BLFQ Collaboration

# Gravitational Form Factors and Mechanical Properties of Quarks in Protons: A
Basis Light-Front Quantization Approach

Sreeraj Nair<EMAIL_ADDRESS>Chandan Mondal<EMAIL_ADDRESS>Siqi Xu
<EMAIL_ADDRESS>Xingbo Zhao<EMAIL_ADDRESS>Institute for Modern
Physics, Chinese Academy of Sciences, Lanzhou-730000, China
School of Nuclear Science and Technology, University of Chinese Academy of
Sciences, Beijing 100049, China
CAS Key Laboratory of High Precision Nuclear Spectroscopy, Institute of Modern
Physics, Chinese Academy of Sciences, Lanzhou 730000, China  Asmita Mukherjee
<EMAIL_ADDRESS>Indian Institute of Technology Bombay, Powai, Mumbai
400076, India  James P. Vary<EMAIL_ADDRESS>Department of Physics and
Astronomy, Iowa State University, Ames, Iowa 50011, USA

###### Abstract

We compute the gravitational form factors (GFFs) and study their applications
for the description of the mechanical properties such as the pressure, shear
force distributions, and the mechanical radius of the proton from its light-
front wave functions (LFWFs) based on basis light-front quantization (BLFQ).
The LFWFs of the proton are given by the lowest eigenvector of a light-front
effective Hamiltonian that incorporates a three-dimensional confining
potential and a one-gluon exchange interaction with fixed coupling between the
constituent quarks solved in the valence Fock sector. We find acceptable
agreement between our BLFQ computations and the lattice QCD for the GFFs. Our
$D$-term form factor also agrees well with the extracted data from the deeply
virtual Compton scattering experiments at Jefferson Lab, and the results of
different phenomenological models. The distributions of pressures and shear
forces are similar to those from different models.

††preprint: APS/123-QED

##  I Introduction

Nucleons are confined systems of partons (quarks and gluons) with pressure and
energy inside, dictated by the strong interaction. However, understanding the
color confinement and the mass generation of the nucleon in quantum
chromodynamics (QCD) theory remains one of the central questions in modern
particle and nuclear physics. As a composite system, various distribution
functions, such as form factors, parton distribution functions (PDFs),
generalized parton distributions (GPDs), transverse momentum dependent parton
distributions (TMDs), etc., are often used to describe the structure of the
nucleon. The form factors of the nucleon give critical information about many
fundamental aspects of its structure. The charge and magnetization
distributions are encoded in the electromagnetic form factors Miller:2007uy ;
Carlson:2007xd , while the mechanical properties of the nucleon, namely the
mass, spin, pressure and shear force are well encoded in the gravitational
form factors (GFFs) Polyakov:2018zvc ; Lorce:2018egm ; Burkert:2018bqq ;
Shanahan:2018nnv . Specifically, one can gain a deeper knowledge on how the
nucleon is mechanically shaped by its partons by studying the Druck term
(D-term) GFFs Burkert:2018bqq . The GFFs are parameterized as the matrix
elements of the energy-momentum tensor (EMT) in the nucleon states. The
components of the EMT provide how matter interacts to the gravitational field.
Thus these form factors can be extracted by direct measurement of the
interaction of the nucleon with a strong gravitational field such as a neutron
star. However, due to weak gravitational interaction, the direct access to
them experimentally is very difficult. An indirect way to extract the GFFs
experimentally is from hard exclusive processes, for example, deeply virtual
Compton scattering (DVCS) and deeply-virtual meson production (DVMP), which
are sensitive to GPDs. The GFFs are defined as the second Mellin moments of
the GPDs. The quark GPDs in the nucleon have been constrained in limited
kinematic regions by DVCS and DVMP experiments at JLab CLAS:2001wjj ;
CLAS:2007clm ; CLAS:2020yqf , COMPASS dHose:2004usi , HERA H1:1999pji ;
H1:2001nez ; ZEUS:2003pwh ; ZEUS:2003pwh , and HERMES HERMES:2001bob . The
world datasets have been summarized in Refs. Favart:2015umi ; dHose:2016mda ;
Kumericki:2016ehc . Ongoing investigations at COMPASS and within the $12$ GeV
program at JLab will significantly improve accesses to these quantities. The
first extraction of the quark D-term form factors from DVCS and the pressure
distribution inside the proton was reported in Ref. Burkert:2018bqq . However,
the DVCS is nearly insensitive to the gluon and thus, the gluon D-term form
factor is rarely extracted. The gluon contributions to the nucleon D-term form
factor will be accessed at the future high-luminosity electron-ion colliders
(EICs) AbdulKhalek:2021gbh ; Anderle:2021wcy .

The nucleon matrix element of the EMT involves four GFFs, namely $A(Q^{2})$,
$B(Q^{2})$, $D(Q^{2})$, and $\bar{C}(Q^{2})$, where $Q^{2}$ is the squared
momentum transfer from the initial to final nucleon. They encode information
on the distributions of energy density, angular momentum, and internal forces
in the interior of the nucleon. The GFFs $A$ and $B$ are linked to the mass
and spin of the nucleon. The Ji’s sum rule Ji:1996ek  relates them to the
partonic contribution to the total angular momentum $J$. They are again
related to the generators of the Poincare group, which provides constraints on
them at $Q^{2}=0$, which facilitates accessing these form factors from the
experimental data. The form factor $D(Q^{2})$ is related to the basic
mechanical properties of the nucleon such as the pressure and stress
distributions Polyakov:2018zvc ; Lorce:2018egm ; Burkert:2018bqq ;
Shanahan:2018nnv . It contributes to the DVCS process when there is nonzero
momentum transfer in the longitudinal direction. By virtue of energy momentum
conservation, the form factor $\bar{C}(Q^{2})$ contributes to both the quark
and gluon parts with the same magnitude but with opposite signs, therefore
$\sum_{i=q,g}\bar{C}_{i}(Q^{2})=0$. This form factor records the reshuffling
of forces between the quark and gluon subsystems inside the nucleon
Polyakov:2018exb .

There has been significant progress in the theoretical determination of the
nucleon GFFs, in particular through phenomenology, QCD inspired models, and
lattice QCD. The GFFs of the nucleon have been studied via chiral perturbation
theory Chen:2001pva ; Belitsky:2002jp ; Dorati:2007bk , chiral quark soliton
model Schweitzer:2002nm ; Wakamatsu:2006dy ; Goeke:2007fq ; Goeke:2007fp ;
Wakamatsu:2007uc , Bag model Neubelt:2019sou , Skyrme model Cebulla:2007ei ;
Kim:2012ts , light-cone QCD sum rules at leading order Anikin:2019kwi , light-
front quark-diquark model motivated by AdS/QCD Chakrabarti:2015lba ;
Chakrabarti:2020kdc ; Chakrabarti:2021mfd ; Chakrabarti:2016mwn ;
Kumar:2017dbf , dispersion relation Pasquini:2014vua , instanton picture
Polyakov:2018exb , and instant and front form Lorce:2018egm , holographic QCD
Abidin:2009hr ; Mondal:2015fok ; Mamo:2022eui ; Mamo:2021krl , lattice QCD
Hagler:2003jd ; Gockeler:2003jfa ; LHPC:2010jcs ; LHPC:2007blg ; QCDSF-
UKQCD:2007gdl ; Deka:2013zha , strongly coupled scalar theory Cao:2023ohj ,
etc., while the asymptotic behavior of the GFFs has been reported in Refs.
Hatta:2018sqd ; Tanaka:2018nae . On the other hand, the gluon contributions to
the nucleon GFFs are far less well constrained and have so far only been
investigated in holographic light-front QCD framework deTeramond:2021lxc ,
dressed quark model More:2023pcy  and lattice QCD Alexandrou:2020sml ;
Shanahan:2018pib ; Yang:2018bft ; Yang:2018nqn ; Alexandrou:2017oeh ;
Pefkou:2021fni . To date, no experimental constraints on gluon GFFs of the
nucleon have been achieved. However, they are accessible via
photo/leptoproduction of $J/\psi$ and $\Upsilon$ Mamo:2019mka ; Hatta:2018ina
; Boussarie:2020vmu ; $J/\psi$ production is investigated in experiments,
which are ongoing at JLab GlueX:2019mkq , whereas $\Upsilon$ production are
proposed at the upcoming EICs AbdulKhalek:2021gbh ; Anderle:2021wcy . The
quark GFFs for the nucleon have been extracted at JLab from DVCS
Burkert:2018bqq  and the constraints on the quark GFFs can be further improved
via the GPDs accessible at several existing and upcoming experimental
facilities, which include the PANDA experiment at the Facility for Antiproton
and Ion Research (FAIR) PANDA:2009yku , the proposed EICs AbdulKhalek:2021gbh
; Anderle:2021wcy , the nuclotron-based ion collider facility (NICA)
MPD:2022qhn , the International Linear Collider (ILC), and the Japan proton
accelerator complex (J-PARC) Kumano:2022cje .

Our theoretical framework to study the nucleon properties is based on basis
light-front quantization (BLFQ) Vary:2009gt , which provides a Hamiltonian
formalism for solving the relativistic many-body bound state problem in
quantum field theories Zhao:2014xaa ; Nair:2022evk ; Wiecki:2014ola ;
Li:2015zda ; Jia:2018ary ; Lan:2019vui ; Lan:2019rba ; Tang:2018myz ;
Tang:2019gvn ; Mondal:2019jdg ; Xu:2021wwj ; Lan:2019img ; Qian:2020utg ;
Lan:2021wok ; Zhu:2023lst ; Xu:2022abw ; Peng:2022lte ; Kuang:2022vdy . In
this paper, we compute the quark GFFs of the proton and investigate their
applications for the description of the mechanical properties, i.e., the
distributions of pressures and shear forces inside the proton from its light-
front wave functions (LFWFs) based on the BLFQ with only the valence Fock
sector of the proton considered. The LFWFs feature all three active quarks’
flavor, spin, and three dimensional spatial information on the same footing.
Our effective Hamiltonian incorporates a three-dimensional confinement
potential embodying the light-front holography in the transverse direction
Brodsky:2014yha  and a complimentary longitudinal confinement Li:2015zda . The
Hamiltonian also includes a one-gluon exchange interaction with fixed coupling
to account for the spin structure Mondal:2019jdg . The nonperturbative
solutions for the three-body LFWFs generated by the recent BLFQ study of the
nucleon Xu:2021wwj  have been applied successfully to generate the
electromagnetic and axial form factors, radii, PDFs, GPDs, TMDs and many other
properties of the nucleon Mondal:2019jdg ; Xu:2021wwj ; Hu:2022ctr ;
Liu:2022fvl ; Mondal:2021wfq ; GPD_SK ; GPD_yiping . Here, we extend those
studies to calculate the proton GFFs and their application for the description
of the mechanical properties.

The rest of the paper is organized as follows. In Sec. II, we briefly
summarize the BLFQ framework for the nucleon. The proton GFFs are evaluated
within BLFQ and discussed in Sec. III. We present the numerical results for
the GFFs and explore the mechanical properties of proton, e.g., the pressures,
energy density distributions, shear forces, and the mechanical radius in Sec.
IV. At the end, we provide a brief summary and conclusions in Sec. V.

##  II Proton wavefunction from basis light-front quantization

The LFWFs encoding the structural information of hadronic bound states are
achieved as the eigenfunctions of the eigenvalue problem of the Hamiltonian,
$H_{\rm LF}|\Psi\rangle=M_{\rm h}^{2}|\Psi\rangle,$ with $H_{\rm LF}$ and
$M_{\rm h}^{2}$ being the light-front Hamiltonian and the mass square
eigenvalue of the hadron, respectively. The light-front effective Hamiltonian
for the proton with quarks being the only explicit degree of freedom is given
by Mondal:2019jdg

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

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

In the BLFQ approach, the discretized plane-wave basis is conveniently adopted
in the longitudinal direction, whereas we utilize the 2D-HO function for the
transverse direction Vary:2009gt ; Zhao:2014xaa . Solving the eigenvalue
equation of the Hamiltonian, Eq. (II), in the chosen basis space provides the
eigenvalues as squares of the system masses, and the eigenfunctions that
describe the LFWFs. The lowest eigenfunction with the relevant symmetries is
naturally specified as the proton state. The LFWFs of the proton are expressed
in terms of the basis function 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)

with
$\psi^{\Lambda}_{\\{x_{i},n_{i},m_{i},\lambda_{i}\\}}=\braket{P,{\Lambda}}{\\{x_{i},n_{i},m_{i},\lambda_{i}\\}}$
representing the LFWF in the BLFQ basis obtained by diagnalizing Eq. (II)
numerically, where $\ket{P,{\Lambda}}$ defines the proton state with $P$ and
$\Lambda$ being the momentum and the helicity of the state. The 2D-HO function
we employ as the transverse basis function is given by

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

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

The parameters in the effective Hamiltonian are determined to reproduce the
nucleon mass and its electromagnetic properties Xu:2021wwj . The model LFWFs
have demonstrated significant effectiveness in analyzing a broad range of
nucleon properties, including electromagnetic and axial form factors, radii,
PDFs, quark helicity asymmetries, GPDs, TMDs, and angular momentum
distributions, achieving notable success across various metrics.
Mondal:2019jdg ; Xu:2021wwj ; Mondal:2021wfq .

##  III Gravitational form factors

The gauge invariant symmetric form of the QCD EMT is given by
Harindranath:1997kk

$\displaystyle\theta^{\mu\nu}=\frac{1}{2}\overline{\psi}\
i\left[\gamma^{\mu}D^{\nu}+\gamma^{\nu}D^{\mu}\right]\psi-F^{\mu\lambda
a}F_{\lambda a}^{\nu}+\frac{1}{4}g^{\mu\nu}\left(F_{\lambda\sigma
a}\right)^{2}-g^{\mu\nu}\overline{\psi}\left(i\gamma^{\lambda}D_{\lambda}-m\right)\psi\,,$
(4)

with $\psi$ and $A^{\mu}$ being the fermion and boson fields, respectively.
$F^{\mu\nu}_{a}$ is the field strength tensor for non-Abelian gauge theory,
which is expressed as

$\displaystyle
F^{\mu\nu}_{a}=\partial^{\mu}A^{\nu}_{a}-\partial^{\nu}A^{\mu}_{a}+g\
f^{abc}A^{\mu}_{b}A^{\nu}_{c}\,,$ (5)

where the covariant derivative
$iD^{\mu}=i\overleftrightarrow{\partial}^{\mu}+gA^{\mu}$ such that
$\alpha(i\overleftrightarrow{\partial}^{\mu})\beta=\frac{i}{2}\alpha\left(\partial^{\mu}\beta\right)-\frac{i}{2}\left(\partial^{\mu}\alpha\right)\beta$.
In this work, we focus only on the fermionic part of the EMT given in Eq. (4).
Note that the last term in Eq. (4) vanishes owing to the equation of motion
and thus, we have the following fermionic contribution to the EMT:

$\displaystyle\theta^{\mu\nu}_{\textsf{\scriptsize
q}}=\frac{1}{2}\overline{\psi}i\left[\gamma^{\mu}D^{\nu}+\gamma^{\nu}D^{\mu}\right]\psi\,.$
(6)

The matrix elements of local operators such as electromagnetic current and EMT
have a precise representation using LFWFs of bound systems such as hadrons.
The GFFs are linked to the matrix elements of the EMT, $\theta^{\mu\nu}$,
whereas the second Mellin moment of the GPDs also gives the GFFs. For a spin
$1/2$ composite system, the standard parameterization of the symmetric EMT
$\theta^{\mu\nu}$ involving the GFFs reads Ji:2012vj ; Harindranath:2013goa

$\displaystyle\langle
P^{\prime},\Lambda^{\prime}|\,\theta^{\mu\nu}_{i}(0)\,|P,\Lambda\rangle$
$\displaystyle=$
$\displaystyle\overline{u}(P^{\prime},\Lambda^{\prime})\bigg{[}-B_{i}(Q^{2})\frac{\overline{P}^{\mu}\
\overline{P}^{\nu}}{M}+\left(A_{i}(Q^{2})+B_{i}(Q^{2})\right)\frac{1}{2}(\gamma^{\mu}\overline{P}^{\nu}+\gamma^{\nu}\overline{P}^{\mu})$
(7) $\displaystyle+$ $\displaystyle
C_{i}(Q^{2})\frac{q^{\mu}q^{\nu}-q^{2}g^{\mu\nu}}{M}+\overline{C}_{i}(Q^{2})M\
g^{\mu\nu}\bigg{]}u(P,\Lambda),$

where $\overline{u}(P^{\prime},\Lambda^{\prime})$, $u(P,\Lambda)$ are the
Dirac spinors and $\overline{P}^{\mu}=\frac{1}{2}(P^{\prime}+P)^{\mu}$ is the
average four momentum of the system. $M$ is the mass of the system and
$\Lambda\left(\Lambda^{\prime}\right)$ is the helicity of the initial (final)
state of the system such that
$(\Lambda,\Lambda^{\prime})\equiv\\{\uparrow,\downarrow\\}$. Here
$\uparrow(\downarrow)$ represents the positive (negative) spin projection
along $z-$ axis. The Lorentz index $(\mu,\nu)\ \equiv\\{+,-,1,2\\}$.
$Q^{2}=-q^{2}$ is the square of the momentum transfer. We consider the
symmetric Drell-Yan frame such that the longitudinal momentum transfer
$q^{+}=0$ and the average transverse momentum $\overline{P}^{\perp}=0$. The
form factors $A_{\textsf{\scriptsize q}}(Q^{2})$ and $B_{\textsf{\scriptsize
q}}(Q^{2})$ are extracted form the $\theta^{++}_{\textsf{\scriptsize q}}$
component of the EMT. In light front dynamics the conserved Noether current
associated with the conserved 4-momentum is
$\theta^{+\mu}_{\textsf{\scriptsize q}}$ Li:2023izn . Furthermore, the
operator $\theta^{++}_{q}$ removes a quark (or antiquark) possessing momentum
$k^{\prime}$ (or $k$) and spin projection $\lambda$ along the $z$-axis, and
then generates a quark (or antiquark) with identical spin and momentum
$k^{\prime}$ (or $k$) Brodsky:2008pf . The GFF $A_{\textsf{\scriptsize
q}}(Q^{2})$ is analogous to the Dirac form factor since it is obtained by
summing over the helicity conserving states, whereas $B_{\textsf{\scriptsize
q}}(Q^{2})$, which is obtained by summing over the helicity flip states, is
analogous to the Pauli form factor. The GFFs $C_{\textsf{\scriptsize
q}}(Q^{2})$ and $\overline{C}_{\textsf{\scriptsize q}}(Q^{2})$ are extracted
from the $\theta^{jk}_{\textsf{\scriptsize q}}$ transverse component of the
EMT such that $(j,k)\equiv(1,2)$. We compactly express the matrix elements of
the EMT required to extract the four GFFs as

$\displaystyle\mathcal{M}^{\mu\nu}_{\Lambda\Lambda^{\prime}}=\frac{1}{2}\left[\langle
P^{\prime},\Lambda^{\prime}|\,\theta^{\mu\nu}_{\textsf{\scriptsize
q}}(0)\,|P,\Lambda\rangle\right]\,,$ (8)

where the proton state with momentum $P$ and helicity $\Lambda$ within the
valence Fock sector can be written in terms of three-particle LFWFs,

$\displaystyle\mid
P,\Lambda\rangle=\int\prod_{i=1}^{3}\left[\frac{dx_{i}d^{2}\vec{k}_{i\perp}}{\sqrt{x}16\pi^{3}}\right]16\pi^{3}\delta\left(1-\sum_{i=1}^{3}x_{i}\right)\delta^{2}\left(\sum_{i=1}^{3}\vec{k}_{i\perp}\right)\Psi^{\Lambda}_{\\{x_{i},\vec{k}_{i\perp},\lambda_{i}\\}}\Bigl{|}\left\\{x_{i}P^{+},\vec{k}_{i\perp}+x_{i}\vec{P}_{\perp},\lambda_{i}\right\\}\Bigl{\rangle}\,.$
(9)

The GFFs $A_{\textsf{\scriptsize q}}(Q^{2})$ and $B_{\textsf{\scriptsize
q}}(Q^{2})$ can then be obtained using Eq. (8) as follows

$\displaystyle\mathcal{M}^{++}_{\uparrow\uparrow}+\mathcal{M}^{++}_{\downarrow\downarrow}$
$\displaystyle=$ $\displaystyle 2\ (P^{+})^{2}A_{\textsf{\scriptsize
q}}(Q^{2})\,,$ (10)
$\displaystyle\mathcal{M}^{++}_{\uparrow\downarrow}+\mathcal{M}^{++}_{\downarrow\uparrow}$
$\displaystyle=$ $\displaystyle\frac{iq^{(2)}}{M}\
(P^{+})^{2}B_{\textsf{\scriptsize q}}(Q^{2})\,,$ (11)

while the GFF $C_{\textsf{\scriptsize q}}(Q^{2})$ is computed from the
transverse components using

$\displaystyle\mathcal{M}^{12}_{\uparrow\uparrow}+\mathcal{M}^{12}_{\downarrow\downarrow}$
$\displaystyle=$ $\displaystyle 4C_{\textsf{\scriptsize
q}}(Q^{2})q^{(1)}q^{(2)}.$ (12)

The GFF $\overline{C}_{\textsf{\scriptsize q}}(Q^{2})$ can be computed from
the non-conservation of the partial EMT Lorce:2018egm  which gives us:

$\displaystyle
q_{\mu}\mathcal{M}^{\mu\nu}_{\uparrow\downarrow}=\overline{C}_{\textsf{\scriptsize
q}}(Q^{2})q^{\nu}M\left(-q^{(1)}-iq^{(2)}\right).$ (13)

The GFFs $A_{\textsf{\scriptsize q}}(Q^{2})$ and $B_{\textsf{\scriptsize
q}}(Q^{2})$ can be written in terms of the overlap of LFWFs upon substituting
the proton valence Fock state as shown in Eq. 9 into the matrix element of the
energy-momentum tensor in Eq. 8 and we get the following expression:

$\displaystyle A_{\textsf{\scriptsize q}}(Q^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{\\{\lambda_{i}\\}}\int\left[\mathop{}\\!\mathrm{d}\mathcal{X}\mathop{}\\!\mathrm{d}\mathcal{P}_{\perp}\right]x_{1}\Big{\\{}\Psi^{\uparrow*}_{\\{x^{\prime}_{i},\vec{k}^{\prime}_{i\perp},\lambda_{i}\\}}\Psi^{\uparrow}_{\\{x_{i},\vec{k}_{i\perp},\lambda_{i}\\}}+\Psi^{\downarrow*}_{\\{x^{\prime}_{i},\vec{k}^{\prime}_{i\perp},\lambda_{i}\\}}\Psi^{\downarrow}_{\\{x_{i},\vec{k}_{i\perp},\lambda_{i}\\}}\Big{\\}},$
$\displaystyle iq^{(2)}B_{\textsf{\scriptsize q}}(Q^{2})$ $\displaystyle=$
$\displaystyle
M\sum_{\\{\lambda_{i}\\}}\int\left[\mathop{}\\!\mathrm{d}\mathcal{X}\mathop{}\\!\mathrm{d}\mathcal{P}_{\perp}\right]x_{1}\Big{\\{}\Psi^{\uparrow*}_{\\{x^{\prime}_{i},\vec{k}^{\prime}_{i\perp},\lambda_{i}\\}}\Psi^{\downarrow}_{\\{x_{i},\vec{k}_{i\perp},\lambda_{i}\\}}+\Psi^{\downarrow*}_{\\{x^{\prime}_{i},\vec{k}^{\prime}_{i\perp},\lambda_{i}\\}}\Psi^{\uparrow}_{\\{x_{i},\vec{k}_{i\perp},\lambda_{i}\\}}\Big{\\}},$
(14)

where the longitudinal and transverse momenta of the struck quark are
$x^{\prime}_{1}=x_{1}$ and
${k^{\prime}}_{1\perp}={k}_{1\perp}+(1-x_{1}){q}_{\perp}$ respectively. The
spectator momenta are $x^{\prime}_{i}=x_{i}$ and
${k^{\prime}}_{i\perp}={k}_{i\perp}+x_{i}{q}_{\perp}$ with $i=(2,3)$. The
shorthand notation used for the integration measure is as follows:

$\displaystyle\left[\mathop{}\\!\mathrm{d}\mathcal{X}\mathop{}\\!\mathrm{d}\mathcal{P}_{\perp}\right]\equiv\prod_{i=1}^{3}\left[\frac{\mathop{}\\!\mathrm{d}x_{i}\mathop{}\\!\mathrm{d}^{2}k_{i\perp}}{16\pi^{3}}\right]16\pi^{3}\delta\left(1-\sum_{i=1}^{3}x_{i}\right)\times\delta^{2}\left(\sum_{i=1}^{3}k_{i\perp}\right).$
(15)

Similarly the expression for $C_{\textsf{\scriptsize q}}(Q^{2})$ and
$\overline{C}_{\textsf{\scriptsize q}}(Q^{2})$ can be extracted from the
equations below:

$\displaystyle q^{(1)}q^{(2)}C_{\textsf{\scriptsize q}}(Q^{2})$
$\displaystyle=$
$\displaystyle\frac{1}{8}\sum_{\\{\lambda_{i}\\}}\int\left[\mathop{}\\!\mathrm{d}\mathcal{X}\mathop{}\\!\mathrm{d}\mathcal{P}_{\perp}\right]\mathcal{O}^{1}\mathcal{O}^{2}\Big{\\{}\Psi^{\uparrow*}_{\\{x^{\prime}_{i},\vec{k}^{\prime}_{i\perp},\lambda_{i}\\}}\Psi^{\uparrow}_{\\{x_{i},\vec{k}_{i\perp},\lambda_{i}\\}}+\Psi^{\downarrow*}_{\\{x^{\prime}_{i},\vec{k}^{\prime}_{i\perp},\lambda_{i}\\}}\Psi^{\downarrow}_{\\{x_{i},\vec{k}_{i\perp},\lambda_{i}\\}}\Big{\\}},$
$\displaystyle\left(-q^{(1)}-iq^{(2)}\right)q^{(1)}\overline{C}_{\textsf{\scriptsize
q}}(Q^{2})$ $\displaystyle=$
$\displaystyle-\frac{1}{M}\sum_{\\{\lambda_{i}\\}}\int\left[\mathop{}\\!\mathrm{d}\mathcal{X}\mathop{}\\!\mathrm{d}\mathcal{P}_{\perp}\right]\left(q^{(1)}\mathcal{O}^{1}\mathcal{O}^{1}+q^{(2)}\mathcal{O}^{2}\mathcal{O}^{1}\right)\Big{\\{}\Psi^{\downarrow*}_{\\{x^{\prime}_{i},\vec{k}^{\prime}_{i\perp},\lambda_{i}\\}}\Psi^{\uparrow}_{\\{x_{i},\vec{k}_{i\perp},\lambda_{i}\\}}\Big{\\}},$
(16)

where the operator $\mathcal{O}^{j}=2k_{\perp}^{(j)}+(1-x)q_{\perp}^{(j)}$.
Identities relating the overlap of HO wavefunction pertaining to the
calculation of Eq. III is shown in Appendix B. The LFWF with negative helicity
($\Lambda=\downarrow$) is obtained from the wavefunction with positive
helicity ($\Lambda=\uparrow$) using the mirror parity symmetry
PhysRevD.73.036007  which gives the following relation:

$\displaystyle\psi^{\downarrow}_{\\{x_{i},n_{i},m_{i},\lambda_{i}\\}}=(-1)^{\sum_{i}m_{i}+1}\psi^{\uparrow}_{\\{x_{i},n_{i},-m_{i},-\lambda_{i}\\}}.$
(17)

##  IV Numerical Results

In our calculations, the quark mass influences both the kinetic energy and the
one-gluon exchange interaction (OGE) terms. We opted for different quark mass
values in these terms due to the distinct physics they represent.
Specifically, the kinetic mass term captures long-distance physics, while the
OGE term encapsulates short-distance physics through the derived interaction
arising from single-gluon exchanges between quarks. This effective OGE
compensates for fluctuations spanning from the valence Fock sector to higher
Fock sectors.

Guided by the mass evolution of the renormalization group, the quark mass
linked to gluon dynamics diminishes due to contributions from higher momentum
scales Xu:2021wwj . Consequently, our choice differentiates the quark masses
in the kinetic energy ($m_{q/k}$) and OGE ($m_{q/g}$) terms PhysRevD.44.3857 ;
PhysRevD.50.971 ; PhysRevD.58.096015 . Specifically, we ensure
$m_{q/k}>m_{q/g}$, with the selected values detailed in Table 1. Besides
these, the model includes two additional parameters: the confinement potential
strength ($\kappa$) and the coupling constant ($\alpha_{s}$), both listed in
Table 1. Our results employ truncation parameters with values
$N_{\mathrm{max}}=10$ and $K=16.5$. We calibrated these parameters against
existing experimental data on the Dirac and Pauli form factors Xu:2021wwj .
Using this setup, we derive the proton’s four gravitational form factors of
the symmetric energy-momentum tensor and depict the mechanical properties,
including the quark pressure and shear distributions inside the proton.

Table 1: Parameters of the model with a transverse basis truncation set at $N_{\mathrm{max}}=10$ and a longitudinal truncation at $K=16.5$. $m_{q/k}$ |  $m_{q/g}$ |  $\kappa$ |  $\alpha_{s}$ |
---|---|---|---|---
$0.3~{}\mathrm{GeV}$ |  $0.2~{}\mathrm{GeV}$ |  $0.34~{}\mathrm{GeV}$ |  $1.1\pm 0.1$ |

Figure 1: Plots illustrating GFFs for both $A_{q}(Q^{2})$ and $B_{q}(Q^{2})$
at two scales, plotted against $Q^{2}$. These include: (a) $A_{q}(Q^{2})$ at
an initial scale $\mu_{0}^{2}=0.195\pm 0.020$ GeV2, (b) $A_{q}(Q^{2})$ evolved
to $\mu^{2}=4$ GeV2, (c) $B_{q}(Q^{2})$ at the same initial scale, and (d)
$B_{q}(Q^{2})$ evolved to $\mu^{2}=4$ GeV2. In the plots, solid magenta lines
with shaded areas signify combined $u$ and $d$ quark contributions, while
individual $d$ and $u$ quark contributions are shown with dashed blue and red
lines, respectively.The shaded areas represent uncertainties: $10\%$ in the
$\alpha_{s}$ coupling at the initial scale and a combined $10\%$ in
$\mu_{0}^{2}$ and $\alpha_{s}$ at the evolved scale, calculated in quadrature.
The evolved GFFs are compared against lattice data at $\mu^{2}=4$ GeV2, shown
as red circles and black squares from Ref. Hagler:2007xi , and cyan triangles
from Ref. Hackett:2023rif .

Figure 2: Plots illustrating GFFs for both $D_{q}(Q^{2})$ and
$\bar{C}_{q}(Q^{2})$ across different scales: (a) $D_{q}(Q^{2})$ at an initial
scale of $\mu_{0}^{2}=0.195\pm 0.020$ GeV2, (b) $D_{q}(Q^{2})$ at a final
evolved scale of $\mu^{2}=4$ GeV2, (c) $\bar{C}_{q}(Q^{2})$ at the initial
scale, and (d) $\bar{C}_{q}(Q^{2})$ at the final evolved scale, all versus
$Q^{2}$. The solid magenta lines with bands in each plot denote the combined
contributions of the $u$ and $d$ quarks, while the dashed blue and red lines
represent individual contributions from the $d$ and $u$ quarks, respectively.
The bands across the plots at the initial scale signify a $10\%$ uncertainty
in the $\alpha_{s}$ coupling constant whereas the bands across the evolved
results are due to $10\%$ uncertainty in $\mu_{0}^{2}$ and the coupling
constant $\alpha_{s}$. In plot (b) the red circles are the data points taken
from Shanahan:2018nnv  where they have rescaled the Jefferson Lab experimental
data used in Burkert:2018bqq  to $\mu^{2}=4$ GeV2. Lattice data are
represented by cyan triangles and black squares Hagler:2007xi . Additionally,
the evolved $D^{u+d}_{q}(Q^{2})$ results are compared with lattice data at
$\mu^{2}=4$ GeV2 as depicted by blue diamond from Ref. Hackett:2023rif .

In Fig. 1, we present our calculations for the GFFs $A(Q^{2})$ and $B(Q^{2})$,
along with with predictions from lattice QCD Hagler:2007xi . The contributions
from the $u$ quark and $d$ quark are individually presented for both
$A(Q^{2})$ and $B(Q^{2})$ at both the initial and final scales. As the
existing lattice data is at a higher scale of $\mu^{2}=4$ GeV2, we’ve evolved
our results for a consistent comparison. For the scale evolution, we employed
the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equations of QCD
Dokshitzer:1977sg ; Gribov:1972ri ; Altarelli:1977zs , specifically utilizing
the next-to-next-to-leading order (NNLO) variant. The evolution is done at
each value of $Q^{2}$ for the GPDs $xH^{q}(x,Q^{2})$ and $xE^{q}(x,Q^{2})$ as
their moments corresponds to the GFFs $A(Q^{2})$ and $B(Q^{2})$ respectively.
The GFFs were evolved from an initial scale of $\mu_{0}^{2}=0.195\pm 0.020$
GeV2 to the lattice scale using the higher order perturbative parton evolution
toolkit (HOPPET) Salam:2008qg . The initial scale is chosen so as to match the
moment of the valence quark PDFs Xu:2021wwj . Our analyses indicate that at
the initial scale, the values for $A^{u+d}(0)$ and $B^{u+d}(0)$ are 1 and 0
respectively. This is expected as a consequence of the conservation of
momentum and total angular momentum. The value of $B^{u+d}(0)=0$ is referred
to as the vanishing of the total anomalous gravitomagnetic moment
Brodsky:2000ii . We find that $A^{u}>A^{d}$ and $B^{u}$ is positive while
$B^{d}$ is negative. Following the evolution process, the magnitudes of both
functions exhibit a decline. We find that the lattice results for
$A^{u+d}(Q^{2})$ significantly exceed our evolved result at $\mu^{2}=4$ GeV2
especially at higher $Q^{2}$.

In Fig. 2, the $D$-term form factor and the GFF $\bar{C}(Q^{2})$ are depicted.
Panels (a) and (b) of Fig. 2 display our findings for $D(Q^{2})$ at the
initial and evolved final scales, respectively. The $D$-term for both $d$ and
$u$ quarks is observed to be negative, with the magnitude being larger for the
$u$ quark. Following scale evolution, the characteristics of our $D$-term, as
shown in Fig. 2 (b), correspond well with the lattice results Hagler:2007xi ;
Hackett:2023rif  and JLab experimental data Burkert:2018bqq 111The notation
for the D-term used in Burkert:2018bqq  is
$d_{q}(Q^{2})=\frac{4}{5}D_{q}(Q^{2})$ and have
$\mu^{2}=1.4~{}\mathrm{GeV}^{2}$. A negative $D$-term is indicative of a
stable bound system, and our evolved results for the $D$-term show a
reasonable alignment with the JLab data at $\mu^{2}=4$ GeV2.

Panels (c) and (d) of Fig. 2 present the $\bar{C}(Q^{2})$ form factor
following our BLFQ framework. We observe $\bar{C}^{u}(Q^{2})>0$ and
$\bar{C}^{d}(Q^{2})<0$, with the aggregate $\bar{C}^{u+d}(Q^{2})$ remaining
near zero for most $Q^{2}$ values. According to the sum rule, the total
$\bar{C}^{u+d}(Q^{2})$ should equal zero for all values of $Q^{2}$. Within our
BLFQ approach, focusing on the valence Fock sector of the proton state, this
condition is partially met, with $\bar{C}^{u+d}(0)$ being $0.0063$ at the
initial scale and $0.0024$ at the evolved scale of $\mu^{2}=4$ GeV2.

In Table 2, we provide a detailed overview of the GFFs at $Q^{2}=0$. We
compare them with predictions from various phenomenological models, lattice
QCD, and the current experimental data for $D_{q}(0)$. For the form factors
$A_{q}(0)$ and $J_{q}(0)$, our estimates are generally consistent, albeit
slightly lower than the projections from Refs. Hagler:2003jd ;
Gockeler:2003jfa ; LHPC:2010jcs ; Hagler:2007xi ; QCDSF-UKQCD:2007gdl ;
Deka:2013zha ; Dorati:2007bk ; Lorce:2018egm  at a renormalization scale of
$\mu^{2}=4$ GeV2. It’s noteworthy that the QCD sum rule (QCDSR) (I & II)
suggests a higher value for $A_{q}(0)$, likely due to its use of a reduced
scale, $\mu^{2}=1$ GeV2 Azizi:2019ytx . In the $\chi$QSM and Skyrme models,
only quarks and antiquarks contribute to the nucleon’s angular momentum,
accounting for the entirety of it. This results in the relationship
$2J_{q}(0)=A_{q}(0)=1$. As for the AdS/QCD models, the results are given at
the model’s inherent scale, with $u$ and $d$ quarks collectively accounting
for approximately $90\%$ of the nucleon’s momentum.

Regarding the form factor $D_{q}(0)$, we observe the following trends. Our
estimate aligns more closely with the lattice QCD results Gockeler:2003jfa ;
LHPC:2010jcs ; Hagler:2007xi , although it’s slightly on the higher side.
Comparatively, the predictions from QCDSR-I Azizi:2019ytx  and QCDSR-II
Azizi:2019ytx  are in the same ballpark as our estimate, but with slight
variations. On the other hand, model-based predictions such as those from the
Skyrme model Cebulla:2007ei ; Kim:2012ts , along with the $\chi$QSM
predictions Goeke:2007fp ; Jung:2014jja ; Wakamatsu:2007uc , deviate more
significantly from our results. Interestingly, the $\chi$PT model
Dorati:2007bk  and IFF model Lorce:2018egm  provide results that resonate
closely with our findings. The KM15 fit Anikin:2017fwu  and the JLab data
Burkert:2018bqq  offer values that hover around our prediction, yet show minor
differences. Examining the $\bar{C}^{u+d}_{q}(0)$ values in Table 2, our
result has the lowest magnitude.

All GFF outcomes were fitted using a two-parameter dipole function.
Comprehensive details of this function and the derived parameter values can be
found in Appendix A.

Table 2: The GFFs for the valence quark combination at $Q^{2}=0$ are compared with predictions from other models and data from JLab. Predictions from the Skyrme and $\chi$QSM models take into account contributions from both quarks and gluons and are not dependent on scale. The notation $(\mu_{0}\to\mu)$ indicates the evolution from the starting scale $\mu_{0}$ to the final scale $\mu$.  Approaches/Models |  $A^{u+d}_{q}(0)$ |  $J_{q}(0)=\frac{1}{2}[A^{u+d}_{q}(0)+B^{u+d}_{q}(0)]$ |  $D^{u+d}(0)=4C^{u+d}(0)$ |  $\bar{C}^{u+d}_{q}(0)$
---|---|---|---|---
This work ($\sqrt{0.195}$ $\to$ $2$ GeV) | 0.420 $\pm$ 0.032 | 0.210 $\pm$ 0.016 | -1.925 $\pm$ 0.398 | 0.0024 $\pm$ 0.0034
LQCD ($2$ GeV) Hagler:2003jd  | 0.675 | 0.34 | - | -
LQCD ($2$ GeV) Gockeler:2003jfa  | 0.547 | 0.33 | -0.80 | -
LQCD ($2$ GeV) LHPC:2010jcs  | 0.553 | 0.238 | -1.02 | -
LQCD ($2$ GeV) Hagler:2007xi  | 0.520 | 0.213 | -1.07 | -
LQCD ($2$ GeV) QCDSF-UKQCD:2007gdl  | 0.572 | 0.226 | - | -
LQCD ($2$ GeV) Deka:2013zha  | 0.565 | 0.314 | - | -
$\chi$PT ($2$ GeV) Dorati:2007bk  | 0.538 | 0.24 | -1.44 | -
IFF ($2$ GeV) Lorce:2018egm  | 0.55 | 0.24 | -1.28 | -0.11
Asymptotic ($\infty$ GeV) Hatta:2018sqd  | - | 0.18 | - | -0.15
QCDSR-I (1 GeV) Azizi:2019ytx  | 0.79 | 0.36 | -1.832 | -2.1 $\times 10^{-2}$
QCDSR-II (1 GeV) Azizi:2019ytx  | 0.74 | 0.30 | -1.64 | -2.5 $\times 10^{-2}$
Skyrme Cebulla:2007ei  | 1 | 0.5 | -3.584 | -
Skyrme Kim:2012ts  | 1 | 0.5 | -2.832 | -
$\chi$QSM Goeke:2007fp  | 1 | 0.5 | -1.88 | -
$\chi$QSM Jung:2014jja  | 1 | 0.5 | -4.024 |
$\chi$QSM Wakamatsu:2007uc  | - | - | -3.88 | -
AdS/QCD Model I Mondal:2015fok  | 0.917 | 0.415 | - | -
AdS/QCD Model II Mondal:2015fok  | 0.8742 | 0.392 | - | -
LCSR-LO Anikin:2019kwi  | - | - | -2.104 | -
KM15 fit Anikin:2017fwu  | - | - | -1.744 | -
DR Pasquini:2014vua  | - | - | -1.36 | -
JLab data Burkert:2018bqq  | - | - | $-1.688$ | -
IP Polyakov:2018exb  | - | - | - | $1.4\times 10^{-2}$

###  IV.1 Mechanical properties

Table 3: The mechanical properties: pressure $p_{0}$, energy density ${\cal E}$, and mechanical radius squared $\langle{r^{2}_{\text{mech}}\rangle}$ of nucleon. Approaches/Models |  $p_{0}$ [GeV/fm3] |  ${\cal E}$ [GeV/fm3] |  $\langle{r^{2}_{\text{mech}}\rangle}$ [fm2]
---|---|---|---
This work ($\sqrt{0.195}$ GeV $\to$ $2$ GeV) | $0.47\pm 0.10$ | $1.86\pm 0.13$ | $0.73\pm 0.01$
QCDSR set-I (1 GeV) Azizi:2019ytx  | $0.67$ | $1.76$ | $0.54$
QCDSR set-II (1 GeV) Azizi:2019ytx  | $0.62$ | $1.74$ | $0.52$
Skyrme model Cebulla:2007ei  | 0.47 | 2.28 | -
modified Skyrme model Kim:2012ts  | 0.26 | 1.45 | -
$\chi$QSM Goeke:2007fp  | 0.23 | 1.70 | -
Soliton model Jung:2014jja  | 0.58 | 3.56 | -
LCSM-LO Anikin:2019kwi  | 0.84 | 0.92 | 0.54

The $D$-term can be directly related to the pressure $p_{0}$ in the center of
the nucleon and the mechanical radius squared $r^{2}_{\text{mech}}$ as
Polyakov:2018zvc

$\displaystyle p_{0}$
$\displaystyle=-\frac{1}{24\pi^{2}M_{n}}\int^{\infty}_{0}{\rm
d}Q^{2}~{}Q^{3}~{}{D}(Q^{2}),$ $\displaystyle\langle
r^{2}_{\text{mech}}\rangle$ $\displaystyle=6D(0)\Big{[}\int^{\infty}_{0}{\rm
d}Q^{2}~{}D(Q^{2})\Big{]}^{-1}.$ (18)

The energy density $\cal E$ can be obtained as follows:

$\displaystyle{\cal E}$
$\displaystyle=\frac{M_{n}}{4\pi^{2}}\int^{\infty}_{0}{\rm
d}Q^{2}\left(A(Q^{2})+\frac{Q^{2}}{4M^{2}_{n}}D(Q^{2})\right),$ (19)

Here, $M_{n}$ denotes the mass of nucleon. In Table 3, we list our findings
alongside those from various models and approaches. For the pressure, $p_{0}$,
our results are closest to those of the Skyrme model Cebulla:2007ei  and the
soliton model Jung:2014jja . As for the energy density, ${\cal E}$, our value
is less than that of the Skyrme model Cebulla:2007ei  and the soliton model
Jung:2014jja  but more than most of the other predictions, including the QCDSR
sets Azizi:2019ytx  and LCSM-LO Anikin:2019kwi . Finally, concerning the
mechanical radius squared, $\langle{r^{2}_{\text{mech}}\rangle}$, our
prediction surpasses the values reported in Refs. Azizi:2019ytx ;
Anikin:2019kwi .

####  IV.1.1 Pressure and Shear forces

In our study, the $D$-term is derived from the transverse components of the
EMT, which have connections to mechanical properties like the pressure and
shear distributions Polyakov:2002yz ; Polyakov:2018zvc ; Polyakov:2018exb . A
deeper understanding of these quantities can be achieved by undertaking a 2D
Fourier transformation of the $D$-term with respect to the transverse momentum
transfer thereby transitioning from momentum space to impact parameter space.

The FT ($\tilde{D}({\bm{b}}^{\perp})$) of the $D$-term can be expressed using
the Bessel function of the zeroth order ($J_{0}$) as follows:

$\displaystyle\tilde{D}(b)$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{2}}~{}\int d^{2}{\bm{q}}^{\perp}\
e^{-i{\bm{q}}^{\perp}{\bm{b}}^{\perp}}D(q^{2}),$ (20) $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{0}^{\infty}d{\bm{q}}^{\perp
2}J_{0}\left({\bm{q}}^{\perp}{\bm{b}}^{\perp}\right)D(q^{2}),$

where, $b=|{\vec{b}_{\perp}}|$ represents the impact parameter.

The 2D pressure $p(b)$ and shear distributions $s(b)$ can thus be defined in
the impact parameter space as follows Freese:2021czn :

$\displaystyle p(b)$ $\displaystyle=$
$\displaystyle\frac{1}{2M}\frac{1}{b}\frac{d}{d\ b}\left[b\frac{d}{d\
b}\tilde{D}_{q}(b)\right],$ (21) $\displaystyle s(b)$ $\displaystyle=$
$\displaystyle-\frac{1}{M}b\frac{d}{d\
b}\left[\frac{1}{b}\frac{d}{db}\tilde{D}_{q}(b)\right].$ (22)

A spherical shell of radius $b$ experiences normal and tangential forces,
which are defined by the combined effects of pressure and shear
Polyakov:2018zvc .

$\displaystyle F_{n}(b)=$ $\displaystyle\left(p(b)+\frac{1}{2}s(b)\right),$
(23) $\displaystyle F_{t}(b)=$
$\displaystyle\left(p(b)-\frac{1}{2}s(b)\right).$ (24)

Figure 3: Plots of (a) the pressure distribution $2\pi p(b)$, and (b) the
shear force distribution $2\pi s(b)$ as a function of $b$. Our results are
compared with results based on light-cone sum rule (LCSR) evaluated in Ref.
Anikin:2019kwi  (purple dot-dashed lines) and using the fitting function of
$D(Q^{2})$ based on JLab experimental data Burkert:2018bqq  (red dashed
lines). The solid magenta lines with magenta bands and the solid blue lines
with blue bands represent the results at the initial scale
$\mu_{0}^{2}=0.195\pm 0.020$ GeV2 and the final evolved scale of
$\mu^{2}=4.00$ GeV2, respectively. The systematic uncertainties presented in
the experimental extraction of the D-term Burkert:2018bqq  were added in
quadrature.

Figure 4: Plots of (a) the normal forces $F_{n}$, and (b) the tangential
forces $F_{t}$ as a function of $b$. The legends are same as mentioned in Fig.
3.

In Fig. 3(a), we show the distribution $2\pi b~{}p(b)$ as a function of $b$.
We compare our result with the analysis from the light-cone sum rule
Anikin:2019kwi  and the experimental data from JLab using fitting functions
for $D(Q^{2})$ Burkert:2018bqq . The pressure distribution $p(b)$ must follow
the von Laue condition which is $\int_{0}^{\infty}db\,b^{2}\,p(b)=0$. This
rule comes from the need to balance internal forces in a system
Polyakov:2018zvc ; Polyakov:2002yz . This means the distribution should have a
point where it changes sign to satisfy the von Laue condition. Our data shows
a positive region followed by a negative region. This suggests that inside the
nucleon, there are forces pushing out and forces pulling in. For our results,
the point where the pressure changes sign is around $b\approx 0.6$ fm for both
scales. For the light-cone sum rule Anikin:2019kwi , this point is at
$b\approx 0.51$ fm, and for the JLab data Burkert:2018bqq , it’s around
$b\approx 0.45$ fm. Even though the zero-crossings in our model occur somewhat
later than those in the comparative datasets, the overarching characteristics
and mechanical implications of the pressure distribution align across
different models and empirical findings Shanahan:2018nnv ; Anikin:2019kwi ;
Goeke:2007fp ; Kim:2012ts ; Jung:2014jja ; Cebulla:2007ei . In Fig. 3(b), we
display the shear force distribution $2\pi bs(b)$. This distribution, $s(b)$,
is associated with attributes like surface tension and surface energy, which
are typically positive in stable hydrostatic systems Polyakov:2018zvc . In
line with previous studies, our results confirm that $s(b)$ remains positive
across all $b$ values. Furthermore, the behavior of our shear force
distribution is consistent with outcomes from other methodologies
Shanahan:2018nnv ; Anikin:2019kwi ; Goeke:2007fp ; Kim:2012ts ; Jung:2014jja ;
Cebulla:2007ei . It seems to be a coincidence that our initial scale results
for both pressure and shear align more closely with the light-cone sum rule
Anikin:2019kwi  and the JLab data Burkert:2018bqq  than the evolved results.
Additionally, the peak position of our results at the final scale occurs at a
lower value of $b$.

In Fig 4(a) and Fig 4(b), we present our results for the normal and tangential
forces attributed to the valence quark combination, respectively. A salient
observation is the consistently positive nature of $F_{n}(b)$. On the other
hand, $F_{t}(b)$ showcases a dual character: a positive core, indicative of
repulsive forces, and a subsequent negative domain, signifying attractive
forces, with the transition (zero-crossing) located around $b\approx 0.4$ fm.
The peak of the repulsive force occurs close to $b\approx 0.3$ fm, while the
maximum attractive force, which plays a pivotal role in binding, is more
pronounced around $b\approx 0.6$ fm. It’s worth noting that this binding force
exhibits a greater magnitude than the repulsive counterpart. Thus within the
BLFQ framework, the key features of these forces exhibit good agreement with
insights from the light-cone sum rule Anikin:2019kwi , the distributions
inferred from JLab’s $D(Q^{2})$ fitting function Burkert:2018bqq , and the
predictions of the chiral quark-soliton model Goeke:2007fp .

####  IV.1.2 The Galilean energy density and pressure distributions

Within the context of nucleonic internal structures, understanding energy
density and pressure distributions offers invaluable insights. To this end,
we’ve drawn upon the Galilean framework to study these distributions, as
described in Ref. Lorce:2018egm . We represent the Galilean energy density
$\mu(b)$, radial pressure $\sigma^{r}(b)$, tangential pressure
$\sigma^{t}(b)$, isotropic pressure $\sigma(b)$, and pressure anisotropy
$\Pi(b)$ through the equations below:

$\displaystyle\mu(b)$ $\displaystyle=$ $\displaystyle
M\left[\frac{A(b)}{2}+\overline{C}(b)+\frac{1}{4M^{2}}\frac{1}{b}\frac{d}{db}\left(b\frac{d}{db}\left[\frac{B(b)}{2}-4C(b)\right]\right)\right],$
(25) $\displaystyle\sigma^{r}(b)$ $\displaystyle=$ $\displaystyle
M\left[-\overline{C}(b)+\frac{1}{M^{2}}\frac{1}{b}\frac{dC(b)}{db}\right],$
(26) $\displaystyle\sigma^{t}(b)$ $\displaystyle=$ $\displaystyle
M\left[-\overline{C}(b)+\frac{1}{M^{2}}\frac{d^{2}C(b)}{d(b)^{2}}\right],$
(27) $\displaystyle\sigma(b)$ $\displaystyle=$ $\displaystyle
M\left[-\overline{C}(b)+\frac{1}{2}\frac{1}{M^{2}}\frac{1}{b}\frac{d}{db}\left(b\frac{d\
C(b)}{db}\right)\right],$ (28) $\displaystyle\Pi(b)$ $\displaystyle=$
$\displaystyle
M\left[-\frac{1}{M^{2}}b\frac{d}{db}\left(\frac{1}{b}\frac{dC(b)}{db}\right)\right].$
(29)

In Fig. 5, we present our findings for $2\pi b~{}\mu$ which encompasses all
four GFFs as per Eq. 25. The energy density’s peak at the initial scale is
discerned around $b\approx 0.4$ fm. However, when evolved, this peak not only
diminishes in magnitude but also repositions closer, at $b\approx 0.3$ fm.

Turning to Fig. 6, we elaborate on the results for the pressures defined in
Eqs. 26-29. The radial pressure remains consistently positive, indicating a
repulsive nature. The tangential pressure, on the other hand, displays a more
varied behavior: its positive (repulsive) region is located at smaller impact
parameters, while the negative (attractive) region spans towards the larger
values of $b$. The isotropic pressure and the pressure anisotropy are
essentially linear combinations of the radial and tangential pressures.
Specifically, they are defined by the relations
$\sigma=\frac{(\sigma_{r}+\sigma_{t})}{2}$ and $\Pi=\sigma_{r}-\sigma_{t}$. We
observe that the isotropic pressure largely mirrors the behavior of the radial
pressure, though it turns slightly negative for larger $b$ values,
specifically beyond $1.0$ fm. The pressure anisotropy remains positive
everywhere, implying that the radial pressure consistently exceeds the
tangential one.

Our projections for these energy and pressure distributions align well with a
straightforward multipole model as proposed in Ref. Lorce:2018egm . An
interesting coincidence is that our results at the scale $\mu^{2}=4.00$ GeV2
align more closely with the multipole model than those at the initial scale of
$\mu_{0}^{2}=0.195$ GeV2.

Figure 5: Plot of the two-dimensional Galilean energy density $2\pi
b~{}\mu_{q}(b)$. The solid magenta line with magenta bands and the solid blue
line with blue bands represent the results at the initial scale
$\mu_{0}^{2}=0.195\pm 0.020$ GeV2 and $\mu^{2}=4$ GeV2, respectively. Our
results are compared with the results in a multipole model (red dashed line)
Lorce:2018egm  and the red band denotes a $10\%$ uncertainty in the parameter
$F_{q}(0)$ as defined in Lorce:2018egm .

Figure 6: Plots of the two-dimensional radial pressure (a) $2\pi
b~{}\sigma_{r}$, (b) tangential pressure $2\pi b~{}\sigma_{t}$, (c) isotropic
pressure $2\pi b~{}\sigma$ and (d) the pressure anisotropy $2\pi b~{}\Pi$. The
legends are same as mentioned in Fig. 5.

##  V Conclusion

In this study, we delved into the gravitational form factors (GFFs) and the
mechanical attributes of quarks in the proton using the basis light-front
quantization (BLFQ) theoretical approach. Utilizing an effective light-front
Hamiltonian, we integrated confinement across both transverse and longitudinal
planes, coupled with a one-gluon exchange interaction tailored for valence
quarks, making it apt for low-resolution applications. By treating the
Hamiltonian as a relativistic three-quark problem within the BLFQ framework,
we deduced the nucleon light-front wave functions (LFWFs) as its eigenvectors.
These LFWFs then served as the foundation to derive the quark GFFs.

We evaluated the four GFFs and studied their $Q^{2}$ dependence at both the
initial scale of $\mu_{0}^{2}=0.195$ GeV2 and final evolved scale of
$\mu^{2}=4.00$ GeV2. The $A(Q^{2})$ and $B(Q^{2})$ GFFs are comparable with
lattice QCD results Hagler:2007xi  although our result for the $A(Q^{2})$ is
slightly lower compared to lattice QCD result. We have observed that our
result for $D(Q^{2})$ is in qualitative accord with the experimental data
extracted from DVCS process at JLab Burkert:2018bqq  and lattice QCD
predictions Hagler:2007xi . We fitted our GFFs to a dipole function with two
parameters as discussed in Appendix A. We have compared the values of the GFFs
at $Q^{2}=0$ with the existing theoretical predictions and the data from JLab.
Our estimates for $A_{q}(0)$ and $J_{q}(0)$ align reasonably with multiple
references, though they tend to be slightly lower at a renormalization scale
of $\mu^{2}=4$ GeV2. For the $D_{q}(0)$ form factor, our results are largely
in sync with lattice QCD. Our $\bar{C}^{u+d}_{q}(0)$ value is found to be
close to zero such that $\bar{C}^{u}_{q}(0)>0$ and $\bar{C}^{u}_{q}(0)<0$
since both have the same shapes, opposite signs and nearly the same
magnitudes.

Using GFFs in the BLFQ framework, we assessed the proton’s internal pressure,
energy density, and mechanical radius, comparing with other theoretical
models. The central proton pressure, $p_{0}$, aligns best with Skyrme
Cebulla:2007ei  and soliton models Jung:2014jja . Our energy density, ${\cal
E}$, situates between LCSM-LO Anikin:2019kwi  and other predictions like QCDSR
Azizi:2019ytx . Despite varying theoretical predictions on $p_{0}$ and
$\mathcal{E}$, our mechanical radius, $\langle{r^{2}_{\text{mech}}\rangle}$,
is notably larger than in Refs. Azizi:2019ytx ; Anikin:2019kwi .

In the BLFQ framework, we analyzed the internal pressure $p(b)$ and shear
force $s(b)$ distributions within the proton. We observed a positive core and
a negative tail for $p(b)$, while $s(b)$ remained consistently positive,
aligning with both experimental data and other theoretical models.
Additionally, the normal force $F_{n}(b)$ is uniformly repulsive, whereas the
tangential force $F_{t}(b)$ shows repulsion at the center but attraction
towards the periphery. These patterns generally concur with experimental and
theoretical expectations. Moreover, our computations of the two-dimensional
Galilean energy density, various pressure metrics, and anisotropy in the BLFQ
framework match well qualitatively with a multipole model, especially for
evolved results at higher scales.

The results showcased validate the efficacy of our effective Hamiltonian
method in the BLFQ framework, encouraging its use for other hadrons. Our
future plans will focus on calculating the gluon GFFs within the proton system
by expanding the Fock space to include an additional dynamical gluon.

##  VI ACKNOWLEDGMENTS

J. P. V. acknowledges useful discussions with Yang Li. S. N. is supported by
the Senior Scientist Program funded by Gansu Province, Grant No. 23JDKA0005.
CM thanks the Chinese Academy of Sciences President’s International Fellowship
Initiative for the support via Grants No. 2021PM0023. C. M. is also supported
by new faculty start up funding by the Institute of Modern Physics, Chinese
Academy of Sciences, Grant No. E129952YR0. 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, by the Foundation for Key Talents of
Gansu Province, by the Central Funds Guiding the Local Science and Technology
Development of Gansu Province, Grant No. 22ZY1QA006, by international
partnership program of the Chinese Academy of Sciences, Grant No.
016GJHZ2022103FN, by the National Natural Science Foundation of China under
Grant No. 12375143, by National Key R&D Program of China, Grant No.
2023YFA1606903, and by the Strategic Priority Research Program of the Chinese
Academy of Sciences, Grant No. XDB34000000. A. M. would like to thank SERB
MATRICS (MTR/2021/000103) for funding. J. P. V. is supported in part by the US
Department of Energy, Division of Nuclear Physics, Grant No. DE-SC0023692. A
portion of the computational resources were also provided by Gansu Computing
Center.

##  Appendix A GFFs fitted with a dipole function

We fit both the initial and final scale GFFs with the following function:

$\displaystyle f(Q^{2})=\frac{a_{0}}{(1+a_{1}Q^{2})^{2}}.$ (30)

The fit parameters for the form factors $A(Q^{2})$ and $B(Q^{2})$ are listed
in Table 4. Parameters for $D(Q^{2})$ and $\bar{C}(Q^{2})$ are provided in
Table 5. It should be emphasized that the fit for $\bar{C}(Q^{2})$ is only
reliable for $Q^{2}>0.2$, as its fitting function does not faithfully
represent $\bar{C}(Q^{2}\approx 0)$. The efficacy of the fit is evaluated
through the $\chi^{2}$ values presented for each GFF in Tables 4 and 5,
calculated using Eq. 31. In this equation, $N$ denotes the number of data
points, $N_{p}=2$ represents the count of parameters, $O_{i}$ are the
calculated GFF values according to the BLFQ framework, and $C_{i}$ are the
values obtained from the fitting procedure. The summation in Eq. 31
encompasses all data points for which $0\leq Q^{2}\leq 2$ for the GFFs, except
for $\bar{C}(Q^{2})$, where the condition is $0.2\leq Q^{2}\leq 2$.

$\displaystyle\chi^{2}=\frac{1}{\left(N-N_{p}-1\right)}\sum_{i}\frac{\left(O_{i}-C_{i}\right)^{2}}{|C_{i}|}.$
(31)

Table 4: Fit parameters for $A(Q^{2})$ and $B(Q^{2})$ for $u$ and $d$ quark flavor. The unprimed parameters ($a_{i}$) are for the initial scale of $\mu_{0}^{2}=0.195$ GeV2 and the primed parameters ($a^{\prime}_{i}$) for the final scale of $\mu^{2}=2$ GeV2. The upper and lower bounds denote the error bands around the main value. The $\chi^{2}$ values indicate the goodness of fit calculated using Eq. 31. For the calculation of each $\chi^{2}$ value, we made use of 29 distinct points across the interval $0\leq Q^{2}\leq 2$. GFF | Bound | $a_{0}$ | $a_{1}$ | $\chi^{2}$ | $a_{0}^{\prime}$ | $a_{1}^{\prime}$ | $\chi^{2}$
---|---|---|---|---|---|---|---
$A^{u}(Q^{2})$ | central | 0.6731 | 0.7762 | 0.00010378 | 0.2837 | 0.7764 | 0.00005875
$A^{u}(Q^{2})$ | upper | 0.6787 | 0.6116 | 0.00015951 | 0.3034 | 0.6924 | 0.00011753
$A^{u}(Q^{2})$ | lower | 0.6703 | 0.9167 | 0.00012235 | 0.2643 | 0.8863 | 6.5513 $\times 10^{-6}$
$A^{d}(Q^{2})$ | central | 0.3188 | 0.9112 | 0.00001585 | 0.1338 | 0.9134 | 0.00001145
$A^{d}(Q^{2})$ | upper | 0.3099 | 0.7733 | 0.00003394 | 0.1435 | 0.8714 | 0.00003479
$A^{d}(Q^{2})$ | lower | 0.3220 | 1.0310 | 0.00002023 | 0.1241 | 0.9640 | 3.9597 $\times 10^{-7}$
$B^{u}(Q^{2})$ | central | 0.3289 | 1.1102 | 2.1633 $\times 10^{-6}$ | 0.1398 | 1.1169 | 3.0792 $\times 10^{-7}$
$B^{u}(Q^{2})$ | upper | 0.3195 | 0.9266 | 9.8522 $\times 10^{-6}$ | 0.1494 | 1.0434 | 9.6802 $\times 10^{-6}$
$B^{u}(Q^{2})$ | lower | 0.3285 | 1.2743 | 5.0886 $\times 10^{-6}$ | 0.1303 | 1.2067 | 0.00002466
$B^{d}(Q^{2})$ | central | -0.3279 | 1.0171 | 6.2102 $\times 10^{-6}$ | -0.1393 | 1.0227 | 3.8685 $\times 10^{-6}$
$B^{d}(Q^{2})$ | upper | -0.3185 | 0.8624 | 0.00003352 | -0.1296 | 1.0929 | 3.5263 $\times 10^{-6}$
$B^{d}(Q^{2})$ | lower | -0.3274 | 1.1574 | 3.7700 $\times 10^{-6}$ | -0.1489 | 0.9649 | 0.00002398
Table 5: Fit parameters for $D(Q^{2})$ and $\bar{C}(Q^{2})$ for both $u$ and
$d$ quark flavor. The unprimed parameters ($a_{i}$) are for the initial scale
of $\mu_{0}^{2}=0.195$ GeV2 and the primed parameter ($a^{\prime}_{i}$) are
for the final scale of $\mu^{2}=2$ GeV2. The upper and lower bounds denote the
error bands around the main value. The $\chi^{2}$ values indicate the goodness
of fit calculated using Eq. 31. To determine the $\chi^{2}$ values for the GFF
$D(Q^{2})$, we utilized a set of 29 discrete data points spanning the range
$0\leq Q^{2}\leq 2$. For the GFF $\bar{C}(Q^{2})$, we calculated using 20
discrete data points within the interval $0.2\leq Q^{2}\leq 2$.
GFF | Bound | $a_{0}$ | $a_{1}$ | $\chi^{2}$ | $a_{0}^{\prime}$ | $a_{1}^{\prime}$ | $\chi^{2}$
---|---|---|---|---|---|---|---
$D^{u}(Q^{2})$ | central | -2.8330 | 3.1549 | 0.00008020 | -1.1460 | 2.9875 | 0.00011587
$D^{u}(Q^{2})$ | upper | -3.4741 | 3.1525 | 0.00106679 | -0.9571 | 3.0757 | 0.00054817
$D^{u}(Q^{2})$ | lower | -2.5569 | 3.2365 | 0.00034558 | -1.3347 | 2.9243 | 0.00010448
$D^{d}(Q^{2})$ | central | -1.9610 | 3.5540 | 0.00011851 | -0.7822 | 3.3392 | 0.00017370
$D^{d}(Q^{2})$ | upper | -2.6775 | 3.687 | 0.00020343 | -0.5869 | 3.2820 | 0.00041175
$D^{d}(Q^{2})$ | lower | -1.6696 | 3.5211 | 0.00040762 | -0.9776 | 3.3753 | 0.00010373
$\bar{C}^{u}(Q^{2}\geq 0.2)$ | central | 0.4924 | 0.7259 | 0.00016296 | 0.2126 | 0.7298 | 0.00023901
$\bar{C}^{u}(Q^{2}\geq 0.2)$ | upper | 0.5856 | 0.6157 | 0.00006947 | 0.2393 | 0.6349 | 0.00023820
$\bar{C}^{u}(Q^{2}\geq 0.2)$ | lower | 0.4732 | 0.8198 | 0.00013749 | 0.1892 | 0.8969 | 0.00029638
$\bar{C}^{d}(Q^{2}\geq 0.2)$ | central | -0.4257 | 0.6935 | 0.00011733 | -0.1839 | 0.7011 | 0.00015369
$\bar{C}^{d}(Q^{2}\geq 0.2)$ | upper | -0.5426 | 0.6545 | 0.00003208 | -0.1531 | 0.7846 | 0.00017627
$\bar{C}^{d}(Q^{2}\geq 0.2)$ | lower | -0.3988 | 0.7485 | 0.00010404 | -0.2154 | 0.6509 | 0.00014935

##  Appendix B HO integrals used for calculating $C_{\textsf{\scriptsize
q}}(Q^{2})$ and $\overline{C}_{\textsf{\scriptsize q}}(Q^{2})$

The recurrence relation for the HO wavefunction is as follows Wiecki:2014ola :

$\displaystyle
k~{}\phi_{n,m}(\vec{k}_{\perp})=b\begin{cases}\sqrt{n+|m|+1}\phi_{n,m+1}(\vec{k}_{\perp})-H(n-1)\sqrt{n}\phi_{n-1,m+1}(\vec{k}_{\perp})&;m\geq
0\\\
\sqrt{n+|m|}\phi_{n,m+1}(\vec{k}_{\perp})-\sqrt{n+1}\phi_{n+1,m+1}(\vec{k}_{\perp})&;m<0\end{cases}$
(32)

$\displaystyle
k^{*}~{}\phi_{n,m}(\vec{k}_{\perp})=b\begin{cases}\sqrt{n+|m|+1}\phi_{n,m-1}(\vec{k}_{\perp})-H(n-1)\sqrt{n}\phi_{n-1,m-1}(\vec{k}_{\perp})&;m<0\\\
\sqrt{n+|m|}\phi_{n,m-1}(\vec{k}_{\perp})-\sqrt{n+1}\phi_{n+1,m-1}(\vec{k}_{\perp})&;m\geq
0\end{cases}$ (33)

where $k$ is the complex representation of $\vec{k}_{\perp}$ such that
$k=\vec{k}_{\perp}^{(1)}+i\vec{k}_{\perp}^{(2)}$ and the HO wavefuntion is as
defined in Eq. 3. $H(n)$ is the unit step function. The following integrals
were utilized when evaluating the GFFs $C_{\textsf{\scriptsize q}}(Q^{2})$ and
$\overline{C}_{\textsf{\scriptsize q}}(Q^{2})$ and they are derived form the
recurrence relation shown above.

$\displaystyle\mathcal{I}_{1}$
$\displaystyle=\int_{-\infty}^{\infty}d^{2}k_{\perp}~{}k_{\perp}^{(1)}\times\phi^{*}_{n,m}(\vec{k}_{\perp},b_{1})\phi_{n^{\prime},m^{\prime}}(\vec{k^{\prime}}_{\perp},b_{2})=\frac{1}{2}\Big{(}\mathcal{OI}_{1}(x,b_{1},n,m,b_{2},n^{\prime},m^{\prime},q^{(1)},q^{(2)})$
$\displaystyle+~{}\mathcal{OI}_{1}^{*}(x,b_{1},n,m,b_{2},n^{\prime},m^{\prime},q^{(1)},q^{(2)})\Big{)},$

$\displaystyle\mathcal{I}_{2}$
$\displaystyle=\int_{-\infty}^{\infty}k_{\perp}^{(2)}\times\phi^{*}_{n,m}(\vec{k}_{\perp},b_{1})\phi_{n^{\prime},m^{\prime}}(\vec{k^{\prime}}_{\perp},b_{2})=-\frac{i}{2}\Big{(}\mathcal{OI}_{1}(x,b_{1},n,m,b_{2},n^{\prime},m^{\prime},q^{(1)},q^{(2)})$
$\displaystyle-~{}\mathcal{OI}_{1}^{*}(x,b_{1},n,m,b_{2},n^{\prime},m^{\prime},q^{(1)},q^{(2)})\Big{)},$

$\displaystyle\mathcal{I}_{12}$
$\displaystyle=\int_{-\infty}^{\infty}k_{\perp}^{(1)}k_{\perp}^{(2)}\times\phi^{*}_{n,m}(\vec{k}_{\perp},b_{1})\phi_{n^{\prime},m^{\prime}}(\vec{k^{\prime}}_{\perp},b_{2})=-\frac{i}{4}\Big{(}\mathcal{OI}_{2}(x,b_{1},n,m,b_{2},n^{\prime},m^{\prime},q^{(1)},q^{(2)})$
$\displaystyle-~{}\mathcal{OI}_{2}^{*}(x,b_{1},n,m,b_{2},n^{\prime},m^{\prime},q^{(1)},q^{(2)})\Big{)},$

$\displaystyle\mathcal{I}_{11}$
$\displaystyle=\int_{-\infty}^{\infty}k_{\perp}^{(1)}k_{\perp}^{(1)}\times\phi^{*}_{n,m}(\vec{k}_{\perp},b_{1})\phi_{n^{\prime},m^{\prime}}(\vec{k^{\prime}}_{\perp},b_{2})=\frac{1}{4}\Big{(}\mathcal{OI}_{2}(x,b_{1},n,m,b_{2},n^{\prime},m^{\prime},q^{(1)},q^{(2)})$
$\displaystyle+~{}\mathcal{OI}_{2}^{*}(x,b_{1},n,m,b_{2},n^{\prime},m^{\prime},q^{(1)},q^{(2)})+2~{}\mathcal{OI}_{12}(x,b_{1},n,m,b_{2},n^{\prime},m^{\prime},q^{(1)},q^{(2)})\Big{)},$

$\displaystyle\mathcal{I}_{22}$
$\displaystyle=\int_{-\infty}^{\infty}k_{\perp}^{(2)}k_{\perp}^{(2)}\times\phi^{*}_{n,m}(\vec{k}_{\perp},b_{1})\phi_{n^{\prime},m^{\prime}}(\vec{k^{\prime}}_{\perp},b_{2})=-\frac{1}{4}\Big{(}\mathcal{OI}_{2}(x,b_{1},n,m,b_{2},n^{\prime},m^{\prime},q^{(1)},q^{(2)})$
$\displaystyle+~{}\mathcal{OI}_{2}^{*}(x,b_{1},n,m,b_{2},n^{\prime},m^{\prime},q^{(1)},q^{(2)})-2~{}\mathcal{OI}_{12}(x,b_{1},n,m,b_{2},n^{\prime},m^{\prime},q^{(1)},q^{(2)})\Big{)},$

where $\vec{k}_{\perp}^{{}^{\prime}}=\vec{k}_{\perp}+(1-x)q_{\perp}$ and
$k_{\perp}^{(j)}$ with $j=(1,2)$ are the transverse components of
$\vec{k}_{\perp}$ such that
$\left(k_{\perp}^{(1)}\right)^{2}+\left(k_{\perp}^{(2)}\right)^{2}=\left(k_{\perp}\right)^{2}$.
We take $b_{1}=b_{2}=b$. The functions $\mathcal{OI}$ are defined as follows:

$\displaystyle\mathcal{OI}_{1}(x,b_{1},n,m,b_{2},n^{\prime},m^{\prime},q^{(1)},q^{(2)})=$
$\displaystyle
b\sqrt{x}\times\begin{cases}\sqrt{n^{\prime}+|m^{\prime}|+1}\times\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime},m^{\prime}+1)-H(n^{\prime}-1)\times\sqrt{n^{\prime}}\times\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime}-1,m^{\prime}+1)&\text{if
}m^{\prime}\geq 0\\\
\sqrt{n^{\prime}+|m^{\prime}|}\times\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime},m^{\prime}+1)-\sqrt{n^{\prime}+1}\times\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime}+1,m^{\prime}+1)&\text{otherwise}\end{cases}$

$\displaystyle\mathcal{OI}^{*}_{1}(x,b_{1},n,m,b_{2},n^{\prime},m^{\prime},q^{(1)},q^{(2)})=$
$\displaystyle
b\sqrt{x}\times\begin{cases}\sqrt{n^{\prime}+|m^{\prime}|+1}\times\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime},m^{\prime}-1)-H(n^{\prime}-1)\times\sqrt{n^{\prime}}\times\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime}-1,m^{\prime}-1)&\text{if
}m^{\prime}\leq 0\\\
\sqrt{n^{\prime}+|m^{\prime}|}\times\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime},m^{\prime}-1)-\sqrt{n^{\prime}+1}\times\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime}+1,m^{\prime}-1)&\text{otherwise}\end{cases}$

$\displaystyle\mathcal{OI}_{2}(x,b_{1},n,m,b_{2},n^{\prime},m^{\prime},q^{(1)},q^{(2)})=b^{2}~{}x~{}f(n,m,n^{\prime},m^{\prime}).$
(39) $\displaystyle f(n,m,n^{\prime},m^{\prime})$
$\displaystyle=\begin{aligned}
&\sqrt{n^{\prime}+|m^{\prime}|+1}\bigg{(}\sqrt{n^{\prime}+|m^{\prime}+1|+1}\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime},m^{\prime}+2)\\\
&-H(n^{\prime}-1)\sqrt{n^{\prime}}\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime}-1,m^{\prime}+2)\bigg{)}\\\
&-H(n^{\prime}-1)\sqrt{n^{\prime}}\bigg{(}\sqrt{n^{\prime}-1+|m^{\prime}+1|+1}\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime}-1,m^{\prime}+2)\\\
&-H(n^{\prime}-2)\sqrt{n^{\prime}-1}\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime}-2,m^{\prime}+2)\bigg{)}\end{aligned}$
$\displaystyle\text{if }m^{\prime}\geq 0$ (40) $\displaystyle
f(n,m,n^{\prime},m^{\prime})$ $\displaystyle=\begin{aligned}
&\sqrt{n^{\prime}+|m^{\prime}|}\bigg{(}\sqrt{n^{\prime}+|m^{\prime}+1|+1}\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime},m^{\prime}+2)\\\
&-H(n^{\prime}-1)\sqrt{n^{\prime}}\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime}-1,m^{\prime}+2)\bigg{)}\\\
&-\sqrt{n^{\prime}+1}\bigg{(}\sqrt{n^{\prime}+1+|m^{\prime}+1|+1}\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime}+1,m^{\prime}+2)\\\
&-H(n^{\prime})\sqrt{n^{\prime}+1}\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime},m^{\prime}+2)\bigg{)}\end{aligned}$
$\displaystyle\text{if }m^{\prime}=-1$ (41) $\displaystyle
f(n,m,n^{\prime},m^{\prime})$ $\displaystyle=\begin{aligned}
&\sqrt{n^{\prime}+|m^{\prime}|}\bigg{(}\sqrt{n^{\prime}+|m^{\prime}+1|}\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime},m^{\prime}+2)\\\
&-\sqrt{n^{\prime}+1}\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime}+1,m^{\prime}+2)\bigg{)}\\\
&-\sqrt{n^{\prime}+1}\bigg{(}\sqrt{n^{\prime}+1+|m^{\prime}+1|}\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime}+1,m^{\prime}+2)\\\
&-\sqrt{n^{\prime}+2}\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime}+2,m^{\prime}+2)\bigg{)}\end{aligned}$
otherwise (42)

$\displaystyle\mathcal{OI}_{2}^{*}(x,b_{1},n,m,b_{2},n^{\prime},m^{\prime},q^{(1)},q^{(2)})=b^{2}~{}x~{}f^{*}(n,m,n^{\prime},m^{\prime}).$
(43) $\displaystyle f^{*}(n,m,n^{\prime},m^{\prime})$
$\displaystyle=\begin{aligned}
&\sqrt{n^{\prime}+|m^{\prime}|+1}\bigg{(}\sqrt{n^{\prime}+|m^{\prime}-1|+1}\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime},m^{\prime}-2)\\\
&-H(n^{\prime}-1)\sqrt{n^{\prime}}\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime}-1,m^{\prime}-2)\bigg{)}\\\
&-H(n^{\prime}-1)\sqrt{n^{\prime}}\bigg{(}\sqrt{n^{\prime}+|m^{\prime}-1|}\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime}-1,m^{\prime}-2)\\\
&-H(n^{\prime}-2)\sqrt{n^{\prime}-1}\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime}-2,m^{\prime}-2)\bigg{)}\end{aligned}$
$\displaystyle\text{if }m^{\prime}\leq 0$ (44) $\displaystyle
f^{*}(n,m,n^{\prime},m^{\prime})$ $\displaystyle=\begin{aligned}
&\sqrt{n^{\prime}+|m^{\prime}|}\bigg{(}\sqrt{n^{\prime}+|m^{\prime}-1|+1}\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime},m^{\prime}-2)\\\
&-H(n^{\prime}-1)\sqrt{n^{\prime}}\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime}-1,m^{\prime}-2)\bigg{)}\\\
&-\sqrt{n^{\prime}+1}\bigg{(}\sqrt{n^{\prime}+1+|m^{\prime}-1|+1}\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime}+1,m^{\prime}-2)\\\
&-H(n^{\prime})\sqrt{n^{\prime}+1}\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime},m^{\prime}-2)\bigg{)}\end{aligned}$
$\displaystyle\text{if }m^{\prime}=1$ (45) $\displaystyle
f^{*}(n,m,n^{\prime},m^{\prime})$ $\displaystyle=\begin{aligned}
&\sqrt{n^{\prime}+|m^{\prime}|}\bigg{(}\sqrt{n^{\prime}+|m^{\prime}-1|}\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime},m^{\prime}-2)\\\
&-\sqrt{n^{\prime}+1}\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime}+1,m^{\prime}-2)\bigg{)}\\\
&-\sqrt{n^{\prime}+1}\bigg{(}\sqrt{n^{\prime}+1+|m^{\prime}-1|}\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime}+1,m^{\prime}-2)\\\
&-\sqrt{n^{\prime}+2}\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime}+2,m^{\prime}-2)\bigg{)}\end{aligned}$
otherwise (46)

$\displaystyle\mathcal{OI}_{12}(x,b_{1},n,m,b_{2},n^{\prime},m^{\prime},q^{(1)},q^{(2)})=b^{2}~{}x~{}g(n,m,n^{\prime},m^{\prime}).$
(47) $\displaystyle g(n,m,n^{\prime},m^{\prime})$
$\displaystyle=\begin{aligned}
&\sqrt{n^{\prime}+|m^{\prime}|+1}\bigg{(}\sqrt{n^{\prime}+|m^{\prime}+1|}\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime},m^{\prime})\\\
&-\sqrt{n^{\prime}+1}\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime}+1,m^{\prime})\bigg{)}\\\
&-H(n^{\prime}-1)\sqrt{n^{\prime}}\bigg{(}\sqrt{n^{\prime}-1+|m^{\prime}+1|}\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime}-1,m^{\prime})\\\
&-\sqrt{n^{\prime}}\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime},m^{\prime})\bigg{)}\end{aligned}$
$\displaystyle\text{if }m^{\prime}\geq 0$ (48) $\displaystyle
g(n,m,n^{\prime},m^{\prime})$ $\displaystyle=\begin{aligned}
&\sqrt{n^{\prime}+|m^{\prime}|}\bigg{(}\sqrt{n^{\prime}+|m^{\prime}+1|+1}\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime},m^{\prime})\\\
&-H(n^{\prime}-1)\sqrt{n^{\prime}}\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime}-1,m^{\prime})\bigg{)}\\\
&-\sqrt{n^{\prime}+1}\bigg{(}\sqrt{n^{\prime}+1+|m^{\prime}+1|+1}\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime}+1,m^{\prime})\\\
&-H(n^{\prime})\sqrt{n^{\prime}+1}\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}(n,m,n^{\prime},m^{\prime})\bigg{)}\end{aligned}$
otherwise (49)

The function $\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}$ represents the
overlap of the HO wavefunction and its approximate analytic expression is
shown in the following sub-section.

###  B.1 The HO overlap

$\displaystyle H(n)$ $\displaystyle:\text{unit step function}$ (50)
$\displaystyle\theta$
$\displaystyle=\tan^{-1}\left(\frac{b_{2}}{b_{1}}\right)$ (51) $\displaystyle
l_{i}$
$\displaystyle=q^{(i)}\times\sin(2\theta)\times\frac{\sqrt{b_{1}^{2}+b_{2}^{2}}}{2}\quad(i=1,2)$
(52) $\displaystyle l$ $\displaystyle=\sqrt{l_{1}^{2}+l_{2}^{2}}$ (53)
$\displaystyle\mathcal{C}(b_{1},b_{2},\tilde{q})$
$\displaystyle=\begin{cases}1&\text{if }b_{1}=b_{2}\text{ and }\tilde{q}=0\\\
\cos^{\tilde{q}}(2\theta)&\text{otherwise}\end{cases}$ (54)

We represent by
$\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}\left(n_{1},m_{1},n_{2},m_{2}\right)$
the integral of the overlap between the harmonic oscillator wavefunction such
that

$\displaystyle\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}\left(n_{1},m_{1},n_{2},m_{2}\right)=\int_{-\infty}^{\infty}d^{2}k_{\perp}~{}\phi^{*}_{n_{1},m_{1}}(\vec{k}_{\perp},b_{1})\phi_{n_{2},m_{2}}(\vec{k^{\prime}}_{\perp},b_{2}),$
(55)

where $\vec{k}_{\perp}^{{}^{\prime}}=\vec{k}_{\perp}+(1-x)q_{\perp}$.

The approximate analytic expression for the above overlap can be represented
in the following form:

$\displaystyle\mathcal{OL}^{b_{1},b_{2}}_{q^{(1)},q^{(2)}}\left(n_{1},m_{1},n_{2},m_{2}\right)$
$\displaystyle=$ $\displaystyle
H(n_{1})H(n_{2})\times\left(\frac{b_{1}b_{2}}{b_{1}^{2}+b_{2}^{2}}\right)\times\sqrt{2n_{1}!(n_{1}+|m_{1}|)!2n_{2}!(n_{2}+|m_{2}|)!}\times(-1)^{n_{1}+n_{2}}\times$
(56)
$\displaystyle\mathcal{OL}^{\prime}\left(\frac{1}{b_{1}},n_{1},m_{1},\frac{1}{b_{2}},n_{2},m_{2},q^{(1)},q^{(2)}\right).$

For the case $l\neq 0$ we have

$\mathcal{OL}^{\prime}\left(b_{1},n_{1},m_{1},b_{2},n_{2},m_{2},q^{(1)},q^{(2)}\right)=\exp\left(-\frac{l^{2}}{2}\right)\sum_{r=r_{\text{min}}}^{r_{\text{max}}}\left(\sum_{v=v_{\text{min}}}^{v_{\text{max}}}\left(\sum_{w=w_{\text{min}}}^{w_{\text{max}}}\left(\sum_{z=z_{\text{min}}}^{z_{\text{max}}}\mathcal{EV}_{0}(r,v,w,z)\right)\right)\right),$
(57)

$\displaystyle\mathcal{EV}_{0}(r,v,w,z)$ $\displaystyle=$
$\displaystyle(-1)^{\tilde{q}-u+t}\cdot
l^{s+t}\cdot\frac{1}{p!}\cdot\binom{\tilde{q}+r+s+t}{\tilde{q},r,s,t}\cdot\binom{\tilde{q}}{u}\cdot\binom{r}{v}\cdot\binom{s}{w}\cdot\binom{t}{z}$
$\displaystyle\cdot\mathcal{C}(b_{1},b_{2},\tilde{q})\cdot\sin^{r}(2\theta)\cdot\sin^{s}(\theta)\cdot\cos^{t}(\theta)\cdot
e^{-i\cdot(s-2w+t-2z)\cdot\tan^{-1}\left(l_{2}/l_{1}\right)},$

$\displaystyle r_{\text{min}}$ $\displaystyle=0,$ $\displaystyle
r_{\text{max}}$
$\displaystyle=n_{1}+n_{2}+\frac{|m_{1}|-m_{1}}{2}+\frac{|m_{2}|+m_{2}}{2},$
(59) $\displaystyle v_{\text{min}}$
$\displaystyle=\max\left(-n_{2}-\frac{|m_{2}|+m_{2}}{2}+r,0\right),$
$\displaystyle v_{\text{max}}$
$\displaystyle=\min\left(n_{1}+\frac{|m_{1}|-m_{1}}{2},r\right),$
$\displaystyle w_{\text{min}}$ $\displaystyle=\max\left(0,r-m_{1}-2v\right),$
$\displaystyle w_{\text{max}}$
$\displaystyle=\min\left(r_{\text{max}}-r,n_{1}+\frac{|m_{1}|-m_{1}}{2}-v\right),$
$\displaystyle z_{\text{min}}$
$\displaystyle=\begin{cases}\max\left(m_{2}+2v-r,0,r_{\text{max}}-r-w\right)&\text{if
}b_{1}=b_{2}\\\ \max\left(m_{2}+2v-r,0\right)&\text{otherwise}\end{cases},$
$\displaystyle z_{\text{max}}$
$\displaystyle=\min\left(r_{\text{max}}-r-w,n_{2}+\frac{|m_{2}|+m_{2}}{2}-r+v\right),$
$\displaystyle s$ $\displaystyle=m_{1}+2v+2w-r,$ $\displaystyle t$
$\displaystyle=-m_{2}-2v+2z+r,$ $\displaystyle\tilde{q}$
$\displaystyle=n_{1}+n_{2}+\frac{|m_{1}|-m_{1}}{2}+\frac{|m_{2}|+m_{2}}{2}-r-w-z,$
$\displaystyle u$ $\displaystyle=n_{1}+\frac{|m_{1}|-m_{1}}{2}-v-w,$
$\displaystyle p$ $\displaystyle=\tilde{q}+r+s+t.$

For the case $l=0$ we have

$\displaystyle\mathcal{OL}^{\prime}\left(b_{1},n_{1},m_{1},b_{2},n_{2},m_{2},q^{(1)},q^{(2)}\right)=\sum_{v=0}^{v_{\mathrm{max}}}(-1)^{\tilde{q}-u}\times\frac{1}{p!}\times\mathcal{C}(b_{1},b_{2},\tilde{q})\times\sin(2\theta)^{r}\binom{p}{\tilde{q}}\times\binom{r}{v}\times\binom{\tilde{q}}{u},$
(60)

where

$\displaystyle p$ $\displaystyle=\tilde{q}+r$ $\displaystyle\tilde{q}$
$\displaystyle=n_{1}+n_{2}-2v$ $\displaystyle r$ $\displaystyle=|m_{1}|+2v$
$\displaystyle u$ $\displaystyle=n_{1}-v$ $\displaystyle v_{\mathrm{max}}$
$\displaystyle=\min\left(\frac{n_{1}+n_{2}}{2},n_{1}\right)$

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