BLFQ Collaboration

# Generalized parton distributions and spin structures of light mesons from a
light-front Hamiltonian approach

Lekha Adhikari<EMAIL_ADDRESS>Department of Physics and Astronomy, Iowa
State University, Ames, IA 50011, U.S.A.  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  CAS Key Laboratory of High Precision
Nuclear Spectroscopy, Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  Sreeraj Nair<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  CAS Key Laboratory of High Precision Nuclear
Spectroscopy, Institute of Modern Physics, Chinese Academy of Sciences,
Lanzhou 730000, 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  CAS Key Laboratory of High Precision Nuclear Spectroscopy,
Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000,
China  Shaoyang Jia<EMAIL_ADDRESS>Department of Physics and Astronomy, Iowa
State University, Ames, IA 50011, U.S.A.  Physics Division, Argonne National
Laboratory, Argonne, IL 60439, U.S.A.  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  CAS Key Laboratory of High Precision
Nuclear Spectroscopy, Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  James P. Vary<EMAIL_ADDRESS>Department
of Physics and Astronomy, Iowa State University, Ames, IA 50011, U.S.A.

###### Abstract

We present the generalized parton distributions (GPDs) for the valence quarks
of the pion and the kaon in both momentum space and position space within the
basis light-front quantization framework. These GPDs are obtained from the
eigenvectors of a light-front effective Hamiltonian consisting of the
holographic quantum chromodynamics (QCD) confinement potential, a
complementary longitudinal confinement potential, and the color-singlet Nambu-
Jona–Lasinio interactions for the valence quarks of mesons. We then calculate
the generalized form factors of the pion and the kaon from the moments of
these GPDs. Combining the tensor form factors with the electromagnetic form
factors, we subsequently evaluate the impact parameter dependent probability
density of transversely polarized quarks inside the pion and the kaon. The
numerical results for the generalized form factors, corresponding charges, as
well as those for the probability densities and the transverse shift of the
polarized densities are consistent with lattice QCD simulations and with
chiral quark models.

##  I Introduction

The description of the nonperturbative structure of hadrons using generalized
parton distributions (GPDs) is related to phenomenology, therefore attracting
numerous dedicated experimental and theoretical efforts Ji:PRL ;
Polyakov:1999gs ; Diehl:2003ny ; Belitsky:2005qn ; Goeke:2001tz ; Choi:2001fc
; Kaur:2018ewq ; deTeramond:2018ecg ; Ji:2006ea ; Fanelli:2016aqc ;
Chouika:2017rzs ; Mezrag:2014jka ; Broniowski:2007si ; Zhang:2021mtn ;
Zhang:2021shm ; Hwang:2007tb ; Mondal:2017wbf ; Chakrabarti:2014cwa ;
Adhikari:2016idg ; Adhikari:2018umb ; Meissner:2007rx ; Alexandrou:2020zbe ;
Chen:2019lcm ; Dupre:2016mai ; Kriesten:2021sqc . These GPDs are
experimentally accessible through exclusive processes including deeply virtual
Compton scattering (DVCS) and deeply virtual meson production (DVMP). The GPDs
present an attractive testing ground for comparing theory with experiment
since they encode a wealth of information about the spatial structure of the
hadron as well as the partonic distribution of spin and orbital angular
momenta. Unlike the parton distribution functions (PDFs), which are solely
functions of longitudinal momentum fraction ($x$) carried by the active
parton, the GPDs are functions of $x$, the skewness ($\zeta$) which represents
the longitudinal momentum transfer, and the square of total momentum transfer
($t$) to the hadrons.

The GPDs provide a picture that unites PDFs with form factors (FFs), where the
former describe the longitudinal momentum distribution of partons within a
hadron while the latter characterize the spatial extent. One obtains the FFs,
charge distributions, PDFs, etc. from the GPDs by marginalizing guidal-ff ;
nikkhoo-ff ; miller-charge . Additionally, in the absence of the longitudinal
momentum transfer ($\zeta=0$), the GPDs are converted to the impact parameter
dependent parton distributions via Fourier transform with respect to the
transverse momentum transfer. Unlike the GPDs themselves, the impact parameter
dependent parton distribution is the probability density of partons at a given
combination of the longitudinal momentum fraction and the transverse distance
from the center of the hadron Burkardt:2002hr ; Burkardt:2000za ;
Ralston:2001xs ; Broniowski:2003rp . For different polarizations of the
partons, spin densities can be expressed in terms of the polarized impact
parameter dependent GPDs Brommel:2007xd ; Gockeler:2006zu ; Pasquini:2007xz ;
Maji:2017ill ; Diehl:2005jf .

For many years, DVCS and DVMP data have been accumulated by J-PARC, Hall-A and
Hall-B of JLab by the CLAS collaboration and by COMPASS at CERN gpd-exp ; gpd-
exp1 ; gpd-exp2 ; gpd-exp3 ; gpd-exp4 ; gpd-exp6 ; gpd-exp7 . Recently, JLab
has also started a positron initiated DVCS experiment Accardi:2020swt ,
COMPASS at CERN will start to collect more DVCS data, while future Electron-
Ion Colliders AbdulKhalek:2021gbh ; Anderle:2021wcy  are planned to explore
the GPDs through DVCS. However, experimental extractions of the GPDs are not
straightforward. In particular, fitting of DVCS data does not provide direct
information about the GPDs but, instead, provides some weighted integrals of
the GPDs. Since nonperturbative QCD predictions are not yet possible from the
first principles, model predictions of the GPDs are useful for constraining
the GPDs and data fitting in order to develop insights into GPDs from DVCS
data.

Among known hadrons, the pion plays a leading role for comparing theory with
experiment. From the Drell-Yan process with pion beams Drell:1970wh ;
Christenson:1970um , we can access the partonic structure of the pion by
colliding them with nuclear targets Peng:2014hta ; Chang:2013opa ;
Reimer:2007iy ; McGaughey:1999mq . Chiral symmetry is dynamically broken in
QCD leading to generation of the Goldstone bosons (pions) having a small mass
when compared to other hadrons. On the one hand, the pions are salient in
providing the force that binds the neutrons and the protons inside the nuclei
and they also affect the properties of the isolated nucleons. Hence one can
safely say that our understanding of visible (baryonic) matter is incomplete
without detailed knowledge of the structure and interactions of the pion. On
the other hand, the pseudoscalar kaons, counterparts of the pions with one
strange valence quark, play a critical role in our understanding of Charge and
Parity (CP) symmetry violation Woods:1988za ; Barr:1993rx ; Gibbons:1993zq .
In this paper, we investigate the partonic structure of the pions and the
kaons in terms of their GPDs. As background, we note that different
theoretical analyses have provided useful insights regarding the pion GPDs,
_e.g_. Refs. Broniowski:2003rp ; Polyakov:1999gs ; Frederico:2009fk ;
Kaur:2018ewq ; Theussl:2002xp ; Dalley:2003sz ; Broniowski:2007si ;
Mezrag:2014jka ; Fanelli:2016aqc ; Kumano:2017lhr ; Ma:2019agv ; Zhang:2020ecj
; Shi:2020pqe ; Gutsche:2015 ; Gutsche:2013zia ; deTeramond:2018ecg ;
Chang:2020kjj ; Brommel:2007xd ; Hagler:2009ni ; dalley ; dalley1 ;
Chen:2019lcm ; Zhang:2021mtn ; Roberts:2021nhw ; Kaur:2020vkq ; Raya:2021zrz ,
while for the kaon, foundations are just being laid and several significant
analyses can be found in Refs. Kaur:2020vkq ; Nam:2011yw ; Xu:2018eii ;
Kock:2020frx ; Kaur:2019jow ; Zhang:2021mtn ; Zhang:2021tnr ; Raya:2021zrz .

Another salient issue is the transversity of the hadrons Barone:2001sp , which
provides access to their spin structures. Due to transversity’s chiral-odd
nature, it is challenging to measure experimentally. Nevertheless, the
transverse spin asymmetry in Drell-Yan processes in $p\bar{p}$ reactions
Anselmino:2004ki ; Pasquini:2006iv  and the azimuthal single spin asymmetry in
semi-inclusive deep inelastic scattering (SIDIS) Anselmino:2007fs  can be used
to extract valuable information on the transversity of the nucleon. While the
transversity of the nucleon is nonzero and has now been well determined
Radici:2018iag , it vanishes for the spin-zero hadrons. However, the chiral-
odd GPDs defined as off-forward matrix elements of the tensor current are
nonzero and much less information is available for them in the case of the
pion and the kaon.

From the perspective of theory, the QCDSF/UKQCD Collaboration has reported the
first result for the pion’s chiral-odd GPD using lattice QCD Brommel:2007xd .
They have also presented the probability density of the polarized quarks
inside the pion and found that their spatial distribution is strongly
distorted when the quarks are transversely polarized. The distortion in the
density occurs due to the pion tensor FF. The lattice QCD results have
triggered various theoretical studies on the pion and the kaon tensor FFs. The
models for such results include constituent quark models Frederico:2009fk ;
Fanelli:2016aqc , the Nambu–Jona-Lasinio (NJL) model with Pauli-Villars
regularization Broniowski:2010nt ; Dorokhov:2011ew , and the nonlocal chiral
quark model (N$\chi$QM) from the instanton vacuum Nam:2010pt ; Nam:2011yw .

In this paper, we evaluate the GPDs of the light pseudoscalar mesons using the
light-front wave functions (LFWFs) based on the theoretical framework of basis
light front quantization (BLFQ) Vary:2009gt , with only the valence Fock
sector of mesons considered. The effective Hamiltonian incorporates the
confining potential adopted from the light-front holography in the transverse
direction Brodsky:2014yha , a longitudinal confinement Li:2015zda ; Li:2017mlw
, and the color-singlet NJL interactions Klimt:1989pm ; Shigetani:1993dx  to
account for the dynamical chiral symmetry breaking of QCD. The nonperturbative
solutions for the LFWFs are given by the recent BLFQ study of light mesons
Jia:2018ary . These LFWFs have been applied successfully to predict the decay
constants, electromagnetic form factors (EMFFs), charge radii, PDFs, and many
other quantities of the pion and the kaon Jia:2018ary ; Lan:2019rba ;
Lan:2019vui ; Mondal:2021czk . Here, we extend those investigations to study
the pion and the kaon GPDs and their QCD evolution. We use the Dokshitzer-
Gribov-Lipatov-Altarelli-Parisi (DGLAP) equation of QCD Dokshitzer:1977sg ;
Gribov:1972ri ; Altarelli:1977zs  up to the next-to-next-to-leading order
(NNLO) for the evolution of the valence quark GPDs. We also calculate the pion
and the kaon tensor FFs in the space-like region. Combining the result of the
tensor FFs with the EMFFs, which have been evaluated previously in Ref.
Jia:2018ary  within the BLFQ-NJL framework, we then compute the probability
density of transversely polarized quarks inside the pion and the kaon. We
further calculate the $x$-dependent squared radius of the quark density in the
transverse plane that describes the transverse size of the hadron.

We organize the main results of this paper in the following sequence. We
briefly summarize the BLFQ-NJL formalism for the light mesons in Sec. II. We
then present a detailed description of the GPDs and the associated
distributions in Sec. III. Sec. IV details our numerical results for the GPDs,
electromagnetic and gravitational FFs, impact parameter dependent GPDs, and
spin densities of the pion and the kaons. We summarize the outcomes in Sec. V.

##  II BLFQ-NJL model for the light mesons

In this section, we provide an overview of the BLFQ-NJL model for the light
mesons following Ref. Jia:2018ary . The BLFQ approach represents the dynamics
of bound state constituents in quantum field theory through a light-front
quantum many-body Hamiltonian Vary:2009gt ; Zhao:2014xaa ; Wiecki:2014ola ;
Li:2015zda ; Jia:2018ary ; Tang:2018myz ; Tang:2019gvn ; Xu:2019xhk ;
Lan:2021wok . The structures of the bound states are encoded in the LFWFs
achievable as the eigenfunctions of the light-front eigenvalue equation

$H_{\mathrm{eff}}|\Psi\rangle=M^{2}|\Psi\rangle,$ (1)

where $H_{\mathrm{eff}}=P^{+}P^{-}$ with $P^{\pm}=P^{0}\pm P^{3}$ being the
light-front Hamitonian ($P^{-}$) and the longitudinal momentum ($P^{+}$) of
the system, respectively. The mass squared, $M^{2}$, is the corresponding
eigenvalue of the state $|\Psi\rangle$. In the constituent quark-antiquark
representation, our adopted effective light-front Hamiltonian for the light
mesons with non-singlet flavor wave functions is written as

$\displaystyle H_{\mathrm{eff}}=$ $\displaystyle\frac{\vec{k}^{\perp
2}+m_{q}^{2}}{x}+\frac{\vec{k}^{\perp
2}+m_{\bar{q}}^{2}}{1-x}+\kappa^{4}\vec{\zeta}^{\perp 2}$
$\displaystyle-\frac{\kappa^{4}}{(m_{q}+m_{\bar{q}})^{2}}\partial_{x}\big{(}x(1-x)\partial_{x}\big{)}+H^{\rm
eff}_{\rm NJL}.$ (2)

The first two terms in Eq. (2) are the light-front kinetic energy for the
quark and the antiquark, where $m_{q}$ ($m_{\bar{q}}$) is the mass of the
quark (antiquark), $x=k^{+}/P^{+}$ is the longitudinal momentum fraction
carried by the valence quark, and $\vec{k}^{\perp}$ is its transverse
momentum. The third and the fourth terms are respectively the confining
potential in the transverse direction based on the light-front holographic QCD
Brodsky:2014yha  and a longitudinal confining potential Li:2015zda . The
parameter $\kappa$ is the strength of the confinement. The holographic
variable is defined as $\vec{\zeta}^{\perp}\equiv\sqrt{x(1-x)}\vec{r}^{\perp}$
Brodsky:2014yha , where $\vec{r}^{\perp}$ is the transverse separation between
the quark and antiquark and is conjugated to $\vec{k}^{\perp}$. The
$x$-derivative is defined as $\partial_{x}f(x,\vec{\zeta}^{\perp})=\partial
f(x,\vec{\zeta}^{\perp})/\partial x|_{\vec{\zeta}^{\perp}}$. The last term in
the effective Hamiltonian, $H_{\mathrm{NJL}}^{\mathrm{eff}}$, represents the
color-singlet NJL interaction to account for the chiral dynamics Klimt:1989pm
.

For the positively-charged pion, the NJL interaction is given by Jia:2018ary ,

$\displaystyle
H_{\mathrm{NJL},\pi}^{\mathrm{eff}}=G_{\pi}\,\big{\\{}\overline{u}_{\mathrm{u}s1^{\prime}}(p_{1}^{\prime})u_{\mathrm{u}s1}(p_{1})\,\overline{v}_{\mathrm{d}s2}(p_{2})v_{\mathrm{d}s2^{\prime}}(p_{2}^{\prime})$
$\displaystyle\quad\quad+\overline{u}_{\mathrm{u}s1^{\prime}}(p_{1}^{\prime})\gamma_{5}u_{\mathrm{u}s1}(p_{1})\,\overline{v}_{\mathrm{d}s2}(p_{2})\gamma_{5}v_{\mathrm{d}s2^{\prime}}(p_{2}^{\prime})$
$\displaystyle\quad\quad+2\,\overline{u}_{\mathrm{u}s1^{\prime}}(p_{1}^{\prime})\gamma_{5}v_{\mathrm{d}s2^{\prime}}(p_{2}^{\prime})\,\overline{v}_{\mathrm{d}s2}(p_{2})\gamma_{5}u_{\mathrm{u}s1}(p_{1})\big{\\}}.$
(3)

While, for the positively charged kaon, the interaction is given by

$\displaystyle
H^{\mathrm{eff}}_{\mathrm{NJL},K}=G_{K}\,\big{\\{}-2\,\overline{u}_{\mathrm{u}s1^{\prime}}(p_{1}^{\prime})v_{\mathrm{s}s2^{\prime}}(p_{2}^{\prime})\,\overline{v}_{\mathrm{s}s2}(p_{2})u_{\mathrm{u}s1}(p_{1})$
$\displaystyle\quad\quad+2\,\overline{u}_{\mathrm{u}s1^{\prime}}(p_{1}^{\prime})\gamma_{5}v_{\mathrm{s}s2^{\prime}}(p_{2}^{\prime})\,\overline{v}_{\mathrm{s}s2}(p_{2})\gamma_{5}u_{\mathrm{u}s1}(p_{1})\big{\\}}.$
(4)

Equations (3) and (4) are obtained from the NJL Lagrangian after the Legendre
transform in the two and three flavor NJL model, respectively Klimt:1989pm ;
Vogl:1989ea ; Vogl:1991qt ; Klevansky:1992qe . Here, ${u_{\mathrm{f}s}(p)}$
and ${v_{\mathrm{f}s}(p)}$ are the Dirac spinors with the nonitalic subscripts
representing the flavors and the italic subscripts denoting the spins.
Meanwhile, $p_{1}$ and $p_{2}$ are the momenta of the valence quark and the
valence antiquark, respectively. The coefficients $G_{\pi}$ and $G_{K}$ are
independent coupling constants of the theory. In the interactions, we only
include the combinations of Dirac bilinears relevant to the valence Fock
sector LFWFs of the systems. The instantaneous terms due to the NJL
interactions have been omitted. The explicit expressions and the detailed
calculations of the matrix elements of the NJL interactions in the BLFQ
formalism can be found in Ref. Jia:2018ary .

In the leading Fock sector, the eigenstate for the mesons reads

$\displaystyle\big{|}\Psi(P^{+},\vec{P}^{\perp})\big{\rangle}=\sum_{r,s}\int_{0}^{1}\dfrac{dx}{4\pi
x(1-x)}\int\dfrac{d\vec{\kappa}^{\perp}}{(2\pi)^{2}}\,$
$\displaystyle\quad\quad\times\,\psi_{rs}(x,\vec{\kappa}^{\perp})\,b_{r}^{\dagger}(xP^{+},\vec{\kappa}^{\perp}+x\vec{P}^{\perp})$
$\displaystyle\quad\quad\times\,d_{s}^{\dagger}((1-x)P^{+},-\vec{\kappa}^{\perp}+(1-x)\vec{P}^{\perp})\,|0\rangle,$
(5)

where $P$ is the momentum of the meson. The relative transverse momentum of
the valence quark is $\vec{\kappa}^{\perp}=\vec{k}^{\perp}-x\vec{P}^{\perp}$.
The coefficients of the expansion, $\psi_{rs}(x,\vec{\kappa}^{\perp})$, are
the valence sector LFWFs with $r$($s$) representing the spin of the
quark(antiquark). To compute the Hamiltonian matrix, one needs to construct
the BLFQ basis. The two-dimensional (2D) harmonic oscillator (HO) basis
functions are adopted in the transverse direction, which are defined as
Vary:2009gt ; Li:2015zda :

$\displaystyle\phi_{nm}\left(\vec{q}^{\perp};b_{h}\right)$
$\displaystyle=\dfrac{1}{b_{h}}\sqrt{\dfrac{4\pi
n!}{(n+|m|)!}}\left(\dfrac{|\vec{q}^{\perp}|}{b_{h}}\right)^{|m|}$
$\displaystyle\times\,\exp\left(-\dfrac{\vec{q}^{\perp
2}}{2b_{h}^{2}}\right)L_{n}^{|m|}\left(\dfrac{\vec{q}^{\perp
2}}{b_{h}^{2}}\right)\,e^{im\varphi},$ (6)

with $\tan(\varphi)=q^{2}/q^{1}$, $b_{h}$ is the HO basis scale parameter with
dimension of mass, $n$ and $m$ are the radial and the angular quantum numbers,
$L_{n}^{|m|}(z)$ is the associated Laguerre polynomial. Meanwhile, in the
longitudinal direction, the basis functions are defined as Li:2015zda

$\displaystyle\chi_{l}(x;\alpha,\beta)$
$\displaystyle=\sqrt{4\pi(2l+\alpha+\beta+1)}$
$\displaystyle\times\,\sqrt{\dfrac{\Gamma(l+1)\Gamma(l+\alpha+\beta+1)}{\Gamma(l+\alpha+1)\Gamma(l+\beta+1)}}$
$\displaystyle\times\,x^{\beta/2}(1-x)^{\alpha/2}\,P_{l}^{(\alpha,\beta)}(2x-1),$
(7)

where $P_{l}^{(\alpha,\beta)}(z)$ is the Jacobi polynomial and the
dimensionless parameters
${\alpha=2m_{\overline{q}}(m_{q}+m_{\overline{q}})/\kappa^{2}}$,
${\beta=2m_{q}(m_{q}+m_{\overline{q}})/\kappa^{2}}$ and $l=0,~{}1,~{}2,...$.
The valence LFWFs are then expanded in the orthonormal bases given in Eqs. (6)
and (7):

$\displaystyle\psi_{rs}(x,\vec{\kappa}^{\perp})$
$\displaystyle=\sum_{n,m,l}\langle n,m,l,r,s|\psi\rangle~{}$
$\displaystyle\times\,\phi_{nm}\left(\dfrac{\vec{\kappa}^{\perp}}{\sqrt{x(1-x)}};b_{h}\right)\chi_{l}(x),$
(8)

where the coefficients $\langle n,m,l,r,s|\psi\rangle$ are obtained in the
BLFQ basis space by diagonalizing the truncated Hamiltonian matrix. The
infinite dimensional basis is truncated to a finite dimension by restricting
the quantum numbers using

$0\leq n\leq N_{\mathrm{max}},\quad-2\leq m\leq 2,\quad 0\leq l\leq
L_{\mathrm{max}},$ (9)

where $N_{\text{max}}$ controls the transverse momentum covered by 2D HO
functions and $L_{\text{max}}$ provides the basis resolution in the
longitudinal direction. Note that we have a natural truncation for $m$ as the
NJL interactions do not couple to $|m|\geq 3$ basis states Jia:2018ary . The
LFWF $\psi_{rs}(x,\vec{\kappa}^{\perp})$ is normalized as

$\displaystyle\sum_{r,s}\int_{0}^{1}\\!\\!\frac{dx}{2x(1-x)}\\!\int\\!\frac{d^{2}\vec{\kappa}^{\perp}}{(2\pi)^{3}}\big{|}\psi_{rs}(x,\vec{\kappa}^{\perp})\big{|}^{2}\\!\\!=\\!\\!1.$
(10)

Parameters in the BLFQ-NJL model are fixed to reproduce the ground state
masses of the light pseudoscalar and vector mesons as well as the experimental
charge radii of the $\pi^{+}$ and the $K^{+}$ Jia:2018ary . The LFWFs in this
model have been successfully applied to compute the parton distribution
amplitudes and the EMFFs Jia:2018ary , PDFs for the pion and the kaon and
pion-nucleus induced Drell-Yan cross sections Lan:2019vui ; Lan:2019rba .

Table 1: Summary of the model parameters Jia:2018ary .  Valence flavor | $N_{\text{max}}$ | $L_{\text{max}}$ | $\kappa(\text{MeV})$ | $m_{q}(\text{MeV})$ | $m_{\bar{q}}(\text{MeV})$ |
---|---|---|---|---|---|---
$u\bar{d}$ | 8 | 8$-$32 | 227 | 337 | 337 |
$u\bar{s}$ | 8 | 8$-$32 | 276 | 308 | 445 |

---
(a)

---
(b)

Figure 1: The valence ($u$ or $\bar{d}$) quark GPDs of the pion: (a)
$H(x,0,t)$ and (b) $E_{T}(x,0,t)$ as functions of $x$ and the invariant
momentum transfer $-t$. The GPDs are evaluated with $N_{\text{max}}=8$ and
$L_{\text{max}}=32$ in the BLFQ-NJL model.

---
(a)

---
(b)

---
(c)

---
(d)

Figure 2: The valence quark GPDs of the kaon: (a) $H(x,0,t)$ and (c)
$E_{T}(x,0,t)$ are for the valence $u$ quark; (b) and (d) are same as (a) and
(c), respectively, but for the valence $\bar{s}$ quark as functions of $x$ and
the invariant momentum transfer $-t$. These GPDs are evaluated with
$N_{\text{max}}=8$ and $L_{\text{max}}=32$ in the BLFQ-NJL model.

##  III generalized parton distributions: KINEMATICS AND FORMALISM

At leading twist, there are two independent GPDs for a spin-0 meson. One of
them is chirally even, while the other is chirally odd. Those GPDs are defined
as off-forward matrix elements of the bilocal operator of light-front
correlation functions of vector and tensor currents, respectively as
Diehl:2003ny ; Hagler:2009ni ,

$\displaystyle
H^{\mathcal{P}}(x,\zeta,t)=\int\frac{dz^{-}}{4\pi}e^{ixP^{+}z^{-}/2}$
$\displaystyle\quad\times\,\langle\mathcal{P}(P^{\prime})|\bar{\Psi}_{q}(0)\gamma^{+}\Psi_{q}(z)|\mathcal{P}(P)\rangle|_{z^{+}=\textbf{z}^{\perp}=0},$
(11)
$\displaystyle\frac{i\epsilon_{ij}^{\perp}q_{i}^{\perp}}{2M_{\mathcal{P}}}E_{T}^{\mathcal{P}}(x,\zeta,t)=\int\frac{dz^{-}}{4\pi}e^{ixP^{+}z^{-}/2}$
$\displaystyle\quad\times\,\langle\mathcal{P}(P^{\prime})|\bar{\Psi}_{q}(0)i\sigma^{j+}\gamma_{5}\Psi(z)_{q}|\mathcal{P}(P)\rangle|_{z^{+}=\textbf{z}^{\perp}=0},$
(12)

where $\Psi_{q}(z)$ is the quark field operator and $P(P^{\prime})$ denotes
the meson momentum of initial (final) state of the meson $(\mathcal{P})$.
$M_{\mathcal{P}}$ defines the mass of the meson; $\epsilon_{ij}^{\perp}$ is
the anti-symmetric tensor in the transverse plane and
$\sigma^{j+}=\frac{i}{2}[\gamma^{j},\gamma^{+}]$ with $j=1,~{}2$ as transverse
index. The $H$, called the unpolarized quark GPD, is chirally even, while the
transversely polarized quark GPD, $E_{T}$, is chirally odd. The GPD $E_{T}$ is
responsible for the distortion in the spatial distribution of a transversely
polarized quark, revealing a nontrivial spin structure of the meson
Brommel:2007xd . The moments of the GPD $E_{T}$ can be linked to the Boer-
Mulders function, which describes the correlation between transverse spin and
intrinsic transverse momentum of the quark in the meson Burkardt:2003uw ;
Burkardt:2002ks ; Ahmady:2019yvo . Recently, the limits of validity of this
relationship have been discussed in Ref. Pasquini:2019evu . In the symmetric
frame, the kinematical variables are

$\bar{P}^{\mu}=\frac{(P+P^{\prime})^{\mu}}{2},~{}\Delta^{\mu}=P^{\prime\mu}-P^{\mu},~{}\zeta=-\Delta^{+}/2\bar{P}^{+},$
(13)

and $t=\Delta^{2}$. Here, we choose the light cone gauge $A^{+}=0$ implying
that the gauge link between the quark fields in Eqs. (11) and (12) is unity
therefore omitted.

By inserting the initial and the final states of the meson, Eq. (5), in above
Eqs. (11) and (12), one obtains the quark GPDs $H$ and $E_{T}$ in terms of
overlaps of LFWFs. We restrict ourselves to the kinematical region: $0<x<1$ at
zero skewness. This domain corresponds to the situation where a quark is
removed from the initial meson with light-front longitudinal momentum $xP^{+}$
and reinserted into the final meson with the same longitudinal momentum.
Therefore, the change in momentum occurs purely in the transverse direction.
The particle number $(n_{p})$ remains conserved in this kinematical domain
describing the diagonal $n_{p}\to n_{p}$ overlaps. The GPDs, $H$ and $E_{T}$,
at zero skewness, in the diagonal $2\to 2$ overlap representation in terms of
LFWFs are given by

$\displaystyle H(x,\zeta=0,t)$
$\displaystyle=\dfrac{1}{4\pi\,x(1-x)}\sum_{rs}\int\dfrac{d^{2}\vec{\kappa}^{\perp}}{(2\pi)^{2}}\,$
$\displaystyle\times\,\psi^{*}_{rs}(x,\vec{\kappa}^{\prime\perp})\,\psi_{rs}(x,\vec{\kappa}^{\perp}),$
(14)
$\displaystyle\frac{i\Delta^{\perp}_{j}}{2M_{\mathcal{P}}}E_{T}(x,\zeta=0,t)$
$\displaystyle=\dfrac{1}{4\pi\,x(1-x)}\sum_{s}\int\dfrac{d^{2}\vec{\kappa}^{\perp}}{(2\pi)^{2}}\,$
$\displaystyle\times\,\Big{[}(-i)^{j}\psi^{*}_{\uparrow
s}(x,\vec{\kappa}^{\prime\perp})\,\psi_{\downarrow s}(x,\vec{\kappa}^{\perp})$
$\displaystyle+(i)^{j}\psi^{*}_{\downarrow
s}(x,\vec{\kappa}^{\prime\perp})\,\psi_{\uparrow
s}(x,\vec{\kappa}^{\perp})\Big{]},$ (15)

where, for the struck quark,
$\vec{\kappa}^{\prime\perp}=\vec{\kappa}^{\perp}+(1-x){\vec{\Delta}}^{\perp}$
and for the spectator,
$\vec{\kappa}^{\prime\perp}=\vec{\kappa}^{\perp}-x{\vec{\Delta}}^{\perp}$ and
the total momentum transferred to the meson is $t=-{\vec{\Delta}}^{\perp 2}$.

Note that integrating the bilocal matrix element in Eq. (11) over the momentum
fraction $x$ yields the local matrix elements that provide FFs. In the Drell-
Yan frame, the expressions for the GPDs are very similar to those for FFs,
except that the longitudinal momentum fraction $x$ of the struck parton is not
integrated out. Therefore, GPDs defined in Eqs. (14) and (15) are also known
as momentum-dissected FFs and measure the contribution of the struck parton
with momentum fraction $x$ to the corresponding FFs. Consequently, the first
moments of the GPDs can be related to the FFs for the spin-0 hadrons by the
sum rules on the light-front as Ji:1998pc

$\displaystyle F(t)$ $\displaystyle=\int dx\,H(x,\zeta,t),$ $\displaystyle
F_{T}(t)$ $\displaystyle=\int dx\,E_{T}(x,\zeta,t).$ (16)

Meanwhile, the gravitational FFs which are expressed as the matrix elements of
the energy-momentum tensor, are linked to GPDs through the second-moment as
Ji:1998pc

$\displaystyle A(t)$ $\displaystyle=\int dx\,x\,H(x,\zeta,t),$ $\displaystyle
B_{T}(t)$ $\displaystyle=\int dx\,x\,E_{T}(x,\zeta,t).$ (17)

Aside from these FFs, the impact parameter dependent GPDs are defined as the
Fourier transform of the GPDs with respect to the momentum transfer along the
transverse direction $\vec{\Delta}^{\perp}$ Burkardt:2002hr :

$\displaystyle q(x,{\vec{b}}^{\perp})$
$\displaystyle=\int\frac{d^{2}{\vec{\Delta}}^{\perp}}{(2\pi)^{2}}e^{-i{\vec{\Delta}}^{\perp}\cdot{\vec{b}}^{\perp}}H(x,0,-{\vec{\Delta}}^{\perp
2}),$ (18) $\displaystyle q_{T}(x,{\vec{b}}^{\perp})$
$\displaystyle=\int\frac{d^{2}{\vec{\Delta}}^{\perp}}{(2\pi)^{2}}e^{-i{\vec{\Delta}}^{\perp}\cdot{\vec{b}}^{\perp}}E_{T}(x,0,-{\vec{\Delta}}^{\perp
2}),$ (19)

where ${\vec{b}}^{\perp}$ is the Fourier conjugate to the momentum transfer
${\vec{\Delta}}^{\perp}$. The impact parameter
${b}^{\perp}=|{\vec{b}}^{\perp}|$ corresponds to the transverse displacement
of the struck parton from the center of momentum of the hadron. For zero
skewness, $\vec{b}^{\perp}$ provides a measure of the transverse distance of
the struck parton from the center of momentum of the hadron. The variable
$\vec{b}^{\perp}$ follows the condition $\sum_{i}x_{i}\vec{b}^{\perp}_{i}=0$,
where the sum runs over the number of partons. The relative distance between
the center of momentum of the spectator and the struck parton is
${{b}^{\perp}/(1-x})$, therefore providing an estimate of the transverse size
of the hadron Diehl:2003ny .

Following the standard formulation Miller:2007uy , one can further define the
transverse charge density $\rho({\vec{b}}^{\perp})$ by

$\displaystyle\rho({\vec{b}}^{\perp})$
$\displaystyle=\int\frac{d^{2}{\vec{\Delta}}^{\perp}}{(2\pi)^{2}}e^{-i{\vec{\Delta}}^{\perp}\cdot{\vec{b}}^{\perp}}F(-{\vec{\Delta}}^{\perp
2})$ $\displaystyle=\int_{0}^{1}dx\,q(x,{\vec{b}}^{\perp}),$ (20)

while the longitudinal momentum density for a given transverse seperation is
given by Abidin:2008sb ; Chakrabarti:2015lba ; Mondal:2015fok ; Kumar:2017dbf
; Mondal:2016xsm

$\displaystyle p({\vec{b}}^{\perp})$
$\displaystyle=\int\frac{d^{2}{\vec{\Delta}}^{\perp}}{(2\pi)^{2}}e^{-i{\vec{\Delta}}^{\perp}\cdot{\vec{b}}^{\perp}}A(-{\vec{\Delta}}^{\perp
2})$ $\displaystyle=\int_{0}^{1}dx\,x\,q(x,{\vec{b}}^{\perp}).$ (21)

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(b)

Figure 3: Scale evolution of the valence ($u$ or $\bar{d}$) quark GPDs of the
pion: (a) $H(x,0,t)$ and (b) $E_{T}(x,0,t)$ as functions of $x$ and fixed
value of $-t=0.11$ GeV2. The GPDs are evaluated with $N_{\text{max}}=8$ and
$L_{\text{max}}=32$ within the BLFQ-NJL model. The GPDs are evolved from our
model scale for the pion $\mu_{0\pi}^{2}=0.240$ GeV2 to the final scales
$\mu^{2}=1,~{}10,~{}100$ GeV2.

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Figure 4: Scale evolution of the valence quark GPDs of the kaon: (a)
$H(x,0,t)$ and (c) $E_{T}(x,0,t)$ are for the valence $u$ quark: (b) and (d)
are same as (a) and (c), respectively, but for the valence $\bar{s}$ quark as
functions of $x$ and fixed value of $-t=0.11$ GeV2. The GPDs are evaluated
with $N_{\text{max}}=8$ and $L_{\text{max}}=32$ within the BLFQ-NJL model. The
GPDs are evolved from our model scale for the kaon $\mu_{0K}^{2}=0.246$ GeV2
to the final scales $\mu^{2}=1,~{}10,~{}100$ GeV2.

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Figure 5: The first two Mellin moments of the valence quark GPDs of the pion:
(a) $-tA^{\pi,u(\bar{d})}_{n0}(t)$ and (b) $-tB^{\pi,u(\bar{d})}_{Tn0}(t)$ for
$n=1$ (black lines) and $n=2$ (blue lines) as functions of $-t$. The
electromagnetic form factor $A_{10}(t)$ of the pion is compared with the
experimental data Amendolia:1986wj ; Bebek:1974iz ; Bebek:1974ww ;
Bebek:1977pe ; Volmer:2000ek ; Horn:2006tm  and the lattice QCD result
Brommel:2006ww . The gravitational FF $A_{20}(t)$ is compared with the
parameterization of lattice QCD simulations at $\mu^{2}=4$ GeV2, while
$B_{T10}(t)$ and $B_{T20}(t)$ are compared with lattice QCD and the $\chi$QM
results at the same scale $\mu^{2}=4$ GeV2. The lines with circle and triangle
symbols correspond to the results calculated in the BLFQ-NJL model (present
work). The dashed ($n=1$) and dotted ($n=2$) lines represent the lattice QCD
results Brommel:2007xd , whereas the dash-dotted ($n=1$) and solid ($n=2$)
lines in (b) represent the $\chi$QM Nam:2010pt  results. The experimental
results in (a) are for the EMFF only.

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Figure 6: The first two Mellin moments of the valence quark GPDs of the kaon:
(a) $-tA^{K,u(\bar{s})}_{n0}(t)$, (b) $-tA^{K}_{n0}(t)$, and (c)
$-tB^{K,u(\bar{s})}_{Tn0}(t)$ for $n=1$ (black lines) and $n=2$ (blue lines)
as functions of $-t$. The electromagnetic form factor $A_{10}(t)$ of the kaon
in (b) is compared with the experimental data Dally:1980dj ; Amendolia:1986ui
. The inset in (b) provides an expanded view of the EMFF at low $-t$. The
tensor FFs in (c) $B_{T10}(t)$ and $B_{T20}(t)$ are compared with the $\chi$QM
Nam:2011yw  at $\mu^{2}=4$ GeV2. The lines with circle and triangle symbols
correspond to the results calculated in the BLFQ-NJL model (this work). The
lines without symbols in (c) represent the $\chi$QM results. The experimental
results in (b) are for the EMFF only.

##  IV numerical results and discussion

###  IV.1 GPDs and generalised form factors

The LFWFs of the valence quarks in the pion and the kaon Jia:2018ary  have
been solved in the BLFQ framework using the NJL interactions as briefly
discussed in the Subsection II. We insert the valence wave functions given by
Eq. (8) into Eqs. (14) and (15) to calculate the GPDs for the pion and the
kaon. We employ the wave functions obtained at the basis truncation
$N_{\text{max}}=8$ and $L_{\text{max}}=32$ with other model parameters given
in Table 1. We illustrate the valence GPDs $H^{q}$ and $E_{T}^{q}$ ($q\equiv
u$ or $\bar{d}$) as functions of $x$ and $-t$ for the pion in Fig. 1. In the
forward limit ($-t=0$), the unpolarized GPD $H$ reduces to the ordinary PDF,
which peaks at $x=0.5$ for the pion, reflecting the symmetry between the
valence quark and the valence antiquark. Unlike the unpolarized GPD $H$, the
chiral-odd GPD, $E_{T}$ in the pion has its peak located below the central
value of $x$ and is asymmetric under $x\leftrightarrow(1-x)$ even when $-t=0$.
This is due to the fact that $E_{T}$ involves the overlaps of the
wavefunctions with different orbital angular momentum $L_{z}=0$ and $L_{z}=\pm
1$. The peaks of these GPDs shift towards higher values of $x$ and the
magnitudes of distributions decrease with increasing the value of $-t$.

The valence quark GPDs for the kaon are shown in Fig. 2. The up quark GPD
$H(x,0,t)$ in the kaon, unlike the valence quark GPD $H(x,0,t)$ in the pion,
has the maximum at lower $x$ ($<0.5$) when $t=0$, whereas, due to its heavy
mass, the peak in the strange quark distribution appears at higher $x$
($>0.5$). Meanwhile, the peaks along $x$ get shifted to larger values of $x$
with increasing $-t$ similar to that observed in the pion GPD. This seems to
be a model independent behavior of the GPDs which has been noticed in other
phenomenological models for the pion Kaur:2018ewq  as well as for the nucleon
Chakrabarti:2013gra ; Mondal:2015uha ; Chakrabarti:2015ama ; Xu:2021wwj . We
also notice that the GPD $E_{T}$ for the up quark in the kaon exhibits a
behavior similar to that observed in the pion, however, the magnitude of
$E_{Tu}$ in the kaon is larger than that in the pion. Meanwhile, $E_{Ts}$
displays a different behavior compared to $E_{Tu}$ in the kaon: $E_{Ts}$ is
broader along $x$ and falls slower at large $x$ compared to $E_{Tu}$. As $-t$
increases, $E_{Tu}$ also falls faster than $E_{Ts}$ in the kaon. One can also
observe oscillations in the GPDs along $x$ in Fig. 1 and Fig. 2, which are
numerical artifacts due to longitudinal cutoff $L_{\rm max}$. The amplitudes
of the oscillations decrease with increasing $L_{\rm max}$ Lan:2019rba .

By performing the QCD evolution, the valence quark GPDs at high $\mu^{2}$
scale can be obtained with the input GPDs at the model scale $\mu_{0}^{2}$. We
adopt the DGLAP equations Dokshitzer:1977sg ; Gribov:1972ri ; Altarelli:1977zs
of QCD with NNLO for this scale evolution. Explicitly, we evolve our input
GPDs to the relevant experimental scales with independently adjustable initial
scales of the pion and the kaon GPDs utilizing the higher order perturbative
parton evolution toolkit (HOPPET) Salam:2008qg . We adopt
${\mu_{0\pi}^{2}=0.240\pm 0.024~{}\rm{GeV}^{2}}$ for the initial scale of the
pion GPDs and ${\mu_{0K}^{2}=0.246\pm 0.024~{}\rm{GeV}^{2}}$ for the initial
scale of the kaon GPDs which we determined by requiring the results after NNLO
DGLAP evolution to fit both the pion and the kaon PDFs results from the
experiments Lan:2019rba . We show the valence quark GPDs in the pion and the
kaon for a fixed value of $-t$ at different $\mu^{2}$ evolved from the
corresponding initial scales in Fig. 3 and Fig. 4, respectively. We observe
that the peaks of the distributions move to lower $x$ as we evolve the GPDs to
higher scales. The moments of the distributions decrease uniformly as the
scale $\mu^{2}$ increases. The qualitative behavior of the evolved GPDs is
similar in both the pion and the kaon.

The Mellin moments of the valence GPDs give the generalized FFs. The Mellin
moments are defined as Brommel:2007xd

$\displaystyle A^{q}_{n0}(t)=\int_{0}^{1}~{}dx\,x^{n-1}\,H^{q}(x,0,t),$ (22)
$\displaystyle B^{q}_{Tn0}(t)=\int_{0}^{1}~{}dx\,x^{n-1}\,E_{T}^{q}(x,0,t),$
(23)

where the index $n=1,2,3\dots$, and the second subscript corresponds to the
fact that the moments are evaluated at zero skewness ($\zeta=0$). The first
moments of the unpolarized GPD ${H}^{q}(x,0,t)$ give the electromagnetic FF,
$F^{q}(t)=A^{q}_{10}(t)$ of an unpolarized quark, while in the forward limit,
i.e, $t=0$, the FF $F^{q}(0)$ gives the number of the valence quarks of flavor
$q$. The first moment of chiral-odd GPD ${E}^{q}_{T}(x,0,t)$ provides the
tensor FF $B_{T}^{q}(t)$ when the quark is transversely polarized. The second
moments of these GPDs correspond to the gravitational FFs of the quarks.
Meanwhile, the third moments of the GPDs provide the FFs of a twist-two
operator having two covariant derivatives Diehl:2003ny ; Belitsky:2005qn  and
the higher order moments produce the FFs of higher-twist operators.

Table 2: BLFQ-NJL model predictions for $A^{\pi(K),q}_{20}(0)$, $B^{\pi(K),q}_{T10}(0)$ and $B^{\pi(K),q}_{T20}(0)$ at the scale 4 GeV2. We compare our results with the available lattice QCD simulations Brommel:2007xd , the $\chi$QMs Nam:2010pt ; Nam:2011yw , and the CCQM Fanelli:2016aqc  at the same scale 4 GeV2. The errors in our results correspond to the QCD evolution from the initial scales ${\mu_{0\pi}^{2}=0.240\pm 0.024~{}\rm{GeV}^{2}}$ for the pion and ${\mu_{0K}^{2}=0.246\pm 0.024~{}\rm{GeV}^{2}}$ for the kaon. Quantity |  BLFQ-NJL |  Lattice QCD Brommel:2007xd  |  $\chi$QM Nam:2010pt  |  $\chi$QM Nam:2011yw  |  $\chi$QM Nam:2011yw  |  CCQM Fanelli:2016aqc  |
---|---|---|---|---|---|---|---
|  (this work) |  |  |  (model I) |  (model II) |  |
$A^{\pi,q}_{20}(0)$ | $0.244\pm 0.018$ | $0.27\pm 0.01$ | … | … | … | $0.248$ |
$A^{K,u}_{20}(0)$ | $0.235\pm 0.018$ | … | … | … | … | … |
$A^{K,\bar{s}}_{20}(0)$ | $0.265\pm 0.020$ | … | … | … | … | … |
$B^{\pi,q}_{T10}(0)$ | $0.229\pm 0.004$ | $0.216\pm 0.034$ | $0.216$ | … | … | $0.126$ |
$B^{K,u}_{T10}(0)$ | $0.821\pm 0.014$ | … | … | $0.783$ | $0.611$ | … |
$B^{K,\bar{s}}_{T10}(0)$ | $0.706\pm 0.010$ | … | … | $0.676$ | $0.421$ | … |
$B^{\pi,q}_{T20}(0)$ | $0.045\pm 0.004$ | $0.039\pm 0.010$ | $0.032$ | … | … | $0.028$ |
$B^{K,u}_{T20}(0)$ | $0.152\pm 0.011$ | … | … | $0.139$ | $0.090$ | … |
$B^{K,\bar{s}}_{T20}(0)$ | $0.152\pm 0.011$ | … | … | $0.100$ | $0.076$ | … |

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Figure 7: The valence ($u$ or $\bar{d}$) quark GPDs for the pion in the
transverse impact parameter space: (a) $q(x,{\vec{b}}^{\perp})$ and (b)
$q_{T}(x,{\vec{b}}^{\perp})$ as functions of $x$ and $b^{\perp}$, where
$b^{\perp}$ is the transverse distance of the active quark from the center of
momentum of the hadron.

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(b)

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Figure 8: The valence quark GPDs of the kaon in the transverse impact
parameter space: (a) $q(x,{\vec{b}}^{\perp})$ and (c)
$q_{T}(x,{\vec{b}}^{\perp})$ are for the valence $u$ quark; (b) and (d) are
same as (a) and (c), respectively, but for the valence $\bar{s}$ quark as
functions of $x$ and $b^{\perp}$, where $b^{\perp}$ is the transverse distance
of the active quark from the center of momentum of the hadron.

In Fig. 5(a), we present the first two moments of the GPD ${H}^{q}(x,0,t)$ of
the pion. The EMFF of the pion is given by
$F^{\pi}(t)=e_{u}A^{\pi,u}_{10}(t)+e_{\bar{d}}A^{\pi,\bar{d}}_{10}(t)$, where
$e_{q}$ denotes the charge of the quark $q$. We find that the pion EMFF within
our BLFQ-NJL model is in good agreement with the experimental data and with
the lattice QCD simulations. The second moment of the GPD ${H}^{q}(x,0,t)$ is
the gravitational FF $A^{q}_{20}(t)$, which, at $t=0$, provides the momentum,
$\langle x\rangle_{q}$ carried by the quark. For the pion
$A^{\pi,u}_{20}(0)=A^{\pi,\bar{d}}_{20}(0)=0.5$ at the model scale. To compare
with lattice QCD, we evolve the GPD to the relevant scale. As summarized in
Table 2, we obtain that at $\mu^{2}=4$ GeV2, $A^{\pi,q}_{20}(0)=0.244\pm
0.018$, which is compatible with the result from the covariant constituent
quark model (CCQM) model Fanelli:2016aqc , while the lattice QCD provides the
value of $0.27\pm 0.01$ Brommel:2007xd . In addition, substantial difference
between our BLFQ-NJL model and lattice QCD for $A^{\pi,q}_{20}(t)$ is observed
when $-t$ is nonzero with disagreement increasing as $-t$ increases, as can be
seen in Fig. 5(a). We show the tensor FFs of the pion in Fig. 5(b), where we
also compare the FFs $B^{\pi,q}_{T10}(t)$ and $B^{\pi,q}_{T20}(t)$ with the
lattice QCD results evaluated at the physical pion mass Brommel:2007xd . At
$\mu^{2}=4$ GeV2, we obtain: $B_{T10}^{\pi,q}(0)=0.229\pm 0.004$ and
$B_{T20}^{\pi,q}(0)=0.045\pm 0.004$, which reasonably agree with the lattice
QCD simulations within the uncertainty: $B_{T10}^{\pi,q}(0)=0.216\pm 0.034$
and $B_{T20}^{\pi,q}(0)=0.039\pm 0.010$, respectively. It is notable that
$B_{T10}^{\pi,q}(0)$ in the CCQM Fanelli:2016aqc  differs significantly from
our result. The qualitative behavior of the tensor FFs $B^{\pi,q}_{T10}(t)$
and $B^{\pi,q}_{T20}(t)$ is also found to be comparable with the lattice QCD
calculations and the chiral quark model ($\chi$QM) Nam:2010pt  as shown in
Fig. 5(b).

Fig. 6 shows the moments of the kaon GPDs. As can be seen from Fig. 6(a), the
magnitude of $-tA^{K,u}_{n0}(t)$ is lower than that for $\bar{s}$ quark,
implying the faster fall-off of the $u$ quark EMFF compared to the $\bar{s}$
quark in the kaon as $-t$ increases. The EMFF of the kaon,
$F^{K}(t)=e_{u}A^{K,u}_{10}(t)+e_{\bar{s}}A^{K,\bar{s}}_{10}(t)$, is in good
agreement with the experimental data as shown in Fig. 6(b). This is expected
because model parameters of the BLFQ-NJL model are partially determined based
on the experimental charge radii. On the other hand, we obtain
$A^{K,u}_{20}(0)=0.43$ and $A^{K,\bar{s}}_{20}(0)=0.57$ at the model scale,
whereas at $\mu^{2}=4$ GeV2, the corresponding values are
$A^{K,u}_{20}(0)=0.235\pm 0.018$ and $A^{K,\bar{s}}_{20}(0)=0.265\pm 0.020$ as
summarized in Table 2. We also illustrate the $t$ dependence of the kaon
gravitational FFs $A^{K,u}_{20}(t)$ and $A^{K,\bar{s}}_{20}(t)$ at $\mu^{2}=4$
GeV2 in Fig. 6(a). The tensor FFs for the kaon in our BLFQ-NJL model are
presented in Fig. 6(c), in comparison with that of the $\chi$QM calculations
(model I in Ref. Nam:2011yw ). The qualitative behavior of
$-tB^{K,q}_{Tn0}(t)$ in those models agree. At large $-t$, $-tB_{Tn0}(t)$ for
the $\bar{s}$ quark is larger than that for the $u$ quark in the BLFQ-NJL
model, while in the $\chi$QM one observes the opposite. We also compare the
quark tensor FFs at $t=0$ in the kaon with the $\chi$QM in Table 2.

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Figure 9: The valence quark probability density in the pion in the transverse
impact parameter plane; (a) when the quark is unpolarized
$\rho({\vec{b}}^{\perp})=\rho^{n=1}({\vec{b}}^{\perp},{\vec{s}}^{\perp}={\vec{0}})$
and (b) when quark is transversely polarized along-$x$ direction
$\rho_{T}({\vec{b}}^{\perp})=\rho^{n=1}({\vec{b}}^{\perp},{\vec{s}}^{\perp}=(1,0))$.
Our corresponding results are compared with lattice QCD and the $\chi$QM model
in (c) and (d), respectively. The solid, dotted and dash-dotted lines
correspond to the BLFQ-NJL model, lattice QCD Brommel:2007xd  and the $\chi$QM
Nam:2010pt , respectively.

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Figure 10: The valence quarks’ probability densities in the kaon in the
transverse impact parameter plane; (a) for unpolarized $u$ quark
$\rho({\vec{b}}^{\perp})=\rho^{n=1}({\vec{b}}^{\perp},{\vec{s}}^{\perp}=\vec{0})$
and (b) for polarized $u$ quark
$\rho_{T}({\vec{b}}^{\perp})=\rho^{n=1}({\vec{b}}^{\perp},{\vec{s}}^{\perp}=(1,0))$.
(c) and (d) are same as (a) and (b), respectively but for $\bar{s}$ quark in
the kaon.

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Figure 11: The valence quarks’ probability densities in the kaon in our BLFQ-
NJL model (solid lines) are compared with $\chi$QM predictions Nam:2011yw
(dashed lines).

###  IV.2 Spin densities of the pion and the kaon

The GPDs in the transverse impact parameter space at zero skewness can be
interpreted as the densities of quarks with longitudinal momentum fraction $x$
and transverse location ${\vec{b}}^{\perp}$ with respect to the center of
momentum of the hadron independent of the polarization. On the one hand, the
density $\rho(x,{\vec{b}}^{\perp},\lambda)$ of quarks with helicity $\lambda$
in the pion (kaon) is determined by the unpolarized density,
$2\rho(x,{\vec{b}}^{\perp},\lambda)=q(x,{\vec{b}}^{\perp})$, where the latter
is the ${\vec{b}}^{\perp}$-dependent GPD at zero skewness given by Eq. (18).
On the other hand, the density of quarks with transverse spin
$\vec{s}^{\perp}$, $\rho(x,{\vec{b}}^{\perp},{\vec{s}}^{\perp})$, in the pion
(kaon) can be expressed in a combination of the GPDs $q(x,{\vec{b}}^{\perp})$
and $q_{T}(x,{\vec{b}}^{\perp})$ as

$\displaystyle\rho(x,{\vec{b}}^{\perp},{\vec{s}}^{\perp})=\frac{1}{2}\left[q(x,{\vec{b}}^{\perp})-\frac{{\vec{s}}^{\perp}_{i}\epsilon_{ij}^{\perp}{\vec{b}}^{\perp}_{j}}{M_{\mathcal{P}}}\,q_{T}^{\prime}(x,{\vec{b}}^{\perp})\,\right]\,,$
(24)

where
$q_{T}^{\prime}(x,{\vec{b}}^{\perp})=\frac{\partial}{\partial(b^{\perp})^{2}}\,q_{T}(x,{\vec{b}}^{\perp})$.
The quark spin densities have been investigated in Refs. Diehl:2005jf ;
Gockeler:2006zu ; Pasquini:2007xz ; Maji:2017ill  for quarks with transverse
spin ${\vec{s}}^{\perp}$ in the nucleon having transverse spin
(${\vec{S}}^{\perp}$). The corresponding expression for transversely polarized
quarks in the pseudoscalar mesons is achieved by setting ${\vec{S}}^{\perp}=0$
in the nucleon densities Gockeler:2006zu . One finds that the result is much
simpler but still involves a dipole term
$\propto{\vec{s}}^{\perp}_{i}\epsilon_{ij}^{\perp}{\vec{b}}^{\perp}_{j}$
leading to a dependence on the direction of ${\vec{b}}^{\perp}$ for fixed
${\vec{s}}^{\perp}$. The $x$-moments of quark spin densities are then given by
Brommel:2007xd

$\displaystyle\rho^{n}({\vec{b}}^{\perp},{\vec{s}}^{\perp})$
$\displaystyle=\int_{0}^{1}dx\,x^{n-1}\rho(x,{\vec{b}}^{\perp},{\vec{s}}^{\perp})$
$\displaystyle=\frac{1}{2}\left[\mathcal{A}^{q}_{n0}({\vec{b}}^{\perp})-\frac{{\vec{s}}^{\perp}_{i}\epsilon_{ij}^{\perp}{\vec{b}}^{\perp}_{j}}{M_{\mathcal{P}}}\,\mathcal{B}_{Tn0}^{q\prime}({\vec{b}}^{\perp})\,\right]\,,$
(25)

where the ${\vec{b}}^{\perp}$-dependent vector and tensor generalized FFs,
$\mathcal{A}^{q}_{n0}$ and $\mathcal{B}^{q}_{Tn0}$, are obtained by performing
the Fourier transform of the FFs $A^{q}_{n0}(t)$ and $B^{q}_{Tn0}(t)$ with
respect to $\vec{\Delta}^{\perp}$ or equivalently by taking the $x$ moments of
the impact parameter dependent GPDs $q(x,{\vec{b}}^{\perp})$ and
$q_{T}(x,{\vec{b}}^{\perp})$, respectively:

$\displaystyle\mathcal{A}^{q}_{n0}({\vec{b}}^{\perp})$
$\displaystyle=\int\frac{d^{2}{\vec{\Delta}}^{\perp}}{(2\pi)^{2}}e^{-i{\vec{\Delta}}^{\perp}\cdot{\vec{b}}^{\perp}}{A}^{q}_{n0}(-{\vec{\Delta}}^{\perp
2})$ $\displaystyle=\int_{0}^{1}dx\,x^{n-1}q(x,{\vec{b}}^{\perp})\,,$
$\displaystyle\mathcal{B}^{q}_{Tn0}({\vec{b}}^{\perp})$
$\displaystyle=\int\frac{d^{2}{\vec{\Delta}}^{\perp}}{(2\pi)^{2}}e^{-i{\vec{\Delta}}^{\perp}\cdot{\vec{b}}^{\perp}}{B}^{q}_{Tn0}(-{\vec{\Delta}}^{\perp
2})$ $\displaystyle=\int_{0}^{1}dx\,x^{n-1}q_{T}(x,{\vec{b}}^{\perp})\,.$ (26)

The impact parameter dependent GPDs $q(x,{\vec{b}}^{\perp})$ and
$q_{T}(x,{\vec{b}}^{\perp})$ for the pion are presented in Fig. 7. We find
that both distributions have sharp peaks located at the center of the pion
($b^{\perp}=0$) when the quark carries large longitudinal momentum.
Nevertheless, the magnitude of the unpolarized distribution is much higher
compared to that of the polarized distribution. A substantial difference is
also observed between $q(x,{\vec{b}}^{\perp})$ and
$q_{T}(x,{\vec{b}}^{\perp})$ at large $x$. We also notice that the qualitative
behavior of the GPDs $q(x,{\vec{b}}^{\perp})$ and $q_{T}(x,{\vec{b}}^{\perp})$
for the kaon, shown in Fig. 8, is very similar to those for the pion. However,
due to the heavier mass of the $\bar{s}$ quark, its distributions are narrower
than those distributions for the $u$ quark in the kaon. Another interesting
feature is that the width of all the GPDs in the transverse impact parameter
space decrease as $x$ increases. This indicates that the distributions are
more localized near the center of the momentum ($b_{\perp}=0$) when quarks are
carrying higher longitudinal momentum. This characteristic of the GPDs in the
transverse impact parameter space is reassuring since the distributions in the
momentum space become broader in $-t$ with increasing $x$, as can be seen from
Figs. 1 and 2. On the light-front, this is understood as the larger the
momentum fraction, the lower the kinetic energy carried by the quarks. As the
total kinetic energy remains limited, the distribution in the transverse
momentum is required to become broader to carry a larger portion of the
kinetic energy. This model-independent property of the GPDs is also observed
in the case of the nucleon Chakrabarti:2013gra ; Mondal:2015uha ;
Chakrabarti:2015ama ; Maji:2017ill .

We present the first moment of the quark-spin probability density
$\rho^{n=1}({\vec{b}}^{\perp},{\vec{s}}^{\perp})$, defined in Eq. (25), in
Fig. 9. When the quark is unpolarized (${\vec{s}}^{\perp}=\vec{0}$), only
$\mathcal{A}^{q}_{10}({\vec{b}}^{\perp})$ contributes in the probability
density, which is rotationally symmetric in the two-dimensional impact
parameters $(b_{x},b_{y})$ plane as shown in Fig. 9(a) and hence, one does not
see any interesting structures from this. We now turn our attention to the
case when the quark is transversely polarized. Without loss of generality, we
consider the quark polarized along the $x$-axis, i.e.,
${\vec{s}}^{\perp}=(+1,0)$ and show the numerical results as functions of
$b_{x}$ and $b_{y}$. The probability density becomes distorted when the quark
inside the pion is transversely polarized as can be seen from Fig. 9(b)
indicating the spin structure inside the pion. The second term in Eq. (25)
provides the distortion, and one can clearly observe the deviation from
rotational symmetry of the unpolarized density due to the polarization. We
also find that the present results are very similar to those given by the
lattice QCD calculation Brommel:2007xd . For instance, in the lower panel of
Fig. 9 we show the probability densities as a function of $b_{y}$ at fixed
$b_{x}=0.15$ fm, comparing those with that of the lattice QCD simulations and
the $\chi$QM Nam:2010pt . The BLFQ-NJL model results are found to be
consistent with the results of lattice QCD and the $\chi$QM. The spin
densities of the $u$ and $\bar{s}$ quarks in the kaon are shown in Fig. 10,
where we notice the similar patterns of the quark-spin probability densities
as observed in the pion. It is however interesting to note that $\bar{s}$
quark densities, due to the heavier $\bar{s}$ mass, are more localized near
the origin compared to the $u$-quark densities in the kaon. We also find that
the qualitative behavior of the present results in the BLFQ-NJL model is
compatible with the results obtained in the $\chi$QM Nam:2011yw  as shown in
Fig. 11, where we plot the probability densities as a function of $b_{y}$ at
fixed $b_{x}=0.15$ fm.

###  IV.3 Average transverse shift and transverse squared radius

It is also interesting to examine the average transverse shift of the peak
position of the probability density along the $b_{y}$ direction for a
transverse quark spin in the $x$-direction, which is defined as Brommel:2007xd

$\displaystyle\langle b_{y}^{\perp}\rangle_{n}$ $\displaystyle=\frac{\int
d^{2}\vec{b}^{\perp}\,b_{y}^{\perp}\,\rho^{n}({\vec{b}}^{\perp},{\vec{s}}^{\perp})}{\int
d^{2}\vec{b}^{\perp}\,\rho^{n}({\vec{b}}^{\perp},{\vec{s}}^{\perp})}=\frac{1}{2M_{\mathcal{P}}}\,\frac{B^{q}_{Tn0}(0)}{A^{q}_{n0}(0)}.$
(27)

Our BLFQ-NJL model results for the pion give $\langle
b_{y}^{\perp}\rangle_{1}=0.162\pm 0.003$ fm and $\langle
b_{y}^{\perp}\rangle_{2}=0.131\pm 0.003$ fm, while the lattice simulations
provide Brommel:2007xd  $\langle b_{y}^{\perp}\rangle_{1}=0.151(24)$ fm and
$\langle b_{y}^{\perp}\rangle_{2}=0.106(28)$ fm. Our results for $\langle
b_{y}^{\perp}\rangle_{n=1,2}$ for the pion and the kaon are compared with the
lattice QCD, the $\chi$QM, CCQM, and NJL model in Table 3.

Table 3: BLFQ-NJL model predictions for average transverse shift $\langle b_{y}^{\perp}\rangle_{1,2}^{q}$ in the pion and the kaon. Our results are compared with the available lattice simulations Brommel:2007xd , the $\chi$QM Nam:2010pt ; Nam:2011yw , CCQM Fanelli:2016aqc , and NJL model Zhang:2021tnr .  Approach |  $\langle b_{y}^{\perp}\rangle_{1}^{q,\pi}$ fm |  $\langle b_{y}^{\perp}\rangle_{2}^{q,\pi}$ fm |  $\langle b_{y}^{\perp}\rangle_{1}^{u,K}$ fm |  $\langle b_{y}^{\perp}\rangle_{1}^{\bar{s},K}$ fm |  $\langle b_{y}^{\perp}\rangle_{2}^{u,K}$ fm |  $\langle b_{y}^{\perp}\rangle_{2}^{\bar{s},K}$ fm |
---|---|---|---|---|---|---|---
BLFQ-NJL (this work) | $0.162\pm 0.003$ | $0.131\pm 0.003$ | $0.164\pm 0.003$ | $0.141\pm 0.002$ | $0.114\pm 0.002$ | $0.114\pm 0.002$ |
Latiice QCD Brommel:2007xd  | 0.151 $\pm$ 0.024 | 0.106 $\pm$ 0.028 | … | … | … | … |
$\chi$QM Nam:2010pt  | 0.152 | … | … | … | … | … |
$\chi$QM (model I) Nam:2011yw  | … | … | 0.168 | 0.166 | … | … |
$\chi$QM (model II) Nam:2011yw  | … | … | 0.139 | 0.100 | … | … |
CCQM Fanelli:2016aqc  | $0.090\pm 0.001$ | $0.080\pm 0.001$ | … | … | … | … |
NJL model Zhang:2021tnr  | … | … | 0.116 | … | 0.083 | … |
Figure 12: $x$-dependence of $\langle b^{2}_{\perp}\rangle$ for quarks in the
pion (dashed line) and the kaon (solid line) in the BLFQ-NJL model.

One can also define the $x$-dependent squared radius of the quark density in
the transverse plane as Dupre:2016mai :

$\displaystyle\langle b_{\perp}^{2}\rangle^{q}(x)=\frac{\int
d^{2}{\vec{b}^{\perp}}\,({\vec{b}}^{\perp})^{2}q(x,{\vec{b}}^{\perp})}{\int
d^{2}{\vec{b}^{\perp}}\,q(x,{\vec{b}}^{\perp})},$ (28)

which can also be written through the GPD $H(x,0,t)$ as:

$\langle b_{\perp}^{2}\rangle^{q}(x)=4\frac{\partial}{\partial t}\ln
H^{q}(x,0,t)\biggr{|}_{t=0}.$ (29)

For the pion and the kaon the squared radius $\langle b_{\perp}^{2}\rangle(x)$
is obtained as the charge-weighted sum over the valence quarks: $\langle
b_{\perp}^{2}\rangle^{\pi}(x)=e_{u}\langle
b_{\perp}^{2}\rangle^{u}(x)+e_{\bar{d}}\langle
b_{\perp}^{2}\rangle^{\bar{d}}(x)$ and $\langle
b_{\perp}^{2}\rangle^{K}(x)=e_{u}\langle
b_{\perp}^{2}\rangle^{u}(x)+e_{\bar{s}}\langle
b_{\perp}^{2}\rangle^{\bar{s}}(x)$. The $\langle b^{2}_{\perp}\rangle(x)$
describes the transverse size of the hadron and shows an increase of
transverse radius with decreasing value of the quark momentum fraction $x$
Dupre:2016mai . As can be seen from Fig. 12 and as expected, the transverse
size of the kaon is smaller than that of the pion for a fixed value of $x$. We
also compute the pion’s and the kaon’s transverse squared radius through the
following average over $x$ Dupre:2016mai

$\langle
b_{\perp}^{2}\rangle=\sum_{q}e_{q}\frac{1}{N_{q}}\int_{0}^{1}dxH^{q}(x,0,0){\langle
b_{\perp}^{2}\rangle}^{q}(x)\,,$ (30)

with the integrated number of valence quark $N_{q}$ of flavor $q$. We obtain
the squared radius of the pion and the kaon, $\langle
b_{\perp}^{2}\rangle^{\pi}=0.285$ fm2 and $\langle
b_{\perp}^{2}\rangle^{K}=0.223$ fm2, respectively. The quantity $\langle
b_{\perp}^{2}\rangle$ is connected to the conventionally defined squared
radius $\langle r_{c}^{2}\rangle$ from the EMFF by $\langle
b_{\perp}^{2}\rangle=\frac{2}{3}\langle r_{c}^{2}\rangle$ Dupre:2016mai ;
Li:2017mlw . Our results are close to the experimental data for the pion,
$\langle b_{\perp}^{2}\rangle^{\pi}_{\rm exp}=0.301\pm 0.014$ fm2 and for the
kaon, $\langle b_{\perp}^{2}\rangle^{K}_{\rm exp}=0.209\pm 0.047$ fm2
ParticleDataGroup:2018ovx  and also consistent with the previously computed
charge radii of the pion and the kaon in the BLFQ-NJL model Jia:2018ary .

##  V Summary

We have investigated the valence quark GPDs of the light pseudoscalar mesons
in the framework of BLFQ using a light-front model for light mesons that
incorporates light-front holography, longitudinal confinement, and the color-
singlet Nambu-Jona–Lasinio interactions. The parameters in the BLFQ-NJL model
have previously been adjusted to generate the experimental mass spectrum and
the charge radii of the light mesons Jia:2018ary . We have evaluated the quark
unpolarized GPD $H$ and tensor GPD $E_{T}$ in the pion and the kaon in both
momentum and transverse position space. The generalized form factors for the
pion and the kaon, i.e., vector and tensor form factors from the first two
moments of the quark unpolarized and tensor GPDs have been calculated. We have
verified the agreement of the the electromagnetic form factors resulting from
the unpolarized GPD with the experimental data of the pion and the kaon. The
moments of the tensor GPD $E_{T}$, which give the tensor form factors, have
been found to be comparable with the parameterization of lattice QCD
simulations as well as with the results of the $\chi$QM.

We have subsequently calculated the probability densities of the unpolarized
and polarized quarks inside the pion and the kaon. We have observed that the
spatial distribution of the unpolarized quarks is axially symmetric, while it
strongly distorted when quarks are transversely polarized, revealing a non-
trivial distribution of quark polarization in the pseudoscalar mesons. The
quark probability densities in the BLFQ-NJL model have been found to be in
good agreement to those from lattice QCD. The qualitative nature of the quark
densities in the kaon was also consistent with those in the $\chi$QM. In order
to examine the shift of the peaks of the densities in the $b_{y}$ direction,
we have computed the average value of $b_{y}$, which turned out to be
compatible with the lattice QCD and the $\chi$QM.

We have also evaluated the $x$-dependent squared radius of the quark density
in the transverse plane, which describes the transverse size of the hadron. We
have found that, with increasing quark longitudinal momentum, the transverse
radius of the pion and the kaon decreases. A similar effect has also been
observed in the nucleon Dupre:2016mai . We have noticed that the quarks are
more transversely localized in the kaon than in the pion.

## 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. and S. N.
thank the Chinese Academy of Sciences Presidents International Fellowship
Initiative for the support via Grants No. 2021PM0023 and 2021PM0021,
respectively. 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). S. J. is supported by U.S. Department of Energy, Office of
Science, Office of Nuclear Physics, contract no. DE-AC02-06CH11357. 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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