# The electromagnetic multipole moments of the charged open-flavor
$Z_{\bar{c}q}$ states

K. Azizi<EMAIL_ADDRESS>Department of Physics, Dogus University,
Acibadem-Kadikoy, 34722 Istanbul, Turkey  School of Physics, Institute for
Research in Fundamental Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran  U.
Özdem<EMAIL_ADDRESS>Department of Physics, Dogus University, Acibadem-
Kadikoy, 34722 Istanbul, Turkey

###### Abstract

The electromagnetic multipole moments of the open-flavor $Z_{\bar{c}q}$ states
are investigated by assuming a diquark-antidiquark picture for their internal
structure and quantum numbers $J^{PC}=1^{+-}$ for their spin-parity. In
particular, their magnetic and quadrupole moments are extracted in the
framework of light-cone QCD sum rule by the help of the photon distribution
amplitudes. The electromagnetic multipole moments of the open-flavor
$Z_{\bar{c}q}$ states are important dynamical observables, which encode
valuable information on their underlying structure. The results obtained for
the magnetic moments of different structures are considerably large and can be
measured in future experiments. We obtain very small values for the quadrupole
moments of $Z_{\bar{c}q}$ states indicating a nonspherical charge
distribution.

Tetraquarks, Electromagnetic form factors, Multipole moments, Open-flavor
states

##  I Introduction

Since 2003, there are many non-conventional hadrons discovered experimentally,
such as many XYZ tetraquarks, $P_{c}^{+}(4380)$ and $P_{c}^{+}(4450)$
pentaquarks etc., which could not be described as the conventional hadrons
composed of two or three valence quark/antiquarks. They are called exotic
hadrons. For some reviews on the theoretical and experimental progress on the
properties of these new states see Refs. Nielsen  _et al._ (2010); Swanson
(2006); Voloshin (2008); Klempt and Zaitsev (2007); Godfrey and Olsen (2008);
Faccini  _et al._ (2012); Esposito  _et al._ (2015); Chen  _et al._ (2016);
Ali  _et al._ (2017); Esposito  _et al._ (2016); Olsen  _et al._ (2017). The
greatest achievement with regard to the exotic states was the discovery of the
charged multiquark states. The charged states with a hidden pair of heavy
quark and antiquark such as the $Z_{c}^{\pm}(3900)$ Ablikim  _et al._ (2013a),
$Z_{c}^{\pm}(4020)$ Ablikim  _et al._ (2013b), $Z_{c}^{\pm}(4430)$ Choi  _et
al._ (2008), $Z_{b}^{\pm}(10610,10650)$ Bondar  _et al._ (2012), would be
undoubtedly considered as the exotic resonances, because these charged states
cannot be explained as excited charmonium-like or bottomonium-like states.

Most of the discovered exotic states up to now share a common properties: they
contain a hidden heavy quark-antiquark pair, $\bar{c}c$ or $\bar{b}b$. However
existence of the multiquarks, which do not contain $\bar{c}c$ or $\bar{b}b$
pairs is also possible, because fundamental laws of QCD do not prohibit
existence of such open-flavor multiquark states. It should be noted that they
have not been discovered experimentally, and to our best knowledge, there are
not any candidates to be considered for these states. They may be seen in the
exclusive reactions as the open-charm and open-bottom resonances. In 2003, the
two narrow charm-strange mesons $D^{∗}_{s0}(2317)$ and $D_{s1}(2460)$ were
observed in the $D^{+}_{s}\pi^{0}$ and $D^{*+}_{s}\pi^{0}$ invariant mass
distributions by the BABAR Aubert  _et al._ (2003) and CLEO Besson  _et al._
(2003)] collaborations, are now being considered as candidates to open-charm
tetraquark states. In 2016, the D0 Collaboration reported the observation of a
state with four different quark flavors, the $X(5568)$, and assigned the
quantum numbers $J^{P}=0^{+}$ for it, but they did not exclude the possibility
of $J^{P}=1^{+}$ Abazov  _et al._ (2016). Reported in the $B^{0}_{s}\pi^{\pm}$
final states, the $X(5568)$ meson, if exist, cannot be categorized into the
conventional meson family, and is a good candidate of exotic tetraquark state
with valence quarks of four different flavors such as $su\bar{d}\bar{b}$ or
$sd\bar{u}\bar{b}$. The observation of these states have immediately inspired
extensive discussions on the possibility of their internal structure. For more
information see for instance Refs. Agaev  _et al._ (2016a, b); Chen  _et al._
(2017a) and references therein. In 2017, the D0 Collaboration repeated their
analysis when the $B_{s}$ is reconstructed semileptonically. They reported
evidence for a narrow structure, which was consistent with their previous
measurement in the hadronic decay mode Abazov  _et al._ (2017). However, other
experimental groups, namely the LHCb Aaij  _et al._ (2016), CDF Aaltonen  _et
al._ (2017), CMS Sirunyan  _et al._ (2017) and ATLAS Aaboud  _et al._ (2018)
collaborations could not find this resonance from analysis of their
experimental data.

In order to understand the inner structure of the hadrons in the
nonperturbative regime of QCD, the main challenges are the determination of
the dynamical and statical features of hadrons such as their decay form
factors, masses, electromagnetic multipole moments and so on, both
experimentally and theoretically. Many theoretical models accurately estimate
the mass and decay width of the discovered exotic states, but the inner
structure of these states is still unclear. In other words, the mass and decay
width alone can not distinguish the inner structure of the exotic states.
Recall that the electromagnetic multipole moments are equally important
dynamical observables of the exotic states. The electromagnetic multipole
moments include the spatial distributions of the charge and magnetization in
the hadrons and these parameters are directly related to the spatial
distributions of quarks and gluons inside the hadrons. There are many studies
in the literature devoted to the investigation of the electromagnetic
multipole moments of the standard hadrons, but unfortunately relatively little
are known the electromagnetic multipole moments of the exotic hadrons. There
are a few studies in the literature where the magnetic dipole and quadrupole
moments of the exotic states are studied: see Agamaliev  _et al._ (2017);
Ozdem and Azizi (2017, 2018) for tetraquarks and Kim and Praszalowicz (2004);
Huang  _et al._ (2004); Liu  _et al._ (2004); Wang  _et al._ (2005, 2006,
2016) for pentaquarks. More detailed analyses are needed in order to get
useful knowledge on the charge distribution, electromagnetic multipole moments
and geometric shapes of the non-conventional hadrons. In this study, we are
going to concentrate on the charged open-flavor $[qq][\overline{qc}]$ states
(hereafter we will denote these states as $Z_{\bar{c}q}$) with spin-parity
$J^{PC}=1^{+-}$, and calculate their electromagnetic multipole moments in the
framework of light-cone QCD sum rule (LCSR). In LCSR, the hadronic parameters
are expressed in terms of the vacuum condensates and the light cone
distribution amplitudes (DAs) of the on-shell particles (for more about this
method see, e.g., Chernyak and Zhitnitsky (1990); Braun and Filyanov (1989);
Balitsky  _et al._ (1989) and references therein).

The rest of the paper is organized as follows: In section II, the calculation
of the sum rules in LCSR will be presented. In the last section, we
numerically analyze the sum rules obtained for the multipole moments and
discuss the obtained results. The explicit expressions of the magnetic and
quadrupole moments are moved to the Appendix A.

##  II Formalism

In this section we derive the LCSR for the magnetic and quadrupole moments of
the $Z_{\bar{c}q}$ states. For this aim, we consider a correlation function in
the presence of the external electromagnetic field ($\gamma$),

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

where $J_{\mu}$ is the interpolating current of $Z_{\bar{c}q}$ state with
quantum numbers $J^{PC}=1^{+-}$ in the diquark-antidiquark picture. It is
given in terms of three light quark and one heavy quark fields as Chen  _et
al._ (2017b):

$\displaystyle J_{\mu}^{Z_{\bar{c}q}}(x)$ $\displaystyle=$
$\displaystyle\Big{[}q_{1_{a}}^{T}(x)C\gamma_{\mu}q_{2_{b}}(x)\Big{]}\Big{[}\bar{q}_{3_{a}}(x)\gamma_{5}C\bar{c}_{b}^{T}(x)-\bar{q}_{3_{b}}(x)\gamma_{5}C\bar{c}_{a}^{T}(x)\Big{]},$
(2)

where $q_{1}$ is u, d and/or s-quark, $q_{2}$ and $q_{3}$ are u and/or
d-quark, $C$ is the charge conjugation matrix; and $a$ and $b$ are color
indices.

In order to acquire sum rules for the magnetic and quadrupole moments, we need
to represent the correlation function in two different forms: (1) in terms of
the quark-gluon parameters and distribution amplitudes (DAs) of the photon in
the deep Euclidean region, so called the QCD representation, and (2) in terms
of hadronic properties, so called the hadronic representation.

We start our analysis by calculating the correlation function on Eq. (1) in
terms of quarks and gluon properties in deep Euclidean region. For this
purpose, the interpolating current is inserted into the correlation function
and after the contracting of light and heavy quark pairs using the Wick
theorem the following result is obtained:

$\displaystyle\Pi_{\mu\nu}^{\mathrm{QCD}}(q)=i\int d^{4}xe^{ipx}\langle
0|\Bigg{\\{}\mathrm{Tr}\Big{[}\gamma_{5}\widetilde{S}_{c}^{b^{\prime}b}(-x)\gamma_{5}S_{q_{3}}^{a^{\prime}a}(-x)\Big{]}\mathrm{Tr}\Big{[}\gamma_{\nu}\widetilde{S}_{q_{1}}^{aa^{\prime}}(x)\gamma_{\mu}S_{q_{2}}^{bb^{\prime}}(x)\Big{]}$
$\displaystyle-\mathrm{Tr}\Big{[}\gamma_{5}\widetilde{S}_{c}^{a^{\prime}b}(-x)\gamma_{5}S_{q_{3}}^{b^{\prime}a}(-x)\Big{]}\mathrm{Tr}\Big{[}\gamma_{\nu}\widetilde{S}_{q_{1}}^{aa^{\prime}}(x)\gamma_{\mu}S_{q_{2}}^{bb^{\prime}}(x)\Big{]}$
$\displaystyle-\mathrm{Tr}\Big{[}\gamma_{5}\widetilde{S}_{c}^{b^{\prime}a}(-x)\gamma_{5}S_{q_{3}}^{a^{\prime}b}(-x)\Big{]}\mathrm{Tr}\Big{[}\gamma_{\nu}\widetilde{S}_{q_{1}}^{aa^{\prime}}(x)\gamma_{\mu}S_{q_{2}}^{bb^{\prime}}(x)\Big{]}$
$\displaystyle+\mathrm{Tr}\Big{[}\gamma_{5}\widetilde{S}_{c}^{a^{\prime}a}(-x)\gamma_{5}S_{q_{3}}^{b^{\prime}b}(-x)\Big{]}\mathrm{Tr}\Big{[}\gamma_{\nu}\widetilde{S}_{q_{1}}^{aa^{\prime}}(x)\gamma_{\mu}S_{q_{2}}^{bb^{\prime}}(x)\Big{]}\Bigg{\\}}|0\rangle_{\gamma},$
(3)

where

$\widetilde{S}(x)=CS^{\mathrm{T}}(x)C,$

with $S_{q,c}(x)$ being the quark propagators. The light and heavy propagators
are given as Balitsky and Braun (1989)

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

and

$\displaystyle
S_{c}(x)=S^{free}-\frac{g_{s}m_{c}}{16\pi^{2}}\int_{0}^{1}dv~{}G^{\mu\nu}(vx)\Bigg{[}(\sigma_{\mu\nu}{x\\!\\!\\!/}+{x\\!\\!\\!/}\sigma_{\mu\nu})\frac{K_{1}(m_{c}\sqrt{-x^{2}})}{\sqrt{-x^{2}}}+2\sigma^{\mu\nu}K_{0}(m_{c}\sqrt{-x^{2}})\Bigg{]},$
(5)

where

$\displaystyle
S_{q}^{free}(x)=i\frac{{x\\!\\!\\!/}}{2\pi^{2}x^{4}}-\frac{m_{q}}{4\pi^{2}x^{2}},$
$\displaystyle
S_{c}^{free}(x)=\frac{m_{c}^{2}}{4\pi^{2}}\Bigg{[}\frac{K_{1}(m_{c}\sqrt{-x^{2}})}{\sqrt{-x^{2}}}+i\frac{{x\\!\\!\\!/}~{}K_{2}(m_{c}\sqrt{-x^{2}})}{(\sqrt{-x^{2}})^{2}}\Bigg{]}.$
(6)

Here $K_{1,2}$ are Bessel functions of the second kind.

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

$\displaystyle S^{free}(x)\rightarrow\int d^{4}y\,S^{free}(x-y)\,\hbox
to0.0pt{/\hss}{\\!A}(y)\,S^{free}(y)\,,$ (7)

and the remaining three propagators are taken as the full quark propagators.

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

$\displaystyle
S_{\alpha\beta}^{ab}\rightarrow-\frac{1}{4}(\bar{q}^{a}\Gamma_{i}q^{b})(\Gamma_{i})_{\alpha\beta},$
(8)

and the remaining propagators are taken as the full quark propagators, as well
including the perturbative and the nonperturbative contributions. Once Eq. (8)
is plugged into Eq. (II), there appear matrix elements such as
$\langle\gamma(q)\left|\bar{q}(x)\Gamma_{i}q(0)\right|0\rangle$ and
$\langle\gamma(q)\left|\bar{q}(x)\Gamma_{i}G_{\alpha\beta}q(0)\right|0\rangle$,
representing the nonperturbative contributions. The reader can find some
details about the transformations of Eqs. (7) and (8) in Ref. Ozdem and Azizi
(2018). These matrix elements can be written in terms of the photon DAs with
definite twists, whose expressions all can be found in Ref. Ball  _et al._
(2003). The QCD side of the correlation function can be acquired in terms of
QCD parameters using the Eqs. (II)-(8) and after applying the Fourier
transformation to transfer the calculations to the momentum space.

The next step is to calculate the correlation function in terms of the
hadronic parameters. To this end we insert intermediate states of
$Z_{\bar{c}q}$ with the same quantum numbers as the interpolating current into
Eq. (1). As a result, in zero-width approximation, we get

$\displaystyle\Pi_{\mu\nu}^{Had}(p,q)={\frac{\langle 0\mid
J_{\mu}^{Z_{\bar{c}q}}\mid
Z_{\bar{c}q}(p)\rangle}{[p^{2}-m_{Z_{\bar{c}q}}^{2}]}}\langle
Z_{\bar{c}q}(p)\mid Z_{\bar{c}q}(p+q)\rangle_{\gamma}\frac{\langle
Z_{\bar{c}q}(p+q)\mid{J^{\dagger}}_{\nu}^{Z_{\bar{c}q}}\mid
0\rangle}{[(p+q)^{2}-m_{Z_{\bar{c}q}}^{2}]}+\cdots,$ (9)

where dots stand for the contributions of the higher and continuum states and
$q$ is the momentum of the photon. The matrix element $\langle 0\mid
J_{\mu}^{Z_{\bar{c}q}}\mid Z_{\bar{c}q}\rangle$ is determined as

$\displaystyle\langle 0\mid J_{\mu}^{Z_{\bar{c}q}}\mid
Z_{\bar{c}q}\rangle=\lambda_{Z_{\bar{c}q}}\varepsilon_{\mu}^{\theta}\,,$ (10)

with $\lambda_{Z_{\bar{c}q}}$ being the residue of the $Z_{\bar{c}q}$ state.

The matrix element $\langle Z_{\bar{c}q}(p,\varepsilon^{\theta})\mid
Z_{\bar{c}q}(p+q,\varepsilon^{\delta})\rangle_{\gamma}$ can be written in
terms of three Lorentz invariant form factors as follows Brodsky and Hiller
(1992):

$\displaystyle\langle Z_{\bar{c}q}(p,\varepsilon^{\theta})\mid
Z_{\bar{c}q}(p+q,\varepsilon^{\delta})\rangle_{\gamma}$
$\displaystyle=-\varepsilon^{\tau}(\varepsilon^{\theta})^{\alpha}(\varepsilon^{\delta})^{\beta}\Bigg{[}G_{1}(Q^{2})~{}(2p+q)_{\tau}~{}g_{\alpha\beta}+G_{2}(Q^{2})~{}(g_{\tau\beta}~{}q_{\alpha}-g_{\tau\alpha}~{}q_{\beta})$
$\displaystyle-\frac{1}{2m_{Z_{\bar{c}q}}^{2}}G_{3}(Q^{2})~{}(2p+q)_{\tau}~{}q_{\alpha}q_{\beta}\Bigg{]}\,,$
(11)

where $\varepsilon^{\theta}$ and $\varepsilon^{\delta}$ are the polarization
vectors of the initial and final $Z_{\bar{c}q}$ states and
$\varepsilon^{\tau}$ is the polarization vector of the photon. The Lorentz
invariant form factors $G_{1}(Q^{2})$, $G_{2}(Q^{2})$ and $G_{3}(Q^{2})$ are
related to the charge, magnetic and quadrupole form factors through the
relations

$\displaystyle F_{C}(Q^{2})=G_{1}(Q^{2})+\frac{2}{3}\eta F_{\cal D}(Q^{2})\,,$
$\displaystyle F_{M}(Q^{2})=G_{2}(Q^{2})\,,$ $\displaystyle F_{\cal
D}(Q^{2})=G_{1}(Q^{2})-G_{2}(Q^{2})+(1+\eta)G_{3}(Q^{2})\,,$ (12)

where $\eta=Q^{2}/4m_{Z_{\bar{c}q}}^{2}$ with $Q^{2}=-q^{2}$. At $Q^{2}=0$,
these form factors are corresponding to the electric charge, magnetic moment
$\mu$ and the quadrupole moment ${\cal D}$ as:

$\displaystyle eF_{C}(0)=e\,,$ $\displaystyle
eF_{M}(0)=2m_{Z_{\bar{c}q}}\mu\,,$ $\displaystyle eF_{\cal
D}(0)=m_{Z_{\bar{c}q}}^{2}{\cal D}\,.$ (13)

Using Eqs. (10)-(II) and imposing the condition $q\\!\cdot\\!\varepsilon=0$
the Eq. (9) takes the form,

$\displaystyle\Pi_{\mu\nu}^{Had}$
$\displaystyle=\frac{\lambda_{Z_{\bar{c}q}}^{2}}{[m_{Z_{\bar{c}q}}^{2}-(p+q)^{2}][m_{Z_{\bar{c}q}}^{2}-p^{2}]}\Bigg{[}2(p.\varepsilon)F_{C}(0)\Bigg{(}g_{\mu\nu}-\frac{p_{\mu}q_{\nu}-p_{\nu}q_{\mu}}{m_{Z_{\bar{c}q}}^{2}}\Bigg{)}$
$\displaystyle+F_{M}(0)\Bigg{(}q_{\mu}\varepsilon_{\nu}-q_{\nu}\varepsilon_{\mu}+\frac{1}{m_{Z_{\bar{c}q}}^{2}}(p.\varepsilon)(p_{\mu}q_{\nu}-p_{\nu}q_{\mu})\Bigg{)}-\Bigg{(}F_{C}(0)+F_{\cal
D}(0)\Bigg{)}{\frac{p.\varepsilon}{m_{Z_{\bar{c}q}}^{2}}}q_{\mu}q_{\nu}\Bigg{]}\,,$
(14)

where we inserted

$\displaystyle\sum_{\lambda}\varepsilon_{\mu}(p,\lambda)\varepsilon_{\nu}(p,\lambda)=-g_{\mu\nu}+\frac{p_{\mu}p_{\nu}}{m_{Z_{\bar{c}q}}^{2}}.$
(15)

Equating the QCD and hadronic sides of the correlation function, we obtain the
expression of the electromagnetic multipole moments in LCSR in terms of the
QCD degrees of freedom and the photon DAs. We perform the double Borel
transforms with respect to the variables $p^{2}$ and $(p+q)^{2}$ on both sides
of the correlation function in order to suppress the contributions of the
higher states and continuum, and use the quark-hadron duality assumption. By
matching the coefficients of the structures $q_{\mu}\varepsilon_{\nu}$ and
$(\varepsilon.p)q_{\mu}q_{\nu}$, respectively for the magnetic and quadrupole
moments, we get

$\displaystyle\mu=\frac{e^{m_{Z_{\bar{c}q}}^{2}/M^{2}}}{\lambda_{Z_{\bar{c}q}}^{2}}\Pi_{1}^{QCD},$
$\displaystyle\mathcal{D}=m^{2}_{Z_{\bar{c}q}}\frac{e^{m_{Z_{\bar{c}q}}^{2}/M^{2}}}{\lambda_{Z_{\bar{c}q}}^{2}}\Pi_{2}^{QCD},$
(16)

where explicit expressions of the $\Pi_{1}^{QCD}$ and $\Pi_{2}^{QCD}$ are
given in Appendix A.

##  III Numerical analysis

In this section, we numerically analyze the results of calculations for
magnetic and quadrupole moments. We use $m_{u}=m_{d}=0$,
$m_{s}(2~{}GeV)=0.096^{+0.008}_{-0.004}~{}GeV$,
$\overline{m}_{c}(m_{c})=(1.28\pm 0.03)\,GeV$ Patrignani  _et al._ (2016),
$\langle\bar{u}u\rangle(1~{}GeV)=\langle\bar{d}d\rangle(1~{}GeV)=(-0.24\pm
0.01)^{3}~{}GeV^{3}$ Ioffe (2006), $m_{0}^{2}=0.8\pm 0.1~{}GeV^{2}$, $\langle
g_{s}^{2}G^{2}\rangle=0.88~{}GeV^{4}$ Nielsen  _et al._ (2010),
$\chi(1~{}GeV)=-2.85\pm 0.5~{}GeV^{-2}$ Rohrwild (2007),
$\lambda_{Z_{sq\bar{q}c}}=7.3\pm 1.7\times 10^{-3}~{}GeV^{5}$,
$\lambda_{Z_{qq\bar{q}c}}=7.6\pm 1.8\times 10^{-3}~{}GeV^{5}$,
$m_{Z_{sq\bar{q}c}}=2.826^{+0.134}_{-0.157}~{}GeV$ and
$m_{Z_{qq\bar{q}c}}=2.843^{+0.115}_{-0.139}~{}GeV$ Agaev  _et al._ (2017). The
parameters used in the photon DAs are given in Ref. Ball  _et al._ (2003).

The predictions for the electromagnetic multipole moments depend on two
auxiliary parameters; the Borel mass parameter $M^{2}$ and continuum threshold
$s_{0}$. Complying with the standard procedure of the sum rule method the
predictions on the electromagnetic multipole moments should not depend on
$M^{2}$ and $s_{0}$, but in real computations one can only decrease their
effect to a minimum. The working interval for the continuum threshold is
chosen such that the maximum pole contribution is acquired and the results
relatively weakly depend on its choices. Our numerical computations lead to
the interval [10-12] $GeV^{2}$ for this parameter. The working region for
$M^{2}$ is determined requiring that the contributions of the higher states
and continuum are effectively suppressed. There are two criteria for choosing
working region for the Borel parameter $M^{2}$: Convergence of the operator
product expansion (OPE) and pole dominance. The requirement of the OPE
convergence results in a lower bound, while the constraint of the maximum pole
contribution leads to an upper bound on it. Our numerical calculation shows
that these requirements are satisfied in the region $3~{}GeV^{2}\leq M^{2}\leq
4~{}GeV^{2}$ and, the magnetic and quadrupole moments in this region is
practically independent of $M^{2}$. In Figs. 1-2, we plot the dependencies of
the magnetic and quadrupole moments on $M^{2}$ at several fixed values of the
continuum threshold $s_{0}$. As is seen, the variation of the results with
respect to the continuum threshold causes a change on the results on the
magnetic and quadrupole moments of about 15% and there is a very less
dependence of the quantities under consideration on the Borel parameter in its
working interval. Hence, one can say that the results of the magnetic and
quadrupole moments are almost insensitive to $s_{0}$ and $M^{2}$.

Our final results for the magnetic and quadrupole moments are given in Table
I. The errors in the results come from the variations in the calculations of
the working regions of $M^{2}$ and $s_{0}$ as well as the uncertainties in the
values of the input parameters and the photon DAs. We also would like to note
that in Table I and Figs. 1-2, the absolute values are given since it is not
possible to determine the sign of the residue from the mass sum rules.
Therefore, it is not possible to estimate the signs of the magnetic and
quadrupole moments.

${Z_{\bar{c}q}}$ |  $|\mu_{Z_{\bar{c}q}}|[\mu_{N}]$ |  $|\mathcal{D}_{Z_{\bar{c}q}}|[fm^{2}]$
---|---|---
$[sd][\overline{uc}]$ |  1.12$\pm$ 0.18 |  0.0086 $\pm$ 0.0015
$[sd][\overline{dc}]$ |  0.90$\pm$ 0.13 |  0.0085 $\pm$ 0.0015
$[su][\overline{uc}]$ |  0.51$\pm$ 0.24 |  0.0070 $\pm$ 0.0013
$[dd][\overline{uc}]$ |  1.09$\pm$ 0.17 |  0.0082 $\pm$ 0.0014
$[du][\overline{uc}]$ |  0.84$\pm$ 0.31 |  0.0067 $\pm$ 0.0012
$[dd][\overline{dc}]$ |  0.93$\pm$ 0.13 |  0.0082 $\pm$ 0.0014
$[uu][\overline{dc}]$ |  2.05$\pm$ 0.30 |  0.016$\pm$0.003
Table 1: Results of the magnetic and quadrupole moments of $Z_{\bar{c}q}$
states.

In summary, the electromagnetic multipole moments of the open-flavor
$Z_{\bar{c}q}$ states have been investigated by assuming that these states are
represented as diquark-antidiquark structure with quantum numbers
$J^{PC}=1^{+-}$. Their magnetic and quadrupole moments have been extracted in
the framework of light-cone QCD sum rule. The electromagnetic multipole
moments of the open-flavor $Z_{\bar{c}q}$ states are important dynamical
observables, which would encode important information of their underlying
structure, charge distribution and geometric shape. The results obtained for
the magnetic moments are considerably large and can be measured in future
experiments. We obtain very small values for the quadrupole moments of
$Z_{\bar{c}q}$ states indicating a nonspherical charge distribution. It is
easy to see that $[sd][\overline{uc}]$ and $[dd][\overline{uc}]$ states belong
to a class of doubly charged tetraquarks that the measurements of their
electromagnetic parameters, like those of the $\Delta^{++}$ baryon, are
relatively easy compared to other exotic states. These kind of exotic states
have not been observed so far. We hope our predictions on the electromagnetic
moments of these states together with the results of other theoretical studies
on the spectroscopic parameters of these states will be useful for their
searches in future experiments and will hep us determine exact internal
structures of these non-conventional states.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1:  The dependence of the magnetic and quadrupole moments on the Borel
parameter squared $M^{2}$ at different fixed values of the continuum
threshold: (a) and (b) for the $Z_{sd\bar{u}\bar{c}}$ state, (c) and (d) for
the $Z_{sd\bar{d}\bar{c}}$ state and, (e) and (f) for the
$Z_{su\bar{u}\bar{c}}$ state.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 2:  The dependence of the magnetic and quadrupole moments on the Borel
parameter squared $M^{2}$ at different fixed values of the continuum
threshold: (a) and (b) for the $Z_{dd\bar{u}\bar{c}}$ state, (c) and (d) for
the $Z_{du\bar{u}\bar{c}}$ state, (e) and (f) for the $Z_{dd\bar{d}\bar{c}}$
state and, (g) and (h) for the $Z_{uu\bar{d}\bar{c}}$ state.

##  IV Acknowledgement

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

## Appendix A: Explicit forms of the functions $\Pi_{i}^{QCD}$

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

$\displaystyle\Pi_{1}^{QCD}$
$\displaystyle=\frac{1}{442368\pi^{2}}\Bigg{[}32P_{1}\Bigg{\\{}eq_{3}mcP_{3}+144eq_{1}\pi^{2}P_{2}(m_{0}^{2}-m_{c}^{2})+3eq_{2}\Bigg{(}m_{q_{1}}P_{3}+12\Big{(}m_{c}^{2}m_{q_{1}}+8\pi^{2}m_{c}m_{q_{1}}P_{1}$
$\displaystyle+4\pi^{2}m_{0}^{2}P_{2}-8\pi^{2}m_{c}^{2}P_{2}\Big{)}\Bigg{)}\Bigg{\\}}\mathbb{A}(u_{0})+64\chi
P_{1}\Bigg{\\{}8eq_{1}\pi^{2}P_{2}\Bigg{(}P_{3}+6m_{c}\Big{(}-3m_{c}m_{0}^{2}+2m_{c}^{3}+16\pi^{2}P_{1}\Big{)}\Bigg{)}$
$\displaystyle+eq_{2}\Bigg{(}P_{3}(3m_{c}^{2}m_{q_{1}}-8\pi^{2}P_{2})+48\pi^{2}m_{c}\Big{(}3m_{q_{1}}m_{0}^{2}P_{1}-3m_{c}m_{0}^{2}P_{2}+2m_{c}^{3}P_{2}+16\pi^{2}P_{1}P_{2}\Big{)}\Bigg{)}\Bigg{\\}}\varphi_{\gamma}(u_{0})$
$\displaystyle+32f_{3\gamma}\Bigg{\\{}-eq_{3}m_{c}^{2}P_{3}+3eq_{1}m_{c}\Big{(}m_{c}P_{3}+96\pi^{2}P_{1}(m_{0}^{2}-m_{c}^{2})\Big{)}+3eq_{2}\Bigg{(}-m_{c}^{2}P_{3}-48\pi^{2}\Big{(}m_{0}^{2}(2m_{c}P_{1}+m_{q_{1}}P_{2})$
$\displaystyle-
m_{c}^{2}(2m_{c}P_{1}+3m_{q_{1}}P_{2})\Big{)}\Bigg{)}\Bigg{\\}}\psi^{a}(u_{0})-1536\pi^{2}f_{3\gamma}\Bigg{\\{}3eq_{1}m_{c}m_{0}^{2}P_{1}+eq_{2}\Big{(}-3m_{c}^{2}m_{q_{1}}P_{1}+m_{0}^{2}(3m_{c}P_{1}$
$\displaystyle+3m_{q_{1}}P_{2})\Big{)}\Bigg{\\}}\Bigg{(}\psi^{\nu}(u_{0})+2I_{6}[\psi^{\nu}]\Bigg{)}+192P_{1}\Bigg{\\{}24eq_{1}\pi^{2}P_{2}(m_{0}^{2}-2m_{c}^{2})+eq_{2}\Bigg{(}m_{q_{1}}P_{3}+24\pi^{2}\Big{(}4m_{c}m_{q_{1}}P_{1}$
$\displaystyle+m_{0}^{2}P_{2}-2m_{c}^{2}P_{2}\Big{)}\Bigg{)}\Bigg{\\}}I_{6}[h_{\gamma}]+11eq_{3}m_{c}^{2}f_{3\gamma}P_{3}I_{2}[\mathcal{A}]+(eq_{1}-eq_{2})m_{c}f_{3\gamma}\Big{(}23m_{c}P_{3}+576\pi^{2}P_{1}(m_{0}^{2}-2m_{c}^{2})\Big{)}I_{1}[\mathcal{A}]$
$\displaystyle+576eq_{3}\pi^{2}m_{c}m_{q_{1}}P_{1}P_{2}\Big{(}2I_{5}[\mathcal{S}]+I_{2}[\mathcal{S}]\Big{)}-44eq_{3}m_{c}P_{1}P_{3}\Big{(}I_{4}[\mathcal{T}_{1}]+I_{5}[\mathcal{T}_{1}]\Big{)}$
$\displaystyle-240eq_{3}\pi^{2}m_{q_{1}}f_{3\gamma}P_{2}(m_{0}^{2}-3m_{c}^{2})I_{2}[\mathcal{V}]+(eq_{1}+eq_{2})m_{c}f_{3\gamma}\Big{(}-23m_{c}P_{3}-576\pi^{2}P_{1}(m_{0}^{2}-2m_{c}^{2})\Big{)}I_{1}[\mathcal{V}]\Bigg{]}I_{7}[0]$
$\displaystyle+\frac{1}{221184m_{c}^{2}\pi^{4}}\Bigg{[}-16m_{c}P_{1}\Bigg{\\{}eq_{3}P_{3}+36m_{c}\Big{(}3eq_{2}m_{c}^{2}m_{q_{1}}-8(eq_{1}+eq_{2})\pi^{2}P_{2}\Big{)}\Bigg{\\}}\mathbb{A}(u_{0})$
$\displaystyle+192m_{c}^{2}\chi
P_{1}\Big{\\{}-eq_{2}m_{q_{1}}P_{3}+96eq_{2}\pi^{2}m_{c}m_{q_{1}}P_{1}+24(eq_{1}+eq_{2})\pi^{2}P_{1}(m_{0}^{2}-2m_{c}^{2})\Bigg{\\}}\varphi_{\gamma}(u_{0})$
$\displaystyle+32m_{c}f_{3\gamma}\Bigg{\\{}(eq_{3}-3eq_{1})m_{c}P_{3}-72eq_{1}\pi^{2}P_{1}(m_{0}^{2}-4m_{c}^{2})+3eq_{2}\Bigg{(}m_{c}P_{3}+24\pi^{2}\Big{(}m_{0}^{2}P_{1}-4m_{c}(m_{c}P_{1}$
$\displaystyle+m_{q_{1}}P_{2})\Big{)}\Bigg{)}\Bigg{\\}}\psi^{a}(u_{0})+2304\pi^{2}m_{c}f_{3\gamma}\Bigg{\\{}(eq_{1}+eq_{2})m_{0}^{2}P_{1}-2eq_{2}m_{c}m_{q_{1}}P_{1}\Bigg{\\}}\Bigg{(}\psi^{\nu}(u_{0})+2I_{6}[\psi^{\nu}]\Bigg{)}$
$\displaystyle+96P_{1}\Bigg{\\{}96(eq_{1}+eq_{2})\pi^{2}m_{c}^{2}P_{2}+eq_{2}m_{q_{1}}\Big{(}P_{3}+96\pi^{2}m_{c}P_{1}\Big{)}\Bigg{\\}}I_{6}[h_{\gamma}]+22eq_{3}m_{c}P_{1}P_{3}\Bigg{(}I_{5}[\mathcal{T}_{1}]+I_{5}[\mathcal{T}_{2}]\Bigg{)}$
$\displaystyle-288eq_{3}\pi^{2}m_{c}m_{q_{1}}P_{1}P_{2}I_{2}[\mathcal{S}]-11eq_{3}m_{c}^{2}f_{3\gamma}P_{3}I_{2}[\mathcal{A}]+(eq_{1}-eq_{2})m_{c}^{2}f_{3\gamma}\Big{(}-23P_{3}+1152\pi^{2}m_{c}P_{1}\Big{)}I_{1}[\mathcal{A}]$
$\displaystyle-432eq_{3}\pi^{2}m_{c}^{2}m_{q_{1}}f_{3\gamma}P_{2}I_{2}[\mathcal{V}]-(eq_{1}+eq_{2})m_{c}^{2}f_{3\gamma}\Big{(}-23P_{3}+1152\pi^{2}m_{c}P_{1}\Big{)}I_{1}[\mathcal{V}]\Bigg{]}I_{7}[1]$
$\displaystyle+\frac{1}{442368m_{c}^{2}\pi^{4}}\Bigg{[}3456eq_{2}m_{c}^{2}m_{q_{1}}P_{1}\mathbb{A}(u_{0})+192\chi
P_{1}\Bigg{\\{}96(eq_{1}+eq_{2})\pi^{2}m_{c}^{2}P_{2}+eq_{2}m_{q_{1}}\Big{(}P_{3}+96\pi^{2}m_{c}P_{1}\Big{)}\Bigg{\\}}\varphi_{\gamma}(u_{0})$
$\displaystyle-32f_{3\gamma}\Bigg{\\{}eq_{3}P_{3}+288eq_{1}\pi^{2}m_{c}P_{1}+3eq_{2}\Big{(}P_{3}+48\pi^{2}(2m_{c}P_{1}+msP_{2})\Big{)}\Bigg{\\}}\psi^{a}(u_{0})+11eq_{3}f_{3\gamma}P_{3}I_{2}[\mathcal{A}]$
$\displaystyle-9216(eq_{1}+eq_{2})\pi^{2}P_{1}P_{2}I_{6}[h_{\gamma}]+2304eq_{2}\pi^{2}m_{q_{1}}f_{3\gamma}P_{2}I_{3}[\psi^{a}]+4608eq_{2}\pi^{2}m_{q_{1}}f_{3\gamma}P_{2}\Big{(}\psi^{\nu}(u_{0})+2I_{6}[\psi^{\nu}]\Big{)}$
$\displaystyle+(eq_{1}+eq_{2})f_{3\gamma}\Big{(}23P_{3}+1152\pi^{2}m_{c}P_{1}\Big{)}I_{1}[\mathcal{A}]+(eq_{1}+eq_{2})f_{3\gamma}\Big{(}-23P_{3}+1152\pi^{2}m_{c}P_{1}\Big{)}I_{1}[\mathcal{V}]$
$\displaystyle+144eq_{3}\pi^{2}m_{q_{1}}f_{3\gamma}P_{2}I_{2}[\mathcal{V}]\Bigg{]}I_{7}[2]$

$\displaystyle-\frac{eq_{3}m_{c}m_{q_{1}}P_{1}P_{2}P_{3}}{20736\pi^{4}}\Bigg{[}2\varphi_{\gamma}(u_{0})+I_{3}[\varphi_{\gamma}]\Bigg{]}I_{7}[-1]-\frac{P_{1}}{1152m_{c}^{2}\pi^{4}}\Bigg{[}16(eq_{1}+eq_{2})\pi^{2}\chi
P_{2}\varphi_{\gamma}(u_{0})+3eq_{2}m_{q_{1}}\mathbb{A}(u_{0})\Bigg{]}I_{7}[3]$
$\displaystyle-\frac{m_{c}^{4}}{442368\pi^{4}}\Bigg{[}64eq_{3}m_{c}\chi
P_{1}P_{3}\varphi_{\gamma}(u_{0})+64f_{3\gamma}\Bigg{\\{}-eq3P_{3}+3eq_{1}\Big{(}P_{3}+96\pi^{2}m_{c}P_{1}\Big{)}+3eq_{2}\Big{(}-P_{3}+48\pi^{2}\big{(}2m_{c}P_{1}$
$\displaystyle-
m_{q_{1}}P_{2}\big{)}\Big{)}\Bigg{\\}}\Big{(}2\psi^{a}(u_{0})+I_{3}[\psi^{a}]\Big{)}-192f_{3\gamma}\Bigg{\\{}(eq_{1}+eq_{2})\Big{(}P_{3}+96\pi^{2}m_{c}P_{1}\Big{)}-48eq_{2}\pi^{2}m_{q_{1}}P_{2}\Bigg{\\}}\Big{(}\psi^{\nu}(u_{0})+2I_{6}[\psi^{\nu}]\Big{)}$
$\displaystyle+18432(eq_{1}+eq_{2})\pi^{2}P_{1}P_{2}I_{6}[h_{\gamma}]+32eq_{3}m_{c}\chi
P_{3}P_{1}I_{3}[h_{\gamma}]-11eq_{3}f_{3\gamma}P_{3}I_{2}[\mathcal{A}]-288eq_{3}\pi^{2}m_{q_{1}}f_{3\gamma}P_{2}I_{2}[\mathcal{V}]\Bigg{]}I_{8}[-3,1]$
$\displaystyle+\frac{m_{c}^{2}}{442368\pi^{2}}\Bigg{[}-384eq_{2}m_{q_{1}}\chi
P_{1}\Big{(}P_{3}+96\pi^{2}m_{c}P_{1}\Big{)}\varphi_{\gamma}(u_{0})-64eq_{3}m_{c}\chi
P_{1}P_{3}I_{3}[\varphi_{\gamma}]-192(eq_{1}+eq_{2})f_{3\gamma}\Big{(}P_{3}$
$\displaystyle+96\pi^{2}m_{c}P_{1}\Big{)}\Big{(}\psi^{\nu}(u_{0})+2I_{6}[\psi^{\nu}]\Big{)}+32f_{3\gamma}\Bigg{\\{}(3eq_{1}-3eq_{2}-eq_{3})P_{3}+288(eq_{1}-eq_{2})\pi^{2}m_{c}P_{1}\Bigg{\\}}I_{3}[\psi^{a}]$
$\displaystyle-2(eq_{1}-eq_{2})f_{3\gamma}\Big{(}23P_{3}+1152\pi^{2}m_{c}P_{1}\Big{)}I_{1}[\mathcal{A}]+2(eq_{1}+eq_{2})f_{3\gamma}\Big{(}23P_{3}+1152\pi^{2}m_{c}P_{1}\Big{)}I_{1}[\mathcal{V}]$
$\displaystyle-33eq_{3}f_{3\gamma}P_{3}I_{2}[\mathcal{A}]\Bigg{]}I_{8}[-2,1]$
$\displaystyle+\frac{P_{1}}{110592m_{c}^{2}\pi^{4}}\Bigg{[}8eq_{3}m_{c}P_{3}A(u_{0})+576(eq_{1}-eq_{2})\pi^{2}m_{c}f_{3\gamma}m_{0}^{2}\Big{(}2\psi^{a}(u_{0})+I_{3}[\psi^{a}]\Big{)}+144eq_{3}\pi^{2}m_{c}m_{q_{1}}P_{2}I_{2}[\mathcal{S}]$
$\displaystyle-1152(eq_{1}+eq_{2})\pi^{2}m_{c}f_{3\gamma}m_{0}^{2}\Big{(}\psi^{\nu}(u_{0})+2I_{6}[\psi^{\nu}]\Big{)}+48eq_{2}m_{q_{1}}\Big{(}-P_{3}+96\pi^{2}m_{c}P_{1}\Big{)}I_{6}[h_{\gamma}]$
$\displaystyle-11eq_{3}m_{c}P_{3}\Big{(}I_{5}[\mathcal{T}_{1}]-I_{5}[\mathcal{T}_{2}]\Big{)}\Bigg{]}I_{8}[0,0]$
$\displaystyle-\frac{1}{221184m_{c}^{2}\pi^{4}}\Bigg{[}64\chi
P_{1}\Bigg{\\{}P_{3}(2eq_{3}m_{c}-3eq_{2}m_{q_{1}})+288\pi^{2}m_{c}m_{q_{1}}P_{1}\Bigg{\\}}\varphi_{\gamma}(u_{0})+32f_{3\gamma}\Bigg{\\{}3eq_{1}\Big{(}96\pi^{2}m_{c}P_{1}-P_{3}\Big{)}+eq_{3}P_{3}$
$\displaystyle+3eq_{2}\Big{(}P_{3}+48\pi^{2}(2m_{c}P_{1}+m_{q_{1}}P_{2})\Big{)}\Bigg{\\}}\Big{(}\psi^{a}(u_{0})-2I_{3}[\psi^{a}]\Big{)}-4608eq_{2}\pi^{2}m_{q_{1}}f_{3\gamma}P_{2}\Big{(}\psi^{\nu}(u_{0})+2I_{6}[\psi^{\nu}]\Big{)}$
$\displaystyle-9216eq_{2}\pi^{2}P_{1}P_{2}I_{6}[h_{\gamma}]\Bigg{]}I_{8}[0,1]$
$\displaystyle+\frac{1}{1536m_{c}^{2}\pi^{4}}\Bigg{[}32eq_{2}m_{q_{1}}\chi
P_{1}\varphi_{\gamma}(u_{0})+16(eq_{1}+eq_{2})f_{3\gamma}\Big{(}\psi^{\nu}(u_{0})+2I_{6}[\psi^{\nu}]\Big{)}-8(eq_{1}-eq_{2})f_{3\gamma}I_{3}[\psi^{a}]$
$\displaystyle+f{3\gamma}\Big{\\{}2(eq_{1}-eq_{2})I_{1}[\mathcal{A}]-eq_{3}I_{2}[\mathcal{V}]-2(eq_{1}+eq_{2})I_{1}[\mathcal{V}]\Big{\\}}\Bigg{]}I_{8}[0,3]$
$\displaystyle+\frac{m_{c}^{4}}{6144\pi^{4}}\Bigg{[}64eq_{2}m_{q_{1}}\chi
P_{1}\Big{(}m_{c}^{2}F[-4,3]+I_{8}[-3,3]\Big{)}\varphi_{\gamma}(u_{0})-16(eq_{1}-eq_{2})f_{3\gamma}\Big{(}m_{c}^{4}F[-5,3]+I_{8}[-3,3]\Big{)}\psi^{a}(u_{0})$
$\displaystyle+f_{3\gamma}\Bigg{\\{}4(eq_{1}-eq_{2})\Big{(}m_{c}^{2}F[-4,3]+I_{8}[-3,3]\Big{)}\Big{(}I_{4}[\mathcal{A}]-I_{4}[\mathcal{V}]\Big{)}+eq_{3}\Big{(}m_{c}^{4}F[-5,3]-2m_{c}^{2}F[-4,3]+I_{8}[-3,3]\Big{)}I_{2}[\mathcal{V}]$
$\displaystyle+16(eq_{1}+eq_{2})f_{3\gamma}\Big{(}m_{c}^{4}F[-5,3]+2m_{c}^{2}F[-4,3]+I_{8}[-3,3]\Big{)}\Big{(}\psi^{\nu}(u_{0})+2I_{6}[\psi^{\nu}]\Big{)}-8(eq_{1}-eq2)\Big{(}m_{c}^{4}I_{8}[-5,3]$
$\displaystyle+2m_{c}^{2}I_{8}[-4,3]+I_{8}[-3,3]\Big{)}I_{3}[\psi^{a}]\Bigg{\\}}\Bigg{]}$

$\displaystyle-\frac{eq_{3}m_{c}\chi
P_{1}P_{3}}{1382\pi^{4}}\Bigg{[}2\varphi_{\gamma}(u_{0})+I_{3}[\varphi_{\gamma}]\Bigg{]}I_{8}[-1,1]+\frac{eq_{3}m_{c}P_{1}P_{3}}{27648\pi^{4}}I_{3}[\mathbb{A}]I_{8}[1,0]+\frac{eq_{3}m_{c}P_{1}}{1024\pi^{4}}\Big{(}4I_{5}[\mathcal{S}]-I_{2}[\mathcal{S}]\Big{)}I_{8}[-2,2]$
$\displaystyle-\frac{m_{c}^{6}P_{1}}{1024\pi^{4}}\Bigg{[}16eq_{2}m_{q_{1}}I_{6}[h_{\gamma}]+eq_{3}m_{c}I_{1}[\mathcal{S}]\Bigg{]}I_{8}[-4,2]+\frac{P_{1}}{384m_{c}^{2}\pi^{4}}\Bigg{[}3eq_{2}m_{q_{1}}\mathbb{A}(u_{0})+16(eq_{1}+eq_{2})\pi^{2}\chi
P_{2}\varphi_{\gamma}(u_{0})\Bigg{]}I_{8}[0,2]$
$\displaystyle-\frac{m_{c}^{6}m_{q_{1}}P_{1}}{512\pi^{4}}\Bigg{[}4eq_{2}\Big{(}\mathbb{A}(u_{0})-2I_{6}[h_{\gamma}]\Big{)}-eq_{3}\Big{(}I_{2}[\mathcal{S}]-2I_{5}[\mathcal{S}]\Big{)}\Bigg{]}I_{8}[-3,2].$
(17)

and

$\displaystyle\Pi_{2}^{QCD}$
$\displaystyle=-\frac{m_{c}^{3}P_{1}}{55296\pi^{4}}\Bigg{[}11eq_{3}P_{3}\Big{(}I_{5}[\mathcal{T}_{1}]+I_{5}[\mathcal{T}_{2}]\Big{)}-4608(eq_{1}+eq_{2})\pi^{2}f_{3\gamma}m_{0}^{2}I_{6}[\psi^{\nu}]\Bigg{]}I_{7}[-2]$
$\displaystyle-\frac{f_{3\gamma}}{55296\pi^{4}}\Bigg{[}11eq_{3}P_{3}I_{5}[\mathcal{A}]+9216eq_{2}\pi^{2}m_{q_{1}}P_{2}I_{6}[\psi^{\nu}]+(eq_{1}+eq_{2})\Big{(}23P_{3}+1152m_{c}P_{1}\Big{)}I_{4}[\mathcal{A}]\Bigg{]}\Big{(}I_{7}[0]-m_{c}^{2}I_{7}[1]\Big{)}$
$\displaystyle-\frac{f_{3\gamma}}{55296m_{c}^{4}\pi^{4}}\Bigg{[}11eq_{3}P_{3}I_{5}[\mathcal{A}]+(eq_{1}-eq_{2})\Big{(}23P_{3}+1152m_{c}P_{1}\Big{)}I_{4}[\mathcal{A}]+9216eq_{2}\pi^{2}m_{q_{1}}P_{2}I_{6}[\psi^{\nu}]\Bigg{]}I_{8}[0,0]$
$\displaystyle+\frac{f_{3\gamma}m_{c}^{4}}{128\pi^{4}}\Bigg{[}(eq_{1}-eq_{2})\Big{(}m_{c}^{2}I_{8}[-4,2]-I_{8}[-3,2]\Big{)}I_{4}[\mathcal{A}]+8(eq_{1}+eq_{2})\Big{(}m_{c}^{4}I_{8}[-5,2]-2m_{c}^{2}I_{8}[-4,2]$
$\displaystyle-
I_{8}[-3,2]\Big{)}I_{6}[\psi^{\nu}]\Bigg{]}+\frac{11eq_{3}P_{1}P_{3}}{18432\pi^{4}}\Bigg{(}I_{5}[\mathcal{T}_{1}]+I_{5}[\mathcal{T}_{2}]\Bigg{)}I_{7}[-1],$
(18)

where the values of $e_{q_{1}}$, $e_{q_{2}}$, $e_{q_{3}}$, $m_{q_{1}}$,
$P_{1}$, $P_{2}$ and $P_{3}$ corresponding to different states are given in
Table II.

${Z_{\bar{c}q}}$ |  $e_{q_{1}}$ |  $e_{q_{2}}$ |  $e_{q_{3}}$ |  $m_{q_{1}}$ |  $P_{1}$ |  $P_{2}$ |  $P_{3}$ |
---|---|---|---|---|---|---|---|---
$[sd][\overline{uc}]$ |  $e_{s}$ |  $e_{d}$ |  $e_{u}$ |  $m_{s}$ |  $\langle\bar{q}q\rangle$ |  $\langle\bar{s}s\rangle$ |  $\langle g_{s}^{2}G^{2}\rangle$ |
$[sd][\overline{dc}]$ |  $e_{s}$ |  $e_{d}$ |  $e_{d}$ |  $m_{s}$ |  $\langle\bar{q}q\rangle$ |  $\langle\bar{s}s\rangle$ |  $\langle g_{s}^{2}G^{2}\rangle$ |
$[su][\overline{uc}]$ |  $e_{s}$ |  $e_{u}$ |  $e_{u}$ |  $m_{s}$ |  $\langle\bar{q}q\rangle$ |  $\langle\bar{s}s\rangle$ |  $\langle g_{s}^{2}G^{2}\rangle$ |
$[dd][\overline{uc}]$ |  $e_{d}$ |  $e_{d}$ |  $e_{u}$ |  0 |  $\langle\bar{q}q\rangle$ |  $\langle\bar{q}q\rangle$ |  $\langle g_{s}^{2}G^{2}\rangle$ |
$[du][\overline{uc}]$ |  $e_{d}$ |  $e_{u}$ |  $e_{u}$ |  0 |  $\langle\bar{q}q\rangle$ |  $\langle\bar{q}q\rangle$ |  $\langle g_{s}^{2}G^{2}\rangle$ |
$[dd][\overline{dc}]$ |  $e_{d}$ |  $e_{d}$ |  $e_{d}$ |  0 |  $\langle\bar{q}q\rangle$ |  $\langle\bar{q}q\rangle$ |  $\langle g_{s}^{2}G^{2}\rangle$ |
$[uu][\overline{dc}]$ |  $e_{u}$ |  $e_{u}$ |  $e_{d}$ |  0 |  $\langle\bar{q}q\rangle$ |  $\langle\bar{q}q\rangle$ |  $\langle g_{s}^{2}G^{2}\rangle$ |
Table 2: The values of $e_{q_{1}}$, $e_{q_{2}}$, $e_{q_{3}}$, $m_{q_{1}}$,
$P_{1}$, $P_{2}$ and $P_{3}$ related to the expressions of the magnetic and
quadrupole moments in Eqs.(17) and (18).

The functions $I_{1}[\mathcal{A}]$, $I_{2}[\mathcal{A}]$,
$I_{3}[\mathcal{A}]$, $I_{4}[\mathcal{A}]$, $I_{5}[\mathcal{A}]$,
$I_{6}[\mathcal{A}]$, $I_{7}[n]$ and $I_{8}[n,m]$ are defined as:

$\displaystyle I_{1}[\mathcal{A}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{A}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta^{\prime}(\alpha_{q}+\bar{v}\alpha_{g}-u_{0}),$
$\displaystyle I_{2}[\mathcal{A}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{A}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta^{\prime}(\alpha_{\bar{q}}+v\alpha_{g}-u_{0}),$
$\displaystyle I_{3}[\mathcal{A}]$
$\displaystyle=\int_{0}^{1}du~{}A(u)\delta^{\prime}(u-u_{0}),$ $\displaystyle
I_{4}[\mathcal{A}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{A}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta(\alpha_{q}+\bar{v}\alpha_{g}-u_{0}),$
$\displaystyle I_{5}[\mathcal{A}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{A}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta(\alpha_{\bar{q}}+v\alpha_{g}-u_{0}),$
$\displaystyle I_{6}[\mathcal{A}]$ $\displaystyle=\int_{0}^{1}du~{}A(u),$
$\displaystyle I_{7}[n]$
$\displaystyle=\int_{m_{c}^{2}}^{s_{0}}ds~{}e^{-s/M^{2}}~{}s^{n}~{},$
$\displaystyle I_{8}[n,m]$
$\displaystyle=\int_{m_{c}^{2}}^{s_{0}}ds\int_{m_{c}^{2}}^{s}dl~{}e^{-s/M^{2}}~{}\frac{(s-l)^{m}}{l^{n}}.$

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