# Generalized parton distributions of sea quarks in the proton
from nonlocal chiral effective theory

Fangcheng He  CAS Key Laboratory of Theoretical Physics, Institute of
Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
Chueng-Ryong Ji  Department of Physics, North Carolina State University,
Raleigh, North Carolina 27695, USA  W. Melnitchouk  Jefferson Lab, Newport
News, Virginia 23606, USA  A. W. Thomas  CDMPP and CSSM, Department of
Physics, University of Adelaide, Adelaide, SA 5005, Australia  P. Wang
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
China  College of Physics Sciences, University of Chinese Academy of Sciences,
Beijing 100049, China

###### Abstract

We calculate the spin-averaged generalized parton distributions (GPDs) of sea
quarks in the proton at zero skewness from nonlocal covariant chiral effective
theory, including one-loop contributions from intermediate states with
pseudoscalar mesons and octet and decuplet baryons. A relativistic regulator
is generated from the nonlocal Lagrangian where a gauge link is introduced to
guarantee local gauge invariance, with additional diagrams from the expansion
of the gauge link ensuring conservation of electric charge and strangeness.
Flavor asymmetries for sea quarks at zero and finite momentum transfer, as
well as strange form factors, are obtained from the calculated GPDs, and
results compared with phenomenological extractions and lattice QCD.

###### pacs:

13.40.Gp; 12.39.Fe; 14.20.Dh

††preprint: JLAB-THY-22-3553, ADP-22-1/T1172

##  I Introduction

Reconstructing the three-dimensional structure of the nucleon and other
hadrons in terms of their fundamental quark and gluon (or parton) constituents
is one of the defining problems in modern nuclear physics, and one which is a
major driver of experimental programs at facilities such as Jefferson Lab and
the future Electron-Ion Collider (EIC) Accardi  _et al._ (2016); Avakian
(2019). A central element of this endeavor is the extraction of generalized
parton distributions (GPDs), which, as Fourier transforms of nonforward (and
nondiagonal) matrix elements of nonlocal operators, contain rich information
on the partonic structure of the nucleon. GPDs interpolate between exclusive
form factors, when integrated over parton momentum fraction $x$, and parton
distribution functions (PDFs) in the forward limit, and contain considerably
more information about the nucleon’s internal structure than do PDFs or form
factors alone (for reviews of GPDs see, e.g., Refs. Goeke  _et al._ (2001);
Belitsky and Radyushkin (2005)).

The mapping of nucleon GPDs requires a comprehensive program of experimental
studies of hard exclusive processes, such as deeply-virtual Compton scattering
(DVCS) and hard exclusive meson production (HEMP), over a broad kinematic
range. While theoretical tools have been developed to formally factorize GPDs
from the process-dependent, hard scattering amplitudes Ji (1998, 1997);
Collins and Freund (1999), the reconstruction of the full functional
dependence of the GPDs, including their flavor and spin dependence, from
limited experimental data is a formidable challenge Kumericki  _et al._
(2016). Experimental data were obtained at the HERA collider by the H1 Adloff
_et al._ (2001); Aktas  _et al._ (2005) and ZEUS Breitweg  _et al._ (1999);
Chekanov  _et al._ (2003) collaborations and by the HERMES Airapetian  _et
al._ (2001, 2011, 2012) fixed target experiment, as well as by COMPASS at CERN
d’Hose  _et al._ (2004); Fuchey (2015). A rich program of DVCS and HEMP
measurements is also underway at Jefferson Lab with the 12 GeV energy
upgraded, high-luminosity CEBAF accelerator Stepanyan  _et al._ (2001);
Defurne  _et al._ (2015); Jo  _et al._ (2015); Seder  _et al._ (2015).

In addition to the experimental efforts, considerable progress has also been
made on the theoretical front. Because of the complex, nonperturbative
properties of QCD, it is extremely challenging to calculate GPDs from first
principles. Since parton distributions and other light-cone correlation
functions are defined in Minkowski space, it has also been very difficult to
simulate GPDs on the Euclidean lattice. Recent breakthroughs, however, have
enabled the $x$ dependence of PDFs to be inferred from matrix elements of
nonlocal operators on the lattice, in the form of quasi-parton distributions
using the large momentum effective theory Ji (2013), pseudo-PDFs Orginos  _et
al._ (2017), and lattice good cross sections Ma and Qiu (2018a, b).

As with PDFs, the simulation of GPDs on the lattice is still at a relatively
early stage of development. Much of the work on GPDs has focused on finding
effective ways to parametrize their dependence on kinematic variables Guidal
_et al._ (2005). From more phenomenological perspectives, characteristics of
GPDs have been studied within nonperturbative approaches, such as the MIT and
cloudy bag models Ji  _et al._ (1997); Pasquini and Boffi (2007), the
constituent quark model Boffi  _et al._ (2003); Scopetta and Vento (2004), the
NJL model Mineo  _et al._ (2005), the light-front quark model Choi  _et al._
(2001, 2002), the color glass condensate model Goeke  _et al._ (2008), the
chiral quark-soliton model Goeke  _et al._ (2001); Ossmann  _et al._ (2005),
and the Bethe-Salpeter approach Tiburzi and Miller (2002); Theussl  _et al._
(2004).

In addition to the phenomenological models, more systematic approaches using
heavy baryon and relativistic chiral effective field theory (EFT) have been
widely used to study hadron structure at small momentum transfer Fuchs  _et
al._ (2004); Kubis and Meißner (2001a). Historically, most formulations of EFT
have been based on dimensional or infrared regularization. Recently, a
nonlocal chiral effective Lagrangian was proposed Wang (2011); He and Wang
(2018a, b), which makes it possible to extend the range of momentum transfers
over which hadron properties can be described. The method is a relativistic
extension of finite range regularization, which has been applied extensively
to extrapolate lattice QCD calculations of quantities such as the vector meson
mass, magnetic moments, magnetic and strange form factors, charge radii, and
moments of PDFs and GPDs Young  _et al._ (2003); Leinweber  _et al._ (2004);
Wang  _et al._ (2007); Allton  _et al._ (2005); Wang  _et al._ (2009, 2012);
Hall  _et al._ (2014); Shanahan  _et al._ (2013, 2014) from unphysically large
quark masses to the physical region. The nonlocal interaction generates both
the regulator which makes the loop integral convergent and the momentum
dependence of the form factors at tree level. The electromagnetic and strange
form factors of the nucleon obtained in this approach have been found to be in
excellent agreement with experiment up to values of the four-momentum transfer
squared of $\approx 1$ GeV2 He and Wang (2018a, b). Recently, the method has
also been applied to calculate the $\bar{d}-\bar{u}$ flavor asymmetry in the
proton Salamu  _et al._ (2015), the strange–antistrange PDF asymmetry
$s-\bar{s}$ Wang  _et al._ (2016a, b); Salamu  _et al._ (2019a, b), and the
sea quark Sivers function He and Wang (2019) in the proton.

In this paper we apply the nonlocal chiral effective theory for the first time
to GPDs of sea quarks in the proton. The study is timely, given the ongoing
experimental program of DVCS and HEMP measurements at Jefferson Lab, and plans
for future studies of high-$Q^{2}$ exclusive reactions at the EIC. We begin in
Sec. II by introducing the local and nonlocal chiral Lagrangian, including a
derivation of the currents which couple to the external vector field. The GPDs
may be written as convolutions of splitting functions, describing the nucleon
to meson plus baryon process, with the GPDs of the bare hadrons. The one-loop
nucleon $\to$ meson plus octet and decuplet baryon splitting functions are
derived in Sec. III from the full set of rainbow, Kroll-Ruderman, tadpole, and
bubble diagrams. Unlike the one-dimensional splitting functions relevant for
PDFs, the splitting functions for nonforward GPDs also depend on the momentum
transfer squared, in addition to the dependence on the longitudinal momentum
fraction variable. Taking moments of the nonforward splitting functions and
expanding in powers of the pseudoscalar meson mass, in Sec. IV we derive their
nonanalytic behavior, which serves as a model-independent constraint on
phenomenological models. The convolution formalism is discussed in Sec. V,
where we present explicit expressions for the unpolarized electric ($H$) and
magnetic ($E$) GPDs in terms of the splitting functions and GPDs of the
pseudoscalar mesons and intermediate state baryons. Numerical results are
presented in Sec. VI for the nonperturbative sea quark contributions to the
$H$ and $E$ GPDs for light quark and strange flavors, interpolating the
corresponding constraints from the sea quark contributions to PDFs and form
factors. Finally, Sec. VII summarises our results and anticipates future
extensions of this analysis. Explicit formulas for splitting function
integrals are compiled in Appendix A.

##  II Theoretical framework

In this section we introduce the basic chiral Lagrangian which defines the
theoretical basis of our calculations, as well as its nonlocal generalization
which generates the ultraviolet regulator for loop integrals in a natural way,
respecting Lorentz and gauge invariance. The nonlocal formulation relevant for
PDFs was presented in Refs. Salamu  _et al._ (2019a, b); here we generalize
the formalism to the case of nonforward matrix elements needed to compute
GPDs.

###  II.1 Local chiral effective Lagrangian

We begin by introducing the lowest-order local Lagrangian of chiral
SU(3)${}_{L}\times$SU(3)R effective theory that describes the interaction of
pseudoscalar mesons ($\phi$) with octet ($B$) and decuplet ($T_{\mu}$) baryons
Jenkins (1992); Ledwig  _et al._ (2014),

$\displaystyle{\cal L}$ $\displaystyle=$ $\displaystyle{\rm
Tr}\big{[}\bar{B}(i\not{D}-M_{B})B\big{]}-\frac{D}{2}\,{\rm
Tr}\big{[}\bar{B}\gamma^{\mu}\gamma_{5}\\{u_{\mu},B\\}\big{]}-\frac{F}{2}\,{\rm
Tr}\big{[}\bar{B}\gamma^{\mu}\gamma_{5}[u_{\mu},B]\big{]}$ (1)
$\displaystyle+$
$\displaystyle\overline{T}_{\mu}^{ijk}(i\gamma^{\mu\nu\alpha}D_{\alpha}-M_{T}\gamma^{\mu\nu})T_{\nu}^{ijk}-\
\frac{{\cal
H}}{2}\,(\overline{T}_{\mu})^{ijk}\gamma^{\alpha}\gamma_{5}(u_{\alpha})^{kl}\,(T^{\mu})^{ijl}$
$\displaystyle-$ $\displaystyle\frac{{\cal
C}}{2}\big{[}\epsilon^{ijk}\,\overline{T}_{\mu}^{ilm}\Theta^{\mu\nu}(u_{\nu})^{lj}B^{mk}+{\rm
H.c.}\big{]}+\ \frac{f^{2}}{4}{\rm
Tr}\big{[}D_{\mu}U(D^{\mu}U)^{\dagger}\big{]},$

where $M_{B}$ and $M_{T}$ are the octet and decuplet masses, $D$, $F$, $\cal
C$ and $\cal H$ are the baryon-meson coupling constants, and $f=93$ MeV is the
pseudoscalar decay constant. The octet–decuplet transition operator
$\Theta^{\mu\nu}$ is given by

$\Theta^{\mu\nu}=g^{\mu\nu}-\Big{(}Z+\frac{1}{2}\Big{)}\gamma^{\mu}\gamma^{\nu},$
(2)

where $Z$ is the decuplet off-shell parameter, chosen here to be 1/2 Nath  _et
al._ (1971). We define the tensors
$\gamma^{\mu\nu}=\frac{1}{2}[\gamma^{\mu},\gamma^{\nu}]=-i\sigma^{\mu\nu}$ and
$\gamma^{\mu\nu\alpha}=\frac{1}{2}\\{\gamma^{\mu\nu},\gamma^{\alpha}\\}$, and
$\epsilon^{ijk}$ is the antisymmetric tensor in flavor space. The SU(3) baryon
octet fields $B^{ij}$ and decuplet fields $T^{ijk}_{\mu}$ are represented by
the matrix

$\displaystyle
B=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}}\Sigma^{0}+\frac{1}{\sqrt{6}}\Lambda&\Sigma^{+}&p\\\
\Sigma^{-}&-\frac{1}{\sqrt{2}}\Sigma^{0}+\frac{1}{\sqrt{6}}\Lambda&n\\\
\Xi^{-}&\Xi^{0}&-\frac{2}{\sqrt{6}}\Lambda\end{array}\right)$ (6)

and by symmetric tensors with components

$\displaystyle T^{111}=\Delta^{++},\ \ T^{112}=\frac{1}{\sqrt{3}}\Delta^{+},\
\ T^{122}=\frac{1}{\sqrt{3}}\Delta^{0},\ \ T^{222}=\Delta^{-},$ $\displaystyle
T^{113}=\frac{1}{\sqrt{3}}\Sigma^{*+},\ \
T^{123}=\frac{1}{\sqrt{6}}\Sigma^{*0},\ \
T^{223}=\frac{1}{\sqrt{3}}\Sigma^{*-},$ $\displaystyle
T^{133}=\frac{1}{\sqrt{3}}\Xi^{*0},\ \ T^{233}=\frac{1}{\sqrt{3}}\Xi^{*-},$
$\displaystyle T^{333}=\Omega^{-},$

respectively. The operator $U$ is defined in terms of the matrix of
pseudoscalar meson fields $\phi$,

$U\equiv u^{2}=\exp\bigg(i\frac{\sqrt{2}\phi}{f}\bigg{missing}),$ (8)

where the matrix

$\displaystyle\phi=\left({\begin{array}[]{*{20}{c}}\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta&\pi^{+}&K^{+}\\\
\pi^{-}&-\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta&K^{0}\\\
K^{-}&\overline{K}^{0}&-\frac{2}{\sqrt{6}}\eta\end{array}}\right)$ (12)

represents the $\pi$, $K$ and $\eta$ mesons. The covariant derivatives of the
octet and decuplet baryon fields in Eq. (1) are given by Hemmert  _et al._
(1998, 1999)

$\displaystyle D_{\mu}B$ $\displaystyle=$
$\displaystyle\partial_{\mu}B+[\Gamma_{\mu},B]-i\langle\lambda^{0}\rangle\upsilon_{\mu}^{0}\,B,$
(13) $\displaystyle D_{\mu}T_{\nu}^{ijk}$ $\displaystyle=$
$\displaystyle\partial_{\mu}T_{\nu}^{ijk}+(\Gamma_{\mu},T_{\nu})^{ijk}-i\langle\lambda^{0}\rangle\upsilon_{\mu}^{0}\,T_{\nu}^{ijk},$
(14)

respectively, where $\upsilon_{\mu}^{0}$ denotes an external singlet vector
field, $\lambda^{0}$ is the unit matrix, and $\langle\,\cdots\rangle$
represents a trace in flavor space. For the covariant derivative of the
decuplet field, we employ the shorthand notation

$(\Gamma_{\mu},T_{\nu})^{ijk}=(\Gamma_{\mu})_{l}^{i}\,T_{\nu}^{ljk}+(\Gamma_{\mu})_{l}^{j}\,T_{\nu}^{ilk}+(\Gamma_{\mu})_{l}^{k}\,T_{\nu}^{ijl}.$
(15)

For the meson fields, the covariant derivarive is given by

$\displaystyle D_{\mu}U$ $\displaystyle=$
$\displaystyle\partial_{\mu}U+(iU\lambda^{a}-i\lambda^{a}U)\,\upsilon_{\mu}^{a}.$
(16)

The mesons couple to the baryon fields through the vector and axial vector
combinations

$\displaystyle\Gamma_{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(u\partial_{\mu}u^{\dagger}+u^{\dagger}\partial_{\mu}u\right)-\frac{i}{2}\left(u\lambda^{a}u^{\dagger}+u^{\dagger}\lambda^{a}u\right)\upsilon_{\mu}^{a},$
(17) $\displaystyle u_{\mu}$ $\displaystyle=$ $\displaystyle
i\left(u^{\dagger}\partial_{\mu}u-u\partial_{\mu}u^{\dagger}\right)+\left(u^{\dagger}\lambda^{a}u-u\lambda^{a}u^{\dagger}\right)\upsilon_{\mu}^{a},$
(18)

where $\upsilon_{\mu}^{a}$ corresponds to an external octet vector field, and
$\lambda^{a}$ ($a=1,\ldots,8$) are the Gell-Mann matrices.

While the unpolarized twist-two GPD $H$ receives contributions from each quark
flavor from the lowest-order Lagrangian in Eq. (1), to compute the effects of
meson loops on the magnetic GPD $E$ requires an additional contribution to the
Lagrangian for the magnetic interaction, which enters at a higher order. The
magnetic Lagrangian for the octet, decuplet and octet-decuplet transition
interaction is given by He and Wang (2018a, b); Yang and Wang (2020); Jones
and Scadron (1973); Geng  _et al._ (2009)

$\displaystyle{\cal L}_{\rm mag}$ $\displaystyle=$
$\displaystyle\frac{1}{4M_{B}}\Big{(}c_{1}{\rm
Tr}\left[\bar{B}\sigma^{\mu\nu}\left\\{F^{+}_{\mu\nu},B\right\\}\right]+c_{2}{\rm
Tr}\left[\bar{B}\sigma^{\mu\nu}\left[F^{+}_{\mu\nu},B\right]\right]+c_{3}{\rm
Tr}\left[\bar{B}\sigma^{\mu\nu}B\right]{\rm
Tr}\left[F^{+}_{\mu\nu}\right]\Big{)}$ (19) $\displaystyle+$
$\displaystyle\frac{i}{4M_{B}}c_{4}F_{\mu\nu}\Big{(}\epsilon_{ijk}(\lambda_{q})^{i}_{l}\bar{B}^{j}_{m}\gamma^{\mu}\gamma_{5}(T^{\nu})^{klm}+\epsilon^{ijk}(\lambda_{q})^{l}_{i}(\overline{T}^{\mu})_{klm}\gamma^{\nu}\gamma_{5}B^{m}_{j}\Big{)}$
$\displaystyle+$
$\displaystyle\frac{F_{2}^{T}}{2M_{T}}(\overline{T}_{\mu})^{abc}\sigma^{\rho\sigma}\partial_{\sigma}\upsilon_{\rho}^{q}(\lambda_{q})^{a}_{e}(T^{\mu})^{ebc},$

where we adopt the notation $c_{1}$, $c_{2}$ and $c_{3}$ for the octet baryon
interaction from Ref. Yang and Wang (2020) and $c_{4}$ for the octet-decuplet
transition, which corresponds to the constant $\mu_{T}$ in Refs. He and Wang
(2018a, b), and following Refs. He and Wang (2018a, b) denote by $F_{2}^{T}$
the coupling for the decuplet interaction. In Eq. (19) the electromagnetic
interaction with the individual quark flavors is introduced by the field
strength tensor

$F^{+}_{\mu\nu}=\frac{1}{2}\left(u^{\dagger}F^{q}_{\mu\nu}\lambda_{q}u+uF^{q}_{\mu\nu}\lambda_{q}u^{\dagger}\right),$
(20)

where
$F^{q}_{\mu\nu}=\partial_{\mu}\upsilon^{q}_{\nu}-\partial_{\nu}\upsilon^{q}_{\mu}$
for the external field $\upsilon^{q}_{\mu}$ interacting with the quark flavor
$q=u,d,s$ with unit charge, and the matrix $\lambda_{q}$ is the diagonal quark
flavor matrix defined as $\lambda_{q}={\rm
diag}\\{\delta_{qu},\delta_{qd},\delta_{qs}\\}$. At this order, the magnetic
Lagrangian ${\cal L}_{\rm mag}$ generates the following quark flavor
decomposition for the proton anomalous magnetic moment, given by the proton’s
Pauli form factor $F_{2}^{p}(t)$ at $t=0$,

$\displaystyle F^{p(u)}_{2}(0)$ $\displaystyle=$ $\displaystyle
c_{1}+c_{2}+c_{3},$ (21a) $\displaystyle F^{p(d)}_{2}(0)$ $\displaystyle=$
$\displaystyle c_{3},$ (21b) $\displaystyle F^{p(s)}_{2}(0)$ $\displaystyle=$
$\displaystyle c_{1}-c_{2}+c_{3}.$ (21c)

Since at tree level there is no strange quark contribution to the proton, we
take $c_{3}=c_{2}-c_{1}$. Furthermore, from SU(3) symmetry one also obtains
relationships between the octet and decuplet constants He and Wang (2018a, b),

$\displaystyle c_{4}$ $\displaystyle=$ $\displaystyle 4c_{1},$ (22a)
$\displaystyle F_{2}^{T}$ $\displaystyle=$ $\displaystyle c_{1}+3c_{2}.$ (22b)

Within the flavor SU(3) framework, the magnetic moments of the octet and
decuplet baryons, and the transition moments between the octet and decuplet
baryons, can be expressed in terms of quark magnetic moments, $\mu_{q}$. For
example, for the proton and neutron one would have
$\mu_{p}=\frac{4}{3}\mu_{u}-\frac{1}{3}\mu_{d}$ and
$\mu_{n}=\frac{4}{3}\mu_{d}-\frac{1}{3}\mu_{u}$, respectively, while for the
$\Delta^{++}$ baryon $\mu_{\Delta^{++}}=3\mu_{u}$.

If we include the higher order magnetic Lagrangian ${\cal L}_{\rm mag}$ in Eq.
(19), for consistency in the power counting we also need to consider the next-
to-leading order Lagrangian for the baryon-meson interaction. Generalizing Eq.
(1), and using the notation from Ref. Kubis and Meißner (2001b), we therefore
include the additional baryon contribution involving two derivatives Kubis and
Meißner (2001b)

$\displaystyle\\!\\!{\cal L}_{B\phi}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\sigma^{\mu\nu}\Big{(}b_{9}\,{\rm
Tr}\left[\bar{B}u_{\mu}\right]{\rm Tr}\left[u_{\nu}B\right]+b_{10}\,{\rm
Tr}\left[\bar{B}\\{[u_{\mu},u_{\nu}],B\\}\right]+b_{11}\,{\rm
Tr}\left[\bar{B}[[u_{\mu},u_{\nu}],B]\right]\\!\Big{)},~{}~{}~{}$ (23)

where the values of the coefficients have been determined to be $b_{9}=1.36$
GeV-1, $b_{10}=1.24$ GeV-1, and $b_{11}=0.46$ GeV-1 Kubis and Meißner (2001b).
Expanding the Lagrangians $\cal L$ in Eq. (1) and ${\cal L}_{B\phi}^{\prime}$
in Eq. (23), the lowest order baryon-meson interaction involving the proton
can then be written as

$\begin{split}{\cal L}_{\rm
int}&=\frac{(D+F)}{2f}\left(\bar{p}\,\gamma^{\mu}\gamma_{5}p\,\partial_{\mu}\pi^{0}+\sqrt{2}\,\bar{p}\,\gamma^{\mu}\gamma_{5}n\,\partial_{\mu}\pi^{+}\right)-\frac{(D-3F)}{\sqrt{12}f}\bar{p}\,\gamma^{\mu}\gamma_{5}p\,\partial_{\mu}\eta\\\
&+\frac{(D-F)}{2f}\left(\sqrt{2}\,\bar{p}\,\gamma^{\mu}\gamma_{5}\Sigma^{+}\,\partial_{\mu}K^{0}+\bar{p}\,\gamma^{\mu}\gamma_{5}\Sigma^{0}\,\partial_{\mu}K^{+}\right)-\frac{(D+3F)}{\sqrt{12}f}\bar{p}\,\gamma^{\mu}\gamma_{5}\Lambda\,\partial_{\mu}K^{0}\\\
&+\frac{\cal
C}{\sqrt{12}f}\left(-2\,\bar{p}\,\Theta^{\nu\mu}\Delta_{\mu}^{+}\,\partial_{\nu}\pi^{0}-\sqrt{2}\,\bar{p}\,\Theta^{\nu\mu}\Delta_{\mu}^{0}\,\partial_{\nu}\pi^{+}+\sqrt{6}\,\bar{p}\,\Theta^{\nu\mu}\Delta_{\mu}^{++}\,\partial_{\nu}\pi^{-}\right.\\\
&\hskip
51.21504pt\left.-\bar{p}\,\Theta^{\nu\mu}\Sigma_{\mu}^{*0}\,\partial_{\nu}K^{+}+\sqrt{2}\,\bar{p}\,\Theta^{\nu\mu}\Sigma_{\mu}^{*+}\,\partial_{\nu}K^{0}+{\rm
H.c.}\right)\\\
&+\frac{i}{4f^{2}}\bar{p}\,\gamma^{\mu}p\Big{[}(\pi^{+}\partial_{\mu}\pi^{-}-\pi^{-}\partial_{\mu}\pi^{+})+2(K^{+}\partial_{\mu}K^{-}-K^{-}\partial_{\mu}K^{+})+(K^{0}\partial_{\mu}\bar{K}^{0}-\bar{K}^{0}\partial_{\mu}K^{0})\Big{]}\\\
&+\frac{i}{f^{2}}\bar{p}\,\sigma^{\mu\nu}p\Big{(}2(b_{10}+b_{11})\partial_{\mu}\pi^{+}\partial_{\nu}\pi^{-}+(4b_{11}+b_{9})\partial_{\mu}K^{+}\partial_{\nu}K^{-}+2(b_{10}-{b_{11}})\partial_{\mu}K^{0}\partial_{\nu}\bar{K}^{0}\Big{)}.\end{split}$
(24)

For the interactions with the external field $\upsilon_{\mu}^{a}$, from the
Lagrangian $\cal L$ in Eq. (1) one can obtain the vector current

$\displaystyle J^{\mu}_{a}$ $\displaystyle=$ $\displaystyle\frac{1}{2}{\rm
Tr}\big{[}\bar{B}\gamma^{\mu}\left[u\lambda^{a}u^{\dagger}+u^{\dagger}\lambda^{a}u,B\right]+\frac{D}{2}{\rm
Tr}\big{[}\bar{B}\gamma^{\mu}\gamma_{5}\left\\{u\lambda^{a}u^{\dagger}-u^{\dagger}\lambda^{a}u,B\right\\}\big{]}$
(25) $\displaystyle+$ $\displaystyle\frac{F}{2}{\rm
Tr}\big{[}\bar{B}\gamma^{\mu}\gamma_{5}\left[u\lambda^{a}u^{\dagger}-u^{\dagger}\lambda^{a}u,B\right]\big{]}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\,\overline{T}_{\nu}\gamma^{\nu\alpha\mu}\left(u\lambda^{a}u^{\dagger}+u^{\dagger}\lambda^{a}u,T_{\alpha}\right)+\frac{\cal
C}{2}\left[\overline{T}_{\nu}\Theta^{\nu\mu}(u\lambda^{a}u^{\dagger}-u^{\dagger}\lambda^{a}u)B+{\rm
H.c.}\right]$ $\displaystyle+$ $\displaystyle\frac{f^{2}}{4}{\rm
Tr}\big{[}\partial^{\mu}U(U^{\dagger}i\lambda^{a}-i\lambda^{a}U^{\dagger})+(Ui\lambda^{a}-i\lambda^{a}U)\partial^{\mu}U^{\dagger}\big{]}.$

For the SU(3) flavor singlet case, the current coupling to the external field
$\upsilon_{\mu}^{0}$ can be written

$\displaystyle J^{\mu}_{0}$ $\displaystyle=$
$\displaystyle\langle\lambda^{0}\rangle\,{\rm
Tr}[\bar{B}\gamma^{\mu}B]+\langle\lambda^{0}\rangle\,\overline{T}_{\nu}\gamma^{\nu\alpha\mu}\,T_{\alpha}.$
(26)

The magnetic current coupling to the external field $\upsilon_{\mu}^{q}$ can
be obtained from the magnetic Lagrangian in Eq. (19),

$\displaystyle J_{q,\rm mag}^{\mu}$ $\displaystyle=$
$\displaystyle\frac{\partial_{\nu}}{4M_{B}}\Big{(}c_{1}{\rm
Tr}\bar{B}\sigma^{\mu\nu}\left\\{u^{\dagger}\lambda_{q}u+u\lambda_{q}u^{\dagger},B\right\\}+c_{2}{\rm
Tr}\bar{B}\sigma^{\mu\nu}\left[u^{\dagger}\lambda_{q}u+u\lambda_{q}u^{\dagger},B\right]$
(27) $\displaystyle\qquad+\,c_{3}{\rm Tr}\bar{B}\sigma^{\mu\nu}B\,{\rm
Tr}(u^{\dagger}\lambda_{q}u+u\lambda_{q}u^{\dagger})\Big{)}-\
\frac{F_{2}^{T}}{2M_{T}}\partial_{\sigma}\Big{(}(\overline{T}_{\nu})^{abc}\sigma^{\mu\sigma}(\lambda_{q})^{a}_{e}(T^{\nu})^{ebc}\Big{)}$
$\displaystyle-$
$\displaystyle\frac{ic_{4}}{4M_{B}}(\partial^{\sigma}g^{\mu\nu}-\partial^{\nu}g^{\mu\sigma})\Big{(}\epsilon_{ijk}(\lambda_{q})^{i}_{l}\,\bar{B}^{j}_{m}\gamma_{\sigma}\gamma_{5}T_{\nu}^{klm}+\epsilon^{ijk}(\lambda_{q})^{l}_{i}\,\overline{T}_{\sigma,klm}\gamma_{\nu}\gamma_{5}B^{m}_{j}\Big{)},$

which satisfies current conservation, $\partial_{\mu}J_{q,\rm mag}^{\mu}=0$.
The quark flavor currents can be written in terms of the SU(3) singlet ($a=0$)
and octet ($a=3,8$), and quark magnetic currents,

$\displaystyle J^{\mu}_{u}$ $\displaystyle=$
$\displaystyle\frac{1}{3}J^{\mu}_{0}+\frac{1}{2}J^{\mu}_{3}+\frac{1}{2\sqrt{3}}J^{\mu}_{8}+J_{u,\rm
mag}^{\mu},$ (28a) $\displaystyle J^{\mu}_{d}$ $\displaystyle=$
$\displaystyle\frac{1}{3}J^{\mu}_{0}-\frac{1}{2}J^{\mu}_{3}+\frac{1}{2\sqrt{3}}J^{\mu}_{8}+J_{d,\rm
mag}^{\mu},$ (28b) $\displaystyle J^{\mu}_{s}$ $\displaystyle=$
$\displaystyle\frac{1}{3}J^{\mu}_{0}-\frac{1}{\sqrt{3}}J^{\mu}_{8}+J_{s,\rm
mag}^{\mu}.$ (28c)

Using Eqs. (25), (26) and (28), the quark flavor currents can be written more
explicitly in the form

$\displaystyle J_{u}^{\mu}$ $\displaystyle=$ $\displaystyle
2\bar{p}\gamma^{\mu}p+\bar{n}\gamma^{\mu}n+\bar{\Lambda}\gamma^{\mu}\Lambda+2\overline{\Sigma}^{+}\gamma^{\mu}\Sigma^{+}+{\overline{\Sigma}}^{0}\gamma^{\mu}\Sigma^{0}-\frac{1}{2f^{2}}\,\bar{p}\gamma^{\mu}p\,\big{(}\pi^{+}\pi^{-}+2K^{+}K^{-}\big{)}$
$\displaystyle+$ $\displaystyle
3\overline{\Delta}_{\alpha}^{++}\gamma^{\alpha\beta\mu}\Delta_{\beta}^{++}+2\overline{\Delta}_{\alpha}^{+}\gamma^{\alpha\beta\mu}\Delta_{\beta}^{+}+\overline{\Delta}_{\alpha}^{0}\gamma^{\alpha\beta\mu}\Delta_{\beta}^{0}+2\overline{\Sigma}_{\alpha}^{*+}\gamma^{\alpha\beta\mu}\Sigma_{\beta}^{*+}+\overline{\Sigma}_{\alpha}^{*0}\gamma^{\alpha\beta\mu}\Sigma_{\beta}^{*0}$
$\displaystyle+$ $\displaystyle
i\left(\pi^{-}\partial^{\mu}\pi^{+}-\pi^{+}\partial^{\mu}\pi^{-}\right)+i\left(K^{-}\partial^{\mu}K^{+}-K^{+}\partial^{\mu}K^{-}\right)$
$\displaystyle-$
$\displaystyle\frac{i(D+F)}{\sqrt{2}f}\bar{p}\gamma^{\mu}\gamma_{5}n\,\pi^{+}+\frac{i(D+3F)}{\sqrt{12}f}\bar{p}\gamma^{\mu}\gamma_{5}\Lambda\,K^{+}-\frac{i(D-F)}{2f}\bar{p}\gamma^{\mu}\gamma_{5}\Sigma^{0}\,K^{+}$
$\displaystyle+$ $\displaystyle\frac{i\,\cal
C}{\sqrt{12}f}\left(\sqrt{6}\,\bar{p}\,\Theta^{\mu\nu}\Delta_{\nu}^{++}\,\pi^{-}+\sqrt{2}\,\bar{p}\,\Theta^{\mu\nu}\Delta_{\nu}^{0}\,\pi^{+}+\,\bar{p}\,\Theta^{\mu\nu}\Sigma_{\nu}^{*0}\,K^{+}+{\rm
H.c.}\right)$ $\displaystyle+$
$\displaystyle\frac{1}{4M_{B}}\partial_{\nu}(\bar{p}\sigma^{\mu\nu}p)\bigg{[}4c_{2}\Big{(}1-\frac{1}{2f^{2}}K^{+}K^{-}\Big{)}-\frac{(c_{1}+c_{2})}{f^{2}}\pi^{+}\pi^{-}\bigg{]}+\frac{c_{2}-c_{1}}{2M_{B}}\partial_{\nu}(\bar{n}\sigma^{\mu\nu}n)$
$\displaystyle+$
$\displaystyle\frac{3c_{2}-2c_{1}}{6M_{B}}\partial_{\nu}(\bar{\Lambda}\sigma^{\mu\nu}\Lambda)+\frac{c_{1}}{2\sqrt{3}M_{B}}\partial_{\nu}(\overline{\Lambda}\sigma^{\mu\nu}\Sigma^{0})+\frac{c_{2}}{M_{B}}\partial_{\nu}(\overline{\Sigma}^{+}\sigma^{\mu\nu}\Sigma^{+})+\frac{c_{2}}{2M_{B}}\partial_{\nu}(\overline{\Sigma}^{0}\sigma^{\mu\nu}\Sigma^{0})$
$\displaystyle+$
$\displaystyle\frac{ic_{4}}{4\sqrt{3}M_{B}}\partial^{\nu}\bigg{[}\bar{p}(\gamma_{\nu}\gamma_{5}\Delta^{+\mu}-\gamma^{\mu}\gamma_{5}\Delta^{+}_{\nu})+\bar{n}(\gamma_{\nu}\gamma_{5}\Delta^{0\mu}-\gamma^{\mu}\gamma_{5}\Delta^{0}_{\nu})-\overline{\Sigma}^{+}(\gamma_{\nu}\gamma_{5}\Sigma^{*+\mu}-\gamma^{\mu}\gamma_{5}\Sigma^{*+}_{\nu})$
$\displaystyle\hskip
56.9055pt-\frac{\sqrt{3}}{2}\bar{\Lambda}(\gamma_{\nu}\gamma_{5}\Sigma^{*0\mu}-\gamma^{\mu}\gamma_{5}\Sigma^{*0}_{\nu})+\frac{1}{2}\overline{\Sigma}^{0}(\gamma_{\nu}\gamma_{5}\Sigma^{*0\mu}-\gamma^{\mu}\gamma_{5}\Sigma^{*0}_{\nu})\bigg{]}$
$\displaystyle-$
$\displaystyle\frac{F_{2}^{T}}{6M_{T}}\partial_{\nu}\Big{[}3\overline{\Delta}^{++}_{\alpha}\sigma^{\mu\nu}\Delta^{++\alpha}+2\overline{\Delta}^{+}_{\alpha}\sigma^{\mu\nu}\Delta^{+\alpha}+\overline{\Delta}^{0}_{\alpha}\sigma^{\mu\nu}\Delta^{0\alpha}+2\overline{\Sigma}^{*+}_{\alpha}\sigma^{\mu\nu}\Sigma^{*+\alpha}+\overline{\Sigma}^{*0}_{\alpha}\sigma^{\mu\nu}\Sigma^{*0\alpha}\Big{]},$
$\displaystyle J^{\mu}_{d}$ $\displaystyle=$
$\displaystyle\bar{p}\gamma^{\mu}p+2\bar{n}\gamma^{\mu}n+2\overline{\Sigma}^{-}\gamma^{\mu}\Sigma^{-}+\overline{\Sigma}^{0}\gamma^{\mu}\Sigma^{0}+\bar{\Lambda}\gamma^{\mu}\Lambda+\frac{1}{2f^{2}}\,\bar{p}\gamma^{\mu}p\,\big{(}\pi^{+}\pi^{-}-\overline{K}^{0}K^{0}\big{)}$
$\displaystyle+$
$\displaystyle\overline{\Delta}_{\alpha}^{+}\gamma^{\alpha\beta\mu}\Delta_{\beta}^{+}+2\overline{\Delta}_{\alpha}^{0}\gamma^{\alpha\beta\mu}\Delta_{\beta}^{0}+3\overline{\Delta}_{\alpha}^{-}\gamma^{\alpha\beta\mu}\Delta_{\beta}^{-}+\overline{\Sigma}_{\alpha}^{*0}\gamma^{\alpha\beta\mu}\Sigma_{\beta}^{*0}+2\overline{\Sigma}_{\alpha}^{*0-}\gamma^{\alpha\beta\mu}\Sigma_{\beta}^{*-}$
$\displaystyle-$ $\displaystyle
i(\pi^{-}\partial^{\mu}\pi^{+}-\pi^{+}\partial^{\mu}\pi^{-})+i(\overline{K}^{0}\partial^{\mu}K^{0}-K^{0}\partial^{\mu}\overline{K}^{0})$
$\displaystyle+$
$\displaystyle\frac{i(D+F)}{\sqrt{2}f}\bar{p}\gamma^{\mu}\gamma_{5}n\,\pi^{+}-\frac{i(D-F)}{\sqrt{2}f}\bar{p}\gamma^{\mu}\gamma_{5}\Sigma^{+}\,K^{0}$
$\displaystyle-$ $\displaystyle\frac{i\,\cal
C}{\sqrt{6}f}\left(\sqrt{3}\,\bar{p}\,\Theta^{\mu\nu}\Delta_{\nu}^{++}\,\pi^{-}+\bar{p}\,\Theta^{\mu\nu}\Delta_{\nu}^{0}\,\pi^{+}+\bar{p}\,\Theta^{\mu\nu}\Sigma_{\nu}^{*+}\,K^{0}+{\rm
H.c.}\right)$ $\displaystyle+$
$\displaystyle\frac{1}{4M_{B}}\partial_{\nu}(\bar{p}\sigma^{\mu\nu}p)\bigg{[}(c_{2}-c_{1})\Big{(}2-\frac{1}{f^{2}}\overline{K}^{0}K^{0}\Big{)}+\frac{(c_{1}+c_{2})}{f^{2}}\pi^{+}\pi^{-}\bigg{]}+\frac{c_{2}}{M_{B}}\partial_{\nu}(\bar{n}\sigma^{\mu\nu}n)$
$\displaystyle+$
$\displaystyle\frac{3c_{2}-2c_{1}}{6M_{B}}\partial_{\nu}(\bar{\Lambda}\sigma^{\mu\nu}\Lambda)+\frac{c_{2}}{M_{B}}\partial_{\nu}(\overline{\Sigma}^{-}\sigma^{\mu\nu}\Sigma^{-})+\frac{c_{2}}{2M_{B}}\partial_{\nu}(\overline{\Sigma}^{0}\sigma^{\mu\nu}\Sigma^{0})-\frac{c_{1}}{2\sqrt{3}M_{B}}\partial_{\nu}(\bar{\Lambda}\sigma^{\mu\nu}\Sigma^{0})$
$\displaystyle-$
$\displaystyle\frac{ic_{4}}{4\sqrt{3}M_{B}}\partial^{\nu}\bigg{[}\bar{p}(\gamma_{\nu}\gamma_{5}\Delta^{+\mu}-\gamma^{\mu}\gamma_{5}\Delta^{+}_{\nu})+\bar{n}(\gamma_{\nu}\gamma_{5}\Delta^{0\mu}-\gamma^{\mu}\gamma_{5}\Delta^{0}_{\nu})-\overline{\Sigma}^{-}(\gamma_{\nu}\gamma_{5}\Sigma^{*-\mu}-\gamma^{\mu}\gamma_{5}\Sigma^{*-}_{\nu})$
$\displaystyle\hskip
56.9055pt-\frac{\sqrt{3}}{2}\bar{\Lambda}(\gamma_{\nu}\gamma_{5}\Sigma^{*0\mu}-\gamma^{\mu}\gamma_{5}\Sigma^{*0}_{\nu})-\frac{1}{2}\overline{\Sigma}^{0}(\gamma_{\nu}\gamma_{5}\Sigma^{*0\mu}-\gamma^{\mu}\gamma_{5}\Sigma^{*0}_{\nu})\bigg{]}$
$\displaystyle-$
$\displaystyle\frac{F_{2}^{T}}{6M_{T}}\partial_{\nu}\Big{[}3\overline{\Delta}^{-}_{\alpha}\sigma^{\mu\nu}\Delta^{-\alpha}+2\overline{\Delta}^{0}_{\alpha}\sigma^{\mu\nu}\Delta^{0\alpha}+\overline{\Delta}^{+}_{\alpha}\sigma^{\mu\nu}\Delta^{+\alpha}+2\overline{\Sigma}^{*-}_{\alpha}\sigma^{\mu\nu}\Sigma^{*-\alpha}+\overline{\Sigma}^{*0}_{\alpha}\sigma^{\mu\nu}\Sigma^{*0\alpha}\Big{]},$
$\displaystyle J^{\mu}_{s}$ $\displaystyle=$
$\displaystyle\overline{\Sigma}^{+}\gamma^{\mu}\Sigma^{+}+\overline{\Sigma}^{0}\gamma^{\mu}\Sigma^{0}+\bar{\Lambda}\gamma^{\mu}\Lambda+\frac{1}{2f^{2}}\,\bar{p}\gamma^{\mu}p\,\big{(}2K^{+}K^{-}+\overline{K}^{0}K^{0}\big{)}$
(29c) $\displaystyle+$
$\displaystyle\overline{\Sigma}_{\alpha}^{*+}\gamma^{\alpha\beta\mu}\Sigma_{\beta}^{*+}+\overline{\Sigma}_{\alpha}^{*0}\gamma^{\alpha\beta\mu}\Sigma_{\beta}^{*0}-i(K^{-}\partial^{\mu}K^{+}-K^{+}\partial^{\mu}K^{-})-i(\overline{K}^{0}\partial^{\mu}K^{0}-K^{0}\partial^{\mu}\overline{K}^{0})$
$\displaystyle+$
$\displaystyle\frac{i(D-F)}{\sqrt{2}f}\bar{p}\gamma^{\mu}\gamma_{5}\Sigma^{+}\,K^{0}+\frac{i(D-F)}{2f}\bar{p}\gamma^{\mu}\gamma_{5}\Sigma^{0}\,K^{+}-\frac{i(D+3F)}{\sqrt{12}f}\bar{p}\gamma^{\mu}\gamma_{5}\Lambda\,K^{+}$
$\displaystyle-$ $\displaystyle\frac{i\,\cal
C}{\sqrt{12}f}\left(\bar{p}\,\Theta^{\mu\nu}\Sigma_{\nu}^{*0}\,K^{+}-\sqrt{2}\,\bar{p}\,\Theta^{\mu\nu}\Sigma_{\nu}^{*+}\,K^{0}+{\rm
H.c.}\right)$ $\displaystyle+$
$\displaystyle\frac{1}{4M_{B}f^{2}}\partial_{\nu}(\bar{p}\sigma^{\mu\nu}p)\bigg{[}2c_{2}K^{+}K^{-}+(c_{2}-c_{1})\overline{K}^{0}K^{0}\bigg{]}+\frac{(c_{1}+3c_{2})}{6M_{B}}\partial_{\nu}(\bar{\Lambda}\sigma^{\mu\nu}\Lambda)$
$\displaystyle+$
$\displaystyle\frac{(c_{2}-c_{1})}{2M_{B}}\partial_{\nu}(\overline{\Sigma}^{+}\sigma^{\mu\nu}\Sigma^{+})+\frac{(c_{2}-c_{1})}{2M_{B}}\partial_{\nu}(\overline{\Sigma}^{-}\sigma^{\mu\nu}\Sigma^{-})+\frac{(c_{2}-c_{1})}{2M_{B}}\partial_{\nu}(\overline{\Sigma}^{0}\sigma^{\mu\nu}\Sigma^{0})$
$\displaystyle-$
$\displaystyle\frac{\,ic_{4}}{4\sqrt{3}M_{B}}\partial^{\nu}\bigg{[}\overline{\Sigma}^{0}(\gamma_{\nu}\gamma_{5}\Sigma^{*0\mu}-\gamma^{\mu}\gamma_{5}\Sigma^{*0}_{\nu})+\overline{\Sigma}^{-}(\gamma_{\nu}\gamma_{5}\Sigma^{*-\mu}-\gamma^{\mu}\gamma_{5}\Sigma^{*-}_{\nu})$
$\displaystyle\hskip
56.9055pt-\overline{\Sigma}^{+}(\gamma_{\nu}\gamma_{5}\Sigma^{*+\mu}-\gamma^{\mu}\gamma_{5}\Sigma^{*+}_{\nu})\bigg{]},$
$\displaystyle-$
$\displaystyle\frac{F_{2}^{T}}{6M_{T}}\partial_{\nu}\Big{[}\overline{\Sigma}^{*-}_{\alpha}\sigma^{\mu\nu}\Sigma^{*-\alpha}+\overline{\Sigma}^{*0}_{\alpha}\sigma^{\mu\nu}\Sigma^{*0\alpha}+\overline{\Sigma}^{*+}_{\alpha}\sigma^{\mu\nu}\Sigma^{*+\alpha}\Big{]},$

for the $u$, $d$ and $s$ quark flavors, respectively. As in Ref. Salamu  _et
al._ (2019a), terms involving the doubly-strange $\Xi^{0,-}$ and $\Xi^{*0,-}$
hyperons and the triply-strange $\Omega^{-}$ baryon do not couple directly to
proton states and not included here.

###  II.2 Nonlocal chiral Lagrangian

In this section we outline the generalization of the effective local chiral
Lagrangian to the case of nonlocal interactions. Taking the traces in Eqs.
(1), (19) and (23) in Sec. II.1, we can write the local Lagrangian density in
the form

$\displaystyle{\cal L}^{\rm(local)}(x)$ $\displaystyle=$
$\displaystyle\bar{B}(x)(i\gamma^{\mu}\mathscr{D}_{\mu}-M_{B})B(x)+\frac{C_{B\phi}}{f}\big{[}\bar{p}(x)\gamma^{\mu}\gamma_{5}B(x)\,\mathscr{D}_{\mu}\phi(x)+{\rm
H.c.}\big{]}$ (30) $\displaystyle+$
$\displaystyle\overline{T}_{\mu}(x)(i\gamma^{\mu\nu\alpha}\mathscr{D}_{\alpha}-M_{T}\gamma^{\mu\nu})\,T_{\nu}(x)+\frac{C_{T\phi}}{f}\left[\,\bar{p}(x)\Theta^{\mu\nu}T_{\nu}(x)\,\mathscr{D}_{\mu}\phi(x)+{\rm
H.c.}\right]$ $\displaystyle+$
$\displaystyle\mathscr{D}_{\mu}\phi(x)(\mathscr{D}_{\mu}\phi)^{\dagger}(x)\,+\frac{iC_{\phi\phi^{\dagger}}}{2f^{2}}\bar{p}(x)\gamma^{\mu}p(x)\left[\phi(x)(\mathscr{D}_{\mu}\phi)^{\dagger}(x)-\mathscr{D}_{\mu}\phi(x)\phi^{\dagger}(x)\right]$
$\displaystyle+$
$\displaystyle\frac{iC_{\phi\phi^{\dagger}}^{\prime}}{2f^{2}}\bar{p}(x)\sigma^{\mu\nu}p(x)\mathscr{D}_{\mu}\phi(x)(\mathscr{D}_{\nu}\phi)^{\dagger}(x)$
$\displaystyle+$ $\displaystyle\frac{C_{\phi\phi^{\dagger}}^{\rm
mag}}{4M_{B}f^{2}}\bar{p}(x)\sigma^{\mu\nu}p(x)F_{\mu\nu}(x)\phi(x)\phi^{\dagger}(x)+\frac{C_{B}^{\rm
mag}}{4M_{B}}\bar{B}(x)\sigma^{\mu\nu}B(x)F_{\mu\nu}(x)$ $\displaystyle+$
$\displaystyle\frac{iC_{BT}^{\rm
mag}}{4M_{B}}\bar{B}(x)\gamma^{\mu}\gamma_{5}T^{\nu}(x)F_{\mu\nu}(x)-\frac{C_{T}^{\rm
mag}}{4M_{T}}\overline{T}_{\alpha}(x)\sigma^{\mu\nu}T^{\alpha}(x)F_{\mu\nu}(x)+\cdots,$

where the dependence on the space-time coordinate $x$ is shown explicitly, and
for the interaction part we show only those terms that contribute to the
proton GPDs. The covariant derivatives in Eq. (30) are given by

$\displaystyle\mathscr{D}_{\mu}B(x)$ $\displaystyle=$
$\displaystyle\left[\partial_{\mu}-ie^{q}_{B}\,\mathscr{A_{\mu}}(x)\right]B(x),$
(31a) $\displaystyle\mathscr{D}_{\mu}T^{\nu}(x)$ $\displaystyle=$
$\displaystyle\left[\partial_{\mu}-ie^{q}_{T}\,\mathscr{A_{\mu}}(x)\right]T^{\nu}(x),$
(31b) $\displaystyle\mathscr{D}_{\mu}\phi(x)$ $\displaystyle=$
$\displaystyle\left[\partial_{\mu}-ie^{q}_{\phi}\,\mathscr{A_{\mu}}(x)\right]\phi(x),$
(31c)

where $\mathscr{A_{\mu}}$ is the electromagnetic gauge field, and $e^{q}_{B}$,
$e^{q}_{T}$ and $e^{q}_{\phi}$ denote, respectively, the quark flavor charges
of the octet and decuplet baryons, $B$ and $T$, and meson $\phi$. In the case
of the proton, for instance, one has the flavor charges
$e^{u}_{p}=2e^{d}_{p}=2$, $e^{s}_{p}=0$, while for the $\Sigma^{+}$ hyperon
$e^{u}_{\Sigma^{+}}=2e^{s}_{\Sigma^{+}}=2$, $e^{d}_{\Sigma^{+}}=0$, and
similarly for the other baryons. For the pion and kaon, the flavor charges are
$e^{u}_{\pi^{+}}=-e^{d}_{\pi^{+}}=1$, $e^{q}_{\pi^{0}}=0$ for all $q$, and
$e^{u}_{K^{+}}=-e^{s}_{K^{+}}=1$, $e^{d}_{K^{+}}=0$, with the values for other
mesons obtained by charge conjugation. The coefficients $C_{B\phi}$,
$C_{T\phi}$, $C_{\phi\phi^{\dagger}}$, $C_{\phi\phi^{\dagger}}^{\prime}$,
$C_{B}^{\rm mag}$, $C_{BT}^{\rm mag}$, $C_{T}^{\rm mag}$ and
$C_{\phi\phi^{\dagger}}^{\rm mag}$ in Eq. (30) are given explicitly in Table 1
for the various processes discussed in this work.

Following Ref. Salamu  _et al._ (2019a), we sketch here the derivation of the
nonlocal Lagrangian from Eq. (30). Details of the methodology used here can be
found in Refs. Terning (1991); Holdom (1992); Fäßler  _et al._ (2003); Wang
and Banerjee (1996); He and Wang (2018a, b). The nonlocal analog of the local
Lagrangian (30) can be written as

$\displaystyle{\cal L}^{\rm(nonloc)}(x)$ $\displaystyle=$
$\displaystyle\bar{B}(x)(i\gamma^{\mu}\mathscr{D}_{\mu}-M_{B})B(x)+\overline{T}_{\mu}(x)(i\gamma^{\mu\nu\alpha}\mathscr{D}_{\alpha}-M_{T}\gamma^{\mu\nu})T_{\nu}(x)$
(32) $\displaystyle+\
\bar{p}(x)\left[\frac{C_{B\phi}}{f}\gamma^{\mu}\gamma_{5}B(x)\,+\frac{C_{T\phi}}{f}\Theta^{\mu\nu}T_{\nu}(x)\right]\mathscr{D}_{\mu}\\!\int\\!\differential[4]{a}\,{\cal
G}_{\phi}^{q}(x,x+a)F(a)\phi(x+a)+{\rm H.c.}$ $\displaystyle+\
\frac{iC_{\phi\phi^{\dagger}}}{2f^{2}}\,\bar{p}(x)\gamma^{\mu}p(x)\\!\int\\!\differential[4]{a}\,{\cal
G}_{\phi}^{q}(x,x+a)F(a)\phi(x+a)\,\mathscr{D}_{\mu}\\!\\!\int\\!\differential[4]{b}\,{\cal
G}_{\phi}^{q}(x+b,x)F(b)\phi^{\dagger}(x+b)$ $\displaystyle+\
\frac{iC_{\phi\phi^{\dagger}}^{\prime}}{2f^{2}}\,\bar{p}(x)\sigma^{\mu\nu}p(x)\,\mathscr{D}_{\mu}\\!\int\\!\differential[4]{a}\,{\cal
G}_{\phi}^{q}(x,x+a)F(a)\phi(x+a)\,\mathscr{D}_{\nu}\\!\int\\!\differential[4]{b}\,{\cal
G}_{\phi}^{q}(x+b,x)F(b)\phi^{\dagger}(x+b)$ $\displaystyle+\ \frac{C_{B}^{\rm
mag}}{4M_{B}}\,\bar{B}(x)\sigma^{\mu\nu}B(x)F_{\mu\nu}(x)+\frac{iC_{BT}^{\rm
mag}}{4M_{B}}\bar{B}(x)\gamma^{\mu}\gamma_{5}T^{\nu}(x)F_{\mu\nu}(x)-\frac{C_{T}^{\rm
mag}}{4M_{T}}\overline{T}_{\alpha}(x)\sigma^{\mu\nu}T^{\alpha}(x)F_{\mu\nu}(x)$
$\displaystyle+\ \frac{C_{\phi\phi^{\dagger}}^{\rm
mag}}{4M_{B}f^{2}}\,\bar{p}(x)\sigma^{\mu\nu}p(x)\int\\!\differential[4]{a}\\!\int\\!\differential[4]{b}\,F_{\mu\nu}(x)\,{\cal
G}_{\phi}^{q}(x+b,x+a)F(a)F(b)\,\phi(x+a)\phi^{\dagger}(x+b)$ $\displaystyle+\
\mathscr{D}_{\mu}\phi(x)(\mathscr{D}_{\mu}\phi)^{\dagger}(x)+\ \cdots,$

where the gauge link ${\cal G}_{\phi}^{q}$ is introduced for local gauge
invariance,

${\cal
G}_{\phi}^{q}(x,y)=\exp\left[-ie^{q}_{\phi}\int_{x}^{y}\differential{z}^{\mu}\mathscr{A}_{\mu}(z)\right],$
(33)

and $F(a)$ is the meson–baryon vertex form factor in coordinate space. Note
that both the nonlocal Lagrangian in Eq. (32) and the local Lagrangian in Eq.
(30) are invariant under the gauge transformations,

$\displaystyle B(x)$ $\displaystyle\to$ $\displaystyle
B^{\prime}(x)=B(x)\exp[ie^{q}_{B}\,\theta(x)\big{]},$ (34a) $\displaystyle
T_{\mu}(x)$ $\displaystyle\to$ $\displaystyle
T_{\mu}^{\prime}(x)=T_{\mu}(x)\exp[ie^{q}_{T}\,\theta(x)\big{]},$ (34b)
$\displaystyle\phi(x)$ $\displaystyle\to$
$\displaystyle\phi^{\prime}(x)=\phi(x)\exp[ie^{q}_{\phi}\,\theta(x)\big{]},$
(34c) for the baryon and meson fields, and
$\mathscr{A}^{\mu}(x)\to\mathscr{A}^{\prime\mu}(x)=\mathscr{A}^{\mu}(x)+\partial^{\mu}\theta(x)$
(34d)

for the electromagnetic field, where $\theta(x)$ is an auxiliary function.

Table 1: Coupling constants $C_{B\phi}$ and $C_{T\phi}$ for the $pB\phi$ and $pT\phi$ interactions, respectively, and $C_{\phi\phi^{\dagger}}$ and $C^{\prime}_{\phi\phi^{\dagger}}$ for the $pp\phi\phi^{\dagger}$ coupling, and the tree level magnetic moments $C_{B}^{\rm mag}$, $C_{T}^{\rm mag}$, $C_{BT}^{\rm mag}$ and $C_{\phi\phi^{\dagger}}^{\rm mag}$, respectively, for all the allowed flavor channels. $\bm{B}$ |  $\bm{p}$ |  $\bm{n}$ |  $\bm{\Sigma}^{+}$ |  $\bm{\Sigma}^{0}$ |  $\bm{\Sigma}^{-}$ |  $\bm{\Lambda}$ |  $\bm{\Lambda\Sigma^{0}}$
---|---|---|---|---|---|---|---
$C_{B}^{\text{mag}}$ |  $\frac{1}{3}c_{1}+c_{2}$ |  $-\frac{2}{3}c_{1}$ |  $\frac{1}{3}c_{1}+c_{2}$ |  $\frac{1}{3}c_{1}$ |  $\frac{1}{3}c_{1}-c_{2}$ |  $-\frac{1}{3}c_{1}$ |  $\frac{1}{\sqrt{3}}c_{1}$
$\bm{T}$ |  $\bm{\Delta}^{++}$ |  $\bm{\Delta}^{+}$ |  $\bm{\Delta}^{0}$ |  $\bm{\Delta}^{-}$ |  $\bm{\Sigma}^{*+}$ |  $\bm{\Sigma}^{*0}$ |  $\bm{\Sigma}^{*-}$
$C_{T}^{\text{mag}}$ |  $\frac{2}{3}F_{2}^{T}$ |  $\frac{1}{3}F_{2}^{T}$ |  $0$ |  $-\frac{1}{3}F_{2}^{T}$ |  $\frac{1}{3}F_{2}^{T}$ |  $0$ |  $-\frac{1}{3}F_{2}^{T}$
$\bm{BT}$ |  $\bm{p\Delta^{+}}$ |  $\bm{\Delta^{0}}$ |  $\bm{\Sigma^{+}\Sigma^{*+}}$ |  $\bm{\Sigma^{0}\Sigma^{*0}}$ |  $\bm{\Lambda\Sigma^{*0}}$ |  $\bm{\Sigma^{-}\Sigma^{*-}}$ |
$C_{BT}^{\text{mag}}$ |  $-\frac{1}{\sqrt{3}}c_{4}$ |  $-\frac{1}{\sqrt{3}}c_{4}$ |  $\frac{1}{\sqrt{3}}c_{4}$ |  $\frac{1}{2\sqrt{3}}c_{4}$ |  $\frac{1}{2}c_{4}$ |  $0$ |
$\bm{B\phi}$ |  $\bm{p\pi^{0}}$ |  $\bm{n\pi^{+}}$ |  $\bm{\Sigma^{+}K^{0}}$ |  $\bm{\Sigma^{0}K^{+}}$ |  $\bm{\Lambda K^{+}}$ |  |
$C_{B\phi}$ |  $\frac{1}{2}(D+F)$ |  $\frac{1}{\sqrt{2}}(D+F)$ |  $\frac{1}{\sqrt{2}}(D-F)$ |  $\frac{1}{2}(D-F)$ |  $-\frac{1}{\sqrt{12}}(D+3F)$ |  |
$\bm{T\phi}$ | $\bm{\Delta^{0}\pi^{+}}$ | $\bm{\Delta^{+}\pi^{0}}$ | $\bm{\Delta^{++}\pi^{-}}$ | $\bm{\Sigma^{*+}K^{0}}$ | $\bm{\Sigma^{*0}K^{+}}$ |  |
$C_{T\phi}$ | $-\frac{1}{\sqrt{6}}{\cal C}$ | $-\frac{1}{\sqrt{3}}{\cal C}$ | $\frac{1}{\sqrt{2}}{\cal C}$ | $\frac{1}{\sqrt{6}}{\cal C}$ | $-\frac{1}{\sqrt{12}}{\cal C}$ |  |
$\bm{\phi\phi^{\dagger}}$ | $\bm{\pi^{+}\pi^{-}}$ | $\bm{K^{0}\overline{K}^{0}}$ | $\bm{K^{+}K^{-}}$ |  |  |  |
$C_{\phi\phi^{\dagger}}$ | $\frac{1}{2}$ | $\frac{1}{2}$ | 1 |  |  |  |
$C_{\phi\phi^{\dagger}}^{\prime}$ | $4(b_{10}+b_{11})$ | $4(b_{11}-b_{10})$ | $8b_{11}+2b_{9}$ |  |  |  |
$C_{\phi\phi^{\dagger}}^{\rm mag}$ | $-\frac{1}{2}(c_{1}+c_{2})$ | $0$ | $-c_{2}$ |  |  |  |

The gauge link (33) can next be expanded in powers of the charge
$e_{\phi}^{q}$,

$\displaystyle{\cal G}^{q}_{\phi}(x+b,x+a)$ $\displaystyle=$
$\displaystyle\exp[-ie^{q}_{\phi}\,(a-b)^{\mu}\int_{0}^{1}\differential{t}\,\mathscr{A}_{\mu}\big{(}x+at+b(1-t)\big{)}\Big{]}$
(35) $\displaystyle=$ $\displaystyle 1\ +\ \delta{\cal G}^{q}_{\phi}\ +\
\cdots,$

using a change of variables $z^{\mu}\to x^{\mu}+a^{\mu}\,t+b^{\mu}\,(1-t)$,

$\displaystyle\delta{\cal G}^{q}_{\phi}$ $\displaystyle=$ $\displaystyle-\
ie^{q}_{\phi}\,(a-b)^{\mu}\int_{0}^{1}\differential{t}\,\mathscr{A}_{\mu}\big{(}x+at+b(1-t)\big{)}.$
(36)

This allows the nonlocal Lagrangian ${\cal L}^{\rm(nonloc)}$ in Eq. (32) to be
further decomposed into free and interacting parts, with the latter consisting
of purely hadronic (${\cal L}^{\rm(nonloc)}_{\rm had}$), electromagnetic
(${\cal L}^{\rm(nonloc)}_{\rm em}$), and gauge link (${\cal
L}^{\rm(nonloc)}_{\rm link}$) contributions. The hadronic and electromagnetic
interaction parts of ${\cal L}^{\rm(nonloc)}$ are obtained from the first term
in Eq. (35), and given by

$\displaystyle{\cal L}^{\rm(nonloc)}_{\rm had}(x)$ $\displaystyle=$
$\displaystyle\bar{p}(x)\left[\frac{C_{B\phi}}{f}\,\gamma^{\mu}\gamma_{5}B(x)+\frac{C_{T\phi}}{f}\,\Theta^{\mu\nu}T_{\nu}(x)\right]\\!\int\\!\differential[4]{a}\,F(a)\,\partial_{\mu}\phi(x+a)+{\rm
H.c.}$ $\displaystyle+\
\frac{iC_{\phi\phi^{\dagger}}}{2f^{2}}\bar{p}(x)\gamma^{\mu}p(x)\int\\!\differential[4]{a}\\!\int\\!\differential[4]{b}\
F(a)F(b)\left[\phi(x+a)\partial_{\mu}\phi^{\dagger}(x+b)-\partial_{\mu}\phi(x+a)\phi^{\dagger}(x+b)\right]$
$\displaystyle+\
\frac{iC_{\phi\phi^{\dagger}}^{\prime}}{2f^{2}}\bar{p}(x)\sigma^{\mu\nu}p(x)\int\\!\differential[4]{a}\\!\int\\!\differential[4]{b}\
F(a)F(b)\left[\partial_{\mu}\phi(x+a)\partial_{\nu}\phi^{\dagger}(x+b)-\partial_{\mu}\phi(x+a)\partial_{\nu}\phi^{\dagger}(x+b)\right],$

and

$\displaystyle{\cal L}^{\rm(nonloc)}_{\rm em}(x)$ $\displaystyle=$
$\displaystyle e^{q}_{B}\,\bar{B}(x)\gamma^{\mu}B(x)\,\mathscr{A}_{\mu}(x)\ +\
e^{q}_{T}\,\overline{T}_{\mu}(x)\gamma^{\mu\nu\alpha}T_{\nu}(x)\,\mathscr{A}_{\alpha}(x)$
(38)
$\displaystyle+ie^{q}_{\phi}\left[\partial^{\mu}\phi(x)\phi^{\dagger}(x)-\phi(x)\partial^{\mu}\phi^{\dagger}(x)\right]\mathscr{A}_{\mu}(x)$
$\displaystyle-
ie^{q}_{\phi}\,\bar{p}(x)\left[\frac{C_{B\phi}}{f}\,\gamma^{\mu}\gamma_{5}B(x)+\frac{C_{T\phi}}{f}\,\Theta^{\mu\nu}T_{\nu}(x)\right]\int\\!\differential[4]{a}\,F(a)\,\phi(x+a)\mathscr{A}_{\mu}(x)+{\rm
H.c.}$
$\displaystyle-\frac{e^{q}_{\phi}C_{\phi\phi^{\dagger}}}{2f^{2}}\,\bar{p}(x)\gamma^{\mu}p(x)\int\\!\differential[4]{a}\int\\!\differential[4]{b}\
F(a)F(b)\,\phi(x+a)\phi^{\dagger}(x+b)\mathscr{A}_{\mu}(x)$
$\displaystyle-\frac{C_{\phi\phi^{\dagger}}^{\prime}}{f^{2}}\,\bar{p}(x)\sigma^{\mu\nu}p(x)\int\\!\differential[4]{a}\int\\!\differential[4]{b}\
F(a)F(b)\,\phi(x+a)\partial_{\nu}\phi^{\dagger}(x+b)\mathscr{A}_{\mu}(x)$
$\displaystyle+\frac{C_{B}^{\rm
mag}}{4M_{B}}\bar{B}(x)\sigma^{\mu\nu}B(x)F_{\mu\nu}(x)+\frac{iC_{BT}^{\rm
mag}}{4M_{B}}\bar{B}(x)\gamma^{\mu}\gamma_{5}T^{\nu}(x)F_{\mu\nu}(x)-\frac{C_{T}^{\rm
mag}}{4M_{T}}\overline{T}_{\alpha}(x)\sigma^{\mu\nu}T^{\alpha}(x)F_{\mu\nu}(x)$
$\displaystyle+\frac{C_{\phi\phi^{\dagger}}^{\rm
mag}}{4M_{B}f^{2}}\,\bar{p}(x)\sigma^{\mu\nu}p(x)\int\\!\differential[4]{a}\int\\!\differential[4]{b}\,F(a)F(b)\,\phi(x+a)\phi^{\dagger}(x+b)F_{\mu\nu}(x),$

respectively. The second term in Eq. (35), which explicitly depends on the
gauge link, gives rise to an additional contribution to the Lagrangian density
that can be expanded as

$\displaystyle{\cal L}^{\rm(nonloc)}_{\rm link}(x)$ $\displaystyle=$
$\displaystyle-
ie^{q}_{\phi}\,\bar{p}(x)\left[\frac{C_{B\phi}}{f}\,\gamma^{\rho}\gamma_{5}B(x)+\frac{C_{T\phi}}{f}\,\Theta^{\rho\nu}T_{\nu}(x)\right]$
(39) $\displaystyle\hskip
8.5359pt\times\int_{0}^{1}\differential{t}\int\\!\differential[4]{a}\,F(a)a^{\mu}\,\partial_{\rho}(\phi(x+a)\mathscr{A}_{\mu}\big{(}x+at)\big{)}+{\rm
H.c.}$ $\displaystyle+$
$\displaystyle\frac{e^{q}_{\phi}C_{\phi\phi^{\dagger}}}{2f^{2}}\,\bar{p}(x)\gamma^{\rho}p(x)\int_{0}^{1}\differential{t}\int\\!\differential[4]{a}\\!\int\\!\differential[4]{b}\,F(a)\,F(b)\,(a-b)^{\mu}\,$
$\displaystyle\hskip
8.5359pt\times\left[\phi(x+a)\partial_{\rho}\phi^{\dagger}(x+b)-\partial_{\rho}\phi(x+a)\phi^{\dagger}(x+b)\right]\mathscr{A}_{\mu}\big{(}x+at+b(1-t)\big{)}.$

Finally, the quark current for the nonlocal theory can be written as a sum of
two terms, from the usual electromagnetic current obtained from Eq. (38) with
minimal substitution, $J_{q,\rm em}^{\mu}$, and from the additional term
associated with the gauge link, $\delta J_{q}^{\mu}$,

$J_{q,\rm em}^{\mu}(x)\equiv\frac{\delta\int\\!\differential[4]{y}\,{\cal
L}_{\rm em}^{\rm(nonloc)}(y)}{\delta\mathscr{A_{\mu}}(x)},\qquad\delta
J_{q}^{\mu}(x)\equiv\frac{\delta\int\\!\differential[4]{y}\,{\cal
L}^{\rm(nonloc)}_{\rm link}(y)}{\delta\mathscr{A_{\mu}}(x)},$ (40)

where, explicitly,

$\displaystyle J_{q,\rm em}^{\mu}(x)$ $\displaystyle=$ $\displaystyle
e^{q}_{B}\,\bar{B}(x)\gamma^{\mu}B(x)+e^{q}_{T}\,\overline{T}_{\alpha}(x)\gamma^{\alpha\nu\mu}T_{\nu}(x)+ie^{q}_{\phi}\big{[}\partial^{\mu}\phi(x)\phi^{\dagger}(x)-\phi(x)\partial^{\mu}\phi^{\dagger}(x)\big{]}$
(41) $\displaystyle-$ $\displaystyle
ie^{q}_{\phi}\,\bigg{(}\int\\!\differential[4]{a}F(a)\,\bar{p}(x)\left[\frac{C_{B\phi}}{f}\,\gamma^{\mu}\gamma_{5}B(x)+\frac{C_{T\phi}}{f}\,\Theta^{\mu\nu}T_{\nu}(x)\right]\phi(x+a)\
+\ {\rm H.c.}\bigg{)}$ $\displaystyle-$
$\displaystyle\frac{e^{q}_{\phi}C_{\phi\phi^{\dagger}}}{2f^{2}}\int\\!\differential[4]{a}\int\\!\differential[4]{b}F(a)F(b)\,\bar{p}(x)\gamma^{\mu}p(x)\,\phi(x+a)\phi^{\dagger}(x+b)$
$\displaystyle+$ $\displaystyle\frac{C_{B}^{\rm
mag}}{2M_{B}}\,\int\\!\differential[4]{a}F(a)\,\partial_{\nu}\big{(}\bar{p}(x)\sigma^{\mu\nu}p(x)\big{)}-\frac{C_{T}^{\rm
mag}}{2M_{T}}\,\int\\!\differential[4]{a}F(a)\,\partial_{\nu}\big{(}\overline{T}_{\alpha}(x)\sigma^{\mu\nu}T^{\alpha}(x)\big{)}$
$\displaystyle+$ $\displaystyle\frac{iC_{BT}^{\rm
mag}}{4M_{B}}\,(\partial_{\nu}\big{(}\bar{p}(x)\gamma^{\nu}\gamma_{5}T^{\mu}(x)\big{)}-\partial_{\nu}\big{(}\bar{p}(x)\gamma^{\mu}\gamma_{5}T^{\nu}(x)\big{)}$
$\displaystyle+$ $\displaystyle\frac{C_{\phi\phi^{\dagger}}^{\rm
mag}}{2M_{B}f^{2}}\,\int\\!\differential[4]{a}\int\\!\differential[4]{b}F(a)F(b)\,\partial_{\nu}\big{(}\bar{p}(x)\sigma^{\mu\nu}p(x)\phi(x+a)\phi^{\dagger}(x+b)\big{)},$
$\displaystyle\delta J_{q}^{\mu}(x)$ $\displaystyle=$ $\displaystyle
ie^{q}_{\phi}\int_{0}^{1}\differential{t}\int\\!\differential[4]{a}F(a)\,a^{\mu}\,$
(42)
$\displaystyle\times\partial_{\rho}\left(\bar{p}(x-at)\left[\frac{C_{B\phi}}{f}\,\gamma^{\rho}\gamma_{5}B(x-at)\,+\,\frac{C_{T\phi}}{f}\,\Theta^{\rho\nu}T_{\nu}(x-at)\right]\right)\phi(x+a\bar{t}\big{)}+{\rm
H.c.}$ $\displaystyle-$
$\displaystyle\frac{e^{q}_{\phi}C_{\phi\phi^{\dagger}}}{2f^{2}}\int_{0}^{1}\differential{t}\\!\int\\!\differential[4]{a}\\!\int\\!\differential[4]{b}F(a)F(b)\,(a-b)^{\mu}\,$
$\displaystyle\times\bigg{[}\partial_{\rho}\Big{(}\bar{p}\big{(}x-at-b\bar{t}\big{)}\gamma^{\rho}p\big{(}x-at-b\bar{t}\big{)}\phi\,\big{(}x+(a-b)\bar{t}\big{)}\Big{)}\phi^{\dagger}\big{(}x-(a-b)t)$
$\displaystyle\hskip
5.69046pt-\partial_{\rho}\Big{(}\bar{p}\big{(}x-at-b\bar{t}\big{)}\gamma^{\rho}p\big{(}x-at-b\bar{t}\big{)}\phi^{\dagger}\big{(}x-(a-b)t\big{)}\Big{)}\phi\big{(}x+(a-b)\bar{t}\big{)}\bigg{]},$

with $\bar{t}\equiv 1-t$. Compared with the local theory, Eqs. (24) and (29),
the nonlocal formulation in Eqs. (LABEL:eq:Lnonloc_had)–(42) includes the
regulator function $F(a)$. In the limit where $F(a)\to\delta^{(4)}(a)$, which
corresponds to taking the momentum space form factor to unity, the local limit
can be obtained from the nonlocal result. In the next section we will apply
the nonlocal interaction derived here to compute the hadronic splitting
functions for protons transitioning to baryons and pseudoscalar mesons.

##  III Nonforward splitting functions

The GPDs for a quark flavor $q$ in a proton with initial momentum $p$ and
final momentum $p^{\prime}$ are defined by the Fourier transform of the matrix
elements of the quark bilocal field operators $\psi_{q}$ as

$\displaystyle\int_{-\infty}^{\infty}\frac{\differential{\lambda}}{2\pi}e^{-ix\lambda}\langle
p^{\prime}|\bar{\psi}_{q}(\tfrac{1}{2}\lambda
n)\not{n}\,\psi_{q}(-\tfrac{1}{2}\lambda n)|p\rangle$ $\displaystyle=$
$\displaystyle\bar{u}(p^{\prime})\Big{[}\not{n}H^{q}(x,\xi,t)+\frac{i\sigma^{\mu\nu}n_{\mu}\Delta_{\nu}}{2M}\,E^{q}(x,\xi,t)\Big{]}u(p),$

where $n_{\mu}$ is the light-cone vector which projects the “plus” component
of momenta and $\lambda$ is a dimensionless parameter. From Lorentz
invariance, the Dirac ($H^{q}$) and Pauli ($E^{q}$) GPDs are typically written
as functions of the light-cone momentum fraction $x$ of the proton carried by
the initial quark with momentum $k_{q}$ and the skewness parameter $\xi$,
which are defined as

$x\,\equiv\,\frac{k_{q}^{+}}{P^{+}},\qquad\xi\,\equiv\,-\frac{\Delta^{+}}{2P^{+}},$
(44)

where

$P\,=\,\frac{1}{2}\big{(}p+p^{\prime}\big{)},\qquad\Delta\,=\,p^{\prime}-p,$
(45)

are the average of the initial and final proton momenta and the momentum
difference, respectively. The light-cone components $k^{+}$ and $k^{-}$ of any
four-vector $k^{\mu}$ are defined as $k^{+}=\frac{1}{\sqrt{2}}(k^{0}+k^{3})$
and $k^{-}=\frac{1}{\sqrt{2}}(k^{0}-k^{3})$. The GPDs are also functions of
the hadronic four-momentum transfer squared, $t\equiv\Delta^{2}$. The
dependence of the GPDs on the fourth variable, typically taken to be the four-
momentum transfer squared from the incident lepton, $Q^{2}$, is suppressed.

Integrating the $H^{q}$ and $E^{q}$ GPDs over $x$, one obtains the Dirac and
Pauli form factors for a given quark flavor $q$, respectively,

$\displaystyle
F_{1}^{q}(t)=\int_{-1}^{1}\,\differential{x}H^{q}(x,\xi,t),\qquad
F_{2}^{q}(t)=\int_{-1}^{1}\,\differential{x}E^{q}(x,\xi,t),$ (46)

and summing over the quark flavors gives the nucleon Dirac, $F_{1}^{N}$, and
Pauli, $F_{1}^{N}$, form factors,

$\displaystyle F_{1,2}^{N}(t)=\sum_{q}e_{q}\,F_{1,2}^{\,q}(t).$ (47)

The combination of these form factors can generate the usual Sachs electric
and magnetic form factors as

$\displaystyle G_{E}^{N}(t)=F_{1}^{N}(t)+{t\over{4M^{2}}}F_{2}^{N}(t),\qquad
G_{M}^{N}(t)=F_{1}^{N}(t)+F_{2}^{N}(t).$ (48)

In our calculation, we set $\xi=0$ and calculate the GPDs in the
$\Delta^{+}=0$ frame Brodsky  _et al._ (1998), in which the hadron momenta are
parametrized as

$\displaystyle p^{\mu}$ $\displaystyle=$
$\displaystyle\Big{(}P^{+},P^{-},-\frac{1}{2}{\bm{\Delta}_{\perp}}\Big{)},\qquad
p^{\prime\mu}\,=\,\Big{(}P^{+},P^{-},\frac{1}{2}{\bm{\Delta}_{\perp}}\Big{)},$
(49)

where the momentum transfer $\Delta^{\mu}$ is purely in the transverse
direction.

In the application of our nonlocal EFT framework to GPDs, we need to compute
the nonforward splitting functions defined by the matrix elements of the
hadronic level currents derived in the previous section. The electromagnetic
vertex is given by

$\langle
N(p^{\prime})|J^{\mu}|N(p)\rangle\,=\,\bar{u}(p^{\prime})\Big{[}\gamma^{\mu}F_{1}^{N}(t)+\frac{i\sigma^{\mu\nu}\Delta_{\nu}}{2M}F_{2}^{N}(t)\Big{]}u(p)\,\equiv\int\\!\differential[4]{k}\widetilde{\Gamma}^{\mu}(k),$
(50)

where the integrand $\widetilde{\Gamma}^{\mu}(k)$ depends on the internal
meson momentum, $k$. Defining the light-cone momentum fraction of the target
nucleon carried by the interacting hadron, $y=k^{+}/P^{+}$, the Dirac-like
splitting function $f(y,t)$ and Pauli-like splitting function $g(y,t)$ are
related to the vertex by

$\bar{u}(p^{\prime})\Big{[}\gamma^{+}f(y,t)+\frac{i\sigma^{+\nu}\Delta_{\nu}}{2M}\,g(y,t)\Big{]}u(p)=\int\differential[4]{k}\widetilde{\Gamma}^{+}(k)\,\delta\Big{(}y-\frac{k^{+}}{P^{+}}\Big{)}\,\equiv\,\Gamma^{+}.$
(51)

One can easily verify that the integral of the splitting functions over $y$
leads to the corresponding form factors in (50).

Figure 1: One-loop diagrams for the proton to pseudoscalar meson (dashed
lines) and octet baryon (solid lines) or decuplet baryon (double solid lines)
splitting functions: (a)–(c) octet baryon rainbow diagrams, (d)–(g) octet
baryon Kroll-Ruderman diagrams, (h)–(j) tadpole diagrams, (k)–(l) bubble
diagrams, (m)–(o) decuplet baryon rainbow diagrams, (p)–(q) octet-decuplet
transition rainbow diagrams, (r)–(u) decuplet baryon Kroll-Ruderman diagrams.
The crossed circles ($\otimes$) represent the interaction with external vector
field from the minimal substitution, filled circles ($\bullet$) denote
additional gauge link interaction with the external field, black squares
($\blacksquare$) represent the magnetic interaction in Eq. (19), and gray
squares (${{\color[rgb]{.5,.5,.5}\blacksquare}}$) denote the interaction in
Eq. (23).

The diagrams that are relevant for the calculation of the one-meson loop
contributions to GPDs are shown in Fig. 1. In the following we outline the
calculation of the corresponding splitting functions, beginning with the
diagrams involving only octet baryons [Fig. 1(a)–1(l)], and then presenting
results for contributions with intermediate decuplet baryons. Since the final
results, after integration, for the latter are rather lengthy, we collect the
complete expressions for these in Appendix A.

###  III.1 Octet baryon intermediate states

Starting with the octet baryon rainbow diagram in Fig. 1(a), in which the
external field couples to the meson, the contribution of this diagram to the
matrix element $\Gamma^{+}$ in Eq. (51) is given by

$\displaystyle\Gamma_{(\rm a)}^{+}$ $\displaystyle=$
$\displaystyle\bar{u}(p^{\prime})\frac{C^{2}_{B\phi}}{f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}(\not{k}+\not{\Delta})\gamma_{5}\,\widetilde{F}(k+\Delta)\frac{i}{D_{\phi}(k+\Delta)}\,2k^{+}$
(52) $\displaystyle\hskip
65.44142pt\times\frac{i}{D_{\phi}(k)}\frac{i(\not{p}-\not{k}+M_{B})}{D_{B}(p-k)}\gamma_{5}\not{k}\,\widetilde{F}(k)\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)}\,u(p)$
$\displaystyle\equiv$
$\displaystyle\bar{u}(p^{\prime})\bigg{[}\gamma^{+}f_{\phi B}^{({\rm
rbw})}(y,t)+\frac{i\sigma^{+\nu}\Delta_{\nu}}{2M}g_{\phi B}^{({\rm
rbw})}(y,t)\bigg{]}u(p),$ (53)

where the propagator factors $D_{\phi}(k)$ and $D_{B}(p)$ are defined as

$D_{\phi}(k)=k^{2}-m_{\phi}^{2}+i\epsilon,\qquad
D_{B}(p)=p^{2}-M_{B}^{2}+i\epsilon,$ (54)

and $m_{\phi}$ is the meson mass. The function $\widetilde{F}$ regulates the
ultraviolet divergence in the loop integration, and for simplicity is chosen
to be a function of the meson momentum only (see Sec. VI below). After
simplifying the combinations of Dirac $\gamma$ matrices in Eq. (52), the
“Dirac” $f_{\phi B}^{({\rm rbw})}(y,t)$ and “Pauli” $g_{\phi B}^{({\rm
rbw})}(y,t)$ splitting functions can be written as

$\displaystyle f_{\phi B}^{({\rm rbw})}(y,t)$ $\displaystyle=$
$\displaystyle\frac{C_{B\phi}^{2}}{f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\frac{-iF_{\phi
B}^{({\rm
rbw})}}{D_{B}(p-k)D_{\phi}(k+\Delta)D_{\phi}(k)}\widetilde{F}(k)\widetilde{F}(k+\Delta)\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)},$
$\displaystyle g_{\phi B}^{({\rm rbw})}(y,t)$ $\displaystyle=$
$\displaystyle\frac{C_{B\phi}^{2}}{f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\frac{-iG_{\phi
B}^{({\rm
rbw})}}{D_{B}(p-k)D_{\phi}(k+\Delta)D_{\phi}(k)}\widetilde{F}(k)\widetilde{F}(k+\Delta)\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)},$

where the trace factors in the numerator of the integrand are given by

$\displaystyle F_{\phi B}^{({\rm rbw})}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,yk^{2}\,\big{(}4k\cdot
P+4M\overline{M}_{\\!B}+y\,t\big{)}-\,y\,\big{(}2(k\cdot
p)^{2}-M\overline{M}_{\\!B}\,(2k\cdot\Delta+y\,t)\big{)}$ (56a)
$\displaystyle+\,k\cdot p\,\big{(}2k\cdot p^{\prime}+y\,t\big{)},$
$\displaystyle G_{\phi B}^{({\rm rbw})}$ $\displaystyle=$
$\displaystyle-2yM\,\big{(}2yM^{2}\overline{M}_{\\!B}+(\overline{M}_{\\!B}+2yM)(k^{2}-2k\cdot
p)\big{)},$ (56b)

with $\overline{M}_{\\!B}\equiv M+M_{B}$.

For the baryon rainbow diagram in Fig. 1(b), the photon couples to the
intermediate octet baryon through an electric vertex. The contribution of this
diagram to the nonforward matrix element $\Gamma^{+}$ is given by

$\displaystyle\Gamma_{\rm(b)}^{+}$ $\displaystyle=$
$\displaystyle\bar{u}(p^{\prime})\,\frac{C^{2}_{B\phi}}{f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\,\not{k}\gamma_{5}\,\widetilde{F}(k)\frac{i}{D_{\phi}(k)}\frac{i(\not{p^{\prime}}-\not{k}+M_{B})}{D_{B}(p^{\prime}-k)}\gamma^{+}\frac{i(\not{p}-\not{k}+M_{B})}{D_{B}(p-k)}$
(57)
$\displaystyle\qquad\qquad\qquad\quad\times\,\gamma_{5}\not{k}\,\widetilde{F}(k)\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)}\,u(p).$

Using a similar projection as in Eq. (53), the splitting functions
$f_{B\phi}^{({\rm rbw})}(y,t)$ and $g_{B\phi}^{({\rm rbw})}(y,t)$ are obtained
as

$\displaystyle f_{B\phi}^{({\rm rbw})}(y,t)$ $\displaystyle=$
$\displaystyle\frac{C^{2}_{B\phi}}{f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\frac{-iF_{B\phi}^{({\rm
rbw})}}{D_{B}(p^{\prime}-k)D_{B}(p-k)D_{\phi}(k)}\,\widetilde{F}^{2}(k)\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)},$
(58a) $\displaystyle g_{B\phi}^{({\rm rbw})}(y,t)$ $\displaystyle=$
$\displaystyle\frac{C^{2}_{B\phi}}{f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\frac{-iG_{B\phi}^{({\rm
rbw})}}{D_{B}(p^{\prime}-k)D_{B}(p-k)D_{\phi}(k)}\,\widetilde{F}^{2}(k)\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)},$
(58b)

where the trace factors in the integrand are given by

$\displaystyle F_{B\phi}^{({\rm rbw})}$ $\displaystyle=$ $\displaystyle-k\cdot
p\,\big{(}4k\cdot
p^{\prime}+y\overline{M}_{\\!B}\Delta_{B}\big{)}+k^{2}\big{(}4k\cdot
P+\overline{M}_{\\!B}(\overline{M}_{\\!B}-2yM)\big{)}$ (59a)
$\displaystyle-\,\frac{1}{2}y\overline{M}_{\\!B}\big{(}y\overline{M}_{\\!B}\,t+2\Delta_{B}\,k\cdot
p^{\prime}\big{)}-k^{4},$ $\displaystyle G_{B\phi}^{({\rm rbw})}$
$\displaystyle=$
$\displaystyle-\frac{2M\overline{M}_{\\!B}}{t}\Big{(}2(k\cdot\Delta)^{2}+y\,t\,\big{(}4k\cdot
p+k\cdot p^{\prime}-k^{2}-yM\overline{M}_{\\!B}\big{)}\Big{)},$ (59b)

with $\Delta_{B}\equiv M_{B}-M$.

The diagram in Fig. 1(c) involves the magnetic photon-baryon interaction. The
contribution of this diagram to $\Gamma^{+}$ can be written as

$\displaystyle\Gamma_{\rm(c)}^{+}$ $\displaystyle=$
$\displaystyle\bar{u}(p^{\prime})\,\frac{C^{2}_{B\phi}}{f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\,\not{k}\gamma_{5}\widetilde{F}(k)\frac{i(\not{p^{\prime}}-\not{k}+M_{B})}{D_{B}(p^{\prime}-k)}\frac{i\sigma^{+\nu}\Delta_{\nu}}{2M_{B}}\frac{i(\not{p}-\not{k}+M_{B})}{D_{B}(p-k)}\frac{i}{D_{\phi}(k)}$
(60)
$\displaystyle\qquad\qquad\qquad\quad\times\,\gamma_{5}\not{k}\,\widetilde{F}(k)\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)}\,u(p).$

The splitting functions $f_{B\phi}^{({\rm rbw\,mag})}(y,t)$ and
$g_{B\phi}^{({\rm rbw\,mag})}(y,t)$ in this case are given by

$\displaystyle f_{B\phi}^{({\rm rbw\,mag})}(y,t)$ $\displaystyle=$
$\displaystyle\frac{C^{2}_{B\phi}}{2f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\frac{-iF_{B\phi}^{({\rm
rbw\,mag})}}{D_{B}(p^{\prime}-k)D_{B}(p-k)D_{\phi}(k)}\widetilde{F}^{2}(k)\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)},$
(61a) $\displaystyle g_{B\phi}^{({\rm rbw\,mag})}(y,t)$ $\displaystyle=$
$\displaystyle\frac{C^{2}_{B\phi}}{2f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\frac{-iG_{B\phi}^{({\rm
rbw\,mag})}}{D_{B}(p^{\prime}-k)D_{B}(p-k)D_{\phi}(k)}\widetilde{F}^{2}(k)\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)},$
(61b)

where

$\displaystyle F_{B\phi}^{({\rm rbw\,mag})}$ $\displaystyle=$
$\displaystyle\frac{\overline{M}_{\\!B}}{M_{B}}\Big{(}2(k\cdot\Delta)^{2}+y\,t\big{(}2k\cdot
P-k^{2}-y\,M\overline{M}_{\\!B}\big{)}\Big{)},$ (62a) $\displaystyle
G_{B\phi}^{({\rm rbw\,mag})}$ $\displaystyle=$
$\displaystyle-\frac{2M}{M_{B}\,t}\Big{(}2\overline{M}_{\\!B}^{2}\,(k\cdot\Delta)^{2}+2y\overline{M}_{\\!B}(2M+\overline{M}_{\\!B})\,t\,k\cdot
P-2y^{2}M^{2}\overline{M}_{\\!B}^{2}\,t$ (62b)
$\displaystyle\qquad\qquad-\,4\,t\,k\cdot p~{}k\cdot
p^{\prime}+\,tk^{2}\big{(}4k\cdot
P-\overline{M}_{\\!B}(\overline{M}_{\\!B}+2yM)+k^{2}\big{)}\Big{)}.$

The magnetic coupling constant $C_{B}^{\rm mag}$ does not appear in the
splitting functions, but rather is included in the input GPDs.

For the Kroll-Ruderman diagrams in Figs. 1(d) and 1(e), the contributions to
the matrix element $\Gamma^{+}$ is given by

$\displaystyle\Gamma_{\rm(d)+(e)}^{+}$ $\displaystyle=$
$\displaystyle\bar{u}(p^{\prime})\,\frac{C^{2}_{B\phi}}{f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\bigg{[}\not{k}\gamma_{5}\frac{i(\not{p^{\prime}}-\not{k}+M_{B})}{D_{B}(p^{\prime}-k)}i\gamma_{5}\gamma^{+}-i\gamma^{+}\gamma_{5}\frac{i(\not{p}-\not{k}+M_{B})}{D_{B}(p-k)}\gamma_{5}\not{k}\bigg{]}$
(63) $\displaystyle\hskip
91.04872pt\times\frac{i}{D_{\phi}(k)}\,\widetilde{F}^{2}(k)\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)}\,u(p).$

The corresponding Kroll-Ruderman splitting functions $f^{({\rm
KR})}_{B\phi}(y,t)$ and $g^{({\rm KR})}_{B\phi}(y,t)$ are then defined as

$\displaystyle f^{({\rm KR})}_{B\phi}(y,t)$ $\displaystyle=$
$\displaystyle\frac{C^{2}_{B\phi}}{f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\bigg{[}\frac{-iF^{({\rm
KR\,1})}_{B\phi}}{D_{B}(p^{\prime}-k)}+\frac{-iF^{({\rm
KR\,2})}_{B\phi}}{D_{B}(p-k)}\bigg{]}\frac{1}{D_{\phi}(k)}\widetilde{F}^{2}(k)\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)},$
(64a) $\displaystyle g^{({\rm KR})}_{B\phi}(y,t)$ $\displaystyle=$
$\displaystyle\frac{C^{2}_{B\phi}}{f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\bigg{[}\frac{-iG^{({\rm
KR\,1})}_{B\phi}}{D_{B}(p^{\prime}-k)}+\frac{-iG^{({\rm
KR\,2})}_{B\phi}}{D_{B}(p-k)}\bigg{]}\frac{1}{D_{\phi}(k)}\widetilde{F}^{2}(k)\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)},$
(64b)

where the trace factors in the integrands are given by

$\displaystyle F^{({\rm KR\,1})}_{B\phi}$ $\displaystyle=$ $\displaystyle
k^{2}-2k\cdot p^{\prime}+yM\overline{M}_{\\!B},$ (65a) $\displaystyle F^{({\rm
KR\,2})}_{B\phi}$ $\displaystyle=$ $\displaystyle k^{2}-2k\cdot
p\,+yM\overline{M}_{\\!B},$ (65b) $\displaystyle G^{({\rm KR\,1})}_{B\phi}$
$\displaystyle=$
$\displaystyle+\frac{M\overline{M}_{\\!B}}{t}\big{(}2k\cdot\Delta-y\,t\big{)},$
(65c) $\displaystyle G^{({\rm KR\,2})}_{B\phi}$ $\displaystyle=$
$\displaystyle-\frac{M\overline{M}_{\\!B}}{t}\big{(}2k\cdot\Delta+y\,t\big{)}.$
(65d)

The additional Kroll-Ruderman diagrams generated from the expansion of the
gauge link terms are shown in Fig. 1(f) and 1(g), and are important to ensure
that the renormalized charge of the proton (neutron) is 1 (0). The
contribution of these two additional diagrams with intermediate octet baryons
is expressed as

$\displaystyle\Gamma_{\rm(f)+(g)}^{+}$ $\displaystyle=$
$\displaystyle\bar{u}(p^{\prime})\frac{C^{2}_{B\phi}}{f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\,\widetilde{F}(k)$
$\displaystyle\times\bigg{[}\not{k}\gamma_{5}\frac{i(\not{p^{\prime}}-\not{k}+M_{B})}{D_{B}(p^{\prime}-k)}\frac{i}{D_{\phi}(k)}(\not{k}-\not{\Delta})\gamma_{5}\frac{2ik^{+}}{2k\cdot\Delta-t}\,\big{(}\widetilde{F}(k-\Delta)-\widetilde{F}(k)\big{)}$
$\displaystyle+\,(\not{k}+\not{\Delta})\gamma_{5}\frac{-2ik^{+}}{2k\cdot\Delta+t}\frac{i(\not{p}-\not{k}+M_{B})}{D_{B}(p-k)}\frac{i}{D_{\phi}(k)}\not{k}\gamma_{5}\,\big{(}\widetilde{F}(k+\Delta)-\widetilde{F}(k)\big{)}\bigg{]}\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)}\,u(p).$

The respective splitting functions for these gauge link diagrams, $\delta
f^{({\rm KR})}_{B\phi}(y,t)$ and $\delta g^{({\rm KR})}_{B\phi}(y,t)$, can be
written as

$\displaystyle\delta f^{({\rm KR})}_{B\phi}(y,t)$ $\displaystyle=$
$\displaystyle\frac{C^{2}_{B\phi}}{f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\,\widetilde{F}(k)\frac{1}{D_{\phi}(k)}\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)}$
(67a) $\displaystyle\times\bigg{[}\frac{i\delta F^{({\rm
KR\,1})}_{B\phi}}{D_{B}(p^{\prime}-k)}\frac{\widetilde{F}(k-\Delta)-\widetilde{F}(k)}{-2k\cdot\Delta+t}\,+\,\frac{i\delta
F^{({\rm
KR\,2})}_{B\phi}}{D_{B}(p-k)}\frac{\widetilde{F}(k)-\widetilde{F}(k+\Delta)}{2k\cdot\Delta+t}\bigg{]},\hskip
28.45274pt$ $\displaystyle\delta g^{({\rm KR})}_{B\phi}(y,t)$ $\displaystyle=$
$\displaystyle\frac{C^{2}_{B\phi}}{f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\,\widetilde{F}(k)\frac{1}{D_{\phi}(k)}\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)}$
(67b) $\displaystyle\times\bigg{[}\frac{i\delta G^{({\rm
KR\,1})}_{B\phi}}{D_{B}(p^{\prime}-k)}\frac{\widetilde{F}(k-\Delta)-\widetilde{F}(k)}{-2k\cdot\Delta+t}\,+\,\frac{i\delta
G^{({\rm
KR\,2})}_{B\phi}}{D_{B}(p-k)}\frac{\widetilde{F}(k)-\widetilde{F}(k+\Delta)}{2k\cdot\Delta+t}\bigg{]},\hskip
28.45274pt$

where the numerator factors are given by

$\displaystyle\delta F^{({\rm KR\,1})}_{B\phi}$ $\displaystyle=$
$\displaystyle\hskip 9.38945pt\frac{y}{2}\Big{(}4(k\cdot
p)^{2}-k^{2}\big{(}4k\cdot P+4M\overline{M}_{\\!B}+y\,t\big{)}$ (68a)
$\displaystyle\hskip
22.76228pt+\,2M\overline{M}_{\\!B}\big{(}2k\cdot\Delta-y\,t\big{)}+2k\cdot
p^{\prime}\,\big{(}2k\cdot p+y\,t\big{)}\Big{)},$ $\displaystyle\delta
F^{({\rm KR\,2})}_{B\phi}$ $\displaystyle=$
$\displaystyle-\frac{y}{2}\Big{(}4(k\cdot p)^{2}-k^{2}\big{(}4k\cdot
P+4M\overline{M}_{\\!B}+y\,t\big{)}$ (68b) $\displaystyle\hskip
22.76228pt-\,2M\overline{M}_{\\!B}\big{(}2k\cdot\Delta+y\,t\big{)}+2k\cdot
p\,\big{(}2k\cdot p^{\prime}+y\,t\big{)}\Big{)},$ $\displaystyle\delta
G^{({\rm KR\,1})}_{B\phi}$ $\displaystyle=$
$\displaystyle-2My\,\Big{(}\overline{M}_{\\!B}\big{(}2k\cdot
p^{\prime}-2yM^{2}\big{)}-k^{2}\big{(}\overline{M}_{\\!B}+yM\big{)}+2yMk\cdot
p^{\prime}\Big{)},$ (68c) $\displaystyle\delta G^{({\rm KR\,2})}_{B\phi}$
$\displaystyle=$ $\displaystyle\hskip
9.38945pt2My\,\Big{(}\overline{M}_{\\!B}\big{(}2k\cdot
p\,-2yM^{2}\big{)}-k^{2}\big{(}\overline{M}_{\\!B}+yM\big{)}+2yMk\cdot
p\Big{)}.$ (68d)

The contribution to $\Gamma^{+}$ from the tadpole diagram in Fig. 1(h) is
given by a relatively simple expression,

$\Gamma_{\rm(h)}^{+}=\bar{u}(p^{\prime})\,\frac{C_{\phi\phi}}{f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\frac{i}{D_{\phi}(k)}\gamma^{+}\widetilde{F}^{2}(k)\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)}u(p).$
(69)

The splitting function $f^{({\rm tad})}_{\phi}(y,t)$ in this case can be
easily read off from Eq. (69), and is given by

$f^{({\rm
tad})}_{\phi}(y,t)=\frac{C_{\phi\phi}}{f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\frac{i}{D_{\phi}(k)}\widetilde{F}^{2}(k)\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)}.$
(70)

There is no contribution to a Pauli-like splitting function $g^{({\rm
tad})}_{\phi}(y,t)$ from this diagram. The related tadpole diagram that is
associated with the gauge link in Fig. 1(i) makes a contribution to the
$\Gamma^{+}$ matrix element that can be written as

$\displaystyle\Gamma_{\rm(i)}^{+}$ $\displaystyle=$
$\displaystyle\bar{u}(p^{\prime})\frac{C_{\phi\phi}}{f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\,2\not{k}\,\widetilde{F}(k)\frac{i}{D_{\phi}(k)}\frac{2k^{+}}{-2k\cdot\Delta+t}\,\big{(}\widetilde{F}(k-\Delta)-\widetilde{F}(k)\big{)}\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)}u(p),$

with the corresponding splitting function $\delta f^{({\rm tad})}_{\phi}(y,t)$
given by

$\displaystyle\delta f^{({\rm tad})}_{\phi}(y,t)$ $\displaystyle=$
$\displaystyle\frac{iC_{\phi\phi}}{f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\,\widetilde{F}(k)\frac{y\,(4k\cdot
P+y\,t)}{D_{\phi}(k)(-2k\cdot\Delta+t)}\big{(}\widetilde{F}(k-\Delta)-\widetilde{F}(k)\big{)}\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)}.~{}~{}$
(72)

The tadpole diagram in Fig. 1(j) associated with the magnetic interaction
makes a contribution

$\Gamma_{\rm(j)}^{+}=\bar{u}(p^{\prime})\,\frac{C^{{\rm
mag}}_{\phi\phi}}{f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\frac{i}{D_{\phi}(k)}\frac{i\sigma^{+\nu}\Delta_{\nu}}{2M}\widetilde{F}^{2}(k)\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)}\,u(p),$
(73)

which gives rise to a Pauli-like magnetic tadpole splitting function given by,

$\displaystyle g^{({\rm tad\,mag})}_{\phi}(y,t)=\frac{C^{{\rm
mag}}_{\phi\phi}}{f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\frac{i}{D_{\phi}(k)}\widetilde{F}^{2}(k)\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)}\,u(p).$
(74)

The contributions of the bubble diagrams are illustrated in Figs. 1(k) and
1(l). For the regular bubble diagram in Fig. 1(k), one has

$\displaystyle\Gamma_{\rm(k)}^{+}$ $\displaystyle=$
$\displaystyle-\bar{u}(p^{\prime})\,\frac{C_{\phi\phi}}{2f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\,i(2\not{k}+\not{\Delta})\,\widetilde{F}(k+\Delta)\widetilde{F}(k)\,\frac{i}{D_{\phi}(k+\Delta)}2k^{+}\frac{i}{D_{\phi}(k)}\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)}\,u(p).$

As with the tadpole diagrams, this also generates only a Dirac-like splitting
function, $f^{({\rm bub})}_{\phi}(y,t)$, which is expressed as

$\displaystyle f^{({\rm
bub})}_{\phi}(y,t)=\frac{iC_{\phi\phi}}{4f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\frac{y\,(4k\cdot
P+y\,t)}{D_{\phi}(k+\Delta)D_{\phi}(k)}\,\widetilde{F}(k+\Delta)\,\widetilde{F}(k)\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)}.$
(76)

Finally, for the bubble diagram derived from the Lagrangian ${\cal
L}_{B\phi}^{\prime}$ in Eq. (23) and illustrated in Fig. 1(l), the
contribution can be written as

$\displaystyle\Gamma_{\rm(l)}^{+}$ $\displaystyle=$
$\displaystyle\bar{u}(p^{\prime})\,\frac{C^{\prime}_{\phi\phi}}{2f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\,i\sigma^{\lambda\nu}\Delta_{\lambda}k_{\nu}\,\widetilde{F}(k+\Delta)\widetilde{F}(k)\frac{i}{D_{\phi}(k+\Delta)}2k^{+}\frac{i}{D_{\phi}(k)}\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)}\,u(p),$

and produces a Pauli-like magnetic bubble splitting function given by

$\displaystyle g^{\prime({\rm
bub})}_{\phi}(y,t)=\frac{iC^{\prime}_{\phi\phi}}{2f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\frac{yM\,(4k\cdot
P+y\,t)}{D_{\phi}(k+\Delta)D_{\phi}(k)}\,\widetilde{F}(k+\Delta)\,\widetilde{F}(k)\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)}.$
(78)

###  III.2 Decuplet baryon intermediate states

The splitting functions for the diagrams involving decuplet baryons in the
intermediate state in Figs. 1(m)–1(u) are computed in a similar way, although
because of the higher spin of the decuplet baryons the expressions are
typically somewhat more involved. Here we give the basic expressions for the
contributions from each diagram to the matrix element $\Gamma^{+}$, with the
full results for the numerator trace factors in the decuplet splitting
functions given in Appendix A.

Beginning with the decuplet baryon rainbow diagram in Fig. 1(m), the
contribution to $\Gamma^{+}$ is expressed in a similar form to that for the
octet baryon rainbow diagram Fig. 1(a) in Eqs. (52)–(53),

$\displaystyle\Gamma_{\rm(m)}^{+}$ $\displaystyle=$
$\displaystyle-\bar{u}(p^{\prime})\,\frac{C_{T\phi}^{2}}{f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\,(k+\Delta)_{\lambda}\,\Theta^{\lambda\alpha}\widetilde{F}(k+\Delta)\frac{i}{D_{\phi}(k+\Delta)}\,2k^{+}$
(79) $\displaystyle\hskip
62.59596pt\times\frac{i}{D_{\phi}(k)}\frac{i}{\not{p}-\not{k}-M_{T}}S_{\alpha\beta}(p-k)\,\Theta^{\beta\rho}k_{\rho}\,\widetilde{F}(k)\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)}\,u(p)$
$\displaystyle\equiv$
$\displaystyle\bar{u}(p^{\prime})\bigg{[}\gamma^{+}f_{\phi T}^{({\rm
rbw})}(y,t)+\frac{i\sigma^{+\nu}\Delta_{\nu}}{2M}g_{\phi T}^{({\rm
rbw})}(y,t)\bigg{]}u(p),$ (80)

where the octet-decuplet transition operator $\Theta^{\mu\nu}$ is defined in
Eq. (2), and the spin-3/2 projection operator $S_{\alpha\beta}$ for a particle
of momentum $k$ is given by

$S_{\alpha\beta}(k)=-g_{\alpha\beta}+\frac{\gamma_{\alpha}\gamma_{\beta}}{3}+\frac{2k_{\alpha}\,k_{\beta}}{3M_{T}^{2}}+\frac{\gamma_{\alpha}\,k_{\beta}-\gamma_{\beta}\,k_{\alpha}}{3M_{T}}.$
(81)

The corresponding Dirac and Pauli decuplet rainbow splitting functions,
$f^{({\rm rbw})}_{\phi T}(y,t)$ and $g^{({\rm rbw})}_{\phi T}(y,t)$, can be
written as

$\displaystyle f^{({\rm rbw})}_{\phi T}(y,t)$ $\displaystyle=$
$\displaystyle\frac{C_{T\phi}^{2}}{f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\frac{iF_{\phi
T}^{({\rm
rbw})}}{D_{T}(p-k)D_{\phi}(k+\Delta)D_{\phi}(k)}\,\widetilde{F}(k+\Delta)\,\widetilde{F}(k)\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)},~{}~{}~{}~{}~{}~{}$
(82a) $\displaystyle g^{({\rm rbw})}_{\phi T}(y,t)$ $\displaystyle=$
$\displaystyle\frac{C_{T\phi}^{2}}{f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\frac{iG_{\phi
T}^{({\rm
rbw})}}{D_{T}(p-k)D_{\phi}(k+\Delta)D_{\phi}(k)}\,\widetilde{F}(k+\Delta)\,\widetilde{F}(k)\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)},$
(82b)

respectively. The explicit expressions for the numerator factors $F_{\phi
T}^{({\rm rbw})}$ and $G_{\phi T}^{({\rm rbw})}$ are given in Eqs. (147) of
Appendix A.

In Fig. 1(n), the photon couples to the decuplet baryon in the intermediate
state with an electric vertex. The contribution of this diagram is given by

$\displaystyle\Gamma_{\rm(n)}^{+}$ $\displaystyle=$
$\displaystyle-\bar{u}(p^{\prime})\,\frac{C^{2}_{T\phi}}{f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\,k_{\lambda}\Theta^{\lambda\sigma}\,\widetilde{F}(k)\,\frac{i}{D_{K}(k)}\,\frac{i}{\not{p^{\prime}}-\not{k}-M_{T}}S_{\sigma\alpha}(p^{\prime}-k)\,\gamma^{\alpha\beta+}$
(83) $\displaystyle\hskip
91.04872pt\times\frac{i}{\not{p}-\not{k}-M_{T}}\,S_{\beta\rho}(p-k)\,\Theta^{\rho\nu}k_{\nu}\,\widetilde{F}(k)\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)}\,u(p),~{}~{}~{}$

where the corresponding splitting functions $f^{({\rm rbw})}_{T\phi}(y,t)$ and
$g^{({\rm rbw})}_{T\phi}(y,t)$ are

$\displaystyle f^{({\rm rbw})}_{T\phi}(y,t)$ $\displaystyle=$
$\displaystyle\frac{C_{T\phi}^{2}}{f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\frac{iF_{T\phi}^{({\rm
rbw})}}{D_{T}(p^{\prime}-k)D_{T}(p-k)D_{\phi}(k)}\,\widetilde{F}^{2}(k)\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)},$
(84a) $\displaystyle g^{({\rm rbw})}_{T\phi}(y,t)$ $\displaystyle=$
$\displaystyle\frac{C_{T\phi}^{2}}{f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\frac{iG_{T\phi}^{({\rm
rbw})}}{D_{T}(p^{\prime}-k)D_{T}(p-k)D_{\phi}(k)}\,\widetilde{F}^{2}(k)\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)}.$
(84b)

The factors $F_{T\phi}^{({\rm rbw})}$ and $G_{T\phi}^{({\rm rbw})}$ in the
numerators here are given in Eqs. (148). For the magnetic photon-decuplet
baryon interaction in Fig. 1(o), the contribution to $\Gamma^{+}$ is expressed
as

$\displaystyle\Gamma_{\rm(o)}^{+}$ $\displaystyle=$
$\displaystyle-\frac{C^{2}_{T\phi}}{f^{2}}\bar{u}(p^{\prime})\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\,k_{\lambda}\Theta^{\lambda\sigma}\widetilde{F}(k)\frac{i}{D_{K}(k)}\frac{i}{\not{p^{\prime}}-\not{k}-M_{T}}\,S_{\sigma}^{\
\,\alpha}(p^{\prime}-k)\frac{-i}{2M_{T}}\sigma^{+\beta}\Delta_{\beta},$ (85)
$\displaystyle\hskip
91.04872pt\times\frac{i}{\not{p}-\not{k}-M_{T}}\,S_{\alpha\rho}(p-k)\,\Theta^{\rho\nu}k_{\nu}\,\widetilde{F}(k)\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)}\,u(p),$

where the corresponding splitting functions $f^{\rm(rbw\,mag)}_{T\phi}(y,t)$
and $g^{\rm(rbw\,mag)}_{T\phi}(y,t)$ are given by

$\displaystyle f^{\rm(rbw\,mag)}_{T\phi}(y,t)$ $\displaystyle=$
$\displaystyle\frac{C_{T\phi}^{2}}{f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\frac{iF_{T\phi}^{\rm(rbw\,mag)}}{D_{T}(p^{\prime}-k)D_{T}(p-k)D_{\phi}(k)}\widetilde{F}^{2}(k)\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)},$
(86a) $\displaystyle g^{\rm(rbw\,mag)}_{T\phi}(y,t)$ $\displaystyle=$
$\displaystyle\frac{C_{T\phi}^{2}}{f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\frac{iG_{T\phi}^{\rm(rbw\,mag)}}{D_{T}(p^{\prime}-k)D_{T}(p-k)D_{\phi}(k)}\,\widetilde{F}^{2}(k)\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)},$
(86b)

and the factors $F_{T\phi}^{\rm(rbw\,mag)}$ and $G_{T\phi}^{\rm(rbw\,mag)}$
are written out in Eqs. (149).

In Figs. 1(p) and 1(q), the photon couples to the octet-decuplet transition
vertex, whose contribution to the matrix element $\Gamma^{+}$ can be written
as

$\displaystyle\Gamma_{\rm(p)+(q)}^{+}$ $\displaystyle=$
$\displaystyle\frac{C_{B\phi}C_{T\phi}}{4M_{B}f^{2}}\,\bar{u}(p^{\prime})\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\,\widetilde{F}^{2}(k)$
(87)
$\displaystyle\times\bigg{[}-\\!\not{k}\gamma_{5}\frac{i}{\not{p^{\prime}}-\not{k}-M_{B}}\not{\Delta}\gamma_{5}\frac{i}{\not{p}-\not{k}-M_{T}}S^{+}_{\
\,\rho}(p-k)\,\Theta^{\rho\nu}k_{\nu}$ $\displaystyle\hskip
17.07182pt+\,\not{k}\gamma_{5}\frac{i}{\not{p^{\prime}}-\not{k}-M_{B}}\gamma^{+}\gamma_{5}\Delta^{\lambda}\frac{i}{\not{p}-\not{k}-M_{T}}S_{\lambda\rho}(p-k)\,\Theta^{\rho\nu}k_{\nu}$
$\displaystyle\hskip
17.07182pt-\,k_{\lambda}\Theta^{\lambda\nu}\frac{i}{\not{p^{\prime}}-\not{k}-M_{T}}S_{\nu\rho}(p^{\prime}-k)\Delta^{\rho}\gamma^{+}\gamma_{5}\frac{i}{\not{p}-\not{k}-M_{B}}\not{k}\gamma_{5}$
$\displaystyle\hskip
17.07182pt+\,k_{\lambda}\Theta^{\lambda\nu}\frac{i}{\not{p^{\prime}}-\not{k}-M_{T}}S_{\nu}^{\
\,+}(p^{\prime}-k)\not{\Delta}\gamma_{5}\frac{1}{\not{p}-\not{k}-M_{B}}\not{k}\gamma_{5}\bigg{]}$
$\displaystyle\times\,\frac{i}{D_{\phi}(k)}\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)}\,u(p).$

The corresponding Dirac and Pauli splitting functions
$f^{\rm(rbw\,mag)}_{BT}(y,t)$ and $g^{\rm(rbw\,mag)}_{BT}(y,t)$ are in this
case given by

$\displaystyle f^{\rm(rbw\,mag)}_{BT}(y,t)$ $\displaystyle=$
$\displaystyle\frac{C_{B\phi}C_{T\phi}}{4f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\,\widetilde{F}^{2}(k)\,\frac{1}{D_{\phi}(k)}\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)}\,$
(88a) $\displaystyle\hskip
45.52458pt\times\,\bigg{[}\frac{iF^{\rm(rbw\,mag\,1)}_{BT}}{D_{B}(p^{\prime}-k)D_{T}(p-k)}+\frac{iF^{\rm(rbw\,mag\,2)}_{BT}}{D_{T}(p^{\prime}-k)D_{B}(p-k)}\bigg{]},$
$\displaystyle g^{\rm(rbw\,mag)}_{BT}(y,t)$ $\displaystyle=$
$\displaystyle\frac{C_{B\phi}C_{T\phi}}{4f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\,\widetilde{F}^{2}(k)\,\frac{1}{D_{\phi}(k)}\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)}\,$
(88b) $\displaystyle\hskip
45.52458pt\times\,\bigg{[}\frac{iG^{\rm(rbw\,mag\,1)}_{BT}}{D_{B}(p^{\prime}-k)D_{T}(p-k)}+\frac{iG^{\rm(rbw\,mag\,2)}_{BT}}{D_{T}(p^{\prime}-k)D_{B}(p-k)}\bigg{]},~{}~{}~{}~{}$

with the four numerator factors $F^{\rm(rbw\,mag\,1,2)}_{BT}$ and
$G^{\rm(rbw\,mag\,1,2)}_{BT}$ given in Eqs. (150).

For the KR diagrams with decuplet baryon intermediate states in Fig. 1(r) and
1(s), the contribution to $\Gamma^{+}$ is given by the expression

$\displaystyle\Gamma_{\rm(r)+(s)}^{+}$ $\displaystyle=$
$\displaystyle-\frac{C^{2}_{T\phi}}{f^{2}}\,\bar{u}(p^{\prime})\,\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\,\widetilde{F}^{2}(k)$
(89)
$\displaystyle\times\bigg{[}k_{\nu}\Theta^{\nu\sigma}\frac{i}{\not{p^{\prime}}-\not{k}-M_{T}}\,S_{\sigma\rho}(p^{\prime}-k)\,i\Theta^{\rho+}\,-\
i\Theta^{+\sigma}\frac{i}{\not{p}-\not{k}-M_{T}}\,S_{\sigma\rho}(p-k)\,\Theta^{\rho\nu}k_{\nu}\bigg{]}~{}~{}$
$\displaystyle\times\frac{i}{D_{\phi}(k)}\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)}\,u(p),$

where the corresponding KR splitting functions $f^{(\rm KR)}_{T\phi}(y,t)$ and
$g_{T\phi}^{(\rm KR)}(y,t)$ are given by

$\displaystyle f^{\rm(KR)}_{T\phi}(y,t)$ $\displaystyle=$
$\displaystyle\frac{C^{2}_{T\phi}}{f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\bigg{[}\frac{iF^{\rm(KR\,1)}_{T\phi}}{D_{T}(p^{\prime}-k)}+\frac{iF^{\rm(KR\,2)}_{T\phi}}{D_{T}(p-k)}\bigg{]}\widetilde{F}^{2}(k)\,\frac{1}{D_{\phi}(k)}\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)},$
(90a) $\displaystyle g^{({\rm KR})}_{T\phi}(y,t)$ $\displaystyle=$
$\displaystyle\frac{C^{2}_{T\phi}}{f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\bigg{[}\frac{iG^{\rm(KR\,1)}_{T\phi}}{D_{T}(p^{\prime}-k)}+\frac{iG^{\rm(KR\,2)}_{T\phi}}{D_{T}(p-k)}\bigg{]}\widetilde{F}^{2}(k)\,\frac{1}{D_{\phi}(k)}\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)},$
(90b)

and the numerator functions $F^{\rm(rbw\,mag\,1,2)}_{BT}$ and
$G^{\rm(rbw\,mag\,1,2)}_{BT}$ are in Eqs. (151). Similarly, the additional KR
diagrams generated from the gauge link terms with decuplet intermediate states
are shown in Fig. 1(t) and 1(u), and their contribution can be written as

$\displaystyle\Gamma_{\rm(t)+(u)}$ $\displaystyle=$
$\displaystyle\frac{C^{2}_{\phi
T}}{f^{2}}\,\bar{u}(p^{\prime})\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}$
(91) $\displaystyle\times$
$\displaystyle\bigg{[}k_{\nu}\Theta^{\nu\sigma}\frac{i}{\not{p^{\prime}}-\not{k}-M_{T}}S_{\sigma\rho}(p^{\prime}-k)\,\Theta^{\rho\lambda}(k-\Delta)_{\lambda}\frac{2ik^{+}}{-2k\cdot\Delta+\Delta^{2}}\big{(}\widetilde{F}(k-\Delta)-\widetilde{F}(k)\big{)}$
$\displaystyle+\,(k+\Delta)_{\nu}\Theta^{\nu\sigma}\frac{-2ik^{+}}{2k\cdot\Delta+\Delta^{2}}\big{(}\widetilde{F}(k)-\widetilde{F}(k+\Delta)\big{)}\frac{i}{\not{p}-\not{k}-M_{T}}S_{\sigma\rho}(p-k)\,\Theta^{\rho\lambda}k_{\lambda}\bigg{]}$
$\displaystyle\times$
$\displaystyle\frac{i}{D_{\phi}(k)}\,\widetilde{F}(k)\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)}\,u(p).$

where the gauge link dependent KR splitting functions $\delta
f^{\rm(KR)}_{T\phi}(y,t)$ and $\delta g_{T\phi}^{\rm(KR)}(y,t)$ are given by

$\displaystyle\delta f^{({\rm KR})}_{T\phi}(y,t)$ $\displaystyle=$
$\displaystyle-\frac{C^{2}_{T\phi}}{f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\,\widetilde{F}(k)\,\frac{1}{D_{\phi}(k)}\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)}$
(92a) $\displaystyle\times\bigg{[}\frac{i\delta
F^{\rm(KR\,1)}_{T\phi}}{D_{T}(p^{\prime}-k)}\frac{\widetilde{F}(k)-\widetilde{F}(k-\Delta)}{2k\cdot\Delta-\Delta^{2}}+\frac{i\delta
F^{\rm(KR\,2)}_{T\phi}}{D_{T}(p-k)}\frac{\widetilde{F}(k)-\widetilde{F}(k+\Delta)}{2k\cdot\Delta+\Delta^{2}}\bigg{]},$
$\displaystyle\delta g^{({\rm KR})}_{T\phi}(y,t)$ $\displaystyle=$
$\displaystyle-\frac{C^{2}_{T\phi}}{f^{2}}\int\\!\frac{\differential[4]{k}}{(2\pi)^{4}}\,\widetilde{F}(k)\,\frac{1}{D_{\phi}(k)}\,\delta\Big{(}y-\frac{k^{+}}{p^{+}}\Big{)}$
(92b) $\displaystyle\times\bigg{[}\frac{i\delta
G^{\rm(KR\,1)}_{T\phi}}{D_{T}(p^{\prime}-k)}\frac{\widetilde{F}(k)-\widetilde{F}(k-\Delta)}{2k\cdot\Delta-\Delta^{2}}+\frac{i\delta
G^{\rm(KR\,2)}_{T\phi}}{D_{T}(p-k)}\frac{\widetilde{F}(k)-\widetilde{F}(k+\Delta)}{2k\cdot\Delta+\Delta^{2}}\Bigg{]},~{}~{}~{}~{}$

The complete expressions for the numerator functions $\delta
F^{\rm(KR\,1,2)}_{T\phi}$ and $\delta G^{\rm(KR\,1,2)}_{T\phi}$ are presented
in Eq. (152) of Appendix A.

##  IV Nonanalytic behavior of splitting functions

While certain features of the hadronic splitting functions derived in the
previous section depend on short-distance physics that in the current
framework is controlled by the regulator function $\widetilde{F}$, some
aspects of their calculation are in fact model independent. In particular, the
moments of the splitting functions, which can be expanded as a series in the
pseudoscalar meson mass, $m_{\phi}$, contain terms that do not depend on the
regularization method Thomas  _et al._ (2000); Arndt and Savage (2002); Chen
and Ji (2001); Salamu  _et al._ (2019a). These are the coefficients of the
leading nonanalytic (LNA) terms, that are determined by the low-energy
properties of the nucleon, such as the hadronic couplings and masses. Since
the LNA behavior is derived solely from the long-distance characteristics of
the chiral effective theory, understanding these can place constraints on
models of the splitting functions consistent with the symmetries of QCD.

To explore the LNA terms further, we define the lowest moments of the
splitting functions $f^{\rm(x)}(y,t)$ and $g^{\rm(x)}(y,t)$ for the diagram in
Fig. 1(“x”), where “x” = “a”, $\ldots$ “u”, as

$\displaystyle F^{\rm(x)}_{1}(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\differential{y}f^{\rm(x)}(y,t),$ (93a)
$\displaystyle F^{\rm(x)}_{2}(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\differential{y}g^{\rm(x)}(y,t),$ (93b)

which correspond to the Dirac and Pauli electromagnetic form factors of the
nucleon, respectively. Taking the values at zero four-momentum transfer
squared, $t=0$, we expand the form factors $F^{\rm(x)}_{1,2}(t\\!=\\!0)$ in
$m_{\phi}$, keeping only the nonanalytic (NA) terms in $m_{\phi}^{2}$, from
which the LNA behavior can be extracted.

For the rainbow diagram in Fig. 1(a), the NA contributions can be written as

$\displaystyle F_{1}^{\rm(a)}(0)\Big{|}_{\rm NA}$ $\displaystyle=$
$\displaystyle\frac{3C^{2}_{B\phi}}{(4\pi
f)^{2}}\bigg{[}\big{(}m_{\phi}^{2}-2\Delta_{B}^{2}\big{)}\log
m_{\phi}^{2}+2\Delta_{B}R_{B}\log\frac{\Delta_{B}-R_{B}}{\Delta_{B}+R_{B}}\bigg{]},$
(94a) $\displaystyle F_{2}^{\rm(a)}(0)\Big{|}_{\rm NA}$ $\displaystyle=$
$\displaystyle\frac{4C^{2}_{B\phi}}{(4\pi
f)^{2}}\bigg{[}\big{(}M\Delta_{B}+2\Delta^{2}_{B}+2R_{B}^{2}\big{)}\log
m_{\phi}^{2}$ (94b) $\displaystyle\hskip
105.2751pt-\,\big{(}M+4\Delta_{B}\big{)}R_{B}\,\log\frac{\Delta_{B}-R_{B}}{\Delta_{B}+R_{B}}\bigg{]},$

where $R_{B}=\sqrt{\Delta_{B}^{2}-m_{\phi}^{2}}$ . If $\Delta_{B}<m_{\phi}$,
$R_{B}$ will be an imaginary number and the log term will become an
arctangent, according to its definition Salamu  _et al._ (2019a). The NA terms
of $F_{1}^{\rm(a)}$ are the same as those reported in Eq. (62) of Ref. Salamu
_et al._ (2019a) if one sums the NA contributions from the on-shell and
$\delta$ terms of Eqs. (114) and (117) of Salamu  _et al._ (2019a).

For the baryon rainbow diagram in Fig. 1(b), the NA behavior is given by

$\displaystyle F_{1}^{\rm(b)}(0)\Big{|}_{\rm NA}$ $\displaystyle=$
$\displaystyle~{}~{}\frac{3C^{2}_{B\phi}}{(4\pi
f)^{2}}\bigg{[}\big{(}m_{\phi}^{2}-2\Delta_{B}^{2}\big{)}\log
m_{\phi}^{2}+2\Delta_{B}R_{B}\log\frac{\Delta_{B}-R_{B}}{\Delta_{B}+R_{B}}\bigg{]},$
(95a) $\displaystyle F_{2}^{\rm(b)}(0)\Big{|}_{\rm NA}$ $\displaystyle=$
$\displaystyle-\frac{4C^{2}_{B\phi}}{(4\pi
f)^{2}}\bigg{[}\big{(}m_{\phi}^{2}-2\Delta_{B}^{2}\big{)}\log
m_{\phi}^{2}+2\Delta_{B}R_{B}\log\frac{\Delta_{B}-R_{B}}{\Delta_{B}+R_{B}}\bigg{]}.$
(95b)

Note that the NA terms of $F_{1}^{\rm(b)}(0)$ are identical to those of
$F_{1}^{\rm(a)}(0)$. For the diagram in Fig. 1(c), there is no NA term for
$F_{1}^{\rm(c)}(0)$, while the NA part of $F_{2}^{\rm(c)}(0)$ can be written
as

$\displaystyle F_{2}^{\rm(c)}(0)\Big{|}_{\rm NA}$ $\displaystyle=$
$\displaystyle-\frac{C^{2}_{B\phi}}{(4\pi
f)^{2}}\bigg{[}\big{(}m_{\phi}^{2}-2\Delta_{B}^{2}\big{)}\log
m_{\phi}^{2}+2\Delta_{B}R_{B}\log\frac{\Delta_{B}-R_{B}}{\Delta_{B}+R_{B}}\bigg{]}.$
(96)

For the KR diagrams in Figs. 1(d) and 1(e), the NA contributions to the Dirac
form factor is given by

$\displaystyle F_{1}^{\rm(d)+(e)}(0)\Big{|}_{\rm NA}$ $\displaystyle=$
$\displaystyle\frac{2C^{2}_{B\phi}}{(4\pi
f)^{2}}\frac{R_{B}^{3}}{M}\log\frac{\Delta_{B}-R_{B}}{\Delta_{B}+R_{B}}.$ (97)

There is no NA contribution from these two diagrams to the Pauli form factor.
For the tadpole diagrams in Fig. 1(h) and 1(j), the NA terms contain only the
LNA contributions,

$\displaystyle F_{1}^{\rm(h)}(0)\Big{|}_{\rm LNA}$ $\displaystyle=$
$\displaystyle\frac{C_{\phi\phi^{\dagger}}}{(4\pi f)^{2}}\,m_{\phi}^{2}\log
m_{\phi}^{2},$ (98) $\displaystyle F_{2}^{\rm(j)}(0)\Big{|}_{\rm LNA}$
$\displaystyle=$ $\displaystyle\frac{C^{\rm mag}_{\phi\phi^{\dagger}}}{(4\pi
f)^{2}}\,m_{\phi}^{2}\log m_{\phi}^{2},$ (99)

for the Dirac an Pauli form factors, respectively. For the bubble diagram in
Fig. 1(k), the NA behavior also reflects the simple LNA form for the Dirac
form factor,

$\displaystyle F_{1}^{\rm(k)}(0)\Big{|}_{\rm LNA}$ $\displaystyle=$
$\displaystyle\frac{C_{\phi\phi^{\dagger}}}{(4\pi f)^{2}}\,m_{\phi}^{2}\log
m_{\phi}^{2},$ (100)

and is in fact identical to the LNA contributions from the tadpole diagram,
Eq. (98), as required by gauge invariance. The bubble diagram with the
magnetic interaction in Fig. 1(l) gives rise to a contribution to the Pauli
form factor given by

$\displaystyle F_{2}^{\rm(l)}(0)\Big{|}_{\rm LNA}$ $\displaystyle=$
$\displaystyle-\frac{C^{\prime}_{\phi\phi^{\dagger}}}{(4\pi
f)^{2}}\,Mm_{\phi}^{2}\log m_{\phi}^{2}.$ (101)

Note that the mass dimensions of the couplings
$C^{\prime}_{\phi\phi^{\dagger}}$ here are inverse mass, in contrast to all
the other couplings which are dimensionless (see Table 1 in Sec. II above). As
discussed in Ref. Salamu  _et al._ (2019a), the additional diagrams in Fig.
1(f), 1(g) and 1(i) generated from the gauge link yield no NA terms for either
the Dirac or Pauli form factors.

The NA terms for the splitting functions involving intermediate decuplet
baryon states can be obtained in a similar manner. For the rainbow diagram
Fig. 1(m), the NA behavior is given by

$\displaystyle F_{1}^{\rm(m)}(0)\Big{|}_{\rm NA}$ $\displaystyle=$
$\displaystyle\ \ \ \frac{2C^{2}_{T\phi}}{(4\pi
f)^{2}}\bigg{[}\big{(}m_{\phi}^{2}-2\Delta_{T}^{2}\big{)}\log
m_{\phi}^{2}+2\Delta_{T}R_{T}\log\frac{\Delta_{T}-R_{T}}{\Delta_{T}+R_{T}}\bigg{]},$
(102a) $\displaystyle F_{2}^{\rm(m)}(0)\Big{|}_{\rm NA}$ $\displaystyle=$
$\displaystyle\\!-\frac{C^{2}_{T\phi}}{3(4\pi
f)^{2}}\bigg{[}\big{(}11m_{\phi}^{2}+4M\Delta_{T}-6\Delta_{T}^{2}\big{)}\log
m_{\phi}^{2}-4MR_{T}\log\frac{\Delta_{T}-R_{T}}{\Delta_{T}+R_{T}}\bigg{]}.~{}~{}~{}~{}$
(102b)

The NA behavior for the decuplet baryon coupling rainbow diagram in Fig. 1(n)
can be analogously written as

$\displaystyle F_{1}^{\rm(n)}(0)\Big{|}_{\rm NA}$ $\displaystyle=$
$\displaystyle\ \ \ \frac{2C^{2}_{T\phi}}{(4\pi
f)^{2}}\bigg{[}\big{(}m_{\phi}^{2}-2\Delta_{T}^{2}\big{)}\log
m_{\phi}^{2}+2\Delta_{T}R_{T}\log\frac{\Delta_{T}-R_{T}}{\Delta_{T}+R_{T}}\bigg{]},$
(103a) $\displaystyle F_{2}^{\rm(n)}(0)\Big{|}_{\rm NA}$ $\displaystyle=$
$\displaystyle\\!-\frac{8C^{2}_{T\phi}}{9(4\pi
f)^{2}}\bigg{[}\big{(}m_{\phi}^{2}-2\Delta_{T}^{2}\big{)}\log
m_{\phi}^{2}+2\Delta_{T}R_{T}\log\frac{\Delta_{T}-R_{T}}{\Delta_{T}+R_{T}}\bigg{]}.~{}~{}~{}$
(103b)

As in the octet case, the NA term for the Dirac form factor contribution from
Fig. 1(n) is identical to that from Fig. 1(m). For the additional magnetic
interaction decuplet baryon rainbow diagram in Fig. 1(o), the contribution to
the $F_{2}$ form factor is given by

$\displaystyle F_{2}^{\rm(o)}(0)\Big{|}_{\rm NA}$ $\displaystyle=$
$\displaystyle\frac{10\,C^{2}_{T\phi}}{9(4\pi
f)^{2}}\bigg{[}\big{(}m_{\phi}^{2}-2\Delta_{T}^{2}\big{)}\log
m_{\phi}^{2}+2\Delta_{T}R_{T}\log\frac{\Delta_{T}-R_{T}}{\Delta_{T}+R_{T}}\bigg{]},$
(104)

while there is no NA term for the Dirac $F_{1}$ form factor.

For the magnetic octet-decuplet baryon transition diagrams in Fig. 1(p) and
1(q), the combined NA contribution to the Pauli form factor can be written as

$\displaystyle F_{2}^{\rm(p)+(q)}(0)\Big{|}_{\rm NA}$ $\displaystyle=$
$\displaystyle\frac{C_{B\phi}C_{T\phi}}{6(4\pi
f)^{2}}\bigg{[}\big{(}8m_{\phi}^{2}-5\Delta_{B}^{2}-6\Delta_{B}\Delta_{T}-5\Delta_{T}^{2}\big{)}\,\log
m_{\phi}^{2}$ (105) $\displaystyle\hskip
42.67912pt+\,R_{T}\frac{6m_{\phi}^{2}-\Delta_{T}(\Delta_{B}+5\Delta_{T})}{\Delta_{B}-\Delta_{T}}\log\frac{\Delta_{T}-R_{T}}{\Delta_{T}+R_{T}}$
$\displaystyle\hskip
42.67912pt-\,R_{B}\frac{6m_{\phi}^{2}-\Delta_{B}(\Delta_{T}+5\Delta_{B})}{\Delta_{B}-\Delta_{T}}\log\frac{\Delta_{B}-R_{B}}{\Delta_{B}+R_{B}}\bigg{]},$

with again no NA contributions to $F_{1}(0)$. Finally, for the KR diagrams
with decuplet intermediate states in Fig. 1(r) and 1(s) the NA behavior of the
Dirac and Pauli form factors is given by

$\displaystyle F_{1}^{\rm(r)+(s)}(0)\Big{|}_{\rm NA}$ $\displaystyle=$
$\displaystyle\ \ \frac{4C^{2}_{T\phi}}{3M(4\pi
f)^{2}}\bigg{[}\Delta_{T}\,\big{(}\,m_{\phi}^{2}\,-\,2\Delta_{T}^{2}\big{)}\log
m_{\phi}^{2}+2R_{T}^{3}\log\frac{\Delta_{T}-R_{T}}{\Delta_{T}+R_{T}}\bigg{]},$
(106a) $\displaystyle F_{2}^{\rm(r)+(s)}(0)\Big{|}_{\rm NA}$ $\displaystyle=$
$\displaystyle\\!-\frac{16C^{2}_{T\phi}}{9M(4\pi
f)^{2}}\bigg{[}\Delta_{T}\,\big{(}3m_{\phi}^{2}-2\Delta_{T}^{2}\big{)}\log
m_{\phi}^{2}+2R_{T}^{3}\log\frac{\Delta_{T}-R_{T}}{\Delta_{T}+R_{T}}\bigg{]}.~{}~{}~{}~{}~{}$
(106b)

As for the octet baryon case, the additional diagrams derived from the gauge
link for the decuplet baryons do not give rise to any NA terms. Our results
for the LNA terms of the various loop diagrams are consistent with the results
for the Dirac form factor discussed in Ref. Salamu  _et al._ (2019a).

##  V Convolution formalism

Having derived the full set of splitting functions for the diagrams in Fig. 1
involving the SU(3) octet and decuplet baryon intermediate states, in this
section we discuss the calculation of the GPDs in the proton arising from
these contributions. We derive expressions for the GPDs in terms of
convolutions of the splitting functions and GPDs of quarks in the various
hadronic configurations. Using flavor symmetry constraints, we discuss
relations for the GPDs in the hadronic configurations among the various SU(3)
baryons.

###  V.1 GPDs as convolutions

The $n$-th Mellin moments of the generalized quark distributions
$H^{q}(x,\xi,t)$ and $E^{q}(x,\xi,t)$ are defined as

$\displaystyle
H_{q}^{(n)}(\xi,t)\equiv\int_{-1}^{1}\differential{x}x^{n-1}H^{q}(x,\xi,t)$
$\displaystyle=$ $\displaystyle\sum_{i=0,\,{\rm
even}}^{n-1}\big{(}\\!-2\xi\big{)}^{i}A^{(n)i}_{q}(t)+\big{(}\\!-2\xi\big{)}^{n}C_{q}^{(n)}(t)\Big{|}_{n\,{\rm
even}},~{}~{}~{}~{}~{}~{}~{}~{}$ (107a) $\displaystyle
E_{q}^{(n)}(\xi,t)\equiv\int_{-1}^{1}\differential{x}x^{n-1}E^{q}(x,\xi,t)$
$\displaystyle=$ $\displaystyle\sum_{i=0,\,{\rm
even}}^{n-1}\big{(}\\!-2\xi\big{)}^{i}B^{(n)i}_{q}(t)-\big{(}\\!-2\xi\big{)}^{n}C_{q}^{(n)}(t)\Big{|}_{n\,{\rm
even}},~{}~{}~{}~{}~{}~{}~{}~{}$ (107b)

where $A_{q}^{(n)i}(t)$, $B_{q}^{(n)i}(t)$ and $C_{q}^{(n)}(t)$ are the
generalized Compton form factors of rank $n$. The generalized form factors can
be related to the matrix elements of local twist-2 operators ${\cal
O}_{q}^{\\{\mu\mu_{1}\cdots\mu_{n-1}\\}}$ between nucleon states,

$\displaystyle\langle
N(p^{\prime})|\mathcal{O}_{q}^{\\{\mu\mu_{1}\cdots\mu_{n-1}\\}}|N(p)\rangle$
$\displaystyle=$ $\displaystyle\bar{u}(p^{\prime})\bigg{[}\ \sum_{i=0,\,\rm
even}^{n-1}A_{q}^{(n)i}(t)\,\gamma^{\\{\mu}\Delta^{\mu_{1}}\cdots\Delta^{\mu_{i}}P^{\mu_{i}+1}\cdots
P^{\mu_{n-1}\\}}$ (108) $\displaystyle\hskip
5.69046pt-\frac{i}{2M}\sum_{i=0,\,\rm
even}^{n-1}B_{q}^{(n)i}(t)\,\Delta_{\nu}\sigma^{\nu\\{\mu}\Delta^{\mu_{1}}\cdots\Delta^{\mu_{i}}P^{\mu_{i}+1}\cdots
P^{\mu_{n-1}\\}}~{}~{}~{}~{}~{}$ $\displaystyle\hskip 5.69046pt+\frac{1}{M}\
C_{q}^{(n)}(t)\Big{|}_{n\,{\rm even}}\
\Delta^{\\{\mu}\cdots\Delta^{\mu_{n-1}\\}}\bigg{]}u(p),$

where the symmetric and traceless operators are defined as

$\mathcal{O}^{\\{\mu\mu_{1}\cdots\mu_{n-1}\\}}_{q}=i^{n-1}\,\bar{\psi}_{q}\,\gamma^{\\{\mu}\overleftrightarrow{D}^{\mu_{1}}\cdots\overleftrightarrow{D}^{\mu_{n-1}\\}}\,\psi_{q},$
(109)

with
$\overleftrightarrow{D}=\frac{1}{2}\big{(}\overrightarrow{D}-\overleftarrow{D}\big{)}$.
The braces $\\{\cdots\\}$ represent symmetrization over the indices
$\mu\mu_{1}\cdots\mu_{n}$ and subtraction of traces.

In an effective field theory, these quark operators are matched to hadronic
operators with the corresponding set of quantum numbers Chen and Ji (2001),

$\mathcal{O}^{\\{\mu\mu_{1}\cdots\mu_{n-1}\\}}_{q}=\sum_{j}Q^{(n-1)}_{j}\,\mathcal{O}^{\\{\mu\mu_{1}\cdots\mu_{n-1}\\}}_{j},$
(110)

where the subscript $j$ labels different types of hadronic operators. The
coefficients $Q^{(n-1)}_{j}$ can be defined through the $n$-th moments of the
generalized parton distributions in the hadronic configuration $j$ (see Sec.
V.2 below). Matrix elements of the hadronic operators
$\mathcal{O}^{\\{\mu\mu_{1}\cdots\mu_{n-1}\\}}_{j}$ can be used to define the
moments of the Dirac-like and Pauli-like hadronic splitting functions $f_{j}$
and $g_{j}$, respectively, introduced in Eq. (51),

$\displaystyle f^{(n)}_{j}$ $\displaystyle=$
$\displaystyle\int^{1}_{-1}\differential{y}y^{(n-1)}f_{j}(y,t),$ (111a)
$\displaystyle g^{(n)}_{j}$ $\displaystyle=$
$\displaystyle\int^{1}_{-1}\differential{y}y^{(n-1)}g_{j}(y,t),$ (111b)

where $y$ is the light-cone momentum fraction of the nucleon carried by the
hadronic state $j$. Taking the matrix elements of the matched operators in Eq.
(110) between nucleon states with unequal initial and final momenta, and
contracting both sides with light-cone vector $n_{\mu}n_{\mu_{1}}\cdots
n_{\mu_{n-1}}$, we then arrive at a relation for the GPD moments in terms of
the moments of the hadronic splitting functions,

$\displaystyle
H_{q}^{(n)}\,\bar{u}(p^{\prime})\not{n}u(p)+\frac{i}{2M}E_{q}^{(n)}\,\bar{u}(p^{\prime})n_{\mu}\sigma^{\mu\nu}\Delta_{\nu}\,u(p)$
$\displaystyle=\sum_{j}Q^{(n-1)}_{j}\bigg{[}f^{(n)}_{j}\,\bar{u}(p^{\prime})\not{n}u(p)+\frac{i}{2M}\,g^{(n)}_{j}\
\bar{u}(p^{\prime})n_{\mu}\sigma^{\mu\nu}\Delta_{\nu}\,u(p)\bigg{]}.$ (112)

Since Eq. (V.1) is valid for all moments $n$, we deduce that the corresponding
convolution relation exists also for the GPDs as a function of $x$.
Specializing to the case of zero skewness, $\xi=0$, we can write the
contributions to the $H^{q}$ and $E^{q}$ GPDs from the hadronic configurations
in the form

$\displaystyle H^{q}(x,t)\,\equiv\,H^{q}(x,\xi\\!=\\!0,t)$ $\displaystyle=$
$\displaystyle\sum_{j}\big{[}f_{j}\otimes q_{j}^{v}\big{]}(x,t)$ (113a)
$\displaystyle\equiv$
$\displaystyle\sum_{j}\int_{0}^{1}\differential{y}\int_{0}^{1}\differential{z}\delta(x-yz)\,f_{j}(y,t)\,q_{j}^{v}(z,t),$
$\displaystyle E^{q}(x,t)\,\equiv\,E^{q}(x,\xi\\!=\\!0,t)$ $\displaystyle=$
$\displaystyle\sum_{j}\big{[}g_{j}\otimes q_{j}^{v}\big{]}(x,t)$ (113b)
$\displaystyle\equiv$
$\displaystyle\sum_{j}\int_{0}^{1}\differential{y}\int_{0}^{1}\differential{z}\delta(x-yz)\,g_{j}(y,t)\,q_{j}^{v}(z,t),$

where we define $q^{v}_{j}(x,t)\equiv
q_{j}(x,\xi\\!=\\!0,t)-\bar{q}_{j}(x,\xi\\!=\\!0,t)$ as the GPD of the valence
quark $q$ in the hadronic configuration $j$ evaluated at zero skewness. Since
the total $H^{q}$ and $E^{q}$ GPDs can each receive contributions from both
Dirac-like and Pauli-like GPDs of the hadronic configurations, the sum over
$j$ in Eqs. (113) includes both electric and magnetic couplings, $q_{j}\to
H^{q}_{j}$ or $E^{q}_{j}$. Note also that crossing symmetry,
$q_{j}(-x,\xi\\!=\\!0,t)=-\bar{q}_{j}(x,\xi\\!=\\!0,t)$, in direct analogy to
that for forward ($t=0$) parton distributions, has been used to write the
integrals in Eqs. (113) over the interval 0 to 1.

To illustrate the application of Eqs. (113) to chiral loop contributions to
sea quark GPDs in the proton, in this paper we focus on the asymmetries
between the GPDs for $\bar{d}$ and $\bar{u}$ quarks, and between the $s$ and
$\bar{s}$ quark flavors. Assuming that the intermediate state octet and
decuplet baryons are flavor symmetric, with mesons $\phi$ the only source of
antiquarks, the convolution form for the antiquark electric and magnetic GPDs
in the proton involves contributions only from the diagrams in Fig. 1(a), (k),
(l) and (m). Specifically, for the $H^{\bar{q}}$ and $E^{\bar{q}}$ GPDs at
zero skewness, we have

$\displaystyle H^{\bar{q}}(x,t)$ $\displaystyle=$ $\displaystyle\sum_{\phi
BT}\Big{[}\big{(}f_{\phi B}^{(\rm rbw)}+f_{\phi T}^{(\rm rbw)}+f^{(\rm
bub)}_{\phi}\big{)}\otimes H^{\bar{q}}_{\phi}\Big{]}(x,t),$ (114a)
$\displaystyle E^{\bar{q}}(x,t)$ $\displaystyle=$ $\displaystyle\sum_{\phi
BT}\Big{[}\big{(}g_{\phi B}^{(\rm rbw)}+g_{\phi T}^{(\rm rbw)}+g^{\prime(\rm
bub)}_{\phi}\big{)}\otimes H^{\bar{q}}_{\phi}\Big{]}(x,t),$ (114b)

where $H^{\bar{q}}_{\phi}$ is the electric GPD for quark flavor $\bar{q}$ in
the meson $\phi$. The splitting functions $f_{\phi B}^{(\rm rbw)}$ and
$g_{\phi B}^{(\rm rbw)}$ for Fig. 1(a) are given in Eqs. (55) and (55),
respectively, the decuplet recoil splitting functions $f_{\phi T}^{(\rm rbw)}$
and $g_{\phi T}^{(\rm rbw)}$ for Fig. 1(m) are given in Eqs. (82a) and (82b),
respectively, and the functions $f^{(\rm bub)}_{\phi}$ and $g^{\prime(\rm
bub)}_{\phi}$ for Figs. 1(k) and (l) are given in Eqs. (76) and (78),
respectively.

The convolution form for the quark GPDs received contributions from all other
diagrams in Fig. 1, and so has a more complicated structure,

$\displaystyle H^{q}(x,t)$ $\displaystyle=$ $\displaystyle
Z_{2}\,H^{q}_{0}(x,t)\ +\ \sum_{\phi BT}\Big{[}\big{(}f_{\phi B}^{(\rm
rbw)}+f_{\phi T}^{(\rm rbw)}+f^{(\rm bub)}_{\phi}\big{)}\otimes H^{q}_{\phi}$
$\displaystyle\hskip 93.89418pt+\ \bar{f}_{B\phi}^{(\rm rbw)}\otimes
H^{q}_{B}\ +\ \bar{f}^{(\rm KR)}_{B\phi}\otimes H^{q({\rm KR})}_{B}\ +\
\delta\bar{f}^{(\rm KR)}_{B}\otimes H^{q({\rm KR})}_{B}\ $
$\displaystyle\hskip 93.89418pt+\ \bar{f}_{T\phi}^{(\rm rbw)}\otimes
H^{q}_{T}\ +\ \bar{f}^{(\rm KR)}_{T\phi}\otimes H^{q({\rm KR})}_{T}\ +\
\delta\bar{f}^{(\rm KR)}_{T\phi}\otimes H^{q({\rm KR})}_{T}\ $
$\displaystyle\hskip 93.89418pt+\ \bar{f}_{B\phi}^{(\rm rbw\,mag)}\otimes
E^{q}_{B}\ +\ \bar{f}_{T\phi}^{(\rm rbw\,mag)}\otimes E^{q}_{T}\ +\
\bar{f}_{BT}^{(\rm rbw\,mag)}\otimes E^{q}_{BT}~{}~{}~{}~{}~{}~{}$
$\displaystyle\hskip 93.89418pt+\ \bar{f}^{(\rm tad)}_{\phi}\otimes H^{q({\rm
tad})}_{\phi\phi^{\dagger}}\ +\ \delta\bar{f}^{(\rm tad)}_{\phi}\otimes
H^{q({\rm tad})}_{\phi\phi^{\dagger}}\Big{]}(x,t),$ $\displaystyle E^{q}(x,t)$
$\displaystyle=$ $\displaystyle Z_{2}\,E^{q}_{0}(x,t)\ +\ \sum_{\phi
BT}\Big{[}\big{(}g_{\phi B}^{(\rm rbw)}+g_{\phi T}^{(\rm rbw)}+g^{\prime(\rm
bub)}_{\phi}\big{)}\otimes H^{q}_{\phi}$ (115b) $\displaystyle\hskip
93.89418pt+\ \bar{g}_{B\phi}^{(\rm rbw)}\otimes H^{q}_{B}\ +\ \bar{g}^{(\rm
KR)}_{B\phi}\otimes H^{q({\rm KR})}_{B}\ +\ \delta\bar{g}^{(\rm
KR)}_{B}\otimes H^{q({\rm KR})}_{B}\ $ $\displaystyle\hskip 93.89418pt+\
\bar{g}_{T\phi}^{(\rm rbw)}\otimes H^{q}_{T}\ +\ \bar{g}^{(\rm
KR)}_{T\phi}\otimes H^{q({\rm KR})}_{T}\ +\ \delta\bar{g}^{(\rm
KR)}_{T\phi}\otimes H^{q({\rm KR})}_{T}\ $ $\displaystyle\hskip 93.89418pt+\
\bar{g}_{B\phi}^{(\rm rbw\,mag)}\otimes E^{q}_{B}\ +\ \bar{g}_{T\phi}^{(\rm
rbw\,mag)}\otimes E^{q}_{T}\ +\ \bar{g}_{BT}^{(\rm rbw\,mag)}\otimes
E^{q}_{BT}~{}~{}~{}~{}~{}~{}$ $\displaystyle\hskip 93.89418pt+\ \bar{g}^{(\rm
tad\,mag)}_{\phi}\otimes E^{q({\rm tad})}_{\phi\phi^{\dagger}}\Big{]}(x,t),$

where $H^{q}_{0}$ and $E^{q}_{0}$ are the quark GPDs of the bare proton, and
$Z_{2}$ is the wave function renormalization constant associated with the
dressing of the bare proton by the meson loops in Fig. 1. As shorthand, in
Eqs. (115) we use the notation $\bar{f}_{j}(y)\equiv f_{j}(1-y)$ and
$\bar{g}_{j}(y)\equiv g_{j}(1-y)$ for the electric and magnetic splitting
functions involving couplings to baryons. Note that both the electric and
magnetic operators contribute to $H^{q}(x,t)$ and $E^{q}(x,t)$ at zero and
finite momentum transfer. At zero momentum transfer, however, there is no
contribution from the magnetic term to the matrix element, even though the GPD
$E^{q}(x,0)$ itself is nonzero.

The expressions for the quark and antiquark GPDs in Eqs. (114) and (115) form
the basis for our calculations of the meson loop contributions to the GPD
flavor asymmetries. For the case of $u$ and $d$ quarks, the intermediate
states include the nucleon and $\Delta$ baryons and $\pi$ mesons. For the
strange quark, on the other hand, the intermediate states that contribute are
the hyperons $\Lambda$, $\Sigma$ and $\Sigma^{*}$ and $K$ mesons. To compute
the quark and antiquark GPDs numerically, we require information on the GPDs
for the various hadronic configurations that contribute in Eqs. (114)–(115),
which we turn to next.

###  V.2 GPDs of hadronic configurations

The twist-two operators associated with the PDFs of the hadronic
configurations have been discussed in detail in Refs. Wang  _et al._ (2016b);
Salamu  _et al._ (2019b). For GPDs at finite momentum transfer $t$, the
operators are somewhat more complicated, and we consider first the
intermediate octet baryons as the hadronic configuration as an example. The
relevant operator here can be written as

$\displaystyle{\cal O}_{q\,B}^{\mu\mu_{1}\ldots\mu_{n-1}}$ $\displaystyle=$
$\displaystyle\sum_{i=0,\rm
even}^{n-1}\frac{1}{2}\Big{(}\alpha_{A}^{(n)i}\,{\rm
Tr}\left[\bar{B}\gamma^{\mu}\left\\{(u^{\dagger}\lambda_{q}u+u\lambda_{q}u^{\dagger}),B\right\\}\right]$
(116) $\displaystyle\hskip 20.77039pt+\ \beta_{A}^{(n)i}\,{\rm
Tr}\left[\bar{B}\gamma^{\mu}\left[(u^{\dagger}\lambda_{q}u+u\lambda_{q}u^{\dagger}),B\right]\right]$
$\displaystyle\hskip 20.77039pt+\ \sigma_{A}^{(n)i}\,{\rm
Tr}\left[\bar{B}\gamma^{\mu}B\right]{\rm
Tr}\left[(u^{\dagger}\lambda_{q}u+u\lambda_{q}u^{\dagger})\right]\Big{)}\,\Delta^{\mu_{1}}\cdots\Delta^{\mu_{i}}P^{\mu_{i+1}}\cdots
P^{\mu_{n-1}}$ $\displaystyle+$ $\displaystyle\sum_{i=0,\rm
even}^{n-1}\frac{i}{4M_{B}}\Big{(}\alpha_{B}^{(n)i}\,{\rm
Tr}\left[\bar{B}\sigma^{\mu\nu}\left\\{(u^{\dagger}\lambda_{q}u+u\lambda_{q}u^{\dagger}),B\right\\}\right]$
$\displaystyle\hskip 20.77039pt+\ \beta_{B}^{(n)i}\,{\rm
Tr}\left[\bar{B}\sigma^{\mu\nu}\left[(u^{\dagger}\lambda_{q}u+u\lambda_{q}u^{\dagger}),B\right]\right]$
$\displaystyle\hskip 20.77039pt+\ \sigma_{B}^{(n)i}\,{\rm
Tr}\left[\bar{B}\sigma^{\mu\nu}B\right]{\rm
Tr}\left[(u^{\dagger}\lambda_{q}u+u\lambda_{q}u^{\dagger})\right]\Big{)}\,\Delta^{\nu}\Delta^{\mu_{1}}\cdots\Delta^{\mu_{i}}P^{\mu_{i+1}}\cdots
P^{\mu_{n-1}}$ $\displaystyle+$
$\displaystyle\frac{1}{2M_{B}}\Big{(}\alpha_{C}^{(n)}\big{|}_{n=\rm
even}\,{\rm
Tr}\left[\bar{B}\left\\{(u^{\dagger}\lambda_{q}u+u\lambda_{q}u^{\dagger}),B\right\\}\right]$
$\displaystyle\hskip 20.77039pt+\ \beta_{C}^{(n)}\big{|}_{n=\rm even}\,{\rm
Tr}\left[\bar{B}\left[(u^{\dagger}\lambda_{q}u+u\lambda_{q}u^{\dagger}),B\right]\right]$
$\displaystyle\hskip 20.77039pt+\ \sigma_{C}^{(n)}\big{|}_{n=\rm even}\,{\rm
Tr}\left[\bar{B}B\right]{\rm
Tr}\left[(u^{\dagger}\lambda_{q}u+u\lambda_{q}u^{\dagger})\right]\Big{)}\,\Delta^{\mu}\Delta^{\mu_{1}}\cdots\Delta^{\mu_{n-1}}.$

Contracting both sides of Eq. (116) with the light-front unit vectors
$n_{\mu}n_{\mu_{1}}\cdots n_{\mu_{n-1}}$ then gives

$\displaystyle n_{\mu}n_{\mu_{1}}\cdots n_{\mu_{n-1}}{\cal
O}_{q\,B}^{\mu\mu_{1}\ldots\mu_{n-1}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\alpha^{(n)}\,{\rm
Tr}\left[\bar{B}\not{n}\left\\{(u^{\dagger}\lambda_{q}u+u\lambda_{q}u^{\dagger}),B\right\\}\right]$
(117) $\displaystyle+$ $\displaystyle\frac{1}{2}\beta^{(n)}\,{\rm
Tr}\left[\bar{B}\not{n}\left[(u^{\dagger}\lambda_{q}u+u\lambda_{q}u^{\dagger}),B\right]\right]$
$\displaystyle+$ $\displaystyle\frac{1}{2}\sigma^{(n)}\,{\rm
Tr}\left[\bar{B}\not{n}B\right]{\rm
Tr}\left[(u^{\dagger}\lambda_{q}u+u\lambda_{q}u^{\dagger})\right]$
$\displaystyle+$ $\displaystyle\frac{i\alpha_{\rm mag}^{(n)}}{8M_{B}}\,{\rm
Tr}\left[\bar{B}n_{\mu}\sigma^{\mu\nu}\Delta_{\nu}\left\\{(u^{\dagger}\lambda_{q}u+u\lambda_{q}u^{\dagger}),B\right\\}\right]$
$\displaystyle+$ $\displaystyle\frac{i\beta_{\rm mag}^{(n)}}{8M_{B}}\,{\rm
Tr}\left[\bar{B}n_{\mu}\sigma^{\mu\nu}\Delta_{\nu}\left[(u^{\dagger}\lambda_{q}u+u\lambda_{q}u^{\dagger}),B\right]\right]$
$\displaystyle+$ $\displaystyle\frac{i\sigma_{\rm mag}^{(n)}}{8M_{B}}\,{\rm
Tr}\left[\bar{B}n_{\mu}\sigma^{\mu\nu}\Delta_{\nu}B\right]{\rm
Tr}\left[(u^{\dagger}\lambda_{q}u+u\lambda_{q}u^{\dagger})\right],$

where for shorthand we define by $X^{(n)}$ and $X_{\rm mag}^{(n)}$, where
$X=\alpha$, $\beta$ or $\sigma$, the following combinations,

$\displaystyle X^{(n)}$ $\displaystyle=$ $\displaystyle\sum_{i=0,{\rm
even}}^{n-1}(-2\xi)^{i}X_{A}^{(n)i}+(-2\xi)^{n}X_{C}^{(n)}\Big{|}_{n={\rm
even}},$ (118a) $\displaystyle X_{\rm mag}^{(n)}$ $\displaystyle=$
$\displaystyle\sum_{i=0,{\rm
even}}^{n-1}(-2\xi)^{i}X_{B}^{(n)i}-(-2\xi)^{n}X_{C}^{(n)}\Big{|}_{n={\rm
even}}.$ (118b)

The $n$-th moments of the GPDs in octet baryons can then be related to the
coefficients $X^{(n)}$ and $X^{(n)}_{\rm mag}$. By expanding the matrices in
Eq. (117), one can derive relations between the quark GPDs of different
flavors in the octet baryons. Since such relations are independent of the
momentum transfer and structure of the $\gamma$-matrices of the operator, the
relations between the Dirac-like GPDs $H^{q}(x,t)$ will be the same as those
for PDFs obtained in Refs. Wang  _et al._ (2016b); Salamu  _et al._ (2019b).
In the following, we therefore focus on the derivation of the relations for
the spin-flip GPDs, $E^{q}(x,t)$, and examine whether the relations between
the different flavors are the same as those for the PDFs.

For the magnetic operator, the contraction with the light-front unit vectors
gives

$\displaystyle n_{\mu}n_{\mu_{1}}\cdots n_{\mu_{n-1}}{\cal O}_{q\,\rm
mag}^{\mu\mu_{1}\ldots\mu_{n-1}}$ $\displaystyle=$
$\displaystyle\frac{i\alpha_{\rm mag}^{(n)}}{4M_{B}}\,{\rm
Tr}\left[\bar{B}n_{\mu}\sigma^{\mu\nu}\Delta_{\nu}\left\\{(u^{\dagger}\lambda_{q}u+u\lambda_{q}u^{\dagger}),B\right\\}\right]$
(119) $\displaystyle+$ $\displaystyle\frac{i\beta_{\rm
mag}^{(n)}}{4M_{B}}\,{\rm
Tr}\left[\bar{B}n_{\mu}\sigma^{\mu\nu}\Delta_{\nu}\left[(u^{\dagger}\lambda_{q}u+u\lambda_{q}u^{\dagger}),B\right]\right]$
$\displaystyle+$ $\displaystyle\frac{i\sigma_{\rm mag}^{(n)}}{4M_{B}}\,{\rm
Tr}\left[\bar{B}n_{\mu}\sigma^{\mu\nu}\Delta_{\nu}B\right]{\rm
Tr}\left[(u^{\dagger}\lambda_{q}u+u\lambda_{q}u^{\dagger})\right]$
$\displaystyle-$ $\displaystyle\frac{i\theta_{\rm
mag}^{(n)}}{2M_{T}}\,\overline{T}_{\alpha}^{abc}n_{\mu}\sigma^{\mu\nu}\Delta_{\nu}(\lambda_{q}^{+})_{ae}T_{\alpha}^{ebc}$
$\displaystyle-$ $\displaystyle\frac{i\rho_{\rm
mag}^{(n)}}{2M_{T}}\,\overline{T}_{\alpha}^{abc}n_{\mu}\sigma^{\mu\nu}\Delta_{\nu}T_{\alpha}^{abc}\,{\rm
Tr}[\lambda_{q}^{+}]$ $\displaystyle+$ $\displaystyle\frac{\omega_{\rm
mag}^{(n)}}{4M_{B}}\,\Big{[}\epsilon_{ijk}(\lambda_{q}^{+})_{il}\bar{B}^{j}_{m}\not{\Delta}\gamma_{5}n_{\mu}T^{\mu,klm}$
$\displaystyle\hskip
25.6073pt-\epsilon_{ijk}(\lambda_{q}^{+})_{il}\bar{B}^{j}_{m}\not{n}\gamma_{5}\Delta_{\nu}T^{\nu,klm}\Big{]}+{\rm
H.c.}$

The coefficients $\big{\\{}\alpha_{\rm mag}^{(n)}$, $\beta_{\rm mag}^{(n)}$,
$\sigma_{\rm mag}^{(n)}$, $\theta_{\rm mag}^{(n)}$, $\rho_{\rm mag}^{(n)}$,
$\omega_{\rm mag}^{(n)}\big{\\}}$ are related to the $n$-th magnetic moments
of the quark distributions in the corresponding hadronic configurations. With
the simplification of the flavor matrices, the contracted magnetic operator
can be rewritten in the form

$\displaystyle n_{\mu}n_{\mu_{1}}\cdots n_{\mu_{n-1}}{\cal O}_{q\,\rm
mag}^{\mu\mu_{1}\cdots\mu_{n-1}}$ $\displaystyle\hskip 19.91684pt=\,Q_{B\,\rm
mag}^{(n-1)}\,{\cal O}_{B\,\rm mag}\,+\,Q_{T\,\rm mag}^{(n-1)}\,{\cal
O}_{T\,\rm mag}\,+\,Q_{BT\,\rm mag}^{(n-1)}\,{\cal O}_{BT\,\rm
mag}\,+\,Q_{\phi\phi^{\dagger}\,\rm mag}^{(n-1)}\,{\cal
O}_{\phi\phi^{\dagger}\,\rm mag},$ (120)

where the hadronic operators are given by

$\displaystyle{\cal O}_{B\,\rm mag}$ $\displaystyle=$
$\displaystyle\frac{i}{2M_{B}}\,\bar{B}n_{\mu}\sigma^{\mu\nu}\Delta_{\nu}B,$
(121a) $\displaystyle{\cal O}_{T\,\rm mag}$ $\displaystyle=$
$\displaystyle-\frac{i}{2M_{T}}\,\overline{T}_{\alpha}n_{\mu}\sigma^{\mu\nu}\Delta_{\nu}T_{\alpha},$
(121b) $\displaystyle{\cal O}_{BT\,\rm mag}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{3}}{2\sqrt{2}M_{B}}\,\big{(}\bar{B}\not{\Delta}\gamma_{5}n_{\mu}T^{\mu}-\bar{B}\not{n}\gamma_{5}\Delta_{\nu}T^{\nu}\big{)},$
(121c) $\displaystyle{\cal O}_{\phi\phi^{\dagger}\,\rm mag}$ $\displaystyle=$
$\displaystyle-\frac{i}{2M_{B}f^{2}}\,\bar{B}n_{\mu}\sigma^{\mu\nu}\Delta_{\nu}B\,\phi\phi^{\dagger}.$
(121d)

The coefficients $Q^{(n-1)}_{j\,\rm mag}$ of each of the operators are defined
in terms of the Mellin moments of the corresponding distribution functions in
the intermediate hadron states,

$\displaystyle Q^{(n-1)}_{B\,\rm mag}$ $\displaystyle=$
$\displaystyle\int_{-1}^{1}\differential{x}x^{n-1}\,E^{q}_{B}(x,t),$ (122a)
$\displaystyle Q^{(n-1)}_{T\,\rm mag}$ $\displaystyle=$
$\displaystyle\int_{-1}^{1}\differential{x}x^{n-1}\,E^{q}_{T}(x,t),$ (122b)
$\displaystyle Q^{(n-1)}_{BT\,\rm mag}$ $\displaystyle=$
$\displaystyle\int_{-1}^{1}\differential{x}x^{n-1}\,E^{q}_{BT}(x,t),$ (122c)
$\displaystyle Q^{(n-1)}_{\phi\phi^{\dagger}\,\rm mag}$ $\displaystyle=$
$\displaystyle\int_{-1}^{1}\differential{x}x^{n-1}\,E^{q}_{\phi\phi^{\dagger}}(x,t),$
(122d)

where the GPDs correspond to those appearing in the convolution expressions on
the right-hand-sides of Eqs. (115).

Table 2: Moments $Q^{(n-1)}_{B\,\rm mag}$ of the $u$, $d$ and $s$ quark GPDs
in octet baryons arising from the magnetic interaction.
$B$ | $U^{(n-1)}_{B\,\rm mag}$ | $D^{(n-1)}_{B\,\rm mag}$ | $S^{(n-1)}_{B\,\rm mag}$
---|---|---|---
$p$ |  $\alpha_{\rm mag}^{(n)}+\beta_{\rm mag}^{(n)}+\sigma_{\rm mag}^{(n)}$ | $\sigma_{\rm mag}^{(n)}$ |  $\alpha_{\rm mag}^{(n)}-\beta_{\rm mag}^{(n)}+\sigma_{\rm mag}^{(n)}$
$n$ | $\sigma_{\rm mag}^{(n)}$ |  $\alpha_{\rm mag}^{(n)}+\beta_{\rm mag}^{(n)}+\sigma_{\rm mag}^{(n)}$ | $\alpha_{\rm mag}^{(n)}-\beta_{\rm mag}^{(n)}+\sigma_{\rm mag}^{(n)}$
$\Sigma^{+}$ | $\alpha_{\rm mag}^{(n)}+\beta_{\rm mag}^{(n)}+\sigma_{\rm mag}^{(n)}$ | $\alpha_{\rm mag}^{(n)}-\beta_{\rm mag}^{(n)}+\sigma_{\rm mag}^{(n)}$ | $\sigma_{\rm mag}^{(n)}$
$\Sigma^{0}$ | $\alpha_{\rm mag}^{(n)}+\sigma_{\rm mag}^{(n)}$ | $\alpha_{\rm mag}^{(n)}+\sigma_{\rm mag}^{(n)}$ | $\sigma_{\rm mag}^{(n)}$
$\Sigma^{-}$ | $\alpha_{\rm mag}^{(n)}-\beta_{\rm mag}^{(n)}+\sigma_{\rm mag}^{(n)}$ | $\alpha_{\rm mag}^{(n)}+\beta_{\rm mag}^{(n)}+\sigma_{\rm mag}^{(n)}$ | $\sigma_{\rm mag}^{(n)}$
$\Lambda$ | $\frac{1}{3}\alpha_{\rm mag}^{(n)}+\sigma_{\rm mag}^{(n)}$ | $\frac{1}{3}\alpha_{\rm mag}^{(n)}+\sigma_{\rm mag}^{(n)}$ | $\frac{4}{3}\alpha_{\rm mag}^{(n)}+\sigma_{\rm mag}^{(n)}$
$\Lambda\Sigma^{0}$ | $\frac{1}{\sqrt{3}}\alpha_{\rm mag}^{(n)}$ | $-\frac{1}{\sqrt{3}}\alpha_{\rm mag}^{(n)}$ | $0$

The moments $U_{B\,\rm mag}^{(n-1)}$, $D_{B\,\rm mag}^{(n-1)}$ and $S_{B\,\rm
mag}^{(n-1)}$ of the $u$, $d$ and $s$ quark GPDs in the octet baryons can be
expressed in terms of the coefficients $\alpha_{\rm mag}^{(n)}$, $\beta_{\rm
mag}^{(n)}$ and $\sigma_{\rm mag}^{(n)}$, as listed in Table 2. Solving for
the coefficients, one can write these as linear combinations of the quark GPD
moments in the proton,

$\displaystyle\alpha_{\rm mag}^{(n)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{(}U_{p\,\rm mag}^{(n-1)}+S_{p\,\rm
mag}^{(n-1)}\Big{)}-D_{p\,\rm mag}^{(n-1)},$ (123a) $\displaystyle\beta_{\rm
mag}^{(n)}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\Big{(}U_{p\,\rm
mag}^{(n-1)}-S_{p\,\rm mag}^{(n-1)}\Big{)},$ (123b) $\displaystyle\sigma_{\rm
mag}^{(n)}$ $\displaystyle=$ $\displaystyle D_{p\,\rm mag}^{(n-1)}.$ (123c)

Furthermore, assuming the strangeness in the bare nucleon state to be zero, we
have

$\sigma_{\rm mag}^{(n)}=\beta_{\rm mag}^{(n)}-\alpha_{\rm mag}^{(n)}.$ (124)

In particular, for $n=1$ the relation Eq. (124) is consistent with the
Lagrangian for the magnetic interaction, Eq. (19), where $c_{3}=c_{2}-c_{1}$.

Table 3: Moments $Q^{(n-1)}_{T\,\rm mag}$ of the $u$, $d$ and $s$ quark GPDs
in decuplet baryons arising from the magnetic interaction.
$T$ | $U^{(n-1)}_{T\,\rm mag}$ | $D^{(n-1)}_{T\,\rm mag}$ | $S^{(n-1)}_{T\,\rm mag}$
---|---|---|---
$\Delta^{++}$ | $\theta_{\rm mag}^{(n)}+\rho_{\rm mag}^{(n)}$ | $\rho_{\rm mag}^{(n)}$ | $\rho_{\rm mag}^{(n)}$
$\Delta^{+}$ | $\frac{2}{3}\theta_{\rm mag}^{(n)}+\rho_{\rm mag}^{(n)}$ | $\frac{1}{3}\theta_{\rm mag}^{(n)}+\rho_{\rm mag}^{(n)}$ | $\rho_{\rm mag}^{(n)}$
$\Delta^{0}$ | $\frac{1}{3}\theta_{\rm mag}^{(n)}+\rho_{\rm mag}^{(n)}$ | $\frac{2}{3}\theta_{\rm mag}^{(n)}+\rho_{\rm mag}^{(n)}$ | $\rho_{\rm mag}^{(n)}$
$\Delta^{-}$ | $\rho_{\rm mag}^{(n)}$ | $\theta_{\rm mag}^{(n)}+\rho_{\rm mag}^{(n)}$ | $\rho_{\rm mag}^{(n)}$
$\Sigma^{*+}$ | $\frac{2}{3}\theta_{\rm mag}^{(n)}+\rho_{\rm mag}^{(n)}$ | $\rho_{\rm mag}^{(n)}$ | $\frac{1}{3}\theta_{\rm mag}^{(n)}+\rho_{\rm mag}^{(n)}$
$\Sigma^{*0}$ |  $\frac{1}{3}\theta_{\rm mag}^{(n)}+\rho_{\rm mag}^{(n)}$ |  $\frac{1}{3}\theta_{\rm mag}^{(n)}+\rho_{\rm mag}^{(n)}$ |  $\frac{1}{3}\theta_{\rm mag}^{(n)}+\rho_{\rm mag}^{(n)}$
$\Sigma^{*-}$ | $\rho_{\rm mag}^{(n)}$ | $\frac{2}{3}\theta_{\rm mag}^{(n)}+\rho_{\rm mag}^{(n)}$ | $\frac{1}{3}\theta_{\rm mag}^{(n)}+\rho_{\rm mag}^{(n)}$

The moments of the GPDs in the decuplet baryons, $Q_{T\,\rm mag}^{(n-1)}$, for
$u$, $d$ and $s$ quarks can be expressed in terms of the coefficients
$\theta_{\rm mag}^{(n)}$ and $\rho_{\rm mag}^{(n)}$, which are listed in Table
3. Solving for the coefficients $\theta_{\rm mag}^{(n)}$ and $\rho_{\rm
mag}^{(n)}$ in terms of the GPD moments in the $\Delta^{+}$ baryon, one has

$\displaystyle\theta_{\rm mag}^{(n)}$ $\displaystyle=$ $\displaystyle
3\Big{(}D_{\Delta^{+}}^{(n-1)}-S_{\Delta^{+}}^{(n-1)}\Big{)}=\frac{3}{2}\Big{(}U_{\Delta^{+}}^{(n-1)}-S_{\Delta^{+}}^{(n-1)}\Big{)},$
(125a) $\displaystyle\rho_{\rm mag}^{(n)}$ $\displaystyle=$ $\displaystyle
S_{\Delta^{+}}^{(n-1)}.$ (125b)

Proceeding in analogy with the nucleon case, we assume no strange quark
contribution in the undressed $\Delta$ baryons, which enables the decuplet
coefficients to be written in terms of the $u$ and $d$ quark GPD moments in
the $\Delta^{+}$,

$\displaystyle\theta_{\rm mag}^{(n)}$ $\displaystyle=$ $\displaystyle
3D_{\Delta^{+}}^{(n-1)}\,=\,\frac{3}{2}U_{\Delta^{+}}^{(n-1)},\qquad\rho_{\rm
mag}^{(n)}\,=\,0.$ (126)

This is also consistent with the effective Lagrangian of Eq. (19), where only
one term for the decuplet magnetic interaction is included.

Table 4: Moments $Q^{(n-1)}_{BT\,\rm mag}$ of the $u$, $d$ and $s$ quark
distributions arising from the magnetic interactions for the octet-decuplet
transition.
$BT$ | $U^{(n-1)}_{BT\,\rm mag}$ | $D^{(n-1)}_{BT\,\rm mag}$ | $S^{(n-1)}_{BT\,\rm mag}$
---|---|---|---
$~{}~{}p\Delta^{+}$ | $\frac{1}{3\sqrt{2}}\,\omega_{\rm mag}^{(n)}$ |  $-\frac{1}{3\sqrt{2}}\,\omega_{\rm mag}^{(n)}$ | $0$
$~{}~{}n\Delta^{0}$ | $\frac{1}{3\sqrt{2}}\,\omega_{\rm mag}^{(n)}$ | $-\frac{1}{3\sqrt{2}}\,\omega_{\rm mag}^{(n)}$ | $0$
$\Sigma^{+}\Sigma^{*+}$ |  $-\frac{1}{3\sqrt{2}}\,\omega_{\rm mag}^{(n)}$ | $0$ | $\frac{1}{3\sqrt{2}}\,\omega_{\rm mag}^{(n)}$
$\Sigma^{0}\Sigma^{*0}$ | $\frac{1}{6\sqrt{2}}\,\omega_{\rm mag}^{(n)}$ | $\frac{1}{6\sqrt{2}}\,\omega_{\rm mag}^{(n)}$ | $-\frac{1}{3\sqrt{2}}\,\omega_{\rm mag}^{(n)}$
$\Sigma^{-}\Sigma^{*-}$ | $0$ | $\frac{1}{3\sqrt{2}}\,\omega_{\rm mag}^{(n)}$ | $-\frac{1}{3\sqrt{2}}\,\omega_{\rm mag}^{(n)}$
$\Lambda\Sigma^{*0}$ |  $-\frac{1}{\sqrt{24}}\,\omega_{\rm mag}^{(n)}$ | $\frac{1}{\sqrt{24}}\,\omega_{\rm mag}^{(n)}$ | $0$

The moments of quark GPDs for the octet-decuplet transition, $Q_{BT\,\rm
mag}^{(n-1)}$, can be expressed in terms of the coefficient $\omega_{\rm
mag}^{(n)}$ defined in Eq. (119), and are given in Table 4 for the allowed
configurations. One can write $\omega_{\rm mag}^{(n)}$ in terms of the
proton–$\Delta^{+}$ transition GPD moments as

$\omega_{\rm
mag}^{(n)}=3\sqrt{2}\,U_{p\Delta^{+}}^{(n-1)}=-3\sqrt{2}\,D_{p\Delta^{+}}^{(n-1)},$
(127)

with relations for the other octet-decuplet transitions in Table 4.

The moments of the distributions generated by the tadpole vertex are listed in
Table 5, where the corresponding moments $Q^{(n-1)}_{\phi\phi^{\dagger}\,\rm
mag}$ of the $u$, $d$ and $s$ quark GPDs are expressed in terms of the
$\alpha_{\rm mag}^{(n)}$ and $\beta_{\rm mag}^{(n)}$ coefficients in Eq.
(123). Note that combinations involving $K^{0}\overline{K}^{0}$ mesons do not
contribute to the $u$-quark moments, while those involving $K^{+}K^{-}$ do not
contribute to the $d$-quark moments, and contributions from $\pi^{+}\pi^{-}$
configurations to the $s$-quark moments also vanish.

Table 5: Moments $Q^{(n-1)}_{B\phi\phi\,\rm mag}$ of the $u$, $d$ and $s$
quark GPDs arising from the magnetic interactions involving the $BB\phi\phi$
tadpole vertex. For short-hand we define $\Gamma_{\pm}^{(n)}\equiv\alpha_{\rm
mag}^{(n)}\pm\beta_{\rm mag}^{(n)}$.
| $U^{(n-1)}_{\phi\phi\,\rm mag}$ | $D^{(n-1)}_{\phi\phi\,\rm mag}$ | $S^{(n-1)}_{\phi\phi\,\rm mag}$
---|---|---|---
$B$ |  |  |  |  |  |
| $\pi^{+}\pi^{-}$ | $K^{+}K^{-}$ | $\pi^{+}\pi^{-}$ | $K^{0}\overline{K}^{0}$ | $K^{0}\overline{K}^{0}$ | $K^{+}K^{-}$
$p$ |  $\frac{1}{2}\Gamma_{+}^{(n)}$ | $\beta_{\rm mag}^{(n)}$ | $-\frac{1}{2}\Gamma_{+}^{(n)}$ | $-\frac{1}{2}\Gamma_{-}^{(n)}$ | $\frac{1}{2}\Gamma_{-}^{(n)}$ | $-\beta_{\rm mag}^{(n)}$
$n$ | $-\frac{1}{2}\Gamma_{+}^{(n)}$ | $-\frac{1}{2}\Gamma_{-}^{(n)}$ | $\frac{1}{2}\Gamma_{+}^{(n)}$ | $\beta_{\rm mag}^{(n)}$ | $-\beta_{\rm mag}^{(n)}$ | $\frac{1}{2}\Gamma_{-}^{(n)}$
$\Sigma^{+}$ | $\beta_{\rm mag}^{(n)}$ | $\frac{1}{2}\Gamma_{+}^{(n)}$ | $-\beta_{\rm mag}^{(n)}$ | $\frac{1}{2}\Gamma_{-}^{(n)}$ | $-\frac{1}{2}\Gamma_{-}^{(n)}$ | $-\frac{1}{2}\Gamma_{+}^{(n)}$
$\Sigma^{0}$ | $0$ | $\frac{1}{2}\alpha_{\rm mag}^{(n)}$ | $0$ | $\frac{1}{2}\alpha_{\rm mag}^{(n)}$ | $-\frac{1}{2}\alpha_{\rm mag}^{(n)}$ | $-\frac{1}{2}\alpha_{\rm mag}^{(n)}$
$\Sigma^{-}$ | $-\beta_{\rm mag}^{(n)}$ | $\frac{1}{2}\Gamma_{-}^{(n)}$ | $\beta_{\rm mag}^{(n)}$ | $\frac{1}{2}\Gamma_{+}^{(n)}$ | $-\frac{1}{2}\Gamma_{+}^{(n)}$ | $-\frac{1}{2}\Gamma_{-}^{(n)}$
$\Lambda$ | $0$ | $-\frac{1}{2}\alpha_{\rm mag}^{(n)}$ | $0$ | $-\frac{1}{2}\alpha_{\rm mag}^{(n)}$ | $\frac{1}{2}\alpha_{\rm mag}^{(n)}$ | $\frac{1}{2}\alpha_{\rm mag}^{(n)}$
$\Lambda\Sigma^{0}$ |  $\frac{1}{\sqrt{3}}\,\alpha_{\rm mag}^{(n)}$ |  $\frac{1}{2\sqrt{3}}\,\alpha_{\rm mag}^{(n)}$ |  $-\frac{1}{\sqrt{3}}\,\alpha_{\rm mag}^{(n)}$ |  $-\frac{1}{2\sqrt{3}}\,\alpha_{\rm mag}^{(n)}$ |  $\frac{1}{2\sqrt{3}}\,\alpha_{\rm mag}^{(n)}$ |  $-\frac{1}{2\sqrt{3}}\,\alpha_{\rm mag}^{(n)}$

Since the above relations for the GPD moments are valid for all values of $n$,
one can derive from the moments explicit relations between the input valence
GPDs for different quark flavors in various hadrons. Focusing still on the
magnetic GPD $E^{q}$, we relate the valence GPDs in the octet baryons to the
GPDs in the proton, denoted by $E^{q}\equiv E^{q}_{p}(x,t)\equiv
E^{q}_{p}(x,\xi=0,t)$, using the results from Table 2,

$\displaystyle E^{u}_{n}\ =E^{d},$ $\displaystyle E^{d}_{n}\ =E^{u},$
$\displaystyle E^{s}_{n}\ =E^{s},$ (128a) $\displaystyle
E^{u}_{\Sigma^{+}}\\!=E^{u},$ $\displaystyle E^{d}_{\Sigma^{+}}\\!=E^{s},$
$\displaystyle E^{s}_{\Sigma^{+}}\\!=E^{d},$ (128b) $\displaystyle
E^{u}_{\Sigma^{0}}=\frac{1}{2}\big{(}E^{u}+E^{s}\big{)},$ $\displaystyle
E^{d}_{\Sigma^{0}}=E^{u}_{\Sigma^{0}},$ $\displaystyle
E^{s}_{\Sigma^{0}}=E^{d},$ (128c) $\displaystyle E^{u}_{\Sigma^{-}}\\!=E^{s},$
$\displaystyle E^{d}_{\Sigma^{-}}\\!=E^{u},$ $\displaystyle
E^{s}_{\Sigma^{-}}\\!=E^{d},$ (128d) $\displaystyle E^{u}_{\Lambda}\
=\frac{1}{6}\big{(}E^{u}+4E^{d}+E^{s}\big{)},$ $\displaystyle E^{d}_{\Lambda}\
=E^{u}_{\Lambda},$ $\displaystyle E^{s}_{\Lambda}\
=\frac{1}{3}\big{(}2E^{u}-E^{d}+2E^{s}\big{)},$ (128e) $\displaystyle
E^{u}_{\Lambda\Sigma^{0}}\\!=\frac{1}{2\sqrt{3}}\big{(}E^{u}-2E^{d}+E^{s}\big{)},~{}~{}~{}~{}~{}$
$\displaystyle
E^{d}_{\Lambda\Sigma^{0}}\\!=-E^{u}_{\Lambda\Sigma^{0}},~{}~{}~{}~{}~{}$
$\displaystyle E^{s}_{\Lambda\Sigma^{0}}\\!=0.$ (128f)

For the GPDs in the SU(3) decuplet baryons, using the relations in Table 3
these can be written in terms of the GPDs in the $\Delta^{+}$ baryon, which we
denote by $E^{q}_{\Delta}\equiv E^{q}_{\Delta^{+}}(x,t)\equiv
E^{q}_{\Delta^{+}}(x,\xi=0,t)$,

$\displaystyle
E^{u}_{\Delta^{++}}\\!=E^{u}_{\Delta}+E^{d}_{\Delta}-E^{s}_{\Delta},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle
E^{d}_{\Delta^{++}}\\!=E^{s}_{\Delta},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle
E^{s}_{\Delta^{++}}\\!=E^{s}_{\Delta},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(129a) $\displaystyle E^{u}_{\Delta^{0}}\,\>=E^{d}_{\Delta},$ $\displaystyle
E^{d}_{\Delta^{0}}\,\>=E^{u}_{\Delta},$ $\displaystyle
E^{s}_{\Delta^{0}}\,\>=E^{s}_{\Delta},$ (129b) $\displaystyle
E^{u}_{\Sigma^{*+}}=E^{u}_{\Delta},$ $\displaystyle
E^{d}_{\Sigma^{*+}}=E^{s}_{\Delta},$ $\displaystyle
E^{s}_{\Sigma^{*+}}=E^{d}_{\Delta},$ (129c) $\displaystyle
E^{u}_{\Sigma^{*0}}\,=E^{d}_{\Delta},$ $\displaystyle
E^{d}_{\Sigma^{*0}}\,=E^{d}_{\Delta},$ $\displaystyle
E^{s}_{\Sigma^{*0}}\,=E^{d}_{\Delta},$ (129d) $\displaystyle
E^{u}_{\Sigma^{*-}}=E^{s}_{\Delta},$ $\displaystyle
E^{d}_{\Sigma^{*-}}=E^{u}_{\Delta},$ $\displaystyle
E^{s}_{\Sigma^{*-}}=E^{d}_{\Delta}.$ (129e)

Interestingly, the relations between the $E^{q}$ GPDs between different
flavors in the SU(3) octet and decuplet baryon configurations are identical to
those for the PDFs derived in Refs. Wang  _et al._ (2016b); Salamu  _et al._
(2019b). Similar relations will therefore follow also for the electric $H^{q}$
GPDs.

For the octet-decuplet transition GPDs, from Table 4 one can write each of the
transition GPDs in terms of the $u$-quark GPD for the $p$–$\Delta^{+}$
transition,

$\displaystyle E^{u}_{n\Delta^{0}}\;\ =E^{u}_{p\Delta^{+}},$ $\displaystyle
E^{d}_{n\Delta^{0}}\;\ =-E^{u}_{p\Delta^{+}},$ $\displaystyle
E^{s}_{n\Delta^{0}}\;\ =0,$ (130a) $\displaystyle
E^{u}_{\Sigma^{+}\Sigma^{*+}}\\!\\!=-E^{u}_{p\Delta^{+}},$ $\displaystyle
E^{d}_{\Sigma^{+}\Sigma^{*+}}\\!\\!=0,$ $\displaystyle
E^{s}_{\Sigma^{+}\Sigma^{*+}}\\!\\!=E^{u}_{p\Delta^{+}},$ (130b)
$\displaystyle E^{u}_{\Sigma^{0}\Sigma^{*0}}=\frac{1}{2}E^{u}_{p\Delta^{+}},$
$\displaystyle E^{d}_{\Sigma^{0}\Sigma^{*0}}=\frac{1}{2}E^{u}_{p\Delta^{+}},$
$\displaystyle E^{s}_{\Sigma^{0}\Sigma^{*0}}=-E^{u}_{p\Delta^{+}},$ (130c)
$\displaystyle E^{u}_{\Sigma^{-}\Sigma^{*-}}\\!\\!=0,$ $\displaystyle
E^{d}_{\Sigma^{-}\Sigma^{*-}}\\!\\!=E^{u}_{p\Delta^{+}},$ $\displaystyle
E^{s}_{\Sigma^{-}\Sigma^{*-}}\\!\\!=-E^{u}_{p\Delta^{+}},$ (130d)
$\displaystyle
E^{u}_{\Lambda\Sigma^{*0}}\,\>=-\frac{\sqrt{3}}{2}E^{u}_{p\Delta^{+}},~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle
E^{d}_{\Lambda\Sigma^{*0}}\,\>=\frac{\sqrt{3}}{2}E^{u}_{p\Delta^{+}},~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle E^{s}_{\Lambda\Sigma^{*0}}\,\>=0.$ (130e)

Finally, for the GPDs associated with the tadpoles, the distributions can be
expressed in terms of GPDs in the proton. For the case of the nucleon, from
Table 5 we have

$\displaystyle E^{u(\rm
tad)}_{\pi^{+}\pi^{-}}\,=\frac{1}{2}\big{(}E^{u}-E^{d}\big{)},~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle E^{d(\rm
tad)}_{\pi^{+}\pi^{-}}\,=\frac{1}{2}\big{(}E^{d}-E^{u}\big{)},~{}~{}~{}~{}~{}~{}$
$\displaystyle E^{s(\rm tad)}_{\pi^{+}\pi^{-}}\,=0,~{}~{}$ (131a)
$\displaystyle E^{u(\rm
tad)}_{K^{+}K^{-}}\\!=\frac{1}{2}\big{(}E^{u}-E^{s}\big{)},$ $\displaystyle
E^{d(\rm tad)}_{K^{+}K^{-}}\\!=0,$ $\displaystyle E^{s(\rm
tad)}_{K^{+}K^{-}}\\!=\frac{1}{2}\big{(}E^{s}-E^{u}\big{)},$ (131b)
$\displaystyle E^{u(\rm tad)}_{K^{0}\overline{K}^{0}}\,=0,$ $\displaystyle
E^{d(\rm
tad)}_{K^{0}\overline{K}^{0}}\,=\frac{1}{2}\big{(}E^{d}-E^{s}\big{)},$
$\displaystyle E^{s(\rm
tad)}_{K^{0}\overline{K}^{0}}\,=\frac{1}{2}\big{(}E^{s}-E^{d}\big{)},$ (131c)

he tadpole GPDs for the other baryons can also be derived from the relations
in Table 5.

Turning now to the Dirac-like $H^{q}_{j}(x,t)$ GPDs for the various hadronic
configurations $j$, we observe that these have the same relationships as for
PDFs Wang  _et al._ (2016b); Salamu  _et al._ (2019b). Taking the strange
quark flavor as an example, for the strange GPDs in octet baryons we have

$\displaystyle H^{s}_{\Sigma^{+}}\\!$ $\displaystyle=$ $\displaystyle
H^{s}_{\Sigma^{0}}=H^{s}_{\Sigma^{-}}=H^{d},$ (132a) $\displaystyle
H^{s}_{\Lambda}\,$ $\displaystyle=$
$\displaystyle\frac{1}{3}\big{(}2H^{u}-H^{d}+2H^{s}\big{)},$ (132b)

while for the strange GPDs in decuplet baryons,

$\displaystyle
H^{s}_{\Sigma^{*+}}\\!=H^{s}_{\Sigma^{*0}}\\!=H^{s}_{\Sigma^{*-}}\\!=H^{d}_{\Delta},$
(133)

with the strange GPDs in all other baryons vanishing.

For the strange GPDs associated with the tadpole diagram, we find

$\displaystyle H^{s(\rm