark
TMDs have been proposed in a series of papers in Refs. Ralston:1979ys ;
Brodsky:2002cx . Gluon TMDs have been introduced in Ref. Mulders:2000sh  and
later considered in Ref. Meissner:2007rx -Kaur:2020pvc .

The calculation of partonic densities in hadrons, using soft-wall AdS/QCD, can
be done indirectly using an integral representation for the hadronic form
factors or normalization conditions for the hadronic wave functions
Vega:2009zb -Lyubovitskij:2020otz . One should stress that the study of parton
densities in the soft-wall approach is closely related to other important
problems such as the construction of hadronic effective wave functions
Brodsky:2014yha ; Brodsky:2007hb ; Vega:2009zb ; Branz:2010ub ; Brodsky:2011xx
; Lyubovitskij:2020otz ; Gutsche:2014zua ; deTeramond:2018ecg ;
Lyubovitskij:2013ski ; Gutsche:2013zia ; Gutsche:2014yea ; Gutsche:2016gcd ;
Brodsky:2020ajy ; Vega:2020ctz . First results in soft-wall AdS/QCD on parton
densities – quark GPDs – have been obtained in Ref. Vega:2009zb . The idea for
the extraction of GPDs in Ref. Vega:2009zb  was based on the use of the
integral representation of the hadronic form factor with twist $\tau$
Brodsky:2007hb ; Vega:2010ns ; Gutsche:2011vb . It can also be written in
closed form in terms of the beta function $B(\alpha,\beta)$

$\displaystyle
F_{\tau}(Q^{2})=\int\limits_{0}^{1}dy\,(\tau-1)\,(1-y)^{\tau-2}\,y^{a}=(\tau-1)\,B(\tau-1,a+1)$
(10)

Upon identification of the $y$ variable with the light-cone momentum fraction
$x$, this gives an expression for both the PDFs $q_{\tau}(x)$ and the GPDs
${\cal H}_{\tau}(x,Q^{2})$ Brodsky:2007hb ; Vega:2010ns ; Gutsche:2013zia :

$\displaystyle q_{\tau}(x)=(\tau-1)\,(1-x)^{\tau-2}\,,\quad{\cal
H}_{\tau}(x,Q^{2})=q_{\tau}(x)\,x^{a}\,.$ (11)

Nevertheless, this $x$ dependence of the PDFs and GPDs contradicts model-
independent results: the DY inclusive counting rules for $q_{\tau}(x)$ at
$x\to 1$ Drell:1969km ; Blankenbecler:1974tm ; Yuan:2003fs  and the prediction
of pQCD for GPDs — pion ${\cal H}_{q}^{\pi}(x,Q^{2})$ and nucleon ${\cal
H}_{q}^{N}(x,Q^{2})$, ${\cal E}_{q}^{N}(x,Q^{2})$ at large $x$ and finite
$Q^{2}$ Yuan:2003fs .

To solve the problem of large-$x$ scaling of parton densities in soft-wall
AdS/QCD, it was found in Ref. Lyubovitskij:2013ski  that the interpretation of
the variable $y$ in the integral representation (32) as light-cone variable is
not truly correct and that one should propose a generalized $x$ dependent
light-cone variable $y(x)$. With this assumption the power behavior of
hadronic PDFs and GPDs at large $x$ can be made consistent with the model-
independent results of Refs. Drell:1969km ; Blankenbecler:1974tm ; Yuan:2003fs
, provided that an appropriate choice of the $x$ dependence of the function
$y(x)$ is imposed. In particular, the simplest choice for the function $y(x)$
was found to be:

$\displaystyle y_{N}(x)=\exp\Big{[}-\log(1/x)(1-x)^{2/(N-1)}\Big{]}$ (12)

leading to the correct large-$x$ scaling of PDFs and GPDs in mesons

$\displaystyle q_{\tau}^{M}(x)\sim{\cal
H}_{\tau}^{M}(x,Q^{2})\sim(1-x)^{2\tau-2}$ (13)

at $N=2\tau-2$ and in baryons

$\displaystyle q_{\tau}^{B}(x)\sim{\cal
H}_{\tau}^{B}(x,Q^{2})\sim(1-x)^{2\tau-3}$ (14)

at $N=2\tau-3$. The function $y_{\tau}(x)$ obeys the following boundary
conditions $y_{\tau}(0)=0$ and $y_{\tau}(1)=1$. Extension of ideas proposed in
Ref. Lyubovitskij:2013ski  were further developed in Ref. Lyubovitskij:2020otz
. In particular, it was explicitly demonstrated how to correctly define
hadronic parton distributions (PDFs, TMDs, and GPDs) in the soft-wall AdS/QCD
approach, in order for them to be consistent with quark counting rules and
Drell-Yan-West duality. All parton distributions are defined in terms of
profile functions universal for each specific hadron.

Recently a similar idea was considered in the framework of light-front
holographic QCD (LFHQCD) deTeramond:2018ecg ; Brodsky:2020ajy ; Chang:2020kjj
. In particular, a function [named as $w(x)$] was introduced in the integral
representation of the form factor deTeramond:2018ecg ; Brodsky:2020ajy :

$\displaystyle
F_{\tau}(Q^{2})=\frac{1}{N_{\tau}}\,\int\limits_{0}^{1}dx\,w^{\prime}(x)\,[w(x)]^{Q^{2}/4\lambda-1/2}\,[1-w(x)]^{\tau-2}$
(15)

In fact, both mathematical extensions considered in Refs. Lyubovitskij:2013ski
and deTeramond:2018ecg ; Brodsky:2020ajy  are equivalent. The only difference
is that in Refs. deTeramond:2018ecg ; Brodsky:2020ajy  an extra power $-1/2$
was included in the $[w(x)]^{Q^{2}/4\lambda-1/2}$, while in the soft-wall
model of Brodsky:2007hb ; Vega:2010ns ; Gutsche:2011vb  the factor is
$[w(x)]^{Q^{2}/4\lambda}$. In other words, the soft-wall model Brodsky:2007hb
; Vega:2010ns ; Gutsche:2011vb  and LFHQCD Brodsky:2007hb ; Vega:2010ns ;
Gutsche:2011vb  deal with slightly different analytical expressions for the
hadronic form factors: $F_{\tau}(Q^{2})\sim B(\tau-1,1+Q^{2}/4\lambda)$ in the
soft-wall AdS/QCD and $F_{\tau}(Q^{2})\sim B(\tau-1,1/2+Q^{2}/4\lambda)$ in
LFHQCD.

The main objective of this paper is to extend the ideas proposed in Refs.
Lyubovitskij:2013ski ; Lyubovitskij:2020otz , applying them to gluon parton
densities. In particular, we will derive results for gluon parton
distributions (PDFs, TMDs, and GPDs) in hadrons with arbitrary quark content
and spin. The obtained power behavior of parton distributions at large values
of light-cone variable $x$ are then also consistent with quark counting rules
and DYW duality. All parton distributions are defined in terms of profile
functions depending on the light-cone coordinate and are fixed from PDFs and
electromagnetic form factors.

The paper is organized as follows. In Sec. II we present an overview of our
approach and consider the derivation of PDFs, TMDs, and GPDs using quarks as
an example. In Sec. III we consider the derivation of gluon parton densities
(PDFs, TMDs, and GPDs). In Sec. IV we present numerical applications of our
analytical results for gluon parton densities using different parametrizations
for $G(x)$ and $\Delta G(x)$ PDFs. Finally, Sec. V contains our summary.

##  II Basic notions of soft-wall AdS/QCD approach

Here we briefly overview the soft-wall AdS/QCD approach (for more details see,
e.g., Refs. Vega:2010ns ; Gutsche:2011vb ). The framework for studies of the
dynamics of boson $\Phi_{M_{1}\cdots M_{J}}(x,z)$ and fermion
$\Psi_{M_{1}\cdots M_{J-1/2}}(x,z)$ fields with spin $J$ (dual to mesons and
baryons, respectively) in a five-dimensional AdS space, is specified by the
metric

$\displaystyle
ds^{2}=g_{MN}dx^{M}dx^{N}=\eta_{ab}\,e^{2A(z)}\,dx^{a}dx^{b}=e^{2A(z)}\,(\eta_{\mu\nu}dx^{\mu}dx^{\nu}-dz^{2})\,,\hskip
28.45274pt\eta_{\mu\nu}={\rm diag}(1,-1,\ldots,-1)\,,$ (16)

where $M$ and $N=0,1,\cdots,d$ are the space-time (base manifold) indices,
$a=(\mu,z)$ and $b=(\nu,z)$ are the local Lorentz (tangent) indices, $g_{MN}$
and $\eta_{ab}$ are curved and flat metric tensors, which are related by the
vielbein $\epsilon_{M}^{a}(z)=e^{A(z)}\,\delta_{M}^{a}$ as
$g_{MN}=\epsilon_{M}^{a}\epsilon_{N}^{b}\eta_{ab}$. Here $z$ is the
holographic coordinate, $R$ is the AdS radius, and $g=|{\rm
det}g_{MN}|=e^{10A(z)}$. We restrict ourselves to a conformal-invariant metric
with $A(z)=\log(R/z)$.

The boson action is written as:

$\displaystyle S_{B}$ $\displaystyle=$ $\displaystyle\frac{(-)^{J}}{2}\int
d^{d}xdz\sqrt{g}\,e^{-\varphi(z)}\biggl{[}g^{MN}g^{M_{1}N_{1}}\cdots
g^{M_{J}N_{J}}\,\partial_{M}\Phi_{M_{1}\cdots
M_{J}}(x,z)\,\partial_{N}\Phi_{N_{1}\cdots N_{J}}(x,z)$ (17) $\displaystyle-$
$\displaystyle(\mu_{J}^{2}+V_{J}(z))\,g^{M_{1}N_{1}}\cdots
g^{M_{J}N_{J}}\Phi_{M_{1}\cdots M_{J}}(x,z)\,\Phi_{N_{1}\cdots
N_{J}}(x,z)\biggr{]}$

where the bosonic spin-$J$ field $\Phi_{M_{1}\cdots M_{J}}(x,z)$ is described
by a symmetric, traceless tensor, satisfying the conditions

$\displaystyle\partial^{M_{1}}\Phi_{M_{1}M_{2}\cdots M_{J}}=0\,,\quad\quad
g^{M_{1}M_{2}}\Phi_{M_{1}M_{2}\cdots M_{J}}=0\,,$ (18)

Here $V_{J}(z)=e^{-2A(z)}U_{J}(z)$, where $U_{J}(z)$ is the effective dilaton
potential

$\displaystyle
U_{J}(z)=\frac{1}{2}\varphi^{\prime\prime}(z)\,+\,(d-1-2J)\,\varphi^{\prime}(z)A^{\prime}(z)$
(19)

and

$\displaystyle\mu_{J}^{2}R^{2}=(\Delta-J)(\Delta+J-4)$ (20)

is the bulk mass. The quadratic dilaton field $\varphi(z)$ is specified as
$\varphi(z)=\kappa^{2}z^{2}$, where $\kappa$ is the dimensional parameter. The
dimension of the boson AdS fields $\Delta$ is identified with twist $\tau$ as
$\Delta=\tau=N+L$, where $N$ is the number of partons and $L$ is the orbital
angular momentum.

Restricting to the axial gauge $\Phi_{\cdots z\cdots}(x,z)=0$ and performing
the Kaluza-Klein expansion

$\displaystyle\Phi^{\mu_{1}\cdots\mu_{J}}(x,z)=\sum\limits_{n}\
\Phi^{\mu_{1}\cdots\mu_{J}}_{n}(x)\ \Phi_{n}(z)$ (21)

one can derive the equation of motion (EOM) for the profile function
$\phi_{n\tau}(z)=e^{3A(z)/2}\,\Phi_{n}(z)$:

$\displaystyle\Big{[}-\frac{d^{2}}{dz^{2}}+\frac{4(\tau-2)^{2}-1}{4z^{2}}+U_{J}(z)\Big{]}\phi_{n\tau}(z)=M^{2}_{n\tau\\!J}\phi_{n\tau}(z)$
(22)

with analytical solutions for the bulk profile

$\displaystyle\phi_{n\tau}(z)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{\Gamma(\tau-1)}}\,\kappa^{\tau-1}\,z^{\tau-3/2}\,e^{-\kappa^{2}z^{2}/2}\,L_{n}^{\tau-2}(\kappa^{2}z^{2})$
(23)

and mass spectrum

$\displaystyle
M^{2}_{n\tau\\!J}=4\kappa^{2}\Big{(}n+\frac{\tau+J}{2}-1\Big{)}\,.$ (24)

Here $L_{n}^{m}(x)$ is the generalized Laguerre polynomial.

In the case of fermion fields $\Psi_{K_{1}\cdots K_{J-1/2}}(x,z)$ with spin
$J$, the soft-wall AdS/QCD action reads Gutsche:2011vb :

$\displaystyle S_{F}$ $\displaystyle=$ $\displaystyle\int
d^{d}xdz\,\sqrt{g}\,e^{-\varphi(z)}\,g^{K_{1}N_{1}}\cdots
g^{K_{J-1/2}N_{J-1/2}}\,\biggl{[}\frac{i}{2}\bar{\Psi}_{K_{1}\cdots
K_{J-1/2}}(x,z)\epsilon_{a}^{M}\Gamma^{a}{\cal D}_{M}\Psi_{N_{1}\cdots
N_{J-1/2}}(x,z)$ (25) $\displaystyle-$ $\displaystyle\frac{i}{2}({\cal
D}_{M}\Psi_{K_{1}\cdots
K_{J-1/2}}(x,z))^{\dagger}\Gamma^{0}\epsilon_{a}^{M}\Gamma^{a}\Psi_{N_{1}\cdots
N_{J-1/2}}(x,z)-\bar{\Psi}_{K_{1}\cdots
K_{J-1/2}}(x,z)\Big{(}\mu+V_{F}(z)\Big{)}\Psi_{N_{1}\cdots
N_{J-1/2}}(x,z)\biggr{]}\,,$

where $V_{F}(z)=\varphi(z)/R$ is the dilaton potential, ${\cal D}_{M}$ is the
covariant derivative acting on the spin-tensor field $\Psi^{\pm}_{N_{1}\cdots
N_{J-1/2}}$ as:

$\displaystyle{\cal D}_{M}\Psi_{N_{1}\cdots
N_{J-1/2}}=\partial_{M}\Psi_{N_{1}\cdots
N_{J-1/2}}-\frac{1}{8}\omega_{M}^{ab}[\Gamma_{a},\Gamma_{b}]\Psi_{N_{1}\cdots
N_{J-1/2}}\,,$ (26)

where
$\omega_{M}^{ab}=A^{\prime}(z)\,(\delta^{a}_{z}\delta^{b}_{M}-\delta^{b}_{z}\delta^{a}_{M})$
is the spin connection term, and $\Gamma^{a}=(\gamma^{\mu},-i\gamma^{5})$ are
the Dirac matrices.

After expanding the fermion field in left- and right-chirality components
$\Psi^{L/R}=(1\mp\gamma^{5})/2\,\Psi$ and performing a KK expansion for the
$\Psi^{L/R}(x,z)$ fields $\Psi^{L/R}(x,z)=\sum\limits_{n}\ \Psi^{L/R}_{n}(x)\
F_{n}^{L/R}(z)$, one can obtain decoupled Schrödinger EOMs for the fermion
bulk profiles $f_{n}^{L/R}(z)=e^{2A(z)}\,F_{n}^{L/R}(z)$:

$\displaystyle\biggl{[}-\partial_{z}^{2}+\kappa^{4}z^{2}+2\kappa^{2}\Big{(}m\mp\frac{1}{2}\Big{)}+\frac{m(m\pm
1)}{z^{2}}\biggr{]}f_{n\tau}^{L/R}(z)=M_{n\tau}^{2}\,f_{n\tau}^{L/R}(z)\,,$
(27)

where $m=\tau-3/2=L+3/2$ and

$\displaystyle f_{n\tau}^{L}(z)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2\Gamma(n+1)}{\Gamma(n+\tau)}}\ \kappa^{\tau}\
z^{\tau-1/2}\ e^{-\kappa^{2}z^{2}/2}\ L_{n}^{\tau-1}(\kappa^{2}z^{2})\,,$
$\displaystyle f_{n\tau}^{R}(z)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2\Gamma(n+1)}{\Gamma(n+\tau-1)}}\ \kappa^{\tau-1}\
z^{\tau-3/2}\ e^{-\kappa^{2}z^{2}/2}\ L_{n}^{\tau-2}(\kappa^{2}z^{2})$ (28)

and

$\displaystyle
M_{n\tau}^{2}=4\kappa^{2}\Big{(}n+\tau-1\Big{)}=4\kappa^{2}\Big{(}n+L+2\Big{)}\,.$
(29)

In order to study electromagnetic properties of hadrons we need to obtain the
vector bulk-to-boundary propagator $V(q,z)$, dual to the $q^{2}$-dependent
electromagnetic current:

$\displaystyle\partial_{z}\biggl{(}\frac{e^{-\varphi(z)}}{z}\,\partial_{z}V(-q^{2},z)\biggr{)}+q^{2}\frac{e^{-\varphi(z)}}{z}\,V(-q^{2},z)=0\,.$
(30)

The latter equation is solved analytically in terms of the gamma $\Gamma(n)$
and Tricomi $U(a,b,z)$ functions, with the result:

$\displaystyle V(Q^{2},z)=\Gamma(1+a)\,U(a,0,\kappa^{2}z^{2})\,,$ (31)

where $Q^{2}=-q^{2}$ and $a=Q^{2}/(4\kappa^{2})$. It is convenient to use the
integral representation for $V(Q,z)$ Grigoryan:2007my

$\displaystyle
V(Q^{2},z)=\kappa^{2}z^{2}\int\limits_{0}^{1}\frac{dy}{(1-y)^{2}}\,y^{a}\,e^{-\kappa^{2}z^{2}\frac{y}{1-y}}\,.$
(32)

The expression for the hadron form factors is given in the soft-wall AdS/QCD
by

$\displaystyle
F_{n\tau}(Q^{2})=\int\limits_{0}^{\infty}dz\,\phi_{n\tau}^{2}(z)\,V(Q^{2},z)\,,$
(33)

where the integrand contains the square of the holographic wave function in
fifth dimension $z$ (dual to hadron wave function), multiplied with the vector
bulk-to-boundary propagator $V(Q^{2},z)$.

Next we illustrate the derivation of PDFs in soft-wall AdS/QCD for hadrons
with arbitrary partonic content (twist) Lyubovitskij:2020otz . In the
following, for simplicity, we restrict our discussion to a consideration of
ground states of hadrons with $n=0$. We start with the hadronic wave function
normalization condition, which depends on the holographic variable $z$:

$\displaystyle 1=\int\limits_{0}^{1}dz\,\phi_{\tau}^{2}(z)$ (34)

where $\phi_{\tau}(z)$ is the AdS bulk profile function (for simplicity we
restrict here to the bosonic case; extension to the fermion case is
straightforward Lyubovitskij:2020otz ). In the next step we use the integral
representation for unity

$\displaystyle
1=-e^{\kappa^{2}z^{2}}\,\int\limits_{0}^{1}d\biggl{[}f_{\tau}(x)\,e^{-\kappa^{2}z^{2}/(1-x)^{2}}\biggr{]}=e^{\kappa^{2}z^{2}}\,\int\limits_{0}^{1}dx\,\biggl{[}\frac{2f_{\tau}(x)\,\kappa^{2}z^{2}}{(1-x)^{3}}\,-f^{\prime}_{\tau}(x)\biggr{]}\,e^{-\kappa^{2}z^{2}/(1-x)^{2}}$
(35)

and insert it into Eq. (34). Here $x$ is the light-cone coordinate and
$f_{\tau}(x)$ is the profile function with boundary condition $f_{\tau}(0)=1$,
which is specific for a particular hadron and fixed from its PDF. The
functions $f_{\tau}(x)$ and $y_{\tau}(x)$ [see Eq. (12)] are related by:

$\displaystyle\Big{(}1-y_{\tau}(x)\Big{)}^{\tau-1}=f_{\tau}(x)\,(1-x)^{2(\tau-1)}$
(36)

or

$\displaystyle
y_{\tau}(x)=1-\Big{[}f_{\tau}(x)\Big{]}^{\frac{1}{\tau-1}}\,(1-x)^{2}\,.$ (37)

At $x=0$ the functions $y_{\tau}(x)$ and $f_{\tau}(x)$ obey the boundary
conditions $y_{\tau}(0)=0$ and $f_{\tau}(0)=1$. At $x=1$ function $f_{\tau}$
is finite and its value depends on the specific choice of twist $\tau$ (see
below), while $y_{\tau}(1)=1$ is independent on twist.

After integration over the variable $z$ one gets

$\displaystyle
1=\int\limits_{0}^{1}dx\,(1-x)^{2\tau-3}\,\biggl{[}2f_{\tau}(x)(\tau-1)-f^{\prime}_{\tau}(x)(1-x)\biggr{]}\,.$
(38)

Here and in the following the superscript $(^{\prime})$ means derivative with
respect to variable $x$. Using the general definition for the hadronic PDF
$q_{\tau}(x)$, in the form of an integral representation (zero moment) over
$x$ (here for simplicity we normalize the generic PDF to 1, but for specific
hadrons the corresponding normalization is understood, e.g., 1 for valence PDF
in pion, 2 and 1 for valence $u$ and $d$ quark PDF in the nucleon
respectively, etc.):

$\displaystyle 1=\int\limits_{0}^{1}dx\,q_{\tau}(x)$ (39)

we get:

$\displaystyle
q_{\tau}(x)=(1-x)^{2\tau-3}\,\biggl{[}2f_{\tau}(x)(\tau-1)-f^{\prime}_{\tau}(x)(1-x)\biggr{]}=\biggl{[}-f_{\tau}(x)(1-x)^{2\tau-2}\biggr{]}^{\prime}\,.$
(40)

We require that the hadronic PDF $q_{\tau}(x)$ must have the correct scaling
at large $x$ and this behavior is governed by the profile function
$f_{\tau}(x)$. In particular, we derived Lyubovitskij:2020otz ) the following
results for the PDFs of pions

$\displaystyle
q_{\pi}(x)=(1-x)^{2}\,\biggl{[}\frac{2f_{\pi}(x)}{1-x}-f^{\prime}_{\pi}(x)\biggr{]}=[-f_{\pi}(x)(1-x)^{2}]^{\prime}$
(41)

and nucleons

$\displaystyle u_{v}(x)$ $\displaystyle=$
$\displaystyle\biggl{[}-f_{u}(x)(1-x)^{4}\,\Big{(}1+2\eta_{u}+(1-x)^{2}(1-4\eta_{u})+2\eta_{u}(1-x)^{4}\Big{)}\biggr{]}^{\prime}\,,$
$\displaystyle d_{v}(x)$ $\displaystyle=$
$\displaystyle\biggl{[}-f_{d}(x)(1-x)^{4}\,\Big{(}\frac{1}{2}+2\eta_{d}+(1-x)^{2}\Big{(}\frac{1}{2}-4\eta_{d}\Big{)}+2\eta_{d}(1-x)^{4}\Big{)}\biggr{]}^{\prime}\,,$
$\displaystyle{\cal E}_{v}^{q}(x)$ $\displaystyle=$ $\displaystyle
k^{q}\,\Big{[}-f_{q}(x)\,(1-x)^{6}\Big{]}^{\prime}\,.$ (42)

where $f_{\pi}(x)$, $f_{u}(x)$, and $f_{d}(x)$ are the profile functions,
which define the distribution of quarks in the pion, and of $u$ and $d$ quarks
in the nucleon, respectively. Here $\eta_{u}=2\eta_{p}+\eta_{n}$ and
$\eta_{d}=2\eta_{n}+\eta_{p}$ are linear combinations of the nucleon couplings
with the vector field, related to nucleon anomalous magnetic moments $k_{N}$
and fixed as Abidin:2009hr ; Vega:2010ns :
$\eta_{N}=k_{N}\kappa/(2M_{N}\sqrt{2})$, where $M_{N}$ is the nucleon mass.

We have shown that the profile functions $f_{\pi}$, $f_{u}$, and $f_{d}$ can
be fixed using results of extractions of PDFs using world data. In particular,
using the pQCD prediction for the pion PDF Aicher:2010cb , at the initial
scale $\mu_{0}=0.63$ GeV

$\displaystyle
q_{\pi}(x,\mu_{0})=N_{\pi}x^{\alpha-1}\,(1-x)^{\beta}\,(1+\gamma
x^{\delta})\,,$ (43)

where $N_{\pi}$ is the normalization constant, $\alpha=0.70$, $\beta=2.03$,
$\gamma=13.8$, $\delta=2$ we found Lyubovitskij:2020otz

$\displaystyle
f_{\pi}(x)(1-x)^{2}=1-N_{\pi}\,x^{\alpha}\biggl{[}\frac{1}{\alpha}-\frac{2x}{\alpha+1}+\frac{x^{2}}{\alpha+2}+\gamma
x^{\delta}\,\biggl{(}\frac{1}{\alpha+\delta}-\frac{2x}{\alpha+\delta+1}+\frac{x^{2}}{\alpha+\delta+2}\biggr{)}\biggr{]}\,.$
(44)

In the case of nucleons we used the Martin-Stirling-Thornem-Watt (MSTW) 2008
LO global analysis of PDFs Martin:2009iq :

$\displaystyle u_{v}(x,\mu_{0})$ $\displaystyle=$ $\displaystyle
A_{u}\,x^{\alpha_{u}-1}\,(1-x)^{\beta_{u}}\,(1+\epsilon_{u}\sqrt{x}+\gamma_{u}x)\,,$
$\displaystyle d_{v}(x,\mu_{0})$ $\displaystyle=$ $\displaystyle
A_{d}\,x^{\alpha_{d}-1}\,(1-x)^{\beta_{d}}\,(1+\epsilon_{d}\sqrt{x}+\gamma_{d}x)\,,$
(45)

where $\mu_{0}=1$ GeV is the initial scale. The normalization constants
$A_{q}$ and the constants $\alpha_{q}$, $\beta_{q}$, $\epsilon_{q}$,
$\gamma_{q}$ were fixed as

$\displaystyle A_{u}=1.4335\,,\quad A_{d}=5.0903\,,$
$\displaystyle\alpha_{u}=0.45232\,,\quad\alpha_{d}=0.71978\,,$
$\displaystyle\beta_{u}=3.0409\simeq 3\,,\quad\beta_{d}=5.1244\simeq 5\,,$
(46) $\displaystyle\epsilon_{u}=-2.3737\,,\quad\epsilon_{d}=-4.3654\,,$
$\displaystyle\gamma_{u}=8.9924\,,\quad\gamma_{d}=7.4730\,.$

Using the MSTW results Martin:2009iq  we predict the $u$ and $d$ profile
functions in the nucleon as Lyubovitskij:2020otz

$\displaystyle f_{u}(x)\,(1-x)^{4}$ $\displaystyle=$ $\displaystyle
1-A_{u}x^{\alpha_{u}}\,\Big{[}B_{u}(x,0)+\epsilon_{u}\sqrt{x}B_{u}(x,1/2)+\gamma_{u}xB_{u}(x,1)\Big{]}\,,$
$\displaystyle f_{d}(x)\,(1-x)^{6}$ $\displaystyle=$ $\displaystyle
1-A_{d}x^{\alpha_{d}}\,\Big{[}B_{d}(x,0)+\epsilon_{d}\sqrt{x}D_{d}(x,1/2)+\gamma_{d}xB_{d}(x,1)\Big{]}\,,$
(47)

where

$\displaystyle
B_{u}(x,n)=\sum\limits_{k=0}^{3}\,\frac{C_{3}^{k}\,(-x)^{k}}{\alpha_{u}+n+k}\,,\quad
B_{d}(x,n)=\sum\limits_{k=0}^{5}\,\frac{C_{5}^{k}\,(-x)^{k}}{\alpha_{d}+n+k}\,.$
(48)

Here $C_{m}^{k}=\frac{m!}{k!(m-k)!}$ are the binomial coefficients.

TMDs were derived using a normalization condition involving integration over
light-cone $x$ and transverse momentum ${\bf k}_{\perp}$ coordinates:

$\displaystyle 1$ $\displaystyle=$
$\displaystyle-e^{\kappa^{2}z^{2}}\,\int\limits_{0}^{1}d\biggl{[}f_{\tau}(x)e^{-\kappa^{2}z^{2}/(1-x)^{2}}\biggr{]}\,\int
d^{2}{\bf k}_{\perp}\,\frac{D_{\tau}(x)}{\pi\kappa^{2}}\,e^{-{\bf
k}_{\perp}^{2}D_{\tau}(x)/\kappa^{2}}$ (49) $\displaystyle=$
$\displaystyle\frac{e^{\kappa^{2}z^{2}}}{\pi\kappa^{2}}\,\int\limits_{0}^{1}dx\,\int
d^{2}{\bf
k}_{\perp}\,\biggl{[}\frac{2f_{\tau}(x)\,\kappa^{2}z^{2}}{(1-x)^{3}}\,-f^{\prime}_{\tau}(x)\biggr{]}\,D_{\tau}(x)\,e^{-\kappa^{2}z^{2}/(1-x)^{2}}\,e^{-{\bf
k}_{\perp}^{2}D_{\tau}(x)/\kappa^{2}}\,,$

where $D_{\tau}(x)$ is the longitudinal factor, which is related to the
profile function $f_{\tau}(x)$ (or function $y_{\tau}(x)$ as

$\displaystyle
D_{\tau}(x)=\frac{\log[1/y_{\tau}(x)]}{(1-x)^{2}}=-\frac{\log\biggl{[}1-\Big{[}f_{\tau}(x)\Big{]}^{\frac{1}{\tau-1}}\,(1-x)^{2}\biggr{]}}{(1-x)^{2}}\,.$
(50)

For large $x$ the function $D_{\tau}(x)$ behaves as

$\displaystyle D_{\tau}(x)=\Big{[}f_{\tau}(x)\Big{]}^{\frac{1}{\tau-1}}\,,$
(51)

Notice that the explicit form of the function $D_{\tau}(x)$ was obtained via
matching the expression for the hadronic form factors in two approaches —
soft-wall AdS/QCD and LF QCD.

As illustration we present the result for the unpolarized quark TMD in the
nucleon (more details on our results on quark TMDs see in Lyubovitskij:2020otz
):

$\displaystyle f_{1}^{q_{v}}(x,{\bf k}_{\perp}^{2})$ $\displaystyle=$
$\displaystyle\biggl{[}q_{v}^{+}(x)+q_{v}^{-}(x)\,\frac{{\bf
k}_{\perp}^{2}\,D_{q}(x)}{\kappa^{2}}\biggr{]}\,\frac{D_{q}(x)}{2\pi\kappa^{2}}\,e^{-{\bf
k}_{\perp}^{2}D_{q}(x)/\kappa^{2}}$ (52)

where $q_{v}^{\pm}(x)=q_{v}(x)\pm\delta q_{v}(x)$, $q_{v}(x)$ and $\delta
q_{v}(x)$ are the helicity-independent and helicity-dependent valence quark
parton distributions. One can see that in our approach TMDs and PDFs are
related.

Finally, we present results for hadron GPDs ${\cal
H}_{\tau}(y_{\tau}(x),Q^{2})$ and form factor $F_{\tau}(Q^{2})$ with arbitrary
twist $\tau$ Gutsche:2013zia :

$\displaystyle{\cal
H}_{\tau}(y_{\tau}(x),Q^{2})=(\tau-1)\,(1-y_{\tau}(x))^{\tau-2}\,\Big{[}y_{\tau}(x)\Big{]}^{a}\,,\quad
a=\frac{Q^{2}}{4\kappa^{2}}\,.$ (53)

and

$\displaystyle F_{\tau}(Q^{2})=\int\limits_{0}^{1}dx{\cal
H}_{\tau}(y_{\tau}(x),Q^{2})\,.$ (54)

The GPD can be written in more convenient form in terms of the PDF:

$\displaystyle{\cal
H}_{\tau}(x,Q^{2})=q_{\tau}(x)\,\Big{[}y_{\tau}(x)\Big{]}^{a}=q_{\tau}(x)\,\exp\Big{(}-a\log\Big{[}1/y_{\tau}(x)\Big{]}\Big{)}\,.$
(55)

One can see that all parton distributions (PDFs, TMDs, GPDs) and form factors
are related to each other.

##  III Gluon parton distribution in soft-wall AdS/QCD

In this section we extend our formalism for parton distributions from quarks
to gluons. Such extension is straightforward. We just take into account that
in case of gluons we should use specific values of twist and the available
normalization conditions for gluon parton densities.

###  III.1 Gluon PDFs

We start with gluon PDFs. We will base our discussion on the QCD predictions
for the gluon PDFs derived in Refs. Brodsky:1989db ; Brodsky:1994kg . The
results of Refs. Brodsky:1989db  and Brodsky:1994kg  we will call,
respectively, as QCDI and QCDII. As stressed in Refs. Brodsky:1989db ;
Brodsky:1994kg , the gluon PDFs in nonexotic hadrons (e.g., pion, nucleon,
etc.) must fall off at large $x$ by at least one power faster than the
respective quark PDFs. It means that the helicity-nonflip gluon PDFs in pion
and nucleon should fall off at large $x$ as $(1-x)^{3}$ and $(1-x)^{4}$,
respectively. Hence, the EOM for the bulk wave function of the gluon content
in a hadron with twist $\tau$ is deduced from the general formula (40) for the
profile of parton density with arbitrary twist $f_{\tau}(x)$ by shifting the
twist as $\tau\to\tau+1/2$:

$\displaystyle\Big{[}-\frac{d^{2}}{dz^{2}}+\frac{4(\tau-3/2)^{2}-1}{4z^{2}}+U_{J}(z)\Big{]}\phi_{n\tau}(z)=M^{2}_{n\tau\\!J}\phi_{n\tau}(z)$
(56)

This EOM gives the following analytical solution for the gluon bulk profile

$\displaystyle\phi_{n\tau}(z)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{\Gamma(\tau-1/2)}}\,\kappa^{\tau-1/2}\,z^{\tau-1}\,e^{-\kappa^{2}z^{2}/2}\,L_{n}^{\tau-3/2}(\kappa^{2}z^{2})$
(57)

Using Eq. (57) we derive the differential equations for the gluon profile
functions $f_{\tau}^{G}(x)$ and $f_{\tau}^{\Delta G}(x)$, characterizing the
unpolarized and polarized gluon PDFs $G(x)$ and $\Delta G(x)$, respectively.
Using the corresponding normalization unpolarized and polarized gluon PDFs,
and QCD predictions for the moments of gluon PDFs (7) and (9) we derive the
following differential equations for $f_{\tau}^{G}(x)$ and $f_{\tau}^{\Delta
G}(x)$:

QCDI Brodsky:1989db

$\displaystyle\biggl{[}-f_{\tau}^{G}(x)(1-x)^{2\tau-1}\biggr{]}^{\prime}=xG(x)=N_{1}(1-x)^{4}(2+2x+x^{2})\,,$
$\displaystyle\biggl{[}-f_{\tau}^{\Delta
G}(x)(1-x)^{2\tau-1}\biggr{]}^{\prime}=\Delta G(x)=N_{1}(1-x)^{4}(6-x)\,,$
(58)

where $N_{1}=\frac{21}{10}\,\langle x_{g}\rangle=\frac{6}{7}\,\Delta G$.

QCDII Brodsky:1994kg

$\displaystyle\biggl{[}-f_{\tau}^{G}(x)(1-x)^{2\tau-1}\biggr{]}^{\prime}=xG(x)=N_{2}(1-x)^{4}(1+(1+x)^{2})\,,$
$\displaystyle\biggl{[}-f_{\tau}^{\Delta
G}(x)(1-x)^{2\tau-1}\biggr{]}^{\prime}=\Delta G(x)=N_{2}(1-x)^{4}(2-x)\,.$
(59)

where $N_{2}=\frac{35}{12}\,\langle x_{g}\rangle=\frac{30}{11}\,\Delta G$.

Taking $\tau=3$ as leading twist value of gluon PDF and integrating over $x$
with boundary conditions $f_{G}(0)=\langle x_{g}\rangle$ and $f_{\Delta
G}(0)=\Delta G$ one gets:

QCDI Brodsky:1989db

$\displaystyle f_{G}(x)$ $\displaystyle=$ $\displaystyle\langle
x_{g}\rangle\,\biggl{[}1+\frac{4}{5}x+\frac{3}{10}x^{2}\biggr{]}\,,$
$\displaystyle f_{\Delta G}(x)$ $\displaystyle=$ $\displaystyle\Delta
G\,\biggl{[}1-\frac{x}{7}\biggr{]}\,.$ (60)

QCDII Brodsky:1994kg

$\displaystyle f_{G}(x)$ $\displaystyle=$ $\displaystyle\langle
x_{g}\rangle\,\biggl{[}1-\frac{5}{6}x+\frac{5}{12}x^{2}\biggr{]}\,,$
$\displaystyle f_{\Delta G}(x)$ $\displaystyle=$ $\displaystyle\Delta
G\,\biggl{[}1-\frac{5}{11}x\biggr{]}\,.$ (61)

Note that at large $x$ the profile functions approach a constant and
degenerate for each QCD-based framework version:

QCDI Brodsky:1989db

$\displaystyle f_{g}=f_{G}(1)=f_{\Delta G}(1)=\frac{21}{10}\,\langle
x_{g}\rangle=N_{1}\,.$ (62)

QCDII Brodsky:1994kg

$\displaystyle f_{g}=f_{G}(1)=f_{\Delta G}(1)=\frac{7}{12}\,\langle
x_{g}\rangle=\frac{N_{2}}{5}\,.$ (63)

We derive also the profile function defining the gluon distribution in pions.
Choosing $\tau=2$ we get the differential equation for the gluon PDF in a
pion:

$\displaystyle\biggl{[}-f_{\pi}^{G}(x)(1-x)^{3}\biggr{]}^{\prime}=xG_{\pi}(x)=N_{G_{\pi}}\,(1-x)^{3}$
(64)

with boundary condition

$\displaystyle
f_{\pi}^{G}(0)=\int\limits_{0}^{1}dx\biggl{[}-f_{\pi}^{G}(x)(1-x)^{3}\biggr{]}^{\prime}=\int\limits_{0}^{1}dx\,x\,G_{\pi}(x)=\langle
x_{g}^{\pi}\rangle\,.$ (65)

As in the nucleon case, the value of the profile function $f_{\pi}^{G}(x)$ at
$x=0$ is related to the first moment of the gluon PDF $G_{\pi}(x)$. On the
other hand, $\langle x_{g}^{\pi}\rangle$ can be uniquely fixed from the
energy-momentum sum rule Gluck:1991ey ; Novikov:2020snp :

$\displaystyle\int\limits_{0}^{1}dx\,x\,\Big{[}2q_{\pi}(x)+S_{\pi}(x)+G_{\pi}(x)\Big{]}=2\langle
x^{\pi}\rangle+\langle x^{\pi}_{s}\rangle+\langle x^{\pi}_{g}\rangle=1\,,$
(66)

where $\langle x^{\pi}\rangle$, $\langle x^{\pi}_{s}\rangle$, and $\langle
x^{\pi}_{g}\rangle$ are the first moments of pion PDFs (valence, total sea,
and gluon contributions). Using the xFitter Developers Team Novikov:2020snp
parametrization for the gluon PDF in the pion

$\displaystyle xG_{\pi}(x)=\langle
x^{\pi}_{g}\rangle\,(1+C_{g})\,(1-x)^{C_{g}}$ (67)

where $C_{g}=3\pm 1$ at the initial scale squared $\mu_{0}^{2}=1.9$ GeV2 is
the parameter determining the large $x$ behavior of the PDF $G_{\pi}(x)$.
Solving the differential equation (64) one gets:

$\displaystyle f_{\pi}^{G}(x)=\langle x^{\pi}_{g}\rangle\,(1-x)^{C_{g}-2}$
(68)

Note that the solution (68) obeys the boundary condition
$f_{\pi}^{G}(0)=\langle x^{\pi}_{g}\rangle$.

###  III.2 Gluon TMDs

Now we are in a position to derive the soft-wall AdS/QCD prediction for the
T-even gluon TMDs in a hadron. In the following we use light-cone kinematics,
which is specified by two light-light vectors $n_{\pm}$ as Mulders:2000sh :

$\displaystyle n_{+}^{\mu}=(1,0,{\bf 0}_{\perp})\,,\quad n_{-}^{\mu}=(0,1,{\bf
0}_{\perp})$ (69)

obeying the conditions $n_{+}\,n_{-}=1$ and $n_{\pm}^{2}=0$. Any four momentum
$p$ is expanded through $n_{\pm}$ as

$\displaystyle
p^{\mu}=p^{+}n_{+}^{\mu}+p^{-}n_{-}^{\mu}+p^{\mu}_{\perp}=(p^{+},p^{-},{\bf
p}_{\perp})\,.$ (70)

We work in the hadron rest frame, where the nucleon $P$ and parton $k$ momenta
are specified as

$\displaystyle P^{\mu}$ $\displaystyle=$ $\displaystyle\Big{(}P^{+},P^{-},{\bf
0}_{\perp}\Big{)}\,,\quad P^{-}=\frac{M^{2}}{2P^{+}}\,,$ $\displaystyle
k^{\mu}$ $\displaystyle=$ $\displaystyle\Big{(}xP^{+},\frac{k^{2}+{\bf
k}_{\perp}^{2}}{2xP^{+}},{\bf k}_{\perp}\Big{)}\,,$ (71)

where $M$ is a hadron mass, $x=k^{+}/P^{+}$ is the longitudinal momentum
fraction carried by the gluon. The spin vector of the hadron is expanded into
one-dimensional longitudinal $S_{L}$ (helicity) and two-dimensional transverse
${\bf S}_{T}$ components in a manifestly covariant way as

$\displaystyle S^{\mu}=S_{L}^{\mu}+S_{T}^{\mu}\,,$ (72)

where

$\displaystyle S_{L}^{\mu}$ $\displaystyle=$ $\displaystyle
S_{L}\,\frac{Pn_{-}}{M}\,n_{+}^{\mu}-S_{L}\,\frac{Pn_{+}}{M}\,n_{-}^{\mu}=S_{L}\,\biggl{(}\frac{P^{+}}{M},-\frac{P^{-}}{M},{\bf
0}\biggr{)}\,,$ $\displaystyle S_{T}^{\mu}$ $\displaystyle=$
$\displaystyle\Big{(}0,0,{\bf S}_{T}\Big{)}$ (73)

with $S_{L}^{2}+{\bf S}_{T}^{2}=1$. Note that in the infinite momentum frame
$P^{+}\to\infty$ the $P^{-}$ component and components of all four-vectors
proportional to $P^{-}$ vanish.

Next we specify the two symmetric $g^{\mu\nu}_{T}$ and $\eta^{\mu\nu}_{T}$ and
antisymmetric $\epsilon^{\mu\nu}_{T}$ transverse tensors, using the
$n_{\pm}^{\mu}$ vectors Mulders:2000sh :

$\displaystyle g^{\mu\nu}_{T}$ $\displaystyle=$ $\displaystyle
g^{\mu\nu}-n^{\mu}_{+}n^{\nu}_{-}-n^{\mu}_{-}n^{\nu}_{+}={\rm
diag}(0,0,-1,-1)\,,$ $\displaystyle\eta^{\mu\nu}_{T}$ $\displaystyle=$
$\displaystyle g^{\mu\nu}_{T}+\frac{2{\bf k}_{\perp}^{\mu}{\bf
k}_{\perp}^{\nu}}{{\bf k}_{\perp}^{2}}=\left(\begin{array}[]{cc}\mbox{\large
0}&\mbox{\large 0}\\\\[2.84526pt] \mbox{\large 0}&\begin{array}[]{cr}\cos
2\phi_{k}&\sin 2\phi_{k}\\\ \sin 2\phi_{k}&-\cos 2\phi_{k}\\\
\end{array}\end{array}\right)\,,$ (78) $\displaystyle\epsilon^{\mu\nu}_{T}$
$\displaystyle=$
$\displaystyle\epsilon^{\alpha\beta\mu\nu}\,n_{+\alpha}\,n_{-\beta}=\epsilon^{n_{+}n_{-}\mu\nu}=\epsilon^{-+\mu\nu}=\left(\begin{array}[]{cc}\mbox{\large
0}&\ \ \ \mbox{\large 0}\\\\[2.84526pt] \mbox{\large
0}&\begin{array}[]{rc}0&1\\\ -1&0\\\ \end{array}\end{array}\right)\,,$ (83)

where ${\bf k}_{\perp}^{\mu}=(0,0,{\bf k}_{\perp})=\sqrt{{\bf
k}_{\perp}^{2}}\,(0,0,\cos\phi_{k},\sin\phi_{k})$ and $\phi_{k}$ is the
azimuthal angle, defining an orientation of ${\bf k}_{\perp}$ in the
transverse plane. These three tensors play a fundamental role in the
classification of TMDs Mulders:2000sh  and obey the following normalization
and orthogonality conditions:

$\displaystyle g^{\mu\nu}_{T}\ g_{\mu\nu,T}=\eta^{\mu\nu}_{T}\
\eta_{\mu\nu,T}=\epsilon^{\mu\nu}_{T}\ \epsilon_{\mu\nu,T}=2\,,$
$\displaystyle
g^{\mu\nu}_{T}\,\eta_{\mu\nu,T}=g^{\mu\nu}_{T}\,\epsilon_{\mu\nu,T}=\eta^{\mu\nu}_{T}\,\epsilon_{\mu\nu,T}=0\,.$
(84)

Gluon polarization vectors $\epsilon^{\mu}_{\lambda}$ read

$\displaystyle\epsilon^{\mu}_{\pm}=(0,0,{\bm{\epsilon}}_{\pm})=\frac{1}{\sqrt{2}}\,(0,0,\mp
1,-i)\quad$ (85)

in the case of circular polarization and

$\displaystyle\epsilon^{\mu}_{x}=(0,0,{\bm{\epsilon}}_{x})=(0,0,1,0)\,,\quad\epsilon^{\mu}_{y}=(0,0,{\bm{\epsilon}}_{y})=(0,0,0,1)\,.$
(86)

in the case of linear polarization. The two sets of polarization vectors are
related as:

$\displaystyle\epsilon^{\mu}_{\pm}=\mp\frac{1}{\sqrt{2}}\,\biggl{[}\epsilon^{\mu}_{x}\pm
i\epsilon^{\mu}_{y}\biggr{]}\,.$ (87)

In both cases the polarization vectors obey the completeness and
orthonormality conditions:

$\displaystyle-g^{\mu\nu}_{T}=\sum\limits_{\lambda}\,\epsilon^{\mu}_{\lambda}\,\epsilon^{\dagger\nu}_{\lambda}\,,\qquad\epsilon^{\dagger\mu}_{\lambda}\,\epsilon_{\mu\lambda^{\prime}}=-\delta_{\lambda\lambda^{\prime}}\,.$
(88)

The antisymmetric tensor $\epsilon^{\mu\nu}_{T}$ is expressed in terms of
polarization vectors for the case of linear and circular polarizations as

$\displaystyle-i\epsilon^{\mu\nu}_{T}=-i\,\Big{[}\epsilon^{\mu}_{x}\,\epsilon^{\dagger\nu}_{y}-\epsilon^{\mu}_{y}\,\epsilon^{\dagger\nu}_{x}\Big{]}=\epsilon^{\mu}_{+}\,\epsilon^{\dagger\nu}_{+}-\epsilon^{\mu}_{-}\,\epsilon^{\dagger\nu}_{-}\,.$
(89)

The unpolarized gluon TMD in the nucleon is defined in analogy with the
corresponding quark TMD:

$\displaystyle f_{1}^{g}(x,{\bf k}_{\perp}^{2})$ $\displaystyle=$
$\displaystyle\biggl{[}G(x)+G^{-}(x)\,\alpha_{+}(x)\biggl{(}\frac{{\bf
k}_{\perp}^{2}\,D_{g}^{2}(x)}{\kappa^{2}}-1\biggr{)}\biggr{]}\,\frac{D_{g}(x)}{\pi\kappa^{2}}\,e^{-{\bf
k}_{\perp}^{2}D_{g}(x)/\kappa^{2}}\,,$ (90)

where $G^{\pm}=(G\pm\Delta G)/2$ are the helicity-aligned and helicity-
antialigned gluon PDFs, $D_{g}(x)>0$ is the profile function, and

$\displaystyle\alpha_{\pm}(x)=\frac{1\pm(1-x)^{2}}{(1-x)^{2}}\,.$ (91)

It is clear that the integration of the $f_{1}^{g}(x,{\bf k}_{\perp})$ over
${\bf k}_{\perp}$ gives the unpolarized gluon PDF:

$\displaystyle f_{1}^{g}(x)=\int d^{2}{\bf k}_{\perp}\,f_{1}^{g}(x,{\bf
k}_{\perp}^{2})=G(x)\,.$ (92)

Notice that that Gaussian form of the ${\bf k}_{\perp}^{2}$ dependence of TMD
in our approach [see, e.g., Eq. (90) is dictated by the Gaussian dependence of
the light-front wave functions (LFWFs), defined below in Eq. (III.2). The
reason is that the Gaussian ansatz for the LFWFs is dictated by the Gaussian
$Q^{2}$-dependence of the form factors [see Eq.(III.3)], calculated using
holographic wave functions of hadrons. Since all parton densities and form
factors in our approach are related, we can guarantee that LFWFs produce the
same functional behavior of form factors.

Following Refs. Brodsky:2000ii  and Lu:2016vqu ; Kaur:2020pvc  one can set up
the light-front wave function (LFWF) for the nucleon as a composite system of
a spin-1 gluon and spin-$\frac{1}{2}$ three-quark spectator. This is the so-
called spectator model, which was applied also in Refs. Meissner:2007rx ;
Bacchetta:2008af ; Bacchetta:2020vty  for the derivation of quark and gluon
TMDs. In our case we derive LFWF model without refering to additional
parameters such as spectator mass (a similar model was considered by us for
the nucleon as bound state of an active quark and a scalar diquark spectator
Gutsche:2013zia ; Gutsche:2014yea ; Gutsche:2016gcd ; Lyubovitskij:2020otz .
We introduce the notation of LFVW
$\psi^{\lambda_{N}}_{\lambda_{g}\lambda_{X}}(x,{\bf k}_{\perp})$, describing
the bound state of a gluon $(g)$ and a three-quark spectator $(X)$, where
$\lambda_{N}$, $\lambda_{g}$, and $\lambda_{X}$ are the helicities of nucleon,
gluon, and three-quark spectator, respectively. For a nucleon with up helicity
$\lambda_{N}=+\frac{1}{2}$ (we use shorten notation $+$), four possible LFWFs
are available, taking into account the different helicities $\lambda_{g}=\pm
1$ and $\lambda_{X}=\pm\frac{1}{2}$

$\displaystyle\psi^{+}_{+1+\frac{1}{2}}(x,{\bf k}_{\perp})$ $\displaystyle=$
$\displaystyle\frac{k^{1}-ik^{2}}{M_{N}}\,\varphi^{(2)}(x,{\bf
k}_{\perp}^{2})\,,$ $\displaystyle\psi^{+}_{+1-\frac{1}{2}}(x,{\bf
k}_{\perp})$ $\displaystyle=$ $\displaystyle\varphi^{(1)}(x,{\bf
k}_{\perp}^{2})\,,$ $\displaystyle\psi^{+}_{-1+\frac{1}{2}}(x,{\bf
k}_{\perp})$ $\displaystyle=$
$\displaystyle-\frac{k^{1}+ik^{2}}{M_{N}}\,(1-x)\,\varphi^{(2)}(x,{\bf
k}_{\perp}^{2})\,,$ $\displaystyle\psi^{+}_{-1-\frac{1}{2}}(x,{\bf
k}_{\perp})$ $\displaystyle=$ $\displaystyle 0\,.$ (93)

The LFWFs with down helicity $\lambda_{N}=-\frac{1}{2}$ (we use shorten
notation $-$) are

$\displaystyle\psi^{-}_{+1+\frac{1}{2}}(x,{\bf k}_{\perp})$ $\displaystyle=$
$\displaystyle 0\,,$ $\displaystyle\psi^{-}_{+1-\frac{1}{2}}(x,{\bf
k}_{\perp})$ $\displaystyle=$
$\displaystyle-\Big{[}\psi^{+}_{-1+\frac{1}{2}}(x,{\bf
k}_{\perp})\Big{]}^{*}=\frac{k^{1}-ik^{2}}{M_{N}}\,(1-x)\,\varphi^{(2)}(x,{\bf
k}_{\perp})\,,$ $\displaystyle\psi^{-}_{-1+\frac{1}{2}}(x,{\bf k}_{\perp})$
$\displaystyle=$ $\displaystyle+\Big{[}\psi^{+}_{+1-\frac{1}{2}}(x,{\bf
k}_{\perp})\Big{]}^{*}=\varphi^{(1)}(x,{\bf k}_{\perp}^{2})\,,$
$\displaystyle\psi^{-}_{-1-\frac{1}{2}}(x,{\bf k}_{\perp})$ $\displaystyle=$
$\displaystyle-\Big{[}\psi^{+}_{+1+\frac{1}{2}}(x,{\bf
k}_{\perp})\Big{]}^{*}=-\frac{k^{1}+ik^{2}}{M_{N}}\,\varphi^{(2)}(x,{\bf
k}_{\perp}^{2})\,.$ (94)

where the functions $\varphi^{(1)}(x,{\bf k}_{\perp})$ and
$\varphi^{(2)}(x,{\bf k}_{\perp})$ can be expressed through the gluon PDF
functions $G^{\pm}(x)$ as

$\displaystyle\varphi^{(1)}(x,{\bf k}_{\perp}^{2})$ $\displaystyle=$
$\displaystyle\frac{4\pi}{\kappa}\,\sqrt{G^{+}(x)}\,\beta(x)\,\sqrt{D_{g}(x)}\,\exp\biggl{[}-\frac{{\bf
k}_{\perp}^{2}}{2\kappa^{2}}\,D_{g}(x)\biggr{]}\,,$
$\displaystyle\frac{1}{M_{N}}\,\varphi^{(2)}(x,{\bf k}_{\perp}^{2})$
$\displaystyle=$
$\displaystyle\frac{4\pi}{\kappa^{2}}\,\sqrt{G^{-}(x)}\,\frac{D_{g}(x)}{1-x}\,\exp\biggl{[}-\frac{{\bf
k}_{\perp}^{2}}{2\kappa^{2}}\,D_{g}(x)\biggr{]}\,,$ (95)

where

$\displaystyle\beta(x)=\sqrt{1-\frac{G^{-}(x)}{G^{+}(x)(1-x)^{2}}}\,\,.$ (96)

One can see that the $\varphi^{(2)}(x,{\bf k}_{\perp}^{2})$ function is
expressed in terms of the $G^{-}(x)$ PDF, while the $\varphi^{(1)}(x,{\bf
k}_{\perp}^{2})$ is expressed through the combination of functions $G^{+}(x)$
and $G^{-}(x)$. Using Eq. (III.2) we get useful formula:

$\displaystyle\Big{[}\varphi^{(1)}(x,{\bf
k}_{\perp}^{2})\Big{]}^{2}+\frac{{\bf
k}_{\perp}^{2}}{M_{N}^{2}}\,\Big{[}\varphi^{(2)}(x,{\bf
k}_{\perp}^{2})\Big{]}^{2}=\frac{16\pi^{2}}{\kappa^{2}}\,D_{g}(x)\,\biggl{[}G^{+}(x)+\frac{G^{-}(x)}{(1-x)^{2}}\,\biggl{(}\frac{{\bf
k}_{\perp}^{2}}{\kappa^{2}}D_{g}(x)-1\biggr{)}\biggr{]}\exp\biggl{[}-\frac{{\bf
k}_{\perp}^{2}}{\kappa^{2}}\,D_{g}(x)\biggr{]}\,.$ (97)

Note that in Eq. (III.2) the LFWFs $\varphi^{(i)}(x,{\bf k}_{\perp}^{i})$ are
defined in terms of a single profile function $D_{g}(x)$. A generalization to
the case when the LFWFs $\varphi^{(i)}(x,{\bf k}_{\perp}^{i})$ are accompanied
by two different functions $D_{g_{1}}(x)$ and $D_{g_{2}}(x)$, respectively, is
straightforward. Generalized version is considered in Appendix A, where we
present the corresponding list of the LFWFs and TMDs.

Note that from the positivity of the argument of the square root in the
definition of the $\varphi^{(1)}(x,{\bf k}_{\perp}^{2})$ function, it follows
that the gluon PDF $G^{+}(x)$ and $G^{-}(x)$ should satisfy the bound

$\displaystyle\frac{G^{+}(x)(1-x)^{2}}{G^{-}(x)}\geq 1$ (98)

i.e. $\beta(x)$ defined in Eq. (96) must be real. The positivity bound (98) is
fulfilled in both versions of the QCD derivation of gluon PDFs Brodsky:1989db
; Brodsky:1994kg . In particular, QCDI Brodsky:1989db ; Brodsky:1994kg  and
QCDII Brodsky:1994kg  give, respectively

$\displaystyle\frac{G^{+}(x)(1-x)^{2}}{G^{-}(x)}=1+4x\geq 1$ (99)

and

$\displaystyle\frac{G^{+}(x)(1-x)^{2}}{G^{-}(x)}=1\,.$ (100)

Note that our wave functions $\varphi^{(1)}(x,{\bf k}_{\perp}^{2})$ and
$\varphi^{(2)}(x,{\bf k}_{\perp}^{2})$ are generalizations of the LFWF
$\varphi(x,{\bf k}_{\perp}^{2})$ used in Refs. Brodsky:2000ii  and Lu:2016vqu
; Kaur:2020pvc :

$\displaystyle\varphi^{(1)}(x,{\bf
k}_{\perp}^{2})\longrightarrow\varphi(x,{\bf
k}_{\perp}^{2})\,\sqrt{2}\,\frac{M_{N}(1-x)-M_{X}}{1-x}\,,\qquad\varphi^{(2)}(x,{\bf
k}_{\perp}^{2})\longrightarrow\varphi(x,{\bf
k}_{\perp}^{2})\,\sqrt{2}\,\frac{M_{N}}{x(1-x)}\,,$ (101)

where $M_{X}$ is the mass of the three-quark spectator. Also, the functions
$\varphi^{(1)}(x,{\bf k}_{\perp}^{2})$ and $\varphi^{(2)}(x,{\bf
k}_{\perp}^{2})$ can be considered as extensions of the quark-gluon coupling
in the quark target model Meissner:2007rx  and the form factor $g_{1}(k^{2})$,
parametrizing the minimal nucleon-gluon coupling in the spectator model
proposed in Ref. Bacchetta:2020vty

$\displaystyle\varphi^{(1)}(x,{\bf k}_{\perp}^{2})\longrightarrow
2g\sqrt{\frac{2}{3x}}\,\frac{mx^{2}}{m^{2}x^{2}+{\bf k}_{\perp}^{2}}\,,$
$\displaystyle\varphi^{(2)}(x,{\bf k}_{\perp}^{2})\longrightarrow
2g\sqrt{\frac{2}{3x}}\,\frac{M_{N}}{m^{2}x^{2}+{\bf k}_{\perp}^{2}}\,.$ (102)

and

$\displaystyle\varphi^{(1)}(x,{\bf
k}_{\perp}^{2})\longrightarrow\frac{g_{1}(k^{2})}{k^{2}}\,\sqrt{2x}\,\frac{M_{N}(1-x)-M_{X}}{1-x}\,,$
$\displaystyle\varphi^{(2)}(x,{\bf
k}_{\perp}^{2})\longrightarrow\frac{g_{1}(k^{2})}{k^{2}}\,\sqrt{2x}\,\frac{M_{N}}{x(1-x)}\,.$
(103)

It is clear that LFWF $\varphi(x,{\bf k}_{\perp}^{2})$ in Refs. Brodsky:2000ii
; Lu:2016vqu ; Kaur:2020pvc  is related with form factor $g_{1}(k^{2})$ from
Ref. Bacchetta:2020vty  as

$\displaystyle\varphi(x,{\bf
k}_{\perp}^{2})=\frac{g_{1}(k^{2})}{k^{2}}\,\sqrt{x}\,.$ (104)

In terms of LFWFs (III.2) the unpolarized gluon TMD in the nucleon reads (see
also Refs. Lu:2016vqu ; Kaur:2020pvc )

$\displaystyle f_{1}^{g}(x,{\bf k}_{\perp}^{2})$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi^{3}}\,\frac{1}{2}\,g^{\mu\nu}_{T}\,\sum\limits_{\lambda_{N}\lambda_{g}\lambda_{g}^{\prime}\lambda_{X}}\,\epsilon^{\dagger\lambda_{g}^{\prime}}_{\mu}\,\epsilon^{\lambda_{g}}_{\nu}\,\,\psi^{*\lambda_{N}}_{\lambda_{g}^{\prime}\lambda_{X}}(x,{\bf
k}_{\perp})\,\psi^{\lambda_{N}}_{\lambda_{g}\lambda_{X}}(x,{\bf k}_{\perp})$
(105) $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{3}}\,\frac{1}{2}\,\sum\limits_{\lambda_{N}\lambda_{g}\lambda_{X}}\,|\psi^{\lambda_{N}}_{\lambda_{g}\lambda_{X}}(x,{\bf
k}_{\perp})|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{3}}\,\biggl{[}|\psi^{+}_{+1+\frac{1}{2}}(x,{\bf
k}_{\perp})|^{2}+|\psi^{+}_{+1-\frac{1}{2}}(x,{\bf
k}_{\perp})|^{2}+|\psi^{+}_{-1+\frac{1}{2}}(x,{\bf k}_{\perp})|^{2}\biggr{]}$
$\displaystyle=$
$\displaystyle\frac{1}{16\pi^{3}}\,\biggl{[}\Big{(}\varphi^{(1)}(x,{\bf
k}_{\perp}^{2})\Big{)}^{2}+\frac{{\bf
k}_{\perp}^{2}}{M_{N}^{2}}\,\Big{[}1+(1-x)^{2}\Big{]}\,\Big{(}\varphi^{(2)}(x,{\bf
k}_{\perp}^{2})\Big{)}^{2}\biggr{]}$ $\displaystyle=$
$\displaystyle\frac{D_{g}(x)}{\pi\kappa^{2}}\,\biggl{[}G(x)+G^{-}(x)\,\alpha_{+}(x)\,\biggl{(}\frac{{\bf
k}_{\perp}^{2}}{\kappa^{2}}\,D_{g}(x)-1\biggr{)}\biggr{]}\,\exp\biggl{[}-\frac{{\bf
k}_{\perp}^{2}}{\kappa^{2}}\,D_{g}(x)\biggr{]}\,.$

The LFWF describing the nucleon, with both transverse and longitudinal
polarizations, is constructed as a superposition of two LFWFs with definite
nucleon helicity $\lambda_{N}=+1/2$ and $\lambda_{N}=-1/2$ Diehl:2005jf :

$\displaystyle\psi^{{\bf S},S^{z}}_{\lambda_{g}\lambda_{X}}(x,{\bf
k}_{\perp})$ $\displaystyle=$
$\displaystyle\cos\frac{\theta}{2}\,\psi^{+}_{\lambda_{g}\lambda_{X}}(x,{\bf
k}_{\perp})+\sin\frac{\theta}{2}\,e^{i\phi}\,\psi^{-}_{\lambda_{g}\lambda_{X}}(x,{\bf
k}_{\perp})\,.$ (106)

Here the three-dimensional nucleon spin is expressed in terms of two spherical
angles $\phi$ and $\theta$ as

$\displaystyle\vec{S}=({\bf S},S^{z})\,,\quad{\bf
S}=(\sin\theta\cos\phi,\sin\theta\sin\phi)\,,\quad S^{z}=\cos\theta\,.$ (107)

The limiting cases are:

(1) Nucleon has only transverse polarization with ${\bf
S}_{T}=(\cos\phi,\sin\phi)$ and $S_{L}=0$, i.e. $\psi^{{\bf
S},S^{z}}_{\lambda_{g}\lambda_{X}}\to\psi^{{\bf
S}_{T}}_{\lambda_{g}\lambda_{X}}$ at $\theta=90^{\circ}$;

(2) Nucleon has only longitudinal polarization with ${\bf S}_{T}=0$ and
$S_{L}=\pm 1$, i.e. $\psi^{{\bf
S},S^{z}}_{\lambda_{g}\lambda_{X}}\to\psi^{S_{L}}_{\lambda_{g}\lambda_{X}}\equiv\psi^{\lambda_{N}}_{\lambda_{g}\lambda_{X}}$
at $\theta=0^{\circ}$ or $180^{\circ}$.

The helicity TMD $g_{1L}^{g}(x,{\bf k}_{\perp}^{2})$, describing a
distribution of a circularly polarized gluon in a longitudinally polarized
nucleon, is defined by analogy with the quark case Bacchetta:2008af  as

$\displaystyle g_{1L}^{g}(x,{\bf k}_{\perp}^{2})$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi^{3}}\,\frac{1}{S_{L}}i\epsilon^{\mu\nu}_{T}\,\sum\limits_{\lambda_{g}\lambda_{g}^{\prime}\lambda_{X}}\,\epsilon^{\dagger\lambda_{g}^{\prime}}_{\mu}\,\epsilon^{\lambda_{g}}_{\nu}\,\psi^{*S_{L}}_{\lambda_{g}^{\prime}\lambda_{X}}(x,{\bf
k}_{\perp})\,\psi^{S_{L}}_{\lambda_{g}\lambda_{X}}(x,{\bf k}_{\perp})$ (108)
$\displaystyle=$
$\displaystyle\frac{1}{16\pi^{3}}\,\frac{1}{S_{L}}\sum\limits_{\lambda_{X}}\biggl{[}|\psi^{S_{L}}_{+1\lambda_{X}}(x,{\bf
k}_{\perp})|^{2}-|\psi^{S_{L}}_{-1\lambda_{X}}(x,{\bf
k}_{\perp})|^{2}\biggr{]}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{3}}\,\biggl{[}|\psi^{+}_{+1+\frac{1}{2}}(x,{\bf
k}_{\perp})|^{2}+|\psi^{+}_{+1-\frac{1}{2}}(x,{\bf
k}_{\perp})|^{2}-|\psi^{+}_{-1+\frac{1}{2}}(x,{\bf k}_{\perp})|^{2}\biggr{]}$
$\displaystyle=$
$\displaystyle\frac{1}{16\pi^{3}}\,\biggl{[}(\varphi^{(1)}(x,{\bf
k}_{\perp})\Big{)}^{2}+\frac{{\bf
k}_{\perp}^{2}}{M_{N}^{2}}\,\Big{[}1-(1-x)^{2}\Big{]}\,\Big{(}\varphi^{(2)}(x,{\bf
k}_{\perp})\Big{)}^{2}\biggr{]}$ $\displaystyle=$
$\displaystyle\frac{D_{g}(x)}{\pi\kappa^{2}}\,\biggl{[}\Delta
G(x)+G^{-}(x)\,\alpha_{-}(x)\,\biggl{(}\frac{{\bf
k}_{\perp}^{2}}{\kappa^{2}}\,D_{g}(x)-1\biggr{)}\biggr{]}\,\exp\biggl{[}-\frac{{\bf
k}_{\perp}^{2}}{\kappa^{2}}\,D_{g}(x)\biggr{]}\,.$

The T worm-gear TMD $g_{1T}^{g}(x,{\bf k}_{\perp}^{2})$, describing the
distribution of a circularly polarized gluon in a transversally polarized
nucleon, is defined as

$\displaystyle g_{1T}^{g}(x,{\bf k}_{\perp}^{2})$ $\displaystyle=$
$\displaystyle-\frac{M_{N}}{{\bf S}_{T}\,{{\bf
k}_{\perp}}}\,\frac{1}{16\pi^{3}}\,i\epsilon^{\mu\nu}_{T}\,\sum\limits_{\lambda_{g}\lambda_{g}^{\prime}\lambda_{X}}\,\epsilon^{\dagger\lambda_{g}^{\prime}}_{\mu}\,\epsilon^{\lambda_{g}}_{\nu}\,\psi^{*{\bf
S}_{T}}_{\lambda_{g}^{\prime}\lambda_{X}}(x,{\bf k}_{\perp})\,\psi^{{\bf
S}_{T}}_{\lambda_{g}\lambda_{X}}(x,{\bf k}_{\perp})$ (109) $\displaystyle=$
$\displaystyle\frac{M_{N}}{{\bf S}_{T}\,{{\bf
k}_{\perp}}}\,\frac{1}{16\pi^{3}}\sum\limits_{\lambda_{X}}\biggl{[}|\psi^{{\bf
S}_{T}}_{+1\lambda_{X}}(x,{\bf k}_{\perp})|^{2}-|\psi^{{\bf
S}_{T}}_{-1\lambda_{X}}(x,{\bf k}_{\perp})|^{2}\biggr{]}$ $\displaystyle=$
$\displaystyle\frac{M_{N}}{{\bf S}_{T}\,{{\bf
k}_{\perp}}}\,\frac{1}{16\pi^{3}}\,{\bf S_{T}}\,{\bf M_{T}}(x,{\bf
k}_{\perp})=\frac{1}{8\pi^{3}}\,\varphi^{(1)}(x,{\bf
k}_{\perp}^{2})\,\varphi^{(2)}(x,{\bf k}_{\perp}^{2})\,(1-x)$ $\displaystyle=$
$\displaystyle\frac{D_{g}^{3/2}(x)M_{N}}{\pi\kappa^{3}}\,\sqrt{G^{2}(x)-\Delta
G^{2}(x)}\ \beta(x)\,\exp\biggl{[}-\frac{{\bf
k}_{\perp}^{2}}{\kappa^{2}}\,D_{g}(x)\biggr{]}\,,$

where ${\bf M_{T}}=({\bf M}^{x},{\bf M}^{y})$ and

$\displaystyle{\bf M}^{x}(x,{\bf k}_{\perp})$ $\displaystyle=$
$\displaystyle\psi^{*+}_{+1-\frac{1}{2}}(x,{\bf
k}_{\perp})\psi^{-}_{+1-\frac{1}{2}}(x,{\bf k}_{\perp})+{\rm
H.c.}=\frac{2k^{1}}{M_{N}}\,\varphi^{(1)}(x,{\bf
k}_{\perp}^{2})\,\varphi^{(2)}(x,{\bf k}_{\perp}^{2})\,(1-x)\,,$
$\displaystyle{\bf M}^{y}(x,{\bf k}_{\perp})$ $\displaystyle=$ $\displaystyle
i\,\psi^{*+}_{+1-\frac{1}{2}}(x,{\bf
k}_{\perp})\psi^{-}_{+1-\frac{1}{2}}(x,{\bf k}_{\perp})+{\rm
H.c.}=\frac{2k^{2}}{M_{N}}\,\varphi^{(1)}(x,{\bf
k}_{\perp}^{2})\,\varphi^{(2)}(x,{\bf k}_{\perp}^{2})\,(1-x)\,.$ (110)

The Boer-Mulders TMD $h_{1\perp}^{g}(x,{\bf k}_{\perp})$, describing a
linearly polarized gluon inside an unpolarized nucleon, are

$\displaystyle h_{1}^{\perp g}(x,{\bf k}_{\perp}^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{3}}\,\frac{M_{N}^{2}}{{\bf
k}_{\perp}^{2}}\,\eta^{\mu\nu}_{T}\,\sum\limits_{\lambda_{N}\lambda_{g}\neq\lambda_{g}^{\prime}\lambda_{X}}\,\epsilon^{\dagger\lambda_{g}^{\prime}}_{\mu}\,\epsilon^{\lambda_{g}}_{\nu}\,\psi^{*\lambda_{N}}_{\lambda_{g}^{\prime}\lambda_{X}}(x,{\bf
k}_{\perp})\,\psi^{\lambda_{N}}_{\lambda_{g}\lambda_{X}}(x,{\bf k}_{\perp})$
(111) $\displaystyle=$
$\displaystyle-\frac{1}{16\pi^{3}}\,\frac{M_{N}^{2}}{{\bf
k}_{\perp}^{4}}\,\sum\limits_{\lambda_{N}\lambda_{X}}\,\biggl{[}(k_{-})^{2}\,\psi^{*\lambda_{N}}_{+\lambda_{X}}(x,{\bf
k}_{\perp})\,\psi^{\lambda_{N}}_{-\lambda_{X}}(x,{\bf
k}_{\perp})+(k_{+})^{2}\,\psi^{*\lambda_{N}}_{-\lambda_{X}}(x,{\bf
k}_{\perp})\,\psi^{\lambda_{N}}_{+\lambda_{X}}(x,{\bf k}_{\perp})\biggr{]}$
$\displaystyle=$ $\displaystyle\frac{1}{4\pi^{3}}\Big{[}\varphi^{(2)}(x,{\bf
k}_{\perp}^{2})\Big{]}^{2}\,(1-x)=\frac{2D_{g}^{2}(x)M_{N}^{2}}{\pi\kappa^{4}}\,\frac{G(x)-\Delta
G(x)}{1-x}\ \,\exp\biggl{[}-\frac{{\bf
k}_{\perp}^{2}}{\kappa^{2}}\,D_{g}(x)\biggr{]}\,,$

where $k_{\pm}=k^{1}\pm ik^{2}$.

From decomposition of our gluon TDMs in terms of LFWFs one can see that they
obey the sum rule:

$\displaystyle\biggl{[}f_{1}^{g}(x,{\bf
k}_{\perp}^{2})\biggr{]}^{2}=\biggl{[}g_{1L}^{g}(x,{\bf
k}_{\perp}^{2})\biggr{]}^{2}\,+\,\biggl{[}\frac{|{\bf
k}_{\perp}|}{M_{N}}\,g_{1T}^{g}(x,{\bf
k}_{\perp}^{2})\biggr{]}^{2}\,+\,\biggl{[}\frac{{\bf
k}_{\perp}^{2}}{2M_{N}^{2}}h_{1}^{g}(x,{\bf k}_{\perp}^{2})\biggr{]}^{2}\,.$
(112)

This sum rule is very important because it establishes the relation between
square of unpolarized TMD [left-hand side (lhs) of Eq. (112)] and the
superposition of squares of 3 polarized TMDs [right-hand side (rhs) of Eq.
(112)]. From this sum rule immediately follow the model-independent Mulders-
Rodrigues inequalities (positivity bounds) Mulders:2000sh , without referring
to a specific functional form of LFWFs $\varphi^{(i)}(x,{\bf k}_{\perp})$,
i.e. independent on a choice of $\varphi^{(i)}(x,{\bf k}_{\perp})$:

$\displaystyle\sqrt{\biggl{[}g_{1L}^{g}(x,{\bf
k}_{\perp}^{2})\biggr{]}^{2}\,+\,\biggl{[}\frac{|{\bf
k}_{\perp}|}{M_{N}}\,g_{1T}^{g}(x,{\bf k}_{\perp}^{2})\biggr{]}^{2}}\leq
f_{1}^{g}(x,{\bf k}_{\perp}^{2})\,,$
$\displaystyle\sqrt{\biggl{[}g_{1L}^{g}(x,{\bf
k}_{\perp}^{2})\biggr{]}^{2}\,+\,\biggl{[}\frac{{\bf
k}_{\perp}^{2}}{2M_{N}^{2}}\,h_{1}^{\perp g}(x,{\bf
k}_{\perp}^{2})\biggr{]}^{2}}\leq f_{1}^{g}(x,{\bf k}_{\perp}^{2})\,,$ (113)
$\displaystyle\sqrt{\biggl{[}\frac{|{\bf
k}_{\perp}|}{M_{N}}\,g_{1T}^{g}(x,{\bf
k}_{\perp}^{2})\biggr{]}^{2}\,+\,\biggl{[}\frac{{\bf
k}_{\perp}^{2}}{2M_{N}^{2}}\,h_{1}^{\perp g}(x,{\bf
k}_{\perp}^{2})\biggr{]}^{2}}\leq f_{1}^{g}(x,{\bf k}_{\perp}^{2})\,.$

From above inequalities follow less stringent MR bounds involving unpolarized
and one of the polarized TMDs Mulders:2000sh :

$\displaystyle|g_{1L}^{g}(x,{\bf k}_{\perp}^{2})|\leq f_{1}^{g}(x,{\bf
k}_{\perp}^{2})\,,$ (114) $\displaystyle|g_{1T}^{g}(x,{\bf
k}_{\perp}^{2})|\leq\frac{M_{N}}{|{\bf k}_{\perp}|}\,f_{1}^{g}(x,{\bf
k}_{\perp}^{2})\,,$ (115) $\displaystyle|h_{1}^{\perp g}(x,{\bf
k}_{\perp}^{2})|\leq\frac{2M_{N}^{2}}{{\bf k}_{\perp}^{2}}\,f_{1}^{g}(x,{\bf
k}_{\perp}^{2})\,.$ (116)

In particular, for any choice of $\varphi^{(i)}(x,{\bf k}_{\perp}^{2})$ and
for all values of $x$ and ${\bf k}_{\perp}$ variables the inequalities
(114)-(116) follow respectively from

$\displaystyle\Big{[}\varphi^{(2)}(x,{\bf
k}_{\perp}^{2})\Big{]}^{2}\,\frac{{\bf
k}_{\perp}^{2}}{M_{N}^{2}}\,(1-x)^{2}\geq 0\,,$
$\displaystyle\biggl{[}\varphi^{(1)}(x,{\bf k}_{\perp}^{2})-\frac{|{\bf
k}_{\perp}|}{M_{N}}\,(1-x)\,\varphi^{(2)}(x,{\bf
k}_{\perp}^{2})\biggr{]}^{2}+\frac{{\bf
k}_{\perp}^{2}}{M_{N}^{2}}\,\biggl{[}\varphi^{(2)}(x,{\bf
k}_{\perp}^{2})\biggr{]}^{2}\geq 0\,,$
$\displaystyle\Big{[}\varphi^{(1)}(x,{\bf
k}_{\perp}^{2})\Big{]}^{2}+\frac{{\bf
k}_{\perp}^{2}}{M_{N}^{2}}\,\Big{[}x\,\varphi^{(2)}(x,{\bf
k}_{\perp}^{2})\Big{]}^{2}\geq 0\,.$ (117)

Note that our approach is similar to three frameworks developed in Refs.
Meissner:2007rx ; Bacchetta:2020vty ; Kaur:2020pvc  as on the level LFWFs, as
on the level of gluon TMDs. In particular, one can see that expressions for
the gluon TMDs in terms of LFWFs are the same in four approaches (in our and
in the ones proposed in Refs. Meissner:2007rx ; Bacchetta:2020vty ;
Kaur:2020pvc ). One should stress that the sum rule derived by us in Eq. (112)
also holds in Ref. Bacchetta:2020vty  (if to restrict to the minimal coupling
of gluon with three-quark spectator) and in Ref. Kaur:2020pvc . In case of
Ref. Meissner:2007rx  a slightly different sum rule occurs (the nucleon mass
$M_{N}$ is replaced by the quark mass $m$):

$\displaystyle\biggl{[}f_{1}^{g}(x,{\bf
k}_{\perp}^{2})\biggr{]}^{2}=\biggl{[}g_{1L}^{g}(x,{\bf
k}_{\perp}^{2})\biggr{]}^{2}\,+\,\biggl{[}\frac{|{\bf
k}_{\perp}|}{m}\,g_{1T}^{g}(x,{\bf
k}_{\perp}^{2})\biggr{]}^{2}\,+\,\biggl{[}\frac{{\bf
k}_{\perp}^{2}}{2m^{2}}h_{1}^{g}(x,{\bf k}_{\perp}^{2})\biggr{]}^{2}\,.$ (118)

Also it important to stress that our consideration of the quark and gluon TMDs
is based on ideas of the quark-scalar diquark model and the gluon-three-quark
spectator model proposed one of us in collaboration with Brodsky, Hwang, and
Ma in Refs. Brodsky:2002cx ; Brodsky:2000ii  and was successfully used, e.g.,
in Refs. Meissner:2007rx ; Gutsche:2016gcd ; Lu:2016vqu ; Kaur:2020pvc ;
Lyubovitskij:2020otz . The similar approaches to the quark and gluon TMDs use
based on quark target model Meissner:2007rx  and spectator model
Bacchetta:2020vty  allow to get description of the TMDs in very good agreement
with us. Note that agreement between our formalism using light-front picture
and spectator model for the case of quark parton densities was proved in Ref.
Bacchetta:2008af . For the case of gluon parton densities we make a conclusion
about argeement in the present paper. The main difference is that we also
guarantee the correct scaling of TMDs and other parton densities at at large
$x$ (see detailed discussion in Sec.IIIc) in consistency with constraints
imposed by QCD, while it is not the case in the calculations presented in
Refs. Meissner:2007rx ; Bacchetta:2020vty ; Kaur:2020pvc . One should stress
that an implementation of large $x$ behavior in frameworks used in Refs.
Meissner:2007rx ; Bacchetta:2020vty ; Kaur:2020pvc  can be done easily — by
taking into account a specific $x$-dependence of the couplings/form factors.
Such idea was proposed in Ref. Jakob:1997wg , where the spectator model was
proposed. In particular, it was suggested that the multipole form factors of
spectator diquark contain a free parameter $\alpha$, which indicates the power
of form factors and can be clearly fixed to fulfill large $x$ counting rules.

The four T-even gluon TMDs, upon integration over $d^{2}{\bf k}_{\perp}$, are
normalized as

$\displaystyle f_{1}^{g}(x)$ $\displaystyle=$ $\displaystyle\int d^{2}{\bf
k}_{\perp}\,f_{1}^{g}(x,{\bf k}_{\perp}^{2})=G(x)\,,$ $\displaystyle
g_{1L}^{g}(x)$ $\displaystyle=$ $\displaystyle\int d^{2}{\bf
k}_{\perp}\,g_{1L}^{g}(x,{\bf k}_{\perp}^{2})=\Delta G(x)\,,$ $\displaystyle
g_{1T}^{g}(x)$ $\displaystyle=$ $\displaystyle\int d^{2}{\bf
k}_{\perp}\,g_{1T}^{g}(x,{\bf
k}_{\perp}^{2})=\frac{D_{g}^{1/2}(x)M_{N}}{\kappa}\,\sqrt{G^{2}(x)-\Delta
G^{2}(x)}\ \beta(x)\,,$ $\displaystyle h_{1}^{\perp g}(x)$ $\displaystyle=$
$\displaystyle\int d^{2}{\bf k}_{\perp}\,h_{1}^{\perp g}(x,{\bf
k}_{\perp}^{2})=\frac{2D_{g}(x)M_{N}^{2}}{\kappa^{2}}\,\frac{G(x)-\Delta
G(x)}{1-x}\,.$ (119)

One can see that they obey the condition

$\displaystyle\frac{\Big{[}g_{1T}^{g}(x)\Big{]}^{2}}{\Big{[}f_{1}^{g}(x)+g_{1L}^{g}(x)\Big{]}\,h_{1}^{\perp
g}(x)}=\frac{1-x}{2}\,\beta^{2}(x)\,.$ (120)

For completeness we now discuss how to construct the T-even gluon correlators
(unpolarized $\Gamma^{\mu\nu}_{U}$, longitudinally polarized
$\Gamma^{\mu\nu}_{L}$, and transversally polarized $\Gamma^{\mu\nu}_{T}$)
derived in Ref. Mulders:2000sh  in terms of our LFWFs. First, we split the
product of the gluon polarization vectors into a sum of antisymmetric
$(AB)^{[\mu\nu]}=A^{\mu}B^{\nu}-A^{\nu}B^{\mu}$ and symmetric
$(AB)^{\\{\mu\nu\\}}=A^{\mu}B^{\nu}+A^{\nu}B^{\mu}$ combinations under
exchange of Lorenz indices $\mu$ and $\nu$:

$\displaystyle\epsilon^{\dagger\mu}_{\lambda_{g}^{\prime}}\,\epsilon^{\nu}_{\lambda_{g}}=\frac{1}{2}\Big{(}\epsilon^{\dagger}_{\lambda_{g}^{\prime}}\,\epsilon_{\lambda_{g}}\Big{)}^{[\mu\nu]}+\frac{1}{2}\Big{(}\epsilon^{\dagger}_{\lambda_{g}^{\prime}}\,\epsilon_{\lambda_{g}}\Big{)}^{\\{\mu\nu\\}}\,.$
(121)

In case of T-even gluon TMDs symmetric combinations of polarization vectors
contribute to the unpolarized $f_{1}^{g}(x,{\bf k}_{\perp}^{2})$ and
$h_{1}^{\perp g}(x,{\bf k}_{\perp}^{2})$ TMDs, while antisymmetric
combinations to the polarized $g_{1L}^{g}$ and $g_{1T}^{g}$ TMDs.

The unpolarized tensor $\Gamma^{\mu\nu}_{U}$ is given by

$\displaystyle\Gamma_{U}^{\mu\nu}(x,{\bf k}_{\perp})$ $\displaystyle=$
$\displaystyle-g^{\mu\nu}_{T}\,f_{1}^{g}(x,{\bf k}_{\perp}^{2})+\frac{{\bf
k}_{\perp}^{2}}{2M_{N}^{2}}\,\eta^{\mu\nu}_{T}\,h_{1}^{\perp g}(x,{\bf
k}_{\perp}^{2})$ (122) $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{3}}\,\frac{1}{2}\,\sum\limits_{\lambda_{N}\lambda_{g}\lambda_{g}^{\prime}\lambda_{X}}\,\Big{(}\epsilon^{\dagger}_{\lambda_{g}^{\prime}}\,\epsilon_{\lambda_{g}}\Big{)}^{\\{\mu\nu\\}}\,\psi^{*\lambda_{N}}_{\lambda_{g}^{\prime}\lambda_{X}}\,\psi^{\lambda_{N}}_{\lambda_{g}\lambda_{X}}\,.$

Next we split the sum over $\lambda_{g}$ and $\lambda_{g}^{\prime}$ in two
terms: $\lambda_{g}=\lambda_{g}^{\prime}$ (it produces the $f_{1}^{g}(x,{\bf
k}_{\perp}^{2})$ TMD) and $\lambda_{g}\neq\lambda_{g}^{\prime}$ (it produces
the $h_{1}^{\perp g}(x,{\bf k}_{\perp}^{2})$ TMD):

$\displaystyle\Gamma_{U}^{\mu\nu}(x,{\bf
k}_{\perp})=\Gamma_{U;f_{1}^{g}}^{\mu\nu}(x,{\bf
k}_{\perp})+\Gamma_{U;h_{1}^{\perp g}}^{\mu\nu}(x,{\bf k}_{\perp})\,,$ (123)

where

$\displaystyle\Gamma_{U;f_{1}^{g}}^{\mu\nu}(x,{\bf k}_{\perp})$
$\displaystyle=$ $\displaystyle-g^{\mu\nu}_{T}\,f_{1}^{g}(x,{\bf
k}_{\perp}^{2})$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi^{3}}\,\frac{1}{2}\,g^{\mu\nu}_{T}\,\sum\limits_{\lambda_{N}\lambda_{g}\lambda_{X}}\,|\psi^{\lambda_{N}}_{\lambda_{g}\lambda_{X}}(x,{\bf
k}_{\perp})|^{2}\,,$ $\displaystyle\Gamma_{U;h_{1}^{\perp g}}^{\mu\nu}(x,{\bf
k}_{\perp})$ $\displaystyle=$ $\displaystyle\frac{{\bf
k}_{\perp}^{2}}{2M_{N}^{2}}\,\eta^{\mu\nu}_{T}\,h_{1}^{\perp g}(x,{\bf
k}_{\perp}^{2})$ (124) $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{3}}\,\frac{1}{2}\,\sum\limits_{\lambda_{N}\lambda_{g}\neq\lambda_{g}^{\prime}\lambda_{X}}\,\Big{(}\epsilon^{\dagger}_{\lambda_{g}^{\prime}}\,\epsilon_{\lambda_{g}}\Big{)}^{\\{\mu\nu\\}}\,\psi^{*\lambda_{N}}_{\lambda_{g}^{\prime}\lambda_{X}}(x,{\bf
k}_{\perp})\,\psi^{\lambda_{N}}_{\lambda_{g}\lambda_{X}}(x,{\bf
k}_{\perp})\,.$

From Eqs. (122)-(III.2) and using the normalization and orthogonality
conditions for the tensors $g^{\mu\nu}_{T}$ and $\eta^{\mu\nu}_{T}$ the master
formulas for the $f_{1}^{g}(x,{\bf k}_{\perp}^{2})$ (105) and $h_{1}^{\perp
g}(x,{\bf k}_{\perp}^{2})$ (111) follow:

$\displaystyle f_{1}^{g}(x,{\bf k}_{\perp}^{2})$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\,g_{\mu\nu;T}\,\Gamma_{U}^{\mu\nu}(x,{\bf
k}_{\perp})=-\frac{1}{2}\,g_{\mu\nu;T}\,\Gamma_{U;f_{1}^{g}}^{\mu\nu}(x,{\bf
k}_{\perp})\,,$ $\displaystyle h_{1}^{\perp g}(x,{\bf k}_{\perp}^{2})$
$\displaystyle=$ $\displaystyle\frac{M_{N}^{2}}{{\bf
k}_{\perp}^{2}}\,\eta_{\mu\nu;T}\,\Gamma_{U}^{\mu\nu}(x,{\bf
k}_{\perp})=\frac{M_{N}^{2}}{{\bf
k}_{\perp}^{2}}\,\eta_{\mu\nu;T}\,\Gamma_{U;h_{1}^{\perp g}}^{\mu\nu}(x,{\bf
k}_{\perp})\,.$ (125)

By analogy, we construct longitudinal $\Gamma^{\mu\nu}_{L}$ and transversal
$\Gamma^{\mu\nu}_{T}$ polarized tensors. Corresponding gluon correlator
tensors read:

$\displaystyle\Gamma^{\mu\nu}_{L}(x,{\bf k}_{\perp})$ $\displaystyle=$
$\displaystyle-i\epsilon^{\mu\nu}_{T}\,S_{L}\,g_{1L}^{g}(x,{\bf
k}_{\perp}^{2})$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi^{3}}\,\sum\limits_{\lambda_{g}\lambda_{g}^{\prime}\lambda_{X}}\,\Big{(}\epsilon^{\dagger}_{\lambda_{g}^{\prime}}\,\epsilon_{\lambda_{g}}\Big{)}^{[\mu\nu]}\,\psi^{*S_{L}}_{\lambda_{g}^{\prime}\lambda_{X}}(x,{\bf
k}_{\perp})\,\psi^{S_{L}}_{\lambda_{g}\lambda_{X}}(x,{\bf k}_{\perp})\,,$
$\displaystyle\Gamma^{\mu\nu}_{T}(x,{\bf k}_{\perp})$ $\displaystyle=$
$\displaystyle-i\epsilon^{\mu\nu}_{T}\,\frac{{\bf S}_{T}\,{{\bf
k}_{\perp}}}{M_{N}}\,g_{1T}^{g}(x,{\bf k}_{\perp}^{2})$ (126) $\displaystyle=$
$\displaystyle-\frac{1}{16\pi^{3}}\,\sum\limits_{\lambda_{g}\lambda_{g}^{\prime}\lambda_{X}}\,\Big{(}\epsilon^{\dagger}_{\lambda_{g}^{\prime}}\,\epsilon_{\lambda_{g}}\Big{)}^{[\mu\nu]}\,\psi^{*{\bf
S}_{T}}_{\lambda_{g}^{\prime}\lambda_{X}}(x,{\bf k}_{\perp})\,\psi^{{\bf
S}_{T}}_{\lambda_{g}\lambda_{X}}(x,{\bf k}_{\perp})\,.$

From Eq. (III.2) follow master formulas for the TMDs $g_{1L}^{g}(x,{\bf
k}_{\perp}^{2})$ (108) and $g_{1T}^{g}(x,{\bf k}_{\perp}^{2})$ (109):

$\displaystyle S_{L}\,g_{1L}^{g}(x,{\bf
k}_{\perp}^{2})=\frac{i}{2}\epsilon_{\mu\nu;T}\,\Gamma_{L}^{\mu\nu}(x,{\bf
k}_{\perp})\,,\qquad\frac{{\bf S}_{T}\,{{\bf
k}_{\perp}}}{M_{N}}\,g_{1T}^{g}(x,{\bf
k}_{\perp}^{2})=\frac{i}{2}\epsilon_{\mu\nu;T}\,\Gamma_{T}^{\mu\nu}(x,{\bf
k}_{\perp})\,.$ (127)

Now we present the results for the T-even gluon TMDs in transverse impact
space ${\bf b}_{\perp}$, performing the Fourier transform of the TMDs with
respect to the ${\bf k}_{\perp}$. For completeness we also display the
decomposition for the gluon impact TMDs in terms of the impact LFWF functions
$\tilde{\varphi}(x,{\bf b}_{\perp})$ and $\tilde{\varphi}(x,{\bf b}_{\perp})$

$\displaystyle\tilde{\varphi}^{(1)}(x,{\bf b}_{\perp}^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\,\sqrt{G^{+}(x)}\
\beta(x)\,\exp\Big{[}-\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{8D_{g}(x)}\Big{]}\,,$
$\displaystyle\tilde{\varphi}^{(2)}(x,{\bf b}_{\perp}^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\,\sqrt{G^{-}(x)}\
\frac{\sqrt{D_{g}(x)}}{1-x}\,\exp\Big{[}-\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{8D_{g}(x)}\Big{]}\,.$ (128)

The list of the T-even gluon TMDs in impact space reads:

Unpolarized gluon TMD in impact space $\tilde{f}_{1}^{g}(x,{\bf
b}_{\perp}^{2})$

$\displaystyle\tilde{f}_{1}^{g}(x,{\bf b}_{\perp}^{2})$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}{\bf k}_{\perp}}{(2\pi)^{2}}\,e^{i{\bf
b}_{\perp}{\bf k}_{\perp}}\,f_{1}^{g}(x,{\bf k}_{\perp}^{2})$ (129)
$\displaystyle=$ $\displaystyle\Big{[}\tilde{\varphi}^{(1)}(x,{\bf
b}_{\perp}^{2})\Big{]}^{2}+\Big{[}\tilde{\varphi}^{(2)}(x,{\bf
b}_{\perp}^{2})\Big{]}^{2}\,\frac{1+(1-x)^{2}}{D_{g}(x)}\,\biggl{[}1+\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{4D_{g}(x)}\biggr{]}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}}\,\biggl{[}G(x)+G^{-}(x)\,\alpha_{+}(x)\,\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{4D_{g}(x)}\biggr{]}\,\exp\Big{[}-\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{4D_{g}(x)}\Big{]}\,.$

Helicity gluon TMD $\tilde{g}_{1L}^{g}(x,{\bf b}_{\perp}^{2})$

$\displaystyle\tilde{g}_{1L}^{g}(x,{\bf b}_{\perp}^{2})$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}{\bf k}_{\perp}}{(2\pi)^{2}}\,e^{i{\bf
b}_{\perp}{\bf k}_{\perp}}\,g_{1L}^{g}(x,{\bf k}_{\perp}^{2})$ (130)
$\displaystyle=$ $\displaystyle\Big{[}\tilde{\varphi}^{(1)}(x,{\bf
b}_{\perp}^{2})\Big{]}^{2}+\Big{[}\tilde{\varphi}^{(2)}(x,{\bf
b}_{\perp}^{2})\Big{]}^{2}\,\frac{1-(1-x)^{2}}{D_{g}(x)}\,\biggl{[}1+\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{4D_{g}(x)}\biggr{]}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}}\,\biggl{[}\Delta
G(x)+G^{-}(x)\,\alpha_{-}(x)\,\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{4D_{g}(x)}\biggr{]}\,\exp\Big{[}-\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{4D_{g}(x)}\Big{]}\,.$

T worm-gear gluon TMD $\tilde{g}_{1T}^{g}(x,{\bf b}_{\perp}^{2})$

$\displaystyle\tilde{g}_{1T}^{g}(x,{\bf b}_{\perp}^{2})$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}{\bf k}_{\perp}}{(2\pi)^{2}}\,e^{i{\bf
b}_{\perp}{\bf k}_{\perp}}\,g_{1T}^{g}(x,{\bf k}_{\perp}^{2})$ (131)
$\displaystyle=$
$\displaystyle\frac{2M_{N}}{\kappa}\,\tilde{\varphi}^{(1)}(x,{\bf
b}_{\perp}^{2})\tilde{\varphi}^{(2)}(x,{\bf b}_{\perp}^{2})(1-x)$
$\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}}\,g_{1T}^{g}(x)\,\exp\Big{[}-\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{4D_{g}(x)}\Big{]}\,.$

Boer-Mulders gluon TMD $\tilde{h}_{1}^{\perp g}(x,{\bf b}_{\perp}^{2})$

$\displaystyle\tilde{h}_{1}^{\perp g}(x,{\bf b}_{\perp}^{2})$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}{\bf k}_{\perp}}{(2\pi)^{2}}\,e^{i{\bf
b}_{\perp}{\bf k}_{\perp}}\,h_{1}^{\perp g}(x,{\bf k}_{\perp}^{2})$ (132)
$\displaystyle=$
$\displaystyle\frac{4M_{N}^{2}}{\kappa^{2}}\,\Big{[}\tilde{\varphi}^{(2)}(x,{\bf
b}_{\perp}^{2})\Big{]}^{2}\,(1-x)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}}\,h_{1}^{\perp g}(x)\,\exp\Big{[}-\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{4D_{g}(x)}\Big{]}$

Gluon TMDs in momentum and impact space obey the following relations:

$\displaystyle{\rm TMD}^{g}(x)=\int d^{2}{\bf k}_{\perp}{\rm TMD}^{g}(x,{\bf
k}_{\perp}^{2})=4\pi^{2}\,{\widetilde{\rm TMD}}^{g}(x,0)\,,$ (133)

where ${\rm TMD}=f_{1},g_{1L},g_{1T},h_{1}^{\perp}$.

The gluon TMDs in the impact space obey the following sum rule:

$\displaystyle\biggl{[}\tilde{f}_{1}^{g}(x,{\bf
b}_{\perp}^{2})\biggr{]}^{2}=\biggl{[}\tilde{g}_{1L}^{g}(x,{\bf
b}_{\perp}^{2})\biggr{]}^{2}+\biggl{[}S(x,{\bf
b}_{\perp}^{2})\,\tilde{g}_{1T}^{g}(x,{\bf
b}_{\perp}^{2})\biggr{]}^{2}+\biggl{[}\frac{S^{2}(x,{\bf
b}_{\perp}^{2})}{2}\,\tilde{h}_{1}^{\perp g}(x,{\bf
b}_{\perp}^{2})\biggr{]}^{2}\,,$ (134)

where

$\displaystyle S(x,{\bf
b}_{\perp}^{2})=\frac{\kappa}{M_{N}\sqrt{D_{g}(x)}}\,\sqrt{1+\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{4D_{g}(x)}}\,.$ (135)

From this sum rule one can deduce the following inequalities:

$\displaystyle\sqrt{\biggl{[}\tilde{g}_{1L}^{g}(x,{\bf
b}_{\perp}^{2})\biggr{]}^{2}\,+\,\biggl{[}S(x,{\bf
b}_{\perp}^{2})\,\tilde{g}_{1T}^{g}(x,{\bf
b}_{\perp}^{2})\biggr{]}^{2}}\leq\tilde{f}_{1}^{g}(x,{\bf b}_{\perp}^{2})\,,$
(136) $\displaystyle\sqrt{\biggl{[}\tilde{g}_{1L}^{g}(x,{\bf
b}_{\perp}^{2})\biggr{]}^{2}\,+\,\biggl{[}\frac{S^{2}(x,{\bf
b}_{\perp}^{2})}{2}\,\tilde{h}_{1}^{\perp g}(x,{\bf
b}_{\perp}^{2})\biggr{]}^{2}}\leq\tilde{f}_{1}^{g}(x,{\bf b}_{\perp}^{2})\,,$
(137) $\displaystyle\sqrt{\biggl{[}S(x,{\bf
b}_{\perp}^{2})\,\tilde{g}_{1T}^{g}(x,{\bf
b}_{\perp}^{2})\biggr{]}^{2}+\biggl{[}\frac{S^{2}(x,{\bf
k}_{\perp}^{2})}{2}\,\tilde{h}_{1}^{\perp g}(x,{\bf
b}_{\perp}^{2})\biggr{]}^{2}}\leq\tilde{f}_{1}^{g}(x,{\bf b}_{\perp}^{2})$
(138)

and

$\displaystyle\tilde{g}_{1L}^{g}(x,{\bf
b}_{\perp}^{2})\leq\tilde{f}_{1}^{g}(x,{\bf b}_{\perp}^{2})\,,$ (139)
$\displaystyle\tilde{g}_{1T}^{g}(x,{\bf
b}_{\perp}^{2})\leq\frac{M_{N}}{\kappa}\,\sqrt{\frac{D_{g}(x)}{1+\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{4D_{g}(x)}}}\,\tilde{f}_{1}^{g}(x,{\bf
b}_{\perp}^{2})\leq\frac{M_{N}}{\kappa}\,\sqrt{D_{g}(x)}\,\tilde{f}_{1}^{g}(x,{\bf
b}_{\perp}^{2})\,,$ (140) $\displaystyle\tilde{h}_{1}^{\perp g}(x,{\bf
b}_{\perp}^{2})\leq\frac{2M_{N}^{2}}{\kappa^{2}}\,\frac{D_{g}(x)}{1+\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{4D_{g}(x)}}\,\tilde{f}_{1}^{g}(x,{\bf
b}_{\perp}^{2})\leq\frac{2M_{N}^{2}}{\kappa^{2}}\,D_{g}(x)\,\tilde{f}_{1}^{g}(x,{\bf
b}_{\perp}^{2})\,.$ (141)

One can see that the above inequalities involve model parameters $\kappa$ and
$D_{g}(x)$. The nucleon mass and the parameter $\kappa$ are related in our
approach, as $M_{N}=2\sqrt{2}\,\kappa$ Gutsche:2011vb ; Gutsche:2019jzh . The
profile function $D_{g}(x)$ can be also related to the profile function
$f_{g}(x)$, using the relations between form factors, GPDs, and TMDs (see
discussion in Sec. III.3). The inequalities between impact gluon TMDs for
generalized case are shown in Appendix A.

###  III.3 Gluon GPDs and form factors

In this section we derive the GPDs and form factors describing the
distribution of gluons in hadrons. Using Eqs. (53)-(55) for specific twist (we
shift $\tau\to\tau+1/2$ and put $\tau=3$) we get four gluon GPDs corresponding
to the four PDFs/TMDs [see Eq. (III.2)]:

$\displaystyle{\cal H}_{G}(x,Q^{2})$ $\displaystyle=$ $\displaystyle\ \ G(x)\
\exp\Big{[}-a\,D_{G}(x)\,(1-x)^{2}\Big{]}\,,$ $\displaystyle{\cal H}_{\Delta
G}(x,Q^{2})$ $\displaystyle=$ $\displaystyle\Delta G(x)\
\exp\Big{[}-a\,D_{\Delta G}(x)\,(1-x)^{2}\Big{]}\,,$ $\displaystyle{\cal
H}_{g_{1T}^{g}}(x,Q^{2})$ $\displaystyle=$ $\displaystyle g_{1T}(x)\
\exp\Big{[}-a\,D_{g_{1T}^{g}}(x)\,(1-x)^{2}\Big{]}\,,$ $\displaystyle{\cal
H}_{h_{1}^{\perp g}}(x,Q^{2})$ $\displaystyle=$ $\displaystyle f_{1}^{\perp
g}(x)\ \exp\Big{[}-a\,D_{h_{1}^{\perp g}}(x)\,(1-x)^{2}\Big{]}\,,$ (142)

where $a=Q^{2}/(4\kappa^{2})$ and $D_{g}$ $(g=G,\Delta
G,g_{1T}^{g},h_{1}^{\perp g})$ is the set of the profile functions relevant
for specific GPD. The corresponding form factors are given by

$\displaystyle F_{g}(Q^{2})=\int\limits_{0}^{1}dx\,{\cal
H}_{g}(x,Q^{2})\,,\qquad g=G,\Delta G,g_{1T}^{g},f_{1}^{\perp g}\,.$ (143)

$D_{g}(x)$ is related to the function $f_{g}(x)$ Lyubovitskij:2020otz

$\displaystyle
D_{g}(x)=-\frac{\log[1-[f_{g}(x)]^{2/5}(1-x)^{2}]}{(1-x)^{2}}\,.$ (144)

On the other hand, the profile function $f_{g}(x)$ is fixed from gluon PDF by
solving differential equation with respect to $x$ variable. Note, within
symmetric version ($D_{g_{1}}=D_{g_{2}}=D_{g}$) and generalized version
($D_{g_{1}}\neq D_{g_{2}}$) of our approach the profile functions $D_{G}(x)$
and $D_{\Delta G}(x)$ degenerate $D_{G}(x)=D_{\Delta G}(x)=D_{g}(x)=D_{g_{1}}$
[see Eqs. (169) and (170) in Appendix A]. Such a scenario can be realized. In
particular, as we showed in Eqs (62) and (63) in both QCD-based approaches the
profile functions for unpolarized $G(x)$ and polarized $\Delta G(x)$ gluon
PDFs degenerate at large $x$ and become constant: $D_{g}=[f_{g}]^{2/5}$, where
$f_{g}=\frac{21}{10}\,\langle x_{g}\rangle$ (QCDI Brodsky:1989db ) and
$f_{g}=\frac{7}{12}\,\langle x_{g}\rangle$ (QCDII Brodsky:1994kg ). Using the
central value of the latest lattice result $\langle x_{g}\rangle=0.427$
Alexandrou:2020sml  one gets: $D_{g_{1}}=D_{g}=0.957\simeq 1$ (QCDI
Brodsky:1989db ) and $D_{g_{1}}=D_{g}=0.574\simeq 0.6$ (QCDII Brodsky:1994kg
). In case of the ${\cal H}_{g_{1T}^{g}}(x,Q^{2})$ and ${\cal H}_{h_{1}^{\perp
g}}(x,Q^{2})$ GPDs the profile functions $D_{g_{1T}^{g}}(x)$ and
$D_{h_{1}^{\perp g}}(x)$ degenerate with profile functions $D_{G}(x)=D_{\Delta
G}(x)$ in the symmetric version of our approach and deviate from them in
generalized version. In the latter case we get the following relations between
two sets $D_{g_{1T}}(x)$, $D_{h_{1}^{\perp g}}(x)$ and $D_{g_{1}}$,
$D_{g_{1}}$:

$\displaystyle D_{g_{1T}^{g}}(x)=\frac{D_{g_{1}}(x)+D_{g_{2}}(x)}{2}\,,\qquad
D_{f_{1}^{\perp g}}(x)=D_{g_{2}}(x)\,.$ (145)

Finally, our profile functions $D_{G}(x)$, $D_{\Delta G}(x)$,
$D_{g_{1T}^{g}}(x)$, $D_{f_{1}^{\perp g}}(x)$ obey the relations:

$\displaystyle D_{g_{1}}(x)=D_{G}(x)=D_{\Delta G}(x)\,,$ $\displaystyle
D_{g_{2}}(x)=D_{f_{1}^{\perp g}}(x)\,,$
$\displaystyle\frac{D_{g_{1}}(x)+D_{g_{2}}(x)}{2}=D_{g_{1T}^{g}}(x)\,,$
$\displaystyle 2D_{g_{1T}^{g}}(x)=D_{G}(x)+D_{f_{1}^{\perp g}}(x)=D_{\Delta
G}(x)+D_{f_{1}^{\perp g}}(x)\,.$ (146)

The main conclusion from above discussion is that the profile functions in the
T-even gluon TMDs/GPDs are related with each other and become constants at
large $x$. It is supported by the corresponding parametrizations of ${\bf
k}_{\perp}$ dependence of the gluon TMDs with constant scale parameters (see,
e.g., Refs. Anselmino:2001js ; Schweitzer:2010tt ; Boer:2011kf ). On the other
hand, we can also explain why the scale parameters are not the same for all
gluon TMDs. In particular, in Ref. Boer:2011kf  two different scales in the
Gaussians parametrizing the ${\bf k}_{\perp}^{2}$ dependence of the
$f_{1}^{g}(x,{\bf k}_{\perp}^{2})$ and $h_{1}^{\perp g}(x,{\bf
k}_{\perp}^{2})$ TMDs have been considered:

$\displaystyle f_{1}^{g}(x,{\bf k}_{\perp}^{2})$ $\displaystyle=$
$\displaystyle\frac{G(x)}{\pi\langle{\bf
k}_{\perp}^{2}\rangle}\,\exp\biggl{[}-\frac{{\bf k}_{\perp}^{2}}{\langle{\bf
k}_{\perp}^{2}\rangle}\biggr{]}\,,$ $\displaystyle h_{1}^{\perp g}(x,{\bf
k}_{\perp}^{2})$ $\displaystyle=$
$\displaystyle\frac{M_{N}^{2}G(x)}{\pi\langle{\bf
k}_{\perp}^{2}\rangle^{2}}\,\,\frac{2e(1-r)}{r}\,\exp\biggl{[}-\frac{{\bf
k}_{\perp}^{2}}{r\langle{\bf k}_{\perp}^{2}\rangle}\biggr{]}\,,$ (147)

where $\langle{\bf k}_{\perp}^{2}\rangle$ is the width, $r$ is a free
parameter fixed as $2/3$. The above ansatz satisfies the model-independent
Mulders-Rodrigues inequality (116) due to holding the inequality $e^{s-1}\geq
s$ (consequence of the well-known Bernoulli inequality), where $s=\frac{{\bf
k}_{\perp}^{2}}{\langle{\bf k}_{\perp}^{2}\rangle}\,\frac{1-r}{r}$. Also two
TMDs in Eq. (III.3) contain two different scale parameters, parametrized by
$\langle{\bf k}_{\perp}^{2}\rangle$ and $r$. As we showed our formalism
supports existence of two different scales in the $f_{1}^{g}(x,{\bf
k}_{\perp}^{2})$ and $h_{1}^{\perp g}(x,{\bf k}_{\perp}^{2})$ TMDs. Two scales
considered in Ref. Boer:2011kf  are related to our parameters $D_{g_{1}}$,
$D_{g_{2}}$, and $\kappa$ as

$\displaystyle\langle{\bf
k}_{\perp}^{2}\rangle=\frac{\kappa^{2}}{D_{g_{1}}}\,,\qquad
r=\frac{D_{g_{1}}}{D_{g_{2}}}\,.$ (148)

Here we neglect by $x$ dependence of the $D_{g_{1}}$ and $D_{g_{2}}$ and by
the second subleading term in our expression for the $f_{1}^{g}(x,{\bf
k}_{\perp}^{2})$ (169) proportional to $G^{-}(x)$.

Now we are on the position to derive the large $x$ and large $Q^{2}$ behavior
of the gluon quantities (LFWFs, PDFs, TMDs, GPDs, and form factors) based on
two version QCD-based approaches to gluon PDFs: QCDI Brodsky:1989db  and QCDII
Brodsky:1994kg . In both approaches the gluon PDFs scale at large $x$ as

$\displaystyle G(x)\sim\Delta G(x)\sim G^{+}(x)\sim(1-x)^{4}\,,\qquad
G^{-}(x)\sim(1-x)^{6}\,.$ (149)

The only difference is in large $x$ scaling of the $\beta(x)$ defined through
$G^{+}(x)$ and $G^{-}(x)$ in Eq. (96):

$\displaystyle\beta(x)\sim\left\\{\begin{array}[]{cl}1,&\ \ \ {\rm QCDI}\\\
{\cal O}(1-x),&\ \ \ {\rm QCDII}\\\ \end{array}\right.$ (152)

We start with large $x$ scaling of the LFWFs:

$\displaystyle\varphi^{(1)}(x,{\bf
k}_{\perp}^{2})\sim\tilde{\varphi}^{(1)}(x,{\bf
b}_{\perp}^{2})\sim\sqrt{G^{+}(x)}\,\beta(x)\sim\left\\{\begin{array}[]{cl}(1-x)^{2},&\
\ \ {\rm QCDI}\\\ (1-x)^{3},&\ \ \ {\rm QCDII}\\\ \end{array}\right.\,,$
$\displaystyle\varphi^{(2)}(x,{\bf
k}_{\perp}^{2})\sim\tilde{\varphi}^{(2)}(x,{\bf
b}_{\perp}^{2})\sim\frac{\sqrt{G^{-}(x)}}{1-x}\sim(1-x)^{2}\,.$ (154)

Gluon PDFs, TMDs, and GPDs scale as

$\displaystyle G(x)\sim f_{1}^{g}(x,{\bf
k}_{\perp}^{2})\sim\tilde{f}_{1}^{g}(x,{\bf b}_{\perp}^{2})\sim{\cal
H}_{G}(x,Q^{2})\sim(1-x)^{4}\,,$ $\displaystyle\Delta G(x)\sim
g_{1L}^{g}(x,{\bf k}_{\perp}^{2})\sim\tilde{g}_{1L}^{g}(x,{\bf
b}_{\perp}^{2})\sim{\cal H}_{\Delta G}(x,Q^{2})\sim(1-x)^{4}\,,$
$\displaystyle g_{1T}^{g}(x)\sim g_{1T}^{g}(x,{\bf
k}_{\perp}^{2})\sim\tilde{g}_{1T}^{g}(x,{\bf b}_{\perp}^{2})\sim{\cal
H}_{g_{1T}^{g}}(x,Q^{2})$ $\displaystyle\hskip
31.2982pt\sim\sqrt{G^{+}(x)G^{-}(x)}\,\beta(x)\sim(1-x)^{5}\,\beta(x)\sim\left\\{\begin{array}[]{cl}(1-x)^{5},&\
\ \ {\rm QCDI}\\\ (1-x)^{6},&\ \ \ {\rm QCDII}\\\ \end{array}\right.\,,$ (157)
$\displaystyle h_{1}^{\perp g}(x)\sim h_{1}^{\perp g}(x,{\bf
k}_{\perp}^{2})\sim\tilde{h}_{1}^{\perp g}(x,{\bf b}_{\perp}^{2})\sim{\cal
H}_{f_{1}^{\perp g}}(x,Q^{2})\sim\frac{G^{-}(x)}{1-x}\sim(1-x)^{5}\,.$ (158)

Finally the gluon form factors scale at large $Q^{2}$ as

$\displaystyle F_{G}(Q^{2})\sim F_{\Delta
G}(x)\sim\int\limits_{0}^{1}dx\,(1-x)^{4}\,\exp\Big{[}-Q^{2}(1-x)^{2}\Big{]}\sim\frac{1}{Q^{5}}\,,$
$\displaystyle
F_{g_{1T}^{g}}(Q^{2})\sim\int\limits_{0}^{1}dx\,(1-x)^{5}\,\beta(x)\,\exp\Big{[}-Q^{2}(1-x)^{2}\Big{]}\sim\left\\{\begin{array}[]{cl}\frac{1}{Q^{6}}\,,&\
\ \ {\rm QCDI}\\\\[5.69054pt] \frac{1}{Q^{7}}\,,&\ \ \ {\rm QCDII}\\\
\end{array}\right.\,,$ (161) $\displaystyle F_{f_{1}^{\perp
g}}(Q^{2})\sim\int\limits_{0}^{1}dx\,(1-x)^{5}\,\exp\Big{[}-Q^{2}(1-x)^{2}\Big{]}\sim\frac{1}{Q^{6}}\,.$
(162)

##  IV Numerical applications

In this section we present numerical applications of our analytical results
for gluon parton densities. We concentrate on the T-even gluon TMDs, which
were analyzed recently in similar approach in Ref. Bacchetta:2020vty . The
main point is that we derive our analytical formulas for the gluon parton
densities in terms of unpolarized $G(x)$ and polarized $\Delta G(x)$ gluon
PDFs, which we take from present-day world data analysis. Recently, in Ref.
Sufian:2020wcv  such set was derived based on ideas of QCD approaches
Brodsky:1989db  and Brodsky:1994kg  and using world data analysis performed by
the NNPDF Collaboration in Refs. Ball:2017nwa ; Nocera:2014gqa . In
particular, the authors of Ref. Sufian:2020wcv  derived three parametrization
based on the NNPDF global analysis. They are more or less are equivalent and
we take the one (so-called “ansatz 3”), which is direct extension of QCD
formalism developed in Refs. Brodsky:1989db ; Brodsky:1994kg  and produce the
factor $\beta\neq 0$ [see definition in Eq. (96)].

The ansatz 3 for the gluon PDFs of Ref. Sufian:2020wcv  reads:

$\displaystyle xG^{+}(x)$ $\displaystyle=$ $\displaystyle
x^{\alpha}\,\Big{[}A(1-x)^{4+\beta}+B(1-x)^{5+\beta}\,\Big{(}1+\gamma\sqrt{x}+\delta
x\Big{)}\,,$ $\displaystyle xG^{-}(x)$ $\displaystyle=$ $\displaystyle
x^{\alpha}\,\Big{[}A(1-x)^{6+\beta^{\prime}}+B(1-x)^{7+\beta^{\prime}}\,\Big{(}1+\gamma^{\prime}\sqrt{x}+\delta^{\prime}x\Big{)}\,,$
(163)

where $A$ and $B$ are the normalization parameters determined from the moments
of the gluon PDFs $\langle x_{g}\rangle$ and $\Delta G$, the set of the
parameters $(\alpha,\beta,\gamma,\delta)$ was fixed using the NNPDF global
analysis as

$\displaystyle\alpha=0.034\pm 0.064\,,\quad\beta=0.54\pm
1.30\,,\quad\gamma=-2.63\pm 0.60\,,\quad\delta=2.54\pm 1.01\,,$
$\displaystyle\beta^{\prime}=0.54\pm 1.30\,,\quad\gamma^{\prime}=-2.63\pm
0.60\,,\quad\delta^{\prime}=2.54\pm 1.01\,.$ (164)

Note, that approach developed in Ref. Bacchetta:2020vty  also used the NNPDF
Collaboration input for the gluon PDFs. Therefore, it will be easy to compare
the predictions of two approaches for gluon TMDs. Before we turn to discussion
of the numerical results for the gluon TMDs we would like to present numerical
results for the electromagnetic form factors of nucleons induced by the set of
quark PDFs. All analytical formulas for quark densities were derived in our
recent paper Lyubovitskij:2020otz  in the same formalism (soft-wall AdS/QCD
approach). In particular, using the GRV/GRSV world data analysis of the quark
PDFs Gluck:1998xa ; Gluck:2000dy  we calculate the electromagnetic form
factors of nucleons induced by valence quark contributions. The results are
given in Figs. 4-4. In particular, in Figs. 4 and 4 we present the results for
the Dirac and Pauli form factors of $u$ and $d$ quarks multiplied with $Q^{4}$
in the Euclidean region up to 30 GeV2. Data points for the quark decomposition
of the nucleon form factors are taken from Refs. Diehl:2013xca ; Cates:2011pz
. In Figs. 4-4 the shaded bands correspond to a variation of single model
parameter $\kappa=383\pm 30$ MeV. In Figs. 4 and 4 we present the results for
the Dirac and Pauli nucleon form factors multiplied with $Q^{4}$ and the ratio
of the Pauli and Dirac proton form factors multiplied with $Q^{2}$ in the
Euclidean region up to 30 GeV2. Note that preliminary analysis of quark parton
densities and quark/nucleon electromagnetic form factors was performed by us
in Refs. Gutsche:2014yea ; Gutsche:2016gcd  using GRV/GRSV Gluck:1998xa ;
Gluck:2000dy  and MSTW Martin:2009iq  parametrizations for the quark PDFs as
input for our analytical formulas. In particular, we did not find a sufficient
preference of one or another parametrization for the quark PDFs.

Now we are in the position to discuss numerical results for the T-even gluon
TMDs and compare them with results of similar approach developed in Ref.
Bacchetta:2020vty . First of all, for completeness in Fig. 8 we plot the
$xg_{1T}^{g}(x)$ and $xh_{1}^{\perp g}(x)$ gluon PDFs. Here and in the
following the shaded band corresponds to the variation of the parameters
defining the set of the gluon PDFs (IV) fixed from world data analysis by the
NNPDF Collaboration in Refs. Ball:2017nwa ; Nocera:2014gqa . The results for
the T-even gluon TMDs are presented in Figs. 8-12. In particular, in Figs.
8-12 we display the results when $x_{0}=0.001$ (left panels) and $x_{0}=0.1$
at ${\bf k}_{\perp}^{2}$ varied from 0 to 1 GeV2. One should stress that all
results (except prediction for $x_{0}g_{1T}^{g}(x_{0},{\bf k}_{\perp}^{2})$ at
$x_{0}=0.001$) in very good agreement with results of Ref. Bacchetta:2020vty .
One can suppose that the difference of two approaches for plot in Fig. 8(a)
can be influenced by a use of an admixutre of minimal and nonminimal couplings
of gluon with three-quark core in Ref. Bacchetta:2020vty . In our case the
results for all four TMDs are cross-checked by Mulders-Rodrigues positivity
bounds Mulders:2000sh  and using the new sum rule (112) between four T-even
gluon TMDs derived in this paper for the first time in literature. Also we
note that we use the sign convention for the $g_{1T}(x,{\bf k}_{\perp}^{2})$
TMD as in Ref. Mulders:2000sh , while Ref. Bacchetta:2020vty  uses opposite
sign.

Figure 1: Dirac $u$ and $d$ quark form factors.

Figure 2: Pauli $u$ and $d$ quark form factors.

Figure 3: Proton and neutron Dirac form factors.

Figure 4: Ratio $Q^{2}F_{2}^{p}(Q^{2})/F_{1}^{p}(Q^{2})$ for proton.

Figure 5: Gluon PDFs $xg_{1T}^{g}(x)$ and $xh_{1}^{\perp g}(x)$.

Figure 6: Gluon TMD $x_{0}f_{1}^{g}(x_{0},{\bf k}_{\perp}^{2})$ for
$x_{0}=0.001$ (left panel) and for $x_{0}=0.1$ (right panel).

Figure 7: Gluon TMD $x_{0}g_{1L}^{g}(x_{0},{\bf k}_{\perp}^{2})$ for
$x_{0}=0.001$ (left panel) and for $x_{0}=0.1$ (right panel).

Figure 8: Gluon TMD $x_{0}g_{1T}^{g}(x_{0},{\bf k}_{\perp}^{2})$ for
$x_{0}=0.001$ (left panel) and for $x_{0}=0.1$ (right panel).

Figure 9: Gluon TMD $x_{0}h_{1}^{\perp g}(x_{0},{\bf k}_{\perp}^{2})$ for
$x_{0}=0.001$ (left panel) and for $x_{0}=0.1$ (right panel).

Figure 10: 3D plots of gluon TMDs $xf_{1}^{g}(x,{\bf k}_{\perp}^{2})$ (left
panel) and $xg_{1L}^{g}(x,{\bf k}_{\perp}^{2})$ (right panel).

Figure 11: 3D plots of gluon TMDs $xg_{1T}^{g}(x,{\bf k}_{\perp}^{2})$ (left
panel) and $xh_{1}^{\perp g}(x,{\bf k}_{\perp}^{2})$ (right panel).

Figure 12: 3D plots of the ratio $R(x,Q^{2})$ for gluon GPDs for $0.2\leq
x\leq 1$ (left panel) and $0.1\leq x\leq 0.2$ (right panel).

Figure 13: $Q^{2}$ dependence of the gluon GPD $x_{0}{\cal
H}_{G}(x_{0},Q^{2})$ for $x_{0}=0.001$ (left panel) and for $x_{0}=0.1$ (right
panel).

Figure 14: $Q^{2}$ dependence of the gluon GPD $x_{0}{\cal H}_{\Delta
G}(x_{0},Q^{2})$ for $x_{0}=0.001$ (left panel) and for $x_{0}=0.1$ (right
panel).

Figure 15: $Q^{2}$ dependence of the gluon GPD $x_{0}{\cal
H}_{g_{1T}^{g}}(x_{0},Q^{2})$ for $x_{0}=0.001$ (left panel) and for
$x_{0}=0.1$ (right panel).

Figure 16: $Q^{2}$ dependence of the gluon GPD $x_{0}{\cal H}_{f_{1}^{\perp
g}}(x_{0},Q^{2})$ for $x_{0}=0.001$ (left panel) and for $x_{0}=0.1$ (right
panel).

Figure 17: $x$ dependence of the gluon GPD $x{\cal H}_{G}(x,Q_{0}^{2})$ for
$Q_{0}^{2}=0.5$ GeV2 (left panel) and for $Q_{0}^{2}=1$ GeV2 (right panel).

Figure 18: $x$ dependence of the gluon GPD $x{\cal H}_{\Delta G}(x,Q_{0}^{2})$
for $Q_{0}^{2}=0.5$ GeV2 (left panel) and for $Q_{0}^{2}=1$ GeV2 (right
panel).

Figure 19: $x$ dependence of the gluon GPD $x{\cal
H}_{g_{1T}^{g}}(x,Q_{0}^{2})$ for $Q_{0}^{2}=0.5$ GeV2 (left panel) and for
$Q_{0}^{2}=1$ GeV2 (right panel).

Figure 20: $x$ dependence of the gluon GPD $x{\cal H}_{h_{1}^{\perp
g}}(x,Q_{0}^{2})$ for $Q_{0}^{2}=0.5$ GeV2 (left panel) and for $Q_{0}^{2}=1$
GeV2 (right panel).

Figure 21: $Q^{2}$ dependence of the gluon form factors $Q^{5}F_{G}(Q^{2})$
(left panel) and $Q^{5}F_{\Delta G}(Q^{2})$ (right panel).

Figure 22: $Q^{2}$ dependence of the gluon form factors
$Q^{6}F_{g_{1T}^{g}}(Q^{2})$ (left panel) and $Q^{6}F_{h_{1}^{\perp
g}}(Q^{2})$ (right panel).

Next, it is interesting to compare the $Q^{2}$ dependence of our GPDs with the
ones derived in Ref. Diehl:1998kh :

$\displaystyle{\cal H}_{G}(x,Q^{2})$ $\displaystyle=$ $\displaystyle\ \ G(x)\
\exp\Big{[}-\frac{a_{g}^{2}}{2}\,\frac{1-x}{x}\,Q^{2}\Big{]}\,,$
$\displaystyle{\cal H}_{\Delta G}(x,Q^{2})$ $\displaystyle=$
$\displaystyle\Delta G(x)\
\exp\Big{[}-\frac{a_{g}^{2}}{2}\,\frac{1-x}{x}\,Q^{2}\Big{]}\,,$ (165)

where $a_{g}=0.75$ GeV-1 is the scale parameter. In both approaches the gluon
GPDs are expressed in terms of gluon PDFs, therefore we just analyze the
ratios of corresponding GPDs derived in Ref. Diehl:1998kh  and by us:

$\displaystyle
R(x,Q^{2})=\frac{\exp\Big{[}-\frac{a_{g}^{2}}{2}\,\frac{1-x}{x}\,Q^{2}\Big{]}}{\exp\Big{[}-D_{g}(1-x)^{2}\frac{Q^{2}}{4\kappa^{2}}\Big{]}}\,,$
(166)

where $D_{g}$ is the same parameter as used in the analysis of the gluon TMDs.
One can see that exponential of the GPDs derived in our formalism has an extra
fall off $(1-x)$, which is consistent with large scaling of the gluon form
factors dictated by quark counting rules. In case of GPDs this extra fall off
is not sufficient at large $x$, because the exponentials in both approaches
becomes 1 at $x\to 1$ and the GPDs are determined by the corresponding PDF. In
Figs. 12(a) and (b) we present a 3D plot of the ratio $R(x,Q^{2})$, as
function of $x$ and $Q^{2}$. It is seen that two predictions for the gluon
GPDs have very good agreement in the region $0.2\leq x\leq 1$, i.e. the ratio
$R(x,Q^{2})\simeq 1$ and are differed at small $x\leq 0.2$. Note that the
small $x$ behavior of the GPDs in our approach is consistent with small $x$
behavior of the PDFs and TMDs, which are in our approach are consistent with
world data Ball:2017nwa ; Nocera:2014gqa  and similar approaches
Bacchetta:2020vty . On the other hand, as we stressed before, the similar
approaches Meissner:2007rx ; Bacchetta:2008af ; Bacchetta:2020vty ;
Kaur:2020pvc  for quark and gluon TMDs and GPDs are consistent with us at
small and intermediate $x$, while they have different large $x$ behavior
inconsistent with counting rules. By the way, they can be easily improved by
taking into account a specific $x$-dependence of the couplings/form factors.
Such idea was proposed in Ref. Jakob:1997wg , where the spectator model was
derived. In particular, it was suggested that the multipole form factors of
spectator diquark contain a free parameter $\alpha$, which indicates the power
of form factors and can be clearly fixed to fulfill large $x$ counting rules.
Therefore, our approach has advantage that it is valid in whole region of $x$
and compatible respectively with approaches developed for specific regions of
$x$.

In Figs. 16-20 we present our results for the GPDs. In particular, in Figs.
16-16 we display the $Q^{2}$ behavior of the gluon GPDs $x_{0}{\cal
H}(x_{0},Q^{2})$ at fixed values of $x_{0}=0.001$ (left panel) and $x_{0}=0.1$
(right panel). In Figs. 20-20 we show the $x$ behavior of the gluon GPDs at
fixed value of transversed momentum squared $Q_{0}^{2}=1$ GeV2. As expected
our curves for the gluon GPDs fall of with increasing $Q^{2}$ and vanish at
$Q^{2}\simeq 2-3$ GeV2. Finally, in Figs. 22 and 22 we show our predictions
for the gluon form factors. In the plots we multiply each form factor with
corresponding inverse power of $Q^{2}$ corresponding to the large $Q^{2}$
asymptotics. In particular, in Figs. 22(a), 22(b), 22(a), and 22(b) we present
the results for $Q^{5}F_{G}(Q^{2})$, $Q^{5}F_{\Delta G}(Q^{2})$,
$Q^{6}F_{g_{1T}^{g}}(Q^{2})$, and $Q^{6}F_{h_{1}^{\perp g}}(Q^{2})$,
respectively. One can see that all form factors approach the corresponding
large $Q^{2}$ asymptotics at $Q^{2}\sim 10$ GeV2.

##  V Summary

In the present paper we have explicitly demonstrated how to correctly define
gluon parton distributions (PDFs, TMDs, and GPDs) and form factors in the
soft-wall AdS/QCD approach based on the use of quadratic dilaton. In the
description of the partonic structure of hadrons our approach based on ideas
of Refs. Brodsky:1989db ; Brodsky:1994kg ; Brodsky:2000ii ; Brodsky:2002cx ;
Lyubovitskij:2020gjz  and consistent with constraints imposed by QCD
Drell:1969km ; Bloom:1970xb ; Brodsky:1973kr ; Blankenbecler:1974tm ;
Yuan:2003fs ; Aicher:2010cb ; Mulders:2000sh . In particular, in the case of
nucleons we derive the expressions for the T-even gluon TMDs in terms of LFWFs
describing the gluon-three quark Fock component in the nucleon as bound state
of struck gluon and three-quark core (spectator). Next using expressions of
the PDFs in terms of LFWfs we can express all gluon parton densities in terms
of the set of the gluon PDFs $G(x)$ and $\Delta G(x)$. We demonstrated that
our formalism is consistent with model-independent derivation of the gluon
correlators proposed in Ref. Mulders:2000sh  and later considered in Ref.
Meissner:2007rx ; Bacchetta:2020vty . We proved that our gluon TDMs obey the
model-independent Mulders-Rodrigues inequalities Mulders:2000sh  without
referring to a choice of model parameters. As a new result we derived the sum
rule involving four T-even TMDs (112), from which the Mulders-Rodrigues
positivity bounds follow immediately. We checked that in similar approaches
this sum rule is also fulfilled. E.g., in the spectator model considered in
Ref. Bacchetta:2020vty  the sum rule holds when the minimal coupling of gluon
with three-quark spectator is used. In the quark target model discussed in
Ref. Meissner:2007rx  the sum rules is slightly different upon replacement of
the nucleon mass by the constituent quark mass. For the first time, we derived
results for the large $x$ behavior of the gluon TMDs, GPDs, and form factors.
All gluon parton distributions are defined in terms of the unpolarized and
polarized gluon PDFs. As numerical application we calculated the T-even gluon
TMDs using as input the gluon PDFs extracted recently in Ref. Sufian:2020wcv
based on ideas of QCD approaches Brodsky:1989db  and Brodsky:1994kg  and using
world data analysis performed by the NNPDF Collaboration in Refs. Ball:2017nwa
; Nocera:2014gqa . We get very good of our results for the gluon TMDs with
results of similar approach developed recently in Ref. Bacchetta:2020vty . To
solidify our approach we calculated the electromagnetic form factors of
nucleons induced by valence quark partonic densities and get perfect agreement
with data Diehl:2013xca ; Cates:2011pz . Finally, we presented our predictions
for the $x$ and $Q^{2}$ dependence of the gluon GPDs and $Q^{2}$ dependence of
the gluon form factors.

In conclusion, we note that our approach for quark and gluon parton densities
is very similar to the approaches developed in Refs. Jakob:1997wg ;
Meissner:2007rx ; Bacchetta:2008af ; Bacchetta:2020vty ; Kaur:2020pvc . In
case of quarks the consistency was shown in Ref. Bacchetta:2008af , while in
case of gluons we discuss it in the present paper. E.g., it is supported by
good agreement of description of the gluon TMDs in Ref. Bacchetta:2020vty  and
in our formalism. As advantage of our framework we mention that we are able to
predict all parton densities using analytical formulas for the TMDs, GPDs, and
form factors in terms of the set of PDFs, which are taken from world data
analysis. Also we fulfill all constraints imposed by QCD including counting
rules at large $x$. We should note that in similar approaches consistency with
counting rules was not yet implemented. However, such possibility exists. In
particular, in Ref. Jakob:1997wg , where the spectator model was proposed, it
was suggested that the multipole form factors of spectator diquark contain a
free parameter $\alpha$. This parameter indicates the power of form factors
and can be clearly fixed to fulfill large $x$ counting rules.

###### Acknowledgements.

This work was funded by BMBF (Germany) “Verbundprojekt 05P2018 - Ausbau von
ALICE am LHC: Jets und partonische Struktur von Kernen” (Förderkennzeichen:
05P18VTCA1), by CONICYT (Chile) under Grants No. 7912010025, No. 1180232, ANID
PIA/APOYO AFB180002 (Chile), by FONDECYT (Chile) under Grant No. 1191103, and
by Tomsk State and Tomsk Polytechnic University Competitiveness Enhancement
Programs (Russia).

##  Appendix A TMDs in the generalized version

Here we list the gluon TMDs in soft-wall AdS/QCD using generalized version for
the LFWFs $\varphi^{(1)}$ and $\varphi^{(2)}$:

$\displaystyle\varphi^{(1)}(x,{\bf k}_{\perp}^{2})$ $\displaystyle=$
$\displaystyle\frac{4\pi}{\kappa}\,\sqrt{G^{+}(x)}\
\beta(x)\,\sqrt{D_{g_{1}}(x)}\,\exp\biggl{[}-\frac{{\bf
k}_{\perp}^{2}}{2\kappa^{2}}\,D_{g_{1}}(x)\biggr{]}\,,$ (167)
$\displaystyle\frac{1}{M_{N}}\,\varphi^{(2)}(x,{\bf k}_{\perp}^{2})$
$\displaystyle=$
$\displaystyle\frac{4\pi}{\kappa^{2}}\,\sqrt{G^{-}(x)}\,\frac{D_{g_{2}}(x)}{1-x}\,\exp\biggl{[}-\frac{{\bf
k}_{\perp}^{2}}{2\kappa^{2}}\,D_{g_{2}}(x)\biggr{]}\,,$ (168)

where $D_{g_{1}}(x)>0$ and $D_{g_{2}}(x)>0$ are the profile functions.

T-even gluon TMDs in the momentum ${\bf k}_{\perp}$ space:

$\displaystyle f_{1}^{g}(x,{\bf k}_{\perp}^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{3}}\,\biggl{[}(\varphi^{(1)}(x,{\bf
k}_{\perp})\Big{)}^{2}+\frac{{\bf
k}_{\perp}^{2}}{M_{N}^{2}}\,\Big{(}1+(1-x)^{2}\Big{)}\,\Big{(}\varphi^{(2)}(x,{\bf
k}_{\perp})\Big{)}^{2}\biggr{]}$ (169) $\displaystyle=$
$\displaystyle\biggl{[}G(x)+G^{-}(x)\,\alpha_{+}(x)\,\biggl{(}\frac{{\bf
k}_{\perp}^{2}\,D_{g_{2}}^{2}(x)}{\kappa^{2}D_{g_{1}}(x)}-1\biggr{)}\,e^{-{\bf
k}_{\perp}^{2}\Delta
D_{g}(x)/\kappa^{2}}\biggr{]}\,\frac{D_{g_{1}}(x)}{\pi\kappa^{2}}\,\exp\biggl{[}-\frac{{\bf
k}_{\perp}^{2}}{\kappa^{2}}D_{g_{1}}(x)\biggr{]}\,,$ $\displaystyle
g_{1L}^{g}(x,{\bf k}_{\perp})$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{3}}\,\biggl{[}(\varphi^{(1)}(x,{\bf
k}_{\perp})\Big{)}^{2}+\frac{{\bf
k}_{\perp}^{2}}{M_{N}^{2}}\,\Big{(}1-(1-x)^{2}\Big{)}\,\Big{(}\varphi^{(2)}(x,{\bf
k}_{\perp})\Big{)}^{2}\biggr{]}$ (170) $\displaystyle=$
$\displaystyle\biggl{[}\Delta
G(x)+G^{-}(x)\,\alpha_{-}(x)\,\biggl{(}\frac{{\bf
k}_{\perp}^{2}\,D_{g_{2}}^{2}(x)}{\kappa^{2}D_{g_{1}}(x)}-1\biggr{)}\,e^{-{\bf
k}_{\perp}^{2}\Delta
D_{g}(x)/\kappa^{2}}\biggr{]}\,\frac{D_{g_{1}}(x)}{\pi\kappa^{2}}\,\exp\biggl{[}-\frac{{\bf
k}_{\perp}^{2}}{\kappa^{2}}D_{g_{1}}(x)\biggr{]}\,,$ $\displaystyle
g_{1T}^{g}(x,{\bf k}_{\perp}^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi^{3}}\,\varphi^{(1)}(x,{\bf
k}_{\perp})\,\varphi^{(2)}(x,{\bf k}_{\perp})\,(1-x)$ (171) $\displaystyle=$
$\displaystyle\sqrt{G^{2}(x)-\Delta
G^{2}(x)}\,\beta(x)\,\frac{\sqrt{D_{g_{1}}(x)}\,D_{g_{2}}(x)M_{N}}{\pi\kappa^{3}}\,\,\exp\biggl{[}-\frac{{\bf
k}_{\perp}^{2}}{2\kappa^{2}}\Big{(}D_{g_{1}}(x)+D_{g_{2}}(x)\Big{)}\biggr{]}$
$\displaystyle=$ $\displaystyle g_{1T}^{g}(x)\
\frac{D_{g_{1}}(x)+D_{g_{2}}(x)}{2\pi\kappa^{2}}\,\exp\biggl{[}-\frac{{\bf
k}_{\perp}^{2}}{2\kappa^{2}}\Big{(}D_{g_{1}}(x)+D_{g_{2}}(x)\Big{)}\biggr{]}\,,$
$\displaystyle h_{1}^{\perp g}(x,{\bf k}_{\perp}^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{3}}\,\Big{[}\varphi_{q}^{(2)}(x,{\bf
k}_{\perp})\Big{]}^{2}\,(1-x)$ (172) $\displaystyle=$
$\displaystyle\frac{G(x)-\Delta
G(x)}{1-x}\,\frac{2D^{2}_{g_{2}}(x)M_{N}^{2}}{\pi\kappa^{4}}\,\exp\biggl{[}-\frac{{\bf
k}_{\perp}^{2}}{\kappa^{2}}\,D_{g_{2}}(x)\biggr{]}=h_{1}^{\perp g}(x)\
\frac{D_{g_{2}}(x)}{\pi\kappa^{2}}\,\exp\biggl{[}-\frac{{\bf
k}_{\perp}^{2}}{\kappa^{2}}D_{g_{2}}(x)\biggr{]}\,,$

where $\Delta D_{g}(x)=D_{g_{2}}(x)-D_{g_{1}}(x)$.

The gluon PDFs are written as:

$\displaystyle f_{1}^{g}(x)$ $\displaystyle=$ $\displaystyle\int d^{2}{\bf
k}_{\perp}\,f_{1}^{g}(x,{\bf k}_{\perp}^{2})=G(x)\,,$ $\displaystyle
g_{1L}^{g}(x)$ $\displaystyle=$ $\displaystyle\int d^{2}{\bf
k}_{\perp}\,g_{1L}^{g}(x,{\bf k}_{\perp}^{2})=\Delta G(x)\,,$ $\displaystyle
g_{1T}^{g}(x)$ $\displaystyle=$ $\displaystyle\int d^{2}{\bf
k}_{\perp}\,g_{1T}^{g}(x,{\bf
k}_{\perp}^{2})=\frac{2\sqrt{D_{g_{1}}(x)}D_{g_{2}}(x)M_{N}}{\Big{[}D_{g_{1}}(x)+D_{g_{2}}(x)\Big{]}\kappa}\,\sqrt{G^{2}(x)-\Delta
G^{2}(x)}\ \beta(x)\,,$ $\displaystyle h_{1}^{\perp g}(x)$ $\displaystyle=$
$\displaystyle\int d^{2}{\bf k}_{\perp}\,h_{1}^{\perp g}(x,{\bf
k}_{\perp}^{2})=\frac{2D_{g_{2}}(x)M_{N}^{2}}{\kappa^{2}}\,\frac{G(x)-\Delta
G(x)}{1-x}\,.$ (173)

They obey the condition

$\displaystyle\frac{\Big{[}g_{1T}^{g}(x)\Big{]}^{2}}{\Big{[}f_{1}^{g}(x)+g_{1L}^{g}(x)\Big{]}\,h_{1}^{\perp
g}(x)}=\frac{2D_{g_{1}}(x)D_{g_{2}}(x)}{\Big{[}D_{g_{1}}(x)+D_{g_{2}}(x)\Big{]}^{2}}\,(1-x)\,\beta^{2}(x)\,.$
(174)

The results for impact TMDs are expressed in terms of generalized impact
LFWFs:

$\displaystyle\tilde{\varphi}^{(1)}(x,{\bf b}_{\perp}^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\,\sqrt{G^{+}(x)}\
\beta(x)\,\exp\Big{[}-\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{8D_{g_{1}}(x)}\Big{]}\,,$ (175)
$\displaystyle\tilde{\varphi}^{(2)}(x,{\bf b}_{\perp}^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\,\sqrt{G^{-}(x)}\,\frac{\sqrt{D_{g_{2}}(x)}}{1-x}\,\exp\Big{[}-\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{8D_{g_{2}}(x)}\Big{]}\,.$ (176)

In this case the expressions for the impact T-even gluon TMDs are given by:

$\displaystyle\tilde{f}_{1}^{g}(x,{\bf b}_{\perp}^{2})$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}{\bf k}_{\perp}}{(2\pi)^{2}}\,e^{i{\bf
b}_{\perp}{\bf k}_{\perp}}\,f_{1}^{g}(x,{\bf k}_{\perp}^{2})$ (177)
$\displaystyle=$ $\displaystyle\Big{[}\tilde{\varphi}^{(1)}(x,{\bf
b}_{\perp}^{2})\Big{]}^{2}+\Big{[}\tilde{\varphi}^{(2)}(x,{\bf
b}_{\perp}^{2})\Big{]}^{2}\,\frac{1+(1-x)^{2}}{D_{g_{2}}(x)}\,\biggl{[}1+\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{4D_{g_{2}}(x)}\biggr{]}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}}\,\biggl{[}G(x)+G^{-}(x)\,\alpha_{+}(x)\,\gamma(x,{\bf
b}_{\perp}^{2})\biggr{]}\,\exp\Big{[}-\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{4D_{g_{1}}(x)}\Big{]}\,,$
$\displaystyle\tilde{g}_{1L}^{g}(x,{\bf b}_{\perp}^{2})$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}{\bf k}_{\perp}}{(2\pi)^{2}}\,e^{i{\bf
b}_{\perp}{\bf k}_{\perp}}\,g_{1L}^{g}(x,{\bf k}_{\perp}^{2})$ (178)
$\displaystyle=$ $\displaystyle\Big{[}\tilde{\varphi}^{(1)}(x,{\bf
b}_{\perp}^{2})\Big{]}^{2}+\Big{[}\tilde{\varphi}^{(2)}(x,{\bf
b}_{\perp}^{2})\Big{]}^{2}\,\frac{1-(1-x)^{2}}{D_{g_{2}}(x)}\,\biggl{[}1+\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{4D_{g_{2}}(x)}\biggr{]}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}}\,\biggl{[}\Delta
G(x)+G^{-}(x)\,\alpha_{-}(x)\,\gamma(x,{\bf
b}_{\perp}^{2})\biggr{]}\,\exp\Big{[}-\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{4D_{g_{1}}(x)}\Big{]}\,,$
$\displaystyle\tilde{g}_{1T}^{g}(x,{\bf b}_{\perp}^{2})$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}{\bf k}_{\perp}}{(2\pi)^{2}}\,e^{i{\bf
b}_{\perp}{\bf k}_{\perp}}\,g_{1T}^{g}(x,{\bf k}_{\perp}^{2})$ (179)
$\displaystyle=$ $\displaystyle\frac{2M_{N}}{\kappa}\,\sigma(x,{\bf
b}_{\perp}^{2})\,\tilde{\varphi}^{(1)}(x,{\bf
b}_{\perp}^{2})\tilde{\varphi}^{(2)}(x,{\bf b}_{\perp}^{2})(1-x)$
$\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}}\,g_{1T}^{g}(x)\,\exp\Big{[}-\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{2(D_{g_{1}}(x)+D_{g_{2}}(x))}\Big{]}\,,$
$\displaystyle\tilde{h}_{1}^{\perp g}(x,{\bf b}_{\perp}^{2})$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}{\bf k}_{\perp}}{(2\pi)^{2}}\,e^{i{\bf
b}_{\perp}{\bf k}_{\perp}}\,h_{1}^{\perp g}(x,{\bf k}_{\perp}^{2})$ (180)
$\displaystyle=$
$\displaystyle\frac{4M_{N}^{2}}{\kappa^{2}}\,\Big{[}\tilde{\varphi}^{(2)}(x,{\bf
b}_{\perp})\Big{]}^{2}\,(1-x)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}}\,h_{1}^{\perp g}(x)\,\exp\Big{[}-\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{4D_{g_{2}}(x)}\Big{]}\,,$

where

$\displaystyle\gamma(x,{\bf b}_{\perp}^{2})$ $\displaystyle=$
$\displaystyle\biggl{[}1+\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{4D_{g_{2}}(x)}\biggr{]}\,\exp\biggl{[}-\frac{{\bf
b}_{\perp}^{2}\kappa}{4D_{g-}(x)}\biggr{]}\,,$
$\displaystyle\frac{1}{D_{g\pm}(x)}$ $\displaystyle=$
$\displaystyle\frac{1}{D_{g_{2}}(x)}\pm\frac{1}{D_{g_{1}}(x)}\,,$
$\displaystyle\sigma(x,{\bf b}_{\perp}^{2})$ $\displaystyle=$
$\displaystyle\frac{2\sqrt{D_{g_{1}}(x)D_{g_{2}}(x)}}{D_{g_{1}}(x)+D_{g_{2}}(x)}\,\exp\biggl{[}-\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{8}\,\frac{D_{g+}(x)}{D_{g-}^{2}(x)}\biggr{]}\,.$
(181)

One should stress again that the Mulders-Rodrigues inequalities Mulders:2000sh
in the momentum ${\bf k}_{\perp}$ space hold in our approach without referring
to a choice of the LFWF functions and in both symmetric
$D_{g_{1}}(x)=D_{g_{2}}=D_{g}(x)$ (III.2) and generalized $D_{g_{1}}(x)\neq
D_{g_{2}}$ (167) case.

In impact ${\bf b}_{\perp}$ space one can also derive the inequalities between
gluon TMDs, which involve profile functions $D_{g}$ functions and scale
parameter $\kappa$. In symmetric case the inequalities were derived in Eqs.
(139)-(141).

In the generalized case the gluon TMDs satisfy the following relations:

$\displaystyle\tilde{g}_{1L}^{g}(x,{\bf
b}_{\perp}^{2})\leq\tilde{f}_{1}^{g}(x,{\bf b}_{\perp}^{2})\,,$ (182)
$\displaystyle\tilde{g}_{1T}^{g}(x,{\bf
b}_{\perp}^{2})\leq\frac{M_{N}}{\kappa}\,\sqrt{\frac{D_{g_{2}}(x)}{1+\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{4D_{g_{2}}(x)}}}\,\tilde{f}_{1}^{g}(x,{\bf
b}_{\perp}^{2})\leq\frac{M_{N}}{\kappa}\,\sqrt{D_{g_{2}}(x)}\,\tilde{f}_{1}^{g}(x,{\bf
b}_{\perp}^{2})\,,$ (183) $\displaystyle\tilde{h}_{1}^{\perp g}(x,{\bf
b}_{\perp}^{2})\leq\frac{2M_{N}^{2}}{\kappa^{2}}\,\frac{D_{g_{2}}(x)}{1+\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{4D_{g_{2}}(x)}}\,\tilde{f}_{1}^{g}(x,{\bf
b}_{\perp}^{2})\leq\frac{2M_{N}^{2}}{\kappa^{2}}\,D_{g_{2}}(x)\,\tilde{f}_{1}^{g}(x,{\bf
b}_{\perp}^{2})\,.$ (184)

One can see that the first inequality (182) between $\tilde{g}_{1L}^{g}(x,{\bf
b}_{\perp}^{2})$ and $\tilde{f}_{1}^{g}(x,{\bf b}_{\perp}^{2})$ is the same in
both symmetric and generalized case and is similar to the one in momentum
${\bf k}_{\perp}$ space. The second (183) and third (184) inequalities involve
the scale parameter $\kappa$ and profile function $D_{g_{2}}(x)$. In the limit
$D_{g_{1}}(x)=D_{g_{2}}(x)=D_{g}(x)$ the inequalities (183) and (184) reduce
to the inequalities (140) and (141) in the symmetric case.

The proof of the inequalities is straightforward. In particular, the
inequality (182) follows from decomposition of $\tilde{f}_{1}^{g}(x,{\bf
b}_{\perp}^{2})$ and $\tilde{g}_{1L}^{g}(x,{\bf b}_{\perp}^{2})$ in terms of
the LFWFs $\tilde{\varphi}^{(i)}(x,{\bf b}_{\perp})$:

$\displaystyle\tilde{f}_{1}^{g}(x,{\bf
b}_{\perp}^{2})-\tilde{g}_{1L}^{g}(x,{\bf
b}_{\perp}^{2})=2\Big{[}\tilde{\varphi}^{(2)}(x,{\bf
b}_{\perp}^{2})\Big{]}^{2}\,\frac{(1-x)^{2}}{D_{g_{2}}(x)}\biggl{[}1+\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{4D_{g_{2}}(x)}\biggr{]}\geq 0\,.$ (185)

To derive the inequality (183) we start with inequality

$\displaystyle\tilde{f}_{1}^{g}(x,{\bf
b}_{\perp}^{2})-\frac{\kappa}{M_{N}\sigma(x,{\bf
b}_{\perp}^{2})\sqrt{D_{g_{2}}(x)}}\,\sqrt{1+\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{4D_{g_{2}}(x)}}\,\tilde{g}_{1T}^{g}(x,{\bf
b}_{\perp}^{2})\geq 0\,,$ (186)

which holds because it can be written in the form

$\displaystyle\biggl{[}\tilde{\varphi}^{(1)}(x,{\bf
b}_{\perp})-\tilde{\varphi}^{(2)}(x,{\bf
b}_{\perp})\,\frac{1-x}{\sqrt{D_{g_{2}}(x)}}\,\sqrt{1+\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{4D_{g_{2}}(x)}}\biggr{]}^{2}+\frac{1}{D_{g_{2}}(x)}\,\biggl{[}\tilde{\varphi}^{(2)}(x,{\bf
b}_{\perp})\biggr{]}^{2}\,\biggl{[}1+\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{4D_{g_{2}}(x)}\biggr{]}\geq 0\,.$ (187)

Using inequalities (186),

$\displaystyle\sigma(x,{\bf
b}_{\perp}^{2})\leq\frac{2\sqrt{D_{g_{1}}(x)D_{g_{2}}(x)}}{D_{g_{1}}(x)+D_{g_{2}}(x)}\leq
1\,,\qquad\frac{1}{\sqrt{1+\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{4D_{g_{2}}(x)}}}\leq 1$ (188)

we arrive at Eq. (183).

To derive the inequality (184) we start with inequality

$\displaystyle\tilde{f}_{1}^{g}(x,{\bf
b}_{\perp}^{2})-\frac{\kappa^{2}}{2M_{N}^{2}D_{g_{2}}(x)}\,\biggl{[}1+\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{4D_{g_{2}}(x)}\biggr{]}\,\tilde{h}_{1}^{\perp
g}(x,{\bf b}_{\perp}^{2})\geq 0\,.$ (189)

The latter inequality holds because it can be rewritten as

$\displaystyle\biggl{[}\tilde{\varphi}^{(1)}(x,{\bf
b}_{\perp})\biggr{]}^{2}+\biggl{[}\frac{x\,\tilde{\varphi}^{(2)}(x,{\bf
b}_{\perp})}{\sqrt{D_{g_{2}}(x)}}\biggr{]}^{2}\,\biggl{[}1+\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{4D_{g_{2}}(x)}\biggr{]}\geq 0\,.$ (190)

Using inequalities (189) and

$\displaystyle\frac{1}{1+\frac{{\bf
b}_{\perp}^{2}\kappa^{2}}{4D_{g_{2}}(x)}}\leq 1$ (191)

we arrive at Eq. (184).

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