# Magnetic and quadrupole moments of the $Z_{c}(3900)$

U. Özdem<EMAIL_ADDRESS>K. Azizi<EMAIL_ADDRESS>Department of
Physics, Doǧuş University, Acıbadem-Kadıköy, 34722 İstanbul, Türkiye

###### Abstract

The electromagnetic properties of the tetraquark state $Z_{c}(3900)$ are
investigated in the diquark-antidiquark picture and its magnetic and
quadrupole moments are extracted. To this end, the light-cone QCD sum rule in
electromagnetic background field is used. The magnetic and quadrupole moments
encode the spatial distributions of the charge and magnetization in the
particle. The result obtained for the magnetic moment is quite large and can
be measured in future experiments. We obtain a nonzero but small value for the
quadrupole moment of $Z_{c}(3900)$ indicating a nonspherical charge
distribution.

Tetraquarks, Electromagnetic form factors, Multipole moments

##  I Introduction

In the conventional quark model, the predicted particles are
mesons($q\bar{q}$), baryons($qqq$) and antibaryons($\bar{q}\bar{q}\bar{q}$).
Hundreds of meson and baryon resonances have been observed till now. However,
the quark model as well as QCD as theory of strong interaction does not
exclude the existence of nonconventional particles. Hence, physicists have
thought that there may be particles in different structures GellMann:1964nj ;
Jaffe:1976ih ; Witten:1979kh . Particles having different quark and gluon
contents such as tetraquarks, pentaquarks, hybrids, glueballs and so on are
called exotic states. To explore the underlying structures of these states,
many exotic structures have been suggested [for instance, see Nielsen:2009uh ;
Swanson:2006st ; Voloshin:2007dx ; Klempt:2007cp ; Godfrey:2008nc ;
Faccini:2012pj ; Esposito:2014rxa ]. Although predicted in the 1970s, there
was not significant experimental evidence of their existence until recently.
Experimentally, the adventure of exotic states began when X(3872) was
discovered by the Belle Collaboration Choi:2003ue  and continued with the
discovery of the Y(4260) by the BABAr Collaboration Aubert:2005rm . At
present, more than twenty exotic states have been discovered in many
experiments, most of which have been classified as the XYZ family (for
details, see Liu:2013waa ). The XYZ family has some decay channels that
severely violate the isospin symmetry and negatively affect the identification
of conventional charmonium/bottomonium states. Because of that these newly
observed XYZ states provide a good platform for studying the nonperturbative
behavior of QCD. The study of the properties of these particles is one of the
most active and interesting branches of particle physics.

One of the most prominent particles among the exotic states is the charged
$Z_{c}(3900)$ tetraquark. The $Z_{c}^{\pm}(3900)$ state discovered by BESIII
in the process $e^{+}e^{-}\rightarrow\pi^{\pm}J/\psi$ Ablikim:2013mio  with a
mass $3899.0\pm 3.6\pm 4.9$ MeV and width $\Gamma=46\pm 10\pm 20MeV$. Almost
at the same time this state was confirmed by the Belle Collaboration
Liu:2013dau , with a mass $3894.5\pm 6.6\pm 4.5$ MeV and width $\Gamma=63\pm
24\pm 26$ MeV. Its existence was also confirmed in Ref. Xiao:2013iha  on the
basis of the CLEO-c data analysis, with mass $3886.0\pm 4.0\pm 2.0$ MeV and
width $\Gamma=37\pm 4\pm 8$ MeV. The decays into $\pi^{\pm}J/\psi$, reveal
that $Z_{c}^{\pm}(3900)$ must be a tetraquark state with constituents
$c\bar{c}u\bar{d}$ or $c\bar{c}d\bar{u}$ Braaten:2014qka . Since the mass of
$Z_{c}^{\pm}(3900)$ is very close to $X(3872)$, it can be advised as the
charged partner of the $X(3872)$ in a tetraquark scenario. The properties of
the $Z_{c}^{\pm}(3900)$ particle have been investigated with different
theoretical models and approaches Dias:2013xfa ; Wang:2010rt ; Wang:2013vex ;
Deng:2014gqa ; Wang:2013daa ; Wilbring:2013cha ; Dong:2013iqa ; Ke:2013gia ;
Guo:2013sya ; Gutsche:2014zda ; Esposito:2014hsa ; Goerke:2016hxf ;
Agaev:2016dev ; Agaev:2017tzv . Although the spectroscopic properties of these
particles have been studied adequately, the internal structure and nature of
the $X(3872)$ and $Z_{c}^{\pm}(3900)$ particles have not been fully understood
yet. For this reason, it is important to study their decay properties as well
as their interactions with other particles. In this context, examining the
interaction of these particles with the photon can play an important role in
understanding of their nature and internal structure.

A detailed study of the electromagnetic structures, such as electromagnetic
multipole moments and electromagnetic form factors, of hadrons not only
provides important information about the nonperturbative nature of QCD but
also the multipole moments of the hadrons are important tools for
understanding their internal structures in terms of quarks and gluons as well
as their geometric shape. The electromagnetic multipole moments encode the
spatial distributions of charge and magnetization in the particle. In hadrons,
quarks are the carriers of the charge, and thus these observables are directly
connected to the spatial distribution of quarks in hadrons, as well as a probe
of the underlying dynamics. The examination of the spatial distri butions of
the charge and magnetism carried by nuclei started in the 1950s. The
electromagnetic properties of the nucleon have been studied in the past
extensively from unpolarized electron scattering experiments-for reviews on
experimental progress, see for instance Refs. Arrington:2011kb ;
Perdrisat:2006hj ; Arrington:2006zm ; HydeWright:2004gh ; Gao:2003ag .

There are many studies in the literature devoted to investigation of the
multipole moments of the standard hadrons. However, unfortunately, almost
nothing is known about the multipole moments of exotic particles and more
detailed analyses are needed in this regard. Since direct experimental
information on the electromagnetic multipole moments of the exotic particles
is very limited, theoretical studies can play an important role in this
respect. In this study, the tetraquark state $Z_{c}(3900)$ is investigated in
the diquark-antidiquark picture and its magnetic and quadrupole moments are
extracted. This is the first theoretical attempt to calculate the
electromagnetic multipole moments of the hidden-charm tetraquark states. To
study the electromagnetic multipole moments, a nonperturbative method is
needed. The light-cone QCD sum rule (LCSR) is one of the nonperturbative
methods that has been successfully applied to study many nonperturbative
properties of hadrons for decades Chernyak:1990ag ; Braun:1988qv ;
Balitsky:1989ry . In the LCSR, the features of the particles under study are
described in terms of the vacuum condensates and the light-cone distribution
amplitudes (DAs). Hence, any uncertainty in these parameters affects the
estimations on the magnetic and quadrupole moments.

The rest of the paper is organized as follows: In Sec. II, the LCSR for the
magnetic and quadrupole moments of the $Z_{c}(3900)$ are derived. Section III
is devoted to the numerical analysis of the obtained sum rules. Section IV
includes our concluding remarks. The explicit expressions of the photon
distribution amplitudes, magnetic and quadrupole moments as well as some
details about calculations are moved to Appendixes A-C.

##  II Formalism

In order to calculate the magnetic and quadrupole moments of the $Z_{c}(3900)$
state in the framework of LCSR, we start from the correlation function

$\Pi_{\mu\nu}(q)=i\int d^{4}xe^{ip\cdot x}\langle
0|\mathcal{T}\\{J_{\mu}^{Z_{c}}(x)J_{\nu}^{Z_{c}\dagger}(0)\\}|0\rangle_{\gamma},$
(1)

where $\gamma$ is the external electromagnetic field and $J_{\mu}$ is the
interpolating current of the $Z_{c}(3900)$ state with quantum numbers
$J^{PC}=1^{+-}$ in the diquark-antidiquark picture. It is given as

$\displaystyle J_{\mu}^{Z_{c}}(x)$ $\displaystyle=$
$\displaystyle\frac{i\epsilon\tilde{\epsilon}}{\sqrt{2}}\left\\{\left[u_{a}^{T}(x)C\gamma_{5}c_{b}(x)\right]\left[\overline{d}_{d}(x)\gamma_{\mu}C\overline{c}_{e}^{T}(x)\right]-\left[u_{a}^{T}(x)C\gamma_{\mu}c_{b}(x)\right]\left[\overline{d}_{d}(x)\gamma_{5}C\overline{c}_{e}^{T}(x)\right]\right\\},$
(2)

where $\epsilon=\epsilon_{abc}$, $\tilde{\epsilon}=\epsilon_{dec}$, $C$ is the
charge conjugation matrix and $a,b,c,d,e$ are color indices.

We start to calculate the correlation function in terms of the hadronic
parameters called the hadronic side. To this end, we insert complete sets of
intermediate states having the same quantum numbers as the interpolating
current of $Z_{c}(3900)$ into the correlation function, and isolate the
contribution of the ground state. As a result the following expression is
obtained:

$\displaystyle\Pi_{\mu\nu}^{Had}(p,q)={\frac{\langle 0\mid J_{\mu}^{Z_{c}}\mid
Z_{c}(p)\rangle}{p^{2}-m_{Z_{c}}^{2}}}\langle Z_{c}(p)\mid
Z_{c}(p+q)\rangle_{\gamma}\frac{\langle
Z_{c}(p+q)\mid{J^{\dagger}}_{\nu}^{Z_{c}}\mid
0\rangle}{(p+q)^{2}-m_{Z_{c}}^{2}}+\cdots,$ (3)

where dots represent the contributions coming from the higher states and
continuum and $q$ is the momentum of the photon. The matrix element $\langle
0\mid J_{\mu}^{Z_{c}}\mid Z_{c}\rangle$ is parametrized as

$\displaystyle\langle 0\mid J_{\mu}^{Z_{c}}\mid
Z_{c}\rangle=\lambda_{Z_{c}}\varepsilon_{\mu}^{\theta}\,,$ (4)

with $\lambda_{Z_{c}}$ being the current coupling constant or residue of the
$Z_{c}(3900)$ state.

In the presence of the electromagnetic background field, the vertex of the two
axial vector mesons can be written in terms of form factors as follows
Brodsky:1992px :

$\displaystyle\langle Z_{c}(p,\varepsilon^{\theta})\mid
Z_{c}(p+q,\varepsilon^{\delta})\rangle_{\gamma}$
$\displaystyle=-\varepsilon^{\tau}(\varepsilon^{\theta})^{\alpha}(\varepsilon^{\delta})^{\beta}\Bigg{[}G_{1}(Q^{2})~{}(2p+q)_{\tau}~{}g_{\alpha\beta}+G_{2}(Q^{2})~{}(g_{\tau\beta}~{}q_{\alpha}-g_{\tau\alpha}~{}q_{\beta})$
$\displaystyle-\frac{1}{2m_{Z_{c}}^{2}}G_{3}(Q^{2})~{}(2p+q)_{\tau}~{}q_{\alpha}q_{\beta}\Bigg{]}\,,$
(5)

where $\varepsilon^{\delta}$ and $\varepsilon^{\theta}$ are the polarization
vectors of the initial and final $Z_{c}(3900)$ mesons and $\varepsilon^{\tau}$
is the polarization vector of the photon. The form factors $G_{1}(Q^{2})$,
$G_{2}(Q^{2})$ and $G_{3}(Q^{2})$ can be written in terms of the charge
$F_{C}(Q^{2})$, magnetic $F_{M}(Q^{2})$ and quadrupole $F_{\cal D}(Q^{2})$
form factors in the following way:

$\displaystyle F_{C}(Q^{2})=G_{1}(Q^{2})+\frac{2}{3}\lambda F_{\cal
D}(Q^{2})\,,$ $\displaystyle F_{M}(Q^{2})=G_{2}(Q^{2})\,,$ $\displaystyle
F_{\cal D}(Q^{2})=G_{1}(Q^{2})-G_{2}(Q^{2})+(1+\lambda)G_{3}(Q^{2})\,,$ (6)

where $\lambda=Q^{2}/4m_{Z_{c}}^{2}$ with $Q^{2}=-q^{2}$. At $Q^{2}=0$, the
form factors $F_{C}(Q^{2}=0)$, $F_{M}(Q^{2}=0)$, and $F_{\cal D}(Q^{2}=0)$ are
related to the electric charge, magnetic moment $\mu$, and quadrupole moment
${\cal D}$ in the following way:

$\displaystyle eF_{C}(0)=e\,,$ $\displaystyle eF_{M}(0)=2m_{Z_{c}}\mu\,,$
$\displaystyle eF_{\cal D}(0)=m_{Z_{c}}^{2}{\cal D}\,.$ (7)

Using Eqs. (3)-(II) and imposing the condition, $q\\!\cdot\\!\varepsilon=0$,
and performing summation over polarization vectors, the correlation function
takes the form,

$\displaystyle\Pi_{\mu\nu}^{Had}$
$\displaystyle=\lambda_{Z_{c}}^{2}\frac{\varepsilon^{\tau}}{[m_{Z_{c}}^{2}-(p+q)^{2}][m_{Z_{c}}^{2}-p^{2}]}\Bigg{[}2p_{\tau}F_{C}(0)\Bigg{(}g_{\mu\nu}-\frac{p_{\mu}q_{\nu}-p_{\nu}q_{\mu}}{m_{Z_{c}}^{2}}\Bigg{)}$
$\displaystyle+F_{M}(0)\Bigg{(}q_{\mu}g_{\nu\tau}-q_{\nu}g_{\mu\tau}+\frac{1}{m_{Z_{c}}^{2}}p_{\tau}(p_{\mu}q_{\nu}-p_{\nu}q_{\mu})\Bigg{)}-\Bigg{(}F_{C}(0)+F_{\cal
D}(0)\Bigg{)}{\frac{p_{\tau}}{m_{Z_{c}}^{2}}}q_{\mu}q_{\nu}\Bigg{]}\,.$ (8)

The next step is to calculate the correlation function in Eq. (1) in terms of
quarks and gluon properties in the deep Euclidean region called the QCD side.
For this aim, the interpolating currents are inserted into the correlation
function and after the contracting of quark pairs using the Wick theorem the
following result is obtained:

$\displaystyle\Pi_{\mu\nu}^{\mathrm{QCD}}(q)=-i\frac{\epsilon\tilde{\epsilon}\epsilon^{\prime}\tilde{\epsilon}^{\prime}}{2}\int
d^{4}xe^{ipx}\langle
0|\Bigg{\\{}\mathrm{Tr}\Big{[}\gamma_{5}\widetilde{S}_{u}^{aa^{\prime}}(x)\gamma_{5}S_{c}^{bb^{\prime}}(x)\Big{]}\mathrm{Tr}\Big{[}\gamma_{\mu}\widetilde{S}_{c}^{e^{\prime}e}(-x)\gamma_{\nu}S_{d}^{d^{\prime}d}(-x)\Big{]}$
(9)
$\displaystyle-\mathrm{Tr}\Big{[}\gamma_{\mu}\widetilde{S}_{c}^{e^{\prime}e}(-x)\gamma_{5}S_{d}^{d^{\prime}d}(-x)\Big{]}\mathrm{Tr}\Big{[}\gamma_{\nu}\widetilde{S}_{u}^{aa^{\prime}}(x)\gamma_{5}S_{c}^{bb^{\prime}}(x)]$
$\displaystyle-\mathrm{Tr}\Big{[}\gamma_{5}\widetilde{S}_{u}^{a^{\prime}a}(x)\gamma_{\mu}S_{c}^{b^{\prime}b}(x)\Big{]}\mathrm{Tr}\Big{[}\gamma_{5}\widetilde{S}_{c}^{e^{\prime}e}(-x)\gamma_{\nu}S_{d}^{d^{\prime}d}(-x)\Big{]}$
$\displaystyle+\mathrm{Tr}\Big{[}\gamma_{\nu}\widetilde{S}_{u}^{aa^{\prime}}(x)\gamma_{\mu}S_{c}^{bb^{\prime}}(x)\Big{]}\mathrm{Tr}\Big{[}\gamma_{5}\widetilde{S}_{c}^{e^{\prime}e}(-x)\gamma_{5}S_{d}^{d^{\prime}d}(-x)\Big{]}\Bigg{\\}}|0\rangle_{\gamma},$

where

$\widetilde{S}_{c(q)}^{ij}(x)=CS_{c(q)}^{ij\mathrm{T}}(x)C,$

with $S_{q(c)}(x)$ being the quark propagators. In the $x$-space for the light
quark propagator we use in the $m_{q}\rightarrow 0$ limit

$\displaystyle S_{q}(x)$ $\displaystyle=$ $\displaystyle
i\frac{{x\\!\\!\\!/}}{2\pi^{2}x^{4}}-\frac{\bar{q}q}{12}-\frac{\bar{q}q}{192}m_{0}^{2}x^{2}-\frac{ig_{s}}{16\pi^{2}x^{2}}\int_{0}^{1}dv~{}G^{\mu\nu}(vx)\Bigg{[}\hbox
to0.0pt{/\hss}{x}\sigma_{\mu\nu}+\sigma_{\mu\nu}\hbox
to0.0pt{/\hss}{x}\Bigg{]}.$ (10)

The heavy quark propagator is given, in terms of the second kind Bessel
functions $K_{\nu}(x)$, as

$\displaystyle
S_{c}(x)=\frac{m_{c}^{2}}{4\pi^{2}}\Bigg{[}\frac{K_{1}(m_{c}\sqrt{-x^{2}})}{\sqrt{-x^{2}}}+i\frac{{x\\!\\!\\!/}~{}K_{2}(m_{c}\sqrt{-x^{2}})}{(\sqrt{-x^{2}})^{2}}\Bigg{]}-\frac{g_{s}m_{c}}{16\pi^{2}}\int_{0}^{1}dv~{}G^{\mu\nu}(vx)\Bigg{[}(\sigma_{\mu\nu}{x\\!\\!\\!/}+{x\\!\\!\\!/}\sigma_{\mu\nu})\frac{K_{1}(m_{c}\sqrt{-x^{2}})}{\sqrt{-x^{2}}}$
$\displaystyle+2\sigma^{\mu\nu}K_{0}(m_{c}\sqrt{-x^{2}})\Bigg{]}.$ (11)

The correlation function contains different types of contributions. In the
first part, one of the free quark propagators in Eq. (9) is replaced by

$\displaystyle S^{free}\rightarrow\int d^{4}y\,S^{free}(x-y)\,\hbox
to0.0pt{/\hss}{\\!A}(y)\,S^{free}(y)\,,$ (12)

with $S^{free}$ representing the first term of the light or heavy quark
propagators and the remaining three propagators with the full quark
propagators. In the calculations the Fock-Schwinger gauge, $x_{\mu}A^{\mu}=0$,
is used.

In the second case one of the light quark propagators in Eq. (9) is replaced
by

$\displaystyle
S_{\alpha\beta}^{ab}\rightarrow-\frac{1}{4}(\bar{q}^{a}\Gamma_{i}q^{b})(\Gamma_{i})_{\alpha\beta},$
(13)

and the remaining propagators with the full quark propagators. Here,
$\Gamma_{i}$ are the full set of Dirac matrices. Once Eq. (13) is plugged into
Eq. (9), there appear matrix elements such as
$\langle\gamma(q)\left|\bar{q}(x)\Gamma_{i}q(0)\right|0\rangle$ and
$\langle\gamma(q)\left|\bar{q}(x)\Gamma_{i}G_{\alpha\beta}q(0)\right|0\rangle$,
representing the nonperturbative contributions. These matrix elements can be
expressed in terms of photon wave functions with definite twists.
Additionally, in principle, nonlocal operators such as $\bar{q}G^{2}q$ and
$\bar{q}q\bar{q}q$ are anticipated to appear. In this study, we take into
account operators with only one gluon field and contributions coming from
three particle nonlocal operators and neglect terms with two gluons
$\bar{q}G^{2}q$, and four quarks $\bar{q}q\bar{q}q$. The matrix elements
$\langle\gamma(q)\left|\bar{q}(x)\Gamma_{i}q(0)\right|0\rangle$ and
$\langle\gamma(q)\left|\bar{q}(x)\Gamma_{i}G_{\alpha\beta}q(0)\right|0\rangle$
are expressed in terms of the photon distribution amplitudes whose expressions
are given in Appendix A. The QCD side of the correlation function can be
obtained in terms of quarks and gluon properties using Eqs. (9)-(13) and after
performing the Fourier transformation to transfer the calculations to the
momentum space.

The sum rules are obtained by matching the expression of the correlation
function in terms of quark-gluon properties to its expression in terms of the
hadron properties, using their spectral representation. In order to eliminate
the subtraction terms in the spectral representation of the correlation
function, the Borel transformation with respect to the variables $p^{2}$ and
$(p+q)^{2}$ is carried out. After the transformation, contributions from the
excited and continuum states are also exponentially suppressed. Finally, we
choose the structures $q_{\mu}\varepsilon_{\nu}$ and
$(\varepsilon.p)q_{\mu}q_{\nu}$, respectively for the magnetic and quadrupole
moments and obtain

$\displaystyle\mu=\frac{e^{m_{Z_{c}}^{2}/M^{2}}}{\lambda_{Z_{c}}^{2}}\Bigg{[}\Pi_{1}+\Pi_{2}\Bigg{]},$
$\displaystyle\mathcal{D}=m^{2}_{Z_{c}}\frac{e^{m_{Z_{c}}^{2}/M^{2}}}{\lambda_{Z_{c}}^{2}}\Bigg{[}\Pi_{3}+\Pi_{4}\Bigg{]},$
(14)

where the functions $\Pi_{1}$ and $\Pi_{3}$ indicate that one of the quark
propagators enters the perturbative interaction with the photon and the
remaining three propagators are taken as full propagators. The functions
$\Pi_{2}$ and $\Pi_{4}$ show that one of the light quark propagators enters
the nonperturbative interaction with the photon and the remaining three
propagators are taken as full propagators. Explicit expressions of the
$\Pi_{1}$, $\Pi_{2}$, $\Pi_{3}$ and $\Pi_{4}$ are given in Appendix B. As an
example we show some details of the calculations i.e., Fourier and Borel
transformations as well as the continuum subtraction, for a specific term in
Appendix C.

##  III Numerical analysis

In this section, we numerically analyze the results of calculations for
magnetic and quadrupole moments. We use $m_{Z_{c}}=3899\pm 8.5~{}MeV$,
$f_{3\gamma}=-0.0039~{}GeV^{2}$ Ball:2002ps ,
$\overline{m}_{c}(m_{c})=(1.275\pm 0.025)\,GeV$,
$\langle\bar{u}u\rangle(1\,GeV)=\langle\bar{d}d\rangle(1\,GeV)=(-0.24\pm
0.01)^{3}\,GeV^{3}$ Ioffe:2005ym , $m_{0}^{2}=0.8\pm 0.1~{}GeV^{2}$, $\langle
g_{s}^{2}G^{2}\rangle=0.88~{}GeV^{4}$ Nielsen:2009uh  and
$\lambda_{Z_{c}}=m_{Z_{c}}f_{Z_{c}}=(1.79\pm 0.12)\times 10^{-2}~{}GeV^{5}$
Agaev:2016dev ; Agaev:2017tzv . We also need the value of the magnetic
susceptibility which is obtained in different studies as
$\chi(1\,GeV)=-2.85\pm 0.5~{}GeV^{-2}$ Rohrwild:2007yt ,
$\chi(1\,GeV)=-3.15\pm 0.3~{}GeV^{-2}$ Ball:2002ps  and
$\chi(1\,GeV)=-4.4~{}GeV^{-2}$ Belyaev:1984ic . The parameters used in the
photon distribution amplitudes are also given in Appendix A.

The predictions for the magnetic and quadrupole moments depend on two
auxiliary parameters; the Borel mass parameter $M^{2}$ and continuum threshold
$s_{0}$. According to the standard prescriptions in the method used the
predictions should weakly depend on these helping parameters. The continuum
threshold represents the scale at which, the excited states and continuum
start to contribute to the correlation function. Our analyses show that the
results depend very weakly on $s_{0}$ in the interval
$(m_{Z_{c}}+0.3)^{2}~{}GeV^{2}\leq s_{0}\leq(m_{Z_{c}}+0.7)^{2}~{}GeV^{2}$.
The working region for $M^{2}$ is determined requiring that the contributions
of the higher states and continuum are effectively suppressed. In technique
language, the upper bound on $M^{2}$ is found demanding the maximum pole
contribution. The lower bound is obtained demanding that the contribution of
the perturbative part exceeds the nonperturbative one and series of the
operator product expansion in the obtained sum rules converge. The above
requirements restrict the working region of the Borel parameter to
$5~{}GeV^{2}\leq M^{2}\leq 7~{}GeV^{2}$. It is worth nothing that with these
intervals of $s_{0}$ and $M^{2}$ we receive a $(85-93)\%$ pole contribution,
which nicely satisfies the requirements of the QCD sum rule approach.

In Fig. 1, we plot the dependencies of the magnetic and quadrupole moments on
$M^{2}$ at several fixed values of the continuum threshold $s_{0}$. As is
seen, the variation of the results with respect to the Borel parameters is
considerable, but there is much less dependence of the quantities under
consideration on the continuum threshold in its working interval.

Figure 1:  The dependence of the magnetic and quadrupole moments; on the Borel
parameter squared $M^{2}$ at different fixed values of the continuum
threshold.

In Fig. 2, we show the contributions of $\Pi_{1}$, $\Pi_{2}$, $\Pi_{3}$ and
$\Pi_{4}$ functions to the results obtained at the average value of $s_{0}$
with respect to the Borel mass parameter. In the case of the magnetic moment,
we see that the contribution of $\Pi_{1}$ is the dominant contribution.
$\Pi_{1}$ corresponds to roughly 65% of the result in average, while the
remaining 35% belongs to $\Pi_{2}$. In the case of quadrupole moment, we see
that all contributions come from $\Pi_{4}$ and the contribution of $\Pi_{3}$
is 0.

Our final results for the magnetic and quadrupole moments are

$\displaystyle|\mu_{Z_{c}}|=0.67\pm 0.32~{}\mu_{N}$
$\displaystyle|\mathcal{D}_{Z_{c}}|=0.054\pm 0.018~{}fm^{2},$ (15)

where the errors in the results come from the variations in the calculations
of the working regions of $M^{2}$ and $s_{0}$ as well as the uncertainties in
the values of the input parameters and the photon DAs. We remark that the main
source of uncertainties is the variations with respect to $M^{2}$ and the
results very weakly depend on the choices of the continuum threshold.

##  IV Discussion and concluding remarks

We calculated the magnetic and quadrupole moments of the $Z_{c}(3900)$ state
within the framework of the LCSR method. We obtained a measurable value for
the magnetic dipole moment but a small value for the quadrupole moment
indicating a nonspherical charge distribution. It is useful to note that the
values of the magnetic and quadrupole moments do not depend on the values of
the magnetic susceptibility $\chi$ presented in the previous section. It is
worth mentioning also that there are different Lorentz structures to calculate
the magnetic moment in the correlation function, but our result is almost
independent of these structures. Any experimental measurements of the
electromagnetic multipole moments of the $Z_{c}(3900)$ state and comparison of
the obtained results with the predictions of the present study may serve as
valuable knowledge on the internal structure of the tetraquark states as well
as the nonperturbative nature of the QCD. A comparison of our results on the
electromagnetic multipole moments of the $Z_{c}(3900)$ state with those that
can be obtained via considering different internal structures and
interpolating currents, such as a molecular type one, would be very helpful in
the determination of the internal structure of this multiquark state. A
comparison of the results obtained with the predictions of other approaches,
such as lattice QCD, chiral perturbation theory, quark model, etc., would also
be interesting.

Figure 2:  Comparison of the contributions to the magnetic and quadrupole
moments with respect to $M^{2}$ at average value of $s_{0}$.

##  V Acknowledgements

This work has been supported by the Scientific and Technological Research
Council of Turkey (TÜBİTAK) under the Grant No. 115F183.

## Appendix A: Photon distribution amplitudes

In this appendix, we present the definitions of the matrix elements of the
form $\langle\gamma(q)\left|\bar{q}(x)\Gamma_{i}q(0)\right|0\rangle$ and
$\langle\gamma(q)\left|\bar{q}(x)\Gamma_{i}G_{\mu\nu}q(0)\right|0\rangle$ in
terms of the photon DAs, and the explicit expressions of the photon
distribution amplitudes Ball:2002ps ,

$\displaystyle\langle\gamma(q)|\bar{q}(x)\gamma_{\mu}q(0)|0\rangle=e_{q}f_{3\gamma}\left(\varepsilon_{\mu}-q_{\mu}\frac{\varepsilon
x}{qx}\right)\int_{0}^{1}due^{i\bar{u}qx}\psi^{v}(u)$
$\displaystyle\langle\gamma(q)|\bar{q}(x)\gamma_{\mu}\gamma_{5}q(0)|0\rangle=-\frac{1}{4}e_{q}f_{3\gamma}\epsilon_{\mu\nu\alpha\beta}\varepsilon^{\nu}q^{\alpha}x^{\beta}\int_{0}^{1}due^{i\bar{u}qx}\psi^{a}(u)$
$\displaystyle\langle\gamma(q)|\bar{q}(x)\sigma_{\mu\nu}q(0)|0\rangle=-ie_{q}\langle\bar{q}q\rangle(\varepsilon_{\mu}q_{\nu}-\varepsilon_{\nu}q_{\mu})\int_{0}^{1}due^{i\bar{u}qx}\left(\chi\varphi_{\gamma}(u)+\frac{x^{2}}{16}\mathbb{A}(u)\right)$
$\displaystyle-\frac{i}{2(qx)}e_{q}\bar{q}q\left[x_{\nu}\left(\varepsilon_{\mu}-q_{\mu}\frac{\varepsilon
x}{qx}\right)-x_{\mu}\left(\varepsilon_{\nu}-q_{\nu}\frac{\varepsilon
x}{qx}\right)\right]\int_{0}^{1}due^{i\bar{u}qx}h_{\gamma}(u)$
$\displaystyle\langle\gamma(q)|\bar{q}(x)g_{s}G_{\mu\nu}(vx)q(0)|0\rangle=-ie_{q}\langle\bar{q}q\rangle\left(\varepsilon_{\mu}q_{\nu}-\varepsilon_{\nu}q_{\mu}\right)\int{\cal
D}\alpha_{i}e^{i(\alpha_{\bar{q}}+v\alpha_{g})qx}{\cal S}(\alpha_{i})$
$\displaystyle\langle\gamma(q)|\bar{q}(x)g_{s}\tilde{G}_{\mu\nu}(vx)i\gamma_{5}q(0)|0\rangle=-ie_{q}\langle\bar{q}q\rangle\left(\varepsilon_{\mu}q_{\nu}-\varepsilon_{\nu}q_{\mu}\right)\int{\cal
D}\alpha_{i}e^{i(\alpha_{\bar{q}}+v\alpha_{g})qx}\tilde{\cal S}(\alpha_{i})$
$\displaystyle\langle\gamma(q)|\bar{q}(x)g_{s}\tilde{G}_{\mu\nu}(vx)\gamma_{\alpha}\gamma_{5}q(0)|0\rangle=e_{q}f_{3\gamma}q_{\alpha}(\varepsilon_{\mu}q_{\nu}-\varepsilon_{\nu}q_{\mu})\int{\cal
D}\alpha_{i}e^{i(\alpha_{\bar{q}}+v\alpha_{g})qx}{\cal A}(\alpha_{i})$
$\displaystyle\langle\gamma(q)|\bar{q}(x)g_{s}G_{\mu\nu}(vx)i\gamma_{\alpha}q(0)|0\rangle=e_{q}f_{3\gamma}q_{\alpha}(\varepsilon_{\mu}q_{\nu}-\varepsilon_{\nu}q_{\mu})\int{\cal
D}\alpha_{i}e^{i(\alpha_{\bar{q}}+v\alpha_{g})qx}{\cal V}(\alpha_{i})$
$\displaystyle\langle\gamma(q)|\bar{q}(x)\sigma_{\alpha\beta}g_{s}G_{\mu\nu}(vx)q(0)|0\rangle=e_{q}\langle\bar{q}q\rangle\left\\{\left[\left(\varepsilon_{\mu}-q_{\mu}\frac{\varepsilon
x}{qx}\right)\left(g_{\alpha\nu}-\frac{1}{qx}(q_{\alpha}x_{\nu}+q_{\nu}x_{\alpha})\right)\right.\right.q_{\beta}$
$\displaystyle-\left(\varepsilon_{\mu}-q_{\mu}\frac{\varepsilon
x}{qx}\right)\left(g_{\beta\nu}-\frac{1}{qx}(q_{\beta}x_{\nu}+q_{\nu}x_{\beta})\right)q_{\alpha}-\left(\varepsilon_{\nu}-q_{\nu}\frac{\varepsilon
x}{qx}\right)\left(g_{\alpha\mu}-\frac{1}{qx}(q_{\alpha}x_{\mu}+q_{\mu}x_{\alpha})\right)q_{\beta}$
$\displaystyle+\left.\left(\varepsilon_{\nu}-q_{\nu}\frac{\varepsilon
x}{q.x}\right)\left(g_{\beta\mu}-\frac{1}{qx}(q_{\beta}x_{\mu}+q_{\mu}x_{\beta})\right)q_{\alpha}\right]\int{\cal
D}\alpha_{i}e^{i(\alpha_{\bar{q}}+v\alpha_{g})qx}{\cal T}_{1}(\alpha_{i})$
$\displaystyle+\left[\left(\varepsilon_{\alpha}-q_{\alpha}\frac{\varepsilon
x}{qx}\right)\left(g_{\mu\beta}-\frac{1}{qx}(q_{\mu}x_{\beta}+q_{\beta}x_{\mu})\right)\right.q_{\nu}$
$\displaystyle-\left(\varepsilon_{\alpha}-q_{\alpha}\frac{\varepsilon
x}{qx}\right)\left(g_{\nu\beta}-\frac{1}{qx}(q_{\nu}x_{\beta}+q_{\beta}x_{\nu})\right)q_{\mu}$
$\displaystyle-\left(\varepsilon_{\beta}-q_{\beta}\frac{\varepsilon
x}{qx}\right)\left(g_{\mu\alpha}-\frac{1}{qx}(q_{\mu}x_{\alpha}+q_{\alpha}x_{\mu})\right)q_{\nu}$
$\displaystyle+\left.\left(\varepsilon_{\beta}-q_{\beta}\frac{\varepsilon
x}{qx}\right)\left(g_{\nu\alpha}-\frac{1}{qx}(q_{\nu}x_{\alpha}+q_{\alpha}x_{\nu})\right)q_{\mu}\right]\int{\cal
D}\alpha_{i}e^{i(\alpha_{\bar{q}}+v\alpha_{g})qx}{\cal T}_{2}(\alpha_{i})$
$\displaystyle+\frac{1}{qx}(q_{\mu}x_{\nu}-q_{\nu}x_{\mu})(\varepsilon_{\alpha}q_{\beta}-\varepsilon_{\beta}q_{\alpha})\int{\cal
D}\alpha_{i}e^{i(\alpha_{\bar{q}}+v\alpha_{g})qx}{\cal T}_{3}(\alpha_{i})$
$\displaystyle+\left.\frac{1}{qx}(q_{\alpha}x_{\beta}-q_{\beta}x_{\alpha})(\varepsilon_{\mu}q_{\nu}-\varepsilon_{\nu}q_{\mu})\int{\cal
D}\alpha_{i}e^{i(\alpha_{\bar{q}}+v\alpha_{g})qx}{\cal
T}_{4}(\alpha_{i})\right\\}~{},$

where $\varphi_{\gamma}(u)$ is the leading twist-2, $\psi^{v}(u)$,
$\psi^{a}(u)$, ${\cal A}(\alpha_{i})$ and ${\cal V}(\alpha_{i})$, are the
twist-3, and $h_{\gamma}(u)$, $\mathbb{A}(u)$, ${\cal S}(\alpha_{i})$,
${\cal{\tilde{S}}}(\alpha_{i})$, ${\cal T}_{1}(\alpha_{i})$, ${\cal
T}_{2}(\alpha_{i})$, ${\cal T}_{3}(\alpha_{i})$ and ${\cal T}_{4}(\alpha_{i})$
are the twist-4 photon DAs. The measure ${\cal D}\alpha_{i}$ is defined as

$\displaystyle\int{\cal
D}\alpha_{i}=\int_{0}^{1}d\alpha_{\bar{q}}\int_{0}^{1}d\alpha_{q}\int_{0}^{1}d\alpha_{g}\delta(1-\alpha_{\bar{q}}-\alpha_{q}-\alpha_{g})~{}.$

The expressions of the DAs entering into the above matrix elements are defined
as:

$\displaystyle\varphi_{\gamma}(u)$ $\displaystyle=$ $\displaystyle
6u\bar{u}\left(1+\varphi_{2}(\mu)C_{2}^{\frac{3}{2}}(u-\bar{u})\right),$
$\displaystyle\psi^{v}(u)$ $\displaystyle=$ $\displaystyle
3\left(3(2u-1)^{2}-1\right)+\frac{3}{64}\left(15w^{V}_{\gamma}-5w^{A}_{\gamma}\right)\left(3-30(2u-1)^{2}+35(2u-1)^{4}\right),$
$\displaystyle\psi^{a}(u)$ $\displaystyle=$
$\displaystyle\left(1-(2u-1)^{2}\right)\left(5(2u-1)^{2}-1\right)\frac{5}{2}\left(1+\frac{9}{16}w^{V}_{\gamma}-\frac{3}{16}w^{A}_{\gamma}\right),$
$\displaystyle h_{\gamma}(u)$ $\displaystyle=$
$\displaystyle-10\left(1+2\kappa^{+}\right)C_{2}^{\frac{1}{2}}(u-\bar{u}),$
$\displaystyle\mathbb{A}(u)$ $\displaystyle=$ $\displaystyle
40u^{2}\bar{u}^{2}\left(3\kappa-\kappa^{+}+1\right)+8(\zeta_{2}^{+}-3\zeta_{2})\left[u\bar{u}(2+13u\bar{u})\right.$
$\displaystyle+\left.2u^{3}(10-15u+6u^{2})\ln(u)+2\bar{u}^{3}(10-15\bar{u}+6\bar{u}^{2})\ln(\bar{u})\right],$
$\displaystyle{\cal A}(\alpha_{i})$ $\displaystyle=$ $\displaystyle
360\alpha_{q}\alpha_{\bar{q}}\alpha_{g}^{2}\left(1+w^{A}_{\gamma}\frac{1}{2}(7\alpha_{g}-3)\right),$
$\displaystyle{\cal V}(\alpha_{i})$ $\displaystyle=$ $\displaystyle
540w^{V}_{\gamma}(\alpha_{q}-\alpha_{\bar{q}})\alpha_{q}\alpha_{\bar{q}}\alpha_{g}^{2},$
$\displaystyle{\cal T}_{1}(\alpha_{i})$ $\displaystyle=$
$\displaystyle-120(3\zeta_{2}+\zeta_{2}^{+})(\alpha_{\bar{q}}-\alpha_{q})\alpha_{\bar{q}}\alpha_{q}\alpha_{g},$
$\displaystyle{\cal T}_{2}(\alpha_{i})$ $\displaystyle=$ $\displaystyle
30\alpha_{g}^{2}(\alpha_{\bar{q}}-\alpha_{q})\left((\kappa-\kappa^{+})+(\zeta_{1}-\zeta_{1}^{+})(1-2\alpha_{g})+\zeta_{2}(3-4\alpha_{g})\right),$
$\displaystyle{\cal T}_{3}(\alpha_{i})$ $\displaystyle=$
$\displaystyle-120(3\zeta_{2}-\zeta_{2}^{+})(\alpha_{\bar{q}}-\alpha_{q})\alpha_{\bar{q}}\alpha_{q}\alpha_{g},$
$\displaystyle{\cal T}_{4}(\alpha_{i})$ $\displaystyle=$ $\displaystyle
30\alpha_{g}^{2}(\alpha_{\bar{q}}-\alpha_{q})\left((\kappa+\kappa^{+})+(\zeta_{1}+\zeta_{1}^{+})(1-2\alpha_{g})+\zeta_{2}(3-4\alpha_{g})\right),$
$\displaystyle{\cal S}(\alpha_{i})$ $\displaystyle=$ $\displaystyle
30\alpha_{g}^{2}\\{(\kappa+\kappa^{+})(1-\alpha_{g})+(\zeta_{1}+\zeta_{1}^{+})(1-\alpha_{g})(1-2\alpha_{g})+\zeta_{2}[3(\alpha_{\bar{q}}-\alpha_{q})^{2}-\alpha_{g}(1-\alpha_{g})]\\},$
$\displaystyle\tilde{\cal S}(\alpha_{i})$ $\displaystyle=$
$\displaystyle-30\alpha_{g}^{2}\\{(\kappa-\kappa^{+})(1-\alpha_{g})+(\zeta_{1}-\zeta_{1}^{+})(1-\alpha_{g})(1-2\alpha_{g})+\zeta_{2}[3(\alpha_{\bar{q}}-\alpha_{q})^{2}-\alpha_{g}(1-\alpha_{g})]\\}.$

Numerical values of parameters used in DAs; $\varphi_{2}(1~{}GeV)=0$,
$w^{V}_{\gamma}=3.8\pm 1.8$, $w^{A}_{\gamma}=-2.1\pm 1.0$, $\kappa=0.2$,
$\kappa^{+}=0$, $\zeta_{1}=0.4$, $\zeta_{2}=0.3$, $\zeta_{1}^{+}=0$, and
$\zeta_{2}^{+}=0$.

## Appendix B:

In this appendix, we present the explicit expressions for the functions,
$\Pi_{1}$, $\Pi_{2}$, $\Pi_{3}$ and $\Pi_{4}$:

$\displaystyle\Pi_{1}$
$\displaystyle=\frac{3m_{c}^{4}M^{2}}{64\pi^{6}}\Bigg{[}2(3e_{u}+4e_{d}-2e_{c})N[3,3,0]-3m_{c}(e_{u}+e_{d}-e_{c})N[3,4,1]-e_{c}\Big{(}8N[4,2,0]-m_{c}N[5,2,1]\Big{)}\Bigg{]}$
$\displaystyle-\frac{m_{c}^{3}M^{2}\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle}{12288\pi^{4}}\Big{(}3e_{u}+3e_{d}-2e_{c}\Big{)}N[1,2,1]$
$\displaystyle+\frac{m_{c}^{4}M^{2}\langle
g_{s}^{2}G^{2}\rangle}{147456\pi^{6}}\Big{(}2e_{u}+2e_{d}-e_{c}\Big{)}\Big{(}N[1,3,1]+N[2,2,1]\Big{)}$
$\displaystyle-\frac{m_{c}^{3}M^{2}\langle
g_{s}^{2}G^{2}\rangle}{18432\pi^{6}}\Big{(}2e_{u}+2e_{d}-e_{c}\Big{)}N[1,2,0]$
$\displaystyle-\frac{m_{c}^{2}M^{2}\langle
g_{s}^{2}G^{2}\rangle}{1536\pi^{6}}\Big{(}e_{u}+e_{d}-7e_{c}\Big{)}N[2,2,0]$
$\displaystyle+\frac{m_{c}^{3}M^{2}}{24576\pi^{6}}\Bigg{[}-(17e_{u}+17e_{d}-31e_{c})\langle
g_{s}^{2}G^{2}\rangle+576(3e_{u}+3e_{d}-4e_{c})\pi^{2}m_{c}\langle\bar{q}q\rangle\Bigg{]}N[2,3,1]$
$\displaystyle+\frac{m_{c}^{2}}{13824M^{6}\pi^{6}}\Bigg{[}(e_{u}+e_{d}+e_{c})\langle
g_{s}^{2}G^{2}\rangle
M^{2}-(e_{u}+e_{d})~{}36\pi^{2}m_{c}\langle\bar{q}q\rangle(3m_{0}^{2}+16M^{2})\Bigg{]}\Bigg{(}64~{}m_{c}^{6}FlP[-3,4,0]$
$\displaystyle-48m_{c}^{4}FlP[-2,4,0]+12~{}m_{c}^{2}FlP[-1,4,0]-FlP[0,4,0]\Bigg{)}$
$\displaystyle-\frac{m_{c}\langle\bar{q}q\rangle}{110592M^{8}\pi^{4}}\Bigg{[}3(e_{u}+e_{d})\langle
g_{s}^{2}G^{2}\rangle M^{2}(3m_{0}^{2}+16M^{2})+2e_{c}\Big{(}\langle
g_{s}^{2}G^{2}\rangle
M^{2}(3m_{0}^{2}-4M^{2})+18\pi^{2}m_{c}\langle\bar{q}q\rangle$
$\displaystyle(3m_{0}^{4}-128M^{4})\Big{)}\Bigg{]}\Big{(}16~{}m_{c}^{4}FlP[-1,2,0]-8m_{c}^{2}~{}FlP[0,2,0]+FlP[1,2,0]\Big{)}$
$\displaystyle+\frac{e_{c}m_{c}m_{0}^{2}\langle\bar{q}q\rangle^{2}}{221184M^{6}\pi^{4}}\Big{(}5\langle
g_{s}^{2}G^{2}\rangle+1152\pi^{2}m_{c}\langle\bar{q}q\rangle\Big{)}\Bigg{[}16~{}m_{c}^{4}FlP[1,2,1]-8m_{c}^{2}FlP[2,2,1]-FlP[3,2,1]\Bigg{]}$
$\displaystyle+\frac{e_{c}m_{c}^{3}m_{0}^{2}\langle\bar{q}q\rangle}{48M^{4}\pi^{4}}\Bigg{[}64m_{c}^{4}FlP[-2,3,0]+28m_{c}^{2}FlP[0,3,0]-5FlP[1,3,0]\Bigg{]}$
$\displaystyle+\frac{e_{c}m_{c}^{2}m_{0}^{4}\langle\bar{q}q\rangle^{2}}{6144M^{8}\pi^{4}}\Bigg{[}16~{}m_{c}^{4}FlP[3,2,2]-8~{}m_{c}^{2}FlP[4,2,2]+FlP[5,2,2]\Bigg{]}.$
(16)

$\displaystyle\Pi_{2}=\frac{m_{c}^{4}\langle\bar{q}q\rangle}{128\pi^{4}}\Bigg{[}e_{u}\Big{(}WFD[\mathcal{S},\bar{v}]-2WF[\mathcal{S},\bar{v}]\Big{)}+e_{d}\Big{(}WFD[\mathcal{S},v]-2WF[\mathcal{S},v]\Big{)}\Bigg{]}\Big{(}4N[2,3,0]-M^{2}N[2,3,1]\Big{)}$
$\displaystyle+\frac{m_{c}^{4}\langle\bar{q}q\rangle}{32\pi^{4}}\Bigg{[}e_{u}\Big{(}-2WF[\mathcal{T}_{1},\bar{v}]-2WF[\mathcal{T}_{2},\bar{v}]+2WF[\tilde{S},\bar{v}]+WFD[\mathcal{T}_{1},\bar{v}]+WFD[\mathcal{T}_{2},\bar{v}]-WFD[\tilde{S},\bar{v}]\Big{)}$
$\displaystyle+e_{d}\Big{(}-8WF[\mathcal{T}_{1},v]-2WF[\mathcal{T}_{2},v]+2WF[\mathcal{S},v]+4WFD[\mathcal{T}_{2},v]-WFD[\mathcal{S},v]-WFD[\tilde{S},v]\Big{)}\Bigg{]}N[1,4,0]$
$\displaystyle+\frac{m_{c}^{4}M^{2}\langle\bar{q}q\rangle}{256\pi^{4}}\Bigg{[}e_{u}\Big{(}4WF[\mathcal{T}_{1},\bar{v}]+4WF[\mathcal{T}_{2},\bar{v}]-2WF[\tilde{S},\bar{v}]-2WFD[\mathcal{T}_{1},\bar{v}]-2WFD[\mathcal{T}_{2},\bar{v}]+2WFD[\tilde{S},v]\Big{)}$
$\displaystyle+e_{d}\Big{(}13WF[\mathcal{T}_{1},v]+7WF[\mathcal{T}_{2},v]-WF[\tilde{S},v]-8WFD[\mathcal{T}_{1},v]-2WFD[\mathcal{T}_{2},v]+2WFD[\tilde{S},v]\Big{)}\Bigg{]}N[1,4,1]$
$\displaystyle+\frac{f_{3\gamma}m_{c}^{4}}{64\pi^{4}}\Bigg{[}16(e_{u}-e_{d})WFD[\psi^{a},u]+e_{u}WFD[\mathcal{V},\bar{v}]+e_{d}WFD[\mathcal{V},v]\Bigg{]}N[3,3,0]$
$\displaystyle+\frac{m_{c}^{3}M^{2}}{512\pi^{4}}\Bigg{[}m_{c}\langle\bar{q}q\rangle\Bigg{\\{}e_{u}\Big{(}-2WF[\mathcal{S},\bar{v}]+6WF[\mathcal{T}_{1},\bar{v}]+6WF[\mathcal{T}_{2},\bar{v}]-2WF[\tilde{S},\bar{v}]+3WFD[\mathcal{S},\bar{v}]$
$\displaystyle-3WFD[\mathcal{T}_{1},\bar{v}]-3WFD[\mathcal{T}_{2},\bar{v}]+3WFD[\tilde{S},\bar{v}]\Big{)}+e_{d}\Big{(}-2WF[\mathcal{S},v]+18WF[\mathcal{T}_{1},v]+12WF[\mathcal{T}_{2},v]+4WF[\tilde{S},v]$
$\displaystyle+3WFD[\mathcal{S},v]-12WFD[\mathcal{T}_{1},v]-3WFD[\mathcal{T}_{2},v]+3WFD[\tilde{S},v]\Big{)}\Bigg{\\}}+2e_{u}f_{3\gamma}M^{2}\Big{(}2WFD[\mathcal{A},\bar{v}]+WFD[\mathcal{V},\bar{v}]\Big{)}$
$\displaystyle+2e_{d}f_{3\gamma}M^{2}\Big{(}2WFD[\mathcal{A},v]+3WFD[\mathcal{V},v]\Big{)}\Bigg{]}N[2,3,1]$
$\displaystyle+\frac{m_{c}^{2}M^{2}f_{3\gamma}\langle
g_{s}^{2}G^{2}\rangle}{110592\pi^{4}}\Big{[}-10(e_{u}+e_{d})\psi^{a}(u_{0})+2(e_{u}-4e_{d})\varphi_{\gamma}(u_{0})-5(e_{u}-e_{d})WFD[\psi^{a},u]+4(e_{u}-e_{d})WFD[\psi^{\nu},u]\Big{]}$
$\displaystyle N[1,1,0]$ $\displaystyle+\frac{\langle\bar{q}q\rangle
m_{c}^{4}M^{4}}{2048\pi^{4}}\Bigg{[}e_{u}\Big{(}-2WF[\mathcal{S},\bar{v}]+2WF[\mathcal{T}_{1},\bar{v}]+2WF[\mathcal{T}_{2},v]-2WF[\tilde{S},v]+WFD[\mathcal{S},\bar{v}]-WFD[\mathcal{T}_{1},\bar{v}]$
$\displaystyle-
WFD[\mathcal{T}_{2},\bar{v}]+WFD[\tilde{S},\bar{v}]\Big{)}+e_{d}\Big{(}-2WF[\mathcal{S},v]+8WF[\mathcal{T}_{1},v]+2WF[\mathcal{T}_{2},v]-2WF[\tilde{S},v]+WFD[\mathcal{S},v]-$
$\displaystyle
4WFD[\mathcal{T}_{1},v]-WFD[\mathcal{T}_{2},v]+WFD[\tilde{S},v]\Big{)}\Bigg{]}N[2,3,2]$
$\displaystyle-\frac{f_{3\gamma}m_{c}^{4}}{32\pi^{4}}\Bigg{[}e_{u}WFD[\mathcal{V},\bar{v}]+e_{d}WFD[\mathcal{V},v]\Bigg{]}N[2,4,0]$
$\displaystyle+\frac{m_{c}^{3}}{128\pi^{4}}\Bigg{[}m_{c}\langle\bar{q}q\rangle\Bigg{\\{}e_{u}\Big{(}2WF[\mathcal{S},\bar{v}]+2WF[\mathcal{T}_{1},\bar{v}]-2WF[\mathcal{T}_{2},\bar{v}]+2WF[\tilde{S},\bar{v}]-WFD[\mathcal{S},\bar{v}]+WFD[\mathcal{T}_{1},\bar{v}]$
$\displaystyle+WFD[\mathcal{T}_{2},\bar{v}]-WFD[\tilde{S},\bar{v}]\Big{)}+e_{d}\Big{(}2WF[\mathcal{S},v]-8WF[\mathcal{T}_{1},v]-2WF[\mathcal{T}_{2},v]+2WF[\tilde{S},v]-WFD[\mathcal{S},v]$
$\displaystyle+4WFD[\mathcal{T}_{1},v]+2WFD[\mathcal{T}_{2},v]-WFD[\tilde{S},v]\Big{)}\Bigg{\\}}+f_{3\gamma}M^{2}\Big{(}e_{u}WFD[\mathcal{V},\bar{v}]+e_{d}WFD[\mathcal{V},v]\Big{)}\Bigg{]}N[2,3,0]$
$\displaystyle-\frac{m_{c}^{4}M^{2}f_{3\gamma}}{128\pi^{4}}\Bigg{(}e_{u}WFD[\mathcal{V},\bar{v}]+e_{d}WFD[\mathcal{V},v]\Bigg{)}N[2,4,1]$
$\displaystyle+\frac{m_{c}^{2}M^{4}\langle
g_{s}^{2}G^{2}\rangle}{1769472\pi^{4}}\Bigg{[}-4(4e_{u}-e_{d})\chi
m_{c}\langle\bar{q}q\rangle
WFD[\varphi_{\gamma},u]-16(e_{u}-e_{d})f_{3\gamma}WFD[\psi^{a},u]-11e_{u}f_{3\gamma}WFD[\mathcal{A},\bar{v}]$
$\displaystyle-11e_{d}f_{3\gamma}WFD[\mathcal{A},v]\Bigg{]}N[2,2,2]$

$\displaystyle+\frac{m_{c}^{3}}{6912\pi^{4}}\Bigg{[}-54M^{2}\langle\bar{q}q\rangle\Bigg{\\{}e_{u}\Big{(}2WF[\mathcal{T}_{1},\bar{v}]+2WF[\mathcal{T}_{2},\bar{v}]+3WF[\tilde{S},\bar{v}]-WFD[\mathcal{T}_{1},\bar{v}]-WFD[\mathcal{T}_{2},\bar{v}]+WFD[\tilde{S},\bar{v}]\Big{)}$
$\displaystyle+e_{d}\Big{(}5WF[\mathcal{T}_{1},v]+5WF[\mathcal{T}_{2},v]+WF[\tilde{S},v]-4WFD[\mathcal{T}_{1},v]-WFD[\mathcal{T}_{2},v]+WFD[\tilde{S},v]\Big{)}\Bigg{\\}}$
$\displaystyle-54m_{c}M^{2}f_{3\gamma}\Big{(}e_{u}WFD[\mathcal{A},\bar{v}]+e_{d}WFD[\mathcal{A},v]\Big{)}+(6e_{u}+7e_{d})\chi\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle
WF[\varphi_{\gamma},u]\Bigg{]}N[1,3,0]$
$\displaystyle+\frac{m_{c}^{3}M^{2}\chi\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle}{55296\pi^{4}}\Bigg{[}-13(e_{u}-e_{d})\varphi_{\gamma}(u_{0})-2(7e_{u}-6e_{d})WFD[\varphi_{\gamma},u]\Bigg{]}N[1,3,1]$
$\displaystyle-\frac{m_{c}^{3}M^{4}\chi\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle}{110592\pi^{4}}\Bigg{[}(e_{u}-e_{d})WFD[\varphi_{\gamma},u]\Bigg{]}N[1,3,2]$
$\displaystyle+\frac{m_{c}^{2}\langle
g_{s}^{2}G^{2}\rangle}{110592\pi^{4}}\Bigg{[}52(e_{u}-e_{d})M^{2}\chi\langle\bar{q}q\rangle\varphi_{\gamma}(u_{0})-4(8e_{u}+5e_{d})M^{2}\chi\langle\bar{q}q\rangle
WFD[\varphi_{\gamma},u]-30(e_{u}-e_{d})m_{c}f_{3\gamma}WFD[\psi^{a},u]$
$\displaystyle+11\langle\bar{q}q\rangle\Bigg{\\{}e_{u}\Big{(}6WF[\mathcal{S},\bar{v}]-4WF[\mathcal{T}_{1},\bar{v}]-10WF[\mathcal{T}_{2},\bar{v}]+2WF[\mathcal{T}_{3},\bar{v}]-WF[\mathcal{T}_{4},\bar{v}]+WF[\tilde{S},\bar{v}]-3WFD[\mathcal{S},\bar{v}]$
$\displaystyle+2WFD[\mathcal{T}_{1},\bar{v}]+5WFD[\mathcal{T}_{2},\bar{v}]-WFD[\mathcal{T}_{3},\bar{v}]+WFD[\mathcal{T}_{4},\bar{v}]-2WFD[\tilde{S},\bar{v}]\Big{)}$
$\displaystyle+e_{d}\Big{(}6WF[\mathcal{S},v]-4WF[\mathcal{T}_{1},v]-10WF[\mathcal{T}_{2},v]+2WF[\mathcal{T}_{3},v]-WF[\mathcal{T}_{4},\bar{v}]+WF[\tilde{S},\bar{v}]-3WFD[\mathcal{S},\bar{v}]$
$\displaystyle+2WFD[\mathcal{T}_{1},v]+5WFD[\mathcal{T}_{2},v]-WFD[\mathcal{T}_{3},v]+WFD[\mathcal{T}_{4},v]-2WFD[\tilde{S},v]\Big{)}\Bigg{]}N[1,2,0]$
$\displaystyle+\frac{f_{3\gamma}m_{c}^{4}M^{2}}{512\pi^{4}}\Bigg{[}3e_{u}WFD[\mathcal{V},\bar{v}]+3e_{d}WFD[\mathcal{V},v]+16(e_{u}-e_{d})WFD[\psi^{a},u]\Bigg{]}N[3,3,1]$
$\displaystyle+\frac{m_{c}^{2}M^{2}\langle
g_{s}^{2}G^{2}\rangle}{884736\pi^{4}}\Bigg{[}40m_{c}f_{3\gamma}(e_{u}-e_{d})\psi^{a}(u_{0})-8f_{3\gamma}m_{c}(e_{u}-e_{d})\psi^{\nu}(u_{0})-16m_{c}f_{3\gamma}(4e_{u}-e_{d})WF[\psi^{\gamma},u]$
$\displaystyle+30f_{3\gamma}m_{c}(e_{u}-e_{d})WFD[\psi^{a},u]-2(8e_{u}-5e_{d})\langle\bar{q}q\rangle
WFD[A,u]+11\langle\bar{q}q\rangle\Bigg{\\{}e_{u}\langle\bar{q}q\rangle\Big{(}-6WF[\mathcal{S},\bar{v}]+4WF[\mathcal{T}_{1},\bar{v}]$
$\displaystyle+10WF[\mathcal{T}_{2},\bar{v}]-2WF[\mathcal{T}_{3},\bar{v}]+4WF[\mathcal{T}_{4},\bar{v}]-4WF[\tilde{S},\bar{v}]+3WFD[\mathcal{S},\bar{v}]-2WFD[\mathcal{T}_{1},\bar{v}]-5WFD[\mathcal{T}_{2},\bar{v}]$
$\displaystyle+WFD[\mathcal{T}_{3},\bar{v}]-2WFD[\mathcal{T}_{4},\bar{v}]+2WFD[\tilde{S},\bar{v}]\Big{)}+e_{d}\Big{(}-6WF[\mathcal{S},v]+4WF[\mathcal{T}_{1},v]+10WF[\mathcal{T}_{2},v]-2WF[\mathcal{T}_{3},v]$
$\displaystyle+4WF[\mathcal{T}_{4},v]-4WF[\tilde{S},v]+3WFD[\mathcal{S},v]-2WFD[\mathcal{T}_{1},v]-5WFD[\mathcal{T}_{2},v]+WFD[\mathcal{T}_{3},v]-2WFD[\mathcal{T}_{4},v]$
$\displaystyle+2WFD[\tilde{S},v]\Big{)}\Bigg{\\}}\Bigg{]}N[1,2,1]$
$\displaystyle+\frac{m_{c}^{2}\langle
g_{s}^{2}G^{2}\rangle}{110592\pi^{4}}\Bigg{[}-104(e_{u}-e_{d})m_{c}\chi\langle
g_{s}^{2}G^{2}\rangle\varphi_{\gamma}(u_{0})+e_{u}f_{3\gamma}(11\langle
g_{s}^{2}G^{2}\rangle+432m_{c}^{2}M^{2})WFD[\mathcal{A},\bar{v}]$
$\displaystyle+e_{d}f_{3\gamma}(11\langle
g_{s}^{2}G^{2}\rangle+432m_{c}^{2}M^{2})WFD[\mathcal{A},v]+4(28e_{u}-19e_{d})m_{c}\chi\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle WFD[\varphi_{\gamma},u]$
$\displaystyle+16(e_{u}-e_{d})f_{3\gamma}\langle g_{s}^{2}G^{2}\rangle
WFD[\psi^{a},u]\Bigg{]}N[2,2,1]$ $\displaystyle+\frac{\langle
g_{s}^{2}G^{2}\rangle
m_{c}^{2}M^{4}}{28311552\pi^{4}}\Bigg{[}11e_{d}f_{3\gamma}WFD[\mathcal{A},v]-11f_{3\gamma}e_{u}WFD[\mathcal{A},\bar{v}]+4\chi
m_{c}\langle\bar{q}q\rangle(e_{d}-4e_{u})WFD[\varphi_{\gamma},u]$
$\displaystyle-16f_{3\gamma}(e_{u}-e_{d})WFD[\psi^{a},u]\Bigg{]}N[2,2,2]$
$\displaystyle-\frac{m_{c}^{2}M^{4}f_{3\gamma}}{2048\pi^{4}}\Bigg{[}e_{u}WFD[\mathcal{V},\bar{v}]+e_{d}WFD[\mathcal{V},v]-16(e_{u}-e_{d})WFD[\psi^{a},u]\Bigg{]}N[3,3,2]$

$\displaystyle+\frac{m_{c}^{2}M^{4}\langle
g_{s}^{2}G^{2}\rangle}{3538944\pi^{4}}\Bigg{[}10(e_{u}-e_{d})m_{c}f_{3\gamma}WFD[\psi^{a},u]+2(8e_{u}+5e_{d})\langle\bar{q}q\rangle
WFD[A,u]+11e_{u}\Big{(}6WF[\mathcal{S},\bar{v}]-4WF[\mathcal{T}_{1},\bar{v}]$
$\displaystyle-10WF[\mathcal{T}_{2},\bar{v}]+2WF[\mathcal{T}_{3},\bar{v}]-4WF[\mathcal{T}_{4},\bar{v}]+4WF[\tilde{S},\bar{v}]-3WFD[\mathcal{S},\bar{v}]+2WFD[\mathcal{T}_{1},\bar{v}]+5WFD[\mathcal{T}_{2},\bar{v}]$
$\displaystyle-3WFD[\mathcal{T}_{3},\bar{v}]+2WFD[\mathcal{T}_{4},\bar{v}]-WFD[\tilde{S},\bar{v}]\Big{)}+11e_{d}\Big{(}6WF[\mathcal{S},v]-4WF[\mathcal{T}_{1},v]-10WF[\mathcal{T}_{2},v]+2WF[\mathcal{T}_{3},v]$
$\displaystyle-4WF[\mathcal{T}_{4},v]+4WF[\tilde{S},v]-3WFD[\mathcal{S},v]+2WFD[\mathcal{T}_{1},v]+5WFD[\mathcal{T}_{2},v]-3WFD[\mathcal{T}_{3},v]+2WFD[\mathcal{T}_{4},v]$
$\displaystyle-WFD[\tilde{S},v]\Big{)}\Bigg{]}N[1,2,2]$
$\displaystyle-\frac{m_{c}^{2}}{55296\pi^{4}}\Bigg{[}f_{3\gamma}\Big{(}11\langle
g_{s}^{2}G^{2}\rangle+432m_{c}^{2}M^{2}\Big{)}\Big{(}e_{u}WFD[\mathcal{A},\bar{v}]+e_{d}WFD[\mathcal{A},v]\Big{)}+12m_{c}\langle\bar{q}q\rangle\Bigg{\\{}36M^{2}\Big{(}e_{u}WFD[\mathcal{S},\bar{v}]$
$\displaystyle+e_{d}WFD[\mathcal{S},v]\Big{)}-(4e_{u}-3e_{d})\chi\langle
g_{s}^{2}G^{2}\rangle
WFD[\varphi_{\gamma},u]\Bigg{\\}}+16(e_{u}-e_{d})f_{3\gamma}\langle
g_{s}^{2}G^{2}\rangle WFD[\psi^{a},u]\Bigg{]}N[2,2,0]$
$\displaystyle+\frac{m_{c}\langle\bar{q}q\rangle^{2}}{1990656M^{10}\pi^{2}}\Bigg{[}-(e_{u}-e_{d})\Big{(}5\langle
g_{s}^{2}G^{2}\rangle(23m_{0}^{2}-8M^{2})-1728m_{c}^{2}m_{0}^{2}M^{2}\Big{)}A(u_{0})$
$\displaystyle+4m_{0}^{2}\langle
g_{s}^{2}G^{2}\rangle\Big{(}10(e_{u}-e_{d})M^{2}\chi\varphi_{\gamma}(u_{0})+(7e_{u}+2e_{d})WF[h_{\gamma},u]\Big{)}\Bigg{]}\Bigg{(}16m_{c}^{4}FLNP[2,3,2]-8m_{c}^{2}FlNP[3,3,2]$
$\displaystyle-FlNP[4,3,2]$ $\displaystyle+\frac{m_{c}m_{0}^{2}\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle}{165888M^{8}\pi^{2}}(e_{u}-e_{d})\Bigg{(}4m_{c}^{2}FlNP[4,1,2]-FlNP[5,1,2]\Bigg{)}A(u_{0})$
$\displaystyle-\frac{f_{3\gamma}m_{0}^{2}\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle}{73728M^{8}\pi^{2}}(e_{u}-e_{d})\Bigg{(}16m_{c}^{4}FlNP[3,3,2]-8m_{c}^{2}FlNP[4,2,2]+FlNP[5,2,2]\Bigg{)}\psi^{a}(u_{0})$
$\displaystyle+\frac{5m_{c}m_{0}^{2}\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle^{2}}{995328M^{10}\pi^{2}}(e_{u}-e_{d})\Bigg{(}16m_{c}^{4}FlNP[4,3,3]-8m_{c}^{2}FlNP[5,3,3]+FlNP[6,3,3]\Bigg{)}A(u_{0})$
$\displaystyle+\frac{m_{c}m_{0}^{2}\langle\bar{q}q\rangle}{9216M^{8}\pi^{2}}\Bigg{[}-4(e_{u}-e_{d})m_{c}\psi^{a}(u_{0})-\langle\bar{q}q\rangle\Bigg{\\{}e_{u}\Big{(}4WF[\mathcal{T}_{1},\bar{v}]+WF[\mathcal{T}_{2},\bar{v}]\Big{)}+e_{d}\Big{(}4WF[\mathcal{T}_{1},v]+WF[\mathcal{T}_{2},v]\Big{)}\Big{\\}}\Bigg{]}$
$\displaystyle\Bigg{(}64m_{c}^{6}FlNP[1,4,2]-48m_{c}^{4}FlNP[2,4,2]+12m_{c}^{2}FlNP[3,4,2]-FlNP[4,4,2]\Bigg{)}$
$\displaystyle+\frac{m_{c}\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle^{2}}{82944M^{8}\pi^{2}}\Bigg{[}(e_{u}+e_{d})(3m_{0}^{2}-2M^{2})A(u_{0})-m_{0}^{2}\Big{(}2M^{2}\chi\varphi_{\gamma}(u_{0})+WF[h_{\gamma},u]\Big{)}\Bigg{]}\Bigg{(}4m_{c}^{2}FlNP[2,1,1]$
$\displaystyle-FlNP[3,1,1]\Bigg{)}$
$\displaystyle+\frac{m_{c}^{5}\langle\bar{q}q\rangle}{124416M^{10}\pi^{4}}\Bigg{[}\Bigg{\\{}e_{u}\Bigg{(}-3456\pi^{2}m_{c}^{2}M^{2}\langle\bar{q}q\rangle(3m_{0}^{2}-2M^{2})+\langle
g_{s}^{2}G^{2}\rangle\Big{(}-69m_{c}M^{4}+20\pi^{2}\langle\bar{q}q\rangle(16m_{0}^{2}-11M^{2})\Big{)}\Bigg{)}$
$\displaystyle+e_{d}\Bigg{(}3456\pi^{2}m_{c}^{2}M^{2}\langle\bar{q}q\rangle(3m_{0}^{2}-2M^{2})+5\langle
g_{s}^{2}G^{2}\rangle\Big{(}9m_{c}M^{4}+4\pi^{2}\langle\bar{q}q\rangle(-16m_{0}^{2}+11M^{2})\Big{)}\Bigg{)}\Bigg{\\}}A(u_{0})$
$\displaystyle+4\pi^{2}\langle\bar{q}q\rangle\Bigg{\\{}-(e_{u}-e_{d})M^{2}\chi\Big{(}5\langle
g_{s}^{2}G^{2}\rangle(11m_{0}^{2}-8M^{2})-1728m_{c}^{2}m_{0}^{2}M^{2}\Big{)}\varphi_{\gamma}(u_{0})+4e_{u}\Big{(}216m_{c}^{2}m_{0}^{2}M^{2}$
$\displaystyle+7\langle
g_{s}^{2}G^{2}\rangle(-2m_{0}^{2}+M^{2})\Big{)}+4e_{d}\Big{(}-216m_{c}^{2}m_{0}^{2}M^{2}+\langle
g_{s}^{2}G^{2}\rangle(-4m_{0}^{2}+2M^{2})\Big{)}\Bigg{\\}}WF[h_{\gamma},u]\Bigg{]}FlNP[0,3,1]$

$\displaystyle+\frac{\langle\bar{q}q\rangle}{663552M^{8}\pi^{2}}\Bigg{[}9(e_{u}-e_{d})f_{3\gamma}\langle
g_{s}^{2}G^{2}\rangle(5m_{0}^{2}-4M^{2})\psi^{a}(u_{0})+2(4e_{u}-e_{d})m_{0}^{2}f_{3\gamma}\langle
g_{s}^{2}G^{2}\rangle WF[\psi^{\nu},u]$
$\displaystyle+(e_{u}-e_{d})m_{0}^{2}f_{3\gamma}\langle
g_{s}^{2}G^{2}\rangle\psi^{\nu}(u_{0})+m_{c}m_{0}^{2}M^{2}\langle\bar{q}q\rangle\Bigg{\\{}e_{u}\Big{(}3WF[\mathcal{S},\bar{v}]-2WF[\mathcal{T}_{1},\bar{v}]-2WF[\mathcal{T}_{2},\bar{v}]+2WF[\mathcal{T}_{3},\bar{v}]$
$\displaystyle+2WF[\mathcal{T}_{4},\bar{v}]-2WF[\tilde{S},\bar{v}]\Big{)}+e_{d}\Big{(}3WF[\mathcal{S},v]-2WF[\mathcal{T}_{1},v]-2WF[\mathcal{T}_{2},v]+2WF[\mathcal{T}_{3},v]+2WF[\mathcal{T}_{4},v]$
$\displaystyle-2WF[\tilde{S},v]\Big{)}\Bigg{\\}}\Bigg{]}\Bigg{(}16m_{c}^{4}FlNP[1,2,1]-8m_{c}^{2}FlNP[2,2,1]+FlNP[3,2,1]\Bigg{)}$
$\displaystyle+\frac{1}{1327104M^{8}\pi^{4}}\Bigg{[}27(e_{u}-e_{d})M^{4}\langle
g_{s}^{2}G^{2}\rangle\langle g\bar{q}q\rangle
A(u_{0})+3(e_{u}-e_{d})f_{3\gamma}\langle
g_{s}^{2}G^{2}\rangle\Big{(}5m_{c}M^{4}+6\pi^{2}\langle\bar{q}q\rangle(-m_{0}^{2}+4M^{2})\Big{)}\psi^{a}(u_{0})$
$\displaystyle-\pi^{2}f_{3\gamma}\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle(3m_{0}^{2}-4M^{2})\Big{(}2(e_{u}-4e_{d})\psi^{\nu}(u_{0})+4(4e_{u}-e_{d})WF[\psi^{\nu},u]\Big{)}$
$\displaystyle+\langle\bar{q}q\rangle\Bigg{\\{}\Big{(}-23M^{2}\langle
g_{s}^{2}G^{2}\rangle-864\pi^{2}m_{c}\langle\bar{q}q\rangle(m_{0}^{2}-2M^{2})\Big{)}\Big{(}e_{u}WF[\mathcal{S},\bar{v}]+e_{d}WF[\mathcal{S},v]\Big{)}$
$\displaystyle+\Big{(}17M^{2}\langle
g_{s}^{2}G^{2}\rangle+288\pi^{2}m_{c}\langle\bar{q}q\rangle(m_{0}^{2}-2M^{2})\Big{)}\Big{(}e_{u}WF[\mathcal{T}_{1},\bar{v}]+e_{d}WF[\mathcal{T}_{1},v]\Big{)}$
$\displaystyle+\Big{(}102M^{2}\langle
g_{s}^{2}G^{2}\rangle+1728\pi^{2}m_{c}\langle\bar{q}q\rangle(m_{0}^{2}-2M^{2})\Big{)}\Big{(}e_{u}WF[\mathcal{T}_{2},\bar{v}]+e_{d}WF[\mathcal{T}_{2},v]\Big{)}$
$\displaystyle-\Big{(}36M^{2}\langle
g_{s}^{2}G^{2}\rangle+1728\pi^{2}m_{c}\langle\bar{q}q\rangle(m_{0}^{2}-2M^{2})\Big{)}\Big{(}e_{u}WF[\mathcal{T}_{3},\bar{v}]+e_{d}WF[\mathcal{T}_{3},v]\Big{)}$
$\displaystyle-\Big{(}36M^{2}\langle
g_{s}^{2}G^{2}\rangle+1728\pi^{2}m_{c}\langle\bar{q}q\rangle(m_{0}^{2}-2M^{2})\Big{)}\Big{(}e_{u}WF[\tilde{S},\bar{v}]+e_{d}WF[\tilde{S},v]\Big{)}\Bigg{\\}}\Bigg{]}\Bigg{(}16m_{c}^{4}FlNP[1,2,0]$
$\displaystyle-8m_{c}^{2}FlNP[0,2,0]+FlNP[-1,2,0]\Bigg{)}$
$\displaystyle+\frac{m_{c}}{15925248M^{12}\pi^{4}}\Bigg{[}10368(e_{d}-e_{u})m_{c}\langle\bar{q}q\rangle
M^{8}A(u_{0})+48(e_{d}-e_{u})f_{3\gamma}M^{4}\Big{(}7\langle
g_{s}^{2}G^{2}\rangle M^{2}$
$\displaystyle+144~{}m_{c}~{}\pi^{2}\langle\bar{q}q\rangle(4M^{2}-3m_{0}^{2})\Big{)}\psi^{a}(u_{0})-288(e_{u}-e_{d})f_{3\gamma}M^{4}\Big{(}\langle
g_{s}^{2}G^{2}\rangle
M^{2}+24~{}m_{c}~{}\pi^{2}\langle\bar{q}q\rangle(4M^{2}-3m_{0}^{2})\Big{)}\psi^{\nu}(u_{0})$
$\displaystyle+2592m_{c}~{}M^{8}\langle\bar{q}q\rangle\Big{(}e_{d}WF[\mathcal{S},v]+e_{u}WF[\mathcal{S},\bar{v}]\Big{)}-e_{d}\langle\bar{q}q\rangle
M^{4}\Big{(}10368~{}m_{c}~{}M^{4}+33696\pi^{2}\langle\bar{q}q\rangle
m_{0}^{2}$ $\displaystyle+44928\pi^{2}\langle\bar{q}q\rangle
M^{2}\Big{)}WF[\mathcal{T}_{1},v]-e_{u}\langle\bar{q}q\rangle
M^{4}\Big{(}2592~{}m_{c}~{}M^{4}+10368\pi^{2}\langle\bar{q}q\rangle
m_{0}^{2}-13824\pi^{2}\langle\bar{q}q\rangle
M^{2}\Big{)}WF[\mathcal{T}_{1},\bar{v}]$ $\displaystyle-
e_{d}\langle\bar{q}q\rangle
M^{4}\Big{(}2592~{}m_{c}~{}M^{4}+18144\pi^{2}\langle\bar{q}q\rangle
m_{0}^{2}+24192\pi^{2}\langle\bar{q}q\rangle
M^{2}\Big{)}WF[\mathcal{T}_{2},v]-e_{u}\langle\bar{q}q\rangle
M^{4}\Big{(}2592~{}m_{c}~{}M^{4}$
$\displaystyle+10368\pi^{2}\langle\bar{q}q\rangle
m_{0}^{2}-13824\pi^{2}\langle\bar{q}q\rangle
M^{2}\Big{)}WF[\mathcal{T}_{2},\bar{v}]-4320e_{d}\pi^{2}\langle\bar{q}q\rangle^{2}M^{4}(3m_{0}^{2}-4M^{2})WF[\tilde{S},v]$
$\displaystyle-1728e_{u}\pi^{2}\langle\bar{q}q\rangle^{2}M^{4}(3m_{0}^{2}-4M^{2})WF[\tilde{S},\bar{v}]+576(e_{d}-e_{u})f_{3\gamma}\langle
g_{s}^{2}G^{2}\rangle M^{6}WF[\psi^{\nu},u]$
$\displaystyle+13824(e_{u}-e_{d})\pi^{2}f_{3\gamma}\langle\bar{q}q\rangle
m_{c}~{}M^{4}(3m_{0}^{2}-4M^{2})WF[\psi^{\nu},u]$
$\displaystyle+(39e_{u}-38e_{d})\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle
m_{c}~{}M^{4}WFD[\mathbb{A},u]+8(4e_{u}-e_{d})\pi^{2}\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle^{2}(5m_{0}^{2}-4M^{2})WFD[\mathbb{A},u]$
$\displaystyle-e_{d}f_{3\gamma}M^{4}\Big{(}204\langle g_{s}^{2}G^{2}\rangle
M^{2}-2592\pi^{2}\langle\bar{q}q\rangle
m_{c}~{}m_{0}^{2}+3456\pi^{2}\langle\bar{q}q\rangle
m_{c}~{}M^{2}\Big{)}WFD[\mathcal{A},v]-e_{u}f_{3\gamma}M^{4}\Big{(}204\langle
g_{s}^{2}G^{2}\rangle M^{2}$ $\displaystyle-2592\pi^{2}\langle\bar{q}q\rangle
m_{c}~{}m_{0}^{2}+3456\pi^{2}\langle\bar{q}q\rangle
m_{c}~{}M^{2}\Big{)}WFD[\mathcal{A},\bar{v}]-e_{d}f_{3\gamma}M^{4}\Big{(}138\langle
g_{s}^{2}G^{2}\rangle M^{2}+2592\pi^{2}\langle\bar{q}q\rangle
m_{c}~{}m_{0}^{2}$ $\displaystyle-3456\pi^{2}\langle\bar{q}q\rangle
m_{c}~{}M^{2}\Big{)}WFD[\mathcal{V},v]-e_{u}f_{3\gamma}M^{4}\Big{(}138\langle
g_{s}^{2}G^{2}\rangle M^{2}+2592\pi^{2}\langle\bar{q}q\rangle
m_{c}~{}m_{0}^{2}$ $\displaystyle-3456\pi^{2}\langle\bar{q}q\rangle
m_{c}~{}M^{2}\Big{)}WFD[\mathcal{V},\bar{v}]+32(e_{d}-e_{u})\pi^{2}\chi\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle^{2}M^{2}(m_{0}^{2}-M^{2})WFD[\varphi_{\gamma},u]\Bigg{]}$
$\displaystyle\Big{(}FlNP[0,4,0]-8m_{c}^{2}FlNP[1,4,0]+16m_{c}^{4}FlNP[2,4,0]\Big{)}$

$\displaystyle+\frac{m_{c}\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle}{82944M^{8}\pi^{2}}\Bigg{[}(e_{u}+e_{d})\Big{(}(3m_{0}^{2}-4M^{2})A(u_{0})+4M^{2}\chi(-m_{0}^{2}+2M^{2})\varphi_{\gamma}(u_{0})+(-3m_{0}^{2}+4M^{2})WF[h_{\gamma},u]\Big{)}\Bigg{]}$
$\displaystyle\Bigg{(}-4m_{c}^{2}FlNP[-1,1,0]+FlNP[0,1,0]\Bigg{)}$
$\displaystyle+\frac{m_{c}m_{0}^{2}\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle^{2}}{7962624M^{12}\pi^{4}}\Bigg{[}-(4e_{u}-e_{d})WFD[\mathcal{A},u]\Bigg{]}\Bigg{(}16m_{c}^{4}FlNP[2,4,2]-8m_{c}^{2}FlNP[3,4,2]+FlNP[4,4,2]\Bigg{)}$
$\displaystyle+\frac{m_{c}^{3}f_{3\gamma}}{92160M^{8}\pi^{4}}\Bigg{[}e_{u}WFD[\mathcal{V},\bar{v}]+e_{d}WF[\mathcal{V},v]\Bigg{]}\Bigg{(}64m_{c}^{6}FlNP[4,6,0]-48m_{c}^{4}FlNP[3,6,0]+12m_{c}^{2}FlNP[2,6,0]$
$\displaystyle-FlNP[1,6,0]\Bigg{)}$
$\displaystyle-\frac{f_{3\gamma}m_{0}^{2}\langle\bar{q}q\rangle}{73728M^{8}\pi^{2}}\Bigg{[}e_{d}WFD[\mathcal{V},v]+e_{u}WFD[\mathcal{V},\bar{v}]\Bigg{]}FlNP[3,4,1]$
$\displaystyle+\frac{m_{c}}{3981312M^{12}\pi^{4}}\Bigg{[}-12(e_{d}-e_{u})f_{3\gamma}M^{4}\Big{(}7\langle
g_{s}^{2}G^{2}\rangle+144\pi^{2}\langle\bar{q}q\rangle
m_{c}(4M^{2}-5m_{0}^{2})\Big{)}\psi^{a}(u_{0})$
$\displaystyle+\langle\bar{q}q\rangle\pi^{2}\Bigg{(}1728(e_{d}-e_{u})f_{3\gamma}m_{0}^{2}m_{c}~{}M^{4}\psi^{\nu}(u_{0})+216e_{d}M^{4}\langle\bar{q}q\rangle(45m_{0}^{2}-32M^{2})WF[\mathcal{T}_{1},v]$
$\displaystyle+864e_{u}M^{4}\langle\bar{q}q\rangle(3m_{0}^{2}-2M^{2})WF[\mathcal{T}_{1},\bar{v}]+216e_{d}M^{4}\langle\bar{q}q\rangle(15m_{0}^{2}-8M^{2})WF[\mathcal{T}_{2},v]$
$\displaystyle+864e_{u}M^{4}\langle\bar{q}q\rangle(3m_{0}^{2}-2M^{2})WF[\mathcal{T}_{2},\bar{v}]+216m_{0}^{2}864M^{4}\langle\bar{q}q\rangle\Big{(}5e_{d}WF[\tilde{S},v]+2e_{u}WF[\tilde{S},\bar{v}]\Big{)}$
$\displaystyle+3456(e_{d}-e_{u})f_{3\gamma}m_{0}^{2}m_{c}~{}M^{4}WF[\psi^{\nu},u]+(4e_{u}-e_{d})\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle(2m_{0}^{2}-5M^{2})WFD[\mathbb{A},u]$
$\displaystyle-216~{}e_{d}f_{3\gamma}m_{0}^{2}m_{c}~{}M^{4}\Big{(}WFD[\mathcal{A},v]-WFD[\mathcal{V},v]\Big{)}-216~{}e_{u}f_{3\gamma}m_{0}^{2}m_{c}~{}M^{4}\Big{(}WFD[\mathcal{A},\bar{v}]-WFD[\mathcal{V},\bar{v}]\Big{)}$
$\displaystyle-2(4e_{u}-e_{d})\chi\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle
m_{0}^{2}M^{2}WFD[\varphi_{\gamma},u]\Bigg{)}\Bigg{]}FlNP[2,4,1]$
$\displaystyle+\frac{m_{c}^{3}}{497664M^{12}\pi^{4}}\Bigg{[}18(e_{d}-e_{u})\pi^{2}f_{3\gamma}\langle\bar{q}q\rangle
M^{4}(4M^{2}-5m_{0}^{2})\psi^{a}(u_{0})+\langle\bar{q}q\rangle\pi^{2}\Bigg{(}2592(e_{u}-e_{d})f_{3\gamma}m_{0}^{2}M^{4}\psi^{\nu}(u_{0})$
$\displaystyle-324e_{d}\langle\bar{q}q\rangle
M^{4}(45m_{0}^{2}-32M^{2})WF[\mathcal{T}_{1},v]-324e_{u}\langle\bar{q}q\rangle
M^{4}(12m_{0}^{2}-8M^{2})WF[\mathcal{T}_{1},\bar{v}]$
$\displaystyle-324e_{d}\langle\bar{q}q\rangle
M^{4}(15m_{0}^{2}-8M^{2})WF[\mathcal{T}_{2},v]-324e_{u}\langle\bar{q}q\rangle
M^{4}(12m_{0}^{2}-8M^{2})WF[\mathcal{T}_{2},\bar{v}]$
$\displaystyle-324e_{u}\langle\bar{q}q\rangle
M^{4}\Big{(}5~{}e_{d}m_{0}^{2}WF[\tilde{S},v]-2~{}e_{u}~{}M^{2}WF[\tilde{S},\bar{v}]\Big{)}+5184(e_{u}-e_{d})f_{3\gamma}m_{c}~{}m_{0}^{2}M^{4}WFD[\psi^{\nu},u]$
$\displaystyle+(4e_{u}-e_{d})\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle(5m_{0}^{2}-2M^{2})WFD[\mathbb{A},u]+108e_{d}f_{3\gamma}m_{0}^{2}M^{2}m_{c}\Big{(}2WFD[\mathbb{A},v]+WFD[\mathcal{V},v]\Big{)}$
$\displaystyle+108e_{u}f_{3\gamma}m_{0}^{2}M^{2}m_{c}\Big{(}2WFD[\mathbb{A},\bar{v}]+WFD[\mathcal{V},\bar{v}]\Big{)}-2(4e_{u}-e_{d})\chi\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle
m_{0}^{2}M^{2}WFD[\varphi_{\gamma},u]\Bigg{)}\Bigg{]}$ $\displaystyle
FlNP[1,4,1]$
$\displaystyle+\frac{m_{c}^{5}}{248832M^{12}\pi^{4}}\Bigg{[}-36(e_{u}-e_{d})f_{3\gamma}M^{4}\Big{(}7\langle
g_{s}^{2}G^{2}\rangle
M^{2}+144~{}m_{c}~{}\pi^{2}\langle\bar{q}q\rangle(4M^{2}-5m_{0}^{2})\Big{)}\psi^{a}(u_{0})$
$\displaystyle+\pi^{2}\langle\bar{q}q\rangle\Bigg{(}5184(e_{d}-e_{u})f_{3\gamma}m_{0}^{2}m_{c}~{}M^{2}\psi^{\nu}(u_{0})+648~{}e_{d}\langle\bar{q}q\rangle
M^{4}(45m_{0}^{2}-32M^{2})WF[\mathcal{T}_{1},v]$
$\displaystyle+2592~{}e_{u}\langle\bar{q}q\rangle
M^{4}(3m_{0}^{2}-2M^{2})WF[\mathcal{T}_{1},\bar{v}]+2592~{}e_{d}\langle\bar{q}q\rangle
M^{4}(4m_{0}^{2}-2M^{2})WF[\mathcal{T}_{2},v]$
$\displaystyle+2592~{}e_{u}\langle\bar{q}q\rangle
M^{4}(3m_{0}^{2}-2M^{2})WF[\mathcal{T}_{2},\bar{v}]+648\langle\bar{q}q\rangle
M^{4}\Big{(}5~{}e_{d}~{}m_{0}^{2}WF[\tilde{S},v]+2~{}e_{u}~{}M^{2}WF[\tilde{S},\bar{v}]\Big{)}$
$\displaystyle+(4e_{u}-e_{d})\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle(2M^{2}-5m_{0}^{2})WFD[\mathbb{A},u]-216e_{d}f_{3\gamma}m_{0}^{2}M^{4}m_{c}\Big{(}WFD[\mathbb{A},v]+WFD[\mathcal{V},v]\Big{)}$
$\displaystyle-216e_{u}f_{3\gamma}m_{0}^{2}M^{4}m_{c}\Big{(}WFD[\mathbb{A},\bar{v}]+WFD[\mathcal{V},\bar{v}]\Big{)}+2(4e_{u}-e_{d})\chi\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle
m_{0}^{2}M^{2}WFD[\varphi_{\gamma},u]\Bigg{)}\Bigg{]}FlNP[0,4,1]$

$\displaystyle+\frac{\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle}{1327104M^{10}\pi^{4}}\Bigg{[}2(e_{u}+e_{d})M^{4}A(u_{0})+3(e_{u}-e_{d})\pi^{2}m_{0}^{2}f_{3\gamma}WFD[\psi^{a},u]\Bigg{]}FlNP[3,3,1]$
$\displaystyle+\frac{m_{c}\langle\bar{q}q\rangle}{1990656M^{10}\pi^{4}}\Bigg{[}(e_{u}-e_{d})\Bigg{(}-3456\pi^{2}m_{c}^{2}M^{2}\langle\bar{q}q\rangle(3m_{0}^{2}-2M^{2})+\langle
g_{s}^{2}G^{2}\rangle\Big{(}-21m_{c}M^{4}$
$\displaystyle+20\pi^{2}\langle\bar{q}q\rangle(16m_{0}^{2}-11M^{2})\Big{)}\Bigg{)}A(u_{0})+4\pi^{2}\Bigg{\\{}(e_{d}-e_{u})\chi
M^{2}\langle\bar{q}q\rangle\Big{(}5\langle
g_{s}^{2}G^{2}\rangle(11m_{0}^{2}-8M^{2})+1728m_{0}^{2}m_{c}^{2}M^{2}\Big{)}\varphi_{\gamma}(u_{0})$
$\displaystyle+4(e_{u}+e_{d})\langle\bar{q}q\rangle\Big{(}7\langle
g_{s}^{2}G^{2}\rangle(2m_{0}^{2}-M^{2})-216m_{c}^{2}m_{0}^{2}M^{2}\Big{)}WF[\varphi_{\gamma},u]+9(e_{d}-e_{u})m_{c}m_{0}^{2}f_{3\gamma}\langle
g_{s}^{2}G^{2}\rangle WF[\psi^{a},u]\Bigg{\\}}\Bigg{]}$ $\displaystyle
FlNP[2,3,1]$
$\displaystyle+\frac{m_{c}^{3}\langle\bar{q}q\rangle}{248832M^{10}\pi^{4}}\Bigg{[}\Bigg{(}3(25e_{u}-13e_{d})m_{c}~{}M^{4}\langle
g_{s}^{2}G^{2}\rangle-4(e_{u}-e_{d})\pi^{2}\langle\bar{q}q\rangle\Big{(}-864~{}\pi^{2}m_{c}^{2}\langle\bar{q}q\rangle(3m_{0}^{2}-2M^{2})$
$\displaystyle-\langle
g_{s}^{2}G^{2}\rangle(80m_{0}^{2}+55M^{2})\Big{)}\Bigg{)}A(u_{0})+4(e_{d}-e_{u})\pi^{2}\chi\langle\bar{q}q\rangle
M^{2}\Big{(}5\langle
g_{s}^{2}G^{2}\rangle(11m_{0}^{2}-8M^{2})+1728m_{0}^{2}m_{c}^{2}M^{2}\Big{)}\varphi_{\gamma}$
$\displaystyle+4\pi^{2}\langle\bar{q}q\rangle\Big{(}(2e_{d}-7e_{u})\langle
g_{s}^{2}G^{2}\rangle(M^{2}-2m_{0}^{2})+216(e_{d}-e_{u})m_{0}^{2}m_{c}^{2}M^{2}\Big{)}WF[h_{\gamma},u]$
$\displaystyle+9(e_{d}-e_{u})\pi^{2}f_{3\gamma}\langle g_{s}^{2}G^{2}\rangle
m_{0}^{2}m_{c}~{}WF[\psi^{a},u]\Bigg{]}FlNP[1,3,1]$
$\displaystyle+\frac{m_{c}^{5}}{82944M^{10}\pi^{4}}\Bigg{[}\langle\bar{q}q\rangle\Bigg{\\{}e_{u}\Bigg{(}2304\pi^{2}m_{c}^{2}M^{2}\langle\bar{q}q\rangle(-3m_{0}^{2}+4M^{2})+\langle
g_{s}^{2}G^{2}\rangle\Big{(}-73m_{c}M^{4}+120\pi^{2}\langle\bar{q}q\rangle(m_{0}^{2}-M^{2})\Big{)}\Bigg{)}$
$\displaystyle+e_{d}\Bigg{(}2304\pi^{2}m_{c}^{2}M^{2}\langle\bar{q}q\rangle(3m_{0}^{2}-4M^{2})+\langle
g_{s}^{2}G^{2}\rangle\Big{(}99m_{c}M^{4}+120\pi^{2}\langle\bar{q}q\rangle(m_{0}^{2}M^{2})\Big{)}\Bigg{)}\Bigg{\\}}A(u_{0})$
$\displaystyle-8M^{2}\chi\langle\bar{q}q\rangle\Bigg{\\{}e_{u}\Bigg{(}1152\pi^{2}m_{c}^{2}M^{2}\langle\bar{q}q\rangle(m_{0}^{2}-2M^{2})+\langle
g_{s}^{2}G^{2}\rangle\Big{(}23m_{c}M^{4}+5\pi^{2}\langle\bar{q}q\rangle(-3m_{0}^{2}+4M^{2})\Big{)}\Bigg{)}$
$\displaystyle+e_{d}\Bigg{(}-1152\pi^{2}m_{c}^{2}M^{2}\langle\bar{q}q\rangle(m_{0}^{2}-M^{2})+5\langle
g_{s}^{2}G^{2}\rangle\Big{(}-3m_{c}M^{4}+\pi^{2}\langle\bar{q}q\rangle(3m_{0}^{2}-4M^{2})\Big{)}\Bigg{)}\Bigg{\\}}\varphi_{\gamma}(u_{0})$
$\displaystyle+\langle\bar{q}q\rangle\Bigg{\\{}e_{u}\Bigg{(}576\pi^{2}m_{c}^{2}M^{2}\langle\bar{q}q\rangle(3m_{0}^{2}-4M^{2})+\langle
g_{s}^{2}G^{2}\rangle\Big{(}17m_{c}M^{4}+56\pi^{2}\langle\bar{q}q\rangle(-m_{0}^{2}+M^{2})\Big{)}\Bigg{)}$
$\displaystyle+e_{d}\Bigg{(}288\pi^{2}m_{c}^{2}M^{2}\langle\bar{q}q\rangle(-3m_{0}^{2}+4M^{2})+\langle
g_{s}^{2}G^{2}\rangle\Big{(}-13m_{c}M^{4}+8\pi^{2}\langle\bar{q}q\rangle(-m_{0}^{2}+M^{2})\Big{)}\Bigg{)}\Bigg{\\}}WF[h_{\gamma},u]$
$\displaystyle+4(e_{u}-e_{d})M^{6}f_{3\gamma}\langle
g_{s}^{2}G^{2}\rangle\psi^{a}(u_{0})\Bigg{]}FlNP[2,3,0]$

$\displaystyle+\frac{m_{c}^{3}}{165888M^{10}\pi^{4}}\Bigg{[}3\langle\bar{q}q\rangle\Bigg{\\{}e_{u}\Bigg{(}768\pi^{2}m_{c}^{2}M^{2}\langle\bar{q}q\rangle(-3m_{0}^{2}+4M^{2})+\langle
g_{s}^{2}G^{2}\rangle\Big{(}23m_{c}M^{4}+40\pi^{2}\langle\bar{q}q\rangle(m_{0}^{2}-M^{2})\Big{)}\Bigg{)}$
$\displaystyle+e_{d}\Bigg{(}768\pi^{2}m_{c}^{2}M^{2}\langle\bar{q}q\rangle(3m_{0}^{2}-4M^{2})+\langle
g_{s}^{2}G^{2}\rangle\Big{(}31m_{c}M^{4}+40\pi^{2}\langle\bar{q}q\rangle(m_{0}^{2}-M^{2})\Big{)}\Bigg{)}\Bigg{\\}}A(u_{0})$
$\displaystyle-8M^{2}\chi\langle\bar{q}q\rangle\Bigg{\\{}e_{u}\Bigg{(}1152\pi^{2}m_{c}^{2}M^{2}\langle\bar{q}q\rangle(m_{0}^{2}-2M^{2})+\langle
g_{s}^{2}G^{2}\rangle\Big{(}25m_{c}M^{4}+5\pi^{2}\langle\bar{q}q\rangle(-3m_{0}^{2}+4M^{2})\Big{)}\Bigg{)}$
$\displaystyle+e_{d}\Bigg{(}-1152\pi^{2}m_{c}^{2}M^{2}\langle\bar{q}q\rangle(m_{0}^{2}-M^{2})+\langle
g_{s}^{2}G^{2}\rangle\Big{(}-13m_{c}M^{4}+5\pi^{2}\langle\bar{q}q\rangle(3m_{0}^{2}-4M^{2})\Big{)}\Bigg{)}\Bigg{\\}}\varphi_{\gamma}(u_{0})$
$\displaystyle+4\langle\bar{q}q\rangle\Bigg{\\{}e_{u}\Bigg{(}576\pi^{2}m_{c}^{2}M^{2}\langle\bar{q}q\rangle(3m_{0}^{2}-4M^{2})+\langle
g_{s}^{2}G^{2}\rangle\Big{(}17m_{c}M^{4}+56\pi^{2}\langle\bar{q}q\rangle(-m_{0}^{2}+M^{2})\Big{)}\Bigg{)}$
$\displaystyle+e_{d}\Bigg{(}288\pi^{2}m_{c}^{2}M^{2}\langle\bar{q}q\rangle(-3m_{0}^{2}+4M^{2})+\langle
g_{s}^{2}G^{2}\rangle\Big{(}-13m_{c}M^{4}+8\pi^{2}\langle\bar{q}q\rangle(-m_{0}^{2}+M^{2})\Big{)}\Bigg{)}\Bigg{\\}}WF[h_{\gamma},u]$
$\displaystyle-32(e_{u}-e_{d})M^{6}f_{3\gamma}\langle
g_{s}^{2}G^{2}\rangle\psi^{a}(u_{0})\Bigg{]}FlNP[1,3,0]$
$\displaystyle+\frac{m_{c}}{1327104M^{10}\pi^{4}}\Bigg{[}3\langle\bar{q}q\rangle\Bigg{\\{}e_{u}\Bigg{(}768\pi^{2}m_{c}^{2}M^{2}\langle\bar{q}q\rangle(-3m_{0}^{2}+4M^{2})+\langle
g_{s}^{2}G^{2}\rangle\Big{(}35m_{c}M^{4}+40\pi^{2}\langle\bar{q}q\rangle(m_{0}^{2}-M^{2})\Big{)}\Bigg{)}$
$\displaystyle+e_{d}\Bigg{(}-768\pi^{2}m_{c}^{2}M^{2}\langle\bar{q}q\rangle(3m_{0}^{2}-4M^{2})+\langle
g_{s}^{2}G^{2}\rangle\Big{(}-19m_{c}M^{4}+40\pi^{2}\langle\bar{q}q\rangle(m_{0}^{2}-M^{2})\Big{)}\Bigg{)}\Bigg{\\}}A(u_{0})$
$\displaystyle-8M^{2}\chi\langle\bar{q}q\rangle\Bigg{\\{}e_{u}\Bigg{(}1152\pi^{2}m_{c}^{2}M^{2}\langle\bar{q}q\rangle(m_{0}^{2}-2M^{2})+\langle
g_{s}^{2}G^{2}\rangle\Big{(}31m_{c}M^{4}+5\pi^{2}\langle\bar{q}q\rangle(-3m_{0}^{2}+4M^{2})\Big{)}\Bigg{)}$
$\displaystyle+e_{d}\Bigg{(}-1152\pi^{2}m_{c}^{2}M^{2}\langle\bar{q}q\rangle(m_{0}^{2}-M^{2})+\langle
g_{s}^{2}G^{2}\rangle\Big{(}-7m_{c}M^{4}+5\pi^{2}\langle\bar{q}q\rangle(3m_{0}^{2}-4M^{2})\Big{)}\Bigg{)}\Bigg{\\}}\varphi_{\gamma}(u_{0})$
$\displaystyle+4\langle\bar{q}q\rangle\Bigg{\\{}e_{u}\Bigg{(}576\pi^{2}m_{c}^{2}M^{2}\langle\bar{q}q\rangle(3m_{0}^{2}-4M^{2})+\langle
g_{s}^{2}G^{2}\rangle\Big{(}17m_{c}M^{4}+56\pi^{2}\langle\bar{q}q\rangle(-m_{0}^{2}+M^{2})\Big{)}\Bigg{)}$
$\displaystyle+e_{d}\Bigg{(}288\pi^{2}m_{c}^{2}M^{2}\langle\bar{q}q\rangle(-3m_{0}^{2}+4M^{2})+\langle
g_{s}^{2}G^{2}\rangle\Big{(}-13m_{c}M^{4}+8\pi^{2}\langle\bar{q}q\rangle(-m_{0}^{2}+M^{2})\Big{)}\Bigg{)}\Bigg{\\}}WF[h_{\gamma},u]$
$\displaystyle-24(e_{u}-e_{d})M^{6}f_{3\gamma}\langle
g_{s}^{2}G^{2}\rangle\psi^{a}(u_{0})\Bigg{]}FlNP[0,3,0]$
$\displaystyle+\frac{\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle}{663552M^{10}\pi^{4}}\Bigg{[}(e_{u}+e_{d})M^{4}\Big{(}3M^{2}\chi\varphi_{\gamma}(u_{o})-A(u_{0})\Big{)}-6(e_{u}-e_{d})\pi^{2}f_{3\gamma}(m_{0}^{2}-M^{2})WFD[\psi^{a},u]\Bigg{]}FlNP[-1,3,0]$
$\displaystyle+\frac{m_{c}^{3}}{3317760M^{10}\pi^{4}}\Bigg{[}-3456(e_{d}-e_{u})f_{3\gamma}M^{6}\Big{(}FlNP[1,5,0]-12~{}m_{c}^{2}FlNP[2,5,0]+48~{}m_{c}^{4}FlNP[3,5,0]\Big{)}\psi^{a}(u_{0})$
$\displaystyle+144\pi^{2}\langle\bar{q}q\rangle^{2}(m_{0}^{2}-M^{2})\Bigg{\\{}e_{u}\Big{(}-2WF[\mathcal{T}_{1},\bar{v}]-2WF[\mathcal{T}_{2},\bar{v}]+2WF[\tilde{S},\bar{v}]+WFD[\mathcal{T}_{1},\bar{v}]+WFD[\mathcal{T}_{2},\bar{v}]+WFD[\tilde{S},\bar{v}]\Big{)}$
$\displaystyle+e_{d}\Big{(}-8WF[\mathcal{T}_{1},v]-2WF[\mathcal{T}_{2},v]+2e_{d}WF[\tilde{S},v]+4WFD[\mathcal{T}_{1},v]+WFD[\mathcal{T}_{2},v]-e_{d}WFD[\tilde{S},v]\Big{)}\Bigg{\\}}$
$\displaystyle+(e_{u}-e_{d})f_{3\gamma}\Big{(}7\langle g_{s}^{2}G^{2}\rangle
M^{2}+576\langle\bar{q}q\rangle
m_{c}(M^{2}-m_{0}^{2})\Big{)}WFD[\psi^{a},u]\Bigg{]}\Bigg{(}FlNP[0,5,0]-12~{}m_{c}^{2}FlNP[1,5,0]$
$\displaystyle+48~{}m_{c}^{4}FlNP[2,5,0]-64~{}m_{c}^{6}FlNP[3,5,0]\Bigg{)}.$
(18)

$\displaystyle\Pi_{3}=0.$ (19)
$\displaystyle\Pi_{4}=-\frac{m_{c}^{3}M^{2}\chi\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle}{18432\pi^{4}}(e_{u}+e_{d})\varphi_{\gamma}(u_{0})N[1,3,2]$
$\displaystyle+\frac{5m_{c}^{2}f_{3\gamma}\langle
g_{s}^{2}G^{2}\rangle}{6912\pi^{4}}(e_{u}-e_{d})WF[\psi^{\nu},u]~{}N[1,1,0]$
$\displaystyle+\frac{f_{3\gamma}m_{c}^{4}}{16\pi^{4}}WF[\psi^{\nu},u]~{}N[3,3,1]$
$\displaystyle+\frac{f_{3\gamma}\langle g_{s}^{2}G^{2}\rangle
m_{c}^{2}M^{2}}{884736\pi^{4}}WF[\psi^{\nu},u]~{}N[1,1,1]$
$\displaystyle+\frac{m_{c}^{2}}{2304\pi^{4}}\Bigg{[}-(e_{u}+e_{d})\chi\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle\varphi_{\gamma}(u_{0})+18m_{c}M^{2}f_{3\gamma}\Big{(}2e_{u}WF[\mathcal{A},\bar{v}]+5e_{d}WF[\mathcal{A},v]\Big{)}\Bigg{]}N[1,2,0]$
$\displaystyle+\frac{m_{c}^{3}\langle\bar{q}q\rangle}{128\pi^{4}}\Bigg{[}e_{d}\Big{(}5WF[\mathcal{T}_{1},v]+5WF[\mathcal{T}_{2},v]+3WF[\tilde{S},v]\Big{)}+2e_{u}\Big{(}WF[\mathcal{T}_{1},\bar{v}]+WF[\mathcal{T}_{2},\bar{v}]\Big{)}\Bigg{]}$
$\displaystyle\Bigg{(}3~{}m_{c}~{}N[2,3,1]-4~{}N[1,3,0]+4~{}m_{c}~{}N[1,4,1]\Bigg{)}$
$\displaystyle+\frac{m_{c}^{3}}{2304\pi^{4}}\Bigg{[}(e_{u}+e_{d})\chi\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle\varphi_{\gamma}(u_{0})-9m_{c}M^{2}f_{3\gamma}\Big{(}(5~{}e_{d}WF[\mathcal{A},v]+2~{}e_{u}~{}WF[\mathcal{A},\bar{v}])\Big{)}$
$\displaystyle+9~{}M^{2}\langle\bar{q}q\rangle\Bigg{(}e_{d}\Big{(}5WF[\mathcal{T}_{1},v]+5WF[\mathcal{T}_{2},v]+3WF[\tilde{S},v]\Big{)}+2e_{u}\Big{(}WF[\mathcal{T}_{1},\bar{v}]+WF[\mathcal{T}_{2},\bar{v}]\Big{)}\Bigg{)}\Bigg{]}N[1,3,1]$
$\displaystyle-\frac{m_{c}^{4}M^{2}\langle\bar{q}q\rangle}{1024\pi^{4}}\Bigg{[}e_{d}\Big{(}5WF[\mathcal{T}_{1},v]+5WF[\mathcal{T}_{2},v]+3WF[\tilde{S},v]\Big{)}+2e_{u}\Big{(}WF[\mathcal{T}_{1},\bar{v}]+WF[\mathcal{T}_{2},\bar{v}]\Big{)}\Bigg{]}$
$\displaystyle\Bigg{(}4~{}N[1,4,2]+3~{}N[2,3,2]\Bigg{)}$
$\displaystyle-\frac{m_{c}^{2}\langle
g_{s}^{2}G^{2}\rangle}{221184\pi^{2}}\Bigg{[}\langle\bar{q}q\rangle\Bigg{(}6(e_{u}+e_{d})\Big{(}A(u_{0})+2\chi
M^{2}\varphi_{\gamma}(u_{0})\Big{)}+22e_{d}\Big{(}2WF[\mathcal{T}_{1},v]+5WF[\mathcal{T}_{2},v]-WF[\mathcal{T}_{3},v]$
$\displaystyle+2WF[\mathcal{T}_{4},v]\Big{)}+e_{u}\Big{(}3WF[\mathcal{S},\bar{v}]+44WF[\mathcal{T}_{1},\bar{v}]+113WF[\mathcal{T}_{2},\bar{v}]-25WF[\mathcal{T}_{3},\bar{v}]+44WF[\mathcal{T}_{4},\bar{v}]\Big{)}\Bigg{)}$
$\displaystyle-48(e_{u}+e_{d})\langle\bar{q}q\rangle
WF[f_{\gamma},u]-120(e_{u}-e_{d})f_{3\gamma}m_{c}~{}WF[\psi^{\nu},u]\Bigg{]}N[1,2,1]$
$\displaystyle-\frac{m_{c}^{2}M^{2}\langle
g_{s}^{2}G^{2}\rangle}{1769472\pi^{2}}\Bigg{[}6(e_{u}+e_{d})\langle\bar{q}q\rangle
A(u_{0})+22e_{d}\langle\bar{q}q\rangle\Big{(}2WF[\mathcal{T}_{1},v]+5WF[\mathcal{T}_{2},v]-WF[\mathcal{T}_{3},v]$
$\displaystyle+2WF[\mathcal{T}_{4},v]\Big{)}+e_{u}\langle\bar{q}q\rangle\Big{(}3WF[\mathcal{S},\bar{v}]-44WF[\mathcal{T}_{1},\bar{v}]+113WF[\mathcal{T}_{2},\bar{v}]-25WF[\mathcal{T}_{3},\bar{v}]+44WF[\mathcal{T}_{4},\bar{v}]\Big{)}\Bigg{)}$
$\displaystyle-48(e_{u}+e_{d})\langle\bar{q}q\rangle
WF[f_{\gamma},u]-280(e_{u}-e_{d})f_{3\gamma}m_{c}~{}WF[\psi^{\nu},u]\Bigg{]}N[1,2,2]$
$\displaystyle+\frac{11f_{3\gamma}\langle g_{s}^{2}G^{2}\rangle
m_{c}^{2}}{9216M^{2}\pi^{4}}\Bigg{[}e_{u}\Big{(}2WF[\mathcal{A},\bar{v}]-WFD[\mathcal{A},\bar{v}]\Big{)}+e_{d}\Big{(}2WF[\mathcal{A},v]-WFD[\mathcal{A},v]\Big{)}\Bigg{]}N[2,2,0]$
$\displaystyle+\frac{m_{c}^{2}}{110592\pi^{4}}\Bigg{[}36(e_{u}+e_{d})m_{c}\chi\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle\varphi_{\gamma}(u_{0})+f_{3\gamma}\Bigg{(}4e_{d}\Big{(}11\langle
g_{s}^{2}G^{2}\rangle+540m_{c}^{2}M^{2}\Big{)}WF[\mathcal{A},v]$
$\displaystyle+e_{u}\Big{(}44\langle
g_{s}^{2}G^{2}\rangle+864m_{c}^{2}M^{2}\Big{)}WF[\mathcal{A},\bar{v}]-33\langle
g_{s}^{2}G^{2}\rangle\Big{(}e_{d}WFD[\mathcal{A},v]+e_{u}WFD[\mathcal{A},\bar{v}]\Big{)}\Bigg{)}\Bigg{]}N[2,2,1]$

$\displaystyle+\frac{m_{c}^{2}M^{2}\langle
g_{s}^{2}G^{2}\rangle}{442368\pi^{4}}\Bigg{[}-18(e_{u}+e_{d})m_{c}\chi\langle\bar{q}q\rangle\varphi_{\gamma}(u_{0})+11f_{3\gamma}\Bigg{(}e_{d}\Big{(}WF[\mathcal{A},v]-WFD[\mathcal{A},v]\Big{)}+e_{u}\Big{(}2WF[\mathcal{A},\bar{v}]$
$\displaystyle-WFD[\mathcal{A},\bar{v}]\Big{)}\Bigg{)}\Bigg{]}N[2,2,2]$
$\displaystyle+\frac{11m_{c}^{2}M^{2}f_{3\gamma}\langle
g_{s}^{2}G^{2}\rangle}{3538944\pi^{4}}\Bigg{[}+e_{d}\Big{(}2WF[\mathcal{A},v]-WFD[\mathcal{A},v]\Big{)}+e_{u}\Big{(}WF[\mathcal{A},\bar{v}]+-WFD[\mathcal{A},\bar{v}]\Big{)}\Bigg{]}N[2,2,3]$
$\displaystyle-\frac{m_{c}m_{0}^{2}\langle\bar{q}q\rangle^{2}}{4608M^{10}\pi^{2}}\Bigg{[}5e_{d}\Big{(}WF[\mathcal{T}_{1},v]+WF[\mathcal{T}_{2},v]\Big{)}+2e_{u}\Big{(}WF[\mathcal{T}_{1},\bar{v}]+WF[\mathcal{T}_{2},\bar{v}]\Big{)}\Bigg{]}\Bigg{(}64~{}m_{c}^{6}FlNP[1,4,2]$
$\displaystyle-48~{}m_{c}^{4}FlNP[2,4,2]+12m_{c}^{2}FlNP[3,4,3]-FlNP[4,4,2]\Bigg{)}$
$\displaystyle-\frac{m_{c}m_{0}^{2}\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle^{2}}{331776M^{12}\pi^{2}}(e_{u}+e_{d})\Big{(}A(u_{0})-8WF[h_{\gamma},u]\Big{)}\Bigg{(}16~{}m_{c}^{4}FlNP[2,3,2]-8~{}m_{c}^{2}FlNP[3,3,2]$
$\displaystyle+FlNP[4,3,2]\Bigg{)}$
$\displaystyle+\frac{m_{c}\langle\bar{q}q\rangle}{165888M^{12}\pi^{2}}\Bigg{[}(e_{u}+e_{d})\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle(5m_{0}^{2}-2M^{2})A(u_{0})+2m_{0}^{2}M^{2}\Bigg{(}(e_{u}+e_{d})\chi\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle\varphi_{\gamma}(u_{0})$
$\displaystyle+36m_{c}M^{2}f_{3\gamma}\Big{(}e_{d}WF[\mathcal{A},v]-2e_{u}FW[\mathcal{A},\bar{v}]\Big{)}\Bigg{)}-8(e_{u}+e_{d})\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle(5m_{0}^{2}-2M^{2})WF[h_{\gamma},u]\Bigg{]}$
$\displaystyle\Bigg{(}16~{}m_{c}^{4}FlNP[0,3,1]-8~{}m_{c}^{2}FlNP[1,3,1]+FlNP[2,3,1]\Bigg{)}$
$\displaystyle+\frac{m_{c}}{663552M^{12}\pi^{4}}\Bigg{[}(e_{u}+e_{d})\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle\Big{(}-3m_{c}M^{4}-8\pi^{2}\langle\bar{q}q\rangle(5m_{0}^{2}-4M^{2})\Big{)}A(u_{0})$
$\displaystyle+8~{}M^{2}\Bigg{\\{}4(e_{u}+e_{d})\pi^{2}\chi\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle^{2}(m_{0}^{2}-M^{2})\varphi_{\gamma}(u_{0})+M^{2}f_{3\gamma}\Bigg{(}e_{d}\Big{(}17\langle
g_{s}^{2}G^{2}\rangle
M^{2}+36\pi^{2}m_{c}\langle\bar{q}q\rangle(3m_{0}^{2}-4M^{2})\Big{)}$
$\displaystyle WF[\mathcal{A},v]+e_{u}\Big{(}17\langle g_{s}^{2}G^{2}\rangle
M^{2}-72\pi^{2}m_{c}\langle\bar{q}q\rangle(3m_{0}^{2}-4M^{2})\Big{)}WF[\mathcal{A},\bar{v}]\Bigg{)}\Bigg{\\}}+8(e_{u}+e_{d})\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle\Big{(}3~{}m_{c}~{}M^{4}$
$\displaystyle+8\pi^{2}\langle\bar{q}q\rangle(5m_{0}^{2}-4M^{2})\Big{)}WF[h_{\gamma},u]\Bigg{]}\Bigg{(}16~{}m_{c}^{4}FlNP[2,3,0]-8~{}m_{c}^{2}FlNP[1,3,0]+FlNP[0,3,0]\Bigg{)}$
$\displaystyle-\frac{m_{0}^{2}\langle\bar{q}q\rangle}{165888M^{10}\pi^{2}}\Bigg{[}432m_{c}M^{2}\langle\bar{q}q\rangle\Bigg{(}e_{d}\Big{(}WF[\mathcal{T}_{1},v]+WF[\mathcal{T}_{2},v]-WF[\mathcal{T}_{3},v]-WF[\mathcal{T}_{4},v]\Big{)}+e_{u}\Big{(}WF[\mathcal{T}_{1},\bar{v}]$
$\displaystyle+WF[\mathcal{T}_{2},\bar{v}]-WF[\mathcal{T}_{3},\bar{v}]-WF[\mathcal{T}_{4},\bar{v}]\Big{)}-5(e_{u}-e_{d})f_{3\gamma}\langle
g_{s}^{2}G^{2}\rangle
WF[\psi^{\nu},u]\Bigg{]}\Bigg{(}16~{}m_{c}^{2}4FlNP[1,2,1]$
$\displaystyle+8~{}m_{c}^{2}FlNP[2,2,1]-FlNP[3,2,1]\Bigg{)}$

$\displaystyle+\frac{m_{c}\langle\bar{q}q\rangle}{9216M^{10}\pi^{2}}\Bigg{[}\langle\bar{q}q\rangle(17m_{0}^{2}-8M^{2})\Bigg{(}5e_{d}\Big{(}WF[\mathcal{T}_{1},v]+WF[\mathcal{T}_{2},v]\Big{)}+2e_{u}\Big{(}WF[\mathcal{T}_{1},\bar{v}]+WF[\mathcal{T}_{2},\bar{v}]\Big{)}\bigg{)}$
$\displaystyle+3e_{d}\langle\bar{q}q\rangle
m_{0}^{2}WF[\tilde{S},v]-32(e_{u}-e_{d})f_{3\gamma}m_{c}~{}m_{0}^{2}WF[\psi^{\nu},u]\bigg{]}\Bigg{(}64~{}m_{c}^{6}FlNP[-1,4,1]-48~{}m_{c}^{4}FlNP[0,4,1]$
$\displaystyle-12~{}m_{c}^{2}FLNP[1,4,1]-FlNP[2,4,1]\Bigg{)}$
$\displaystyle+\frac{m_{c}}{27648M^{10}\pi^{4}}\Bigg{[}\langle\bar{q}q\rangle\Bigg{(}3~{}m_{c}~{}M^{4}+28\pi^{2}\langle\bar{q}q\rangle(m_{0}^{2}-M^{2})\Bigg{)}\Bigg{(}15e_{d}\Big{(}WF[\mathcal{T}_{1},v]+WF[\mathcal{T}_{2},v]\Big{)}$
$\displaystyle+6e_{u}\Big{(}WF[\mathcal{T}_{1},\bar{v}]+WF[\mathcal{T}_{2},\bar{v}]\Big{)}\Bigg{)}+12e_{d}\pi^{2}\langle\bar{q}q\rangle^{2}(m_{0}^{2}-M^{2})WF[\tilde{S},v]+4(e_{u}-e_{d})f_{3\gamma}\Big{(}\langle
g_{s}^{2}G^{2}\rangle M^{2}$ $\displaystyle+96\pi^{2}\langle\bar{q}q\rangle
m_{c}(-m_{0}^{2}+M^{2})\Big{)}WF[\psi^{\nu},u]\Bigg{]}\Bigg{(}-64~{}m_{c}^{6}FlNP[3,4,0]+48~{}m_{c}^{4}FlNP[2,4,0]-12~{}m_{c}^{2}FlNP[1,4,0]$
$\displaystyle+FlNP[0,4,0]\Bigg{)}$
$\displaystyle-\frac{\langle\bar{q}q\rangle}{165888M^{10}\pi^{4}}\Bigg{[}3e_{d}M^{2}\Big{(}23\langle
g_{s}^{2}G^{2}\rangle M^{2}+288\pi^{2}\langle\bar{q}q\rangle
m_{c}(4M^{2}-3m_{0}^{2})\Big{)}WF[\mathcal{T}_{1},v]+6e_{u}\Big{(}17\langle
g_{s}^{2}G^{2}\rangle M^{2}$ $\displaystyle+144\pi^{2}\langle\bar{q}q\rangle
m_{c}(4M^{2}-3m_{0}^{2})\Big{)}WF[\mathcal{T}_{1},\bar{v}]+e_{d}M^{2}\Big{(}69\langle
g_{s}^{2}G^{2}\rangle M^{2}+864\pi^{2}\langle\bar{q}q\rangle
m_{c}(4M^{2}-3m_{0}^{2})\Big{)}WF[\mathcal{T}_{2},v]$
$\displaystyle+e_{u}\Big{(}102\langle g_{s}^{2}G^{2}\rangle
M^{2}+864\pi^{2}\langle\bar{q}q\rangle
m_{c}(4M^{2}-3m_{0}^{2})\Big{)}WF[\mathcal{T}_{2},\bar{v}]-e_{d}M^{2}\Big{(}36\langle
g_{s}^{2}G^{2}\rangle M^{2}$ $\displaystyle+864\pi^{2}\langle\bar{q}q\rangle
m_{c}(4M^{2}-3m_{0}^{2})\Big{)}WF[\mathcal{T}_{3},v]-e_{u}\Big{(}36\langle
g_{s}^{2}G^{2}\rangle M^{2}+864\pi^{2}\langle\bar{q}q\rangle
m_{c}(4M^{2}-3m_{0}^{2})\Big{)}WF[\mathcal{T}_{3},\bar{v}]$ $\displaystyle-
e_{d}M^{2}\Big{(}36\langle g_{s}^{2}G^{2}\rangle
M^{2}+864\pi^{2}\langle\bar{q}q\rangle
m_{c}(4M^{2}-3m_{0}^{2})\Big{)}WF[\mathcal{T}_{4},v]-e_{u}\Big{(}36\langle
g_{s}^{2}G^{2}\rangle M^{2}$ $\displaystyle+864\pi^{2}\langle\bar{q}q\rangle
m_{c}(4M^{2}-3m_{0}^{2})\Big{)}WF[\mathcal{T}_{4},\bar{v}]-40(e_{u}-e_{d})\pi^{2}f_{3\gamma}\langle
g_{s}^{2}G^{2}\rangle(m_{0}^{2}-M^{2})WF[\psi^{\nu},u]\Bigg{]}\Bigg{(}16~{}m_{c}^{4}FlNP[1,2,0]$
$\displaystyle-8~{}m_{c}^{2}FlNP[0,2,0]+FlNP[-1,2,0]\Bigg{)}.$ (20)

The functions $N[n,m,k]$, $FlP[n,m,k]$, $FlNP[n,m,k]$,
$WFD[\mathcal{A},\bar{v}]$, $WFD[\mathcal{A},v]$, $WF[\mathcal{A},\bar{v}]$,
$WF[\mathcal{A},v]$, $WFD[\mathcal{A},u]$ and $WF[\mathcal{A},u]$ are defined
as:

$\displaystyle N[n,m,k]$
$\displaystyle=\int_{0}^{\infty}dt\int_{0}^{\infty}dt^{\prime}~{}\frac{e^{-m_{c}/2(t+t^{\prime})}}{t^{n}~{}(\frac{m_{c}}{t}+\frac{m_{c}}{t^{\prime}})^{k}~{}t^{\prime
m}},$ $\displaystyle FlP[n,m,k]$
$\displaystyle=\int_{4m_{c}^{2}}^{s_{0}}ds\int_{4m_{c}^{2}}^{s}dl~{}\frac{e^{-l^{2}/\phi}~{}l^{n}~{}(l-s)^{m}}{(4m^{2}-l)^{2}~{}\phi^{k}},$
$\displaystyle FlNP[n,m,k]$
$\displaystyle=\int_{4m_{c}^{2}}^{s_{0}}ds\int_{4m_{c}^{2}}^{s}dl~{}\frac{e^{-l^{2}/\beta}~{}l^{n}~{}(l-s)^{m}}{(l-2m_{c}^{2})~{}\beta^{k}},$
$\displaystyle WFD[\mathcal{A},\bar{v}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{A}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta^{\prime}(\alpha_{q}+\bar{v}\alpha_{g}-u_{0}),$
$\displaystyle WFD[\mathcal{A},v]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{A}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta^{\prime}(\alpha_{\bar{q}}+v\alpha_{g}-u_{0}),$
$\displaystyle WF[\mathcal{A},\bar{v}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{A}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta(\alpha_{q}+\bar{v}\alpha_{g}-u_{0}),$
$\displaystyle WF[\mathcal{A},v]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{A}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta(\alpha_{\bar{q}}+v\alpha_{g}-u_{0}),$
$\displaystyle WFD[\mathcal{A},u]$
$\displaystyle=\int_{0}^{1}du~{}A(u)\delta^{\prime}(u-u_{0}),$ $\displaystyle
WF[\mathcal{A},u]$ $\displaystyle=\int_{0}^{1}du~{}A(u),$

where

$\displaystyle\beta=4~{}l~{}M^{2}-16m_{c}^{2}M^{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\phi=8~{}l~{}M^{2}-32m_{c}^{2}M^{2}.$

## Appendix C:

In this appendix, we give some details on Fourier and Borel transformations as
well as continuum subtraction. We take a term in the form

$\displaystyle I=\int_{0}^{1}duA(u)\int
d^{4}xe^{i(p+qu)x}\frac{K_{\nu}(m_{Q}\sqrt{-x^{2}})}{\sqrt{-x^{2}}^{\nu}}\frac{K_{\mu}(m_{Q}\sqrt{-x^{2}})}{\sqrt{-x^{2}}^{\mu}},$
(22)

where $K_{\nu}$ comes from the heavy quark propagator. To proceed we apply the
integral representation of the Bessel function of second kind as

$\frac{K_{\nu}\left(m_{Q}\sqrt{-x^{2}}\right)}{\left(\sqrt{-x^{2}}\right)^{\upsilon}}=\frac{1}{2}\int_{0}^{\infty}\frac{dt}{t^{\nu+1}}\exp\left[-\frac{m_{Q}}{2}\left(t-\frac{x^{2}}{t}\right)\right].$

As a result, we get

$\displaystyle I=\int_{0}^{1}duA(u)\int
d^{4}xe^{i(p+qu)x}\int_{0}^{\infty}\frac{dt}{t^{\nu+1}}\exp\left[-\frac{m_{Q}}{2}\left(t-\frac{x^{2}}{t}\right)\right]\int_{0}^{\infty}\frac{dt^{\prime}}{t^{\prime\mu+1}}\exp\left[-\frac{m_{Q}}{2}\left(t^{\prime}-\frac{x^{2}}{t^{\prime}}\right)\right].$
(23)

By applying the Wick rotation we obtain

$\displaystyle
I=\int_{0}^{1}duA(u)\int_{0}^{\infty}\frac{dt}{t^{\nu+1}}\int_{0}^{\infty}\frac{dt^{\prime}}{t^{\prime\mu+1}}\exp\Bigg{[}{-\frac{m_{Q}}{2}(t+t^{\prime})}\Bigg{]}\int
d^{4}x\exp\Bigg{[}-i(p.x+q.x)-ax^{2}\Bigg{]},$ (24)

where $a=(\frac{m_{Q}}{t}+\frac{m_{Q}}{t^{\prime}})$. Taking the four-
dimensional Gaussian integral we get

$\displaystyle
I=\int_{0}^{1}duA(u)\int_{0}^{\infty}\frac{dt}{t^{\nu+1}}\int_{0}^{\infty}\frac{dt^{\prime}}{t^{\prime\mu+1}}\exp\Bigg{[}{-\frac{m_{Q}}{2}(t+t^{\prime})}-\frac{(p+qu)^{2}}{4a}\Bigg{]}\frac{1}{a^{2}}.$
(25)

Now, we apply the Borel transformation over the variables $p^{2}$ and
$(p+q)^{2}$, which results in

$\displaystyle
I=\int_{0}^{1}duA(u)\int_{0}^{\infty}\frac{dt}{t^{\nu+1}}\int_{0}^{\infty}\frac{dt^{\prime}}{t^{\prime\mu+1}}\exp\Bigg{[}{-\frac{m_{Q}}{2}(t+t^{\prime})}\Bigg{]}\frac{M^{2}}{a^{2}}\delta\Big{[}\frac{1}{M^{2}}-\frac{1}{4a}\Big{]}~{}\delta\Big{[}u-u_{0}\Big{]}.$
(26)

After this step, we take the t integral using the corresponding Dirac delta.
To do this, we use the property:

$\displaystyle\delta(g(x))=\frac{\delta(x-x_{0})}{|g^{\prime}(x)|}\theta(x_{0}),$
(27)

and replace t by

$\displaystyle
t\rightarrow\Bigg{(}\frac{2m_{Q}~{}t^{\prime}}{M^{2}t^{\prime}-2m_{Q}}\Bigg{/}\Bigg{|}\frac{2~{}m_{Q}~{}t^{2}}{M^{2}t^{\prime}-2~{}m_{Q}}\Bigg{|}\Bigg{)}~{}\theta\Bigg{(}\frac{2m_{Q}~{}t^{\prime}}{M^{2}t^{\prime}-2m_{Q}}\Bigg{)}.$
(28)

Then, we change the variable $t^{\prime}\rightarrow s$ via

$\displaystyle t^{\prime}\rightarrow\frac{2~{}m_{Q}}{4~{}m_{Q}^{2}M^{2}}~{}s.$
(29)

Meanwhile, for the Borel transformations the following rules are applied:

$\displaystyle
B_{p^{2}}B_{(p+q)^{2}}\exp\Bigg{[}-\frac{(p+qu)^{2}}{4a}\Bigg{]}~{}(p+q~{}u)^{n}~{}(p.q)^{m}\rightarrow
M^{2}~{}(M^{2}/2)^{m}~{}D\Big{[}\frac{1}{M^{2}},n\Big{]}\delta\Big{[}\frac{1}{M^{2}}-\frac{1}{4a}\Big{]}~{}\delta^{\prime}\Big{[}u-u_{0}\Big{]},$
$\displaystyle
B_{p^{2}}B_{(p+q)^{2}}~{}\exp\Bigg{[}-\frac{(p+qu)^{2}}{4a}\Bigg{]}~{}(p.q)^{m}\rightarrow
M^{2}~{}(M^{2}/2)^{m}~{}\delta\Big{[}\frac{1}{M^{2}}-\frac{1}{4a}\Big{]}~{}\delta^{\prime}\Big{[}u-u_{0}\Big{]},$
$\displaystyle
B_{p^{2}}B_{(p+q)^{2}}~{}\exp\Bigg{[}-\frac{(p+qu)^{2}}{4a}\Bigg{]}~{}(p+q~{}u)^{n}\rightarrow
M^{2}~{}D\Big{[}\frac{1}{M^{2}},n\Big{]}\delta\Big{[}\frac{1}{M^{2}}-\frac{1}{4a}\Big{]},$
$\displaystyle
B_{p^{2}}B_{(p+q)^{2}}~{}\exp\Bigg{[}-\frac{(p+qu)^{2}}{4a}\Bigg{]}\rightarrow
M^{2}~{}\delta\Big{[}\frac{1}{M^{2}}-\frac{1}{4a}\Big{]}~{}\delta\Big{[}u-u_{0}\Big{]}.$

where, D represents the derivation and

$\displaystyle
M^{2}=\frac{M_{1}^{2}M_{2}^{2}}{M_{1}^{2}+M_{2}^{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}u_{0}=\frac{M_{1}^{2}}{M_{1}^{2}+M_{2}^{2}}.$
(31)

The following formula for the continuum subtraction is used

$\displaystyle\left(M^{2}\right)^{N}\int_{4m_{Q}^{2}}^{\infty}dse^{-s/M^{2}}f(s)\rightarrow\int_{4m_{Q}^{2}}^{s_{0}}dse^{-s/M^{2}}F_{N}(s),$
(32)

where

$\displaystyle F_{N}(s)$
$\displaystyle=\Big{(}\frac{d}{ds}\Big{)}^{-N}f(s),~{}~{}~{}~{}~{}N\leq 0,$
$\displaystyle F_{N}(s)$
$\displaystyle=\frac{1}{\Gamma(N)}\int_{4m_{Q}^{2}}^{s}dl~{}(s-l)^{N-1}f(l),~{}~{}~{}~{}~{}N>0,$
(33)

as a result of which we obtain the following expression:

$\displaystyle\int_{0}^{1}duA(u)\int_{4m_{Q^{\prime}}}^{s_{0}}ds\int_{4m_{Q^{\prime}}}^{s}dl~{}\exp\Bigg{[}{-\frac{l+m_{Q^{\prime}}\Big{(}-3-\frac{m_{Q}}{m_{Q}^{2}-m_{Q^{\prime}}^{2}}\Big{)}}{M^{2}}}\Bigg{]}\frac{(l-s)^{3}~{}\delta\Big{[}u-u_{0}\Big{]}}{3~{}m_{Q}~{}m^{4}_{Q^{\prime}}~{}M^{12}\Bigg{|}\frac{m_{Q}\Big{(}-2m_{Q}+\frac{2m_{Q^{\prime}}^{2}}{m_{Q}}\Big{)}^{2}}{m^{4}_{Q^{\prime}}}\Bigg{|}},$
(34)

with $m_{Q}$ and $m_{Q^{\prime}}$ being the charm quark mass. Here we face
with the well-known problem in the case of doubly heavy hadrons when we take
$m_{Q}=m_{Q^{\prime}}$. The expression above becomes indeterminate. To get rid
of this problem we take the limit of the expression in the integral, i.e.,

$\displaystyle\int_{0}^{1}duA(u)\int_{4m_{Q^{\prime}}}^{s_{0}}ds\int_{4m_{Q^{\prime}}}^{s}dl~{}\lim_{{m_{Q^{\prime}}}\rightarrow
m_{Q}}\Bigg{[}\exp\Bigg{\\{}{-\frac{l+m_{Q^{\prime}}\Big{(}-3-\frac{m_{Q}}{m_{Q}^{2}-m_{Q^{\prime}}^{2}}\Big{)}}{M^{2}}}\Bigg{\\}}\frac{(l-s)^{3}~{}\delta\Big{[}u-u_{0}\Big{]}}{3~{}m_{Q}~{}m^{4}_{Q^{\prime}}~{}M^{12}\Bigg{|}\frac{m_{Q}\Big{(}-2m_{Q}+\frac{2m_{Q^{\prime}}^{2}}{m_{Q}}\Big{)}^{2}}{m^{4}_{Q^{\prime}}}\Bigg{|}}\Bigg{]},$
(35)

which gives a finite result.

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