# Gravitational transition form factors of $N\rightarrow\Delta$ via QCD light-
cone sum rules

U. Özdem  Health Services Vocational School of Higher Education, Istanbul
Aydin University, Sefakoy-Kucukcekmece, 34295 Istanbul, Türkiye  K. Azizi
Department of Physics, University of Tehran, North Karegar Avenue, Tehran
14395-547, Iran  Department of Physics, Doǧuş University, Dudullu-Ümraniye,
34775 Istanbul, Türkiye

###### Abstract

We present the first direct calculation on the gravitational form factors
(GFFs) of the $N\rightarrow\Delta$ transition using an analytic method, the
QCD light-cone sum rules. The matrix element of the quark part of the energy
momentum tensor current sandwiched between the nucleon and $\Delta$ states are
parameterized in terms of five independent conserved and four independent non-
conserved GFFs, for calculation of which we use the distribution amplitudes
(DAs) of the on-shell nucleon expanded in terms of functions with different
twists. We present the results for two sets of light-cone input parameters.
The results indicate that the behavior of the form factors with respect to
$Q^{2}$ are described by multipole fit functions. Our results may be checked
by other phenomenological models including the Lattice QCD as well as future
related experiments.

Gravitational form factors, Nucleon, $\Delta(1232)$, QCD light-cone sum rules

##  I Introduction

The interaction of composite particles with electromagnetic, QCD energy-
momentum and weak currents are parameterized in terms of the corresponding
form factors (FFs) as the building blocks. Having determined these FFs, one
can construct different observables in terms of these non-perturbative objects
to determine the nature and internal quark-gluon organizations of hadrons. In
the case of the energy-momentum tensor (EMT) or gravitational form factors
(GFFs), which represent the gravitational interaction between a graviton and
the hadron, these FFs provide information on the mechanical properties of the
hadron such as the mass, spin, pressure, shear force, radius, etc.
Understanding the mechanical structure of hadrons is of great importance as it
provides us with the fundamental information on the internal structure and
geometric shapes of the hadrons.

The GFFs for the spin-1/2 particles entering the matrix elements of the EMT
current were first parameterized in Refs. Kobzarev:1962wt ; Pagels:1966zza ;
Kobsarev:1970qm ; Ng:1993vh . Based on these parametrizations, the GFFs of the
nucleon as a basic baryonic structure have been intensively investigated in
various approaches Polyakov:2002wz ; Ji:1997gm ; Schweitzer:2002nm ;
Jung:2013bya ; Hagler:2003jd ; Gockeler:2003jfa ; Pasquini:2007xz ;
Hwang:2007tb ; Abidin:2008hn ; Brodsky:2008pf ; Pasquini:2014vua ;
Chakrabarti:2015lba ; Lorce:2018egm ; Teryaev:2016edw ; Shanahan:2018nnv ;
Shanahan:2018pib ; Neubelt:2019sou ; Anikin:2019kwi ; Alharazin:2020yjv ;
Gegelia:2021wnj ; Varma:2020crx ; Fujita:2022jus ; Mamo:2022eui ;
Pefkou:2021fni ; Azizi:2019ytx ; Polyakov:2018exb ; Freese:2021mzg ;
Freese:2021qtb ; Freese:2021czn ; Burkert:2021ith ; Burkert:2018bqq . Compared
to the nucleon, the GFFs of other members of the baryon octet were much less
studied Ozdem:2020ieh ; Won:2022cyy . The calculations of EMT FFs have
recently been extended to the parity flip transitions, $N^{*}\rightarrow N$
Polyakov:2020rzq ; Azizi:2020jog ; Ozdem:2019pkg . They were also generalized
to the higher-spin particles Cotogno:2019vjb  in a systematic way. For a
spin-1 particle, the corresponding GFFs were studied in Refs. Polyakov:2019lbq
; Cosyn:2019aio ; Kim:2022wkc ; Freese:2022yur ; Freese:2022ibw ;
Freese:2019bhb ; Sun:2020wfo ; Epelbaum:2021ahi ; Pefkou:2021fni  using
different models and approaches. Very recently, the $N\to\Delta$ transition
matrix elements of the EMT current were also parametrized in Ref. Kim:2022bwn
. The GFFs for the spin-3/2 particle were also examined in Refs.
Pefkou:2021fni ; Fu:2022rkn ; Alharazin:2022wjj ; Panteleeva:2020ejw ,
recently.

In our study on the GFFs of the nucleon Azizi:2019ytx , which were calculated
in a large range of transferred momentum square $Q^{2}$, we compared our
predictions on the GFFs $M_{2}^{q}(Q^{2})$, $J^{q}(Q^{2})$ and
$d_{1}^{q}(Q^{2})$ (D-term) at small values of $Q^{2}$ with the existing
Lattice QCD predictions Hagler:2007xi  as well as the JLab data for the D-term
Burkert:2018bqq . For the form factor $M_{2}^{q}(Q^{2})$, our predictions were
consistent with most of the Lattice QCD data points considering the errorbars.
In the case of $J^{q}(Q^{2})$ and $d_{1}^{q}(Q^{2})$ form factors, the Lattice
results suffer from large uncertainties at small values of $Q^{2}$ that should
be computed more accurately. It was also obtained that our predictions
reproduce most of the JLab data at small values of $Q^{2}$. In general, due to
polynomiality, all the GFFs of the nucleon can also be obtained from the
second Mellin moments of the generalized parton distributions (GPDs)
integrated over Bjorken-$x$, for any value of the skewness variable $\xi$.
Only, the $M_{2}(Q^{2})$ and $J(Q^{2})$ GFFs of the nucleon can be calculated
using its GPDs at zero skewness Polyakov:2002yz . Very recently, the
$M_{2}(Q^{2})$ and $J(Q^{2})$ GFFs were obtained at this limit by integration
of the GPDs $H^{q}(x,Q^{2})$ and $E^{q}(x,Q^{2})$ over Bjorken-$x$ in Refs.
Hashamipour:2022noy ; Hashamipour:2021kes  and compared with our previous
light cone QCD sum rules predictions presented in Ref. Azizi:2019ytx . Very
surprisingly, for special sets, the obtained results via the GPDs that
themselves are extracted from the pure experimental data of different
collaborations are in a nice consistency with the predictions of our previous
study, Ref. Azizi:2019ytx , for the GFFs $M_{2}(Q^{2})$ and $J(Q^{2})$ at a
vide range of $Q^{2}$. Our predictions on the GFFs of the negative parity
excited nucleon $N(1535)$ or $N^{*}$ and the transition GFFs of the
$N^{*}\rightarrow N$ Azizi:2020jog ; Ozdem:2019pkg  can be used to be compared
with the future Lattice QCD results as well as experimental data.

In the present article, we study the transition GFFs of the $N\to\Delta$ for
the first time. We use the light cone QCD sum rule formalism to find the five
independent conserved and four independent non-conserved GFFs defining this
transition by considering the nucleon as the on-shell state, which allows us
to use the DAs of the nucleon in terms of wavefunctions of different twists.
We use two sets of parameters entering the wavefunctions to numerically
analyze the obtained sum rules for the FFs. We also give the fit functions of
the transition form factors describing the behavior of the FFs in terms of
$Q^{2}$.

The article is structured as follows. In next section we derive the desired
sum rules for the transition GFFs of the $N\to\Delta$. In section III, we
numerically analyze the sum rules for the form factors to find their $Q^{2}$
behavior and their values at static limit . The last section is reserved for
our concluding remarks.

##  II Formalism

In light-cone sum rule (LCSR) approach, the following two-point correlation
function in the presence of the on-shell nucleon state is responsible for the
calculations of the GFFs of the $N\to\Delta$ transition:

$\Pi_{\alpha\mu\nu}(p,q)=i\int d^{4}xe^{iqx}\langle
0|\mathcal{T}[J_{\alpha}^{\Delta}(0)T_{\mu\nu}^{q}(x)]|N(p)\rangle,$ (1)

where $\mathcal{T}$ is the time ordering operator, $p$ is the nucleon’s four-
momentum, $q$ is the transferred momentum, $J_{\alpha}^{\Delta}(0)$ is the
$\Delta$’s interpolating current and $T_{\mu\nu}^{q}(x)$ is the quark part of
the EMT current at point $x$. We insert a complete set of the intermediate
$\Delta(p^{\prime},s^{\prime})$ with momentum $p^{\prime}$ and $,s^{\prime}$
to the correlation function and perform the four-integral over $x$. This ends
up in

$\displaystyle\Pi_{\alpha\mu\nu}^{Had}(p,q)$
$\displaystyle=\sum_{s{{}^{\prime}}}\frac{\langle
0|J_{\alpha}^{\Delta}|{\Delta(p^{\prime},s^{\prime})}\rangle\langle{\Delta(p^{\prime},s^{\prime})}|T_{\mu\nu}^{q}|N(p,s)\rangle}{m^{2}_{\Delta}-p^{\prime
2}}+...,$ (2)

where dots stand for the contributions of the higher states and continuum. We
choose the continuum threshold $s_{0}$ (coming from the continuum subtraction
procedure) such that the correlation function includes only the ground state
$\Delta$ and the first and higher excited states are included in dots in the
above equation (for the procedure of the continuum subtraction and
determination of the working window for continuum threshold $s_{0}$ see the
sections Appendix and III, respectively). To go further, we need to define the
following matrix element in terms of the residue of the $\Delta$ baryon
($\lambda_{\Delta}$):

$\displaystyle\langle
0|J_{\alpha}^{\Delta}|{\Delta(p^{\prime},s^{\prime})}\rangle$
$\displaystyle=\lambda_{\Delta}u_{\alpha}(p^{\prime},s^{\prime}),$ (3)

where, $u_{\alpha}(p^{\prime},s^{\prime})$ is the Rarita-Schwinger spinor. To
proceed, we define the matrix element of the quark part of the EMT current
sandwiched between the nucleon and $\Delta$ state. As the quark part of the
EMT current solely is not a conserved quantity, the transition matrix element
is decomposed in terms of nine form factors (five independent conserved and
four independent non-conserved) by demanding the criteria of the Lorentz
invariance, discrete space-time symmetries and the related equations of
motions Kim:2022bwn :

$\displaystyle\langle\Delta(p^{\prime},s^{\prime})|T_{\mu\nu}^{q}|N(p,s)\rangle$
$\displaystyle=\bar{u}_{\beta}(p^{\prime},s^{\prime})\Bigg{[}F_{1}^{N\Delta}(Q^{2})\bigg{\\{}g_{\beta\\{\mu}P_{\nu\\}}+\frac{(m^{2}_{\Delta}-m^{2}_{N})}{\Delta^{2}}g_{\mu\nu}\Delta_{\beta}-\frac{(m^{2}_{\Delta}-m^{2}_{N})}{2\Delta^{2}}g_{\beta\\{\mu}\Delta_{\nu\\}}-\frac{\Delta_{\beta}P_{\\{\mu}\Delta_{\nu\\}}}{\Delta^{2}}\bigg{\\}}$
$\displaystyle+\frac{F_{2}^{N\Delta}(Q^{2})}{{\bar{m}}^{2}}\bigg{\\{}P_{\mu}P_{\nu}\Delta_{\beta}+\frac{(m_{\Delta}^{2}-m_{N}^{2})^{2}}{4\Delta^{2}}g_{\mu\nu}\Delta_{\beta}-\frac{(m^{2}_{\Delta}-m^{2}_{N})}{2\Delta^{2}}P_{\\{\mu}\Delta_{\nu\\}}\Delta_{\beta}\bigg{\\}}$
$\displaystyle+\frac{F_{3}^{N\Delta}(Q^{2})}{{\bar{m}}^{2}}\bigg{\\{}(\Delta_{\mu}\Delta_{\nu}-\Delta^{2}g_{\mu\nu})\Delta_{\beta}\bigg{\\}}$
$\displaystyle+F_{4}^{N\Delta}(Q^{2})\bar{m}\bigg{\\{}\gamma_{\\{\mu}g_{\nu\\}\beta}+\frac{2(m_{\Delta}+m_{N})}{\Delta^{2}}g_{\mu\nu}\Delta_{\beta}-\frac{(m_{\Delta}+m_{N})}{\Delta^{2}}g_{\beta\\{\mu}\Delta_{\nu\\}}-\frac{1}{\Delta^{2}}\gamma_{\\{\mu}\Delta_{\nu\\}}\Delta_{\beta}\bigg{\\}}$
$\displaystyle+\frac{F_{5}^{N\Delta}(Q^{2})}{\bar{m}}\bigg{\\{}\gamma_{\\{\mu}P_{\nu\\}}\Delta_{\beta}+\frac{(m^{2}_{\Delta}-m^{2}_{N})(m_{\Delta}+m_{N})}{\Delta^{2}}g_{\mu\nu}\Delta_{\beta}-\frac{(m_{\Delta}+m_{N})}{\Delta^{2}}\Delta_{\\{\mu}P_{\nu\\}}\Delta_{\beta}$
$\displaystyle-\frac{(m^{2}_{\Delta}-m^{2}_{N})}{2\Delta^{2}}\gamma_{\\{\mu}\Delta_{\nu\\}}\Delta_{\beta}\bigg{\\}}+\bar{C}_{1}^{N\Delta}(Q^{2})g_{\mu\nu}\Delta_{\beta}+\frac{\bar{C}_{2}^{N\Delta}(Q^{2})}{{\bar{m}}^{2}}\Delta_{\\{\mu}P_{\nu\\}}\Delta_{\beta}$
$\displaystyle+\frac{\bar{C}_{3}^{N\Delta}(Q^{2})}{{\bar{m}}}\gamma_{\\{\mu}\Delta_{\nu\\}}\Delta_{\beta}+\bar{C}_{4}^{N\Delta}(Q^{2})g_{\beta\\{\mu}\Delta_{\nu\\}}\Bigg{]}\gamma_{5}u_{N}(p,s),$
(4)

where $P=(p^{\prime}+p)/2$, $\Delta=p^{\prime}-p$,
$\bar{m}=(m_{N}+m_{\Delta})/2$,
$X_{\\{\mu}Y_{\nu\\}}=(X_{\mu}Y_{\nu}+X_{\nu}Y_{\mu})/2$ and
$Q^{2}=-\Delta^{2}$. Here, $F_{1}^{N\Delta}(Q^{2})$, $F_{2}^{N\Delta}(Q^{2})$,
$F_{3}^{N\Delta}(Q^{2})$, $F_{4}^{N\Delta}(Q^{2})$, $F_{5}^{N\Delta}(Q^{2})$,
$\bar{C}_{1}^{N\Delta}(Q^{2})$, $\bar{C}_{2}^{N\Delta}(Q^{2})$,
$\bar{C}_{3}^{N\Delta}(Q^{2})$, and $\bar{C}_{4}^{N\Delta}(Q^{2})$ are the
transition GFFs. Note that by introducing $\bar{m}$ into the above definition
at different places we tried to bring all the form factors to the same
dimensions. To further simplify, the summation over the spin of the the
Rarita-Schwinger spinor for the $\Delta$ baryon is introduced:

$\sum_{s^{\prime}}{u_{\alpha}}(p^{\prime},s^{\prime}){\bar{u}_{\beta}}(p^{\prime},s^{\prime})=-(\not\\!p^{\prime}+m_{\Delta})\\{g_{\alpha\beta}-\frac{1}{3}\gamma_{\alpha}\gamma_{\beta}-\frac{2p^{\prime}_{\alpha}p^{\prime}_{\beta}}{3m_{\Delta}^{2}}+\frac{p^{\prime}_{\alpha}\gamma_{\beta}-p^{\prime}_{\beta}\gamma_{\alpha}}{3m_{\Delta}}\\}.$
(5)

Using Eqs. (3), (II) and (5) in Eq. (2), we recast the phenomenological or
physical form of the correlation function in terms of the GFFs and other
corresponding hadronic parameters as:

$\displaystyle\Pi_{\alpha\mu\nu}^{Had}(p,q)$
$\displaystyle=\frac{\lambda_{\Delta}}{m^{2}_{\Delta}-p^{\prime
2}}\Bigg{[}-(\not\\!p^{\prime}+m_{\Delta})\\{g_{\alpha\beta}-\frac{1}{3}\gamma_{\alpha}\gamma_{\beta}-\frac{2p^{\prime}_{\alpha}p^{\prime}_{\beta}}{3m_{\Delta}^{2}}+\frac{p^{\prime}_{\alpha}\gamma_{\beta}-p^{\prime}_{\beta}\gamma_{\alpha}}{3m_{\Delta}}\\}\Bigg{]}$
$\displaystyle\times\Bigg{[}F_{1}^{N\Delta}(Q^{2})\bigg{\\{}g_{\beta\\{\mu}P_{\nu\\}}+\frac{(m^{2}_{\Delta}-m^{2}_{N})}{\Delta^{2}}g_{\mu\nu}\Delta_{\beta}-\frac{(m^{2}_{\Delta}-m^{2}_{N})}{2\Delta^{2}}g_{\beta\\{\mu}\Delta_{\nu\\}}-\frac{\Delta_{\beta}P_{\\{\mu}\Delta_{\nu\\}}}{\Delta^{2}}\bigg{\\}}$
$\displaystyle+\frac{F_{2}^{N\Delta}(Q^{2})}{{\bar{m}}^{2}}\bigg{\\{}P_{\mu}P_{\nu}\Delta_{\beta}+\frac{(m_{\Delta}^{2}-m_{N}^{2})^{2}}{4\Delta^{2}}g_{\mu\nu}\Delta_{\beta}-\frac{(m^{2}_{\Delta}-m^{2}_{N})}{2\Delta^{2}}P_{\\{\mu}\Delta_{\nu\\}}\Delta_{\beta}\bigg{\\}}$
$\displaystyle+\frac{F_{3}^{N\Delta}(Q^{2})}{{\bar{m}}^{2}}\bigg{\\{}(\Delta_{\mu}\Delta_{\nu}-\Delta^{2}g_{\mu\nu})\Delta_{\beta}\bigg{\\}}$
$\displaystyle+F_{4}^{N\Delta}(Q^{2})\bar{m}\bigg{\\{}\gamma_{\\{\mu}g_{\nu\\}\beta}+\frac{2(m_{\Delta}+m_{N})}{\Delta^{2}}g_{\mu\nu}\Delta_{\beta}-\frac{(m_{\Delta}+m_{N})}{\Delta^{2}}g_{\beta\\{\mu}\Delta_{\nu\\}}-\frac{1}{\Delta^{2}}\gamma_{\\{\mu}\Delta_{\nu\\}}\Delta_{\beta}\bigg{\\}}$
$\displaystyle+\frac{F_{5}^{N\Delta}(Q^{2})}{\bar{m}}\bigg{\\{}\gamma_{\\{\mu}P_{\nu\\}}\Delta_{\beta}+\frac{(m^{2}_{\Delta}-m^{2}_{N})(m_{\Delta}+m_{N})}{\Delta^{2}}g_{\mu\nu}\Delta_{\beta}-\frac{(m_{\Delta}+m_{N})}{\Delta^{2}}\Delta_{\\{\mu}P_{\nu\\}}\Delta_{\beta}$
$\displaystyle-\frac{(m^{2}_{\Delta}-m^{2}_{N})}{2\Delta^{2}}\gamma_{\\{\mu}\Delta_{\nu\\}}\Delta_{\beta}\bigg{\\}}+\bar{C}_{1}^{N\Delta}(Q^{2})g_{\mu\nu}\Delta_{\beta}+\frac{\bar{C}_{2}^{N\Delta}(Q^{2})}{{\bar{m}}^{2}}\Delta_{\\{\mu}P_{\nu\\}}\Delta_{\beta}$
$\displaystyle+\frac{\bar{C}_{3}^{N\Delta}(Q^{2})}{{\bar{m}}}\gamma_{\\{\mu}\Delta_{\nu\\}}\Delta_{\beta}+\bar{C}_{4}^{N\Delta}(Q^{2})g_{\beta\\{\mu}\Delta_{\nu\\}}\Bigg{]}\gamma_{5}u_{N}(p,s),$
(6)

where we only kept the contribution of the spin-$3/2$ $\Delta$ baryon and
omitted the contamination coming from the spin-$1/2$ particles. In principle,
the correlation function can also include the contributions from spin-$1/2$
particles. The overlap of the spin-$1/2$ particles with the
$J_{\alpha}^{\Delta}$ current can be written as

$\langle
1/2(p^{\prime})|J^{\Delta}_{\alpha}|0\rangle=\left(Ap^{\prime}_{\alpha}+B\gamma_{\alpha}\right)u(p^{\prime})$
(7)

where $u(p^{\prime})$ is the Dirac spinor describing the spin-$1/2$ particles.
Hence, when the gamma matrices are put into the order
$\gamma_{\alpha}\gamma_{\mu}\gamma_{\nu}q\\!\\!\\!/p\\!\\!\\!/^{\prime}\gamma_{5}$
in the related correlation function, the spin-$1/2$ states contribute only to
the structures which have $\gamma_{\alpha}$ at the beginning or those that are
proportional to $p^{\prime}_{\alpha}$. Then, the contributions of the
spin-$1/2$ states in the correlation function are eliminated by ignoring the
structures proportional to $p^{\prime}_{\alpha}$ and the structures that
contain a $\gamma_{\alpha}$ at the beginning. By this way, only the
contributions from spin-$3/2$ states are kept (see also Refs. Belyaev:1993ss ;
Belyaev:1982cd ).

As a result, one can decompose the hadronic representation of the correlation
function in terms of the various Lorentz structures entering the calculations:

$\displaystyle\Pi_{\alpha\mu\nu}^{Had}(p,q)$
$\displaystyle=\Pi_{1}^{Had}(Q^{2})\,q_{\mu}g_{\alpha\nu}q\\!\\!\\!/\gamma_{5}+\Pi_{2}^{Had}(Q^{2})\,p^{\prime}_{\mu}p^{\prime}_{\nu}q_{\alpha}\gamma_{5}+\Pi_{3}^{Had}(Q^{2})\,q_{\mu}q_{\nu}q_{\alpha}q\\!\\!\\!/\gamma_{5}+\Pi_{4}^{Had}(Q^{2})\,g_{\alpha\mu}\gamma_{\nu}q\\!\\!\\!/\gamma_{5}$
$\displaystyle+\Pi_{5}^{Had}(Q^{2})\,q_{\alpha}q_{\mu}\gamma_{\nu}q\\!\\!\\!/\gamma_{5}+\Pi_{6}^{Had}(Q^{2})\,g_{\mu\nu}q_{\alpha}\gamma_{5}+\Pi_{7}^{Had}(Q^{2})\,p^{\prime}_{\mu}q_{\alpha}q_{\nu}\gamma_{5}+\Pi_{8}^{Had}(Q^{2})\,q_{\alpha}q_{\nu}\gamma_{\mu}\gamma_{5}$
$\displaystyle+\Pi_{9}^{Had}(Q^{2})\,g_{\alpha\mu}q_{\nu}\gamma_{5}+...,$ (8)

where the invariant functions $\Pi_{i}^{Had}(Q^{2})$ are functions of GFFs.

The next step is to calculate the correlation function in quark-gluon language
and in terms of the QCD fundamental degrees of freedom. For this purpose, we
insert the explicit forms of the currents, which are given in terms of the
corresponding quark fields, into the correlation function. These currents are
given as:

$\displaystyle J_{\alpha}^{\Delta}(0)=$
$\displaystyle\frac{1}{\sqrt{3}}\epsilon^{abc}[2(u^{aT}(0)C\gamma_{\alpha}d^{b}(0))u^{c}(0)+(u^{aT}(0)C\gamma_{\alpha}u^{b}(0))d^{c}(0)],$
$\displaystyle T_{\mu\nu}^{q}(x)$
$\displaystyle=\frac{i}{2}\bigg{[}\bar{u}^{d}(x)\overleftrightarrow{D}_{\mu}(x)\gamma_{\nu}u^{d}(x)+\bar{d}^{d}(x)\overleftrightarrow{D}_{\mu}(x)\gamma_{\nu}d^{d}(x)+(\mu\leftrightarrow\nu)\bigg{]},$
(9)

where $C$ is the charge conjugation operator; and $a$, $b$, $c$, $d$ are color
indices. The covariant derivative, $\overleftrightarrow{D}_{\mu}(x)$, is
defined as

$\displaystyle\overleftrightarrow{D}_{\mu}(x)$
$\displaystyle=\frac{1}{2}\Big{[}\overrightarrow{D}_{\mu}(x)-\overleftarrow{D}_{\mu}(x)\Big{]},$
(10)

where

$\displaystyle\overrightarrow{D}_{\mu}(x)$
$\displaystyle=\overrightarrow{\partial}_{\mu}(x)-i\frac{g}{2}\lambda^{e}A^{e}_{\mu}(x),$
(11) $\displaystyle\overleftarrow{D}_{\mu}(x)$
$\displaystyle=\overleftarrow{\partial}_{\mu}(x)+i\frac{g}{2}\lambda^{e}A^{e}_{\mu}(x),$
(12)

with $\lambda^{e}$ and $A^{e}_{\mu}(x)$ ($e$ runs from $1$ to $8$) being the
Gell-Mann matrices and external gluon field, respectively. By using the
explicit forms of the interpolating currents in the correlation function and
doing all the possible contractions among the quark fields via the Wick’s
theorem, we get the QCD side of the correlation function in terms of the
light-quark propagators and DAs of the nucleon:

$\displaystyle(\Pi_{\alpha\mu\nu}^{QCD})_{\lambda\eta}(p,q)$ $\displaystyle=$
$\displaystyle-\frac{1}{8\sqrt{3}}\int
d^{4}xe^{iqx}~{}\bigg{[}(C\gamma_{\alpha})_{\alpha\beta}(\overleftrightarrow{D}_{\mu}(x)\gamma_{\nu})_{\rho\sigma}+(\mu\leftrightarrow\nu)\bigg{]}\bigg{\\{}4\epsilon^{abc}\langle
0|{u}_{\sigma}^{a}(0){u}_{\theta}^{b}(x){d}_{\phi}^{c}(0)|N(p,s)\rangle$ (13)
$\displaystyle\times\bigg{[}2\delta_{\alpha}^{\eta}\delta_{\sigma}^{\theta}\delta_{\beta}^{\phi}S_{q}(-x)_{\lambda\rho}+2\delta_{\lambda}^{\eta}\delta_{\sigma}^{\theta}\delta_{\beta}^{\phi}S_{q}(-x)_{\alpha\rho}+\delta_{\alpha}^{\eta}\delta_{\sigma}^{\theta}\delta_{\lambda}^{\phi}S_{q}(-x)_{\beta\rho}+\delta_{\beta}^{\eta}\delta_{\sigma}^{\theta}\delta_{\phi}^{\lambda}S_{q}(-x)_{\alpha\rho}\bigg{]}$
$\displaystyle-4\epsilon^{abc}\langle
0|{u}_{\sigma}^{a}(0){u}_{\theta}^{b}(0){d}_{\phi}^{c}(x)|N(p,s)\rangle\bigg{[}2\delta_{\alpha}^{\eta}\delta_{\lambda}^{\theta}\delta_{\sigma}^{\phi}S_{q}(-x)_{\beta\rho}+\delta_{\alpha}^{\eta}\delta_{\beta}^{\theta}\delta_{\sigma}^{\phi}S_{q}(-x)_{\lambda\rho}\bigg{]}\bigg{\\}},$

where $S_{q}(x)$ is the light quark propagator defined as

$\displaystyle S_{q}(x)$
$\displaystyle=\frac{1}{2\pi^{2}x^{2}}\Big{(}i\frac{{x\\!\\!\\!/}}{x^{2}}-\frac{m_{q}}{2}\Big{)}-\frac{\langle\bar{q}q\rangle}{12}\Big{(}1-i\frac{m_{q}x\\!\\!\\!/}{4}\Big{)}-\frac{\langle\bar{q}\sigma.Gq\rangle}{192}x^{2}\Big{(}1-i\frac{m_{q}x\\!\\!\\!/}{6}\Big{)}-\frac{ig_{s}}{32\pi^{2}x^{2}}~{}G^{\mu\nu}(x)\bigg{[}\hbox
to0.0pt{/\hss}{x}\sigma_{\mu\nu}+\sigma_{\mu\nu}\hbox
to0.0pt{/\hss}{x}\bigg{]}.$ (14)

We set $m_{q}=0$, and the terms $\sim$ $\langle\bar{q}q\rangle$ and
$\langle\bar{q}\sigma.Gq\rangle$ are omitted following the Borel
transformation, which is applied to suppress the contributions of the higher
states and continuum. Hence, only the first term in the light quark propagator
gives contribution to the calculations. To proceed with the calculation of the
correlation function, the matrix element of the local three-quark operator
$4\epsilon^{abc}\langle
0|q_{1\alpha}^{a}(a_{1}x)q_{2\beta}^{b}(a_{2}x)q_{3\gamma}^{c}(a_{3}x)|N(p,s)\rangle$
is needed. The light-cone distribution amplitudes of the nucleon, which we use
in our study to extract the GFFs, are presented in Ref. Braun:2006hz  up to
twist six on the basis of QCD conformal partial wave expansion. All parameters
inside the DAs are also borrowed from this reference.

By using the DAs of the nucleon and applying the Fourier transformations, the
QCD representation of the correlation function is acquired in terms of
different Lorentz structures in the following form:

$\displaystyle\Pi_{\alpha\mu\nu}^{QCD}(p,q)$
$\displaystyle=\Pi_{1}^{QCD}(Q^{2})\,q_{\mu}g_{\alpha\nu}q\\!\\!\\!/\gamma_{5}+\Pi_{2}^{QCD}(Q^{2})\,p^{\prime}_{\mu}p^{\prime}_{\nu}q_{\alpha}\gamma_{5}+\Pi_{3}^{QCD}(Q^{2})\,q_{\mu}q_{\nu}q_{\alpha}q\\!\\!\\!/\gamma_{5}+\Pi_{4}^{QCD}(Q^{2})\,g_{\alpha\mu}\gamma_{\nu}q\\!\\!\\!/\gamma_{5}$
$\displaystyle+\Pi_{5}^{QCD}(Q^{2})\,q_{\alpha}q_{\mu}\gamma_{\nu}q\\!\\!\\!/\gamma_{5}+\Pi_{6}^{QCD}(Q^{2})\,g_{\mu\nu}q_{\alpha}\gamma_{5}+\Pi_{7}^{QCD}(Q^{2})\,p^{\prime}_{\mu}q_{\alpha}q_{\nu}\gamma_{5}+\Pi_{8}^{QCD}(Q^{2})\,q_{\alpha}q_{\nu}\gamma_{\mu}\gamma_{5}$
$\displaystyle+\Pi_{9}^{QCD}(Q^{2})\,g_{\alpha\mu}q_{\nu}\gamma_{5}+...~{},$
(15)

where $\Pi_{1,2,...,9}^{QCD}(Q^{2})$ are invariant functions corresponding to
the coefficients of the selected structures.

The LCSR for the $N\rightarrow\Delta$ transition GFFs are obtained by matching
the coefficients of the chosen Lorentz structures from both the physical and
QCD representations of the correlation function. To abolish the contributions
coming from the higher states and continuum, Borel transformation as well as
continuum subtraction supplied by the quark-hadron duality assumption are
applied. We should stress that, we use the Lorentz structures
$q_{\mu}g_{\alpha\nu}q\\!\\!\\!/\gamma_{5}$,
$p^{\prime}_{\mu}p^{\prime}_{\nu}q_{\alpha}\gamma_{5}$,
$q_{\mu}q_{\nu}q_{\alpha}q\\!\\!\\!/\gamma_{5}$,
$g_{\alpha\mu}\gamma_{\nu}q\\!\\!\\!/\gamma_{5}$,
$q_{\alpha}q_{\mu}\gamma_{\nu}q\\!\\!\\!/\gamma_{5}$,
$g_{\mu\nu}q_{\alpha}\gamma_{5}$,
$p^{\prime}_{\mu}q_{\alpha}q_{\nu}\gamma_{5}$,
$q_{\alpha}q_{\nu}\gamma_{\mu}\gamma_{5}$ and $g_{\alpha\mu}q_{\nu}\gamma_{5}$
to find the LCSR for the $N\rightarrow\Delta$ transition GFFs,
$F_{1}^{N\Delta}(Q^{2})$, $F_{2}^{N\Delta}(Q^{2})$, $F_{3}^{N\Delta}(Q^{2})$,
$F_{4}^{N\Delta}(Q^{2})$, $F_{5}^{N\Delta}(Q^{2})$,
$\bar{C}_{1}^{N\Delta}(Q^{2})$, $\bar{C}_{2}^{N\Delta}(Q^{2})$,
$\bar{C}_{3}^{N\Delta}(Q^{2})$, and $\bar{C}_{4}^{N\Delta}(Q^{2})$,
respectively. Hence,

$\displaystyle
F_{1}^{N\Delta}(Q^{2})\frac{\,\lambda_{\Delta}}{\,(m_{\Delta}^{2}-{p^{\prime}}^{2})}=-2\,\rho_{1}^{QCD}(v,y,x_{1},x_{2},x_{3}),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}F_{2}^{N\Delta}(Q^{2})\frac{\lambda_{\Delta}(m_{\Delta}+m_{N})}{(m_{\Delta}^{2}-{p^{\prime}}^{2})}=\bar{m}^{2}\rho_{2}^{QCD}(v,y,x_{1},x_{2},x_{3}),$
$\displaystyle
F_{3}^{N\Delta}(Q^{2})\frac{\lambda_{\Delta}}{(m_{\Delta}^{2}-{p^{\prime}}^{2})}=-2\,\bar{m}^{2}\rho_{3}^{QCD}(v,y,x_{1},x_{2},x_{3}),~{}~{}~{}~{}~{}~{}~{}~{}F_{4}^{N\Delta}(Q^{2})\frac{\lambda_{\Delta}}{(m_{\Delta}^{2}-{p^{\prime}}^{2})}=-\frac{3}{\bar{m}}\rho_{4}^{QCD}(v,y,x_{1},x_{2},x_{3}),$
$\displaystyle
F_{5}^{N\Delta}(Q^{2})\frac{\lambda_{\Delta}}{(m_{\Delta}^{2}-{p^{\prime}}^{2})}=2\,\bar{m}\,\rho_{5}^{QCD}(v,y,x_{1},x_{2},x_{3}),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\bar{C}_{1}^{N\Delta}(Q^{2})\frac{\lambda_{\Delta}(m_{\Delta}+m_{N})}{(m_{\Delta}^{2}-{p^{\prime}}^{2})}=\rho_{6}^{QCD}(v,y,x_{1},x_{2},x_{3}),$
$\displaystyle\bar{C}_{2}^{N\Delta}(Q^{2})\frac{(m_{\Delta}+m_{N})\,\lambda_{\Delta}}{\,(m_{\Delta}^{2}-{p^{\prime}}^{2})}=\frac{\bar{m}^{2}}{2}\,\rho_{7}^{QCD}(v,y,x_{1},x_{2},x_{3}),~{}~{}~{}~{}~{}~{}\bar{C}_{3}^{N\Delta}(Q^{2})\frac{\lambda_{\Delta}(m_{\Delta}+m_{N})}{(m_{\Delta}^{2}-{p^{\prime}}^{2})}=\bar{m}\,\rho_{8}^{QCD}(v,y,x_{1},x_{2},x_{3}),$
$\displaystyle\bar{C}_{4}^{N\Delta}(Q^{2})\frac{\lambda_{\Delta}}{(m_{N}+m_{\Delta})\,(m_{\Delta}^{2}-{p^{\prime}}^{2})}=\rho_{9}^{QCD}(v,y,x_{1},x_{2},x_{3}).$
(16)

The explicit expressions of the $\rho_{i}^{QCD}(v,y,x_{1},x_{2},x_{3})$
functions are given in the Appendix.

##  III Numerical Results

The sum rules obtained for the GFFs in the previous section contain different
variables: Hadronic and QCD input parameters, input parameters inside the DAs
of the nucleon, some auxiliary or helping parameters and transferred momentum
square $Q^{2}$. The main purpose in this section is to discuss the $Q^{2}$
behavior of the GFFs. To this end, the input parameters of the quarks, nucleon
and $\Delta$ baryon entering the sum rules are selected as $m_{u}=m_{d}=0$,
$m_{N}=0.94$ GeV, $m_{\Delta}=1.23$ GeV and $\lambda_{\Delta}=0.038$ GeV3
Aliev:2007pi ; Aliev:2004ju . Various input parameters inside the DAs of the
nucleon in two sets are borrowed from Ref. Braun:2006hz  and presented in
table 1.

$\begin{array}[]{|l|c|c|}\hline\cr\hline\cr\mbox{Parameters}&\mbox{Set-I}&\mbox{Set-
II}\\\ \hline\cr\hline\cr~{}~{}~{}~{}~{}f_{N}&(5.0\pm 0.5)\times
10^{-3}~{}\mbox{GeV}^{2}&(5.0\pm 0.5)\times 10^{-3}~{}\mbox{GeV}^{2}\\\
~{}~{}~{}~{}~{}\lambda_{1}&(-2.7\pm 0.9)\times
10^{-2}~{}\mbox{GeV}^{2}&(-2.7\pm 0.9)\times 10^{-2}~{}\mbox{GeV}^{2}\\\
~{}~{}~{}~{}~{}\lambda_{2}&(5.4\pm 1.9)\times 10^{-2}~{}\mbox{GeV}^{2}&(5.4\pm
1.9)\times 10^{-2}~{}\mbox{GeV}^{2}\\\
\hline\cr\hline\cr~{}~{}~{}~{}~{}A_{1}^{u}&0.38\pm 0.15&0\\\
~{}~{}~{}~{}~{}V_{1}^{d}&0.23\pm 0.03&1/3\\\ ~{}~{}~{}~{}~{}f_{1}^{d}&0.40\pm
0.05&1/3\\\ ~{}~{}~{}~{}~{}f_{2}^{d}&0.22\pm 0.05&4/15\\\
~{}~{}~{}~{}~{}f_{1}^{u}&0.07\pm 0.05&1/10\\\ \hline\cr\hline\cr\end{array}$
Table 1: The numerical values of the main input parameters entering the
expressions of the nucleon’s DAs. The upper panel shows the dimensionfull
parameters the nucleon. In the lower panel we list the values of the five
dimensionless parameters that determine the shape of the DAs.

The next step is to fix the auxiliary parameters: The Borel parameter $M^{2}$
and the continuum threshold $s_{0}$. To this end, we refer to the standard
criteria of the sum rule method: The pole dominance over the contributions of
the higher states and continuum as well as the higher the twist of the nucleon
DAs, the lower its contribution. These criteria are satisfied in the regions
that the GFFs, as physical quantities, depend relatively weakly on the
auxiliary parameters and show relatively good stability in their working
windows. We acquire the following working windows for the $s_{0}$ and $M^{2}$
from the analyses:

$\displaystyle 2.0~{}\mbox{GeV}^{2}\leq s_{0}\leq 2.50~{}\mbox{GeV}^{2},$
$\displaystyle 2.0~{}\mbox{GeV}^{2}\leq M^{2}\leq 3.0~{}\mbox{GeV}^{2}.$

Figures 1 and 2 present the dependence of the GFFs for $N-\Delta$ transition
on the auxiliary parameters at $Q^{2}=1.0$ GeV2 for two sets of the nucleon
DAs parameters. These figures demonstrate good stability of the GFFs with
respect to the changes of the auxiliary parameters in their working regions:
The residual dependencies appear as the uncertainties in the results.

| Results of set-I | Results of set-II
---|---|---
Form Factors |  ${f}(0)$(GeV-2)] |  ${\cal M}$(GeV) |  $\alpha$ |  ${f}(0)$(GeV-2) |  ${\cal M}$(GeV) |  $\alpha$
$F_{1}^{N\Delta}$($Q^{2}$) |  $0.80\pm 0.08$ | $1.17\pm 0.12$ | $2.1-2.3$ | $~{}~{}1.10\pm 0.20$ | $1.18\pm 0.10$ | $2.8-3.0$
$F_{2}^{N\Delta}$($Q^{2}$) |  $0.20\pm 0.03$ | $1.11\pm 0.05$ | $2.4-2.6$ | $~{}~{}0.31\pm 0.04$ | $1.26\pm 0.11$ | $2.2-2.4$
$F_{3}^{N\Delta}$($Q^{2}$) | $-1.57\pm 0.26$ | $1.22\pm 0.10$ | $2.3-2.5$ | $-0.97\pm 0.12$ | $1.14\pm 0.11$ | $3.4-3.6$
$F_{4}^{N\Delta}$($Q^{2}$) |  $0.38\pm 0.06$ | $1.17\pm 0.10$ | $2.4-2.6$ | $~{}~{}0.28\pm 0.03$ | $1.20\pm 0.11$ | $2.4-2.6$
$F_{5}^{N\Delta}$($Q^{2}$) | $-0.51\pm 0.05$ | $1.18\pm 0.09$ | $2.2-2.4$ | $-0.71\pm 0.11$ | $1.10\pm 0.13$ | $2.2-2.4$
$\bar{C}_{1}^{N\Delta}$($Q^{2}$) |  $-0.073\pm 0.003$ | $1.24\pm 0.10$ | $2.4-2.6$ | $~{}-0.083\pm 0.003$ | $1.24\pm 0.11$ | $2.4-2.6$
$\bar{C}_{2}^{N\Delta}$($Q^{2}$) |  $0.35\pm 0.03$ | $1.29\pm 0.05$ | $2.3-2.5$ | $~{}~{}0.63\pm 0.07$ | $1.16\pm 0.10$ | $2.1-2.3$
$\bar{C}_{3}^{N\Delta}$($Q^{2}$) |  $0.20\pm 0.02$ | $1.29\pm 0.03$ | $1.8-2.0$ | $~{}~{}0.28\pm 0.03$ | $1.19\pm 0.10$ | $2.7-2.9$
$\bar{C}_{4}^{N\Delta}$($Q^{2}$) |  $-$ | $-$ | $-$ | $~{}~{}-$ | $-$ |  $-$
Table 2: Numerical values of the fitting parameters for the transition GFFs of
$N-\Delta$.

Now, we proceed to discuss the $Q^{2}$-behavior of the GFFs. The light-cone
QCD sum rules give reliable results for $Q^{2}\geq 1$ GeV2. The mass
corrections of the DAs $\sim m^{2}_{N}/Q^{2}$ turn out to be very large for
$Q^{2}<1.0$ GeV2, in other words, the light-cone sum rules become unreliable.
Hence, for the GFFs, we expect the light-cone QCD sum rules to work
effectively in the $1.0$ GeV${}^{2}\leq Q^{2}\leq 10.0$ GeV2 region. Therefore
to find the values of the GFFs at static limit, $Q^{2}=0$, we need to
extrapolate the results to small values of $Q^{2}$. The following
$\alpha$-pole fit functions do this job well and produce all our sum rules
results for $Q^{2}\geq 1$ GeV2:

$\displaystyle
F_{i}^{N\Delta}(Q^{2})[\bar{C}_{i}^{N\Delta}(Q^{2})]=f(0)\bigg{[}1+\frac{Q^{2}}{{\cal
M}^{2}}\bigg{]}^{-\alpha},$ (17)

where the fit parameters including the values of GFFs at static limit, $f(0)$,
are given in table 2. The uncertainties in the presented values belong to the
errors of all input parameters and those related to the determination of
working intervals for the auxiliary parameters. As we previously mentioned,
the quark part of the EMT current alone is not conserved and matrix element of
the EMT current is parameterized in terms of five independent conserved and
four independent non-conserved form factors. The $C_{i}^{N\Delta}(Q^{2})$ form
factors depict the orders of breaking the conservation of quark part of the
EMT current. Our analyses do not give reliable results for
$C_{4}^{N\Delta}(Q^{2})$ for both the sets of nucleon DAs parameters. To our
best knowledge, this is the first direct study using an analytic method in the
literature dedicated to the calculation of the $N\rightarrow\Delta$ transition
GFFs. These GFFs can also be obtained from the transition GPDs. However, the
connection between the transition GPDs and transition energy-momentum tensor
form factors is unknown Kim:2022bwn . One should first study these connections
like those in the case of nucleon discussed previously. The
$N\rightarrow\Delta$ transition GPDs were studied in the large $N_{c}$ limit
in Ref. Goeke:2001tz . It will be possible to extract the general transition
GPDs when the related complete experimental data are available. The CLAS data
on this transition is ongoing Kim:2022bwn ; Proceedings:2020fyd .

Using the above fit functions, the $Q^{2}$-behavior of the GFFs in a wide
range, $Q^{2}\in[0-10]$ GeV2, are depicted in figures 3 and 4. Our results may
be checked by future related experiments, Lattice QCD and other
phenomenological models.

(a)  (b)

(c)  (d)

(e)  (f)

(g)  (h)

Figure 1: The dependence of the $N\rightarrow\Delta$ transition GFFs on
$M^{2}$ at $Q^{2}=1.0$ GeV2 and three fixed values of the $s_{0}$ and set-I
parameters.

(a)  (b)

(c)  (d)

(e)  (f)

(g)  (h)

Figure 2: The dependence of the $N\rightarrow\Delta$ transition GFFs on
$M^{2}$ at $Q^{2}=1.0$ GeV2 and three fixed values of the $s_{0}$ and set-II
parameters.

(a)  (b)

(c)  (d)

(e)  (f)

(g)  (h)

Figure 3: The dependence of the $N\rightarrow\Delta$ transition GFFs on
$Q^{2}$ at fixed values of the $s_{0}$, average $M^{2}$ and set-I parameters.

(a)  (b)

(c)  (d)

(e)  (f)

(g)  (h)

Figure 4: The dependence of the $N\rightarrow\Delta$ transition GFFs on
$Q^{2}$ at fixed values of the $s_{0}$, average $M^{2}$ and set-II parameters.

##  IV Summary and Concluding Remarks

We investigated the transition GFFs of the $N\rightarrow\Delta$. The quark
part of the EMT current are parameterized in terms of nine independent (five
conserved and four non-conserved) form factors. To calculate these GFFs, we
applied the light-cone QCD sum rule technique using the DAs of the on-shell
nucleon. The nine independent Lorentz structures entering to the calculations
in both the hadronic and QCD sides allowed us to extract the sum rules for the
desired GFFs. We numerically analyzed the obtained sum rules for two sets of
input parameters inside the DAs of the nucleon. We used multipole fit
functions to extrapolate the results to the small values of $Q^{2}$, $0\leq
Q^{2}<1$ GeV2, to find the values of the GFFs at static limit. We presented
the $Q^{2}$-behaviors of the form factors in the interval $[0-10]$ GeV2, using
the working windows of the auxiliary parameters.

As we previously mentioned, investigation of the EMT current interactions of
hadrons besides their electromagnetic, weak and strong interactions can give
us valuable information about their mass and spin as well as the pressure and
shear force their inside. The transition GFFs of $N\rightarrow\Delta$
calculated in the present study include useful knowledge on the $N-\Delta$
system. Our results can be checked in future experiments. Comparison of our
results with the future Lattice QCD and other phenomenological predictions
will be of great importance as well. The direct measurement of the $N-\Delta$
GFFs may not be possible with the present facilities. However, in principle,
these GFFs can be extracted from the $N-\Delta$ GPDs by considering them as
the second Mellin moment of the transition GPDs. The project of extracting the
transition GPDs from the CLAS data is ongoing Kim:2022bwn ;
Proceedings:2020fyd .

## ACKNOWLEDGEMENTS

K. Azizi is grateful to Iran Science Elites Federation (Saramadan) for the
partial financial support provided under the grant number ISEF/M/401385.

## Appendix: Explicit forms of the $\rho_{i}^{QCD}(v,y,x_{1},x_{2},x_{3})$
functions

In this appendix, we present the explicit expressions for the
$\rho_{i}^{QCD}(v,y,x_{1},x_{2},x_{3})$ functions:

$\displaystyle\rho_{1}^{QCD}(v,y,x_{1},x_{2},x_{3})$
$\displaystyle=\frac{m_{N}^{3}}{2\sqrt{3}}\int_{0}^{1}\frac{(1+x_{2})}{(q-px_{2})^{4}}dx_{2}\int_{0}^{1-x_{2}}d{x_{1}}\big{[}\big{(}A_{1}^{M}-T_{1}^{M}-2V_{1}^{M}\big{)}(x_{1},x_{2},1-x_{1}-x_{2})\big{]}$
$\displaystyle-\frac{m_{N}^{3}}{2\sqrt{3}}\int_{0}^{1}\frac{(1+x_{3})}{(q-px_{3})^{4}}dx_{3}\int_{0}^{1-x_{3}}d{x_{1}}\big{[}T_{1}^{M}(x_{1},1-x_{1}-x_{3},x_{3})\big{]}$
$\displaystyle-\frac{m_{N}}{4\sqrt{3}}\int_{0}^{1}\frac{(1+x_{2})}{(q-px_{2})^{2}}dx_{2}\int_{0}^{1-x_{2}}d{x_{1}}\big{[}\big{(}2A_{1}+A_{3}-2V_{1}+V_{3}\big{)}(x_{1},x_{2},1-x_{1}-x_{2})\big{]}$
$\displaystyle+\frac{m_{N}}{4\sqrt{3}}\int_{0}^{1}\frac{(1+x_{3})}{(q-px_{3})^{2}}dx_{3}\int_{0}^{1-x_{3}}d{x_{1}}\big{[}\big{(}A_{3}+2T_{1}+V_{3}\big{)}(x_{1},1-x_{1}-x_{3},x_{3})\big{]}$
$\displaystyle+\frac{m_{N}^{3}}{4\sqrt{3}}\int_{0}^{1}\frac{(1+v)}{(q-pv)^{4}}dv\int_{0}^{v}dy\int_{y}^{1}d{x_{2}}\int_{0}^{1-x_{2}}d{x_{1}}\big{[}\big{(}T_{1}-T_{2}-T_{5}+T_{6}-2T_{7}-2T_{8}$
$\displaystyle+V_{1}-V_{2}-V_{3}-V_{4}-V_{5}+V_{6}\big{)}(x_{1},x_{2},1-x_{1}-x_{2})\big{]}$
$\displaystyle+\frac{m_{N}^{3}}{4\sqrt{3}}\int_{0}^{1}\frac{(1+v)}{(q-pv)^{4}}dv\int_{0}^{v}dy\int_{y}^{1}d{x_{3}}\int_{0}^{1-x_{3}}d{x_{1}}\big{[}\big{(}-A_{1}+A_{2}-A_{3}-A_{4}+A_{5}-A_{6}-4T_{1}$
$\displaystyle+2T_{2}+2T_{3}+2T_{4}+2T_{5}-4T_{6}+2T_{7}+2T_{8}+V_{1}-V_{2}-V_{3}-V_{5}+V_{6}\big{)}(x_{1},1-x_{1}-x_{3},x_{3})\big{]}$
$\displaystyle+\frac{m_{N}}{4\sqrt{3}}\int_{0}^{1}\frac{1}{(q-py)^{2}}dy\int_{y}^{1}d{x_{2}}\int_{0}^{1-x_{2}}d{x_{1}}\big{[}\big{(}-2A_{1}+2A_{2}-2A_{3}-T_{1}+T_{2}+2T_{7}+2V_{1}-2V_{2}$
$\displaystyle-2V_{3}\big{)}(x_{1},x_{2},1-x_{1}-x_{2})\big{]}$
$\displaystyle+\frac{m_{N}}{\sqrt{3}}\int_{0}^{1}\frac{1}{(q-py)^{2}}dy\int_{y}^{1}d{x_{3}}\int_{0}^{1-x_{3}}d{x_{1}}\big{[}\big{(}T_{1}+T_{3}-T_{7}\big{)}(x_{1},1-x_{1}-x_{3},x_{3})\big{]},$
(18) $\displaystyle\rho_{2}^{QCD}(v,y,x_{1},x_{2},x_{3})$
$\displaystyle=\frac{1}{\sqrt{3}}\int_{0}^{1}\frac{x_{2}}{(q-px_{2})^{2}}dx_{2}\int_{0}^{1-x_{2}}d{x_{1}}\big{[}\big{(}-A_{1}+V_{1}\big{)}(x_{1},x_{2},1-x_{1}-x_{2})\big{]}$
$\displaystyle+\frac{2}{\sqrt{3}}\int_{0}^{1}\frac{x_{3}}{(q-px_{3})^{2}}dx_{3}\int_{0}^{1-x_{3}}d{x_{1}}\big{[}T_{1}(x_{1},1-x_{1}-x_{3},x_{3})\big{]}$
$\displaystyle+\frac{m_{N}^{2}}{\sqrt{3}}\int_{0}^{1}\frac{x_{2}^{2}}{(q-px_{2})^{4}}dx_{2}\int_{0}^{1-x_{2}}d{x_{1}}\big{[}\big{(}A_{1}^{M}+4T_{1}^{M}-2V_{1}^{M}\big{)}(x_{1},x_{2},1-x_{1}-x_{2})\big{]}$
$\displaystyle-\frac{m_{N}^{2}}{\sqrt{3}}\int_{0}^{1}\frac{x_{3}^{2}}{(q-px_{3})^{4}}dx_{3}\int_{0}^{1-x_{3}}d{x_{1}}\big{[}\big{(}-A_{1}^{M}-2T_{1}^{M}+V_{1}^{M}\big{)}(x_{1},1-x_{1}-x_{3},x_{3})\big{]}$
$\displaystyle+\frac{m_{N}^{2}}{\sqrt{3}}\int_{0}^{1}\frac{y^{2}}{(q-py)^{4}}dy\int_{y}^{1}d{x_{2}}\int_{0}^{1-x_{2}}d{x_{1}}\big{[}\big{(}-A_{1}+A_{2}-A_{3}+V_{1}-V_{2}-V_{3}\big{)}$
$\displaystyle\times(x_{1},x_{2},1-x_{1}-x_{2})\big{]}$
$\displaystyle+\frac{m_{N}^{2}}{2\sqrt{3}}\int_{0}^{1}\frac{y^{2}}{(q-py)^{4}}dy\int_{y}^{1}d{x_{3}}\int_{0}^{1-x_{3}}d{x_{1}}\big{[}\big{(}A_{3}-A_{4}-4T_{1}-2T_{3}-2T_{5}-4T_{7}$
$\displaystyle-4T_{8}-V_{1}+V_{4}+V_{5}\big{)}(x_{1},1-x_{1}-x_{3},x_{3})\big{]}$
$\displaystyle+\frac{2m_{N}^{2}}{\sqrt{3}}\int_{0}^{1}\frac{v^{2}}{(q-pv)^{4}}dv\int_{0}^{v}dy\int_{y}^{1}d{x_{2}}\int_{0}^{1-x_{2}}d{x_{1}}\big{[}\big{(}-T_{2}+T_{3}+T_{5}-T_{5}-T_{7}-T_{8}\big{)}$
$\displaystyle\times(x_{1},x_{2},1-x_{1}-x_{2})\big{]}$
$\displaystyle+\frac{2m_{N}^{2}}{\sqrt{3}}\int_{0}^{1}\frac{v^{2}}{(q-pv)^{4}}dv\int_{0}^{v}dy\int_{y}^{1}d{x_{3}}\int_{0}^{1-x_{3}}d{x_{1}}\big{[}\big{(}-T_{2}+T_{3}+T_{5}-T_{5}-T_{7}-T_{8}\big{)}$
$\displaystyle\times(x_{1},1-x_{1}-x_{3},x_{3})\big{]},$ (19)
$\displaystyle\rho_{3}^{QCD}(v,y,x_{1},x_{2},x_{3})$
$\displaystyle=\frac{m_{N}}{\sqrt{3}}\int_{0}^{1}\frac{(1+2y+y^{2})}{(q-py)^{4}}dy\int_{y}^{1}d{x_{3}}\int_{0}^{1-x_{3}}d{x_{1}}\big{[}\big{(}A_{1}-A_{2}+A_{3}-V_{1}+V_{2}+V_{3}+2T_{1}-2T_{3}$
$\displaystyle-2T_{7}\big{)}(x_{1},1-x_{1}-x_{3},x_{3})\big{]}$
$\displaystyle+\frac{2m_{N}^{3}}{\sqrt{3}}\int_{0}^{1}\frac{v(1+v+v^{2})}{(q-pv)^{6}}dv\int_{0}^{v}dy\int_{y}^{1}d{x_{3}}\int_{0}^{1-x_{3}}d{x_{1}}\big{[}\big{(}A_{1}-A_{2}+A_{3}+A_{4}-A_{5}+A_{6}+2T_{1}$
$\displaystyle-2T_{3}-2T_{4}+2T_{6}-2T_{7}-2T_{8}-V_{1}+V_{2}+V_{3}+V_{5}-V_{6}\big{)}(x_{1},1-x_{1}-x_{3},x_{3})\big{]},$
(20) $\displaystyle\rho_{4}^{QCD}(v,y,x_{1},x_{2},x_{3})$
$\displaystyle=\frac{m_{N}^{2}}{2\sqrt{3}}\int_{0}^{1}\frac{1}{(q-py)^{2}}dy\int_{y}^{1}d{x_{2}}\int_{0}^{1-x_{2}}d{x_{1}}\big{[}\big{(}A_{1}-A_{2}+A_{4}+P_{1}-P_{2}-S_{1}+S_{2}+T_{1}-T_{2}$
$\displaystyle-
T_{7}-V_{1}+V_{2}+V_{4}\big{)}(x_{1},x_{2},1-x_{1}-x_{2})\big{]}$
$\displaystyle+\frac{m_{N}^{2}}{4\sqrt{3}}\int_{0}^{1}\frac{1}{(q-py)^{2}}dy\int_{y}^{1}d{x_{3}}\int_{0}^{1-x_{3}}d{x_{1}}\big{[}\big{(}A_{3}-A_{4}-4T_{1}+4T_{3}+4T_{7}+V_{3}-V_{4}\big{)}$
$\displaystyle\times(x_{1},1-x_{1}-x_{3},x_{3})\big{]},$ (21)
$\displaystyle\rho_{5}^{QCD}(v,y,x_{1},x_{2},x_{3})$
$\displaystyle=\frac{1}{\sqrt{3}}\int_{0}^{1}\frac{1}{(q-px_{2})^{2}}d{x_{2}}\int_{0}^{1-x_{2}}d{x_{1}}\big{[}\big{(}A_{1}-V_{1}\big{)}(x_{1},x_{2},1-x_{1}-x_{2})\big{]}$
$\displaystyle+\frac{1}{\sqrt{3}}\int_{0}^{1}\frac{1}{(q-px_{3})^{2}}d{x_{3}}\int_{0}^{1-x_{3}}d{x_{1}}\big{[}\big{(}A_{1}-V_{1}-T_{1}\big{)}(x_{1},1-x_{1}-x_{3},x_{3})\big{]}$
$\displaystyle+\frac{m_{N}^{2}}{\sqrt{3}}\int_{0}^{1}\frac{(1+x_{2})}{(q-px_{2})^{4}}d{x_{2}}\int_{0}^{1-x_{2}}d{x_{1}}\big{[}\big{(}-A_{1}^{M}+2V_{1}^{M}-2T_{1}^{M}\big{)}(x_{1},x_{2},1-x_{1}-x_{2})\big{]}$
$\displaystyle+\frac{m_{N}^{2}}{\sqrt{3}}\int_{0}^{1}\frac{(1+x_{3})}{(q-px_{3})^{4}}d{x_{3}}\int_{0}^{1-x_{3}}d{x_{1}}\big{[}\big{(}-A_{1}^{M}+V_{1}^{M}-2T_{1}^{M}\big{)}(x_{1},1-x_{1}-x_{3},x_{3})\big{]}$
$\displaystyle+\frac{m_{N}^{2}}{2\sqrt{3}}\int_{0}^{1}\frac{(1+2y+y^{2})}{(q-py)^{4}}dy\int_{y}^{1}d{x_{2}}\int_{0}^{1-x_{2}}d{x_{1}}\big{[}\big{(}2A_{1}-2A_{2}+2A_{3}-S_{1}+S_{2}-2V_{1}+2V_{2}$
$\displaystyle+V_{3}+V_{4}\big{)}(x_{1},x_{2},1-x_{1}-x_{2})\big{]}$
$\displaystyle+\frac{m_{N}^{2}}{2\sqrt{3}}\int_{0}^{1}\frac{(1+2y+y^{2})}{(q-py)^{4}}dy\int_{y}^{1}d{x_{3}}\int_{0}^{1-x_{3}}d{x_{1}}\big{[}\big{(}A_{3}-A_{4}-4T_{1}+4T_{3}+4T_{7}-2V_{1}+2V_{3}$
$\displaystyle+2V_{5}\big{)}(x_{1},1-x_{1}-x_{3},x_{3})\big{]}$
$\displaystyle+\frac{2m_{N}^{2}}{\sqrt{3}}\int_{0}^{1}\frac{(1+v)}{(q-pv)^{4}}dv\int_{0}^{v}dy\int_{y}^{1}d{x_{2}}\int_{0}^{1-x_{2}}d{x_{1}}\big{[}\big{(}T_{2}-T_{3}-T_{4}+T_{5}+T_{6}+T_{7}+T_{8}\big{)}$
$\displaystyle\times(x_{1},x_{2},1-x_{1}-x_{2})\big{]}$
$\displaystyle-\frac{2m_{N}^{2}}{\sqrt{3}}\int_{0}^{1}\frac{(1+v)}{(q-pv)^{4}}dv\int_{0}^{v}dy\int_{y}^{1}d{x_{3}}\int_{0}^{1-x_{3}}d{x_{1}}\big{[}\big{(}T_{2}-T_{3}-T_{4}+T_{5}+T_{6}+T_{7}+T_{8}\big{)}$
$\displaystyle\times(x_{1},1-x_{1}-x_{3},x_{3})\big{]},$ (22)
$\displaystyle\rho_{6}^{QCD}(v,y,x_{1},x_{2},x_{3})$
$\displaystyle=\frac{m_{N}^{2}}{4\sqrt{3}}\int_{0}^{1}\frac{1}{(q-px_{2})^{2}}d{x_{2}}\int_{0}^{1-x_{2}}d{x_{1}}\big{[}\big{(}A_{1}^{M}+2T_{1}^{M}-2V_{1}^{M}\big{)}(x_{1},x_{2},1-x_{1}-x_{2})\big{]}$
$\displaystyle+\frac{m_{N}^{2}}{4\sqrt{3}}\int_{0}^{1}\frac{1}{(q-px_{3})^{2}}d{x_{3}}\int_{0}^{1-x_{3}}d{x_{1}}\big{[}\big{(}A_{1}^{M}-V_{1}^{M}\big{)}(x_{1},1-x_{1}-x_{3},x_{3})\big{]}$

$\displaystyle+\frac{m_{N}^{2}}{4\sqrt{3}}\int_{0}^{1}\frac{(1+y)}{(q-py)^{2}}dy\int_{y}^{1}d{x_{2}}\int_{0}^{1-x_{2}}d{x_{1}}\big{[}\big{(}A_{3}-A_{4}-2P_{1}+2P_{2}+2S_{1}-2S_{2}+2T_{1}-2T_{2}$
$\displaystyle-4T_{7}+V_{3}+V_{4}\big{)}(x_{1},x_{2},1-x_{1}-x_{2})\big{]}$
$\displaystyle+\frac{m_{N}^{2}}{2\sqrt{3}}\int_{0}^{1}\frac{(1+y)}{(q-py)^{2}}dy\int_{y}^{1}d{x_{3}}\int_{0}^{1-x_{3}}d{x_{1}}\big{[}\big{(}A_{1}-A_{2}+A_{4}-2T_{2}+2T_{3}-2T_{7}-V_{1}+V_{2}$
$\displaystyle+V_{4}\big{)}(x_{1},1-x_{1}-x_{3},x_{3})\big{]}$
$\displaystyle+\frac{m_{N}^{2}}{2\sqrt{3}}\int_{0}^{1}\frac{1}{(q-pv)^{2}}dv\int_{0}^{v}dy\int_{y}^{1}d{x_{2}}\int_{0}^{1-x_{2}}d{x_{1}}\big{[}\big{(}T_{2}-T_{3}-T_{4}+T_{5}+T_{6}+T_{7}+T_{8}\big{)}$
$\displaystyle\times(x_{1},x_{2},1-x_{1}-x_{2})\big{]}$
$\displaystyle-\frac{m_{N}^{2}}{2\sqrt{3}}\int_{0}^{1}\frac{1}{(q-pv)^{2}}dv\int_{0}^{v}dy\int_{y}^{1}d{x_{3}}\int_{0}^{1-x_{3}}d{x_{1}}\big{[}\big{(}T_{2}-T_{3}-T_{4}+T_{5}+T_{6}+T_{7}+T_{8}\big{)}$
$\displaystyle\times(x_{1},1-x_{1}-x_{3},x_{3})\big{]}$
$\displaystyle+\frac{m_{N}^{4}}{2\sqrt{3}}\int_{0}^{1}\frac{v(1+v)}{(q-pv)^{4}}dv\int_{0}^{v}dy\int_{y}^{1}d{x_{2}}\int_{0}^{1-x_{2}}d{x_{1}}\big{[}\big{(}T_{1}-T_{2}-T_{5}+T_{6}-2T_{7}-2T_{8}\big{)}$
$\displaystyle\times(x_{1},x_{2},1-x_{1}-x_{2})\big{]}$
$\displaystyle-\frac{m_{N}^{4}}{2\sqrt{3}}\int_{0}^{1}\frac{v(1+v)}{(q-pv)^{4}}dv\int_{0}^{v}dy\int_{y}^{1}d{x_{3}}\int_{0}^{1-x_{3}}d{x_{1}}\big{[}\big{(}A_{1}-A_{2}+A_{3}+A_{4}-A_{5}+A_{6}+T_{1}+T_{2}$
$\displaystyle-2T_{3}-2T_{4}+T_{5}-V_{1}+V_{2}+V_{3}+V_{4}+V_{5}-V_{6}\big{)}(x_{1},1-x_{1}-x_{3},x_{3})\big{]},$
(23) $\displaystyle\rho_{7}^{QCD}(v,y,x_{1},x_{2},x_{3})$
$\displaystyle=\frac{1}{\sqrt{3}}\int_{0}^{1}\frac{(1+2x_{2}+x_{2}^{2})}{(q-px_{2})^{2}}d{x_{2}}\int_{0}^{1-x_{2}}d{x_{1}}\big{[}\big{(}A_{1}-V_{1}\big{)}(x_{1},x_{2},1-x_{1}-x_{2})\big{]}$
$\displaystyle+\frac{2}{\sqrt{3}}\int_{0}^{1}\frac{(1+2x_{3}+x_{3}^{2})}{(q-px_{3})^{2}}d{x_{3}}\int_{0}^{1-x_{3}}d{x_{1}}\big{[}\big{(}V_{1}-A_{1}-T_{1}\big{)}(x_{1},1-x_{1}-x_{3},x_{3})\big{]}$
$\displaystyle+\frac{2m_{N}^{2}}{\sqrt{3}}\int_{0}^{1}\frac{(1+2x_{2}+x_{2}^{2})}{(q-px_{2})^{4}}d{x_{2}}\int_{0}^{1-x_{2}}d{x_{1}}\big{[}\big{(}A_{1}^{M}+T_{1}^{M}-V_{1}^{M}\big{)}(x_{1},x_{2},1-x_{1}-x_{2})\big{]}$
$\displaystyle+\frac{m_{N}^{2}}{\sqrt{3}}\int_{0}^{1}\frac{(1+2x_{3}+x_{3}^{2})}{(q-px_{3})^{4}}d{x_{3}}\int_{0}^{1-x_{3}}d{x_{1}}\big{[}\big{(}-7A_{1}^{M}-8T_{1}^{M}+7V_{1}^{M}\big{)}(x_{1},1-x_{1}-x_{3},x_{3})\big{]}$
$\displaystyle+\frac{4m_{N}^{2}}{\sqrt{3}}\int_{0}^{1}\frac{v(1+v)}{(q-pv)^{4}}dv\int_{0}^{v}dy\int_{y}^{1}d{x_{2}}\int_{0}^{1-x_{2}}d{x_{1}}\big{[}\big{(}T_{2}-T_{3}-T_{4}+T_{5}+T_{6}+T_{7}+T_{8}\big{)}$
$\displaystyle\times(x_{1},x_{2},1-x_{1}-x_{2})\big{]}$
$\displaystyle-\frac{4m_{N}^{2}}{\sqrt{3}}\int_{0}^{1}\frac{v(1+v)}{(q-pv)^{4}}dv\int_{0}^{v}dy\int_{y}^{1}d{x_{3}}\int_{0}^{1-x_{3}}d{x_{1}}\big{[}\big{(}T_{2}-T_{3}-T_{4}+T_{5}+T_{6}+T_{7}+T_{8}\big{)}$
$\displaystyle\times(x_{1},1-x_{1}-x_{3},x_{3})\big{]}$
$\displaystyle+\frac{m_{N}^{2}}{\sqrt{3}}\int_{0}^{1}\frac{y(y+1)}{(q-py)^{4}}dy\int_{y}^{1}d{x_{2}}\int_{0}^{1-x_{2}}d{x_{1}}\big{[}\big{(}A_{1}-A_{2}+A_{3}-V_{1}+V_{2}+V_{3}\big{)}$
$\displaystyle\times(x_{1},x_{2},1-x_{1}-x_{2})\big{]}$
$\displaystyle+\frac{m_{N}^{2}}{\sqrt{3}}\int_{0}^{1}\frac{y(y+1)}{(q-py)^{4}}dy\int_{y}^{1}d{x_{3}}\int_{0}^{1-x_{3}}d{x_{1}}\big{[}\big{(}A_{3}-A_{4}-2T_{1}+4T_{2}-2T_{3}+4T_{5}+2T_{7}+8T_{8}$
$\displaystyle+2V_{1}-2V_{4}+2V_{5}\big{)}(x_{1},1-x_{1}-x_{3},x_{3})\big{]}$
$\displaystyle-\frac{4m_{N}^{4}}{\sqrt{3}}\int_{0}^{1}\frac{v^{2}(1+v)}{(q-pv)^{6}}dv\int_{0}^{v}dy\int_{y}^{1}d{x_{3}}\int_{0}^{1-x_{3}}d{x_{1}}\big{[}\big{(}A_{1}-A_{2}+A_{3}+A_{4}-A_{5}+A_{6}+2T_{1}$
$\displaystyle-2T_{3}-2T_{4}+2T_{6}-2T_{7}-2T_{8}-V_{1}+V_{2}+V_{3}+V_{5}-V_{6}\big{)}(x_{1},1-x_{1}-x_{3},x_{3})\big{]},$
(24)

$\displaystyle\rho_{8}^{QCD}(v,y,x_{1},x_{2},x_{3})$
$\displaystyle=\frac{m_{N}}{\sqrt{3}}\int_{0}^{1}\frac{(1+2x_{2}+x_{2}^{2})}{(q-px_{2})^{2}}d{x_{2}}\int_{0}^{1-x_{2}}d{x_{1}}\big{[}\big{(}A_{1}+A_{3}-V_{1}+V_{2}\big{)}(x_{1},x_{2},1-x_{1}-x_{2})\big{]}$
$\displaystyle+\frac{m_{N}}{2\sqrt{3}}\int_{0}^{1}\frac{(1+2x_{3}+x_{3}^{2})}{(q-px_{3})^{2}}d{x_{3}}\int_{0}^{1-x_{3}}d{x_{1}}\big{[}\big{(}-A_{1}-A_{3}-2T_{1}+V_{1}-2V_{3}\big{)}(x_{1},1-x_{1}-x_{3},x_{3})\big{]}$
$\displaystyle+\frac{m_{N}^{3}}{\sqrt{3}}\int_{0}^{1}\frac{(1+2x_{2}+x_{2}^{2})}{(q-px_{2})^{4}}d{x_{2}}\int_{0}^{1-x_{2}}d{x_{1}}\big{[}\big{(}-T_{1}^{M}-2V_{1}^{M}-A_{1}^{M}\big{)}(x_{1},x_{2},1-x_{1}-x_{2})\big{]}$
$\displaystyle+\frac{m_{N}^{3}}{2\sqrt{3}}\int_{0}^{1}\frac{(1+2x_{3}+x_{3}^{2})}{(q-px_{3})^{4}}d{x_{3}}\int_{0}^{1-x_{3}}d{x_{1}}\big{[}\big{(}A_{1}^{M}-V_{1}^{M}+T_{1}^{M}\big{)}(x_{1},1-x_{1}-x_{3},x_{3})\big{]}$
$\displaystyle+\frac{m_{N}^{3}}{2\sqrt{3}}\int_{0}^{1}\frac{y(1+y+y^{2})}{(q-py)^{4}}dy\int_{y}^{1}d{x_{2}}\int_{0}^{1-x_{2}}d{x_{1}}\big{[}\big{(}4A_{1}+4A_{2}-A_{3}-A_{4}+2S_{1}-2S_{2}+4V_{1}$
$\displaystyle-4V_{2}-2V_{3}-2V_{4}-2V_{5}\big{)}(x_{1},x_{2},1-x_{1}-x_{2})\big{]}$
$\displaystyle+\frac{m_{N}^{3}}{2\sqrt{3}}\int_{0}^{1}\frac{y(1+y+y^{2})}{(q-py)^{4}}dy\int_{y}^{1}d{x_{3}}\int_{0}^{1-x_{3}}d{x_{1}}\big{[}\big{(}-A_{3}-2A_{4}+4T_{1}-4T_{3}-4T_{7}+2V_{1}-2V_{3}$
$\displaystyle-2V_{5}\big{)}(x_{1},1-x_{1}-x_{3},x_{3})\big{]}$
$\displaystyle+\frac{m_{N}^{3}}{\sqrt{3}}\int_{0}^{1}\frac{(1+v+v^{2})}{(q-pv)^{4}}dv\int_{0}^{v}dy\int_{y}^{1}d{x_{2}}\int_{0}^{1-x_{2}}d{x_{1}}\big{[}\big{(}A_{1}-A_{2}+A_{3}+A_{4}-A_{5}+A_{6}+2T_{1}$
$\displaystyle-2T_{3}-2T_{4}+2T_{6}-2T_{7}-2T_{8}-V_{1}+V_{2}+V_{3}+V_{5}-V_{6}\big{)}(x_{1},x_{2},1-x_{1}-x_{2})\big{]}$
$\displaystyle+\frac{2m_{N}^{3}}{\sqrt{3}}\int_{0}^{1}\frac{(1+v+v^{2})}{(q-pv)^{4}}dv\int_{0}^{v}dy\int_{y}^{1}d{x_{3}}\int_{0}^{1-x_{3}}d{x_{1}}\big{[}\big{(}T_{1}-T_{3}-T_{5}+T_{6}-2T_{7}-2T_{8}\big{)}$
$\displaystyle\times(x_{1},1-x_{1}-x_{3},x_{3})\big{]},$ (25)

and,

$\displaystyle\rho_{9}^{QCD}(v,y,x_{1},x_{2},x_{3})$
$\displaystyle=\frac{1}{\sqrt{3}}\int_{0}^{1}\frac{(1+2x_{2}+x_{2}^{2})}{(q-px_{2})^{2}}d{x_{2}}\int_{0}^{1-x_{2}}d{x_{1}}\big{[}\big{(}A_{1}-V_{1}\big{)}(x_{1},x_{2},1-x_{1}-x_{2})\big{]}$
$\displaystyle+\frac{2}{\sqrt{3}}\int_{0}^{1}\frac{(1+2x_{3}+x_{3}^{2})}{(q-px_{3})^{2}}d{x_{3}}\int_{0}^{1-x_{3}}d{x_{1}}\big{[}T_{1}(x_{1},1-x_{1}-x_{3},x_{3})\big{]}$
$\displaystyle+\frac{m_{N}^{2}}{2\sqrt{3}}\int_{0}^{1}\frac{(1+2x_{2}+x_{2}^{2})}{(q-px_{2})^{4}}d{x_{2}}\int_{0}^{1-x_{2}}d{x_{1}}\big{[}\big{(}-A_{1}^{M}+V_{1}^{M}\big{)}(x_{1},x_{2},1-x_{1}-x_{2})\big{]}$
$\displaystyle+\frac{m_{N}^{2}}{2\sqrt{3}}\int_{0}^{1}\frac{(1+2x_{3}+x_{3}^{2})}{(q-px_{3})^{4}}d{x_{3}}\int_{0}^{1-x_{3}}d{x_{1}}\big{[}T_{1}^{M}(x_{1},1-x_{1}-x_{3},x_{3})\big{]}$
$\displaystyle+\frac{2m_{N}^{2}}{\sqrt{3}}\int_{0}^{1}\frac{(1+v+v^{2})}{(q-pv)^{4}}dv\int_{0}^{v}dy\int_{y}^{1}d{x_{2}}\int_{0}^{1-x_{2}}d{x_{1}}\big{[}\big{(}T_{2}+T_{3}+T_{4}-T_{5}-2T_{7}-2T_{8}\big{)}$
$\displaystyle\times(x_{1},x_{2},1-x_{1}-x_{2})\big{]}$
$\displaystyle-\frac{2m_{N}^{2}}{\sqrt{3}}\int_{0}^{1}\frac{(1+v+v^{2})}{(q-pv)^{4}}dv\int_{0}^{v}dy\int_{y}^{1}d{x_{3}}\int_{0}^{1-x_{3}}d{x_{1}}\big{[}\big{(}T_{2}+T_{3}+T_{4}-T_{5}-2T_{7}-2T_{8}\big{)}$
$\displaystyle\times(x_{1},1-x_{1}-x_{3},x_{3})\big{]}.$ (26)

The Borel transformation as well as the continuum subtraction are performed
making use of the following replacements Braun:2006hz :

$\displaystyle\int dx\frac{\rho(x)}{(q-xp)^{2}}$
$\displaystyle\rightarrow-\int_{x_{0}}^{1}\frac{dx}{x}\rho(x)e^{-s(x)/M^{2}},$
$\displaystyle\int dx\frac{\rho(x)}{(q-xp)^{4}}$
$\displaystyle\rightarrow\frac{1}{M^{2}}\int_{x_{0}}^{1}\frac{dx}{x^{2}}\rho(x)e^{-s(x)/M^{2}}+\frac{\rho(x_{0})}{Q^{2}+x_{0}^{2}m_{N}^{2}}e^{-s_{0}/M^{2}},$
$\displaystyle\int dx\frac{\rho(x)}{(q-xp)^{6}}$
$\displaystyle\rightarrow-\frac{1}{2M^{4}}\int_{x_{0}}^{1}\frac{dx}{x^{3}}\rho(x)e^{-s(x)/M^{2}}-\frac{1}{2M^{2}}\frac{\rho(x_{0})}{x_{0}(Q^{2}+x_{0}^{2}m_{N}^{2})}e^{-s_{0}/M^{2}}$
$\displaystyle+\frac{1}{2}\frac{x_{0}^{2}}{Q^{2}+x_{0}^{2}m_{N}^{2}}\bigg{[}\frac{d}{dx_{0}}\frac{\rho(x_{0})}{x_{0}(Q^{2}+x_{0}^{2}m_{N}^{2})}\bigg{]}e^{-s_{0}/M^{2}},$
(27)

where

$\displaystyle s(x)=(1-x)m_{N}^{2}+\frac{1-x}{x}Q^{2},$ (28)

and $M^{2}$ is the Borel mass parameter, which enters in the calculations
after applying the Borel transformation in terms of $p^{\prime 2}$. Here,
$x_{0}$ is found by solving the equation, $s(x)=s_{0}$, i.e.,

$\displaystyle x_{0}$ $\displaystyle=$
$\displaystyle\Big{[}\sqrt{(Q^{2}+s_{0}-m_{N}^{2})^{2}+4m_{N}^{2}Q^{2}}-(Q^{2}+s_{0}-m_{N}^{2})\Big{]}/2m_{N}^{2},$
(29)

with $s_{0}$ being the continuum threshold parameter.

## References

  * (1) I. Y. Kobzarev, L. B. Okun, GRAVITATIONAL INTERACTION OF FERMIONS, Zh. Eksp. Teor. Fiz. 43 (1962) 1904–1909.
  * (2) H. Pagels, Energy-Momentum Structure Form Factors of Particles, Phys. Rev. 144 (1966) 1250–1260.  doi:10.1103/PhysRev.144.1250.
  * (3) I. Y. Kobsarev, V. I. Zakharov, Consequences of the transversality of the graviton emission amplitude, Annals Phys. 60 (1970) 448–463.  doi:10.1016/0003-4916(70)90498-7.
  * (4) K. L. Ng, Gravitational form-factors of the neutrino, Phys. Rev. D 47 (1993) 5187–5190.  arXiv:gr-qc/9305002, doi:10.1103/PhysRevD.47.5187.
  * (5) M. V. Polyakov, A. G. Shuvaev, On’dual’ parametrizations of generalized parton distributions (7 2002).  arXiv:hep-ph/0207153.
  * (6) X.-D. Ji, W. Melnitchouk, X. Song, A Study of off forward parton distributions, Phys. Rev. D 56 (1997) 5511–5523.  arXiv:hep-ph/9702379, doi:10.1103/PhysRevD.56.5511.
  * (7) P. Schweitzer, S. Boffi, M. Radici, Polynomiality of unpolarized off forward distribution functions and the D term in the chiral quark soliton model, Phys. Rev. D 66 (2002) 114004.  arXiv:hep-ph/0207230, doi:10.1103/PhysRevD.66.114004.
  * (8) J.-H. Jung, U. Yakhshiev, H.-C. Kim, Energy–momentum tensor form factors of the nucleon within a $\pi$–$\rho$–$\omega$ soliton model, J. Phys. G 41 (2014) 055107.  arXiv:1310.8064, doi:10.1088/0954-3899/41/5/055107.
  * (9) P. Hagler, J. W. Negele, D. B. Renner, W. Schroers, T. Lippert, K. Schilling, Moments of nucleon generalized parton distributions in lattice QCD, Phys. Rev. D 68 (2003) 034505.  arXiv:hep-lat/0304018, doi:10.1103/PhysRevD.68.034505.
  * (10) M. Gockeler, R. Horsley, D. Pleiter, P. E. L. Rakow, A. Schafer, G. Schierholz, W. Schroers, Generalized parton distributions from lattice QCD, Phys. Rev. Lett. 92 (2004) 042002.  arXiv:hep-ph/0304249, doi:10.1103/PhysRevLett.92.042002.
  * (11) B. Pasquini, S. Boffi, Nucleon spin densities in a light-front constituent quark model, Phys. Lett. B 653 (2007) 23–28.  arXiv:0705.4345, doi:10.1016/j.physletb.2007.07.037.
  * (12) D. S. Hwang, D. Mueller, Implication of the overlap representation for modelling generalized parton distributions, Phys. Lett. B 660 (2008) 350–359.  arXiv:0710.1567, doi:10.1016/j.physletb.2008.01.014.
  * (13) Z. Abidin, C. E. Carlson, Gravitational Form Factors in the Axial Sector from an AdS/QCD Model, Phys. Rev. D 77 (2008) 115021.  arXiv:0804.0214, doi:10.1103/PhysRevD.77.115021.
  * (14) S. J. Brodsky, G. F. de Teramond, Light-Front Dynamics and AdS/QCD Correspondence: Gravitational Form Factors of Composite Hadrons, Phys. Rev. D 78 (2008) 025032.  arXiv:0804.0452, doi:10.1103/PhysRevD.78.025032.
  * (15) B. Pasquini, M. V. Polyakov, M. Vanderhaeghen, Dispersive evaluation of the D-term form factor in deeply virtual Compton scattering, Phys. Lett. B 739 (2014) 133–138.  arXiv:1407.5960, doi:10.1016/j.physletb.2014.10.047.
  * (16) D. Chakrabarti, C. Mondal, A. Mukherjee, Gravitational form factors and transverse spin sum rule in a light front quark-diquark model in AdS/QCD, Phys. Rev. D 91 (11) (2015) 114026.  arXiv:1505.02013, doi:10.1103/PhysRevD.91.114026.
  * (17) C. Lorcé, H. Moutarde, A. P. Trawiński, Revisiting the mechanical properties of the nucleon, Eur. Phys. J. C 79 (1) (2019) 89.  arXiv:1810.09837, doi:10.1140/epjc/s10052-019-6572-3.
  * (18) O. V. Teryaev, Gravitational form factors and nucleon spin structure, Front. Phys. (Beijing) 11 (5) (2016) 111207.  doi:10.1007/s11467-016-0573-6.
  * (19) P. E. Shanahan, W. Detmold, Pressure Distribution and Shear Forces inside the Proton, Phys. Rev. Lett. 122 (7) (2019) 072003.  arXiv:1810.07589, doi:10.1103/PhysRevLett.122.072003.
  * (20) P. E. Shanahan, W. Detmold, Gluon gravitational form factors of the nucleon and the pion from lattice QCD, Phys. Rev. D 99 (1) (2019) 014511.  arXiv:1810.04626, doi:10.1103/PhysRevD.99.014511.
  * (21) M. J. Neubelt, A. Sampino, J. Hudson, K. Tezgin, P. Schweitzer, Energy momentum tensor and the D-term in the bag model, Phys. Rev. D 101 (3) (2020) 034013\.  arXiv:1911.08906, doi:10.1103/PhysRevD.101.034013.
  * (22) I. V. Anikin, Gravitational form factors within light-cone sum rules at leading order, Phys. Rev. D 99 (9) (2019) 094026.  arXiv:1902.00094, doi:10.1103/PhysRevD.99.094026.
  * (23) H. Alharazin, D. Djukanovic, J. Gegelia, M. V. Polyakov, Chiral theory of nucleons and pions in the presence of an external gravitational field, Phys. Rev. D 102 (7) (2020) 076023.  arXiv:2006.05890, doi:10.1103/PhysRevD.102.076023.
  * (24) J. Gegelia, M. V. Polyakov, A bound on the nucleon Druck-term from chiral EFT in curved space-time and mechanical stability conditions, Phys. Lett. B 820 (2021) 136572.  arXiv:2104.13954, doi:10.1016/j.physletb.2021.136572.
  * (25) M. Varma, P. Schweitzer, Effects of long-range forces on the D-term and the energy-momentum structure, Phys. Rev. D 102 (1) (2020) 014047.  arXiv:2006.06602, doi:10.1103/PhysRevD.102.014047.
  * (26) M. Fujita, Y. Hatta, S. Sugimoto, T. Ueda, Nucleon D-term in holographic quantum chromodynamics, PTEP 2022 (9) (2022) 093B06.  arXiv:2206.06578, doi:10.1093/ptep/ptac110.
  * (27) K. A. Mamo, I. Zahed, J/$\psi$ near threshold in holographic QCD: A and D gravitational form factors, Phys. Rev. D 106 (8) (2022) 086004.  arXiv:2204.08857, doi:10.1103/PhysRevD.106.086004.
  * (28) D. A. Pefkou, D. C. Hackett, P. E. Shanahan, Gluon gravitational structure of hadrons of different spin, Phys. Rev. D 105 (5) (2022) 054509.  arXiv:2107.10368, doi:10.1103/PhysRevD.105.054509.
  * (29) K. Azizi, U. Özdem, Nucleon’s energy–momentum tensor form factors in light-cone QCD, Eur. Phys. J. C 80 (2) (2020) 104.  arXiv:1908.06143, doi:10.1140/epjc/s10052-020-7676-5.
  * (30) M. V. Polyakov, H.-D. Son, Nucleon gravitational form factors from instantons: forces between quark and gluon subsystems, JHEP 09 (2018) 156.  arXiv:1808.00155, doi:10.1007/JHEP09(2018)156.
  * (31) A. Freese, G. A. Miller, Unified formalism for electromagnetic and gravitational probes: Densities, Phys. Rev. D 105 (1) (2022) 014003.  arXiv:2108.03301, doi:10.1103/PhysRevD.105.014003.
  * (32) A. Freese, G. A. Miller, Genuine empirical pressure within the proton, Phys. Rev. D 104 (1) (2021) 014024.  arXiv:2104.03213, doi:10.1103/PhysRevD.104.014024.
  * (33) A. Freese, G. A. Miller, Forces within hadrons on the light front, Phys. Rev. D 103 (2021) 094023.  arXiv:2102.01683, doi:10.1103/PhysRevD.103.094023.
  * (34) V. D. Burkert, L. Elouadrhiri, F. X. Girod, Determination of shear forces inside the proton (4 2021).  arXiv:2104.02031.
  * (35) V. D. Burkert, L. Elouadrhiri, F. X. Girod, The pressure distribution inside the proton, Nature 557 (7705) (2018) 396–399.  doi:10.1038/s41586-018-0060-z.
  * (36) U. Özdem, K. Azizi, Gravitational form factors of hyperons in light-cone QCD, Phys. Rev. D 101 (11) (2020) 114026.  arXiv:2003.12588, doi:10.1103/PhysRevD.101.114026.
  * (37) H.-Y. Won, J.-Y. Kim, H.-C. Kim, Gravitational form factors of the baryon octet with flavor SU(3) symmetry breaking, Phys. Rev. D 106 (11) (2022) 114009\.  arXiv:2210.03320, doi:10.1103/PhysRevD.106.114009.
  * (38) M. V. Polyakov, A. Tandogan, Comment on “Gravitational transition form factors of $N(1535)\to N$ ”, Phys. Rev. D 101 (11) (2020) 118501.  doi:10.1103/PhysRevD.101.118501.
  * (39) K. Azizi, U. Özdem, Gravitational form factors of N(1535) in QCD, Nucl. Phys. A 1015 (2021) 122296.  arXiv:2012.06895, doi:10.1016/j.nuclphysa.2021.122296.
  * (40) U. Özdem, K. Azizi, Gravitational transition form factors of $N(1535)\rightarrow N$, Phys. Rev. D 101 (5) (2020) 054031.  arXiv:1912.06375, doi:10.1103/PhysRevD.101.054031.
  * (41) S. Cotogno, C. Lorcé, P. Lowdon, M. Morales, Covariant multipole expansion of local currents for massive states of any spin, Phys. Rev. D 101 (5) (2020) 056016.  arXiv:1912.08749, doi:10.1103/PhysRevD.101.056016.
  * (42) M. V. Polyakov, B.-D. Sun, Gravitational form factors of a spin one particle, Phys. Rev. D 100 (3) (2019) 036003.  arXiv:1903.02738, doi:10.1103/PhysRevD.100.036003.
  * (43) W. Cosyn, S. Cotogno, A. Freese, C. Lorcé, The energy-momentum tensor of spin-1 hadrons: formalism, Eur. Phys. J. C 79 (6) (2019) 476.  arXiv:1903.00408, doi:10.1140/epjc/s10052-019-6981-3.
  * (44) J.-Y. Kim, B.-D. Sun, D. Fu, H.-C. Kim, Mechanical structure of a spin-1 particle (8 2022).  arXiv:2208.01240.
  * (45) A. Freese, W. Cosyn, Spatial densities of momentum and forces in spin-one hadrons, Phys. Rev. D 106 (11) (2022) 114013.  arXiv:2207.10787, doi:10.1103/PhysRevD.106.114013.
  * (46) A. Freese, W. Cosyn, Spatial densities of the photon on the light front, Phys. Rev. D 106 (11) (2022) 114014.  arXiv:2207.10788, doi:10.1103/PhysRevD.106.114014.
  * (47) A. Freese, A. Freese, I. C. Cloët, I. C. Cloët, Gravitational form factors of light mesons, Phys. Rev. C 100 (1) (2019) 015201, [Erratum: Phys.Rev.C 105, 059901 (2022)].  arXiv:1903.09222, doi:10.1103/PhysRevC.100.015201.
  * (48) B.-D. Sun, Y.-B. Dong, Gravitational form factors of $\rho$ meson with a light-cone constituent quark model, Phys. Rev. D 101 (9) (2020) 096008.  arXiv:2002.02648, doi:10.1103/PhysRevD.101.096008.
  * (49) E. Epelbaum, J. Gegelia, U. G. Meißner, M. V. Polyakov, Chiral theory of $\rho$-meson gravitational form factors, Phys. Rev. D 105 (1) (2022) 016018.  arXiv:2109.10826, doi:10.1103/PhysRevD.105.016018.
  * (50) J.-Y. Kim, Parametrization of transition energy-momentum tensor form factors, Phys. Lett. B 834 (2022) 137442.  arXiv:2206.10202, doi:10.1016/j.physletb.2022.137442.
  * (51) D. Fu, B.-D. Sun, Y. Dong, Electromagnetic and gravitational form factors of $\Delta$ resonance in a covariant quark-diquark approach, Phys. Rev. D 105 (9) (2022) 096002.  arXiv:2201.08059, doi:10.1103/PhysRevD.105.096002.
  * (52) H. Alharazin, E. Epelbaum, J. Gegelia, U. G. Meißner, B. D. Sun, Gravitational form factors of the delta resonance in chiral EFT, Eur. Phys. J. C 82 (10) (2022) 907.  arXiv:2209.01233, doi:10.1140/epjc/s10052-022-10882-0.
  * (53) J. Y. Panteleeva, M. V. Polyakov, Quadrupole pressure and shear forces inside baryons in the large $N_{c}$ limit, Phys. Lett. B 809 (2020) 135707.  arXiv:2004.02912, doi:10.1016/j.physletb.2020.135707.
  * (54) P. Hagler, et al., Nucleon Generalized Parton Distributions from Full Lattice QCD, Phys. Rev. D 77 (2008) 094502.  arXiv:0705.4295, doi:10.1103/PhysRevD.77.094502.
  * (55) M. V. Polyakov, Generalized parton distributions and strong forces inside nucleons and nuclei, Phys. Lett. B 555 (2003) 57–62.  arXiv:hep-ph/0210165, doi:10.1016/S0370-2693(03)00036-4.
  * (56) H. Hashamipour, M. Goharipour, K. Azizi, S. V. Goloskokov, Generalized parton distributions at zero skewness (11 2022).  arXiv:2211.09522.
  * (57) H. Hashamipour, M. Goharipour, K. Azizi, S. V. Goloskokov, Determination of the generalized parton distributions through the analysis of the world electron scattering data considering two-photon exchange corrections, Phys. Rev. D 105 (5) (2022) 054002.  arXiv:2111.02030, doi:10.1103/PhysRevD.105.054002.
  * (58) V. M. Belyaev, Delta - isobar magnetic form-factor in QCD (1 1993).  arXiv:hep-ph/9301257.
  * (59) V. M. Belyaev, B. L. Ioffe, Determination of the baryon mass and baryon resonances from the quantum-chromodynamics sum rule. Strange baryons, Sov. Phys. JETP 57 (1983) 716–721.
  * (60) V. M. Braun, A. Lenz, M. Wittmann, Nucleon Form Factors in QCD, Phys. Rev. D 73 (2006) 094019.  arXiv:hep-ph/0604050, doi:10.1103/PhysRevD.73.094019.
  * (61) T. M. Aliev, K. Azizi, A. Ozpineci, Light cone QCD sum rules analysis of the axial N -$>$ Delta transition form-factors, Nucl. Phys. A 799 (2008) 105–126.  arXiv:0707.1592, doi:10.1016/j.nuclphysa.2007.11.006.
  * (62) T. M. Aliev, A. Ozpineci, Radiative decays of decuplet to octet baryons in light cone QCD, Nucl. Phys. B 732 (2006) 291–320.  arXiv:hep-ph/0406331, doi:10.1016/j.nuclphysb.2005.07.038.
  * (63) K. Goeke, M. V. Polyakov, M. Vanderhaeghen, Hard exclusive reactions and the structure of hadrons, Prog. Part. Nucl. Phys. 47 (2001) 401–515.  arXiv:hep-ph/0106012, doi:10.1016/S0146-6410(01)00158-2.
  * (64) S. J. Brodsky, et al., Strong QCD from Hadron Structure Experiments: Newport News, VA, USA, November 4-8, 2019, Int. J. Mod. Phys. E 29 (08) (2020) 2030006\.  arXiv:2006.06802, doi:10.1142/S0218301320300064.