#  Strong interaction of doubly heavy spin-3/2 baryons with light vector
mesons

K. Azizi1,2,4, A. R. Olamaei3,4, S. Rostami1
1 Department of Physics, University of Tehran, North Karegar Ave. Tehran
14395-547, Iran
2 Department of Physics, Doǧuş University, Acibadem-Kadiköy, 34722 Istanbul,
Turkey
3 Department of Physics, Jahrom University, Jahrom, P. O. Box 74137-66171,
Jahrom, Iran
4 School of Particles and Accelerators, Institute for Research in Fundamental
Sciences (IPM),
P. O. Box 19395-5531, Tehran, Iran

We calculate the strong coupling constants among the doubly heavy
spin-$\frac{3}{2}$ baryons $\Xi^{*}_{QQ}$ and $\Omega^{*}_{QQ}$, with $Q$ and
$Q^{\prime}$ being $c$ or $b$ quark, with light vector meson by means of the
light-cone QCD sum rules. The matrix elements defining these vertices are
described by four coupling constants $g_{1}$, $g_{2}$, $g_{3}$, and $g_{4}$.
The unwanted pollution coming from the doubly heavy spin-$\frac{1}{2}$ baryons
are removed by a special ordering of Dirac matrices and selection of
appropriate Lorentz structures. The strong coupling constants are basic
parameters that carry information on the nature of the strong interaction
among hadronic multiplets. Investigation of these parameters may help
physicists in the construction of the strong potentials among the doubly heavy
baryons and light vector mesons. The values obtained for the strong coupling
constants may also help experimental groups in analyses of the data produced
at various hadron colliders.

##  1 Introduction

Studies on the spectroscopic and various decay parameters as well as internal
structures of doubly heavy baryons constitute one of the directions of active
research in particle physics. In the last two decades, many aspects of these
baryons which consist of two heavy and one light quarks, have been studied
both theoretically and experimentally [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31,
32, 33]. From the experimental point of view, a new era began in the studies
of double charm baryons when the SELEX Collaboration at Fermilab reported the
observation of $\Xi_{cc}^{+}$ baryon in the decay modes
$\Lambda_{c}^{+}K^{-}\pi^{+}$ and $pD^{+}K^{-}$ with mass
$M_{\Xi_{cc}^{+}}=(3518.7\pm 1.7)$ MeV [22, 23]. However, in the years that
followed, other collaborations like FOCUS [24], BaBar [25], LHCb [26], and
Belle [27] failed to observe such a state with the properties that the SELEX
observed. Finally, the LHCb Collaboration discovered the double charm state
$\Xi^{++}_{cc}$ in 2017 [28] and confirmed via measuring the decay channel
$\Xi^{++}_{cc}\rightarrow\Xi^{+}_{c}\pi^{+}$ [34]. The extracted mass of the
$\Xi^{++}_{cc}$ state was given as $3621.24\pm 0.65(\text{stat.})\pm
0.31(\text{syst.})~{}\text{MeV}/c^{2}$. Brodsky et al. [35] found that despite
the difference between the LHCb and SELEX values for the mass (about
$100~{}\text{MeV}/c^{2}$), both can be correct with application of
supersymmetric algebra to hadron spectroscopy, together with the intrinsic
heavy-quark QCD mechanism for the hadroproduction of heavy hadrons at large
$x_{F}$.

Recently, with a data sample corresponding to an integrated luminosity of 5.4
$\mbox{fb}^{-1}$, the $\Xi^{0}_{bc}\rightarrow D^{0}pK^{-}$ decay has been
searched for by the LHCb and no evidence for a signal is found [36]. Of
course, the story will not end here and experiments will continue. Future
searches at LHCb with improved trigger conditions, and larger data samples
will further improve the sensitivity of doubly heavy baryon signals.

Given that more doubly heavy baryons are expected to be discovered
experimentally in the future, theoretical studies on different properties of
them are needed to provide many inputs to experiments. We hope that not only
we will be able to identify the ground state doubly heavy baryons predicted by
the quark model [37, 38, 39], but also will find more information on the
resonances beyond the ground state particles. Heretofore, lattice QCD [40],
Quark Spin Symmetry [41, 42], QCD Sum Rules [29, 30, 31, 43, 44] are just some
of the attempts that have been made to predict masses, residues, decay modes,
lifetimes, strong coupling constants, and other properties of doubly heavy
baryons.

In the present work, we aim to study the strong coupling constants among the
doubly heavy spin-3/2 baryons $\Xi^{*}_{QQ^{\prime}},\Omega^{*}_{QQ^{\prime}}$
and light vector mesons $\rho,\omega,\phi,$ and $K^{*}$. The formation of
hadrons takes place at low energy scale, which belongs to the nonperturbative
regime of QCD. The strong coupling constants among the hadrons are basic
quantities that appear when the hadrons meet each other at colliders. For the
calculations of these fundamental parameters, one can not use the fundamental
Lagrangian of QCD and needs some nonperturbative approaches. One of the most
powerful approaches in this respect is the QCD sum rules method [45]. In the
present work, we investigate the strong coupling constants of the doubly heavy
spin-3/2 baryons with light vector mesons within the light cone QCD sum rule’s
(LCSR) framework. The LCSR approach adopts the operator product expansion
(OPE) near the light cone $x^{2}\approx 0$ instead of the short distance
$x\approx 0$, whose matrix elements are parametrized by the hadronic DA’s
which are classified according to their twists [46, 47, 48]. Note that the
strong coupling constants among the spin-1/2 baryons and light mesons are
investigated in [44, 49, 50, 51].

The plan for this article is as follows. In next section, we review the
formalism of driving the sum rules for the strong coupling constants in the
LCSR using both the physical and QCD representations of the correlation
function (CF). In Section 3, the numerical analysis and numerical results for
the strong coupling constants are presented. Section 4 is devoted to a brief
summary and conclusion.

##  2 LCSR for the strong coupling constants

In this section, we aim to construct the LCSR for the strong coupling
constants of the doubly heavy baryons $\Xi^{*}_{QQ}$ and $\Omega^{*}_{QQ}$
with light vector mesons. To this end, the main inputs are the interpolating
currents of the doubly heavy baryons in terms of the quark fields inside them.
In a compact form, the interpolating current for spin-3/2 doubly heavy baryons
can be written as

$\displaystyle\eta_{\mu}(x)$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{abc}}{\sqrt{3}}\Bigg{\\{}[q^{aT}(x)C\gamma_{\mu}Q^{b}(x)]Q^{\prime
c}(x)+[q^{aT}(x)C\gamma_{\mu}Q^{\prime b}(x)]Q^{c}(x)$ $\displaystyle+$
$\displaystyle[Q^{aT}(x)C\gamma_{\mu}Q^{\prime b}(x)]q^{c}(x)\Bigg{\\}},$

where $\epsilon$ stands for the anti-symmetric Levi-Civita tensor with $a,b$
and $c$ being the color indexes. $C$ and $T$ are the charge conjugation and
transposition operators, respectively. The quark contents for the doubly heavy
spin-3/2 baryons family are shown in Table 1.

Baryon | $q$ | $Q$ | $Q^{\prime}$
---|---|---|---
$\Xi^{*}_{QQ^{\prime}}$ |  $u$ or $d$ |  $b$ or $c$ |  $b$ or $c$
$\Omega^{*}_{QQ^{\prime}}$ | $s$ |  $b$ or $c$ |  $b$ or $c$
Table 1: The quark content of the doubly heavy spin-3/2 baryons.

To derive LCSR for the strong vertices of $\Xi^{*}_{QQ^{\prime}}$ or
$\Omega^{*}_{QQ^{\prime}}$ baryons with the light vector mesons
$\rho,K^{*},\omega$ or $\phi$, the starting point is to consider the following
CF:

$\displaystyle\Pi_{\mu\nu}=i\int d^{4}xe^{ip.x}\left<V(q)|{\cal
T}\left\\{\eta_{\mu}(x)\bar{\eta}_{\nu}(0)\right\\}|0\right>~{},$ (2)

where the time-ordered product of two currents located between an on-shell
vector meson state $V(q)$ with four momentum $q$ and the hadronic vacuum. $p$
is the external four-momentum of the outgoing doubly heavy baryon, and
$\eta(x)$ and $\bar{\eta}(0)$ are interpolating currents for doubly heavy
spin-3/2 baryons. This correlation function is calculated both at hadronic
(low energy) and QCD (high energy) levels:

  * $\bullet$

At the hadron level (in the time-like region), the CF is calculated by
inserting complete sets of baryonic states with the same quantum numbers as
the interpolating currents and isolating the ground state contribution. It is
called the phenomenological or physical representation of the CF.

  * $\bullet$

At the QCD level (in the space-like region), the CF is calculated in terms of
QCD degrees of freedom by using the operator product expansion (OPE). It is
known as the QCD or theoretical representation of the CF.

By matching these two representations via a dispersion relation, we can get
the desired sum rules. By applying the Borel transformation and continuum
subtraction procedures one can remove the divergences coming from the
dispersion integrals as well as suppress the contributions of the higher
states and continuum.

Thus, in the hadronic representation, complete sets of baryonic states are
placed in the CF. By performing the Fourier integration over $x$ and isolating
the ground state contribution we get

$\displaystyle\Pi^{\text{Phys.}}_{\mu\nu}(p,q)=\frac{\langle
0|\eta_{\mu}|B^{*}_{2}(p,r)\rangle\langle
B^{*}_{2}(p,r)V(q)|B^{*}_{1}(p+q,s)\rangle\langle
B^{*}_{1}(p+q,s)|\bar{\eta_{\mu}}|0\rangle}{(p^{2}-m_{2}^{2})[(p+q)^{2}-m_{1}^{2}]}+\cdots~{},$
(3)

where dots correspond to the contribution of the higher states and continuum.
$B^{*}_{1}(p+q,s)$ and $B^{*}_{2}(p,r)$ are the initial and final doubly heavy
baryons with spins $s$ and $r$, respectively. To proceed, we introduce

$\displaystyle\langle 0|\eta_{\mu}|B^{*}_{i}(p,s)\rangle$ $\displaystyle=$
$\displaystyle\lambda_{B^{*}_{i}}u_{\mu}(p,s),$ (4)

where $\lambda_{B^{*}_{i}}$ acts as the residue and $u_{\mu}(p,s)$ are the
Rarita–Schwinger spinor for the baryon $B^{*}_{i}$ with momentum $p$ and spin
$s$. Using the Lorentz and parity symmetries one can define the matrix element
$\left<B_{2}^{*}(p,r)V(q)\left|\right.B_{1}^{*}(p+q,s)\right>$ as (see for
instance Refs. [52, 53]):

$\displaystyle\left<B_{2}^{*}(p,r)V(q)\left|\right.B_{1}^{*}(p+q,s)\right>\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\bar{u}_{\alpha}(p,r)\Bigg{\\{}g^{\alpha\beta}\Bigg{[}\hbox
to0.0pt{/\hss}\varepsilon g_{1}+2(p.\varepsilon){g_{2}\over
m_{1}+m_{2}}\Bigg{]}$ (5) $\displaystyle+$
$\displaystyle\\!\\!\\!{q^{\alpha}q^{\beta}\over(m_{1}+m_{2})^{2}}\Bigg{[}\hbox
to0.0pt{/\hss}\varepsilon g_{3}+2(p.\varepsilon){g_{4}\over
m_{1}+m_{2}}\Bigg{]}\Bigg{\\}}u_{\beta}(p+q,s),$

in terms of four strong coupling form factors $g_{1},g_{2},g_{3}$ and $g_{4}$.
Here $\varepsilon_{\mu}$ is the polarization four-vector of the vector meson
$V(q)$.

Inserting Eqs. 5 and 4 into Eq. 3, since the baryons are considered as un-
polarized, the next stage is to sum over the spins of the incoming and
outgoing baryons which can be done using the following completeness relation:

$\displaystyle\sum_{s}u_{\mu}(p,s)\bar{u}_{\nu}(p,s)=({\hbox
to0.0pt{/\hss}p+m})\Bigg{(}g_{\mu\nu}-{1\over
3}\gamma_{\mu}\gamma_{\nu}+{2p_{\mu}p_{\nu}\over
3m^{2}}+{p_{\mu}\gamma_{\nu}-p_{\nu}\gamma_{\mu}\over 3m}\Bigg{)}~{}.$ (6)

Now, it is necessary to note two important points. First, one should note that
the appearing structures are not all independent. The second point is that
there are contributions from spin-1/2 states due to the coupling of the
interpolating current $\eta_{\mu}$ to them via

$\displaystyle\left<0\left|\eta_{\mu}\right|1/2(p)\right>\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!A\left(\gamma_{\mu}-{4\over
m}p_{\mu}\right)u(p)~{}.$ (7)

Therefore one can find out from the above equation that the structures
proportional to $\gamma_{\mu}$ from the left or $\gamma_{\nu}$ from the right,
and also the ones proportional to $p_{\mu}$ and $(p+q)_{\nu}$ receive
contributions from the unwanted spin-1/2 states which have to be removed. Both
requirements can be fulfilled by reordering the Dirac matrices. In this work
we choose $\gamma_{\mu}\hbox to0.0pt{/\hss}\varepsilon\hbox
to0.0pt{/\hss}q\hbox to0.0pt{/\hss}p\gamma_{\nu}$ as the suitable ordering and
remove the spin-1/2 pollution using the above prescription.

Putting all the above things together, the final form of the physical side of
the CF is

$\displaystyle\Pi^{\text{Phys.}}_{\mu\nu}(p,q)\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!{\lambda_{B_{1}^{*}}\lambda_{B_{2}^{*}}\over[(p+q)^{2}-m_{1}^{2})](p^{2}-m_{2}^{2})}\Bigg{\\{}2(\varepsilon.p)g_{\mu\nu}\hbox
to0.0pt{/\hss}q\Bigg{[}g_{1}+g_{2}{m_{2}\over m_{1}+m_{2}}\Bigg{]}$ (8)
$\displaystyle-$ $\displaystyle\\!\\!\\!2(\varepsilon.p)g_{\mu\nu}\hbox
to0.0pt{/\hss}q\hbox to0.0pt{/\hss}p{g_{2}\over
m_{1}+m_{2}}+q_{\mu}q_{\nu}\hbox to0.0pt{/\hss}\varepsilon\hbox
to0.0pt{/\hss}q\hbox
to0.0pt{/\hss}p{g_{3}\over(m_{1}+m_{2})^{2}}-2(\varepsilon.p)q_{\mu}q_{\nu}\hbox
to0.0pt{/\hss}q\hbox to0.0pt{/\hss}p{g_{4}\over(m_{1}+m_{2})^{3}}$
$\displaystyle+$ $\displaystyle\\!\\!\\!\mbox{\rm other
structures}+...\Bigg{\\}}~{}.$

To acquire the sum rules for the strong coupling constants $g_{i}$, we choose
the coefficients of the structures $(\varepsilon.p)g_{\mu\nu}\hbox
to0.0pt{/\hss}q$, $(\varepsilon.p)g_{\mu\nu}\hbox to0.0pt{/\hss}q\hbox
to0.0pt{/\hss}p$, $q_{\mu}q_{\nu}\hbox to0.0pt{/\hss}\varepsilon\hbox
to0.0pt{/\hss}q\hbox to0.0pt{/\hss}p$ and $(\varepsilon.p)q_{\mu}q_{\nu}\hbox
to0.0pt{/\hss}q\hbox to0.0pt{/\hss}p$ from both the hadronic and QCD sides.

The theoretical (QCD) side of the CF is calculated in deep Euclidean region,
where $-p^{2}\rightarrow\infty$, $-(p+q)^{2}\rightarrow\infty$. To separate
the perturbative and non-perturbative contributions we employ the OPE. The
non-perturbative inputs in LCSR are distribution amplitudes (DAs) of the
corresponding light vector mesons which can be found in [54, 55, 56].

To proceed in QCD side, we insert the interpolating current 2 to the CF 2 and
use the Wick theorem to find

$\displaystyle\Big{(}\Pi^{\text{QCD}}_{\mu\nu}\Big{)}_{\rho\tau}(p,q)$
$\displaystyle=$
$\displaystyle\frac{i}{3}\epsilon_{abc}\epsilon_{a^{\prime}b^{\prime}c^{\prime}}\int
d^{4}xe^{ip.x}\langle
V(q)|\bar{q}^{c^{\prime}}_{\alpha}(0)q^{c}_{\beta}(x)|0\rangle\Bigg{\\{}\delta_{\alpha\rho}\delta_{\beta\tau}\text{Tr}\Big{(}\tilde{S}^{aa^{\prime}}_{Q}(x)\gamma_{\mu}S^{bb^{\prime}}_{Q^{\prime}}(x)\gamma_{\nu}\Big{)}$
(9) $\displaystyle+$
$\displaystyle\delta_{\alpha\rho}\Big{(}\gamma_{\nu}\tilde{S}^{aa^{\prime}}_{Q}(x)\gamma_{\mu}S^{bb^{\prime}}_{Q^{\prime}}(x)\Big{)}_{\beta\tau}+\delta_{\alpha\rho}\Big{(}\gamma_{\nu}\tilde{S}^{bb^{\prime}}_{Q^{\prime}}(x)\gamma_{\mu}S^{aa^{\prime}}_{Q}(x)\Big{)}_{\beta\tau}$
$\displaystyle+$
$\displaystyle\delta_{\beta\tau}\Big{(}S^{bb^{\prime}}_{Q^{\prime}}(x)\gamma_{\nu}\tilde{S}^{aa^{\prime}}_{Q}(x)\gamma_{\mu}\Big{)}_{\rho\alpha}+\Big{(}\gamma_{\nu}\tilde{S}^{aa^{\prime}}_{Q}(x)\gamma_{\mu}\Big{)}_{\beta\alpha}\Big{(}S^{bb^{\prime}}_{Q^{\prime}}(x)\Big{)}_{\rho\tau}$
$\displaystyle-$
$\displaystyle\Big{(}C\gamma_{\mu}S^{aa^{\prime}}_{Q}(x)\Big{)}_{\alpha\tau}\Big{(}S^{bb^{\prime}}_{Q^{\prime}}(x)\gamma_{\nu}C\Big{)}_{\rho\beta}-\delta_{\beta\tau}\Big{(}S^{aa^{\prime}}_{Q}(x)\gamma_{\nu}\tilde{S}^{bb^{\prime}}_{Q^{\prime}}(x)\gamma_{\mu}\Big{)}_{\rho\alpha}$
$\displaystyle-$
$\displaystyle\Big{(}C\gamma_{\mu}S^{bb^{\prime}}_{Q^{\prime}}(x)\Big{)}_{\alpha\tau}\Big{(}S^{aa^{\prime}}_{Q}(x)\gamma_{\nu}C\Big{)}_{\rho\beta}+\Big{(}\gamma_{\nu}\tilde{S}^{bb^{\prime}}_{Q^{\prime}}(x)\gamma_{\mu}\Big{)}_{\beta\alpha}\Big{(}S^{aa^{\prime}}_{Q}(x)\Big{)}_{\rho\tau}\Bigg{\\}},$

in terms of the heavy quark propagators,
$S^{aa^{\prime}}_{Q{{}^{(\prime)}}}(x)$. Here, $\tilde{S}=CS^{T}C$ and the
main non-perturbative contributions are packed in the non-local matrix
elements $\langle
V(q)|\bar{q}^{c^{\prime}}_{\alpha}(x)q^{c}_{\beta}(0)|0\rangle$, which can be
expressed in terms of DAs of the light vector meson $V(q)$.

The heavy quark propagator can be written in terms of the free and interacting
parts as

$\displaystyle S_{Q}^{aa^{\prime}}(x)$ $\displaystyle=$
$\displaystyle{m_{Q}^{2}\over
4\pi^{2}}{K_{1}(m_{Q}\sqrt{-x^{2}})\over\sqrt{-x^{2}}}\delta^{aa^{\prime}}-i{m_{Q}^{2}\hbox
to0.0pt{/\hss}{x}\over
4\pi^{2}x^{2}}K_{2}(m_{Q}\sqrt{-x^{2}})\delta^{aa^{\prime}}$ (10)
$\displaystyle-$ $\displaystyle
ig_{s}\int{d^{4}k\over(2\pi)^{4}}e^{-ikx}\int_{0}^{1}du\Bigg{[}{\hbox
to0.0pt{/\hss}k+m_{Q}\over
2(m_{Q}^{2}-k^{2})^{2}}\sigma^{\lambda\theta}G_{\lambda\theta}^{aa^{\prime}}(ux)$
$\displaystyle+$ $\displaystyle{u\over
m_{Q}^{2}-k^{2}}x^{\lambda}\gamma^{\theta}G_{\lambda\theta}^{aa^{\prime}}(ux)\Bigg{]}+\cdots,$

where the first two terms represent the propagation of free quark, in which
$K_{1}$ and $K_{2}$ are the modified Bessel functions of the second kind. The
rest, which $\sim G_{\lambda\theta}^{aa^{\prime}}$, represent the interacting
part. Here $G_{\lambda\theta}^{aa^{\prime}}$ is the gluon strength tensor with
the shorthand notation

$\displaystyle G^{aa^{\prime}}_{\lambda\theta}\equiv
G^{A}_{\lambda\theta}t^{aa^{\prime}}_{A},$ (11)

where $\lambda$ and $\theta$ are Minkowski indices, $t_{A}=\lambda_{A}/2$ and
$\lambda^{aa^{\prime}}_{A}$ is the $aa^{\prime}$th array of the Gell-Mann
tensor $\lambda_{A}$ in which $A=1,2,...,8$.

As the heavy quark propagator 10 consists of both free and gluonic terms, it
leads to several contributions in the CF as said. The leading order
contribution is obtained by replacing both the heavy quark propagators with
their free parts

$\displaystyle S_{Q}^{aa^{\prime}(\text{pert.})}(x)$ $\displaystyle=$
$\displaystyle{m_{Q}^{2}\over
4\pi^{2}}{K_{1}(m_{Q}\sqrt{-x^{2}})\over\sqrt{-x^{2}}}\delta^{aa^{\prime}}-i{m_{Q}^{2}\hbox
to0.0pt{/\hss}{x}\over
4\pi^{2}x^{2}}K_{2}(m_{Q}\sqrt{-x^{2}})\delta^{aa^{\prime}}.$ (12)

It represents a bare loop with no gluon exchange and the non-local matrix
elements in this case can be expressed in terms of two-particle DAs of the
light vector meson $V(q)$. Inserting the free part of one heavy quark
propagator and the gluonic part of the other, which is

$\displaystyle S^{aa^{\prime}(\text{non-p.})}_{Q}(x)$ $\displaystyle=$
$\displaystyle-
ig_{s}\int{d^{4}k\over(2\pi)^{4}}e^{-ikx}\int_{0}^{1}duG^{aa^{\prime}}_{\mu\nu}(ux)\Delta^{\mu\nu}_{Q}(x),$
(13)

where $\Delta^{\mu\nu}_{Q}(x)$ is

$\displaystyle\Delta^{\mu\nu}_{Q}(x)$ $\displaystyle=$
$\displaystyle\dfrac{1}{2(m_{Q}^{2}-k^{2})^{2}}\Big{[}(\hbox
to0.0pt{/\hss}k+m_{Q})\sigma^{\mu\nu}+2u(m_{Q}^{2}-k^{2})x^{\mu}\gamma^{\nu}\Big{]},$
(14)

leads to the contribution which is responsible for the exchange of one gluon
between one of the heavy quarks and the light vector meson and can be
calculated using three-particle DAs of the corresponding vector meson. The
contribution coming from considering the gluonic parts of both the heavy
quarks propagators representing the exchange of two gluons, leads to four-
particle DAs which are not available yet and therefore we ignore it in the
present study. However, we consider the two-gluon condensate contributions.

The non-local matrix elements can be calculated using the Fierz identities.
For the quark part we have

$\displaystyle\bar{q}_{\alpha}^{c^{\prime}}(0)q_{\beta}^{c}(x)\to-\frac{1}{12}(\Gamma_{J})_{\beta\alpha}\delta^{cc^{\prime}}\bar{q}(0)\Gamma^{J}q(x),$
(15)

and we use similar identity for the gluonic part. Here, $\Gamma^{J}$ runs over
all possible $\gamma$-matrices as

$\Gamma^{J}=\mathbf{1,\ }\gamma_{5},\ \gamma_{\mu},\ i\gamma_{5}\gamma_{\mu},\
\sigma_{\mu\nu}/\sqrt{2}.$ (16)

As a result of the above procedure, there appear the following two- and three-
particle matrix elements

$\displaystyle\langle
V(q)|\bar{q}(0)\Gamma_{J}q(x)|0\rangle~{}~{}\mbox{and}~{}~{}\langle
V(q)|\bar{q}(0)\Gamma_{J}G_{\lambda\theta}(ux)q(x)|0\rangle,$ (17)

which can be expressed in terms of DAs of different twists. The expansion of
the above matrix elements in terms of DAs as well as the explicit forms of the
DAs for the vector mesons with different twists can be found in Refs. [54, 55,
56].

Now, we briefly explain how different contributions to the corresponding form
factors are calculated. The leading order contribution of the CF 9, which is
obtained by replacing both the heavy quark propagators with the perturbative
terms, is as follows

$\displaystyle\Big{(}\Pi^{\text{QCD(0)}}_{\mu\nu}\Big{)}_{\rho\tau}(p,q)$
$\displaystyle=$ $\displaystyle\frac{i}{6}\int d^{4}xe^{ip.x}\langle
V(q)|\bar{q}(0)\Gamma^{J}q(x)|0\rangle\Bigg{\\{}\Big{(}\Gamma_{J}\Big{)}_{\rho\tau}\text{Tr}\Big{(}\tilde{S}^{(\text{pert.})}_{Q}(x)\gamma_{\mu}S^{(\text{pert.})}_{Q^{\prime}}(x)\gamma_{\nu}\Big{)}$
$\displaystyle+$
$\displaystyle\Big{(}\Gamma_{J}\gamma_{\nu}\tilde{S}^{(\text{pert.})}_{Q}(x)\gamma_{\mu}S^{(\text{pert.})}_{Q^{\prime}}(x)\Big{)}_{\rho\tau}+\Big{(}\Gamma_{J}\gamma_{\nu}\tilde{S}^{(\text{pert.})}_{Q^{\prime}}(x)\gamma_{\mu}S^{(\text{pert.})}_{Q}(x)\Big{)}_{\rho\tau}$
$\displaystyle+$
$\displaystyle\Big{(}S^{(\text{pert.})}_{Q^{\prime}}(x)\gamma_{\nu}\tilde{S}^{(\text{pert.})}_{Q}(x)\gamma_{\mu}\Gamma_{J}\Big{)}_{\rho\tau}+\text{Tr}\Big{(}\Gamma_{J}\gamma_{\nu}\tilde{S}^{(\text{pert.})}_{Q}(x)\gamma_{\mu}\Big{)}\Big{(}S^{(\text{pert.})}_{Q^{\prime}}(x)\Big{)}_{\rho\tau}$
$\displaystyle-$
$\displaystyle\Big{(}S^{(\text{pert.})}_{Q^{\prime}}(x)\gamma_{\nu}\tilde{\Gamma}_{J}\gamma_{\mu}S^{(\text{pert.})}_{Q}(x)\Big{)}_{\rho\tau}-\Big{(}S^{(\text{pert.})}_{Q}(x)\gamma_{\nu}\tilde{S}^{(\text{pert.})}_{Q^{\prime}}(x)\gamma_{\mu}\Gamma_{J}\Big{)}_{\rho\tau}$
$\displaystyle-$
$\displaystyle\Big{(}S^{(\text{pert.})}_{Q}(x)\gamma_{\nu}\tilde{\Gamma}_{J}\gamma_{\mu}S^{(\text{pert.})}_{Q^{\prime}}(x)\Big{)}_{\rho\tau}+\text{Tr}\Big{(}\Gamma_{J}\gamma_{\nu}\tilde{S}^{(\text{pert.})}_{Q^{\prime}}(x)\gamma_{\mu}\Big{)}\Big{(}S^{(\text{pert.})}_{Q}(x)\Big{)}_{\rho\tau}\Bigg{\\}},$

where the superscript $(0)$ indicates zero gluon exchange. One can obtain the
contribution of one gluon exchange (say between the heavy quark $Q$ and the
vector meson) by replacing one heavy quark propagator ($Q^{\prime}$) with its
perturbative free terms and the other ($Q$) with its nonperturbative gluonic
terms as

$\displaystyle\Big{(}\Pi^{\text{QCD(1)}}_{\mu\nu}\Big{)}_{\rho\tau}(p,q)$
$\displaystyle=$
$\displaystyle\frac{-ig_{s}}{96}\int\frac{d^{4}k}{(2\pi)^{2}}e^{-ik.x}\int_{0}^{1}du\langle
V(q)|\bar{q}(0)\Gamma^{J}G_{\lambda\delta}q(x)|0\rangle\Bigg{\\{}\Big{(}\Gamma_{J}\Big{)}_{\rho\tau}$
$\displaystyle\times$
$\displaystyle\text{Tr}\Big{(}\tilde{\Delta}^{\lambda\delta}_{Q}(x)\gamma_{\mu}S^{(\text{pert.})}_{Q^{\prime}}(x)\gamma_{\nu}\Big{)}+\Big{(}\Gamma_{J}\gamma_{\nu}\tilde{\Delta}^{\lambda\delta}_{Q}(x)\gamma_{\mu}S^{(\text{pert.})}_{Q^{\prime}}(x)\Big{)}_{\rho\tau}$
$\displaystyle+$
$\displaystyle\Big{(}\Gamma_{J}\gamma_{\nu}\tilde{S}^{(\text{pert.})}_{Q^{\prime}}(x)\gamma_{\mu}\Delta^{\lambda\delta}_{Q}(x)\Big{)}_{\rho\tau}+\Big{(}S^{(\text{pert.})}_{Q^{\prime}}(x)\gamma_{\nu}\tilde{\Delta}^{\lambda\delta}_{Q}(x)\gamma_{\mu}\Gamma_{J}\Big{)}_{\rho\tau}$
$\displaystyle+$
$\displaystyle\text{Tr}\Big{(}\gamma_{\nu}\tilde{\Delta}^{\lambda\delta}_{Q}(x)\gamma_{\mu}\Gamma_{J}\Big{)}\Big{(}S^{(\text{pert.})}_{Q^{\prime}}(x)\Big{)}_{\rho\tau}-\Big{(}S^{(\text{pert.})}_{Q^{\prime}}(x)\gamma_{\nu}\tilde{\Gamma}_{J}\gamma_{\mu}\Delta^{\lambda\delta}_{Q}(x)\Big{)}_{\rho\tau}$
$\displaystyle-$
$\displaystyle\Big{(}\Delta^{\lambda\delta}_{Q}(x)\gamma_{\nu}\tilde{S}^{(\text{pert.})}_{Q^{\prime}}(x)\gamma_{\mu}\Gamma_{J}\Big{)}_{\rho\tau}-\Big{(}\Delta^{\lambda\delta}_{Q}(x)\gamma_{\nu}\tilde{\Gamma}_{J}\gamma_{\mu}S^{(\text{pert.})}_{Q^{\prime}}(x)\Big{)}_{\rho\tau}$
$\displaystyle+$
$\displaystyle\text{Tr}\Big{(}\Gamma_{J}\gamma_{\nu}\tilde{S}^{(\text{pert.})}_{Q^{\prime}}(x)\gamma_{\mu}\Big{)}\Big{(}\Delta^{\lambda\delta}_{Q}(x)\Big{)}_{\rho\tau}\Bigg{\\}},$

where the superscript $(1)$ indicates one gluon exchange contribution. The
contribution of exchanging gluon between the heavy quark $Q^{\prime}$ and
light vector meson can be found by simply exchanging $Q$ and $Q^{\prime}$ in
the above equation.

In calculation of the theoretical sides 2 and 2, one may run into some kinds
of configurations which have the general forms of

$\displaystyle T_{[~{}~{},\alpha,\alpha\beta,\alpha\beta\gamma,...]}(p,q)$
$\displaystyle=$ $\displaystyle i\int d^{4}x\int_{0}^{1}dv\int{\cal D}\alpha
e^{ip.x}\big{(}x^{2}\big{)}^{n}[e^{i(\alpha_{\bar{q}}+v\alpha_{g})q.x}\mathcal{G}(\alpha_{i}),e^{iq.x}f(u)]$
(20) $\displaystyle\times$
$\displaystyle[1,x_{\alpha},x_{\alpha}x_{\beta},x_{\alpha}x_{\beta}x_{\gamma},...]K_{n_{1}}(m_{1}\sqrt{-x^{2}})K_{n_{2}}(m_{2}\sqrt{-x^{2}}),$

where the expressions in the brackets indicate different configurations
resulting from calculations. In the first bracket, the first and second
expressions come from the two and three particle DAs respectively. The blank
subscript on the LHS indicates no $x_{\alpha}$ in the corresponding
configuration and the measure ${\cal D}\alpha$ is defined as

$\int\mathcal{D}\alpha=\int_{0}^{1}d\alpha_{q}\int_{0}^{1}d\alpha_{\bar{q}}\int_{0}^{1}d\alpha_{g}\delta(1-\alpha_{q}-\alpha_{\bar{q}}-\alpha_{g}).$

Here we use the cosine representation of the Bessel function as follows:

$K_{n}(m_{Q}\sqrt{-x^{2}})=\frac{\Gamma(n+1/2)~{}2^{n}}{\sqrt{\pi}m_{Q}^{n}}\int_{0}^{\infty}dt~{}\cos(m_{Q}t)\frac{(\sqrt{-x^{2}})^{n}}{(t^{2}-x^{2})^{n+1/2}}.$
(21)

There are several representations of the Bessel functions and it is shown in
[43] that the cosine representation makes the calculations straightforward and
more suitable for the Borel transformation. In this step, we perform the
$x-$integration and use

$\displaystyle(x^{2})^{n}$ $\displaystyle=$
$\displaystyle(-1)^{n}\frac{d^{n}}{d\beta^{n}}\big{(}e^{-\beta
x^{2}}\big{)}\arrowvert_{\beta=0},$ $\displaystyle x_{\alpha}e^{iP.x}$
$\displaystyle=$ $\displaystyle(-i)\frac{d}{dP^{\alpha}}e^{iP.x}.$ (22)

After this stage, one needs to perform the double Borel transformation with
respect to $(p+q)^{2}$ and $p^{2}$ via

${\cal B}_{(p+q)^{2}}(M_{1}^{2}){\cal
B}_{p^{2}}(M_{2}^{2})e^{b(p+uq)^{2}}=M^{2}\delta(b+\frac{1}{M^{2}})\delta(u_{0}-u)e^{\frac{-q^{2}}{M_{1}^{2}+M_{2}^{2}}},$
(23)

where $u_{0}=M_{1}^{2}/(M_{1}^{2}+M_{2}^{2})$ and
$\frac{1}{M^{2}}=\frac{1}{M_{1}^{2}}+\frac{1}{M_{2}^{2}}$. The details of the
calculation can be found in [43]. For example, for the configuration

$\displaystyle{\cal Z}_{\alpha\beta}(p,q)$ $\displaystyle=$ $\displaystyle
i\int d^{4}x\int_{0}^{1}dv\int{\cal D}\alpha
e^{i[p+(\alpha_{\bar{q}}+v\alpha_{g})q].x}\mathcal{G}(\alpha_{i})\big{(}x^{2}\big{)}^{n}$
(24) $\displaystyle\times$ $\displaystyle
x_{\alpha}x_{\beta}K_{\mu}(m_{Q}\sqrt{-x^{2}})K_{\nu}(m_{Q}\sqrt{-x^{2}}),$

we get

$\displaystyle{\cal Z}_{\alpha\beta}(M^{2})$ $\displaystyle=$
$\displaystyle\frac{i\pi^{2}2^{4-\mu-\nu}e^{\frac{-q^{2}}{M_{1}^{2}+M_{2}^{2}}}}{M^{2}m_{Q_{1}}^{2\mu}m_{Q_{2}}^{2\nu}}\int\mathcal{D}\alpha\int_{0}^{1}dv\int_{0}^{1}dz\frac{\partial^{n}}{\partial\beta^{n}}e^{-\frac{m_{1}^{2}\bar{z}+m_{2}^{2}z}{z\bar{z}(M^{2}-4\beta)}}z^{\mu-1}\bar{z}^{\nu-1}(M^{2}-4\beta)^{\mu+\nu-1}$
(25) $\displaystyle\times$
$\displaystyle\delta[u_{0}-(\alpha_{q}+v\alpha_{g})]\Big{[}p_{\alpha}p_{\beta}+(v\alpha_{g}+\alpha_{q})(p_{\alpha}q_{\beta}+q_{\alpha}p_{\beta})+(v\alpha_{g}+\alpha_{q})^{2}q_{\alpha}q_{\beta}$
$\displaystyle+\frac{M^{2}}{2}g_{\alpha\beta}\Big{]}.$

After completing the calculations in the hadronic and QCD sides in Borel
scheme, we match the coefficients of the selected structures to obtain the
desired sum rules as

$\displaystyle g_{1}+g_{2}{m_{2}\over m_{1}+m_{2}}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!{1\over
2\lambda_{B_{Q_{1}}^{*}}\lambda_{B_{Q_{2}}^{*}}}e^{{m_{1}^{2}\over
M_{1}^{2}}+{m_{2}^{2}\over M_{2}^{2}}+{m_{V}^{2}\over
M_{1}^{2}+M_{2}^{2}}}\Pi_{A}~{},$ $\displaystyle g_{2}\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!-{m_{1}+m_{2}\over
2\lambda_{B_{Q_{1}}^{*}}\lambda_{B_{Q_{2}}^{*}}}e^{{m_{1}^{2}\over
M_{1}^{2}}+{m_{2}^{2}\over M_{2}^{2}}+{m_{V}^{2}\over
M_{1}^{2}+M_{2}^{2}}}\Pi_{B}~{},$ $\displaystyle g_{3}\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!-{(m_{1}+m_{2})^{2}\over\lambda_{B_{Q_{1}}^{*}}\lambda_{B_{Q_{2}}^{*}}}e^{{m_{1}^{2}\over
M_{1}^{2}}+{m_{2}^{2}\over M_{2}^{2}}+{m_{V}^{2}\over
M_{1}^{2}+M_{2}^{2}}}\Pi_{C}~{},$ $\displaystyle g_{4}\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!-{(m_{1}+m_{2})^{3}\over
2\lambda_{B_{Q_{1}}^{*}}\lambda_{B_{Q_{2}}^{*}}}e^{{m_{1}^{2}\over
M_{1}^{2}}+{m_{2}^{2}\over M_{2}^{2}}+{m_{V}^{2}\over
M_{1}^{2}+M_{2}^{2}}}\Pi_{D}~{},$ (26)

where the explicit forms of $\Pi_{A}$, $\Pi_{B}$, $\Pi_{C}$ and $\Pi_{D}$
functions are presented in the Appendix Appendix A: Invariant Functions
$\Pi_{A}$, $\Pi_{B}$, $\Pi_{C}$ and $\Pi_{D}$.. As the mass of the initial and
final baryons are the same/close, we set $M_{1}^{2}=M_{2}^{2}=2M^{2}$ and
$u_{0}=1/2$. To further suppress the contributions of the higher states and
continuum we perform the subtraction procedure according to the standard
prescriptions of the method. In this case, it is enough to set

$\displaystyle
e^{-\frac{m_{1}^{2}\bar{z}+m_{2}^{2}z}{M^{2}z\bar{z}}}=e^{-s_{0}/M^{2}},$ (27)

where $s_{0}$ is the continuum threshold. This leads to

$\displaystyle\int_{0}^{1}dz\rightarrow\int_{z_{\text{min}}}^{z_{\text{max}}}dz,$
(28)

where $z_{\text{max}}$ and $z_{\text{min}}$ are given as

$\displaystyle
z_{\text{max}(\text{min})}=\frac{1}{2s_{0}}\Big{[}(s_{0}+m_{1}^{2}-m_{2}^{2})+(-)\sqrt{(s_{0}+m_{1}^{2}-m_{2}^{2})^{2}-4m_{1}^{2}s_{0}}\Big{]}.$
(29)

##  3 Numerical analyses

In this section, we will go over the numerical analyses of the sum rules for
the strong coupling constants of the spin-$3/2$ doubly heavy baryons with
light vector mesons. To this end, two sets of input parameters are needed. The
first one corresponds to nonperturbative parameters coming from distribution
amplitudes (DAs), associated with different twists normalized to the mass
scale $\mu=1$ GeV as well as the mass and decay constants of the light vector
mesons. These parameters are gathered in tables 2 and 3. The second set
corresponds to the masses and residues of the doubly heavy hadrons with
spin-$3/2$. These parameters are taken from [30] and are listed in table 4.

Parameters | Values
---|---
$m_{s}$ | $95^{+9}_{-3}~{}\mbox{MeV}$
$m_{c}$ | $1.275^{+0.025}_{-0.035}~{}\mbox{GeV}$
$m_{b}$ | $4.18^{+0.04}_{-0.03}~{}\mbox{GeV}$
$m_{\rho}$ | $775.26\pm 0.25~{}\mbox{MeV}$
$m_{\omega}$ | $782.65\pm 0.12~{}\mbox{MeV}$
$m_{\phi}$ | $1019.461\pm 0.016~{}\mbox{MeV}$
$m_{K^{*0}}$ | $895.55\pm 0.020~{}\mbox{MeV}$
$m_{\bar{K}^{*0}}$ | $895.55\pm 0.020~{}\mbox{MeV}$
$m_{K^{*\pm}}$ | $891.76\pm 0.025~{}\mbox{MeV}$
Table 2: The masses of quarks and vector mesons [57].

Parameters | Values $[\text{MeV}]$
---|---
$f_{\rho}^{\parallel}$ | $216\pm 3$
$f_{\omega}^{\parallel}$ | $187\pm 5$
$f_{K^{*}}^{\parallel}$ | $220\pm 5$
$f_{\phi}^{\parallel}$ | $215\pm 5$
$f_{\rho}^{\perp}$ | $165\pm 9$
$f_{\omega}^{\perp}$ | $151\pm 9$
$f_{K^{*}}^{\perp}$ | $185\pm 10$
$f_{\phi}^{\perp}$ | $186\pm 9$

Table 3: The longitudinal ($f^{\parallel}$) and transverse ($f^{\perp}$) decay
constants for the vector mesons $\rho$, $\omega$, and $\phi$ [58]

.

Baryon |  Mass $[\text{GeV}]$ | Residue $[\text{GeV}^{3}]$
---|---|---
$\Xi^{*}_{cc}$ | $369\pm 0.16$ | $0.12\pm 0.01$
$\Xi^{*}_{bc}$ | $7.25\pm 0.20$ | $0.15\pm 0.01$
$\Xi^{*}_{bb}$ | $10.14\pm 1.0$ | $0.22\pm 0.03$
$\Omega^{*}_{cc}$ | $3.78\pm 0.16$ | $0.14\pm 0.02$
$\Omega^{*}_{bc}$ | $7.3\pm 0.2$ | $0.18\pm 0.02$
$\Omega^{*}_{bb}$ | $10.5\pm 0.2$ | $0.25\pm 0.03$
Table 4: The masses and residues of spin-$3/2$ doubly heavy baryons [30].

The continuum threshold $s_{0}$ and the Borel parameter $M^{2}$ are the
auxiliary parameters of the sum rules for the strong coupling constants that
are needed to be fixed. Consequently, we need to locate the working region of
these parameters, where the variations of the results concerning the changes
in these parameters should be minimal.

The continuum threshold $s_{0}$ depends on the energy of the first excited
state at each channel. There is no experimental information on the first
excited states in the doubly heavy baryon channels, hence, we pick up the
region $(m_{B^{*}}+0.3)^{2}\leq
s_{0}\leq(m_{B^{*}}+0.7)^{2}~{}\text{GeV}^{2}$, where the variation of the
results with respect to the threshold parameter are mild. Figs. 1, 2 and 3, as
examples, display the dependence of the strong coupling constants $g_{1}$,
$g_{2}$, $g_{3}$ and $g_{4}$ on the parameter $s_{0}$ for the vertices
$\Xi^{*}_{cc}\Xi^{*}_{cc}\rho^{\pm}(\omega)$,
$\Xi^{*}_{bc}\Xi^{*}_{bc}\rho^{\pm}(\omega)$ and
$\Xi^{*}_{bb}\Xi^{*}_{bb}\rho^{\pm}(\omega)$ respectively. These figures
reveal that the values of the strong coupling constants remain roughly
unchanged at the presented region of $s_{0}$.

In this step, we would like to fix the working region of the Borel parameter
$M^{2}$. The lower limit of $M^{2}$ is determined by the requirement of the
OPE convergence. The upper limit of this parameter is obtained demanding the
pole dominance, i.e., the pole contribution should be more than $50\%$ of the
total contribution for each of the selected structures. As a result of these
requirements the Borel windows are determined to be $14$ GeV2 $\leq M^{2}$
$\leq 18$ GeV 2, $8$ GeV2 $\leq M^{2}$ $\leq 11$ GeV 2 and $4$ GeV2 $\leq
M^{2}$ $\leq 6$ GeV 2 for $bb$, $bc$ and $cc$ channels, respectively. Figs. 4,
5 and 6, as examples, illustrate the dependence of the strong coupling
constants $g_{1}$, $g_{2}$, $g_{3}$ and$g_{4}$ on the Borel mass $M^{2}$ for
the vertices $\Xi^{*}_{cc}\Xi^{*}_{cc}\rho^{\pm}(\omega)$,
$\Xi^{*}_{bc}\Xi^{*}_{bc}\rho^{\pm}(\omega)$ and
$\Xi^{*}_{bb}\Xi^{*}_{bb}\rho^{\pm}(\omega)$, respectively at average values
of the continuum threshold. Our analyses show that the variations of the
strong coupling constants with respect to $M^{2}$ are relatively small in the
selected Borel window.

Using all the input values as well as the working intervals for the auxiliary
parameters we find the numerical values for the strong coupling constants
under study from the obtained sum rules for the strong coupling form factors
at $q^{2}=m_{V}^{2}$. The predicted results for $g_{1}$, $g_{2}$, $g_{3}$ and
$g_{4}$ for all the considered vertices in $bb$, $bc$ and $cc$ channels are
demonstrated in Table 5. The errors in the values can be attributed to the
uncertainties in the calculations of the working regions for $M^{2}$ and
$s_{0}$, the errors in the masses and residues of the doubly heavy baryons and
the uncertainties originating from the DAs parameters. Our results may be
checked via different nonperturbative approaches.

Figure 1: The strong couplings $g_{i}$ as functions of continuum threshold $s_{0}$ for the vertices $\Xi^{*}_{cc}\Xi^{*}_{cc}\rho^{\pm}$ and $\Xi^{*}_{cc}\Xi^{*}_{cc}\omega$ at $M^{2}=5$ GeV2. Figure 2: The strong couplings $g_{i}$ as functions of continuum threshold $s_{0}$ for the vertices $\Xi^{*}_{bc}\Xi^{*}_{bc}\rho^{\pm}$ and $\Xi^{*}_{bc}\Xi^{*}_{bc}\omega$ at $M^{2}=9.5$ GeV2. Figure 3: The strong couplings $g_{i}$ as functions of continuum threshold $s_{0}$ for the vertices $\Xi^{*}_{bb}\Xi^{*}_{bb}\rho^{\pm}$ and $\Xi^{*}_{bb}\Xi^{*}_{bb}\omega$ at $M^{2}=16$ GeV2. Figure 4: The strong couplings $g_{i}$ as functions of the Borel parameter $M^{2}$ for the vertices $\Xi^{*}_{cc}\Xi^{*}_{cc}\rho^{\pm}$ and $\Xi^{*}_{cc}\Xi^{*}_{cc}\omega$ at $s_{0}=17.6$ GeV2. Figure 5: The strong couplings $g_{i}$ as functions of the Borel parameter $M^{2}$ for the vertices $\Xi^{*}_{bc}\Xi^{*}_{bc}\rho^{\pm}$ and $\Xi^{*}_{bc}\Xi^{*}_{bc}\omega$ at $s_{0}=60.1$ GeV2. Figure 6: The strong couplings $g_{i}$ as functions of the Borel parameter $M^{2}$ for the vertices $\Xi^{*}_{bb}\Xi^{*}_{bb}\rho^{\pm}$ and $\Xi^{*}_{bb}\Xi^{*}_{bb}\omega$ at $s_{0}=118.9$ GeV2. Vertex | $g_{1}$ | $g_{2}$ | $g_{3}$ | $g_{4}$
---|---|---|---|---
$\Xi^{*}_{bb}\Xi^{*}_{bb}\rho^{0}$ | $45.8^{\>2.5}_{\>2.0}$ | $164.1^{\>5.9}_{\>2.0}$ | $162.6^{\>24}_{\>20}$ | $24.8^{\>4.1}_{\>3.6}$
$\Xi^{*}_{bc}\Xi^{*}_{bc}\rho^{0}$ | $12.3^{\>0.4}_{\>0.3}$ | $35.3^{\>1.0}_{\>0.7}$ | $43.3^{\>8.1}_{\>6.5}$ | $6.9^{\>0.8}_{\>0.8}$
$\Xi^{*}_{cc}\Xi^{*}_{cc}\rho^{0}$ | $4.5^{\>0.5}_{\>0.4}$ | $8.98^{\>1.4}_{\>1.3}$ | $11.5^{\>0.7}_{\>0.5}$ | $1.6^{\>0.04}_{\>0.04}$
$\Xi^{*}_{bb}\Xi^{*}_{bb}\rho^{\pm}$ | $64.8^{\>3.5}_{\>2.8}$ | $232.04^{\>8.4}_{\>5.6}$ | $229.9^{\>34.5}_{\>29.2}$ | $35.2^{\>5.8}_{\>5.2}$
$\Xi^{*}_{bc}\Xi^{*}_{bc}\rho^{\pm}$ | $17.4^{\>0.6}_{\>0.4}$ | $49.8^{\>1.5}_{\>1.1}$ | $61.2^{\>11.4}_{\>9.3}$ | $9.7^{\>1.2}_{\>1.1}$
$\Xi^{*}_{cc}\Xi^{*}_{cc}\rho^{\pm}$ | $6.3^{\>0.7}_{\>0.6}$ | $12.7^{\>2}_{\>1.8}$ | $16.3^{\>1.0}_{\>0.7}$ | $2.3^{\>0.06}_{\>0.06}$
$\Xi^{*}_{bb}\Xi^{*}_{bb}\omega$ | $56.7^{\>2.9}_{\>2.3}$ | $207.5^{\>6.9}_{\>4.5}$ | $200.9^{\>30.1}_{\>25.5}$ | $34.2^{\>6.0}_{\>5.2}$
$\Xi^{*}_{bc}\Xi^{*}_{bc}\omega$ | $15.27^{\>2.9}_{\>2.3}$ | $44.44^{\>1.1}_{\>0.9}$ | $53.46^{\>10}_{\>8}$ | $9.70^{\>1.4}_{\>1.3}$
$\Xi^{*}_{cc}\Xi^{*}_{cc}\omega$ | $5.6^{\>0.7}_{\>0.6}$ | $11.3^{\>1.8}_{\>1.7}$ | $14.2^{\>0.9}_{\>0.6}$ | $2.2^{\>0.06}_{\>0.07}$
$\Omega^{*}_{bb}\Omega^{*}_{bb}\phi$ | $62.7^{\>1.6}_{\>1.2}$ | $215.5^{\>7.2}_{\>4.8}$ | $264.9^{\>44}_{\>37}$ | $33.1^{\>3.3}_{\>3.2}$
$\Omega^{*}_{bc}\Omega^{*}_{bc}\phi$ | $14.2^{\>0.2}_{\>0.1}$ | $37.3^{\>0.5}_{\>0.6}$ | $60.2^{\>12.1}_{\>9.8}$ | $5.2^{\>0.7}_{\>1.2}$
$\Omega^{*}_{cc}\Omega^{*}_{cc}\phi$ | $6.5^{\>0.8}_{\>0.8}$ | $10.1^{\>1.9}_{\>1.8}$ | $18.5^{\>1.7}_{\>1.2}$ | $1.5^{\>0.4}_{\>0.6}$
$\Omega^{*}_{bb}\Xi^{*}_{bb}K^{*0}$ | $71.5^{\>3.3}_{\>2.6}$ | $251.8^{\>9}_{\>6}$ | $262.8^{\>4}_{\>3}$ | $25.3^{\>1.6}_{\>1.7}$
$\Omega^{*}_{bc}\Xi^{*}_{bc}K^{*0}$ | $18.1^{\>0.4}_{\>0.2}$ | $49.0^{\>1.0}_{\>0.9}$ | $64.7^{\>12}_{\>10}$ | $3.9^{\>1.0}_{\>1.6}$
$\Omega^{*}_{cc}\Xi^{*}_{cc}K^{*0}$ | $7.4^{\>0.8}_{\>0.7}$ | $13.1^{\>2.0}_{\>2.0}$ | $18.5^{\>1.4}_{\>1.0}$ | $1.0^{\>0.5}_{\>0.6}$
$\Xi^{*}_{bb}\Omega^{*}_{bb}\bar{K}^{*0}$ | $72.7^{\>3.4}_{\>2.7}$ | $251.6^{\>9.1}_{\>6.2}$ | $262.6^{\>41}_{\>35}$ | $25.3^{\>1.6}_{\>1.7}$
$\Xi^{*}_{bc}\Omega^{*}_{bc}\bar{K}^{*0}$ | $18.2^{\>0.4}_{\>0.2}$ | $49.0^{\>1.1}_{\>0.9}$ | $64.7^{\>12.5}_{\>10.2}$ | $4.0^{\>1.0}_{\>1.6}$
$\Xi^{*}_{cc}\Omega^{*}_{cc}\bar{K}^{*0}$ | $7.6^{\>0.8}_{\>0.8}$ | $13.1^{\>2.1}_{\>2.0}$ | $18.4^{\>1.4}_{\>1.0}$ | $1.1^{\>0.5}_{\>0.6}$
Table 5: Strong coupling costants of the doubly heavy spin-$3/2$ baryons with
light vector mesons obtained from the sum rules at $q^{2}=m_{V}^{2}$.

##  4 Summary and conclusions

After the observation of $\Xi_{cc}^{++}$ by the LHCb collaboration, the
possibility for identification of other doubly heavy baryons predicted by the
quark model has increased. In parallel, many theoretical studies on different
properties of doubly heavy baryons are performed. One of the important issues
in hadron physics is to determine the strong coupling constants among
different hadronic multiplets. In this connection, we studied the strong
vertices of the spin-3/2 doubly heavy baryons with light vector $\rho$,
$K^{*}$, $\omega$ and $\phi$ mesons. In the calculations that were done via
LCSR, we used the interpolating currents of the baryons and DAs of the vector
mesons. The correlation function in spin-3/2 channel receives contributions
also from the spin-1/2 doubly heavy baryons, that were removed by an
appropriate ordering of Dirac matrices and selection of Lorentz structure free
from the unwanted contributions. Such vertices are parameterized in terms of
four strong coupling form factors. The values of these strong form factors at
$q^{2}=m_{V}^{2}$ are the strong coupling form factors, which have been
determined. These strong coupling form factors are basic quantities that carry
information not only on the internal structure of the participating particles
but also on the nature of strong interactions among the doubly heavy baryons
and vector mesons. The values of these coupling constants may be used in the
determination of the strong potentials among the hadronic multiplets. We hope
that we will have more experimental data on different physical quantities of
the doubly heavy baryons. Our results may help experimental groups in analyses
of the related data.

## ACKNOWLEDGEMENTS

K. Azizi and S. Rostami are thankful to Iran Science Elites Federation
(Saramadan) for the financial support provided under the grant number
ISEF/M/99171.

## Appendix A:
Invariant Functions $\Pi_{A}$, $\Pi_{B}$, $\Pi_{C}$ and $\Pi_{D}$.

In this appendix, we give the explicit forms of the $\Pi_{A}$, $\Pi_{B}$,
$\Pi_{C}$ and $\Pi_{D}$ functions. Each $\Pi_{i}$ function can be written as

$\displaystyle\Pi_{i}=\Pi_{i}^{0}+\Pi_{i}^{GG},$ (30)

where the components $\Pi_{A,B,CD}^{0}$ and $\Pi_{A,B,CD}^{GG}$ for the
$\Omega^{*}_{cc}\Xi^{*}_{cc}K^{*}$ vertex (as an example) are given as:

$\displaystyle\Pi_{A}^{0}$ $\displaystyle=$
$\displaystyle\frac{e^{-\frac{q^{2}}{M_{1}^{2}+M_{2}^{2}}}}{24M^{2}\pi^{2}}\Biggl{\\{}m_{c}M^{2}{\cal
A}^{\perp}(u_{0})f_{\rho}^{\perp}m_{\rho}^{2}(-m_{c}^{2}\zeta_{-1,0}(m_{c})-m_{c}^{2}\zeta_{0,-1}(m_{c})+m_{c}^{2}\zeta_{0,0}(m_{c})$
(31) $\displaystyle-$ $\displaystyle
2M^{2}\zeta_{0,0}(m_{c})+m_{c}^{2}\zeta_{1,-1}(m_{c})+2M^{2}\zeta_{1,0}(m_{c}))+2m_{c}M^{2}f_{\rho}^{\perp}\Bigg{[}2M^{4}\phi^{\perp}(u_{0})(\zeta_{0,0}(m_{c})-\zeta_{1,0}(m_{c}))$
$\displaystyle+$ $\displaystyle m_{\rho}^{2}\Bigg{(}-6q^{2}u_{0}i_{1}({\cal
T}_{4}(\alpha_{i}),1)\zeta_{-1,0}(m_{c})-M^{2}i_{2}({\cal
S}(\alpha_{i}),1)\zeta_{-1,0}(m_{c})+3M^{2}i_{2}(\tilde{\cal
S}(\alpha_{i}),1)\zeta_{-1,0}(m_{c})$ $\displaystyle-$ $\displaystyle
M^{2}i_{2}({\cal T}_{3}(\alpha_{i}),1)\zeta_{-1,0}(m_{c})-M^{2}i_{2}({\cal
T}_{4}(\alpha_{i}),1)\zeta_{-1,0}(m_{c})$ $\displaystyle+$ $\displaystyle
4q^{2}u_{0}i_{1}({\cal
T}(\alpha_{i}),1)(\zeta_{-1,0}(m_{c})-\zeta_{0,0}(m_{c}))-16M^{2}u_{0}\psi^{\parallel}(u_{0})\zeta_{0,0}(m_{c})$
$\displaystyle+$ $\displaystyle 4q^{2}u_{0}i_{1}({\cal
T}_{4}(\alpha_{i}),1)\zeta_{0,0}(m_{c})-5M^{2}i_{2}({\cal
S}(\alpha_{i}),1)\zeta_{0,0}(m_{c})$ $\displaystyle+$ $\displaystyle
8M^{2}i_{2}({\cal
S}(\alpha_{i}),v)\zeta_{0,0}(m_{c})-3M^{2}i_{2}(\tilde{S}(\alpha_{i}),1)\zeta_{0,0}(m_{c})+M^{2}i_{2}({\cal
T}_{1}(\alpha_{i}),1)\zeta_{0,0}(m_{c})$ $\displaystyle-$ $\displaystyle
2M^{2}i_{2}({\cal T}_{1}(\alpha_{i}),v)\zeta_{0,0}(m_{c})-M^{2}i_{2}({\cal
T}_{2}(\alpha_{i}),1)\zeta_{0,0}(m_{c})+2M^{2}i_{2}({\cal
T}_{2}(\alpha_{i}),v)\zeta_{0,0}(m_{c})$ $\displaystyle+$ $\displaystyle
2M^{2}i_{2}({\cal T}_{3}(\alpha_{i}),1)\zeta_{0,0}(m_{c})-2M^{2}i_{2}({\cal
T}_{3}(\alpha_{i}),v)\zeta_{0,0}(m_{c})+M^{2}i_{2}({\cal
T}_{4}(\alpha_{i}),1)\zeta_{0,0}(m_{c})$ $\displaystyle+$ $\displaystyle
2M^{2}i_{2}({\cal
T}_{4}(\alpha_{i}),v)\zeta_{0,0}(m_{c})+6q^{2}u_{0}i_{1}({\cal
T}_{3}(\alpha_{i}),1)(-\zeta_{-1,0}(m_{c})+\zeta_{0,0}(m_{c}))$
$\displaystyle+$ $\displaystyle
16M^{2}u_{0}\psi^{\parallel}(u_{0})\zeta_{1,0}(m_{c})\Bigg{)}\Bigg{]}+2f_{\rho}^{\parallel}m_{\rho}\Bigg{[}m_{\rho}^{2}\Bigg{(}12m_{c}^{2}M^{2}i_{1}(\tilde{\phi}(\alpha_{i}),1)\zeta_{-1,0}(m_{c})$
$\displaystyle-$ $\displaystyle
4m_{c}^{2}M^{2}i_{1}(\psi(\alpha_{i}),1)\zeta_{-1,0}(m_{c})+8m_{c}^{2}q^{2}u_{0}^{2}i_{1}(\psi(\alpha_{i}),1)\zeta_{-1,0}(m_{c})+8m_{c}^{2}M^{2}i_{1}(\psi(\alpha_{i}),v)\zeta_{-1,0}(m_{c})$
$\displaystyle-$ $\displaystyle
16m_{c}^{2}q^{2}u_{0}^{2}i_{1}(\psi(\alpha_{i}),v)\zeta_{-1,0}(m_{c})-8m_{c}^{2}q^{2}u_{0}i_{1}(\alpha_{g}\psi(\alpha_{i}),v)\zeta_{-1,0}(m_{c})$
$\displaystyle+$ $\displaystyle
16m_{c}^{2}q^{2}u_{0}i_{1}(\alpha_{g}\psi(\alpha_{i}),v^{2})\zeta_{-1,0}(m_{c})-8m_{c}^{2}q^{2}u_{0}i_{1}(\alpha_{q}\psi(\alpha_{i}),1)\zeta_{-1,0}(m_{c})$
$\displaystyle+$ $\displaystyle
16m_{c}^{2}q^{2}u_{0}i_{1}(\alpha_{q}\psi(\alpha_{i}),v)\zeta_{-1,0}(m_{c})+12m_{c}^{2}M^{2}i_{1}(\tilde{\psi}(\alpha_{i}),1)\zeta_{-1,0}(m_{c})$
$\displaystyle+$ $\displaystyle
40m_{c}^{2}q^{2}u_{0}^{2}i_{1}(\tilde{\psi}(\alpha_{i}),1)\zeta_{-1,0}(m_{c})-40m_{c}^{2}q^{2}u_{0}i_{1}(\alpha_{g}\tilde{\psi}(\alpha_{i}),v)\zeta_{-1,0}(m_{c})$
$\displaystyle-$ $\displaystyle
40m_{c}^{2}q^{2}u_{0}i_{1}(\alpha_{q}\tilde{\psi}(\alpha_{i}),1)\zeta_{-1,0}(m_{c})+3m_{c}^{4}j_{1}({\cal
A}^{\parallel}(u))\zeta_{-1,0}(m_{c})+4m_{c}^{2}M^{2}i_{1}(\tilde{\phi}(\alpha_{i}),1)\zeta_{0,-1}(m_{c})$
$\displaystyle-$ $\displaystyle
4m_{c}^{2}M^{2}i_{1}(\psi(\alpha_{i}),1)\zeta_{0,-1}(m_{c})+8m_{c}^{2}q^{2}u_{0}^{2}i_{1}(\psi(\alpha_{i}),1)\zeta_{0,-1}(m_{c})+8m_{c}^{2}M^{2}i_{1}(\psi(\alpha_{i}),v)\zeta_{0,-1}(m_{c})$
$\displaystyle-$ $\displaystyle
16m_{c}^{2}q^{2}u_{0}^{2}i_{1}(\psi(\alpha_{i}),v)\zeta_{0,-1}(m_{c})-8m_{c}^{2}q^{2}u_{0}i_{1}(\alpha_{g}\psi(\alpha_{i}),v)\zeta_{0,-1}(m_{c})$
$\displaystyle+$ $\displaystyle
16m_{c}^{2}q^{2}u_{0}i_{1}(\alpha_{g}\psi(\alpha_{i}),v^{2})\zeta_{0,-1}(m_{c})-8m_{c}^{2}q^{2}u_{0}i_{1}(\alpha_{q}\psi(\alpha_{i}),1)\zeta_{0,-1}(m_{c})$
$\displaystyle+$ $\displaystyle
16m_{c}^{2}q^{2}u_{0}i_{1}(\alpha_{q}\psi(\alpha_{i}),v)\zeta_{0,-1}(m_{c})+4m_{c}^{2}M^{2}i_{1}(\tilde{\psi}(\alpha_{i}),1)\zeta_{0,-1}(m_{c})$
$\displaystyle+$ $\displaystyle
24m_{c}^{2}q^{2}u_{0}^{2}i_{1}(\tilde{\psi}(\alpha_{i}),1)\zeta_{0,-1}(m_{c})-24m_{c}^{2}q^{2}u_{0}i_{1}(\alpha_{g}\tilde{\psi}(\alpha_{i}),v)\zeta_{0,-1}(m_{c})$
$\displaystyle-$ $\displaystyle
24m_{c}^{2}q^{2}u_{0}i_{1}(\alpha_{q}\tilde{\psi}(\alpha_{i}),1)\zeta_{0,-1}(m_{c})+4m_{c}^{4}j_{1}({\cal
A}^{\parallel}(u))\zeta_{0,-1}(m_{c})+m_{c}^{2}M^{2}u_{0}{\cal
A}^{\parallel}(u_{0})\zeta_{0,0}(m_{c})$ $\displaystyle+$ $\displaystyle
20m_{c}^{2}M^{2}u_{0}\tilde{j}_{1}({\cal
C}(u))\zeta_{0,0}(m_{c})-4m_{c}^{2}M^{2}i_{1}(\tilde{\phi}(\alpha_{i}),1)\zeta_{0,0}(m_{c})+4M^{4}i_{1}(\tilde{\phi}(\alpha_{i}),1)\zeta_{0,0}(m_{c})$
$\displaystyle+$ $\displaystyle
4m_{c}^{2}M^{2}i_{1}(\psi(\alpha_{i}),1)\zeta_{0,0}(m_{c})-12M^{4}i_{1}(\psi(\alpha_{i}),1)\zeta_{0,0}(m_{c})-8m_{c}^{2}q^{2}u_{0}^{2}i_{1}(\psi(\alpha_{i}),1)\zeta_{0,0}(m_{c})$
$\displaystyle+$ $\displaystyle
16M^{2}q^{2}u_{0}^{2}i_{1}(\psi(\alpha_{i}),1)\zeta_{0,0}(m_{c})-8m_{c}^{2}M^{2}i_{1}(\psi(\alpha_{i}),v)\zeta_{0,0}(m_{c})+24M^{4}i_{1}(\psi(\alpha_{i}),v)\zeta_{0,0}(m_{c})$
$\displaystyle+$ $\displaystyle
16m_{c}^{2}q^{2}u_{0}^{2}i_{1}(\psi(\alpha_{i}),v)\zeta_{0,0}(m_{c})-32M^{2}q^{2}u_{0}^{2}i_{1}(\psi(\alpha_{i}),v)\zeta_{0,0}(m_{c})$
$\displaystyle+$ $\displaystyle
8m_{c}^{2}q^{2}u_{0}i_{1}(\alpha_{g}\psi(\alpha_{i}),v)\zeta_{0,0}(m_{c})+12M^{4}i_{1}(\tilde{\psi}(\alpha_{i}),1)\zeta_{0,0}(m_{c})$
$\displaystyle-$ $\displaystyle
16M^{2}q^{2}u_{0}i_{1}(\alpha_{g}\psi(\alpha_{i}),v)\zeta_{0,0}(m_{c})-16m_{c}^{2}q^{2}u_{0}i_{1}(\alpha_{g}\psi(\alpha_{i}),v^{2})\zeta_{0,0}(m_{c})$
$\displaystyle+$ $\displaystyle
32M^{2}q^{2}u_{0}i_{1}(\alpha_{g}\psi(\alpha_{i}),v^{2})\zeta_{0,0}(m_{c})+8m_{c}^{2}q^{2}u_{0}i_{1}(\alpha_{q}\psi(\alpha_{i}),1)\zeta_{0,0}(m_{c})$
$\displaystyle-$ $\displaystyle
16M^{2}q^{2}u_{0}i_{1}(\alpha_{q}\psi(\alpha_{i}),1)\zeta_{0,0}(m_{c})-16m_{c}^{2}q^{2}u_{0}i_{1}(\alpha_{q}\psi(\alpha_{i}),v)\zeta_{0,0}(m_{c})$
$\displaystyle+$ $\displaystyle
32M^{2}q^{2}u_{0}i_{1}(\alpha_{q}\psi(\alpha_{i}),v)\zeta_{0,0}(m_{c})-4m_{c}^{2}M^{2}i_{1}(\tilde{\psi}(\alpha_{i}),1)\zeta_{0,0}(m_{c})$
$\displaystyle-$ $\displaystyle
24m_{c}^{2}q^{2}u_{0}^{2}i_{1}(\tilde{\psi}(\alpha_{i}),1)\zeta_{0,0}(m_{c})+48M^{2}q^{2}u_{0}^{2}i_{1}(\tilde{\psi}(\alpha_{i}),1)\zeta_{0,0}(m_{c})$
$\displaystyle+$ $\displaystyle
24m_{c}^{2}q^{2}u_{0}i_{1}(\alpha_{g}\tilde{\psi}(\alpha_{i}),v)\zeta_{0,0}(m_{c})-48M^{2}q^{2}u_{0}i_{1}(\alpha_{g}\tilde{\psi}(\alpha_{i}),v)\zeta_{0,0}(m_{c})$
$\displaystyle+$ $\displaystyle
24m_{c}^{2}q^{2}u_{0}i_{1}(\alpha_{q}\tilde{\psi}(\alpha_{i}),1)\zeta_{0,0}(m_{c})-48M^{2}q^{2}u_{0}i_{1}(\alpha_{q}\tilde{\psi}(\alpha_{i}),1)\zeta_{0,0}(m_{c})$
$\displaystyle+$ $\displaystyle
4M^{4}i_{2}(\alpha_{g}\phi(\alpha_{i}),v)\zeta_{0,0}(m_{c})-8M^{4}i_{2}(\alpha_{g}\phi(\alpha_{i}),v^{2})\zeta_{0,0}(m_{c})$
$\displaystyle+$ $\displaystyle
4M^{4}i_{2}(\alpha_{q}\phi(\alpha_{i}),1)\zeta_{0,0}(m_{c})-8M^{4}i_{2}(\alpha_{q}\phi(\alpha_{i}),v)\zeta_{0,0}(m_{c})+4M^{4}i_{2}(\alpha_{g}\tilde{\phi}(\alpha_{i}),v)\zeta_{0,0}(m_{c})$
$\displaystyle+$ $\displaystyle
4M^{4}i_{2}(\alpha_{q}\tilde{\phi}(\alpha_{i}),1)\zeta_{0,0}(m_{c})+4M^{4}i_{2}(\alpha_{g}\psi(\alpha_{i}),v)\zeta_{0,0}(m_{c})-8M^{4}i_{2}(\alpha_{g}\psi(\alpha_{i}),v^{2})\zeta_{0,0}(m_{c})$
$\displaystyle+$ $\displaystyle
4M^{4}i_{2}(\alpha_{q}\psi(\alpha_{i}),1)\zeta_{0,0}(m_{c})-8M^{4}i_{2}(\alpha_{q}\psi(\alpha_{i}),v)\zeta_{0,0}(m_{c})+4M^{4}i_{2}(\alpha_{g}\tilde{\psi}(\alpha_{i}),v)\zeta_{0,0}(m_{c})$
$\displaystyle+$ $\displaystyle
4M^{4}i_{2}(\alpha_{q}\tilde{\psi}(\alpha_{i}),1)\zeta_{0,0}(m_{c})-m_{c}^{4}j_{1}({\cal
A}^{\parallel}(u))\zeta_{0,0}(m_{c})+5m_{c}^{2}M^{2}j_{1}({\cal
A}^{\parallel}(u))\zeta_{0,0}(m_{c})$ $\displaystyle+$ $\displaystyle
m_{c}^{4}j_{1}({\cal
A}^{\parallel}(u))\zeta_{1,-2}(m_{c})+m_{c}^{2}M^{2}u_{0}{\cal
A}^{\parallel}(u_{0})\zeta_{1,-1}(m_{c})+12m_{c}^{2}M^{2}u_{0}\tilde{j}_{1}({\cal
C}(u))\zeta_{1,-1}(m_{c})$ $\displaystyle-$ $\displaystyle
4m_{c}^{2}M^{2}i_{1}(\tilde{\phi}(\alpha_{i}),1)\zeta_{1,-1}(m_{c})+4m_{c}^{2}M^{2}i_{1}(\psi(\alpha_{i}),1)\zeta_{1,-1}(m_{c})-8m_{c}^{2}q^{2}u_{0}^{2}i_{1}(\psi(\alpha_{i}),1)\zeta_{1,-1}(m_{c})$
$\displaystyle-$ $\displaystyle
8m_{c}^{2}M^{2}i_{1}(\psi(\alpha_{i}),v)\zeta_{1,-1}(m_{c})+16m_{c}^{2}q^{2}u_{0}^{2}i_{1}(\psi(\alpha_{i}),v)\zeta_{1,-1}(m_{c})$
$\displaystyle+$ $\displaystyle
8m_{c}^{2}q^{2}u_{0}i_{1}(\alpha_{g}\psi(\alpha_{i}),v)\zeta_{1,-1}(m_{c})-16m_{c}^{2}q^{2}u_{0}i_{1}(\alpha_{g}\psi(\alpha_{i}),v^{2})\zeta_{1,-1}(m_{c})$
$\displaystyle+$ $\displaystyle
8m_{c}^{2}q^{2}u_{0}i_{1}(\alpha_{q}\psi(\alpha_{i}),1)\zeta_{1,-1}(m_{c})-16m_{c}^{2}q^{2}u_{0}i_{1}(\alpha_{q}\psi(\alpha_{i}),v)\zeta_{1,-1}(m_{c})$
$\displaystyle-$ $\displaystyle
4m_{c}^{2}M^{2}i_{1}(\tilde{\psi}(\alpha_{i}),1)\zeta_{1,-1}(m_{c})-24m_{c}^{2}q^{2}u_{0}^{2}i_{1}(\tilde{\psi}(\alpha_{i}),1)\zeta_{1,-1}(m_{c})$
$\displaystyle+$ $\displaystyle
24m_{c}^{2}q^{2}u_{0}i_{1}(\alpha_{q}\tilde{\psi}(\alpha_{i}),1)\zeta_{1,-1}(m_{c})-2m_{c}^{4}j_{1}({\cal
A}^{\parallel}(u))\zeta_{1,-1}(m_{c})+3m_{c}^{2}M^{2}j_{1}({\cal
A}^{\parallel}(u))\zeta_{1,-1}(m_{c})$ $\displaystyle-$ $\displaystyle
m_{c}^{2}M^{2}u_{0}{\cal
A}^{\parallel}(u_{0})\zeta_{1,0}(m_{c})+3M^{4}u_{0}{\cal
A}^{\parallel}(u_{0})\zeta_{1,0}(m_{c})-12m_{c}^{2}M^{2}u_{0}\tilde{j}_{1}({\cal
C}(u))\zeta_{1,0}(m_{c})$ $\displaystyle+$ $\displaystyle
48M^{4}u_{0}\tilde{j}_{1}({\cal
C}(u))\zeta_{1,0}(m_{c})-4M^{4}i_{1}(\tilde{\phi}(\alpha_{i}),1)\zeta_{1,0}(m_{c})+12M^{4}i_{1}(\psi(\alpha_{i}),1)\zeta_{1,0}(m_{c})$
$\displaystyle-$ $\displaystyle
16M^{2}q^{2}u_{0}^{2}i_{1}(\psi(\alpha_{i}),1)\zeta_{1,0}(m_{c})-24M^{4}i_{1}(\psi(\alpha_{i}),v)\zeta_{1,0}(m_{c})+32M^{2}q^{2}u_{0}^{2}i_{1}(\psi(\alpha_{i}),v)\zeta_{1,0}(m_{c})$
$\displaystyle+$ $\displaystyle
16M^{2}q^{2}u_{0}i_{1}(\alpha_{g}\psi(\alpha_{i}),v)\zeta_{1,0}(m_{c})+24m_{c}^{2}q^{2}u_{0}i_{1}(\alpha_{g}\tilde{\psi}(\alpha_{i}),v)\zeta_{1,-1}(m_{c})$
$\displaystyle-$ $\displaystyle
32M^{2}q^{2}u_{0}i_{1}(\alpha_{g}\psi(\alpha_{i}),v^{2})\zeta_{1,0}(m_{c})+16M^{2}q^{2}u_{0}i_{1}(\alpha_{q}\psi(\alpha_{i}),1)\zeta_{1,0}(m_{c})$
$\displaystyle-$ $\displaystyle
32M^{2}q^{2}u_{0}i_{1}(\alpha_{q}\psi(\alpha_{i}),v)\zeta_{1,0}(m_{c})-12M^{4}i_{1}(\tilde{\psi}(\alpha_{i}),1)\zeta_{1,0}(m_{c})-48M^{2}q^{2}u_{0}^{2}i_{1}(\tilde{\psi}(\alpha_{i}),1)\zeta_{1,0}(m_{c})$
$\displaystyle+$ $\displaystyle
48M^{2}q^{2}u_{0}i_{1}(\alpha_{g}\tilde{\psi}(\alpha_{i}),v)\zeta_{1,0}(m_{c})+48M^{2}q^{2}u_{0}i_{1}(\alpha_{q}\tilde{\psi}(\alpha_{i}),1)\zeta_{1,0}(m_{c})$
$\displaystyle-$ $\displaystyle
4M^{4}i_{2}(\alpha_{g}\phi(\alpha_{i}),v)\zeta_{1,0}(m_{c})+8M^{4}i_{2}(\alpha_{g}\phi(\alpha_{i}),v^{2})\zeta_{1,0}(m_{c})$
$\displaystyle-$ $\displaystyle
4M^{4}i_{2}(\alpha_{q}\phi(\alpha_{i}),1)\zeta_{1,0}(m_{c})+8M^{4}i_{2}(\alpha_{q}\phi(\alpha_{i}),v)\zeta_{1,0}(m_{c})$
$\displaystyle-$ $\displaystyle
4M^{4}i_{2}(\alpha_{g}\tilde{\phi}(\alpha_{i}),v)\zeta_{1,0}(m_{c})-4M^{4}i_{2}(\alpha_{q}\tilde{\phi}(\alpha_{i}),1)\zeta_{1,0}(m_{c})-4M^{4}i_{2}(\alpha_{g}\psi(\alpha_{i}),v)\zeta_{1,0}(m_{c})$
$\displaystyle+$ $\displaystyle
8M^{4}i_{2}(\alpha_{g}\psi(\alpha_{i}),v^{2})\zeta_{1,0}(m_{c})-4M^{4}i_{2}(\alpha_{q}\psi(\alpha_{i}),1)\zeta_{1,0}(m_{c})+8M^{4}i_{2}(\alpha_{q}\psi(\alpha_{i}),v)\zeta_{1,0}(m_{c})$
$\displaystyle-$ $\displaystyle
4M^{4}i_{2}(\alpha_{g}\tilde{\psi}(\alpha_{i}),v)\zeta_{1,0}(m_{c})-4M^{4}i_{2}(\alpha_{q}\tilde{\psi}(\alpha_{i}),1)\zeta_{1,0}(m_{c})-3m_{c}^{2}M^{2}j_{1}({\cal
A}^{\parallel}(u))\zeta_{1,0}(m_{c})$ $\displaystyle+$ $\displaystyle
3M^{4}j_{1}({\cal
A}^{\parallel}(u))\zeta_{1,0}(m_{c})+8M^{2}i_{1}(\phi(\alpha_{i}),v)(m_{c}^{2}\zeta_{-1,0}(m_{c})+m_{c}^{2}\zeta_{0,-1}(m_{c})-m_{c}^{2}\zeta_{0,0}(m_{c})$
$\displaystyle+$ $\displaystyle
M^{2}\zeta_{0,0}(m_{c})-m_{c}^{2}\zeta_{1,-1}(m_{c})-M^{2}\zeta_{1,0}(m_{c}))+4M^{2}i_{1}(\phi(\alpha_{i}),1)(-m_{c}^{2}\zeta_{-1,0}(m_{c})-m_{c}^{2}\zeta_{0,-1}(m_{c})$
$\displaystyle+$ $\displaystyle
m_{c}^{2}\zeta_{0,0}(m_{c})-M^{2}\zeta_{0,0}(m_{c})+m_{c}^{2}\zeta_{1,-1}(m_{c})+M^{2}\zeta_{1,0}(m_{c}))-m_{c}^{4}j_{1}({\cal
A}^{\parallel}(u))\zeta_{2,-2}(m_{c})$ $\displaystyle-$ $\displaystyle
m_{c}^{2}M^{2}u_{0}{\cal
A}^{\parallel}(u_{0})\zeta_{2,-1}(m_{c})-12m_{c}^{2}M^{2}u_{0}\tilde{j}_{1}({\cal
C}(u))\zeta_{2,-1}(m_{c})-3m_{c}^{2}M^{2}j_{1}({\cal
A}^{\parallel}(u))\zeta_{2,-1}(m_{c})$ $\displaystyle-$ $\displaystyle
3M^{4}(u_{0}{\cal A}^{\parallel}(u_{0})+16u_{0}\tilde{j}_{1}({\cal
C}(u))+j_{1}({\cal
A}^{\parallel}(u)))\zeta_{2,0}(m_{c})\Bigg{)}+M^{4}\Bigg{(}-7q^{2}i_{2}(\alpha_{q}\tilde{{\cal
A}}(\alpha_{i}),1)\zeta_{0,0}(m_{c})$ $\displaystyle+$ $\displaystyle
8q^{2}i_{2}(\alpha_{q}\tilde{{\cal
A}}(\alpha_{i}),v)\zeta_{0,0}(m_{c})+10q^{2}i_{2}(\alpha_{g}{\cal
V}(\alpha_{i}),v)\zeta_{0,0}(m_{c})-12q^{2}i_{2}(\alpha_{g}{\cal
V}(\alpha_{i}),v^{2})\zeta_{0,0}(m_{c})$ $\displaystyle+$ $\displaystyle
10q^{2}i_{2}(\alpha_{q}{\cal
V}(\alpha_{i}),1)\zeta_{0,0}(m_{c})-12q^{2}i_{2}(\alpha_{q}{\cal
V}(\alpha_{i}),v)\zeta_{0,0}(m_{c})-3M^{2}i_{3}(\tilde{{\cal
A}}(\alpha_{i}),1)\zeta_{0,0}(m_{c})$ $\displaystyle+$ $\displaystyle
3M^{2}i_{3}(\tilde{{\cal
A}}(\alpha_{i}),v)\zeta_{0,0}(m_{c})+2M^{2}i_{3}({\cal
V}(\alpha_{i}),1)\zeta_{0,0}(m_{c})-2M^{2}i_{3}({\cal
V}(\alpha_{i}),v)\zeta_{0,0}(m_{c})$ $\displaystyle+$ $\displaystyle
12m_{c}^{2}j_{1}({\cal
B}(u))\zeta_{0,0}(m_{c})-12m_{c}^{2}j_{1}(\phi^{\parallel}(u))\zeta_{0,0}(m_{c})+4m_{c}^{2}j_{1}({\cal
B}(u))\zeta_{1,-1}(m_{c})$ $\displaystyle-$ $\displaystyle
4m_{c}^{2}j_{1}(\phi^{\parallel}(u))\zeta_{1,-1}(m_{c})+8q^{2}i_{2}(\alpha_{g}\tilde{{\cal
A}}(\alpha_{i}),v^{2})(\zeta_{0,0}(m_{c})-\zeta_{1,0}(m_{c}))-4M^{2}u_{0}\phi^{\parallel}(u_{0})\zeta_{1,0}(m_{c})$
$\displaystyle-$ $\displaystyle
2M^{2}\psi_{3}^{\perp}(u_{0})\zeta_{1,0}(m_{c})+7q^{2}i_{2}(\alpha_{q}\tilde{{\cal
A}}(\alpha_{i}),1)\zeta_{1,0}(m_{c})-8q^{2}i_{2}(\alpha_{q}\tilde{{\cal
A}}(\alpha_{i}),v)\zeta_{1,0}(m_{c})$ $\displaystyle-$ $\displaystyle
10q^{2}i_{2}(\alpha_{g}{\cal
V}(\alpha_{i}),v)\zeta_{1,0}(m_{c})+12q^{2}i_{2}(\alpha_{g}{\cal
V}(\alpha_{i}),v^{2})\zeta_{1,0}(m_{c})-10q^{2}i_{2}(\alpha_{q}{\cal
V}(\alpha_{i}),1)\zeta_{1,0}(m_{c})$ $\displaystyle+$ $\displaystyle
12q^{2}i_{2}(\alpha_{q}{\cal
V}(\alpha_{i}),v)\zeta_{1,0}(m_{c})+3M^{2}i_{3}(\tilde{{\cal
A}}(\alpha_{i}),1)\zeta_{1,0}(m_{c})-3M^{2}i_{3}(\tilde{{\cal
A}}(\alpha_{i}),v)\zeta_{1,0}(m_{c})$ $\displaystyle-$ $\displaystyle
2M^{2}i_{3}({\cal V}(\alpha_{i}),1)\zeta_{1,0}(m_{c})+2M^{2}i_{3}({\cal
V}(\alpha_{i}),v)\zeta_{1,0}(m_{c})-4m_{c}^{2}j_{1}({\cal
B}(u))\zeta_{1,0}(m_{c})$ $\displaystyle+$ $\displaystyle 8M^{2}j_{1}({\cal
B}(u))\zeta_{1,0}(m_{c})+4m_{c}^{2}j_{1}(\phi^{\parallel}(u))\zeta_{1,0}(m_{c})-8M^{2}j_{1}(\phi^{\parallel}(u))\zeta_{1,0}(m_{c})$
$\displaystyle+$ $\displaystyle 7q^{2}i_{2}(\alpha_{g}\tilde{{\cal
A}}(\alpha_{i}),v)(-\zeta_{0,0}(m_{c})+\zeta_{1,0}(m_{c}))-4m_{c}^{2}j_{1}({\cal
B}(u))\zeta_{2,-1}(m_{c})$ $\displaystyle+$ $\displaystyle
4m_{c}^{2}j_{1}(\phi^{\parallel}(u))\zeta_{2,-1}(m_{c})+2M^{2}\Big{[}2u_{0}\phi^{\parallel}(u_{0})+\psi_{3}^{\perp}(u_{0})$
$\displaystyle-$ $\displaystyle 4j_{1}({\cal
B}(u))+4j_{1}(\phi^{\parallel}(u))\Big{]}\zeta_{2,0}(m_{c})\Bigg{)}\Bigg{]}\Biggr{\\}}~{},$

$\displaystyle\Pi_{A}^{GG}$ $\displaystyle=$
$\displaystyle-\frac{e^{-\frac{q^{2}}{M_{1}^{2}+M_{2}^{2}}}}{1728m_{c}M^{8}\pi^{2}}g_{s}^{2}\langle
GG\rangle\Biggl{\\{}2m_{c}f_{\rho}^{\parallel}m_{\rho}\Bigg{[}2m_{\rho}^{2}\Biggl{(}-8m_{c}^{4}q^{2}u_{0}i_{1}(\alpha_{g}\tilde{\psi}(\alpha_{i}),v)\zeta_{-3,0}(m_{c})$
$\displaystyle-$ $\displaystyle
8m_{c}^{4}q^{2}u_{0}i_{1}(\alpha_{q}\tilde{\psi}(\alpha_{i}),1)\zeta_{-3,0}(m_{c})+3m_{c}^{6}j_{1}({\cal
A}^{\parallel}(u))\zeta_{-3,0}(m_{c})$ $\displaystyle-$ $\displaystyle
16m_{c}^{4}q^{2}u_{0}i_{1}(\alpha_{g}\tilde{\psi}(\alpha_{i}),v)\zeta_{-2,-1}(m_{c})-16m_{c}^{4}q^{2}u_{0}i_{1}(\alpha_{q}\tilde{\psi}(\alpha_{i}),1)\zeta_{-2,-1}(m_{c})$
$\displaystyle+$ $\displaystyle 9m_{c}^{6}j_{1}({\cal
A}^{\parallel}(u))\zeta_{-2,-1}(m_{c})+m_{c}^{4}M^{2}u_{0}{\cal
A}^{\parallel}(u_{0})\zeta_{-2,0}(m_{c})$ $\displaystyle+$ $\displaystyle
20m_{c}^{4}M^{2}u_{0}\tilde{j}_{1}({\cal
C}(u))\zeta_{-2,0}(m_{c})-2m_{c}^{4}M^{2}i_{1}(\phi(\alpha_{i}),1)\zeta_{-2,0}(m_{c})$
$\displaystyle+$ $\displaystyle
4m_{c}^{4}M^{2}i_{1}(\phi(\alpha_{i}),v)\zeta_{-2,0}(m_{c})-2m_{c}^{4}M^{2}i_{1}(\psi(\alpha_{i}),1)\zeta_{-2,0}(m_{c})$
$\displaystyle+$ $\displaystyle
4m_{c}^{4}q^{2}u_{0}^{2}i_{1}(\psi(\alpha_{i}),1)\zeta_{-2,0}(m_{c})+4m_{c}^{4}M^{2}i_{1}(\psi(\alpha_{i}),v)\zeta_{-2,0}(m_{c})$
$\displaystyle-$ $\displaystyle
8m_{c}^{4}q^{2}u_{0}^{2}i_{1}(\psi(\alpha_{i}),v)\zeta_{-2,0}(m_{c})-4m_{c}^{4}q^{2}u_{0}i_{1}(\alpha_{g}\psi(\alpha_{i}),v)\zeta_{-2,0}(m_{c})$
$\displaystyle+$ $\displaystyle
8m_{c}^{4}q^{2}u_{0}i_{1}(\alpha_{g}\psi(\alpha_{i}),v^{2})\zeta_{-2,0}(m_{c})-4m_{c}^{4}q^{2}u_{0}i_{1}(\alpha_{q}\psi(\alpha_{i}),1)\zeta_{-2,0}(m_{c})$
$\displaystyle+$ $\displaystyle
8m_{c}^{4}q^{2}u_{0}i_{1}(\alpha_{q}\psi(\alpha_{i}),v)\zeta_{-2,0}(m_{c})-4m_{c}^{4}q^{2}u_{0}i_{1}(\alpha_{g}\tilde{\psi}(\alpha_{i}),v)\zeta_{-2,0}(m_{c})$
$\displaystyle+$ $\displaystyle
24m_{c}^{2}M^{2}q^{2}u_{0}i_{1}(\alpha_{g}\tilde{\psi}(\alpha_{i}),v)\zeta_{-2,0}(m_{c})-4m_{c}^{4}q^{2}u_{0}i_{1}(\alpha_{q}\tilde{\psi}(\alpha_{i}),1)\zeta_{-2,0}(m_{c})$
$\displaystyle+$ $\displaystyle
24m_{c}^{2}M^{2}q^{2}u_{0}i_{1}(\alpha_{q}\tilde{\psi}(\alpha_{i}),1)\zeta_{-2,0}(m_{c})-3m_{c}^{6}j_{1}({\cal
A}^{\parallel}(u))\zeta_{-2,0}(m_{c})$ $\displaystyle-$ $\displaystyle
7m_{c}^{4}M^{2}j_{1}({\cal
A}^{\parallel}(u))\zeta_{-2,0}(m_{c})-8m_{c}^{4}q^{2}u_{0}i_{1}(\alpha_{g}\tilde{\psi}(\alpha_{i}),v)\zeta_{-1,-2}(m_{c})$
$\displaystyle-$ $\displaystyle
8m_{c}^{4}q^{2}u_{0}i_{1}(\alpha_{q}\tilde{\psi}(\alpha_{i}),1)\zeta_{-1,-2}(m_{c})+9m_{c}^{6}j_{1}({\cal
A}^{\parallel}(u))\zeta_{-1,-2}(m_{c})$ $\displaystyle+$ $\displaystyle
2m_{c}^{4}M^{2}u_{0}{\cal
A}^{\parallel}(u_{0})\zeta_{-1,-1}(m_{c})+40m_{c}^{4}M^{2}u_{0}\tilde{j}_{1}({\cal
C}(u))\zeta_{-1,-1}(m_{c})$ $\displaystyle-$ $\displaystyle
4m_{c}^{4}M^{2}i_{1}(\phi(\alpha_{i}),1)\zeta_{-1,-1}(m_{c})+8m_{c}^{4}M^{2}i_{1}(\phi(\alpha_{i}),v)\zeta_{-1,-1}(m_{c})$
$\displaystyle-$ $\displaystyle
4m_{c}^{4}M^{2}i_{1}(\psi(\alpha_{i}),1)\zeta_{-1,-1}(m_{c})+8m_{c}^{4}q^{2}u_{0}^{2}i_{1}(\psi(\alpha_{i}),1)\zeta_{-1,-1}(m_{c})$
$\displaystyle+$ $\displaystyle
8m_{c}^{4}M^{2}i_{1}(\psi(\alpha_{i}),v)\zeta_{-1,-1}(m_{c})-16m_{c}^{4}q^{2}u_{0}^{2}i_{1}(\psi(\alpha_{i}),v)\zeta_{-1,-1}(m_{c})$
$\displaystyle-$ $\displaystyle
8m_{c}^{4}q^{2}u_{0}i_{1}(\alpha_{g}\psi(\alpha_{i}),v)\zeta_{-1,-1}(m_{c})+16m_{c}^{4}q^{2}u_{0}i_{1}(\alpha_{g}\psi(\alpha_{i}),v^{2})\zeta_{-1,-1}(m_{c})$
$\displaystyle-$ $\displaystyle
8m_{c}^{4}q^{2}u_{0}i_{1}(\alpha_{q}\psi(\alpha_{i}),1)\zeta_{-1,-1}(m_{c})+16m_{c}^{4}q^{2}u_{0}i_{1}(\alpha_{q}\psi(\alpha_{i}),v)\zeta_{-1,-1}(m_{c})$
$\displaystyle-$ $\displaystyle
8m_{c}^{4}q^{2}u_{0}i_{1}(\alpha_{g}\tilde{\psi}(\alpha_{i}),v)\zeta_{-1,-1}(m_{c})+24m_{c}^{2}M^{2}q^{2}u_{0}i_{1}(\alpha_{g}\tilde{\psi}(\alpha_{i}),v)\zeta_{-1,-1}(m_{c})$
$\displaystyle-$ $\displaystyle
8m_{c}^{4}q^{2}u_{0}i_{1}(\alpha_{q}\tilde{\psi}(\alpha_{i}),1)\zeta_{-1,-1}(m_{c})+24m_{c}^{2}M^{2}q^{2}u_{0}i_{1}(\alpha_{q}\tilde{\psi}(\alpha_{i}),1)\zeta_{-1,-1}(m_{c})$
$\displaystyle-$ $\displaystyle 9m_{c}^{6}j_{1}({\cal
A}^{\parallel}(u))\zeta_{-1,-1}(m_{c})-14m_{c}^{4}M^{2}j_{1}({\cal
A}^{\parallel}(u))\zeta_{-1,-1}(m_{c})-m_{c}^{4}M^{2}u_{0}{\cal
A}^{\parallel}(u_{0})\zeta_{-1,0}(m_{c})$ $\displaystyle+$ $\displaystyle
2m_{c}^{2}M^{4}u_{0}{\cal
A}^{\parallel}(u_{0})\zeta_{-1,0}(m_{c})-20m_{c}^{4}M^{2}u_{0}\tilde{j}_{1}({\cal
C}(u))\zeta_{-1,0}(m_{c})$ $\displaystyle+$ $\displaystyle
28m_{c}^{2}M^{4}u_{0}\tilde{j}_{1}({\cal
C}(u))\zeta_{-1,0}(m_{c})+2m_{c}^{2}M^{4}i_{1}(\phi(\alpha_{i}),1)\zeta_{-1,0}(m_{c})$
$\displaystyle-$ $\displaystyle
4m_{c}^{2}M^{4}i_{1}(\phi(\alpha_{i}),v)\zeta_{-1,0}(m_{c})-2m_{c}^{2}M^{4}i_{1}(\psi(\alpha_{i}),1)\zeta_{-1,0}(m_{c})$
$\displaystyle+$ $\displaystyle
4m_{c}^{2}M^{4}i_{1}(\psi(\alpha_{i}),v)\zeta_{-1,0}(m_{c})+2m_{c}^{2}M^{4}i_{2}(\alpha_{g}\phi(\alpha_{i}),v)\zeta_{-1,0}(m_{c})$
$\displaystyle-$ $\displaystyle
4m_{c}^{2}M^{4}i_{2}(\alpha_{g}\phi(\alpha_{i}),v^{2})\zeta_{-1,0}(m_{c})+2m_{c}^{2}M^{4}i_{2}(\alpha_{q}\phi(\alpha_{i}),1)\zeta_{-1,0}(m_{c})$
$\displaystyle-$ $\displaystyle
4m_{c}^{2}M^{4}i_{2}(\alpha_{q}\phi(\alpha_{i}),v)\zeta_{-1,0}(m_{c})+2m_{c}^{2}M^{4}i_{2}(\alpha_{g}\tilde{\phi}(\alpha_{i}),v)\zeta_{-1,0}(m_{c})$
$\displaystyle+$ $\displaystyle
2m_{c}^{2}M^{4}i_{2}(\alpha_{q}\tilde{\phi}(\alpha_{i}),1)\zeta_{-1,0}(m_{c})+2m_{c}^{2}M^{4}i_{2}(\alpha_{g}\psi(\alpha_{i}),v)\zeta_{-1,0}(m_{c})$
$\displaystyle-$ $\displaystyle
4m_{c}^{2}M^{4}i_{2}(\alpha_{g}\psi(\alpha_{i}),v^{2})\zeta_{-1,0}(m_{c})+2m_{c}^{2}M^{4}i_{2}(\alpha_{q}\psi(\alpha_{i}),1)\zeta_{-1,0}(m_{c})$
$\displaystyle-$ $\displaystyle
4m_{c}^{2}M^{4}i_{2}(\alpha_{q}\psi(\alpha_{i}),v)\zeta_{-1,0}(m_{c})+2m_{c}^{2}M^{4}i_{2}(\alpha_{g}\tilde{\psi}(\alpha_{i}),v)\zeta_{-1,0}(m_{c})$
$\displaystyle+$ $\displaystyle
2m_{c}^{2}M^{4}i_{2}(\alpha_{q}\tilde{\psi}(\alpha_{i}),1)\zeta_{-1,0}(m_{c})+m_{c}^{4}M^{2}j_{1}({\cal
A}^{\parallel}(u))\zeta_{-1,0}(m_{c})$ $\displaystyle-$ $\displaystyle
2m_{c}^{2}M^{4}j_{1}({\cal
A}^{\parallel}(u))\zeta_{-1,0}(m_{c})+3m_{c}^{6}j_{1}({\cal
A}^{\parallel}(u))\zeta_{0,-3}(m_{c})+m_{c}^{4}M^{2}u_{0}{\cal
A}^{\parallel}(u_{0})\zeta_{0,-2}(m_{c})$ $\displaystyle+$ $\displaystyle
20m_{c}^{4}M^{2}u_{0}\tilde{j}_{1}({\cal
C}(u))\zeta_{0,-2}(m_{c})-2m_{c}^{4}M^{2}i_{1}(\phi(\alpha_{i}),1)\zeta_{0,-2}(m_{c})$
$\displaystyle+$ $\displaystyle
4m_{c}^{4}M^{2}i_{1}(\phi(\alpha_{i}),v)\zeta_{0,-2}(m_{c})-2m_{c}^{4}M^{2}i_{1}(\psi(\alpha_{i}),1)\zeta_{0,-2}(m_{c})$
$\displaystyle+$ $\displaystyle
4m_{c}^{4}M^{2}i_{1}(\psi(\alpha_{i}),v)\zeta_{0,-2}(m_{c})-8m_{c}^{4}q^{2}u_{0}^{2}i_{1}(\psi(\alpha_{i}),v)\zeta_{0,-2}(m_{c})$
$\displaystyle-$ $\displaystyle
4m_{c}^{4}q^{2}u_{0}i_{1}(\alpha_{g}\psi(\alpha_{i}),v)\zeta_{0,-2}(m_{c})+8m_{c}^{4}q^{2}u_{0}i_{1}(\alpha_{g}\psi(\alpha_{i}),v^{2})\zeta_{0,-2}(m_{c})$
$\displaystyle-$ $\displaystyle
4m_{c}^{4}q^{2}u_{0}i_{1}(\alpha_{q}\psi(\alpha_{i}),1)\zeta_{0,-2}(m_{c})+8m_{c}^{4}q^{2}u_{0}i_{1}(\alpha_{q}\psi(\alpha_{i}),v)\zeta_{0,-2}(m_{c})$
$\displaystyle-$ $\displaystyle
4m_{c}^{4}q^{2}u_{0}i_{1}(\alpha_{g}\tilde{\psi}(\alpha_{i}),v)\zeta_{0,-2}(m_{c})-4m_{c}^{4}q^{2}u_{0}i_{1}(\alpha_{g}\tilde{\psi}(\alpha_{i}),v)\zeta_{0,-2}(m_{c})$
$\displaystyle-$ $\displaystyle
4m_{c}^{4}q^{2}u_{0}i_{1}(\alpha_{q}\tilde{\psi}(\alpha_{i}),1)\zeta_{0,-2}(m_{c})-9m_{c}^{6}j_{1}({\cal
A}^{\parallel}(u))\zeta_{0,-2}(m_{c})$ $\displaystyle-$ $\displaystyle
7m_{c}^{4}M^{2}j_{1}({\cal
A}^{\parallel}(u))\zeta_{0,-2}(m_{c})-2m_{c}^{4}M^{2}u_{0}{\cal
A}^{\parallel}(u_{0})\zeta_{0,-1}(m_{c})$ $\displaystyle+$ $\displaystyle
2m_{c}^{2}M^{4}u_{0}{\cal
A}^{\parallel}(u_{0})\zeta_{0,-1}(m_{c})-40m_{c}^{4}M^{2}u_{0}\tilde{j}_{1}({\cal
C}(u))\zeta_{0,-1}(m_{c})$ $\displaystyle+$ $\displaystyle
28m_{c}^{2}M^{4}u_{0}\tilde{j}_{1}({\cal
C}(u))\zeta_{0,-1}(m_{c})+2m_{c}^{2}M^{4}i_{1}(\phi(\alpha_{i}),1)\zeta_{0,-1}(m_{c})$
$\displaystyle+$ $\displaystyle
4m_{c}^{2}M^{4}i_{1}(\phi(\alpha_{i}),v)\zeta_{0,-1}(m_{c})-2m_{c}^{2}M^{4}i_{1}(\psi(\alpha_{i}),1)\zeta_{0,-1}(m_{c})$
$\displaystyle+$ $\displaystyle
m_{c}^{2}M^{4}i_{1}(\psi(\alpha_{i}),v)\zeta_{0,-1}(m_{c})+2m_{c}^{2}M^{4}i_{2}(\alpha_{g}\phi(\alpha_{i}),v)\zeta_{0,-1}(m_{c})$
$\displaystyle-$ $\displaystyle
4m_{c}^{2}M^{4}i_{2}(\alpha_{g}\phi(\alpha_{i}),v^{2})\zeta_{0,-1}(m_{c})+2m_{c}^{2}M^{4}i_{2}(\alpha_{q}\phi(\alpha_{i}),1)\zeta_{0,-1}(m_{c})$
$\displaystyle-$ $\displaystyle
4m_{c}^{2}M^{4}i_{2}(\alpha_{q}\phi(\alpha_{i}),v)\zeta_{0,-1}(m_{c})+2m_{c}^{2}M^{4}i_{2}(\alpha_{g}\tilde{\phi}(\alpha_{i}),v)\zeta_{0,-1}(m_{c})$
$\displaystyle+$ $\displaystyle
2m_{c}^{2}M^{4}i_{2}(\alpha_{q}\tilde{\phi}(\alpha_{i}),1)\zeta_{0,-1}(m_{c})+2m_{c}^{2}M^{4}i_{2}(\alpha_{g}\psi(\alpha_{i}),v)\zeta_{0,-1}(m_{c})$
$\displaystyle-$ $\displaystyle
4m_{c}^{2}M^{4}i_{2}(\alpha_{g}\psi(\alpha_{i}),v^{2})\zeta_{0,-1}(m_{c})+2m_{c}^{2}M^{4}i_{2}(\alpha_{q}\psi(\alpha_{i}),1)\zeta_{0,-1}(m_{c})$
$\displaystyle-$ $\displaystyle
4m_{c}^{2}M^{4}i_{2}(\alpha_{q}\psi(\alpha_{i}),v)\zeta_{0,-1}(m_{c})+4m_{c}^{4}q^{2}u_{0}^{2}i_{1}(\psi(\alpha_{i}),1)\zeta_{0,-2}(m_{c})$
$\displaystyle+$ $\displaystyle
2m_{c}^{2}M^{4}i_{2}(\alpha_{g}\tilde{\psi}(\alpha_{i}),v)\zeta_{0,-1}(m_{c})+2m_{c}^{2}M^{4}i_{2}(\alpha_{q}\tilde{\psi}(\alpha_{i}),1)\zeta_{0,-1}(m_{c})$
$\displaystyle+$ $\displaystyle 2m_{c}^{4}M^{2}j_{1}({\cal
A}^{\parallel}(u))\zeta_{0,-1}(m_{c})-2m_{c}^{2}M^{4}j_{1}({\cal
A}^{\parallel}(u))\zeta_{0,-1}(m_{c})-2m_{c}^{2}M^{4}u_{0}{\cal
A}^{\parallel}(u_{0})\zeta_{0,0}(m_{c})$ $\displaystyle+$ $\displaystyle
2M^{6}u_{0}{\cal
A}^{\parallel}(u_{0})\zeta_{0,0}(m_{c})-52m_{c}^{2}M^{4}u_{0}\tilde{j}_{1}({\cal
C}(u))\zeta_{0,0}(m_{c})+40M^{6}u_{0}\tilde{j}_{1}({\cal
C}(u))\zeta_{0,0}(m_{c})$ $\displaystyle+$ $\displaystyle
2M^{6}i_{1}(\phi(\alpha_{i}),1)\zeta_{0,0}(m_{c})-4M^{6}i_{1}(\phi(\alpha_{i}),v)\zeta_{0,0}(m_{c})-2M^{6}i_{1}(\psi(\alpha_{i}),1)\zeta_{0,0}(m_{c})$
$\displaystyle+$ $\displaystyle
4M^{6}i_{1}(\psi(\alpha_{i}),v)\zeta_{0,0}(m_{c})+2M^{6}i_{2}(\alpha_{g}\phi(\alpha_{i}),v)\zeta_{0,0}(m_{c})-4M^{6}i_{2}(\alpha_{g}\phi(\alpha_{i}),v^{2})\zeta_{0,0}(m_{c})$
$\displaystyle+$ $\displaystyle
2M^{6}i_{2}(\alpha_{q}\phi(\alpha_{i}),1)\zeta_{0,0}(m_{c})-4M^{6}i_{2}(\alpha_{q}\phi(\alpha_{i}),v)\zeta_{0,0}(m_{c})+2M^{6}i_{2}(\alpha_{g}\tilde{\phi}(\alpha_{i}),v)\zeta_{0,0}(m_{c})$
$\displaystyle+$ $\displaystyle
2M^{6}i_{2}(\alpha_{q}\tilde{\phi}(\alpha_{i}),1)\zeta_{0,0}(m_{c})+2M^{6}i_{2}(\alpha_{g}\psi(\alpha_{i}),v)\zeta_{0,0}(m_{c})-4M^{6}i_{2}(\alpha_{g}\psi(\alpha_{i}),v^{2})\zeta_{0,0}(m_{c})$
$\displaystyle+$ $\displaystyle
2M^{6}i_{2}(\alpha_{q}\psi(\alpha_{i}),1)\zeta_{0,0}(m_{c})-4M^{6}i_{2}(\alpha_{q}\psi(\alpha_{i}),v)\zeta_{0,0}(m_{c})+2M^{6}i_{2}(\alpha_{g}\tilde{\psi}(\alpha_{i}),v)\zeta_{0,0}(m_{c})$
$\displaystyle+$ $\displaystyle
2M^{6}i_{2}(\alpha_{q}\tilde{\psi}(\alpha_{i}),1)\zeta_{0,0}(m_{c})+2m_{c}^{2}M^{4}j_{1}({\cal
A}^{\parallel}(u))\zeta_{0,0}(m_{c})-2M^{6}j_{1}({\cal
A}^{\parallel}(u))\zeta_{0,0}(m_{c})$ $\displaystyle-$ $\displaystyle
2M^{2}i_{1}(\tilde{\phi}(\alpha_{i}),1)(-2m_{c}^{4}\zeta_{-3,0}(m_{c})-4m_{c}^{4}\zeta_{-2,-1}(m_{c})+m_{c}^{2}((m_{c}^{2}+6M^{2})\zeta_{-2,0}(m_{c})$
$\displaystyle-$ $\displaystyle
2m_{c}^{2}\zeta_{-1,-2}(m_{c})+2m_{c}^{2}\zeta_{-1,-1}(m_{c})+6M^{2}\zeta_{-1,-1}(m_{c})+M^{2}\zeta_{-1,0}(m_{c})+m_{c}^{2}\zeta_{0,-2}(m_{c})$
$\displaystyle+$ $\displaystyle
M^{2}\zeta_{0,-1}(m_{c}))+M^{4}\zeta_{0,0}(m_{c}))+2i_{1}(\tilde{\psi}(\alpha_{i}),1)(2m_{c}^{4}(M^{2}+2q^{2}u_{0}^{2})\zeta_{-3,0}(m_{c})$
$\displaystyle+$ $\displaystyle
4m_{c}^{4}(M^{2}+2q^{2}u_{0}^{2})\zeta_{-2,-1}(m_{c})+m_{c}^{2}(-(m_{c}^{2}(M^{2}-2q^{2}u_{0}^{2})+6M^{2}(M^{2}+2q^{2}u_{0}^{2}))\zeta_{-2,0}(m_{c})$
$\displaystyle+$ $\displaystyle
2m_{c}^{2}(M^{2}+2q^{2}u_{0}^{2})\zeta_{-1,-2}(m_{c})-2m_{c}^{2}M^{2}\zeta_{-1,-1}(m_{c})-6M^{4}\zeta_{-1,-1}(m_{c})+4m_{c}^{2}q^{2}u_{0}^{2}\zeta_{-1,-1}(m_{c})$
$\displaystyle-$ $\displaystyle
12M^{2}q^{2}u_{0}^{2}\zeta_{-1,-1}(m_{c})+M^{4}\zeta_{-1,0}(m_{c})-m_{c}^{2}M^{2}\zeta_{0,-2}(m_{c})+2m_{c}^{2}q^{2}u_{0}^{2}\zeta_{0,-2}(m_{c})$
$\displaystyle+$ $\displaystyle
M^{4}\zeta_{0,-1}(m_{c}))+M^{6}\zeta_{0,0}(m_{c}))-3m_{c}^{6}j_{1}({\cal
A}^{\parallel}(u))\zeta_{1,-3}(m_{c})-m_{c}^{4}M^{2}u_{0}{\cal
A}^{\parallel}(u_{0})\zeta_{1,-2}(m_{c})$ $\displaystyle-$ $\displaystyle
20m_{c}^{4}M^{2}u_{0}\tilde{j}_{1}({\cal
C}(u))\zeta_{1,-2}(m_{c})+m_{c}^{4}M^{2}j_{1}({\cal
A}^{\parallel}(u))\zeta_{1,-2}(m_{c})-2m_{c}^{2}M^{4}u_{0}{\cal
A}^{\parallel}(u_{0})\zeta_{1,-1}(m_{c})$ $\displaystyle-$ $\displaystyle
52m_{c}^{2}M^{4}u_{0}\tilde{j}_{1}({\cal
C}(u))\zeta_{1,-1}(m_{c})+2m_{c}^{2}M^{4}j_{1}({\cal
A}^{\parallel}(u))\zeta_{1,-1}(m_{c})$ $\displaystyle+$ $\displaystyle
2M^{6}(-u_{0}({\cal A}^{\parallel}(u_{0})+32\tilde{j}_{1}({\cal
C}(u)))+j_{1}({\cal A}^{\parallel}(u)))\zeta_{1,0}(m_{c})\Biggr{)}$
$\displaystyle+$ $\displaystyle
M^{4}\Biggl{(}-8m_{c}^{2}M^{2}u_{0}\phi^{\parallel}(u_{0})\zeta_{-1,0}(m_{c})-4m_{c}^{2}M^{2}\psi_{3}^{\perp}(u_{0})\zeta_{-1,0}(m_{c})$
$\displaystyle-$ $\displaystyle
7m_{c}^{2}q^{2}i_{2}(\alpha_{g}\tilde{A}(\alpha_{i}),v)\zeta_{-1,0}(m_{c})+8m_{c}^{2}q^{2}i_{2}(\alpha_{g}\tilde{A}(\alpha_{i}),v^{2})\zeta_{-1,0}(m_{c})$
$\displaystyle-$ $\displaystyle
7m_{c}^{2}q^{2}i_{2}(\alpha_{q}\tilde{A}(\alpha_{i}),1)\zeta_{-1,0}(m_{c})+8m_{c}^{2}q^{2}i_{2}(\alpha_{q}\tilde{A}(\alpha_{i}),v)\zeta_{-1,0}(m_{c})$
$\displaystyle+$ $\displaystyle 10m_{c}^{2}q^{2}i_{2}(\alpha_{g}{\cal
V}(\alpha_{i}),v)\zeta_{-1,0}(m_{c})-12m_{c}^{2}q^{2}i_{2}(\alpha_{g}{\cal
V}(\alpha_{i}),v^{2})\zeta_{-1,0}(m_{c})$ $\displaystyle+$ $\displaystyle
10m_{c}^{2}q^{2}i_{2}(\alpha_{q}{\cal
V}(\alpha_{i}),1)\zeta_{-1,0}(m_{c})-12m_{c}^{2}q^{2}i_{2}(\alpha_{q}{\cal
V}(\alpha_{i}),v)\zeta_{-1,0}(m_{c})$ $\displaystyle-$ $\displaystyle
3m_{c}^{2}M^{2}i_{3}(\tilde{A}(\alpha_{i}),1)\zeta_{-1,0}(m_{c})+3m_{c}^{2}M^{2}i_{3}(\tilde{A}(\alpha_{i}),v)\zeta_{-1,0}(m_{c})$
$\displaystyle+$ $\displaystyle 2m_{c}^{2}M^{2}i_{3}({\cal
V}(\alpha_{i}),1)\zeta_{-1,0}(m_{c})-2m_{c}^{2}M^{2}i_{3}({\cal
V}(\alpha_{i}),v)\zeta_{-1,0}(m_{c})$ $\displaystyle-$ $\displaystyle
8m_{c}^{2}M^{2}u_{0}\phi^{\parallel}(u_{0})\zeta_{0,-1}(m_{c})-4m_{c}^{2}M^{2}\psi_{3}^{\perp}(u_{0})\zeta_{0,-1}(m_{c})$
$\displaystyle-$ $\displaystyle
7m_{c}^{2}q^{2}i_{2}(\alpha_{g}\tilde{A}(\alpha_{i}),v)\zeta_{0,-1}(m_{c})+8m_{c}^{2}q^{2}i_{2}(\alpha_{g}\tilde{A}(\alpha_{i}),v^{2})\zeta_{0,-1}(m_{c})$
$\displaystyle-$ $\displaystyle
7m_{c}^{2}q^{2}i_{2}(\alpha_{q}\tilde{A}(\alpha_{i}),1)\zeta_{0,-1}(m_{c})+8m_{c}^{2}q^{2}i_{2}(\alpha_{q}\tilde{A}(\alpha_{i}),v)\zeta_{0,-1}(m_{c})$
$\displaystyle+$ $\displaystyle 10m_{c}^{2}q^{2}i_{2}(\alpha_{g}{\cal
V}(\alpha_{i}),v)\zeta_{0,-1}(m_{c})-12m_{c}^{2}q^{2}i_{2}(\alpha_{g}{\cal
V}(\alpha_{i}),v^{2})\zeta_{0,-1}(m_{c})$ $\displaystyle+$ $\displaystyle
10m_{c}^{2}q^{2}i_{2}(\alpha_{q}{\cal
V}(\alpha_{i}),1)\zeta_{0,-1}(m_{c})-12m_{c}^{2}q^{2}i_{2}(\alpha_{q}{\cal
V}(\alpha_{i}),v)\zeta_{0,-1}(m_{c})$ $\displaystyle-$ $\displaystyle
3m_{c}^{2}M^{2}i_{3}(\tilde{A}(\alpha_{i}),1)\zeta_{0,-1}(m_{c})+3m_{c}^{2}M^{2}i_{3}(\tilde{A}(\alpha_{i}),v)\zeta_{0,-1}(m_{c})$
$\displaystyle+$ $\displaystyle 2m_{c}^{2}M^{2}i_{3}({\cal
V}(\alpha_{i}),1)\zeta_{0,-1}(m_{c})-2m_{c}^{2}M^{2}i_{3}({\cal
V}(\alpha_{i}),v)\zeta_{0,-1}(m_{c})$ $\displaystyle+$ $\displaystyle
8m_{c}^{2}M^{2}u_{0}\phi^{\parallel}(u_{0})\zeta_{0,0}(m_{c})-16M^{4}u_{0}\phi^{\parallel}(u_{0})\zeta_{0,0}(m_{c})$
$\displaystyle+$ $\displaystyle
4m_{c}^{2}M^{2}\psi_{3}^{\perp}(u_{0})\zeta_{0,0}(m_{c})-8M^{4}\psi_{3}^{\perp}(u_{0})\zeta_{0,0}(m_{c})-7M^{2}q^{2}i_{2}(\alpha_{g}\tilde{A}(\alpha_{i}),v)\zeta_{0,0}(m_{c})$
$\displaystyle+$ $\displaystyle
8M^{2}q^{2}i_{2}(\alpha_{g}\tilde{A}(\alpha_{i}),v^{2})\zeta_{0,0}(m_{c})-7M^{2}q^{2}i_{2}(\alpha_{q}\tilde{A}(\alpha_{i}),1)\zeta_{0,0}(m_{c})$
$\displaystyle+$ $\displaystyle
8M^{2}q^{2}i_{2}(\alpha_{q}\tilde{A}(\alpha_{i}),v)\zeta_{0,0}(m_{c})+10M^{2}q^{2}i_{2}(\alpha_{g}{\cal
V}(\alpha_{i}),v)\zeta_{0,0}(m_{c})$ $\displaystyle-$ $\displaystyle
12M^{2}q^{2}i_{2}(\alpha_{g}{\cal
V}(\alpha_{i}),v^{2})\zeta_{0,0}(m_{c})+10M^{2}q^{2}i_{2}(\alpha_{q}{\cal
V}(\alpha_{i}),1)\zeta_{0,0}(m_{c})$ $\displaystyle-$ $\displaystyle
12M^{2}q^{2}i_{2}(\alpha_{q}{\cal
V}(\alpha_{i}),v)\zeta_{0,0}(m_{c})-3M^{4}i_{3}(\tilde{A}(\alpha_{i}),1)\zeta_{0,0}(m_{c})$
$\displaystyle+$ $\displaystyle
3M^{4}i_{3}(\tilde{A}(\alpha_{i}),v)\zeta_{0,0}(m_{c})+2M^{4}i_{3}({\cal
V}(\alpha_{i}),1)\zeta_{0,0}(m_{c})$ $\displaystyle-$ $\displaystyle
2M^{4}i_{3}({\cal
V}(\alpha_{i}),v)\zeta_{0,0}(m_{c})+8m_{c}^{2}M^{2}u_{0}\phi^{\parallel}(u_{0})\zeta_{1,-1}(m_{c})+4m_{c}^{2}M^{2}\psi_{3}^{\perp}(u_{0})\zeta_{1,-1}(m_{c})$
$\displaystyle+$ $\displaystyle
8M^{4}(2u_{0}\phi^{\parallel}(u_{0})+\psi_{3}^{\perp}(u_{0}))\zeta_{1,0}(m_{c})+8j_{1}({\cal
B}(u))(3m_{c}^{4}\zeta_{-2,0}(m_{c})+6m_{c}^{4}\zeta_{-1,-1}(m_{c})$
$\displaystyle-$ $\displaystyle
3m_{c}^{4}\zeta_{-1,0}(m_{c})-m_{c}^{2}M^{2}\zeta_{-1,0}(m_{c})+3m_{c}^{4}\zeta_{0,-2}(m_{c})-6m_{c}^{4}\zeta_{0,-1}(m_{c})-m_{c}^{2}M^{2}\zeta_{0,-1}(m_{c})$
$\displaystyle-$ $\displaystyle
5m_{c}^{2}M^{2}\zeta_{0,0}(m_{c})-2M^{4}\zeta_{0,0}(m_{c})-3m_{c}^{4}\zeta_{1,-2}(m_{c})-5m_{c}^{2}M^{2}\zeta_{1,-1}(m_{c})-4M^{4}\zeta_{1,0}(m_{c}))$
$\displaystyle-$ $\displaystyle
8j_{1}(\phi^{\parallel}(u))(3m_{c}^{4}\zeta_{-2,0}(m_{c})+6m_{c}^{4}\zeta_{-1,-1}(m_{c})-3m_{c}^{4}\zeta_{-1,0}(m_{c})-m_{c}^{2}M^{2}\zeta_{-1,0}(m_{c})$
$\displaystyle+$ $\displaystyle
3m_{c}^{4}\zeta_{0,-2}(m_{c})-6m_{c}^{4}\zeta_{0,-1}(m_{c})-m_{c}^{2}M^{2}\zeta_{0,-1}(m_{c})-5m_{c}^{2}M^{2}\zeta_{0,0}(m_{c})-2M^{4}\zeta_{0,0}(m_{c})$
$\displaystyle-$ $\displaystyle
3m_{c}^{4}\zeta_{1,-2}(m_{c})-5m_{c}^{2}M^{2}\zeta_{1,-1}(m_{c})-4M^{4}\zeta_{1,0}(m_{c}))\Biggr{)}\Bigg{]}$
$\displaystyle+$ $\displaystyle
M^{2}f_{\rho}^{\perp}\Bigg{[}-4M^{4}\phi^{\perp}(u_{0})\Biggl{(}(-2m_{c}^{4}+3m_{c}^{2}M^{2})\zeta_{-1,0}(m_{c})+3m_{c}^{2}(-m_{c}^{2}+M^{2})\zeta_{0,-1}(m_{c})$
$\displaystyle+$ $\displaystyle
m_{c}^{4}\zeta_{0,0}(m_{c})-8m_{c}^{2}M^{2}\zeta_{0,0}(m_{c})+6M^{4}\zeta_{0,0}(m_{c})-m_{c}^{4}\zeta_{1,-2}(m_{c})+2m_{c}^{4}\zeta_{1,-1}(m_{c})$
$\displaystyle-$ $\displaystyle
7m_{c}^{2}M^{2}\zeta_{1,-1}(m_{c})+4m_{c}^{2}M^{2}\zeta_{1,0}(m_{c})-12M^{4}\zeta_{1,0}(m_{c})+m_{c}^{4}\zeta_{2,-2}(m_{c})$
$\displaystyle+$ $\displaystyle
4m_{c}^{2}M^{2}\zeta_{2,-1}(m_{c})+6M^{4}\zeta_{2,0}(m_{c})\Biggr{)}+{\cal
A}^{\perp}(u_{0})m_{\rho}^{2}\Bigg{(}(-2m_{c}^{6}+3m_{c}^{4}M^{2})\zeta_{-2,0}(m_{c})$
$\displaystyle+$
$\displaystyle(-5m_{c}^{6}+6m_{c}^{4}M^{2})\zeta_{-1,-1}(m_{c})+m_{c}^{6}\zeta_{-1,0}(m_{c})-6m_{c}^{4}M^{2}\zeta_{-1,0}(m_{c})+6m_{c}^{2}M^{4}\zeta_{-1,0}(m_{c})$
$\displaystyle-$ $\displaystyle
4m_{c}^{6}\zeta_{0,-2}(m_{c})+3m_{c}^{4}M^{2}\zeta_{0,-2}(m_{c})+3m_{c}^{6}\zeta_{0,-1}(m_{c})-12m_{c}^{4}M^{2}\zeta_{0,-1}(m_{c})$
$\displaystyle+$ $\displaystyle
6m_{c}^{2}M^{4}\zeta_{0,-1}(m_{c})+3m_{c}^{4}M^{2}\zeta_{0,0}(m_{c})-12m_{c}^{2}M^{4}\zeta_{0,0}(m_{c})+6M^{6}\zeta_{0,0}(m_{c})$
$\displaystyle-$ $\displaystyle
m_{c}^{6}\zeta_{1,-3}(m_{c})+3m_{c}^{6}\zeta_{1,-2}(m_{c})-6m_{c}^{4}M^{2}\zeta_{1,-2}(m_{c})+6m_{c}^{4}M^{2}\zeta_{1,-1}(m_{c})$
$\displaystyle-$ $\displaystyle
12m_{c}^{2}M^{4}\zeta_{1,-1}(m_{c})+6m_{c}^{2}M^{4}\zeta_{1,0}(m_{c})-12M^{6}\zeta_{1,0}(m_{c})+m_{c}^{6}\zeta_{2,-3}(m_{c})+3m_{c}^{4}M^{2}\zeta_{2,-2}(m_{c})$
$\displaystyle+$ $\displaystyle
6m_{c}^{2}M^{4}\zeta_{2,-1}(m_{c})+6M^{6}\zeta_{2,0}(m_{c})\Bigg{)}+2m_{\rho}^{2}\Bigg{(}-8m_{c}^{4}q^{2}u_{0}i_{1}({\cal
T}_{4}(\alpha_{i}),1)\zeta_{-2,0}(m_{c})$ $\displaystyle-$ $\displaystyle
7m_{c}^{4}M^{2}i_{2}({\cal
S}(\alpha_{i}),1)\zeta_{-2,0}(m_{c})+8m_{c}^{4}M^{2}i_{2}({\cal
S}(\alpha_{i}),v)\zeta_{-2,0}(m_{c})$ $\displaystyle+$ $\displaystyle
3m_{c}^{4}M^{2}i_{2}(\tilde{S}(\alpha_{i}),1)\zeta_{-2,0}(m_{c})+m_{c}^{4}M^{2}i_{2}({\cal
T}_{1}(\alpha_{i}),1)\zeta_{-2,0}(m_{c})$ $\displaystyle-$ $\displaystyle
2m_{c}^{4}M^{2}i_{2}({\cal
T}_{1}(\alpha_{i}),v)\zeta_{-2,0}(m_{c})-m_{c}^{4}M^{2}i_{2}({\cal
T}_{2}(\alpha_{i}),1)\zeta_{-2,0}(m_{c})$ $\displaystyle+$ $\displaystyle
2m_{c}^{4}M^{2}i_{2}({\cal
T}_{2}(\alpha_{i}),v)\zeta_{-2,0}(m_{c})-2m_{c}^{4}M^{2}i_{2}({\cal
T}_{3}(\alpha_{i}),v)\zeta_{-2,0}(m_{c})$ $\displaystyle-$ $\displaystyle
m_{c}^{4}M^{2}i_{2}({\cal
T}_{4}(\alpha_{i}),1)\zeta_{-2,0}(m_{c})-m_{c}^{2}M^{4}i_{2}({\cal
T}_{1}(\alpha_{i}),1)\zeta_{0,-1}(m_{c})$ $\displaystyle+$ $\displaystyle
2m_{c}^{4}M^{2}i_{2}({\cal
T}_{4}(\alpha_{i}),v)\zeta_{-2,0}(m_{c})-10m_{c}^{4}q^{2}u_{0}i_{1}({\cal
T}_{4}(\alpha_{i}),1)\zeta_{-1,-1}(m_{c})$ $\displaystyle-$ $\displaystyle
13m_{c}^{4}M^{2}i_{2}({\cal
S}(\alpha_{i}),1)\zeta_{-1,-1}(m_{c})+16m_{c}^{4}M^{2}i_{2}({\cal
S}(\alpha_{i}),v)\zeta_{-1,-1}(m_{c})$ $\displaystyle+$ $\displaystyle
3m_{c}^{4}M^{2}i_{2}(\tilde{S}(\alpha_{i}),1)\zeta_{-1,-1}(m_{c})+2m_{c}^{4}M^{2}i_{2}({\cal
T}_{1}(\alpha_{i}),1)\zeta_{-1,-1}(m_{c})$ $\displaystyle-$ $\displaystyle
4m_{c}^{4}M^{2}i_{2}({\cal
T}_{1}(\alpha_{i}),v)\zeta_{-1,-1}(m_{c})-2m_{c}^{4}M^{2}i_{2}({\cal
T}_{2}(\alpha_{i}),1)\zeta_{-1,-1}(m_{c})$ $\displaystyle+$ $\displaystyle
4m_{c}^{4}M^{2}i_{2}({\cal
T}_{2}(\alpha_{i}),v)\zeta_{-1,-1}(m_{c})+m_{c}^{4}M^{2}i_{2}({\cal
T}_{3}(\alpha_{i}),1)\zeta_{-1,-1}(m_{c})$ $\displaystyle-$ $\displaystyle
4m_{c}^{4}M^{2}i_{2}({\cal
T}_{3}(\alpha_{i}),v)\zeta_{-1,-1}(m_{c})-m_{c}^{4}M^{2}i_{2}({\cal
T}_{4}(\alpha_{i}),1)\zeta_{-1,-1}(m_{c})$ $\displaystyle+$ $\displaystyle
4m_{c}^{4}M^{2}i_{2}({\cal
T}_{4}(\alpha_{i}),v)\zeta_{-1,-1}(m_{c})+4m_{c}^{4}q^{2}u_{0}i_{1}({\cal
T}(\alpha_{i}),1)(\zeta_{-2,0}(m_{c})+\zeta_{-1,-1}(m_{c}))$ $\displaystyle-$
$\displaystyle 6m_{c}^{4}q^{2}u_{0}i_{1}({\cal
T}_{3}(\alpha_{i}),1)(\zeta_{-2,0}(m_{c})+\zeta_{-1,-1}(m_{c}))-32m_{c}^{4}M^{2}u_{0}\psi_{3}^{\parallel}(u_{0})\zeta_{-1,0}(m_{c})$
$\displaystyle+$ $\displaystyle
48m_{c}^{2}M^{4}u_{0}\psi_{3}^{\parallel}(u_{0})\zeta_{-1,0}(m_{c})+2m_{c}^{4}q^{2}u_{0}i_{1}({\cal
T}_{4}(\alpha_{i}),1)\zeta_{-1,0}(m_{c})$ $\displaystyle+$ $\displaystyle
2m_{c}^{2}M^{2}q^{2}u_{0}i_{1}({\cal
T}_{4}(\alpha_{i}),1)\zeta_{-1,0}(m_{c})+6m_{c}^{4}M^{2}i_{2}({\cal
S}(\alpha_{i}),1)\zeta_{-1,0}(m_{c})$ $\displaystyle+$ $\displaystyle
6m_{c}^{2}M^{4}i_{2}({\cal
S}(\alpha_{i}),1)\zeta_{-1,0}(m_{c})-8m_{c}^{4}M^{2}i_{2}({\cal
S}(\alpha_{i}),v)\zeta_{-1,0}(m_{c})$ $\displaystyle-$ $\displaystyle
8m_{c}^{2}M^{4}i_{2}({\cal
S}(\alpha_{i}),v)\zeta_{-1,0}(m_{c})-m_{c}^{4}M^{2}i_{2}({\cal
T}_{1}(\alpha_{i}),1)\zeta_{-1,0}(m_{c})$ $\displaystyle-$ $\displaystyle
m_{c}^{2}M^{4}i_{2}({\cal
T}_{1}(\alpha_{i}),1)\zeta_{-1,0}(m_{c})+2m_{c}^{4}M^{2}i_{2}({\cal
T}_{1}(\alpha_{i}),v)\zeta_{-1,0}(m_{c})$ $\displaystyle+$ $\displaystyle
2m_{c}^{2}M^{4}i_{2}({\cal
T}_{1}(\alpha_{i}),v)\zeta_{-1,0}(m_{c})+m_{c}^{4}M^{2}i_{2}({\cal
T}_{2}(\alpha_{i}),1)\zeta_{-1,0}(m_{c})$ $\displaystyle+$ $\displaystyle
m_{c}^{2}M^{4}i_{2}({\cal
T}_{2}(\alpha_{i}),1)\zeta_{-1,0}(m_{c})-2m_{c}^{4}M^{2}i_{2}({\cal
T}_{2}(\alpha_{i}),v)\zeta_{-1,0}(m_{c})$ $\displaystyle-$ $\displaystyle
2m_{c}^{2}M^{4}i_{2}({\cal
T}_{2}(\alpha_{i}),v)\zeta_{-1,0}(m_{c})-m_{c}^{4}M^{2}i_{2}({\cal
T}_{3}(\alpha_{i}),1)\zeta_{-1,0}(m_{c})$ $\displaystyle-$ $\displaystyle
m_{c}^{2}M^{4}i_{2}({\cal
T}_{3}(\alpha_{i}),1)\zeta_{-1,0}(m_{c})+2m_{c}^{4}M^{2}i_{2}({\cal
T}_{3}(\alpha_{i}),v)\zeta_{-1,0}(m_{c})$ $\displaystyle+$ $\displaystyle
2m_{c}^{2}M^{4}i_{2}({\cal
T}_{3}(\alpha_{i}),v)\zeta_{-1,0}(m_{c})-2m_{c}^{4}M^{2}i_{2}({\cal
T}_{4}(\alpha_{i}),v)\zeta_{-1,0}(m_{c})$ $\displaystyle-$ $\displaystyle
2m_{c}^{2}M^{4}i_{2}({\cal
T}_{4}(\alpha_{i}),v)\zeta_{-1,0}(m_{c})-2m_{c}^{4}q^{2}u_{0}i_{1}({\cal
T}_{4}(\alpha_{i}),1)\zeta_{0,-2}(m_{c})$ $\displaystyle-$ $\displaystyle
6m_{c}^{4}M^{2}i_{2}({\cal
S}(\alpha_{i}),1)\zeta_{0,-2}(m_{c})+8m_{c}^{4}M^{2}i_{2}({\cal
S}(\alpha_{i}),v)\zeta_{0,-2}(m_{c})$ $\displaystyle+$ $\displaystyle
m_{c}^{4}M^{2}i_{2}({\cal
T}_{1}(\alpha_{i}),1)\zeta_{0,-2}(m_{c})-2m_{c}^{4}M^{2}i_{2}({\cal
T}_{1}(\alpha_{i}),v)\zeta_{0,-2}(m_{c})$ $\displaystyle-$ $\displaystyle
m_{c}^{4}M^{2}i_{2}({\cal
T}_{2}(\alpha_{i}),1)\zeta_{0,-2}(m_{c})+2m_{c}^{4}M^{2}i_{2}({\cal
T}_{2}(\alpha_{i}),v)\zeta_{0,-2}(m_{c})$ $\displaystyle+$ $\displaystyle
m_{c}^{4}M^{2}i_{2}({\cal
T}_{3}(\alpha_{i}),1)\zeta_{0,-2}(m_{c})-2m_{c}^{4}M^{2}i_{2}({\cal
T}_{3}(\alpha_{i}),v)\zeta_{0,-2}(m_{c})$ $\displaystyle+$ $\displaystyle
2m_{c}^{4}M^{2}i_{2}({\cal
T}_{4}(\alpha_{i}),v)\zeta_{0,-2}(m_{c})-48m_{c}^{4}M^{2}u_{0}\psi_{3}^{\parallel}(u_{0})\zeta_{0,-1}(m_{c})$
$\displaystyle+$ $\displaystyle
48m_{c}^{2}M^{4}u_{0}\psi_{3}^{\parallel}(u_{0})\zeta_{0,-1}(m_{c})+4m_{c}^{4}q^{2}u_{0}i_{1}({\cal
T}_{4}(\alpha_{i}),1)\zeta_{0,-1}(m_{c})$ $\displaystyle+$ $\displaystyle
2m_{c}^{2}M^{2}q^{2}u_{0}i_{1}({\cal
T}_{4}(\alpha_{i}),1)\zeta_{0,-1}(m_{c})+12m_{c}^{4}M^{2}i_{2}({\cal
S}(\alpha_{i}),1)\zeta_{0,-1}(m_{c})$ $\displaystyle+$ $\displaystyle
6m_{c}^{2}M^{4}i_{2}({\cal
S}(\alpha_{i}),1)\zeta_{0,-1}(m_{c})-16m_{c}^{4}M^{2}i_{2}({\cal
S}(\alpha_{i}),v)\zeta_{0,-1}(m_{c})$ $\displaystyle-$ $\displaystyle
8m_{c}^{2}M^{4}i_{2}({\cal
S}(\alpha_{i}),v)\zeta_{0,-1}(m_{c})-2m_{c}^{4}M^{2}i_{2}({\cal
T}_{1}(\alpha_{i}),1)\zeta_{0,-1}(m_{c})$ $\displaystyle+$ $\displaystyle
4m_{c}^{4}M^{2}i_{2}({\cal
T}_{1}(\alpha_{i}),v)\zeta_{0,-1}(m_{c})+2m_{c}^{2}M^{4}i_{2}({\cal
T}_{1}(\alpha_{i}),v)\zeta_{0,-1}(m_{c})$ $\displaystyle+$ $\displaystyle
2m_{c}^{4}M^{2}i_{2}({\cal
T}_{2}(\alpha_{i}),1)\zeta_{0,-1}(m_{c})+m_{c}^{2}M^{4}i_{2}({\cal
T}_{2}(\alpha_{i}),1)\zeta_{0,-1}(m_{c})$ $\displaystyle-$ $\displaystyle
4m_{c}^{4}M^{2}i_{2}({\cal
T}_{2}(\alpha_{i}),v)\zeta_{0,-1}(m_{c})-2m_{c}^{2}M^{4}i_{2}({\cal
T}_{2}(\alpha_{i}),v)\zeta_{0,-1}(m_{c})$ $\displaystyle-$ $\displaystyle
2m_{c}^{4}M^{2}i_{2}({\cal
T}_{3}(\alpha_{i}),1)\zeta_{0,-1}(m_{c})-m_{c}^{2}M^{4}i_{2}({\cal
T}_{3}(\alpha_{i}),1)\zeta_{0,-1}(m_{c})$ $\displaystyle+$ $\displaystyle
4m_{c}^{4}M^{2}i_{2}({\cal
T}_{3}(\alpha_{i}),v)\zeta_{0,-1}(m_{c})+2m_{c}^{2}M^{4}i_{2}({\cal
T}_{3}(\alpha_{i}),v)\zeta_{0,-1}(m_{c})$ $\displaystyle-$ $\displaystyle
4m_{c}^{4}M^{2}i_{2}({\cal
T}_{4}(\alpha_{i}),v)\zeta_{0,-1}(m_{c})-2m_{c}^{2}M^{4}i_{2}({\cal
T}_{4}(\alpha_{i}),v)\zeta_{0,-1}(m_{c})+16m_{c}^{4}M^{2}u_{0}\psi_{3}^{\parallel}(u_{0})\zeta_{0,0}(m_{c})$
$\displaystyle-$ $\displaystyle
128m_{c}^{2}M^{4}u_{0}\psi_{3}^{\parallel}(u_{0})\zeta_{0,0}(m_{c})+96M^{6}u_{0}\psi_{3}^{\parallel}(u_{0})\zeta_{0,0}(m_{c})+4m_{c}^{2}M^{2}q^{2}u_{0}i_{1}({\cal
T}_{4}(\alpha_{i}),1)\zeta_{0,0}(m_{c})$ $\displaystyle+$ $\displaystyle
2M^{4}q^{2}u_{0}i_{1}({\cal
T}_{4}(\alpha_{i}),1)\zeta_{0,0}(m_{c})+12m_{c}^{2}M^{4}i_{2}({\cal
S}(\alpha_{i}),1)\zeta_{0,0}(m_{c})+6M^{6}i_{2}({\cal
S}(\alpha_{i}),1)\zeta_{0,0}(m_{c})$ $\displaystyle-$ $\displaystyle
16m_{c}^{2}M^{4}i_{2}({\cal
S}(\alpha_{i}),v)\zeta_{0,0}(m_{c})-8M^{6}i_{2}({\cal
S}(\alpha_{i}),v)\zeta_{0,0}(m_{c})-2m_{c}^{2}M^{4}i_{2}({\cal
T}_{1}(\alpha_{i}),1)\zeta_{0,0}(m_{c})$ $\displaystyle-$ $\displaystyle
M^{6}i_{2}({\cal
T}_{1}(\alpha_{i}),1)\zeta_{0,0}(m_{c})+4m_{c}^{2}M^{4}i_{2}({\cal
T}_{1}(\alpha_{i}),v)\zeta_{0,0}(m_{c})+2M^{6}i_{2}({\cal
T}_{1}(\alpha_{i}),v)\zeta_{0,0}(m_{c})$ $\displaystyle+$ $\displaystyle
2m_{c}^{2}M^{4}i_{2}({\cal
T}_{2}(\alpha_{i}),1)\zeta_{0,0}(m_{c})+M^{6}i_{2}({\cal
T}_{2}(\alpha_{i}),1)\zeta_{0,0}(m_{c})-4m_{c}^{2}M^{4}i_{2}({\cal
T}_{2}(\alpha_{i}),v)\zeta_{0,0}(m_{c})$ $\displaystyle-$ $\displaystyle
2M^{6}i_{2}({\cal
T}_{2}(\alpha_{i}),v)\zeta_{0,0}(m_{c})-2m_{c}^{2}M^{4}i_{2}({\cal
T}_{3}(\alpha_{i}),1)\zeta_{0,0}(m_{c})-M^{6}i_{2}({\cal
T}_{3}(\alpha_{i}),1)\zeta_{0,0}(m_{c})$ $\displaystyle+$ $\displaystyle
4m_{c}^{2}M^{4}i_{2}({\cal
T}_{3}(\alpha_{i}),v)\zeta_{0,0}(m_{c})+2M^{6}i_{2}({\cal
T}_{3}(\alpha_{i}),v)\zeta_{0,0}(m_{c})-4m_{c}^{2}M^{4}i_{2}({\cal
T}_{4}(\alpha_{i}),v)\zeta_{0,0}(m_{c})$ $\displaystyle-$ $\displaystyle
2M^{6}i_{2}({\cal
T}_{4}(\alpha_{i}),v)\zeta_{0,0}(m_{c})-16m_{c}^{4}M^{2}u_{0}\psi_{3}^{\parallel}(u_{0})\zeta_{1,-2}(m_{c})+2m_{c}^{4}q^{2}u_{0}i_{1}({\cal
T}_{4}(\alpha_{i}),1)\zeta_{1,-2}(m_{c})$ $\displaystyle+$ $\displaystyle
6m_{c}^{4}M^{2}i_{2}({\cal
S}(\alpha_{i}),1)\zeta_{1,-2}(m_{c})-8m_{c}^{4}M^{2}i_{2}({\cal
S}(\alpha_{i}),v)\zeta_{1,-2}(m_{c})$ $\displaystyle-$ $\displaystyle
m_{c}^{4}M^{2}i_{2}({\cal
T}_{1}(\alpha_{i}),1)\zeta_{1,-2}(m_{c})+2m_{c}^{4}M^{2}i_{2}({\cal
T}_{1}(\alpha_{i}),v)\zeta_{1,-2}(m_{c})$ $\displaystyle+$ $\displaystyle
m_{c}^{4}M^{2}i_{2}({\cal
T}_{2}(\alpha_{i}),1)\zeta_{1,-2}(m_{c})-2m_{c}^{4}M^{2}i_{2}({\cal
T}_{2}(\alpha_{i}),v)\zeta_{1,-2}(m_{c})$ $\displaystyle-$ $\displaystyle
m_{c}^{4}M^{2}i_{2}({\cal
T}_{3}(\alpha_{i}),1)\zeta_{1,-2}(m_{c})+2m_{c}^{4}M^{2}i_{2}({\cal
T}_{3}(\alpha_{i}),v)\zeta_{1,-2}(m_{c})$ $\displaystyle-$ $\displaystyle
2m_{c}^{4}M^{2}i_{2}({\cal
T}_{4}(\alpha_{i}),v)\zeta_{1,-2}(m_{c})+32m_{c}^{4}M^{2}u_{0}\psi_{3}^{\parallel}(u_{0})\zeta_{1,-1}(m_{c})$
$\displaystyle-$ $\displaystyle
112m_{c}^{2}M^{4}u_{0}\psi_{3}^{\parallel}(u_{0})\zeta_{1,-1}(m_{c})+4m_{c}^{2}M^{2}q^{2}u_{0}i_{1}({\cal
T}_{4}(\alpha_{i}),1)\zeta_{1,-1}(m_{c})$ $\displaystyle+$ $\displaystyle
12m_{c}^{2}M^{4}i_{2}({\cal
S}(\alpha_{i}),1)\zeta_{1,-1}(m_{c})-16m_{c}^{2}M^{4}i_{2}({\cal
S}(\alpha_{i}),v)\zeta_{1,-1}(m_{c})$ $\displaystyle-$ $\displaystyle
2m_{c}^{2}M^{4}i_{2}({\cal
T}_{1}(\alpha_{i}),1)\zeta_{1,-1}(m_{c})+4m_{c}^{2}M^{4}i_{2}({\cal
T}_{1}(\alpha_{i}),v)\zeta_{1,-1}(m_{c})$ $\displaystyle+$ $\displaystyle
2m_{c}^{2}M^{4}i_{2}({\cal
T}_{2}(\alpha_{i}),1)\zeta_{1,-1}(m_{c})-4m_{c}^{2}M^{4}i_{2}({\cal
T}_{2}(\alpha_{i}),v)\zeta_{1,-1}(m_{c})$ $\displaystyle-$ $\displaystyle
2m_{c}^{2}M^{4}i_{2}({\cal
T}_{3}(\alpha_{i}),1)\zeta_{1,-1}(m_{c})+4m_{c}^{2}M^{4}i_{2}({\cal
T}_{3}(\alpha_{i}),v)\zeta_{1,-1}(m_{c})$ $\displaystyle-$ $\displaystyle
4m_{c}^{2}M^{4}i_{2}({\cal
T}_{4}(\alpha_{i}),v)\zeta_{1,-1}(m_{c})+64m_{c}^{2}M^{4}u_{0}\psi_{3}^{\parallel}(u_{0})\zeta_{1,0}(m_{c})$
$\displaystyle-$ $\displaystyle
192M^{6}u_{0}\psi_{3}^{\parallel}(u_{0})\zeta_{1,0}(m_{c})+4M^{4}q^{2}u_{0}i_{1}({\cal
T}_{4}(\alpha_{i}),1)\zeta_{1,0}(m_{c})$ $\displaystyle+$ $\displaystyle
12M^{6}i_{2}({\cal S}(\alpha_{i}),1)\zeta_{1,0}(m_{c})-16M^{6}i_{2}({\cal
S}(\alpha_{i}),v)\zeta_{1,0}(m_{c})-2M^{6}i_{2}({\cal
T}_{1}(\alpha_{i}),1)\zeta_{1,0}(m_{c})$ $\displaystyle+$ $\displaystyle
4M^{6}i_{2}({\cal T}_{1}(\alpha_{i}),v)\zeta_{1,0}(m_{c})+2M^{6}i_{2}({\cal
T}_{2}(\alpha_{i}),1)\zeta_{1,0}(m_{c})-4M^{6}i_{2}({\cal
T}_{2}(\alpha_{i}),v)\zeta_{1,0}(m_{c})$ $\displaystyle-$ $\displaystyle
2M^{6}i_{2}({\cal T}_{3}(\alpha_{i}),1)\zeta_{1,0}(m_{c})+4M^{6}i_{2}({\cal
T}_{3}(\alpha_{i}),v)\zeta_{1,0}(m_{c})-4M^{6}i_{2}({\cal
T}_{4}(\alpha_{i}),v)\zeta_{1,0}(m_{c})$ $\displaystyle+$ $\displaystyle
16m_{c}^{4}M^{2}u_{0}\psi_{3}^{\parallel}(u_{0})\zeta_{2,-2}(m_{c})+64m_{c}^{2}M^{4}u_{0}\psi_{3}^{\parallel}(u_{0})\zeta_{2,-1}(m_{c})+96M^{6}u_{0}\psi_{3}^{\parallel}(u_{0})\zeta_{2,0}(m_{c})\Bigg{)}\Bigg{]}\Biggr{\\}}~{},$

$\displaystyle\Pi_{B}^{0}$ $\displaystyle=$
$\displaystyle\frac{e^{-\frac{q^{2}}{M_{1}^{2}+M_{2}^{2}}}}{24\pi^{2}}\Bigg{\\{}4m_{c}f_{\rho}^{\parallel}m_{\rho}\bigg{[}4m_{\rho}^{2}\big{(}i_{1}(\phi(\alpha_{i}),1)+i_{1}(\psi(\alpha_{i}),1)\big{)}\big{(}\zeta_{-1,0}(m_{c})-\zeta_{0,0}(m_{c})\big{)}$
(33) $\displaystyle+$ $\displaystyle
M^{2}\psi_{3}^{\perp}(u_{0})\big{(}-\zeta_{0,0}(m_{c})+\zeta_{1,0}(m_{c})\big{)}\bigg{]}+f_{\rho}^{\perp}\bigg{[}-2M^{2}m_{\rho}^{2}\bigg{(}i_{2}({\cal
S}(\alpha_{i}),1)+3i_{2}(\tilde{{\cal S}}(\alpha_{i}),1)$ $\displaystyle-$
$\displaystyle 4i_{2}(\tilde{{\cal S}}(\alpha_{i}),v)+i_{2}({\cal
T}_{1}(\alpha_{i}),1)-2i_{2}({\cal T}_{1}(\alpha_{i}),v)-i_{2}({\cal
T}_{2}(\alpha_{i}),1)+2i_{2}({\cal T}_{2}(\alpha_{i}),v)$ $\displaystyle+$
$\displaystyle 2i_{2}({\cal T}_{3}(\alpha_{i}),1)-4i_{2}({\cal
T}_{3}(\alpha_{i}),v)-i_{2}({\cal T}_{4}(\alpha_{i}),1)+2i_{2}({\cal
T}_{4}(\alpha_{i}),v)\bigg{)}\big{[}\zeta_{0,0}(m_{c})-\zeta_{1,0}(m_{c})\big{]}$
$\displaystyle+$ $\displaystyle
4M^{4}\phi^{\perp}(u_{0})(-\zeta_{1,0}(m_{c})+\zeta_{2,0}(m_{c}))+{\cal
A}^{\perp}(u_{0})m_{\rho}^{2}(m_{c}^{2}\zeta_{0,0}(m_{c})+m_{c}^{2}\zeta_{1,-1}(m_{c})$
$\displaystyle-$ $\displaystyle
m_{c}^{2}\zeta_{1,0}(m_{c})+3M^{2}\zeta_{1,0}(m_{c})-m_{c}^{2}\zeta_{2,-1}(m_{c})-3M^{2}\zeta_{2,0}(m_{c}))+16\tilde{j}_{1}({\cal
B}^{\perp}(u))m_{\rho}^{2}\bigg{(}3m_{c}^{2}\zeta_{0,0}(m_{c})$
$\displaystyle+$ $\displaystyle
2m_{c}^{2}\zeta_{1,-1}(m_{c})-2\big{[}(m_{c}^{2}-4M^{2})\zeta_{1,0}(m_{c})+m_{c}^{2}\zeta_{2,-1}(m_{c})+4M^{2}\zeta_{2,0}(m_{c})\big{]}\bigg{)}\bigg{]}\Bigg{\\}}~{},$

$\displaystyle\Pi_{B}^{GG}$ $\displaystyle=$
$\displaystyle-\frac{e^{-\frac{q^{2}}{M_{1}^{2}+M_{2}^{2}}}}{864m_{c}M^{6}\pi^{2}}g_{s}^{2}\langle
GG\rangle\Bigg{\\{}m_{c}{\cal
A}^{\perp}(u_{0})f_{\rho}^{\perp}m_{\rho}^{2}\Bigg{[}m_{c}^{4}\zeta_{-2,0}(m_{c})+2m_{c}^{4}\zeta_{-1,-1}(m_{c})$
(34) $\displaystyle-$ $\displaystyle
m_{c}^{4}\zeta_{-1,0}(m_{c})+2m_{c}^{2}M^{2}\zeta_{-1,0}(m_{c})+m_{c}^{4}\zeta_{0,-2}(m_{c})-2m_{c}^{4}\zeta_{0,-1}(m_{c})$
$\displaystyle+$ $\displaystyle
2m_{c}^{2}M^{2}\zeta_{0,-1}(m_{c})-2m_{c}^{2}M^{2}\zeta_{0,0}(m_{c})+2M^{4}\zeta_{0,0}(m_{c})-m_{c}^{4}\zeta_{1,-2}(m_{c})$
$\displaystyle-$ $\displaystyle
2m_{c}^{2}M^{2}\zeta_{1,-1}(m_{c})-2M^{4}\zeta_{1,0}(m_{c})\Bigg{]}+m_{c}f_{\rho}^{\perp}\Bigg{[}16\tilde{j}_{1}({\cal
B}^{\perp}(u))m_{\rho}^{2}\bigg{(}3m_{c}^{4}\zeta_{-2,0}(m_{c})$
$\displaystyle+$ $\displaystyle
6m_{c}^{4}\zeta_{-1,-1}(m_{c})-3m_{c}^{4}\zeta_{-1,0}(m_{c})+5m_{c}^{2}M^{2}\zeta_{-1,0}(m_{c})+3m_{c}^{4}\zeta_{0,-2}(m_{c})$
$\displaystyle-$ $\displaystyle
6m_{c}^{4}\zeta_{0,-1}(m_{c})+5m_{c}^{2}M^{2}\zeta_{0,-1}(m_{c})-8m_{c}^{2}M^{2}\zeta_{0,0}(m_{c})+7M^{4}\zeta_{0,0}(m_{c})$
$\displaystyle-$ $\displaystyle
3m_{c}^{4}\zeta_{1,-2}(m_{c})-8m_{c}^{2}M^{2}\zeta_{1,-1}(m_{c})-10M^{4}\zeta_{1,0}(m_{c})\bigg{)}-M^{2}\bigg{(}m_{\rho}^{2}(i_{2}({\cal
S}(\alpha_{i}),1)$ $\displaystyle+$ $\displaystyle
3i_{2}(\tilde{S}(\alpha_{i}),1)-4i_{2}(\tilde{S}(\alpha_{i}),v)+i_{2}({\cal
T}_{1}(\alpha_{i}),1)-2i_{2}({\cal T}_{1}(\alpha_{i}),v)-i_{2}({\cal
T}_{2}(\alpha_{i}),1)$ $\displaystyle+$ $\displaystyle 2i_{2}({\cal
T}_{2}(\alpha_{i}),v)+2i_{2}({\cal T}_{3}(\alpha_{i}),1)-4i_{2}({\cal
T}_{3}(\alpha_{i}),v)-i_{2}({\cal T}_{4}(\alpha_{i}),1)$ $\displaystyle+$
$\displaystyle 2i_{2}({\cal
T}_{4}(\alpha_{i}),v))(m_{c}^{2}(\zeta_{-1,0}(m_{c})+\zeta_{0,-1}(m_{c}))+M^{2}\zeta_{0,0}(m_{c}))$
$\displaystyle+$ $\displaystyle
4M^{2}\phi^{\perp}(u_{0})(m_{c}^{2}\zeta_{-1,0}(m_{c})+m_{c}^{2}\zeta_{0,-1}(m_{c})-m_{c}^{2}\zeta_{0,0}(m_{c})+2M^{2}\zeta_{0,0}(m_{c})$
$\displaystyle-$ $\displaystyle
m_{c}^{2}\zeta_{1,-1}(m_{c})-2M^{2}\zeta_{1,0}(m_{c}))\bigg{)}\Bigg{]}+2f_{\rho}^{\parallel}m_{\rho}\Bigg{[}4m_{c}^{4}m_{\rho}^{2}(i_{1}(\phi(\alpha_{i}),1)$
$\displaystyle+$ $\displaystyle
i_{1}(\psi(\alpha_{i}),1))(\zeta_{-2,0}(m_{c})+\zeta_{-1,-1}(m_{c}))+M^{2}\psi_{3}^{\perp}(u_{0})\bigg{(}(-2m_{c}^{4}+3m_{c}^{2}M^{2})\zeta_{-1,0}(m_{c})$
$\displaystyle+$ $\displaystyle
3m_{c}^{2}(-m_{c}^{2}+M^{2})\zeta_{0,-1}(m_{c})+m_{c}^{4}\zeta_{0,0}(m_{c})-8m_{c}^{2}M^{2}\zeta_{0,0}(m_{c})+6M^{4}\zeta_{0,0}(m_{c})$
$\displaystyle-$ $\displaystyle
m_{c}^{4}\zeta_{1,-2}(m_{c})+2m_{c}^{4}\zeta_{1,-1}(m_{c})-7m_{c}^{2}M^{2}\zeta_{1,-1}(m_{c})+4m_{c}^{2}M^{2}\zeta_{1,0}(m_{c})$
$\displaystyle-$ $\displaystyle
12M^{4}\zeta_{1,0}(m_{c})+m_{c}^{4}\zeta_{2,-2}(m_{c})+4m_{c}^{2}M^{2}\zeta_{2,-1}(m_{c})+6M^{4}\zeta_{2,0}(m_{c})\bigg{)}\Bigg{]}\Bigg{\\}}~{},$

$\displaystyle\Pi_{C}^{0}$ $\displaystyle=$
$\displaystyle\frac{e^{-\frac{q^{2}}{M_{1}^{2}+M_{2}^{2}}}}{3\pi^{2}}M^{2}f_{\rho}^{\parallel}m_{\rho}\Bigg{\\{}\Bigg{[}i_{2}(\alpha_{g}\tilde{{\cal
A}}(\alpha_{i}),v)-2i_{2}(\alpha_{g}\tilde{{\cal
A}}(\alpha_{i}),v^{2})+i_{2}(\alpha_{q}\tilde{{\cal A}}(\alpha_{i}),1)$ (35)
$\displaystyle-$ $\displaystyle 2i_{2}(\alpha_{q}\tilde{{\cal
A}}(\alpha_{i}),v)-i_{2}(\alpha_{g}{\cal
V}(\alpha_{i}),v)-i_{2}(\alpha_{q}{\cal
V}(\alpha_{i}),1)\Bigg{]}\bigg{(}\zeta_{0,0}(m_{c})-\zeta_{1,0}(m_{c})\bigg{)}$
$\displaystyle+$ $\displaystyle
u_{0}^{2}\psi_{3}^{\perp}(u_{0})\bigg{(}\zeta_{1,0}(m_{c})-\zeta_{2,0}(m_{c})\bigg{)}\Bigg{\\}}~{},$

$\displaystyle\Pi_{C}^{GG}$ $\displaystyle=$
$\displaystyle-\frac{e^{-\frac{q^{2}}{M_{1}^{2}+M_{2}^{2}}}}{216M^{4}\pi^{2}}g_{s}^{2}\langle
GG\rangle f_{\rho}^{\parallel}\Bigg{\\{}\Bigg{(}i_{2}(\alpha_{g}\tilde{{\cal
A}}(\alpha_{i}),v)-2i_{2}(\alpha_{g}\tilde{{\cal
A}}(\alpha_{i}),v^{2})+i_{2}(\alpha_{q}\tilde{{\cal A}}(\alpha_{i}),1)$ (36)
$\displaystyle-$ $\displaystyle 2i_{2}(\alpha_{q}\tilde{{\cal
A}}(\alpha_{i}),v)-i_{2}(\alpha_{g}{\cal
V}(\alpha_{i}),v)-i_{2}(\alpha_{q}{\cal
V}(\alpha_{i}),1)\Bigg{)}\Bigg{[}m_{c}^{2}(\zeta_{-1,0}(m_{c})+\zeta_{0,-1}(m_{c}))$
$\displaystyle+$ $\displaystyle
M^{2}\zeta_{0,0}(m_{c})\Bigg{]}+2u_{0}^{2}\psi_{3}^{\perp}(u_{0})\Bigg{[}m_{c}^{2}\zeta_{-1,0}(m_{c})+m_{c}^{2}\zeta_{0,-1}(m_{c})-m_{c}^{2}\zeta_{0,0}(m_{c})+2M^{2}\zeta_{0,0}(m_{c})$
$\displaystyle-$ $\displaystyle
m_{c}^{2}\zeta_{1,-1}(m_{c})-2M^{2}\zeta_{1,0}(m_{c})\Bigg{]}\Bigg{\\}}~{},$

$\displaystyle\Pi_{D}^{0}$ $\displaystyle=$
$\displaystyle-\frac{2e^{-\frac{q^{2}}{M_{1}^{2}+M_{2}^{2}}}}{3M^{2}\pi^{2}}u_{0}m_{\rho}^{2}\Bigg{\\{}4m_{c}f_{\rho}^{\parallel}m_{\rho}\bigg{(}u_{0}i_{1}(\psi(\alpha_{i}),1)-i_{1}(\alpha_{g}\psi(\alpha_{i}),v)$
(37) $\displaystyle-$ $\displaystyle
i_{1}(\alpha_{q}\psi(\alpha_{i}),1)\bigg{)}\big{[}\zeta_{-1,0}(m_{c})-\zeta_{0,0}(m_{c})\big{]}+M^{2}f_{\rho}^{\perp}\Bigg{[}\bigg{(}i_{1}({\cal
T}(\alpha_{i}),1)$ $\displaystyle-$ $\displaystyle 2\big{[}i_{1}({\cal
T}(\alpha_{i}),v)+i_{1}({\cal T}_{4}(\alpha_{i}),1)-2i_{1}({\cal
T}_{4}(\alpha_{i}),v)\big{]}\bigg{)}\bigg{[}\zeta_{0,0}(m_{c})-\zeta_{1,0}(m_{c})\bigg{]}$
$\displaystyle+$ $\displaystyle 4u_{0}\tilde{j}_{1}({\cal
B}^{\perp}(u))\bigg{(}\zeta_{1,0}(m_{c})-\zeta_{2,0}(m_{c})\bigg{)}\Bigg{]}\Bigg{\\}}~{},$

and

$\displaystyle\Pi_{D}^{GG}$ $\displaystyle=$
$\displaystyle\frac{e^{-\frac{q^{2}}{M_{1}^{2}+M_{2}^{2}}}}{108M^{8}\pi^{2}}g_{s}^{2}\langle
GG\rangle
u_{0}m_{\rho}^{2}\Bigg{\\{}4m_{c}^{3}f_{\rho}^{\parallel}m_{\rho}\bigg{[}u_{0}i_{1}(\psi,1)-i_{1}(\alpha_{g}\psi(\alpha_{i}),v)$
(38) $\displaystyle-$ $\displaystyle
i_{1}(\alpha_{q}\psi(\alpha_{i}),1)\bigg{]}\bigg{[}\zeta_{-2,0}(m_{c})+\zeta_{-1,-1}(m_{c})\bigg{]}+M^{2}f_{\rho}^{\perp}\Bigg{[}\Bigg{(}i_{1}({\cal
T}(\alpha_{i}),1)$ $\displaystyle-$ $\displaystyle 2\bigg{[}i_{1}({\cal
T}(\alpha_{i}),v)+i_{1}({\cal T}_{4}(\alpha_{i}),1)-2i_{1}({\cal
T}_{4}(\alpha_{i}),v)\bigg{]}\Bigg{)}\Bigg{(}m_{c}^{2}\bigg{[}\zeta_{-1,0}(m_{c})+\zeta_{0,-1}(m_{c})\bigg{]}$
$\displaystyle+$ $\displaystyle
M^{2}\zeta_{0,0}(m_{c})\Bigg{)}+8u_{0}\tilde{j}_{1}({\cal
B}^{\perp}(u))\Bigg{(}m_{c}^{2}\zeta_{-1,0}(m_{c})+m_{c}^{2}\zeta_{0,-1}(m_{c})-m_{c}^{2}\zeta_{0,0}(m_{c})$
$\displaystyle+$ $\displaystyle
2M^{2}\zeta_{0,0}(m_{c})-m_{c}^{2}\zeta_{1,-1}(m_{c})-2M^{2}\zeta_{1,0}(m_{c})\Bigg{)}\Bigg{]}\Bigg{\\}}~{}.$

In the above expressions the functions $\zeta_{m,n}(m_{1},m_{2})$ are defined
as

$\displaystyle\zeta_{m,n}(m_{1},m_{2})=\int_{z_{\text{min}}}^{z_{\text{max}}}dzz^{m}\bar{z}^{n}e^{-\frac{m_{1}^{2}}{M^{2}z}-\frac{m_{2}^{2}}{M^{2}\bar{z}}}~{},$
(39)

where $m$ and $n$ are integers and
$\zeta_{m,n}(m_{1},m_{1})=\zeta_{m,n}(m_{1})$. We have also used the following
short-hand notations:

$\displaystyle i_{1}(\phi(\alpha_{i}),f(v))=\int{\cal
D}\alpha_{i}\int_{0}^{1}dv\phi(\alpha_{q},\alpha_{\bar{q}},\alpha_{g})f(v)\theta(k-u_{0})~{},$
(40) $\displaystyle i_{2}(\phi(\alpha_{i}),f(v))=\int{\cal
D}\alpha_{i}\int_{0}^{1}dv\phi(\alpha_{q},\alpha_{\bar{q}},\alpha_{g})f(v)\delta(k-u_{0})~{},$
(41) $\displaystyle i_{3}(\phi(\alpha_{i}),f(v))=\int{\cal
D}\alpha_{i}\int_{0}^{1}dv\phi(\alpha_{q},\alpha_{\bar{q}},\alpha_{g})f(v)\delta^{{}^{\prime}}(k-u_{0})~{},$
(42) $\displaystyle j_{1}(f(u))=\int_{u_{0}}^{1}duf(u)~{},$ (43)

and

$\displaystyle\tilde{j_{1}}(f(u))=\int_{u_{0}}^{1}du(u-u_{0})f(u)~{},$ (44)

where $k=\alpha_{q}-v\alpha_{g}$. For the explicit forms of wave functions of
vector mesons and the functions used in the above expressions, in terms of
vector mesons’ DAS, one can see Refs. [54, 55, 56].

## References

  * [1] W. Wang, F. S. Yu and Z. X. Zhao, “Weak decays of doubly heavy baryons: the $1/2\rightarrow 1/2$ case,”  Eur. Phys. J. C 77, no. 11, 781 (2017).
  * [2] W. Wang, Z. P. Xing and J. Xu, “Weak Decays of Doubly Heavy Baryons: SU(3) Analysis,”  Eur. Phys. J. C 77, no. 11, 800 (2017).
  * [3] T. Gutsche, M. A. Ivanov, J. G. Körner and V. E. Lyubovitskij, “Decay chain information on the newly discovered double charm baryon state $\Xi_{cc}^{++}$,”  Phys. Rev. D 96, no. 5, 054013 (2017).
  * [4] H. S. Li, L. Meng, Z. W. Liu and S. L. Zhu, “Radiative decays of the doubly charmed baryons in chiral perturbation theory,”  Phys. Lett. B 777, 169 (2018).
  * [5] L. Y. Xiao, K. L. Wang, Q. f. Lu, X. H. Zhong and S. L. Zhu, “Strong and radiative decays of the doubly charmed baryons,”  Phys. Rev. D 96, no. 9, 094005 (2017).
  * [6] N. Sharma and R. Dhir, “Estimates of W-exchange contributions to $\Xi_{cc}$ decays,”  Phys. Rev. D 96, no. 11, 113006 (2017).
  * [7] Y. L. Ma and M. Harada, “Chiral partner structure of doubly heavy baryons with heavy quark spin-flavor symmetry,”  J. Phys. G 45, no. 7, 075006 (2018).
  * [8] X. H. Hu, Y. L. Shen, W. Wang and Z. X. Zhao, “Weak decays of doubly heavy baryons: ”decay constants”,”  Chin. Phys. C 42, no. 12, 123102 (2018).
  * [9] Y. J. Shi, W. Wang, Y. Xing and J. Xu, “Weak Decays of Doubly Heavy Baryons: Multi-body Decay Channels,”  Eur. Phys. J. C 78, no. 1, 56 (2018).
  * [10] D. L. Yao, “Masses and sigma terms of doubly charmed baryons up to $\mathcal{O}(p^{4})$ in manifestly Lorentz-invariant baryon chiral perturbation theory,”  Phys. Rev. D 97, no. 3, 034012 (2018).
  * [11] Z. X. Zhao, “Weak decays of doubly heavy baryons: the $1/2\rightarrow 3/2$ case,”  Eur. Phys. J. C 78, no. 9, 756 (2018).
  * [12] M. Z. Liu, Y. Xiao and L. S. Geng, “Magnetic moments of the spin-1/2 doubly charmed baryons in covariant baryon chiral perturbation theory,”  Phys. Rev. D 98, no. 1, 014040 (2018).
  * [13] Z. P. Xing and Z. X. Zhao, “Weak decays of doubly heavy baryons: the FCNC processes,” Phys. Rev. D 98, no. 5, 056002 (2018).
  * [14] R. Dhir and N. Sharma, “Weak decays of doubly heavy charm $\Omega_{cc}^{+}$ baryon,” Eur. Phys. J. C 78, no. 9, 743 (2018).
  * [15] A. V. Berezhnoy, A. K. Likhoded and A. V. Luchinsky, “Doubly heavy baryons at the LHC,” Phys. Rev. D 98, no. 11, 113004 (2018).
  * [16] L. J. Jiang, B. He and R. H. Li, “Weak decays of doubly heavy baryons: $\mathcal{B}_{cc}\rightarrow\mathcal{B}_{c}V$,” Eur. Phys. J. C 78, no. 11, 961 (2018).
  * [17] Q. A. Zhang, “Weak Decays of Doubly Heavy Baryons: W-Exchange,” Eur. Phys. J. C 78, no. 12, 1024 (2018).
  * [18] T. Gutsche, M. A. Ivanov, J. G. Körner, V. E. Lyubovitskij and Z. Tyulemissov, “Ab initio three-loop calculation of the $W$-exchange contribution to nonleptonic decays of double charm baryons,” Phys. Rev. D 99, no. 5, 056013 (2019).
  * [19] Y. J. Shi, Y. Xing and Z. X. Zhao, “Light-cone sum rules analysis of $\Xi_{QQ^{\prime}q}\to\Lambda_{Q^{\prime}}$ weak decays,” Eur. Phys. J. C 79, no. 6, 501 (2019).
  * [20] X. H. Hu and Y. J. Shi, “Light-Cone Sum Rules Analysis of $\Xi_{QQ^{\prime}}\to\Sigma_{Q^{\prime}}$ Weak Decays,” Eur. Phys. J. C 80, no. 1, 56 (2020).
  * [21] S. J. Brodsky, F. K. Guo, C. Hanhart and U.-G. Meißner, “Isospin splittings of doubly heavy baryons,” Phys. Lett. B 698, 251 (2011).
  * [22] M. Mattson et al. [SELEX Collaboration], “First Observation of the Doubly Charmed Baryon $\Xi^{+}_{cc}$,”  Phys. Rev. Lett. 89, 112001 (2002).
  * [23] A. Ocherashvili et al. [SELEX Collaboration], “Confirmation of the double charm baryon $\Xi^{+}_{cc}$(3520) via its decay to $pD^{+}K^{-}$,”  Phys. Lett. B 628, 18 (2005).
  * [24] S. P. Ratti, “New results on c-baryons and a search for cc-baryons in FOCUS,”  Nucl. Phys. Proc. Suppl. 115, 33 (2003).
  * [25] B. Aubert et al. [BaBar Collaboration], “Search for doubly charmed baryons $\Xi_{cc}^{+}$ and $\Xi_{cc}^{++}$ in BABAR,”  Phys. Rev. D 74, 011103 (2006).
  * [26] R. Aaij et al. [LHCb Collaboration], “Search for the doubly charmed baryon $\Xi_{cc}^{+}$,”  JHEP 1312, 090 (2013).
  * [27] Y. Kato et al. [Belle Collaboration], “Search for doubly charmed baryons and study of charmed strange baryons at Belle,”  Phys. Rev. D 89, no. 5, 052003 (2014).
  * [28] R. Aaij et al. [LHCb Collaboration], “Observation of the doubly charmed baryon $\Xi_{cc}^{++}$,”  Phys. Rev. Lett. 119, no. 11, 112001 (2017).
  * [29] T. M. Aliev, K. Azizi and M. Savci, “Doubly Heavy Spin–1/2 Baryon Spectrum in QCD,”  Nucl. Phys. A 895, 59 (2012).
  * [30] T. M. Aliev, K. Azizi and M. Savci, “The masses and residues of doubly heavy spin-3/2 baryons,”  J. Phys. G 40, 065003 (2013).
  * [31] K. Azizi, T. M. Aliev and M. Savci, “Properties of doubly and triply heavy baryons,”  J. Phys. Conf. Ser. 556, no. 1, 012016 (2014).
  * [32] T. M. Aliev, K. Azizi and M. Savcı, “Mixing angle of doubly heavy baryons in QCD,” Phys. Lett. B 715, 149-151 (2012.
  * [33] U. Özdem, “Magnetic dipole moments of the spin-$\frac{3}{2}$ doubly heavy baryons,” Eur. Phys. J. A 56, no.2, 34 (2020).
  * [34] R. Aaij et al. [LHCb Collaboration], “First Observation of the Doubly Charmed Baryon Decay $\Xi_{cc}^{++}\rightarrow\Xi_{c}^{+}\pi^{+}$,”  Phys. Rev. Lett. 121, no. 16, 162002 (2018).
  * [35] S. Brodsky, S. Groote and S. Koshkarev, “Resolving the SELEX–LHCb double-charm baryon conflict: the impact of intrinsic heavy-quark hadroproduction and supersymmetric light-front holographic QCD,” Eur. Phys. J. C 78, no.6, 483 (2018).
  * [36] R. Aaij et al. [LHCb], “Search for the doubly heavy $\mathit{\Xi}_{bc}^{0}$ baryon via decays to $D^{0}pK^{-}$,”  arXiv:2009.02481 [hep-ex].
  * [37] M. Gell-Mann, A schematic model of baryons and mesons Physics Letters 8 (3) (1964) 214 – 215.
  * [38] G. Zweig, An SU(3) model for strong interaction symmetry and its breaking. Version 1 https://inspirehep.net/record/11881/files/CM-P00042883.pdf.
  * [39] D. Lichtenberg, S. P. Rosen, G. Zweig, DEVELOPMENTS IN THE QUARK THEORY OF HADRONS https://inspirehep.net/record/4674/files/cern-th-412.pdf.
  * [40] Z. S. Brown, W. Detmold, S. Meinel and K. Orginos, “Charmed bottom baryon spectroscopy from lattice QCD,” Phys. Rev. D 90, no.9, 094507 (2014).
  * [41] E. Hernandez, J. Nieves and J. M. Verde-Velasco, “Heavy quark symmetry constraints on semileptonic form-factors and decay widths of doubly heavy baryons,” Phys. Lett. B 663 (2008), 234-241.
  * [42] J. M. Flynn, E. Hernandez and J. Nieves, “Triply Heavy Baryons and Heavy Quark Spin Symmetry,” Phys. Rev. D 85 (2012), 014012.
  * [43] K. Azizi, A. R. Olamaei and S. Rostami, “Beautiful mathematics for beauty-full and other multi-heavy hadronic systems,” Eur. Phys. J. A 54, no. 9, 162 (2018).
  * [44] A. R. Olamaei, K. Azizi and S. Rostami, “Strong coupling constants of the doubly heavy $\Xi_{QQ}$ Baryons with $\pi$ Meson,” Eur. Phys. J. C 80, no.7, 613 (2020).
  * [45] M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, “QCD and Resonance Physics. Theoretical Foundations,” Nucl. Phys. B 147, 385-447 (1979).
  * [46] I. Balitsky, V. M. Braun and A. Kolesnichenko, “Radiative Decay $\Sigma^{+}\rightarrow p\gamma$ in Quantum Chromodynamics,” Nucl. Phys. B 312, 509-550 (1989).
  * [47] A. Khodjamirian, R. Ruckl, S. Weinzierl and O. I. Yakovlev, “Perturbative QCD correction to the $B\rightarrow\pi$ transition form-factor,”  Phys. Lett. B 410, 275 (1997)
  * [48] V. M. Braun, “Light cone sum rules,” arXiv:hep-ph/9801222 [hep-ph].
  * [49] S. Rostami, K. Azizi and A. R. Olamaei, “Strong Coupling Constants of the Doubly Heavy Spin-1/2 Baryons with Light Pseudoscalar Mesons,” arXiv:2008.12715 [hep-ph].
  * [50] H. I. Alrebdi, T. M. Aliev and K. Şimşek, “Determination of the strong vertices of doubly heavy baryons with pseudoscalar mesons in QCD,” Phys. Rev. D 102, no.7, 074007 (2020).
  * [51] T. M. Aliev and K. Şimşek, “Strong coupling constants of doubly heavy baryons with vector mesons in QCD,” Eur. Phys. J. C 80, no.10, 976 (2020).
  * [52] T. M. Aliev, K. Azizi and M. Savci, “Vertices of the heavy spin-3/2 sextet baryons with light vector mesons in QCD,” Eur. Phys. J. A 47, 125 (2011)
  * [53] V. Pascalutsa, M. Vanderhaeghen and S. N. Yang, “Electromagnetic excitation of the $\Delta(1232)$-resonance,” Phys. Rept. 437 (2007), 125-232.
  * [54] P. Ball, V. M. Braun, Y. Koike and K. Tanaka, “Higher twist distribution amplitudes of vector mesons in QCD: Formalism and twist - three distributions,” Nucl. Phys. B 529, 323-382 (1998).
  * [55] P. Ball and V. M. Braun, “Higher twist distribution amplitudes of vector mesons in QCD: Twist - 4 distributions and meson mass corrections,” Nucl. Phys. B 543, 201-238 (1999).
  * [56] P. Ball, V. M. Braun and A. Lenz, “Twist-4 distribution amplitudes of the $K^{*}$ and $\phi$ mesons in QCD,” JHEP 08, 090 (2007).
  * [57] M. Tanabashi et al. [Particle Data Group], “Review of Particle Physics,” Phys. Rev. D 98, no. 3, 030001 (2018).
  * [58] P. Ball and G. W. Jones, “Twist-3 distribution amplitudes of $K^{*}$ and $\phi$ mesons,” JHEP 03, 069 (2007).