BLFQ Collaboration

# Angular momentum and generalized parton distributions for the proton with
basis light-front quantization

Yiping Liu<EMAIL_ADDRESS>Institute of Modern Physics, Chinese
Academy of Sciences, Lanzhou 730000, China  School of Nuclear Science and
Technology, University of Chinese Academy of Sciences, Beijing 100049, China
Siqi Xu<EMAIL_ADDRESS>Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  School of Nuclear Science and Technology,
University of Chinese Academy of Sciences, Beijing 100049, China  Chandan
Mondal<EMAIL_ADDRESS>Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  School of Nuclear Science and Technology,
University of Chinese Academy of Sciences, Beijing 100049, China  Xingbo Zhao
<EMAIL_ADDRESS>Institute of Modern Physics, Chinese Academy of Sciences,
Lanzhou 730000, China  School of Nuclear Science and Technology, University of
Chinese Academy of Sciences, Beijing 100049, China  James P. Vary
<EMAIL_ADDRESS>Department of Physics and Astronomy, Iowa State University,
Ames, IA 50011, U.S.A.

###### Abstract

We study the unpolarized and the helicity dependent generalized parton
distributions (GPDs) for the valence quarks of the proton in both momentum
space and position space within the basis light-front quantization (BLFQ)
framework. The GPDs for the valence quarks are computed from the eigenvectors
of a light-front effective Hamiltonian in the valence Fock sector consisting
of a three-dimensional confinement potential and a one-gluon exchange
interaction with fixed coupling. Employing these GPDs, we obtain the spatial
distributions of quark angular momentum inside the proton. In our BLFQ
approach, we explore various definitions of angular momentum density and
illustrate the differences between them arising from terms that integrate to
zero. We also discuss the flavor contributions to the quark angular momentum
densities.

##  I Introduction

The origin of nucleon spin is one of the major puzzles in modern particle
physics. The well known European Muon Collaboration experiment
EuropeanMuon:1987isl  has triggered interest in understanding the nucleon spin
from the contributions of the spin and the orbital angular momentum (OAM) of
each of its constituents. In this context, how the total angular momentum
(TAM) is split into separate quark and gluon (partons) contributions is
intrinsically debatable due to quark-gluon couplings and the non-uniqueness of
the decomposition Leader:2013jra ; Wakamatsu:2014zza ; Liu:2015xha .
Meanwhile, it has become clear that the generalized parton distributions
(GPDs) Diehl:2003ny ; Belitsky:2005qn ; Goeke:2001tz , appearing in the
description of hard exclusive reactions like deeply virtual Compton scattering
or deeply virtual meson production, provide us with essential information
about the spatial distributions and orbital motion of partons inside the
nucleon, and allow us to draw three-dimensional pictures of the nucleon. For
more than two decades, the GPDs have been attracting numerous dedicated
experimental and theoretical efforts as many observables can be connected to
them. The GPDs are functions of three variables, namely, longitudinal momentum
fraction ($x$) of the constituent, the skewness ($\zeta$) or the longitudinal
momentum transferred, and the square of the total momentum transferred ($t$).
Their first moments are linked to the electromagnetic form factors, whereas
they reduce to the ordinary parton distributions in the forward limit ($t=0$).
The second moments of the GPDs correspond to the gravitational form factors,
which are linked to matrix elements of the energy-momentum tensor (EMT). Being
off-forward matrix elements, the GPDs do not have probabilistic
interpretations. Meanwhile, for zero skewness the Fourier transform (FT) of
the GPDs with respect to the momentum transfer in the transverse direction
provides the impact parameter dependent GPDs that do have a probabilistic
interpretation Burkardt:2000za ; Burkardt:2002hr . The impact parameter
dependent GPDs encode the correlations in spatial and momentum distributions
of partons in the nucleon. They contain the information about partonic
distributions in the transverse position space for a given longitudinal
momentum fraction carried by the constituent.

Ji has shown that the partonic contribution to the total angular momentum of
the nucleon can be calculated using the second moment of the GPDs Ji:PRL .
Since, the GPDs provide the spatial distribution of the constituents inside
the nucleon, it is therefore credible that the GPDs carry also the knowledge
about the spatial distribution of angular momentum Lorce:2017wkb ;
Polyakov:2002yz ; Adhikari:2016dir ; Kumar:2017dbf . The angular momentum
distribution in three-dimensional coordinate space was first introduced in
Ref. Polyakov:2002yz . However, there is an issue of relativistic corrections
for the three-dimensional distribution, while this ambiguity can be avoided by
defining the two-dimensional distribution in the infinite momentum frame
Adhikari:2016dir ; Leader:2013jra . Different techniques to calculate the
angular momentum distributions in the transverse plane have been prescribed in
Ref. Adhikari:2016dir  and concluded that none of them agrees at the density
level. Meanwhile, a more detailed discussion on the various definitions of
angular momentum has been reported in Ref. Lorce:2017wkb  and the authors have
identified all the missing terms, which hinder the proper comparison. They
have illustrated explicitly using a scalar diquark model that there is no
discrepancy between the different definitions of angular momentum densities.
Later, the distributions of quark angular momentum in a light-front quark-
diquark model (with both scalar and axial vector diquark) motivated by soft
wall AdS/QCD have been investigated in Ref. Kumar:2017dbf .

In this paper, we investigate the spatial distributions of quark angular
momentum inside the proton from its valence light-front wave functions (LFWFs)
that features all three active quarks’ spin, flavor, and three-dimensional
spatial information on the same footing. Our theoretical framework to explore
the nucleon structure is rooted in basis light front quantization (BLFQ)
Vary:2009gt , which provides a computational framework for solving
relativistic many-body bound state problem in quantum field theories
Vary:2009gt ; Li:2021jqb ; Zhao:2014xaa ; Wiecki:2014ola ; Li:2015zda ;
Li:2017mlw ; Jia:2018ary ; Lan:2019vui ; Lan:2019rba ; Lekha ; Tang:2018myz ;
Tang:2019gvn ; Mondal:2019jdg ; Xu:2021wwj ; Lan:2019img ; Qian:2020utg ;
Lan:2021wok . We evaluate the valence quark GPDs of the proton in both
momentum space and position space using the LFWFs based on the BLFQ with only
the valence Fock sector of the proton considered. The BLFQ provides for a
Hamiltonian formalism that incorporates the advantages of the light-front
dynamics Brodsky:1997de . Our effective Hamiltonian includes a three-
dimensional confinement potential consisting of the light-front holography in
the transverse direction Brodsky:2014yha , a longitudinal confinement
Li:2015zda , and a one-gluon exchange interaction with fixed coupling to
account for the spin structure Mondal:2019jdg . The nonperturbative solutions
for the three-body LFWFs are given by the recent BLFQ study of the nucleon
Mondal:2019jdg . These LFWFs have been applied successfully to predict the
electromagnetic and axial form factors, radii, parton distribution functions
(PDFs), and many other quantities of the nucleon Mondal:2019jdg ; Xu:2021wwj ;
Mondal:2021wfq . Here, we extend those investigations to study the proton GPDs
and their application for the description of angular momentum distributions.

The paper is organized as follows. We briefly summarize the BLFQ formalism for
the nucleon in Sec. II. We then present a detailed description of the angular
momentum and the associated GPDs in Sec. III. Sec. IV details our numerical
results for the GPDs, and different angular momentum densities. At the end, we
provide a brief summary and conclusions in Sec. V.

##  II Light-front effective Hamiltonian for the proton

The LFWFs that encode the structure of hadronic bound states are obtained as
the eigenfunctions of the eigenvalue equation of the Hamiltonian: $H_{\rm
LF}|\Psi\rangle=M_{\rm h}^{2}|\Psi\rangle,$ where $H_{\rm LF}$ represents the
light-front Hamiltonian of the hadron with the mass squared ($M_{\rm h}^{2}$)
eigenvalue. With quarks being the only explicit degree of freedom, the
effective Hamiltonian we employ for the proton includes the two-dimensional
harmonic oscillator (‘2D-HO’) transverse confining potential along with a
longitudinal confinement and an effective one-gluon exchange interaction
Mondal:2019jdg

$\displaystyle H_{\rm eff}=$ $\displaystyle\sum_{a}\frac{{\vec{k}}_{\perp
a}^{2}+m_{a}^{2}}{x_{a}}+\frac{1}{2}\sum_{a\neq
b}\kappa^{4}\Big{[}x_{a}x_{b}({\vec{r}}_{\perp a}-{\vec{r}}_{\perp
b})^{2}-\frac{\partial_{x_{a}}(x_{a}x_{b}\partial_{x_{b}})}{(m_{a}+m_{b})^{2}}\Big{]}$
$\displaystyle+\frac{1}{2}\sum_{a\neq
b}\frac{C_{F}4\pi\alpha_{s}}{Q^{2}_{ab}}\bar{u}(k^{\prime}_{a},s^{\prime}_{a})\gamma^{\mu}{u}(k_{a},s_{a})\bar{u}(k^{\prime}_{b},s^{\prime}_{b})\gamma^{\nu}{u}(k_{b},s_{b})g_{\mu\nu}\,,$
(1)

where $x_{a}$ and ${\vec{k}}_{\perp a}$ represent the longitudinal momentum
fraction and the relative transverse momentum carried by quark $a$. $m_{a}$ is
the mass of the quark a, and $\kappa$ defines the strength of the confinement.
The variable $\vec{r}_{\perp}={\vec{r}}_{\perp a}-{\vec{r}}_{\perp b}$ is the
transverse separation between two quarks. The last term in the effective
Hamiltonian corresponds to the OGE interaction where
$Q^{2}_{ab}=-q^{2}=-(1/2)(k^{\prime}_{a}-k_{a})^{2}-(1/2)(k^{\prime}_{b}-k_{b})^{2}$
is the average momentum transfer squared, $C_{F}=-2/3$ is the color factor,
$\alpha_{s}$ is the coupling constant and $g_{\mu\nu}$ is the metric tensor.
${u}(k_{a},s_{a})$ represents the spinor with momentum $k_{a}$ and spin
$s_{a}$.

For the BLFQ basis representation, the 2D-HO function is adopted for the
transverse direction, while we employ the discretized plane-wave basis in the
longitudinal direction Vary:2009gt ; Zhao:2014xaa . Diagonalizing the
Hamiltonian, Eq. (II), in our chosen basis space gives the eigenvalues as
squares of the bound state eigenmasses, and the eigenstates which specify the
LFWFs. The lowest eigenstate is naturally identified as the nucleon state,
denoted as $\ket{P,{\Lambda}}$, with $P$ and $\Lambda$ being the momentum and
the helicity of the state. In terms of the basis function the LFWFs of the
nucleon are expressed as

$\displaystyle\Psi^{\Lambda}_{\\{x_{i},\vec{k}_{i\perp},\lambda_{i}\\}}=\sum_{\\{n_{i},m_{i}\\}}\psi^{\Lambda}_{\\{x_{i},n_{i},m_{i},\lambda_{i}\\}}\prod_{i}\phi_{n_{i},m_{i}}(\vec{k}_{i\perp};b)\,,$
(2)

where
$\psi^{\Lambda}_{\\{x_{i},n_{i},m_{i},\lambda_{i}\\}}=\braket{P,{\Lambda}}{\\{x_{i},n_{i},m_{i},\lambda_{i}\\}}$
is the LFWF in the BLFQ basis obtained by diagnalizing Eq. (II) numerically.
The 2D-HO function we adopt as the transverse basis function is

$\displaystyle\phi_{n,m}(\vec{k}_{\perp};b)=\frac{\sqrt{2}}{b(2\pi)^{\frac{3}{2}}}\sqrt{\frac{n!}{(n+|m|)!}}e^{-\vec{k}_{\perp}^{2}/(2b^{2})}\left(\frac{|\vec{k}_{\perp}|}{b}\right)^{|m|}L^{|m|}_{n}(\frac{\vec{k}_{\perp}^{2}}{b^{2}})e^{im\theta}\,,$
(3)

with $b$ as its scale parameter; $n$ and $m$ are the principal and orbital
quantum numbers, respectively, and $L^{|m|}_{n}$ is the associated Laguerre
polynomial. In the discretized plane-wave basis, the longitudinal momentum
fraction $x$ is defined as $x_{i}=p_{i}^{+}/P^{+}=k_{i}/K,$ where the
dimensionless quantity signifying the choice of antiperiodic boundary
conditions is $k=\frac{1}{2},\frac{3}{2},\frac{5}{2},...$ and
$K=\sum_{i}k_{i}$. The multi-body basis states have selected values of the
total angular momentum projection
$M_{J}=\sum_{i}\left(m_{i}+\lambda_{i}\right),$ where $\lambda$ is used to
label the quark helicity. The transverse basis truncation is specified by the
dimensionless parameters $N_{\rm max}$, such that
$\sum_{i}(2n_{i}+|m_{i}|+1)\leq N_{\rm max}$. The basis cutoff $N_{\rm max}$
acts implicitly as the ultraviolet (UV) and infrared (IR) regulators for the
LFWFs in the transverse direction, with a UV cutoff $\Lambda_{\rm UV}\approx
b\sqrt{N_{\rm max}}$ and an IR cutoff $\Lambda_{\rm IR}\approx b/\sqrt{N_{\rm
max}}$. The longitudinal basis cutoff $K$ controls the numerical resolution
and regulates the longitudinal direction.

Parameters in the model Hamiltonian are fixed to reproduce the ground state
mass of the nucleon and to fit the Dirac flavor form factors Xu:2021wwj . The
LFWFs in this model have been successfully applied to compute a wide class of
different and related nucleon observables, e.g., the electromagnetic and axial
form factors, radii, PDFs, helicity asymmetries, transverse momentum dependent
parton distribution functions etc., with remarkable overall success
Mondal:2019jdg ; Xu:2021wwj ; Mondal:2021wfq .

##  III Angular momentum distributions

In this section, we introduce our notation and briefly review the derivation
of angular momentum distribution following Ref. Lorce:2017wkb . In field
theory, the generalized angular momentum tensor operator is written as follows

$J^{\mu\alpha\beta}(y)=L^{\mu\alpha\beta}(y)+S^{\mu\alpha\beta}(y)\;.$ (4)

Both of the contributions are antisymmetric under
$\alpha\leftrightarrow\beta$. When $\alpha,\beta$ are spatial components,
$L^{\mu\alpha\beta}(y)$ and $S^{\mu\alpha\beta}(y)$ are identified with the
OAM and spin operators, respectively. The first contribution can be expressed
in terms of the EMT as

$L^{\mu\alpha\beta}(y)=y^{\alpha}T^{\mu\beta}(y)-y^{\beta}T^{\mu\alpha}(y)\;.$
(5)

Note that $T^{\mu\nu}$ is referred to the canonical EMT and it is, in general,
neither gauge invariant nor symmetric. Meanwhile, the TAM can also be
expressed in a pure orbital form,

$J^{\mu\alpha\beta}_{\text{Bel}}(y)=y^{\alpha}T_{\text{Bel}}^{\mu\beta}(y)-y^{\beta}T_{\text{Bel}}^{\mu\alpha}(y)\,,$
(6)

using the Belinfante-improved EMT Belinfante1 ; Belinfante2 ; Rosenfeld ,
which is defined by adding a term to the definition of $T^{\mu\nu}$ as

$\displaystyle T^{\mu\nu}_{\text{Bel}}(y)$
$\displaystyle=T^{\mu\nu}(y)+\partial_{\lambda}G^{\lambda\mu\nu}(y)\,,$ (7)

where $G^{\lambda\mu\nu}$ is given by

$G^{\lambda\mu\nu}(y)=\frac{1}{2}\left[S^{\lambda\mu\nu}(y)+S^{\mu\nu\lambda}(y)+S^{\nu\mu\lambda}(y)\right]=-G^{\mu\lambda\nu}(y)\;.$
(8)

The additional term revises the definition of the local density without
changing the TAM. The Belinfante-improved tensor $T^{\mu\nu}_{\text{Bel}}$ is
conserved, symmetric and gauge invariant. The Belinfante-improved tensors can
be seen as effective densities, where the effects of spin are imitated by a
superpotential contribution to the angular momentum. In order to determine the
angular momentum distributions, we are interested in the matrix elements of
the above mentioned operator densities.

For a spin-$\frac{1}{2}$ target, the matrix elements of the general local
asymmetric $T^{\mu\nu}$ are parametrized in terms of several gravitational
form factors Leader:2013jra :

$\displaystyle\langle P^{\prime},{\bf\Lambda}^{\prime}\lvert
T^{\mu\nu}(0)\rvert
P,{\bf\Lambda}\rangle=\bar{u}(P^{\prime},{\bf\Lambda}^{\prime})\Big{[}\frac{\bar{P}^{\mu}\bar{P}^{\nu}}{M}\,A(t)+\frac{\bar{P}^{\mu}i\sigma^{\nu\lambda}\Delta_{\lambda}}{4M}\,(A+B+D)(t)$
$\displaystyle\quad\quad\quad\quad+\frac{\Delta^{\mu}\Delta^{\nu}-g^{\mu\nu}\Delta^{2}}{M}\,C(t)+Mg^{\mu\nu}\,\bar{C}(t)+\frac{\bar{P}^{\nu}i\sigma^{\mu\lambda}\Delta_{\lambda}}{4M}\,(A+B-D)(t)\Big{]}u(P,{\bf\Lambda})\;,$
(9)

where $\bar{P}=\frac{1}{2}(P^{\prime}+P)$, $\Delta=P^{\prime}-P$,
$t=\Delta^{2}$, $M$ is the system mass, the three-vector
${\bf\Lambda}({\bf\Lambda}^{\prime})$ denotes the rest-frame polarization of
the initial (final) state, and $u(P,\Lambda)$ is the spinor. The gravitational
form factors $A(t)$, $B(t)$ and $C(t)$ can be related to leading-twist GPDs,
which are accessible in exclusive processes Ji:2004gf . Meanwhile, the form
factor $\bar{C}(t)$, obtainable from the trace of the energy-momentum tensor,
is related to the $\sigma_{\pi N}$ term extracted from pion-nucleon scattering
amplitudes Alarcon:2011zs ; Hoferichter:2015dsa .

On the other hand, the matrix elements of the quark spin operator

$\displaystyle
S^{\mu\alpha\beta}_{q}(y)=\frac{1}{2}\,\varepsilon^{\mu\alpha\beta\lambda}\,\overline{\psi}(y)\gamma_{\lambda}\gamma_{5}\psi(y)\,,$
(10)

with $\psi(y)$ and $\bar{\psi}(y)$ being the quark field are parametrized as

$\langle P^{\prime},{\bf\Lambda}^{\prime}\lvert
S^{\mu\alpha\beta}_{q}(0)\rvert
P,{\bf\Lambda}\rangle=\frac{1}{2}\,\varepsilon^{\mu\alpha\beta\lambda}\,\overline{u}(P^{\prime},{\bf\Lambda}^{\prime})\left[\gamma_{\lambda}\gamma_{5}\,G^{q}_{A}(t)+\frac{\Delta_{\lambda}\gamma_{5}}{2M}\,G^{q}_{P}(t)\right]u(P,{\bf\Lambda})\,,$
(11)

where $G^{q}_{A}(t)$ and $G^{q}_{P}(t)$ are the axial vector and pseudoscalar
form factors, respectively and the convention $\varepsilon^{0123}=+1$.
According to Refs. Bakker:2004ib ; Leader:2013jra , the axial form factor is
connected to the gravitational form factor associated with the antisymmetric
part of the quark EMT, $D_{q}(t)=-G^{q}_{A}(t).$ The axial form factor is
measurable from quasi-elastic neutrino scattering and pion electroproduction
processes Bernard:2001rs . The different angular momentum distributions can
thus be defined through the combination of the gravitational form factors and
the axial form factor.

###  III.1 Distributions in the transverse plane on the light front

In the light-front (LF) formalism, the impact-parameter distributions of
kinetic OAM and spin in the Drell-Yan (DY) frame are given by Lorce:2017wkb

$\displaystyle\langle L^{z}\rangle({b}_{\perp})$
$\displaystyle=-i\varepsilon^{3jk}\int\frac{\text{d}^{2}\vec{\Delta}_{\perp}}{(2\pi)^{2}}\,e^{-i\vec{\Delta}_{\perp}\cdot\vec{b}_{\perp}}\left.\frac{\partial\langle
T^{+k}\rangle}{\partial\Delta^{j}_{\perp}}\right|_{\text{DY}}$
$\displaystyle=\Lambda^{z}\int\frac{\text{d}^{2}\vec{\Delta}_{\perp}}{(2\pi)^{2}}\,e^{-i\vec{\Delta}_{\perp}\cdot\vec{b}_{\perp}}\left[L(t)+t\,\frac{\text{d}L(t)}{\text{d}t}\right]_{t=-\vec{\Delta}^{2}_{\perp}},$
(12) $\displaystyle\langle S^{z}\rangle({b}_{\perp})$
$\displaystyle=\frac{1}{2}\,\varepsilon^{3jk}\int\frac{\text{d}^{2}\vec{\Delta}_{\perp}}{(2\pi)^{2}}\,e^{-i\vec{\Delta}_{\perp}\cdot\vec{b}_{\perp}}\left.\langle
S^{+jk}\rangle\right|_{\text{DY}}$
$\displaystyle=\frac{\Lambda^{z}}{2}\int\frac{\text{d}^{2}\vec{\Delta}_{\perp}}{(2\pi)^{2}}\,e^{-i\vec{\Delta}_{\perp}\cdot\vec{b}_{\perp}}G_{A}(-\vec{\Delta}^{2}_{\perp})\;,$
(13)

respectively, where ${2\sqrt{P^{\prime+}P^{+}}}\langle
T^{\mu\nu}\rangle\equiv\langle P^{\prime},S\lvert T^{\mu\nu}(0)\rvert
P,S\rangle$ and $L(t)$ is the combination of energy-momentum form factors and
the axial form factor,

$\displaystyle L(t)$
$\displaystyle=\frac{1}{2}\left[A(t)+B(t)+D(t)\right]=\frac{1}{2}\left[A(t)+B(t)-G_{A}(t)\right]\,.$
(14)

The variable ${\vec{b}}_{\perp}$ is the Fourier conjugate to the transverse
momentum transfer ${\vec{\Delta}}_{\perp}$. The impact parameter ${b}_{\perp}$
corresponds to the transverse displacement of the active quark from the center
of momentum of the nucleon. Meanwhile, the the Belinfante-improved TAM and the
total divergence in the impact-parameter are defined as Lorce:2017wkb

$\displaystyle\langle J^{z}_{\text{Bel}}\rangle({b}_{\perp})$
$\displaystyle=-i\varepsilon^{3jk}\int\frac{\text{d}^{2}\vec{\Delta}_{\perp}}{(2\pi)^{2}}\,e^{-i\vec{\Delta}_{\perp}\cdot\vec{b}_{\perp}}\left.\frac{\partial\langle
T^{+k}_{\text{Bel}}\rangle}{\partial\Delta^{j}_{\perp}}\right|_{\text{DY}}$
$\displaystyle=\Lambda^{z}\int\frac{\text{d}^{2}\vec{\Delta}_{\perp}}{(2\pi)^{2}}\,e^{-i\vec{\Delta}_{\perp}\cdot\vec{b}_{\perp}}\left[J(t)+t\,\frac{\text{d}J(t)}{\text{d}t}\right]_{t=-\vec{\Delta}^{2}_{\perp}},$
(15) $\displaystyle\langle M^{z}\rangle({b}_{\perp})$
$\displaystyle=\frac{1}{2}\,\varepsilon^{3jk}\int\frac{\text{d}^{2}\vec{\Delta}_{\perp}}{(2\pi)^{2}}\,e^{-i\vec{\Delta}_{\perp}\cdot\vec{b}_{\perp}}\,\Delta^{l}_{\perp}\left.\frac{\partial\langle
S^{l+k}\rangle}{\partial\Delta^{j}_{\perp}}\right|_{\text{DY}}$
$\displaystyle=-\frac{\Lambda^{z}}{2}\int\frac{\text{d}^{2}\vec{\Delta}_{\perp}}{(2\pi)^{2}}\,e^{-i\vec{\Delta}_{\perp}\cdot\vec{b}_{\perp}}\left[t\,\frac{\text{d}G_{A}(t)}{\text{d}t}\right]_{t=-\vec{\Delta}^{2}_{\perp}}\,,$
(16)

respectively, where

$\displaystyle J(t)$ $\displaystyle=\frac{1}{2}\left[A(t)+B(t)\right]\,.$ (17)

Using the two-dimensional Fourier transform of the form factors defined as

$\tilde{F}(b_{\perp})=\int\frac{\text{d}^{2}\vec{\Delta}_{\perp}}{(2\pi)^{2}}\,e^{-i\vec{\Delta}_{\perp}\cdot\vec{b}_{\perp}}\,F(-\vec{\Delta}^{2}_{\perp})\;,$
(18)

Eqs. (12)-(16) can be rewritten as

$\displaystyle\langle L^{z}\rangle({b}_{\perp})$
$\displaystyle=-\frac{\Lambda^{z}}{2}\,b_{\perp}\,\frac{\text{d}\tilde{L}(b_{\perp})}{\text{d}b_{\perp}}\;,$
(19) $\displaystyle\langle S^{z}\rangle({b}_{\perp})$
$\displaystyle=\frac{\Lambda^{z}}{2}\,\tilde{G}_{A}(b_{\perp})\;,$ (20)
$\displaystyle\langle J^{z}_{\text{Bel}}\rangle({b}_{\perp})$
$\displaystyle=-\frac{\Lambda^{z}}{2}\,b_{\perp}\,\frac{\text{d}\tilde{J}(b_{\perp})}{\text{d}b_{\perp}}\;,$
(21) $\displaystyle\langle M^{z}\rangle({b}_{\perp})$
$\displaystyle=\frac{\Lambda^{z}}{2}\left[\tilde{G}_{A}(b_{\perp})+\frac{1}{2}\,b_{\perp}\frac{\text{d}\tilde{G}_{A}(b_{\perp})}{\text{d}b_{\perp}}\right]\;.$
(22)

The total angular momentum density $\langle J^{z}\rangle({b}_{\perp})$ is then
given by

$\displaystyle\langle J^{z}\rangle(b_{\perp})=\langle
L^{z}\rangle({b}_{\perp})+\langle S^{z}\rangle({b}_{\perp})=\langle
J^{z}_{\text{Bel}}\rangle({b}_{\perp})+\langle M^{z}\rangle({b}_{\perp})\,,$
(23)

which is different form the “naive” density, which is defined by the two-
dimensional Fourier transform of $J(t)$,

$\displaystyle\langle
J^{z}_{\text{naive}}\rangle(b_{\perp})=\Lambda^{z}\tilde{J}(b_{\perp})\,,$
(24)

by a correction term

$\displaystyle\langle
J^{z}_{\text{corr}}\rangle({b}_{\perp})=-\Lambda^{z}\left[\tilde{L}(b_{\perp})+\frac{1}{2}\,b_{\perp}\,\frac{\text{d}\tilde{L}(b_{\perp})}{\text{d}b_{\perp}}\right]\;.$
(25)

Beside the densities mentioned above, the Belinfante-improved TAM can also be
formulated as the sum of monopole and quadrupole contributions Lorce:2017wkb

$\displaystyle\langle J_{\text{Bel}}^{z(\text{mono})}\rangle(b_{\perp})$
$\displaystyle=\frac{\Lambda^{z}}{3}\left[\tilde{J}(b_{\perp})-b_{\perp}\,\frac{\text{d}\tilde{J}(b_{\perp})}{\text{d}b_{\perp}}\right]\;,$
(26) $\displaystyle\langle J_{\text{Bel}}^{z(\text{quad})}\rangle(b_{\perp})$
$\displaystyle=-\frac{\Lambda^{z}}{3}\left[\tilde{J}(b_{\perp})+\frac{1}{2}\,b_{\perp}\,\frac{\text{d}\tilde{J}(b_{\perp})}{\text{d}b_{\perp}}\right]\;.$
(27)

The monopole contribution, Eq. (26), is the projection of the expression used
by Polyakov and collaborators Polyakov:2002yz ; Goeke:2007fp  onto the
transverse plane. This has later been studied as Polyakov–Goeke distribution
in Ref. Adhikari:2016dir . The quadrupole contribution, Eq. (27), is also the
2D projection of the 3D quadrupole contribution to the Belinfante-improved TAM
Lorce:2017wkb , which arises from the breaking of spherical symmetry down to
axial symmetry due to the polarization of the state.

Note that the total divergence (Eq. (22)), the correction (Eq. (25)), and the
quadrupole (Eq. (27)) terms vanish when they are integrated over
$\vec{b}_{\perp}$. This clarifies how different definitions lead to the same
integrated total angular momentum though they are distinct from each other at
the density level Lorce:2017wkb ; Kumar:2017dbf ; Adhikari:2016dir .

###  III.2 Generalized parton distributions

In general, the GPDs are defined through the off-forward matrix elements of
the bilocal operators between hadronic states. The unpolarized and helicity
dependent quark GPDs for the nucleon are parameterized as Ji:1998pc

$\displaystyle\int\frac{\text{d}y^{-}}{8\pi}e^{ixP^{+}y^{-}/2}\braket{P^{\prime},\Lambda^{\prime}}{\bar{\psi}(0)\gamma^{+}\psi(y)}{P,\Lambda}|_{y^{+}=\vec{y}_{\perp}=0}$
$\displaystyle\quad\quad\quad\quad\quad\quad=\frac{1}{2\bar{P}^{+}}\bar{u}(P^{\prime},\Lambda^{\prime})\left[H^{q}(x,\zeta,t)\gamma^{+}+E^{q}(x,\zeta,t)\frac{i\sigma^{+j}\Delta_{j}}{2M}\right]u(P,\Lambda)\,,$
(28)
$\displaystyle\int\frac{\text{d}y^{-}}{8\pi}e^{ixP^{+}y^{-}/2}\braket{P^{\prime},\Lambda^{\prime}}{\bar{\psi}(0)\gamma^{+}\gamma_{5}\Psi(y)}{P,\Lambda}|_{y^{+}=\vec{y}_{\perp}=0}$
$\displaystyle\quad\quad\quad\quad\quad\quad=\frac{1}{2\bar{P}^{+}}\bar{u}(P^{\prime},\Lambda^{\prime})\Big{[}\widetilde{H}^{q}(x,\zeta,t)\gamma^{+}\gamma_{5}+\widetilde{E}^{q}(x,\zeta,t)\frac{\gamma_{5}\Delta^{+}}{2M}\Big{]}u(P,\Lambda)\,.$
(29)

Here $H$ and $E$ are the unpolarized quark GPDs, whereas $\widetilde{H}$ and
$\widetilde{E}$ represent the helicity dependent GPDs. The kinematical
variables are $\bar{P}=(P^{\prime}+P)/2$, $\Delta=P^{\prime}-P$,
$\zeta=-\Delta^{+}/2\bar{P}^{+}$ and $t=\Delta^{2}$. For $\zeta=0$,
$t=-\vec{\Delta}_{\perp}^{2}$. We consider the light cone gauge $A^{+}=0$,
which indicates that the gauge-link between the quark fields in Eqs. (28) and
(29) is unity. In this paper, we concentrate only on the GPDs relevant to the
angular momentum densities, i.e., $H$, $E$ and $\widetilde{H}$ at the zero
skewness limit. Note that one has to consider nonzero skewness to compute GPD
$\widetilde{E}$, which is not needed for this work.

Substituting the nucleon states within the valence Fock sector

$\displaystyle\ket{P,{\Lambda}}=$
$\displaystyle\int\prod_{i=1}^{3}\left[\frac{{\rm d}x_{i}{\rm
d}^{2}\vec{k}_{i\perp}}{\sqrt{x_{i}}16\pi^{3}}\right]16\pi^{3}\delta\left(1-\sum_{i=1}^{3}x_{i}\right)\delta^{2}\left(\sum_{i=1}^{3}\vec{k}_{i\perp}\right)$
$\displaystyle\times\Psi^{\Lambda}_{\\{x_{i},\vec{k}_{i\perp},\lambda_{i}\\}}\ket{\\{x_{i}P^{+},\vec{k}_{i\perp}+x_{i}\vec{P}_{\perp},\lambda_{i}\\}}\,,$
(30)

and the quark field operators in Eqs. (28) and (29) leads to the GPDs in terms
of the overlap of the LFWFs

$\displaystyle H^{q}(x,0,t)=$
$\displaystyle\sum_{\\{\lambda_{i}\\}}\int\left[{\rm d}\mathcal{X}\,{\rm
d}\mathcal{P}_{\perp}\right]\,\Psi^{\uparrow*}_{\\{x^{\prime}_{i},\vec{k}^{\prime}_{i\perp},\lambda_{i}\\}}\Psi^{\uparrow}_{\\{x_{i},\vec{k}_{i\perp},\lambda_{i}\\}}\delta(x-x_{1})\,,$
(31) $\displaystyle E^{q}(x,0,t)=$
$\displaystyle-\frac{2M}{(q^{1}-iq^{2})}\sum_{\\{\lambda_{i}\\}}\int\left[{\rm
d}\mathcal{X}\,{\rm
d}\mathcal{P}_{\perp}\right]\,\Psi^{\uparrow*}_{\\{x^{\prime}_{i},\vec{k}^{\prime}_{i\perp},\lambda_{i}\\}}\Psi^{\downarrow}_{\\{x_{i},\vec{k}_{i\perp},\lambda_{i}\\}}\delta(x-x_{1})\,,$
(32) $\displaystyle\widetilde{H}^{q}(x,0,t)=$
$\displaystyle\sum_{\\{\lambda_{i}\\}}\int\left[{\rm d}\mathcal{X}\,{\rm
d}\mathcal{P}_{\perp}\right]\,\lambda_{1}\,\Psi^{\uparrow*}_{\\{x^{\prime}_{i},\vec{k}^{\prime}_{i\perp},\lambda_{i}\\}}\Psi^{\uparrow}_{\\{x_{i},\vec{k}_{i\perp},\lambda_{i}\\}}\delta(x-x_{1})\,,$
(33)

where

$\displaystyle\left[{\rm d}\mathcal{X}\,{\rm
d}\mathcal{P}_{\perp}\right]=\prod_{i=1}^{3}\left[\frac{{\rm d}x_{i}{\rm
d}^{2}\vec{k}_{i\perp}}{16\pi^{3}}\right]16\pi^{3}\delta\left(1-\sum_{i=1}^{3}x_{i}\right)\delta^{2}\left(\sum_{i=1}^{3}\vec{k}_{i\perp}\right),$
(34)

and the light-front momenta are $x^{\prime}_{1}=x_{1}$;
$\vec{k}^{\prime}_{1\perp}=\vec{k}_{1\perp}+(1-x_{1})\vec{\Delta}_{\perp}$ for
the struck quark ($i=1$) and
$x^{\prime}_{i}={x_{i}};~{}\vec{k}^{\prime}_{i\perp}=\vec{k}_{i\perp}-{x_{i}}\vec{\Delta}_{\perp}$
for the spectators ($i\neq 1$) and $\lambda_{1}=1~{}(-1)$ for the struck quark
helicity. The proton helicity is designated by $\Lambda=\uparrow(\downarrow)$,
where $\uparrow$ and $\downarrow$ correspond to $+1$ and $-1$, respectively.

Integrating the non-local matrix element that parameterized the GPDs, over $x$
leads to the local matrix elements yielding the form factors. In the Drell-Yan
frame, the expressions for the form factors are very similar to the
expressions for GPDs, except that the longitudinal momentum fraction $x$ of
the struck quark is not integrated out in the GPDs’ expressions. Thus, GPDs
defined in Eqs. (31), (32) and (33) are also known as momentum-dissected form
factors and measure the contribution of the struck quark with momentum
fraction $x$ to the corresponding form factors. The electromagnetic form
factors are related to the first moments of the unpolarized GPDs for the
nucleon by the sum rules on the light-front as

$\displaystyle F_{1}^{q}(t)=$
$\displaystyle\int\text{d}x\,H^{q}(x,0,t)\,;\quad
F_{2}^{q}(t)=\int\text{d}x\,E^{q}(x,0,t)\,,$ (35)

where $F_{1}^{q}(t)$ and $F_{2}^{q}(t)$ are the Dirac (charge) and the Pauli
(magnetic) form factors, respectively whereas, the axial form factor is
connected to the helicity dependent GPD as

$\displaystyle G_{A}^{q}(t)=$ $\displaystyle\int
dx\,\widetilde{H}^{q}(x,0,t)\,.$ (36)

Meanwhile, the gravitational form factors which are parameterized through the
matrix elements of the EMT are linked to the second moment of GPDs

$\displaystyle A^{q}(t)=\int\text{d}x\,x\,H^{q}(x,0,t)\,;\quad\quad
B^{q}(t)=\int\text{d}x\,x\,E^{q}(x,0,t)\,.$ (37)

The transverse impact parameter dependent GPDs are obtained via the FT of the
GPDs with respect to the momentum transfer along the transverse direction
$\vec{\Delta}_{\perp}$ Burkardt:2002hr :

$\displaystyle{F}(x,{b}_{\perp})$
$\displaystyle=\int\frac{\text{d}^{2}{\vec{\Delta}}_{\perp}}{(2\pi)^{2}}e^{-i{\vec{\Delta}}_{\perp}\cdot{\vec{b}}_{\perp}}F(x,0,t)\,,$
(38)

with $F$ being the GPDs $H$, $E$ and $\widetilde{H}$. The $H(x,b_{\perp})$
provides the description of the density of unpolarized quarks in the
unpolarized proton, while $E(x,b_{\perp})$ is responsible for a deformation of
the density in the transversely polarized proton. The transverse distortion
can be linked to Ji’s angular momentum relation. Ji has shown that the TAM of
quarks and gluons can be expressed in terms of GPDs Ji:PRL :

$\displaystyle J^{z}=\frac{1}{2}\int\text{d}x\,x\,[H(x,0,0)+E(x,0,0)]\,.$ (39)

This sum rule is appropriate at the forward limit of the GPDs and relates the
$z$-component of the TAM of the constituents in a nucleon polarized in the
$z$-direction only. Again in the impact parameter space, the sum rule has a
simple interpretation for a transversely polarized nucleon Burkardt:2005hp .
The term involving $E(x,0,0)$ arises due to the transverse deformation of the
distribution in the center of momentum frame, whereas the term containing
$H(x,0,0)$ is an overall transverse shift when going from the transversely
polarized nucleon in instant form to the front form. Meanwhile, the helicity-
dependent GPD $\widetilde{H}$ in the impact parameter space reflects the
difference in the density of parton with helicity equal or opposite to the
nucleon helicity Diehl:2005jf ; Boffi:2007yc ; Pasquini:2007xz ;
Mondal:2017wbf . This GPD has a direct connection with the partonic spin
contribution to the TAM of the nucleon.

We can now rewrite the distributions defined in Eqs. (19)-(27) using the
impact parameter dependent GPDs, where $\tilde{L}(b_{\perp})$,
$\tilde{J}(b_{\perp})$ and $\tilde{G}_{A}(b_{\perp})$ are given by

$\displaystyle\tilde{L}({b}_{\perp})$
$\displaystyle=\frac{1}{2}\int\text{d}x\,\left\\{x\left[{H}(x,b_{\perp})+{E}(x,b_{\perp})\right]-\widetilde{{H}}(x,b_{\perp})\right\\}\,,$
(40) $\displaystyle\tilde{J}({b}_{\perp})$
$\displaystyle=\frac{1}{2}\int\text{d}x\,x\left[{H}(x,b_{\perp})+{E}(x,b_{\perp})\right]\,,$
(41) $\displaystyle\tilde{G}_{A}({b}_{\perp})$
$\displaystyle=\int\text{d}x\,\widetilde{{H}}(x,b_{\perp})\,.$ (42)

---
(a)

---
(b)

---
(c)

---
(d)

---
(e)

---
(f)

Figure 1: The valence quark GPDs of the proton: (a) $H(x,0,t)$, (c)
$E(x,0,t)$, and (e) $\widetilde{H}(x,0,t)$ are for the valence up quark; (b),
(d) and (f) are same as (a), (c), and (e), respectively, but for the valence
down quark as functions of $x$ and $-t$.

---
(a)

---
(b)

---
(c)

---
(d)

---
(e)

---
(f)

Figure 2: The valence quark GPDs of the proton in the transverse impact
parameter space: (a) $H(x,b_{\perp})$, (c) $E(x,b_{\perp})$, and (e)
$\widetilde{H}(x,b_{\perp})$ are for the valence up quark; (b), (d) and (f)
are same as (a), (c), and (e), respectively, but for the valence down quark as
functions of $x$ and $b_{\perp}$.

##  IV Numerical results and discussions

The LFWFs of the valence quarks in the proton have been solved in the BLFQ
framework with the basis truncation $N_{\rm max}=10$ and $K=16.5$ and the
model parameters $\\{m_{\rm q/KE},~{}m_{\rm
q/OGE},~{}\kappa,~{}\alpha_{s}\\}=\\{0.3~{}{\rm GeV},~{}0.2~{}{\rm
GeV},~{}0.34~{}{\rm GeV},~{}1.1\pm 0.1\\}$ and the HO scale parameter $b=0.6$
GeV. The parameters in our model are fixed to fit the nucleon mass and the
flavor Dirac form factors Mondal:2019jdg . We estimate an uncertainty on the
coupling that accounts for the model selections and major fitting
uncertainties. The uncertainty for the $\alpha_{s}$ decreases with increasing
basis cutoffs $N_{\rm{max}}$ Xu:2021wwj . We employ the resulting wave
functions to investigate the GPDs for the proton. We insert the valence wave
functions given by Eq. (2) into Eqs. (31), (32) and (33) to compute the
valence quark GPDs inside the proton.

We show the unpolarized GPDs, $H^{q}$ and $E^{q}$, and the helicity dependent
GPD $\widetilde{H}^{q}$ as functions of $x$ and $-t$ for the proton in Fig. 1.
The GPD $E$ in the proton has its peak located at a lower $x$ than the peaks
in $H$ and $\widetilde{H}$. In addition, the GPD E falls faster than the other
two GPDs at large-$x$. Meanwhile, the GPD $\widetilde{H}$ exhibits the similar
behavior as manifested by the GPD $H$. This is due to the fact that $E$
involves the overlaps of the wave functions with different orbital angular
momentum $L_{z}=0$ and $L_{z}=\pm 1$ and the other two GPDs entail the
overlaps of the wave functions of the same orbital angular momentum. The
magnitudes of distributions decrease and the peaks along $x$ shift towards
larger values of $x$ with increasing momentum transfer $-t$ similar to that
observed in the other phenomenological models for the nucleon Ji:1997gm ;
Scopetta:2002xq ; Petrov:1998kf ; Penttinen:1999th ; Boffi:2002yy ;
Boffi:2003yj ; Vega:2010ns ; Chakrabarti:2013gra ; Mondal:2015uha ;
Chakrabarti:2015ama ; Mondal:2017wbf ; deTeramond:2018ecg ; Xu:2021wwj  as
well as for the light mesons Kaur:2018ewq ; deTeramond:2018ecg ; Zhang:2021mtn
; Kaur:2020vkq .

---
(a)

---
(b)

---
(c)

---
(d)

Figure 3: Angular momentum distributions summing over the up and the down
quark contributions multiplied by $b_{\perp}$ as functions of $b_{\perp}$: (a)
the kinetic TAM density $\langle J^{z}\rangle(b_{\perp})$ (black band) as the
sum of the spin $\langle S^{z}\rangle(b_{\perp})$ in Eq. (13) (red band) and
the kinetic OAM $\langle L^{z}\rangle(b_{\perp})$ in Eq. (12) (blue band)
contributions; (b) the kinetic TAM density $\langle J^{z}\rangle(b_{\perp})$
(black band) resulting from the sum of the naive TAM density $\langle
J^{z}_{\rm naive}\rangle(b_{\perp})$ in Eq. (24) (blue band) and the
corresponding correction term $\langle J^{z}_{\rm corr}\rangle(b_{\perp})$ in
Eq. (25) (red band); (c) the kinetic TAM density $\langle
J^{z}\rangle(b_{\perp})$ (black band) expressed as the sum of the Belinfante-
improved TAM $\langle J^{z}_{\rm Bel}\rangle(b_{\perp})$ density in Eq. (15)
(blue band) and the total divergence term $\langle M^{z}\rangle(b_{\perp})$ in
Eq. (16) (red band); (d) the Belinfante-improved TAM $\langle J^{z}_{\rm
Bel}\rangle(b_{\perp})$ density (black band) decomposed into its monopole
$\langle J_{\text{Bel}}^{\text{mono}}\rangle(b_{\perp})$ in Eq. (26) (blue
band) and quadrupole $\langle J_{\text{Bel}}^{\text{quad}}\rangle(b_{\perp})$
in Eq. (27) (red band) contributions. The bands reflect our $\alpha_{s}$
uncertainty of $10\%$.

---
(a)

---
(b)

---
(c)

---
(d)

---
(e)

---
(f)

---
(g)

---
(h)

Figure 4: Angular momentum densities of the up and the down quarks multiplied
by $b_{\perp}$ as functions of $b_{\perp}$. The left panels i.e., $\\{$(a),
(c), (e), (g)$\\}$ are for the down quark, while the right panels i.e.,
$\\{$(b), (d), (f), (h)$\\}$ are for the up quark. The legends in $\\{$(a),
(c), (e), (g)$\\}$ or $\\{$(b), (d), (f), (h)$\\}$ are the same as described
in $\\{$(a), (b), (c), (d)$\\}$ of Fig. 3 but for quarks.

We illustrate the valence quark GPDs of the proton in the transverse impact
parameter space for zero skewness in Fig. 2. We observe that, except for the
fact that the magnitude of $H(x,b_{\perp})$ for the up quark is larger than
that for the down quark, the overall nature of this distribution is the same
for both the quarks. After integrating over $b_{\perp}$, $H(x,b_{\perp})$
reduces to the ordinary unpolarized PDF $f_{1}(x)$ and satisfies the quark
counting rule when we further integrate over $x$. Meanwhile, $E(x,b_{\perp})$
for the up quark emerges as a positive distribution, whereas it is negative
for the down quark. After integrating $E(x,b_{\perp})$ over $x$ and
$b_{\perp}$, we obtain the following values for the quark anomalous magnetic
moments: $\kappa_{u}=1.481\pm 0.029$ and $\kappa_{d}=-1.367\pm 0.025$
corresponding to the nucleon anomalous magnetic moments:
$\kappa_{p}=\frac{2}{3}\kappa_{u}-\frac{1}{3}\kappa_{d}=1.443\pm 0.027$ and
$\kappa_{n}=-\frac{1}{3}\kappa_{u}+\frac{2}{3}\kappa_{d}=-1.405\pm 0.026$,
which are close to the recent results from lattice simulations:
$\kappa_{p}^{\rm lat}=1.43(9)$ and $\kappa_{n}^{\rm lat}=-1.54(6)$
Alexandrou:2018sjm . The experimental values are $\kappa_{p}^{\rm exp}=1.793$
and $\kappa_{n}^{\rm exp}=-1.913$.

We also observe in Fig. 2 that $E(x,b_{\perp})$ falls faster than
$H(x,b_{\perp})$ at $x\to 1$ similar to what is found in other
phenomenological models Boffi:2007yc ; Chakrabarti:2013gra ; Mondal:2015uha ;
Maji:2017ill . The qualitative behavior of $\widetilde{H}(x,b_{\perp})$ in our
approach is almost the same as in ${H}(x,b_{\perp})$ but the magnitude of
$\widetilde{H}(x,b_{\perp})$ is relatively lower than that for
${H}(x,b_{\perp})$. Meanwhile, $\widetilde{H}(x,b_{\perp})$ for the down quark
has the opposite sign to the up quark distribution. Integrating
$\widetilde{H}(x,b_{\perp})$ over $b_{\perp}$, we obtain the helicity
distribution $g_{1}(x)$, which in our BLFQ approach is consistent with the
experimental data Xu:2021wwj .

We further notice in Fig. 2 that the width of all the GPDs in the transverse
impact parameter space decrease as $x$ increases. This indicates that the
distributions are more concentrated and the quarks are more localized near the
center of momentum ($b_{\perp}=0$) when they are carrying a higher
longitudinal momentum fraction. Meanwhile, the peaks of all the distributions
shift toward to lower values of $x$ when $b_{\perp}$ increases. This
characteristic of the distributions in the $b_{\perp}$-space is reassuring
since the GPDs in the momentum space become broader in $-t$ with increasing
$x$, as can be seen from Fig. 1. On the light-front, this can be understood as
the larger the momentum fraction, the smaller the kinetic energy carried by
the quarks. As the total kinetic energy remains limited, the distribution in
the transverse momentum broadens at higher $x$ reflecting the trend to carry a
larger portion of the kinetic energy. As a consequence, these general features
should be nearly model-independent properties of the GPDs and, indeed, they
are also observed in other theoretical studies of the GPDs Burkardt:2002hr ;
Chakrabarti:2013gra ; Vega:2010ns ; Mondal:2015uha ; Chakrabarti:2015ama ;
Mondal:2017wbf ; Maji:2017ill .

We employ these GPDs to compute the angular momentum distributions defined in
Eqs. (19)-(25) following Eqs. (40)-(42). In Fig. 3, we illustrate different
definitions of the TAM densities summing over the up and the down quark
contributions. We display the distribution $b_{\perp}\langle
J^{z}\rangle(b_{\perp})$ as a function of $b_{\perp}$ and notice that the TAM
is positive over all $b_{\perp}$. It has the peak near $b_{\perp}\sim 0.3$ fm,
falls slowly with increasing $b_{\perp}$ and becomes very small near
$b_{\perp}\sim 1.5$ fm. The error bands in our distributions are due to the
$10\%$ uncertainties in the coupling constant.

Figure 3(a) shows the kinetic TAM density $\langle J^{z}\rangle(b_{\perp})$ as
the sum of the spin $\langle S^{z}\rangle(b_{\perp})$ and the kinetic OAM
$\langle L^{z}\rangle(b_{\perp})$ contributions each multiplied by
$b_{\perp}$. Both the contributions show positive distributions. In contrast
to the results in a quark-diquark model Lorce:2017wkb , where the OAM
component is larger than the spin component of the TAM density, the spin
distribution in our approach strongly dominates over the kinetic OAM density.
The $\langle L^{z}\rangle(b_{\perp})$ is mainly effective over the range
$0.2<b_{\perp}<0.6$ fm. It should be noted that these results are obtained
within the valence Fock representation, while the higher Fock components
$|qqqg\rangle$ and $|qqqq\bar{q}\rangle$ are anticipated to have significant
effects on the spin and OAM distributions. With the inclusion of dynamical
gluons and sea quarks, the quark spin contribution may be suppressed, and the
OAM can play an enhanced role in the TAM density.

Figure 3(b) compares the kinetic TAM $\langle J^{z}\rangle(b_{\perp})$ and the
naive distribution $\langle J^{z}_{\rm naive}\rangle(b_{\perp})$. Their
difference, attributed to the correction term $\langle
J^{z}_{\text{corr}\rangle}(b_{\perp})$ in Eq. (25), is also shown in this
plot. We find that $\langle J^{z}_{\rm naive}\rangle(b_{\perp})$ is close to
$\langle J^{z}\rangle(b_{\perp})$. The $\langle
J^{z}_{\text{corr}\rangle}(b_{\perp})$ exhibits a negative central region
surrounded by a ring of positive distribution, which is in accord with the
behavior observed in the quark-diquark model Lorce:2017wkb .

We present the comparison between the kinetic TAM $\langle
J^{z}\rangle(b_{\perp})$ and the Belinfante-improved density $\langle
J^{z}_{\text{Bel}}\rangle(b_{\perp})$ in Fig. 3(c). In the same plot, we also
show their difference given by the total divergence $\langle
M^{z}\rangle(b_{\perp})$ term in Eq. (22). The $\langle
J^{z}_{\text{Bel}}\rangle(b_{\perp})$ is smaller at $b_{\perp}<0.4$ fm but
larger at $b_{\perp}>0.4$ fm than the $\langle J^{z}\rangle(b_{\perp})$. The
Belinfante-improved density falls slower at higher $b_{\perp}$ than the TAM
density. We also find that the peak of the Belinfante-improved distribution is
lower and appears at higher value of $b_{\perp}$ compared to that for the TAM
density. Also, the Belinfante-improved distribution is broader and falls more
slowly in $b_{\perp}$ compared with the kinetic TAM $\langle
J^{z}\rangle(b_{\perp})$. It can be noticed that the $\langle
M^{z}\rangle(b_{\perp})$ has a positive core and a negative tail. $\langle
M^{z}\rangle(b_{\perp})$ has a significant contribution to the TAM
distribution and this can be ascribed from the fact that it is related to the
spin distribution, which in our model provides the dominating contribution to
the TAM density.

As another illustration of our results, the decomposition of the Belinfante-
improved TAM in term of its monopole $\langle
J_{\text{Bel}}^{z(\text{mono})}\rangle(b_{\perp})$ and quadrupole $\langle
J_{\text{Bel}}^{z(\text{quad})}\rangle(b_{\perp})$ contributions is presented
in Fig. 3(d). One notices that the qualitative behavior of $\langle
J_{\text{Bel}}{z(\text{mono})}\rangle(b_{\perp})$ and $\langle
J_{\text{Bel}}^{z(\text{quad})}\rangle(b_{\perp})$ distributions is similar to
$\langle J^{z}_{\rm naive}\rangle(b_{\perp})$ and $\langle
J^{z}_{\text{corr}}\rangle(b_{\perp})$, respectively. Finally, we observe that
the correction term $\langle J^{z}_{\text{corr}}\rangle(b_{\perp})$, the total
divergence term $\langle M^{z}\rangle(b_{\perp})$, and the quadrupole
contribution $\langle J^{z(\text{quad})}_{\text{Bel}}\rangle(b_{\perp})$
integrate to zero. However, at the density level, we need to take them into
account while comparing different definitions for the angular momentum
distribution. Note that these findings in our BLFQ approach are also supported
by the analysis based on a light-front quark-diquak model Lorce:2017wkb .

In Fig. 4, we demonstrate the angular momentum densities for quark flavors by
considering all the different definitions described above. In Fig. 4(a) and
4(b), we present the kinetic TAM $\langle J^{z}\rangle^{q}(b_{\perp})=\langle
S^{z}\rangle^{q}(b_{\perp})+\langle L^{z}\rangle^{q}(b_{\perp})$ for the up
and the down quarks, respectively multiplied by $b_{\perp}$. In our BLFQ
approach, the contribution in $\langle J^{z}\rangle(b_{\perp})$ from $\langle
S^{z}\rangle(b_{\perp})$ is larger than that from $\langle
L^{z}\rangle(b_{\perp})$ for the up quark, whereas for the down quark,
$\langle S^{z}\rangle(b_{\perp})$ dominates at lower $b_{\perp}$ but $\langle
L^{z}\rangle(b_{\perp})$ is superior at large distance. For the up quark, the
spin and the OAM densities show positive and negative distributions,
respectively, while they are opposite for the down quark. In essence, $\langle
J^{z}\rangle(b_{\perp})$ for the down quark exhibits a negative core near the
center of momentum of the proton and it has a positive tail at large distance.
Meanwhile, $\langle J^{z}\rangle(b_{\perp})$ is almost equivalent to $\langle
S^{z}\rangle(b_{\perp})$ for the up quark and they show positive
distributions.

In Fig. 4(c) and Fig. 4(d), we compare the up and the down quark kinetic TAM
$\langle J^{z}\rangle(b_{\perp})$ with their naive density $\langle J^{z}_{\rm
naive}\rangle(b_{\perp})$, respectively. The naive density $\langle J^{z}_{\rm
naive}\rangle(b_{\perp})$ is positive for the up quark, while for the down
quark, it is also positive at low $b_{\perp}$ but negative at higher
$b_{\perp}$. The difference between them is provided by the correction term
$\langle J^{z}_{\text{corr}}\rangle(b_{\perp})$ in Eq. (25). The correction
term for the up quark is small but for the down quark it is large and close to
$\langle J^{z}\rangle(b_{\perp})$. Meanwhile, the comparison between the
kinetic TAM $\langle J^{z}\rangle(b_{\perp})$ and the Belinfante-improved TAM
$\langle J^{z}_{\text{Bel}}\rangle(b_{\perp})$ for the up and the down quark
is illustrated in Fig. 4(e) and Fig. 4(f), respectively. The difference is
described by the $\langle M^{z}\rangle(b_{\perp})$ term in Eq. (22). For the
down quark, the $\langle J^{z}_{\text{Bel}}\rangle(b_{\perp})$ is very small
compared to $\langle J^{z}\rangle(b_{\perp})$ and the major contribution in
the kinetic TAM is coming from $M^{z}(b_{\perp})$. However, they are
comparable for the up quark. In our BLFQ approach the qualitative behavior of
the naive and the Belinfante-improved densities is very similar for the down
quark but different for the up quark.

Finally, the monopole and quadrupole contributions to the Belinfante-improved
total density for the quarks are shown in Fig. 4(g) and 4(h). The monopole
contribution dominates over the quadrupole contribution for both the quarks.
Note that the integrations of the correction term $\langle
J^{z}_{\text{corr}}\rangle(b_{\perp})$, the total divergence term $\langle
M^{z}\rangle(b_{\perp})$, and the quadrupole contribution $\langle
J^{z(\text{quad})}_{\text{Bel}}\rangle(b_{\perp})$ for the individual quarks
are also zero but all the terms need to be retained while comparing results
among different definitions at the level of distributions.

Summing over the flavors, we have obtained the total spin contributed by the
quarks to the proton spin. Within our model that incorporates only the valence
Fock sector, we have found that the quark spin at the model scale contributes
$\sim 91\%$ to the proton spin, whereas the contribution of quark spin
captures only $\sim 40\%$ as revealed from the experiment Leader:2010rb . This
evident discrepancy suggests the need to extend our model to append the higher
Fock sectors, which have a significant effects on the proton spin. With
dynamical gluons and sea quarks, the quark spin contribution can be reduced
and the OAM can play a substantial role in understanding the nucleon spin.
Simultaneously, the gluon and sea quark contributions to the total spin will
emerge. Meanwhile, the QCD scale evolution also needs to be taken into
account.

##  V SUMMARY

Using a recently proposed light-front model for the proton based on a
Hamiltonian formalism, we studied its valence quark GPDs. The effective
Hamiltonian incorporates light-front holography, longitudinal confinement, and
the one gluon exchange interaction for the valence quarks suitable for low-
resolution properties. We obtained the proton LFWFs as the eigenvectors of
this Hamiltonian by solving its mass eigenstates using BLFQ as a relativistic
three-quark problem. The parameters in this BLFQ model have previously been
adjusted by fitting the nucleon mass and the flavor Dirac form factors. We
then employed the LFWFs to compute the valence quark unpolarized and helicity
dependent GPDs of the proton. We presented results for the GPDs in both
momentum space and position space for zero skewness and we found that the
qualitative behavior of the GPDs in our BLFQ approach bears similarities to
other phenomenological models.

We have employed these GPDs to study the various definitions of angular
momentum at the density level. Within our model that includes only the leading
Fock sector, we have found that the spin contribution to the TAM strongly
dominates over the OAM distribution in contrast to the results in a quark-
diquark model Lorce:2017wkb . On the other hand, the naive angular momentum
density is found to be close to the TAM distribution. Meanwhile, we have
observed that the Belinfante-improved angular momentum density is distinctly
different from the TAM distributions, whereas the difference between them
given by the total divergence term has a positive core surrounded by a
negative tail. When we decomposed the Belinfante-improved angular momentum
density into its monopole and quadrupole components, we noticed that the
dominating contribution comes from the monopole density. In our approach, we
have illustrated explicitly that no discrepancies were found between different
definitions of angular momentum when all the terms integrating to zero are
included in the expressions. These findings in our BLFQ approach are
consistent with the analysis based on a light-front quark-diquak model
Lorce:2017wkb .

We have subsequently presented the up and the down quarks’ TAM densities and
found that the up quark contribution to the TAM distribution is much larger
than that for the down quark. The spin and OAM densities are comparable but
opposite in sign for the down quark, while for the up quark, the spin
distribution is much stronger than the OAM density. The naive distributions
for both the quarks were found to be similar to their TAM densities. Although
the Belinfante-improved and TAM distributions are similar for the down quark,
they are different from each other for the up quark in our current approach.

## ACKNOWLEDGMENTS

C. M. is supported by new faculty start up funding by the Institute of Modern
Physics, Chinese Academy of Sciences, Grant No. E129952YR0. C. M. also thanks
the Chinese Academy of Sciences Presidents International Fellowship Initiative
for the support via Grants No. 2021PM0023. X. Z. is supported by new faculty
startup funding by the Institute of Modern Physics, Chinese Academy of
Sciences, by Key Research Program of Frontier Sciences, Chinese Academy of
Sciences, Grant No. ZDB-SLY-7020, by the Natural Science Foundation of Gansu
Province, China, Grant No. 20JR10RA067 and by the Strategic Priority Research
Program of the Chinese Academy of Sciences, Grant No. XDB34000000. J. P. V. is
supported by the Department of Energy under Grants No. DE-FG02-87ER40371, and
No. DE-SC0018223 (SciDAC4/NUCLEI). This research used resources of the
National Energy Research Scientific Computing Center (NERSC), a U.S.
Department of Energy Office of Science User Facility operated under Contract
No. DE-AC02-05CH11231. A portion of the computational resources were also
provided by Gansu Computing Center.

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