# Electromagnetic form factors of the $B_{c}$-like tetraquarks: molecular and
diquark-antidiquark pictures

Ulaş Özdem<EMAIL_ADDRESS>Health Services Vocational School of Higher
Education, Istanbul Aydin University, Sefakoy-Kucukcekmece, 34295 Istanbul,
Türkiye

###### Abstract

In this study, we use the molecular and diquark-antidiquark tetraquark
pictures to investigate magnetic and quadrupole moments of the $B_{c}$-like
ground state tetraquarks with the QCD light-cone sum rules with quantum
numbers $J^{P}=1^{+}$. In the numerical analysis, to obtain the magnetic and
quadrupole moments of $B_{c}$-like tetraquark states molecular and diquark-
antidiquark forms of interpolating currents, and photon distribution
amplitudes have been used. The magnetic moments are acquired as
$\mu_{Z_{{uc\bar{u}\bar{b}}}}^{Mol}=1.18^{+0.52}_{-0.40}~{}\mu_{N}$,
$\mu_{Z_{{uc\bar{u}\bar{b}}}}^{Di}=3.05^{+1.19}_{-0.95}~{}\mu_{N}$,
$\mu_{Z_{dc\bar{d}\bar{b}}}^{Mol}=0.32^{+0.18}_{-0.10}~{}\mu_{N}$, and
$\mu_{Z_{dc\bar{d}\bar{b}}}^{Di}=2.38^{+0.95}_{-0.75}~{}\mu_{N}$. The hadrons’
magnetic and quadrupole moments are another fundamental observable as their
mass, which provides information on the underlying quark structure and
dynamics. The results obtained in both pictures are quite different from each
other. Any experimental measurement of the magnetic moments can provide an
understanding of the internal structure of these states. We get nonzero but
small values for the quadrupole moments of $B_{c}$-like tetraquark states
showing non-spherical charge distributions. Hopefully, the examinations given
in this study will be helpful to an experimental search of them, which will be
an interesting research subject.

Magnetic and quadrupole moments, electromagnetic form factors, open-flavor
tetraquark states, QCD light-cone sum rules

##  I motivation

The experimental discovery of exotic states began in 2003 with the observation
of the X(3872) [1], though they were suggested long ago that states other than
standard hadrons could exist. Since 2003, scientists have been increasingly
interested in investigating and understanding the nature of exotic states that
differ from standard mesons and baryons. Analysis of exotic states plays a key
role in understanding low-energy QCD and therefore it is quite important to
search for them in experiments. So far, many exotic states have been observed
by different experimental facilities. Many different theoretical
interpretations of these states have been suggested, such as tetraquarks,
pentaquarks, hybrids, glueballs, and so on (see the reviews of Refs. [2, 3, 4,
5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], and references therein, for
further details).

Most of the discovered tetraquark states belong to the class of so-called
hidden-charm or bottom states including the $c\bar{c}$ or $b\bar{b}$ pair.
However, the first principles of QCD do not prohibit the existence of open-
flavor tetraquarks. The $B_{c}$-like tetraquarks belong to another type of
exotic state. Although these states have not been observed experimentally,
they have attracted the attention of physicists [18, 19, 20, 21, 22, 23, 24,
25, 26, 27, 28]. In Refs. [18, 19], Zhang et al., have employed to calculate
the masses for $[Q\bar{q}][\bar{Q}^{(^{\prime})}q]$ and
$[Q\bar{s}][\bar{Q}^{(^{\prime})}s]$ molecular states within the QCD sum
rules, including the contributions of the operators up to dimension six in
OPE. In Ref. [20], Sun et al., have studied the interaction between the S-wave
$D^{(*)}/D^{(*)}_{s}$ meson and S-wave $B^{(*)}/B^{(*)}_{s}$ meson in the one-
boson exchange model and they predicted the existence of many $B_{c}$-like
molecular states. In Ref. [21], Albuquerque et al., have studied the mass of
the exotic $B_{c}$-like molecular states using QCD sum rules and they
predicted for these states masses around $7.0$ GeV. In Ref. [22], Chen et al.,
have performed QCD sum rule analysis and extracted the masses of
$[bc\bar{q}\bar{q}]$, $[bc\bar{s}\bar{s}]$ and $[qc\bar{q}\bar{b}]$ and
$[sc\bar{s}\bar{b}]$ tetraquark states and they estimated that the tetraquark
states $[qc\bar{q}\bar{b}]$ and $sc\bar{s}\bar{b}]$ lie below the thresholds
of $D^{(*)}B^{(*)}$ and $D^{(*)}_{s}B^{(*)}_{s}$ respectively. In Refs. [23,
24], Agaev et al., have studied the spectroscopic parameters for $B_{c}$-like
tetraquark states $[cq][\bar{b}\bar{q}]$ and $[cs][\bar{b}\bar{s}]$ with
quantum numbers $J^{P}=0^{+},1^{+}$ through QCD sum rule technique and their
analyzes indicated that the masses are about $6.97-7.06$ GeV for the
$[cq][\bar{b}\bar{q}]$ tetraquark states and $7.01-7.30$ GeV for the
$[cs][\bar{b}\bar{s}]$ tetraquark states. In addition to the mass calculations
of these states, some strong decay channels were also investigated in these
studies and the relevant coupling constants were obtained. In Ref. [25], Wang
et al., studied the masses for fully open-flavor tetraquark states
$[bc\bar{q}\bar{s}]$ and $[sc\bar{q}\bar{b}]$ with quantum numbers
$J^{P}=0^{+},1^{+}$ in the framework of QCD sum rule method and their
computations showed that the masses are about $7.1-7.2$ GeV for the
$[bc\bar{q}\bar{s}]$ tetraquark states and $7.0-7.1$ GeV for the
$[sc\bar{q}\bar{b}]$ tetraquarks. In Ref. [26], Z.-G. Wang has constructed the
diquark-antidiquark type current operators to study the axial-vector
$B_{c}$-like tetraquark states with the QCD sum rules with quantum numbers
$J^{P}=1^{+-},1^{++}$ and predicted mass of these states given as
$M_{Z_{b\bar{c}}(1^{+-})}=7.30\pm 0.08$ GeV and
$M_{Z_{b\bar{c}}(1^{++})}=7.31\pm 0.08$ GeV, respectively. In Ref. [27], Wu et
al., calculated the spectra of the possible
$Q_{1}q_{2}\bar{Q}_{3}\bar{q}_{4}~{}(Q=b,c$ and $q=n,s$ with $n=u,d)$
tetraquark states by using the chromomagnetic interaction model in the
diquark-antiquark picture. In Ref. [28], Ortega et al., have obtained the
masses of $DB$, $DB^{*}$, $D^{*}B$, and $D^{*}B^{*}$ states through the quark
model and their computations indicated that the masses are about $7.1-7.3$ GeV
for the $B_{c}$-like tetraquark states.

The studying these $B_{c}$-like systems: contrary to $c\bar{c}$ and
$b\bar{b}$, the $B_{c}$-like tetraquarks cannot annihilate into gluons and
therefore these states are very stable, with narrow widths. Because of these
properties, they are quite valuable to study heavy-quark dynamics and
understanding the dynamics of the QCD at a deeper level. To better understand
the internal structures of these states, it is also important to study decay
channels such as strong, radiative, and electromagnetic together with their
spectroscopic parameters. Calculation of the electromagnetic properties of the
particles allows us to obtain important data about the substructure of the
particles under investigation. Since in the past magnetic moments estimations
from this sort of model have been well accomplished, to distinguish among the
possible configurations, it seems to be helpful to investigate also the
magnetic moments of multiquark states. Furthermore, the magnetic moments of
the hadrons are important measurables like their masses, which have
substantial knowledge about the underlying quark configurations, and can be
used to distinguish the preferred quark configurations from different
theoretical approximations and deepen our understanding of the underlying
dynamics. Inspired by the above reasons, in this work, we will study the
magnetic and quadrupole moments for the $B_{c}$-like ($Z_{c\bar{b}}$ for
short) ground state tetraquarks with the quantum numbers $J^{P}=1^{+}$ in the
method of QCD light-cone sum rules (LCSR) which is the powerful quantitative
tool to investigate features of hadrons. The key idea and the defining
characteristic of the LCSR are that the short-distance operator product
expansion is replaced by the light-cone expansion in operators of increasing
twist [29, 30, 31]. Over the past few decades, LCSR has shown to be a very
robust method for investigating non-perturbative hadron properties such as
form factors, coupling constants, and magnetic moments associated with
conventional and unconventional hadron states. The applications of the QCD sum
rules and LCSR to some $c\bar{c}$ and $b\bar{b}$ tetraquark states can be seen
in Refs. [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48,
49, 50, 51, 52, 53, 54, 55, 56].

The article is organized as follows. In Sec. II, we briefly introduce our
notations and apply the LCSR method to evaluate the magnetic and quadrupole
moments of $Z_{c\bar{b}}$ tetraquark states as molecular and diquark-
antidiquark structures. In Sec. III, the numerical analysis and discussions
for the magnetic and quadrupole moments of the $Z_{c\bar{b}}$ tetraquark
states are presented. The obtained results are summarized and discussed in
Sec. IV. The appendixes include explicit expressions of the correlator used in
computations of the magnetic moments of the $Z_{c\bar{b}}$ tetraquark states
and some details about calculations.

##  II Magnetic and quadrupole Moments from LCSR

To evaluate the magnetic and quadrupole moments of the $Z_{c\bar{b}}$ ground
state tetraquarks within the LCSR, we start with the following correlator

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

where $q$ is the momentum of the photon, the $\gamma$ stands for the external
background electromagnetic field and $J_{\mu}(x)$ is the interpolating
currents of the $Z_{c\bar{b}}$ ground state tetraquarks with the quantum
numbers $J^{P}=1^{+}$. The corresponding molecular and diquark-antidiquark
interpolating currents are given by

$\displaystyle J_{\mu}^{Mol}(x)$
$\displaystyle=[\bar{q}_{a}(x)i\gamma_{5}c_{a}(x)][\bar{b}_{b}(x)\gamma_{\mu}q_{b}(x)],$
(2) $\displaystyle J_{\mu}^{Di}(x)$
$\displaystyle=[q_{a}^{T}(x)C\gamma_{5}c_{b}(x)][\overline{q}_{a}(x)\gamma_{\mu}C\overline{b}_{b}^{T}(x)]+[q_{a}^{T}(x)C\gamma_{5}c_{b}(x)][\overline{q}_{b}(x)\gamma_{\mu}C\overline{b}_{a}^{T}(x)],$
(3)

where $C$ is the charge conjugation operator and $q(x)$ denotes one of the
$u(x)$ or $d(x)$ quarks.

To get LCSR for the magnetic and quadrupole moments we follow standard
instructions of the LCSR method and express the correlator $\Pi_{\mu\nu}(p,q)$
in connection with the physical parameters of the $Z_{c\bar{b}}$ tetraquark
states, which results in obtaining $\Pi^{Had}_{\mu\nu}(p,q)$. From another
side, the same correlator should be acquired concerning the quark-gluon
degrees of freedom $\Pi^{QCD}_{\mu\nu}(p,q)$. Matching the coefficients of
various Lorentz structures from two different representations of the same
correlator and performing double Borel transformations and continuum
subtraction to remove the effects of the continuum and higher states, we get
LCSR for the magnetic and quadrupole moments of the $Z_{c\bar{b}}$ tetraquark
states.

We will begin our analysis by calculating the hadronic representation of the
correlator. To do this, the correlator is computed by its fulfillment with the
intermediate $Z_{c\bar{b}}$ tetraquark states where $p^{2}>0$, $(p+q)^{2}>0$.
By applying the four-integral over $x$ we obtain

$\displaystyle\Pi_{\mu\nu}^{Had}(p,q)$ $\displaystyle={\frac{\langle 0\mid
J_{\mu}(x)\mid Z_{c\bar{b}}(p)\rangle}{p^{2}-m_{Z_{c\bar{b}}}^{2}}}\langle
Z_{c\bar{b}}(p)\mid Z_{c\bar{b}}(p+q)\rangle_{\gamma}\frac{\langle
Z_{c\bar{b}}(p+q)\mid J_{\nu}^{\dagger}(0)\mid
0\rangle}{(p+q)^{2}-m_{Z_{c\bar{b}}}^{2}}+\mbox{higher states},$ (4)

The matrix element $\langle 0\mid J_{\mu}(x)\mid Z_{c\bar{b}}(p)\rangle$ is
given as

$\displaystyle\langle 0\mid J_{\mu}(x)\mid
Z_{c\bar{b}}(p)\rangle=\lambda_{Z_{c\bar{b}}}\varepsilon_{\mu}^{\theta}\,,$
(5)

with $\varepsilon_{\mu}^{\theta}\ $ and $\lambda_{Z_{c\bar{b}}}$ being the
polarization vector and residue of the $Z_{c\bar{b}}$ tetraquark states,
respectively.

The matrix element $\langle Z_{c\bar{b}}(p)\mid
Z_{c\bar{b}}(p+q)\rangle_{\gamma}$ can be written in connection with the
Lorentz invariant form factors as follows [57]:

$\displaystyle\langle Z_{c\bar{b}}(p,\varepsilon^{\theta})\mid
Z_{c\bar{b}}(p+q,\varepsilon^{\delta})\rangle_{\gamma}$
$\displaystyle=-\varepsilon^{\tau}(\varepsilon^{\theta})^{\alpha}(\varepsilon^{\delta})^{\beta}\Big{[}G_{1}(Q^{2})~{}(2p+q)_{\tau}~{}g_{\alpha\beta}+G_{2}(Q^{2})~{}(g_{\tau\beta}~{}q_{\alpha}-g_{\tau\alpha}~{}q_{\beta})$
$\displaystyle-\frac{1}{2m_{Z_{c\bar{b}}}^{2}}G_{3}(Q^{2})~{}(2p+q)_{\tau}q_{\alpha}q_{\beta}\Big{]},$
(6)

where the polarization vectors of the initial and final $Z_{c\bar{b}}$
tetraquark states are represented $\varepsilon^{\delta}$ and,
$\varepsilon^{\theta}$ and $\varepsilon^{\tau}$ is the polarization vector of
the photon. Here, $G_{1}(Q^{2})$, $G_{2}(Q^{2})$ and $G_{3}(Q^{2})$ are
invariant form factors, with $Q^{2}=-q^{2}$.

Employing Eqs. (4)-(II), the correlator takes the form,

$\displaystyle\Pi_{\mu\nu}^{Had}(p,q)$
$\displaystyle=\frac{\varepsilon_{\rho}\,\lambda_{Z_{c\bar{b}}}^{2}}{[m_{Z_{c\bar{b}}}^{2}-(p+q)^{2}][m_{Z_{c\bar{b}}}^{2}-p^{2}]}\bigg{\\{}G_{1}(Q^{2})(2p+q)_{\rho}\bigg{[}g_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{m_{Z_{c\bar{b}}}^{2}}-\frac{(p+q)_{\mu}(p+q)_{\nu}}{m_{Z_{c\bar{b}}}^{2}}+\frac{(p+q)_{\mu}p_{\nu}}{2m_{Z_{c\bar{b}}}^{4}}$
$\displaystyle\times(Q^{2}+2m_{Z_{c\bar{b}}}^{2})\bigg{]}+G_{2}(Q^{2})\bigg{[}q_{\mu}g_{\rho\nu}-q_{\nu}g_{\rho\mu}-\frac{p_{\nu}}{m_{Z_{c\bar{b}}}^{2}}\big{(}q_{\mu}p_{\rho}-\frac{1}{2}Q^{2}g_{\mu\rho}\big{)}+\frac{(p+q)_{\mu}}{m_{Z_{c\bar{b}}}^{2}}\big{(}q_{\nu}(p+q)_{\rho}+\frac{1}{2}Q^{2}g_{\nu\rho}\big{)}$
$\displaystyle-\frac{(p+q)_{\mu}p_{\nu}p_{\rho}}{m_{Z_{c\bar{b}}}^{4}}\,Q^{2}\bigg{]}-\frac{G_{3}(Q^{2})}{m_{Z_{c\bar{b}}}^{2}}(2p+q)_{\rho}\bigg{[}q_{\mu}q_{\nu}-\frac{p_{\mu}q_{\nu}}{2m_{Z_{c\bar{b}}}^{2}}Q^{2}+\frac{(p+q)_{\mu}q_{\nu}}{2m_{Z_{c\bar{b}}}^{2}}Q^{2}-\frac{(p+q)_{\mu}q_{\nu}}{4m_{Z_{c\bar{b}}}^{4}}Q^{4}\bigg{]}\bigg{\\}}\,.$
(7)

The magnetic dipole ($F_{M}(Q^{2})$) and quadrupole ($F_{\cal D}(Q^{2})$) form
factors are described in terms of $G_{1}(Q^{2})$, $G_{2}(Q^{2})$ and
$G_{3}(Q^{2})$ form factors as:

$\displaystyle F_{M}(Q^{2})=G_{2}(Q^{2})\,,$ $\displaystyle F_{\cal
D}(Q^{2})=G_{1}(Q^{2})-G_{2}(Q^{2})+(1+\tau)G_{3}(Q^{2})\,,$ (8)

where $\tau=Q^{2}/4m_{Z_{c\bar{b}}}^{2}$ with $Q^{2}=-q^{2}$. In the static
limit, $Q^{2}=0$, the $F_{M}(Q^{2}=0)$ and $F_{\cal D}(Q^{2}=0)$ form factors
are related to the magnetic moment ($\mu$), and quadrupole moment (${\cal D}$)
as follows

$\displaystyle eF_{M}(0)=2m_{Z_{c\bar{b}}}\mu\,,$ $\displaystyle eF_{\cal
D}(0)=m_{Z_{c\bar{b}}}^{2}{\cal D}\,.$ (9)

In QCD representation, the correlator in Eq. (1), is evaluated concerning the
QCD degrees of freedom in deep Euclidean region where $p^{2}<<0$ and
$(p+q)^{2}<<0$. To do this, we insert the interpolating currents in the
correlator and contract the heavy and light quark fields utilizing Wick’s
theorem. As a result of these steps, we get following expressions for the
$Z_{c\bar{b}}$ states

$\displaystyle\Pi_{\mu\nu}^{\mathrm{QCD}-Mol}(p,q)$ $\displaystyle=i\int
d^{4}xe^{ip\cdot x}\langle 0\mid\
\mathrm{Tr}\big{[}\gamma_{\mu}{S}_{q}^{bb^{\prime}}(x)\gamma_{\nu}S_{b}^{b^{\prime}b}(-x)\big{]}\mathrm{Tr}\big{[}\gamma_{5}{S}_{c}^{aa^{\prime}}(x)\gamma_{5}S_{q}^{a^{\prime}a}(-x)\big{]}\mid
0\rangle_{\gamma},$ (10)

$\displaystyle\Pi_{\mu\nu}^{\mathrm{QCD}-Di}(p,q)$ $\displaystyle=i\int
d^{4}xe^{ip\cdot x}\langle
0\mid\bigg{\\{}\mathrm{Tr}\Big{[}\gamma_{\mu}\widetilde{S}_{b}^{b^{\prime}b}(-x)\gamma_{\nu}S_{q}^{a^{\prime}a}(-x)\Big{]}\mathrm{Tr}\Big{[}\gamma_{5}\widetilde{S}_{q}^{aa^{\prime}}(x)\gamma_{5}S_{c}^{bb^{\prime}}(x)\Big{]}$
$\displaystyle+\mathrm{Tr}\Big{[}\gamma_{\mu}\widetilde{S}_{b}^{a^{\prime}b}(-x)\gamma_{\nu}S_{q}^{b^{\prime}a}(-x)\Big{]}\mathrm{Tr}\Big{[}\gamma_{5}\widetilde{S}_{q}^{aa^{\prime}}(x)\gamma_{5}S_{c}^{bb^{\prime}}(x)\Big{]}$
$\displaystyle+\mathrm{Tr}\Big{[}\gamma_{\mu}\widetilde{S}_{b}^{b^{\prime}a}(-x)\gamma_{\nu}S_{q}^{a^{\prime}b}(-x)\Big{]}\mathrm{Tr}\Big{[}\gamma_{5}\widetilde{S}_{q}^{aa^{\prime}}(x)\gamma_{5}S_{c}^{bb^{\prime}}(x)\Big{]}$
$\displaystyle+\mathrm{Tr}\Big{[}\gamma_{\mu}\widetilde{S}_{b}^{a^{\prime}a}(-x)\gamma_{\nu}S_{q}^{b^{\prime}b}(-x)\Big{]}\mathrm{Tr}\Big{[}\gamma_{5}\widetilde{S}_{q}^{aa^{\prime}}(x)\gamma_{5}S_{c}^{bb^{\prime}}(x)\Big{]}\bigg{\\}}\mid
0\rangle_{\gamma},$ (11)

where

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

with $C$ and $T$ being the charge conjugation and transpose of the operator,
respectively. Here, $S_{q}(x)$ and $S_{Q}(x)$ represent the full light and
heavy quark propagators. Throughout our calculations, we use the x-space
expressions for the light and heavy quark propagators [58, 59]:

$\displaystyle S_{q}(x)$
$\displaystyle=i\frac{{x\\!\\!\\!/}}{2\pi^{2}x^{4}}-\frac{\langle\bar{q}q\rangle}{12}\Big{(}1-i\frac{m_{q}x\\!\\!\\!/}{4}\Big{)}-\frac{\langle\bar{q}q\rangle}{192}m_{0}^{2}x^{2}\Big{(}1-i\frac{m_{q}x\\!\\!\\!/}{6}\Big{)}-\frac{ig_{s}}{32\pi^{2}x^{2}}~{}G^{\mu\nu}(x)\Big{[}\hbox
to0.0pt{/\hss}{x}\sigma_{\mu\nu}+\sigma_{\mu\nu}\hbox
to0.0pt{/\hss}{x}\Big{]},$ (12) $\displaystyle S_{Q}(x)$
$\displaystyle=\frac{m_{Q}^{2}}{4\pi^{2}}\Bigg{[}\frac{K_{1}\Big{(}m_{Q}\sqrt{-x^{2}}\Big{)}}{\sqrt{-x^{2}}}+i\frac{{x\\!\\!\\!/}~{}K_{2}\Big{(}m_{Q}\sqrt{-x^{2}}\Big{)}}{(\sqrt{-x^{2}})^{2}}\Bigg{]}-\frac{g_{s}m_{Q}}{16\pi^{2}}\int_{0}^{1}dv\,G^{\mu\nu}(vx)\Bigg{[}\big{(}\sigma_{\mu\nu}{x\\!\\!\\!/}+{x\\!\\!\\!/}\sigma_{\mu\nu}\big{)}$
$\displaystyle\times\frac{K_{1}\Big{(}m_{Q}\sqrt{-x^{2}}\Big{)}}{\sqrt{-x^{2}}}+2\sigma_{\mu\nu}K_{0}\Big{(}m_{Q}\sqrt{-x^{2}}\Big{)}\Bigg{]},$
(13)

where $\langle\bar{q}q\rangle$ is light-quark condensate, $m_{0}$ is defined
through $\langle
0\mid\bar{q}\,g_{s}\,\sigma_{\alpha\beta}\,G^{\alpha\beta}\,q\mid
0\rangle=m_{0}^{2}\,\langle\bar{q}q\rangle$, $G^{\mu\nu}$ is the gluon field
strength tensor, $v$ is line variable and $K_{i}$’s are modified Bessel
functions of the second kind. The first term of the light and heavy quark
propagators correspond to the perturbative or free part and the rest belong to
the interacting parts. The correlator in QCD representation includes two
different contributions: perturbative and nonperturbative. How perturbative
and nonperturbative contributions are calculated in the analysis is given in
Appendix A using Eq. (10) as an example. The QCD degrees of freedom
representation of the correlator can be obtained in terms of the quark-gluon
properties via the photo DAs and after performing an integration over x, the
expression of the correlator in the momentum representation can be calculated
straightforwardly.

To determine LCSR for the magnetic and quadrupole moments, we perform double
Borel transformation and continuum subtraction to suppress the higher states
and continuum effects. As a result, we get

$\displaystyle\mu_{Z_{c\bar{b}}}^{Mol}\,\,\lambda_{Z_{c\bar{b}}}^{2-Mol}=e^{\frac{m_{Z_{c\bar{b}}}^{2}}{M^{2}}}\,\,\Delta_{1}^{QCD}(M^{2},s_{0}),$
(14)
$\displaystyle\mu_{Z_{c\bar{b}}}^{Di}\,\,\lambda_{Z_{c\bar{b}}}^{2-Di}=e^{\frac{m_{Z_{c\bar{b}}}^{2}}{M^{2}}}\,\,\Delta_{2}^{QCD}(M^{2},s_{0}),$
(15)
$\displaystyle\mathcal{D}_{Z_{c\bar{b}}}^{Mol}\,\,\lambda_{Z_{c\bar{b}}}^{2-Mol}=m^{2}_{Z_{c\bar{b}}}e^{\frac{m_{Z_{c\bar{b}}}^{2}}{M^{2}}}\,\,\Delta_{3}^{QCD}(M^{2},s_{0}),$
(16)
$\displaystyle\mathcal{D}_{Z_{c\bar{b}}}^{Di}\,\,\lambda_{Z_{c\bar{b}}}^{2-Di}=m^{2}_{Z_{c\bar{b}}}e^{\frac{m_{Z_{c\bar{b}}}^{2}}{M^{2}}}\,\,\Delta_{4}^{QCD}(M^{2},s_{0}).$
(17)

For the sake of simplicity, only the explicit expressions of the
$\Delta_{1}^{QCD}(M^{2},s_{0})$ and $\Delta_{2}^{QCD}(M^{2},s_{0})$ functions
are presented in Appendix B.

##  III Numerical illustrations and Discussions

In this section, we perform the LCSR analyses for the magnetic and quadrupole
moments of $Z_{c\bar{b}}$ tetraquark states. We use the values of masses,
residues and various QCD condensates as follows: $m_{u}=m_{d}=0$,
$m_{c}=1.67\pm 0.07\,\mbox{GeV}$, $m_{b}=4.78\pm 0.06\,\mbox{GeV}$,
$m_{Z_{c\bar{b}}}^{Mol}=6.85\pm 0.15\,\mbox{GeV}$,
$m_{Z_{c\bar{b}}}^{Di}=7.06\pm 0.78\,\mbox{GeV}$,
$\lambda_{Z_{c\bar{b}}}^{Mol}=0.036\pm 0.011\,\mbox{GeV}^{5}$ [21],
$\lambda_{Z_{c\bar{b}}}^{Di}=0.024\pm 0.08\,\mbox{GeV}^{5}$ [24]
$m_{0}^{2}=0.8\pm 0.1\,\mbox{GeV}^{2}$ [60],
$\langle\bar{u}u\rangle=\langle\bar{d}d\rangle=(-0.24\pm
0.01)^{3}\,\mbox{GeV}^{3}$ [60], $\langle
g_{s}^{2}G^{2}\rangle=0.88~{}\mbox{GeV}^{4}$ [6],
$f_{3\gamma}=-0.0039~{}\mbox{GeV}^{2}$ [61] and $\chi=-2.85\pm
0.5~{}\mbox{GeV}^{-2}$ [62]. To proceed with the numerical calculations
distribution amplitudes of the photon are needed. The explicit expressions of
the photon distribution amplitudes and numerical values of input parameters
are borrowed from Ref. [61].

There are two free parameters in Eqs. (14)-(17), the Borel mass parameter
($M^{2}$) and the continuum threshold ($s_{0}$). By taking into consideration
that the magnetic and quadrupole moments are physical observables they should
be less dependent on the free parameters $M^{2}$, and $s_{0}$. To obtain
reliable working regions for these free parameters, we use three criteria to
constrain them: The convergence of the operator product expansion (OPE), the
pole contribution ($PC$), and the magnetic and quadrupole moments dependence
on $M^{2}$ and $s_{0}$. In the LCSR analysis, the $PC$ requires that it should
exceed $20\%$ of the total contributions, which is typical for the multiquark
states. We also demand that the series of light-cone expansion converge and
contributions of the higher twist and higher condensate terms are less than
$10\%$ of the total contribution. Considering these constraints working
intervals of these helping parameters have been determined as follows:

$\displaystyle 58.0~{}\mbox{GeV}^{2}\leq s_{0}\leq 60.0~{}\mbox{GeV}^{2},$
$\displaystyle 7.5~{}\mbox{GeV}^{2}\leq M^{2}\leq 9.5~{}\mbox{GeV}^{2}.$ (18)

Our numerical computations indicate that by taking into account these working
intervals for the helping parameters, the magnetic and quadrupole moments of
the $Z_{c\bar{b}}$ tetraquark states $PC$ varies within the interval $30\%\leq
PC\leq 59\%$ corresponding to the upper and lower limits of the Borel mass
parameter. When we analyze the OPE convergence, we see that the contribution
of the higher twist and higher dimensional terms in OPE is $3\%$ of the total
and the series shows good convergence. As an example, in Fig. 1, we also show
the variation of the extracted magnetic moment of $Z_{c\bar{b}}$ tetraquark
states with the Borel mass $M^{2}$ using three fixed values of the continuum
threshold $s_{0}$. It can be seen from this figure that the magnetic moment
indicates a weak dependence on $M^{2}$ in its working region. Though the
magnetic moment of $Z_{c\bar{b}}$ tetraquark states show some dependence on
$s_{0}$, it remains inside the limits allowed by the method and constitutes
the main parts of the uncertainties.

Figure 1:  The dependence of magnetic moment of the $Z_{c\bar{b}}$ tetraquark
states on $M^{2}$ at three fixed values of $s_{0}$.

We have determined all necessary parameters to specify the numerical values
for the magnetic and quadrupole moments of $Z_{c\bar{b}}$ tetraquark states.
As a result of our comprehensive numerical calculations, the magnetic moments
of $Z_{c\bar{b}}$ tetraquark states are given in Table 1. It should be noted
here that in numerical calculations we take into account the uncertainties in
the input parameters, uncertainties entering into the photon DAs, as well as
uncertainties because of the variations of Borel mass parameter $M^{2}$ and
continuum threshold $s_{0}$. We would like to point out that roughly 18$\%$ of
the errors in the numerical results are due to the mass of tetraquarks, 15$\%$
belongs to the residue of tetraquarks, 30$\%$ belongs to $s_{0}$, 7$\%$
belongs to $M^{2}$, 13$\%$ belongs to photon DAs and the remaining 17$\%$
corresponds to other input parameters.

Table 1: Magnetic moments of the $Z_{c\bar{b}}$ states (in units of nuclear magnetons $\mu_{N}$). Picture |  $\mu_{Z_{{uc\bar{u}\bar{b}}}}$ |  $\mu_{Z_{dc\bar{d}\bar{b}}}$
---|---|---
Diquark |  $3.05^{+1.19}_{-0.95}$ |  $2.38^{+0.95}_{-0.75}$
Molecule |  $1.18^{+0.52}_{-0.40}$ |  $0.32^{+0.18}_{-0.10}$

The magnitude of the results obtained for the magnetic moments also gives the
possibility to be measured experimentally. It follows from these results that
the magnetic moments of $B_{c}$-like ground state tetraquarks are large enough
to be measured in future experiments. When we compare the numerical results of
the two pictures above, we observe that states with the same quantum numbers
have different magnetic moments, which clearly shows the magnetic moment
strongly depends on the structure of the hadron. This implies that any
possible experimental measurement of magnetic moments can help us understand
the internal structure of these states. Different magnetic moments will affect
both the total and differential cross sections in the photo- or electro-
production of $B_{c}$-like tetraquarks. Thus, information on the $B_{c}$-like
tetraquarks magnetic moments will help us unveil the mysterious curtain over
the $B_{c}$-like tetraquarks and deepen our understanding of the underlying
quark structure and dynamics. Our results on magnetic moments of the
$B_{c}$-like ground state tetraquarks may be checked in the framework of the
alternative phenomenological models. Hopefully, the examinations given in this
study will be helpful to an experimental search of them, which will be an
interesting research subject.

To our best knowledge, this is the first study in the literature committed to
the investigation of $B_{c}$-like tetraquark magnetic and quadrupole moments.
Hence, theoretical predictions or experimental data are not yet existing to
compare them with our numerical values. However, we give the magnetic moment
results of the standard $B_{c}^{*}$ meson so that the reader can better
understand the difference in the results obtained for the $B_{c}$-like ground
state tetraquarks. The magnetic moments of $B_{c}^{*}$ meson have been
extracted using the Bag model [63], extended Bag model [64] and Blankenbecler-
Sugar (BSLT) equation [65]. The obtained results are given as
$\mu_{B_{c}^{*}}=0.20\,\mu_{N}$, $\mu_{B_{c}^{*}}=0.38\,\mu_{N}$ and
$\mu_{B_{c}^{*}}=0.426\,\mu_{N}$ for the Bag model, extended Bag model, and
BSLT, respectively. We would like to point out that only magnetic moment
results for vector $B_{c}^{*}$ mesons have been calculated in the literature
and therefore these results are presented. As we mentioned above, it is given
only to make the results understandable for the reader.

We also present numerical results of quadrupole moments of $B_{c}$-like
tetraquark states, and their numerical values are given in Table 2. We can
notice that just like the magnetic moment results, the quadrupole moment
results obtained using two different configurations are different from each
other. The quadrupole moments of the $B_{c}$-like tetraquark states indicate
non-spherical charge distributions. The sign of quadrupole moments are
negative for $Z_{{uc\bar{u}\bar{b}}}$ tetraquark states and positive for
$Z_{dc\bar{d}\bar{b}}$ tetraquark states, which correspond to the oblate and
prolate charge distributions, respectively.

Table 2: Quadrupole moments of the $Z_{c\bar{b}}$ states (in units of $fm^{2}$). Picture |  $\mathcal{D}_{Z_{{uc\bar{u}\bar{b}}}}(\times 10^{-2})$ |  $\mathcal{D}_{Z_{dc\bar{d}\bar{b}}}(\times 10^{-2})$
---|---|---
Diquark |  $-1.40^{+0.40}_{-0.40}$ |  $0.70^{+0.30}_{-0.30}$
Molecule |  $-0.40^{+0.10}_{-0.10}$ |  $0.20^{+0.05}_{-0.05}$

##  IV summary and Outlook

We have employed the molecular and diquark-antidiquark tetraquark pictures to
extract magnetic and quadrupole moments of the $B_{c}$-like ground state
tetraquarks with the QCD light-cone sum rules with quantum numbers
$J^{P}=1^{+}$. Comparing the numerical results of the above two
configurations, we notice that the magnetic moments of the $B_{c}$-like
tetraquark states with the same quantum numbers differ significantly from each
other, which can be used to identify the underlying structure of these states.
We have also extracted the quadrupole moments of these states. The quadrupole
moments of the $B_{c}$-like tetraquark states indicate non-spherical charge
distributions.

The magnetic moment of hadrons is an essential ingredient in calculations of
the photo- and electro-production cross sections and may be obtained from the
experiments in the future. With the increased luminosity in future runs,
spectroscopic parameters and magnetic moments of $B_{c}$-like tetraquark
states may be extracted from experimental facilities, which may help
distinguish different theoretical approaches and deepen our understanding of
the underlying dynamics governing their formations. It will also be important
to determine the branching ratios of the different decay modes and decay
channels of the $B_{c}$-like tetraquark states. Furthermore, the study of the
$B_{c}$-like tetraquark states in other theoretical models can also be very
interesting.

## Appendix A: perturbative and nonperturbative contributions of QCD side of
the correlation function

In this appendix, we will show in a short example how the perturbative and
nonperturbative contributions, which appear in the analysis of the QCD side of
the correlation function, are calculated.

$\displaystyle\Pi_{\mu\nu}^{\mathrm{QCD}-Mol}(p,q)$ $\displaystyle=i\int
d^{4}xe^{ip\cdot x}\langle 0\mid\
\mathrm{Tr}\big{[}\gamma_{\mu}{S}_{q}^{bb^{\prime}}(x)\gamma_{\nu}S_{b}^{b^{\prime}b}(-x)\big{]}\mathrm{Tr}\big{[}\gamma_{5}{S}_{c}^{aa^{\prime}}(x)\gamma_{5}S_{q}^{a^{\prime}a}(-x)\big{]}\mid
0\rangle_{\gamma},$ (19)

As we mentioned above, the correlator in Eq. (19) includes different types of
contributions: the photon can be emitted both perturbatively or non-
perturbatively.

Practically, perturbative contributions, the photon interacts with one of the
quarks, can be computed by replacing one of the light or heavy-quark
propagators in Eq. (19) by

$\displaystyle S^{free}\rightarrow\int d^{4}y\,S^{free}(x-z)\,\hbox
to0.0pt{/\hss}{\\!A}(z)\,S^{free}(z)\,,$ (20)

the remaining three propagators are taken as full quark propagators. The QCD
light-cone sum rule calculations are usually done on a fixed-point gauge is
also known as Fock-Schwinger gauge. The most important reason for using this
gauge is to express the external background field according to the field
strength tensor and also the use of this gauge is preferred because it
provides some convenience in calculations. For the electromagnetic field, it
is characterized by $x_{\mu}A^{\mu}=0$. In this gauge, the external
electromagnetic potential is given by

$\displaystyle
A_{\alpha}(z)=-\frac{1}{2}F_{\alpha\beta}z^{\beta}=-\frac{1}{2}(\varepsilon_{\alpha}q_{\beta}-\varepsilon_{\beta}q_{\alpha})\,z^{\beta}.$
(21)

Equation (21) is inserted into Eq. (20), we obtain

$\displaystyle
S^{free}\rightarrow-\frac{1}{2}(\varepsilon_{\alpha}q_{\beta}-\varepsilon_{\beta}q_{\alpha})\int\,d^{4}z\,z^{\beta}\,S^{free}(x-z)\,\gamma_{\alpha}\,S^{free}(z)\,,$
(22)

After some computations for $S_{q}^{free}$ and $S_{Q}^{free}$, their final
form becomes:

$\displaystyle
S_{q}^{free}(x)=\frac{e_{q}}{32\pi^{2}x^{2}}\Big{(}\varepsilon_{\alpha}q_{\beta}-\varepsilon_{\beta}q_{\alpha}\Big{)}\Big{(}x\\!\\!\\!/\sigma_{\alpha\beta}+\sigma_{\alpha\beta}x\\!\\!\\!/\Big{)},$
$\displaystyle
S_{Q}^{free}(x)=-i\frac{e_{Q}m_{Q}}{32\pi^{2}}\Big{(}\varepsilon_{\alpha}q_{\beta}-\varepsilon_{\beta}q_{\alpha}\Big{)}\Big{[}2\sigma_{\alpha\beta}K_{0}\Big{(}m_{Q}\sqrt{-x^{2}}\Big{)}+\frac{K_{1}\Big{(}m_{Q}\sqrt{-x^{2}}\Big{)}}{\sqrt{-x^{2}}}\Big{(}x\\!\\!\\!/\sigma_{\alpha\beta}+\sigma_{\alpha\beta}x\\!\\!\\!/\Big{)}\Big{]}.$
(23)

Equation (Appendix A: perturbative and nonperturbative contributions of QCD
side of the correlation function) is inserted into Eq. (19), and as a result
these manipulations for the perturbative contributions we get

$\displaystyle\Pi_{\mu\nu-Pert}^{\mathrm{QCD}-Mol}(p,q)$ $\displaystyle=i\int
d^{4}xe^{ip\cdot
x}\,\mathrm{Tr}\big{[}\gamma_{\mu}{S}_{q}^{free}(x)\gamma_{\nu}S_{b}^{b^{\prime}b}(-x)\big{]}\mathrm{Tr}\big{[}\gamma_{5}{S}_{c}^{aa^{\prime}}(x)\gamma_{5}S_{q}^{a^{\prime}a}(-x)\big{]}\delta^{bb^{\prime}}$
$\displaystyle+\mathrm{Tr}\big{[}\gamma_{\mu}{S}_{q}^{b^{\prime}b}(x)\gamma_{\nu}S_{b}^{free}(-x)\big{]}\mathrm{Tr}\big{[}\gamma_{5}{S}_{c}^{aa^{\prime}}(x)\gamma_{5}S_{q}^{a^{\prime}a}(-x)\big{]}\delta^{bb^{\prime}}$
$\displaystyle+\mathrm{Tr}\big{[}\gamma_{\mu}{S}_{q}^{b^{\prime}b}(x)\gamma_{\nu}S_{b}^{b^{\prime}b}(-x)\big{]}\mathrm{Tr}\big{[}\gamma_{5}{S}_{c}^{free}(x)\gamma_{5}S_{q}^{a^{\prime}a}(-x)\big{]}\delta^{aa^{\prime}}$
$\displaystyle+\mathrm{Tr}\big{[}\gamma_{\mu}{S}_{q}^{b^{\prime}b}(x)\gamma_{\nu}S_{b}^{b^{\prime}b}(-x)\big{]}\mathrm{Tr}\big{[}\gamma_{5}{S}_{c}^{aa^{\prime}}(x)\gamma_{5}S_{q}^{free}(-x)\big{]}\delta^{a^{\prime}a}.$
(24)

It should be noted that all possibilities have been considered in the above
equation. In the first line of Eq. (24), the photon interacts perturbatively
with one of the light quark propagators, while the remaining three quark
propagators are taken as full. Similarly, in the second line of Eq. (24), the
photon interacts perturbatively with one of the heavy quark propagators, while
the other propagators are taken as full, and so on.

Nonperturbative contributions, the photon is radiated at long distances, can
be computed by replacing one of the light quark propagators in Eq. (19) by

$\displaystyle
S_{\alpha\beta}^{ab}\rightarrow-\frac{1}{4}(\bar{q}^{a}\Gamma_{i}q^{b})(\Gamma_{i})_{\alpha\beta},$
(25)

where
$\Gamma_{i}=I,\gamma_{5},\gamma_{\mu},i\gamma_{5}\gamma_{\mu},\sigma_{\mu\nu}/2$
and the remaining three propagators are considered as full quark propagators
including the perturbative as well as the nonperturbative contributions. In
the second case, the correlator takes the form,

$\displaystyle\Pi_{\mu\nu-Nonpert}^{\mathrm{QCD}-Mol}(p,q)$ $\displaystyle=$
$\displaystyle-\frac{i}{4}\int d^{4}xe^{ip\cdot x}\langle 0\mid\
\Bigg{\\{}\mathrm{Tr}\big{[}\gamma_{\mu}\Gamma_{i}\gamma_{\nu}S_{b}^{b^{\prime}b}(-x)\big{]}\Big{(}\bar{q}^{b}(x)\Gamma_{i}q^{b^{\prime}}(0)\Big{)}\,\mathrm{Tr}\big{[}\gamma_{5}{S}_{c}^{aa^{\prime}}(x)\gamma_{5}S_{q}^{a^{\prime}a}(-x)\big{]}$
(26)
$\displaystyle+\mathrm{Tr}\big{[}\gamma_{\mu}{S}_{q}^{bb^{\prime}}(x)\gamma_{\nu}S_{b}^{b^{\prime}b}(-x)\big{]}\mathrm{Tr}\big{[}\gamma_{5}{S}_{c}^{aa^{\prime}}(x)\gamma_{5}\Gamma_{i}\big{]}\Big{(}\bar{q}^{a}(x)\Gamma_{i}q^{a^{\prime}}(0)\Big{)}\Bigg{\\}}\mid
0\rangle_{\gamma},$

By replacing one of the light quark propagators and using the expression
$\bar{q}^{a}(x)\Gamma_{i}q^{a^{\prime}}(0)\rightarrow\frac{1}{3}\delta^{aa^{\prime}}\bar{q}(x)\Gamma_{i}q(0)$,
the Eq. (26) takes the form

$\displaystyle\Pi_{\mu\nu-Nonpert}^{\mathrm{QCD}-Mol}(p,q)$ $\displaystyle=$
$\displaystyle-i\int d^{4}xe^{ip\cdot
x}\Bigg{\\{}\mathrm{Tr}\big{[}\gamma_{\mu}\Gamma_{i}\gamma_{\nu}S_{b}^{b^{\prime}b}(-x)\big{]}\mathrm{Tr}\big{[}\gamma_{5}{S}_{c}^{aa^{\prime}}(x)\gamma_{5}S_{q}^{a^{\prime}a}(-x)\big{]}\delta^{bb^{\prime}}$
(27)
$\displaystyle+\mathrm{Tr}\big{[}\gamma_{\mu}{S}_{q}^{bb^{\prime}}(x)\gamma_{\nu}S_{b}^{b^{\prime}b}(-x)\big{]}\mathrm{Tr}\big{[}\gamma_{5}{S}_{c}^{aa^{\prime}}(x)\gamma_{5}\Gamma_{i}\big{]}\delta^{a^{\prime}a}\Bigg{\\}}\frac{1}{12}\langle\gamma(q)|\bar{q}(x)\Gamma_{i}q(0)|0\rangle.$

In addition to the above computations, when a light quark interacts with a
photon nonperturbatively, a gluon can also be released from one of the
remaining three quark propagators. The expression obtained after performing
these calculations is as follows:

$\displaystyle\Pi_{\mu\nu-Nonpert}^{\mathrm{QCD}-Mol}(p,q)$ $\displaystyle=$
$\displaystyle-i\int d^{4}xe^{ip\cdot
x}\Bigg{\\{}\mathrm{Tr}\big{[}\gamma_{\mu}\Gamma_{i}\gamma_{\nu}S_{b}^{b^{\prime}b}(-x)\big{]}\mathrm{Tr}\big{[}\gamma_{5}{S}_{c}^{aa^{\prime}}(x)\gamma_{5}S_{q}^{a^{\prime}a}(-x)\big{]}\Big{[}\Big{(}\delta^{bb^{\prime}}\delta^{b^{\prime}b}-\frac{1}{3}\delta^{bb^{\prime}}\delta^{b^{\prime}b}\Big{)}$
(28)
$\displaystyle+\Big{(}\delta^{ba}\delta^{b^{\prime}a^{\prime}}-\frac{1}{3}\delta^{bb^{\prime}}\delta^{aa^{\prime}}\Big{)}+\Big{(}\delta^{ba^{\prime}}\delta^{b^{\prime}a}-\frac{1}{3}\delta^{bb^{\prime}}\delta^{a^{\prime}a}\Big{)}\Big{]}$
$\displaystyle+\mathrm{Tr}\big{[}\gamma_{\mu}{S}_{q}^{bb^{\prime}}(x)\gamma_{\nu}S_{b}^{b^{\prime}b}(-x)\big{]}\mathrm{Tr}\big{[}\gamma_{5}{S}_{c}^{aa^{\prime}}(x)\gamma_{5}\Gamma_{i}\big{]}\Big{[}\Big{(}\delta^{ba^{\prime}}\delta^{b^{\prime}a}-\frac{1}{3}\delta^{bb^{\prime}}\delta^{a^{\prime}a}\Big{)}$
$\displaystyle+\Big{(}\delta^{b^{\prime}a^{\prime}}\delta^{ba}-\frac{1}{3}\delta^{b^{\prime}b}\delta^{a^{\prime}a}\Big{)}+\Big{(}\delta^{aa^{\prime}}\delta^{a^{\prime}a}-\frac{1}{3}\delta^{aa^{\prime}}\delta^{a^{\prime}a}\Big{)}\Bigg{\\}}$
$\displaystyle\times\frac{1}{32}\langle\gamma(q)|\bar{q}(x)\Gamma_{i}G_{\mu\nu}(vx)q(0)|0\rangle,$

where we used

$\displaystyle\bar{q}^{a}(x)\Gamma_{i}G_{\mu\nu}^{bb^{\prime}}(vx)q^{a^{\prime}}(0)\rightarrow\frac{1}{8}\Big{(}\delta^{ab}\delta^{a^{\prime}b^{\prime}}-\frac{1}{3}\delta^{aa^{\prime}}\delta^{bb^{\prime}}\Big{)}\bar{q}(x)\Gamma_{i}G_{\mu\nu}(vx)q(0).$
(29)

We observe that matrix elements of the form
$\langle\gamma(q)|\bar{q}(x)\Gamma_{i}q(0)|0\rangle$ and
$\langle\gamma(q)\left|\bar{q}(x)\Gamma_{i}G_{\mu\nu}q(0)\right|0\rangle$ are
required. These matrix elements are parameterized concerning the photon
distribution amplitudes (DAs) and explicit expressions of these DAs are given
in Ref. [61]. In addition to these matrix elements, non-local operators such
as four quarks ($\bar{q}q\bar{q}q$) and two gluons ($\bar{q}GGq$) can also be
seen in the calculations. However it is known that the effects of such
nonlocal operators are small [66, 67], and hence we will neglect them. Using
Eqs. (24), (27) and (28), the calculations of the QCD side of the correlator
of the analysis are obtained.

## Appendix B: Explicit forms of the $\Delta_{i}^{QCD}(M^{2},s_{0})$ functions

In this appendix, we give the explicit expression for the
$\Delta_{1}^{QCD}(M^{2},s_{0})$ and $\Delta_{2}^{QCD}(M^{2},s_{0})$ functions:

$\displaystyle\Delta_{1}^{QCD}(M^{2},s_{0})$
$\displaystyle=-\frac{e_{b}}{15728640\pi^{5}}\Bigg{[}1440m_{b}P_{2}\pi^{2}\Big{(}I[0,4,1,0]-3I[0,4,1,1]+3I[0,4,1,2]-I[0,4,1,3]-3(I[0,4,2,0]$
$\displaystyle-2I[0,4,2,1]+I[0,4,2,2]-I[0,4,3,0]+I[0,4,3,1])-I[0,4,4,0]\Big{)}+480m_{0}^{2}m_{c}P_{2}\pi^{2}\Big{(}I[0,3,2,0]$
$\displaystyle-2I[0,3,2,1]+I[0,3,2,2]-2I[0,3,3,0]+2I[0,3,3,1]+I[0,3,4,0]+6I[1,2,2,1]-3I[1,2,2,2]$
$\displaystyle-6I[1,2,3,1]\Big{)}+10(-P_{1}+48m_{c}P_{2}\pi^{2})\Big{(}I[0,4,2,0]-3I[0,4,2,1]+3I[0,4,2,2]-I[0,4,2,3]$
$\displaystyle-2I[0,4,3,0]+4I[0,4,3,1]-2I[0,4,3,2]+I[0,4,4,0]-I[0,4,4,1]+4I[1,3,2,1]-8I[1,3,2,2]$
$\displaystyle+4I[1,3,2,3]-8I[1,3,3,1]+8I[1,3,3,2]+4I[1,3,4,1]\Big{)}-9\Big{(}2I[0,6,2,1]-7I[0,6,2,2]$
$\displaystyle-5I[0,6,2,4]+I[0,6,2,5]-6I[0,6,3,1]+15I[0,6,3,2]-12I[0,6,3,3]+3I[0,6,3,4]+6I[0,6,4,1]$
$\displaystyle-9I[0,6,4,2]+3I[0,6,4,3]-2I[0,6,5,1]+I[0,6,5,2]+6I[1,5,2,2]-18I[1,5,2,3]+18I[1,5,2,4]$
$\displaystyle-I[1,5,2,5]-18I[1,5,3,2]-18I[1,5,3,3]+18I[1,5,3,4]-18I[1,5,4,2]+18I[1,5,4,3]$
$\displaystyle-I[1,5,5,2]\Big{)}\Bigg{]}$
$\displaystyle+\frac{5e_{c}P_{1}}{56623104\pi^{5}}\Bigg{[}12I[0,4,2,1]-4I[0,4,2,0]-12I[0,4,2,2]+4I[0,4,2,3]+5I[0,4,3,0]-10I[0,4,3,1]$
$\displaystyle+5I[0,4,3,2]+2I[0,4,4,0]-2I[0,4,4,1]-3I[0,4,5,0]-32m_{b}\Bigg{(}4P_{2}\pi^{2}\Big{(}I[0,2,1,0]-2I[0,2,1,1]$
$\displaystyle+I[0,2,1,2]-2I[0,2,2,0]+2I[0,2,2,1]+I[0,2,3,0]-2I[1,1,1,0]-4I[1,1,1,1]+2I[1,1,1,2]$
$\displaystyle-4I[1,1,2,0]+4I[1,1,2,1]+2I[1,1,3,0]\Big{)}\Bigg{)}-16I[1,3,2,1]-32I[1,3,2,2]+16I[1,3,2,3]$
$\displaystyle-8I[1,3,3,1]+20I[1,3,3,2]-8I[1,3,4,1]\Bigg{]}$
$\displaystyle+\frac{e_{q}}{5242880\pi^{5}}\Bigg{[}5P_{1}\Big{(}I[0,4,3,0]-2I[0,4,3,1]+I[0,4,3,2]-2I[0,4,4,0]+2I[0,4,4,1]+I[0,4,5,0]$
$\displaystyle+8I[1,3,3,1]-4I[1,3,3,2]-8I[1,3,4,1]\Big{)}-240m_{c}P_{2}\pi^{2}\Big{(}I[0,4,3,0]-2I[0,4,3,1]+I[0,4,3,2]$
$\displaystyle-2I[0,4,4,0]+2I[0,4,4,1]+I[0,4,5,0]+8I[1,3,3,1]-4I[1,3,3,2]-8I[1,3,4,1]\Big{)}-9\Big{(}I[0,6,3,0]$
$\displaystyle-4I[0,6,3,1]+6I[0,6,3,2]-4I[0,6,3,3]+I[0,6,3,4]-3I[0,6,4,0]+9I[0,6,4,1]-9I[0,6,4,2]$
$\displaystyle+3I[0,6,4,3]+3I[0,6,5,0]-6I[0,6,5,1]+3I[0,6,5,2]-I[0,6,6,0]+I[0,6,6,1]+6I[1,5,3,1]$
$\displaystyle-18I[1,5,3,2]+18I[1,5,3,3]-6I[1,5,3,4]-18I[1,5,4,1]+36I[1,5,4,2]-18I[1,5,4,3]$
$\displaystyle+18I[1,5,5,1]-18I[1,5,5,2]-6I[1,5,6,1]\Big{)}\Bigg{]}$
$\displaystyle+m_{0}^{2}\Big{(}17e_{q}I_{4}[\mathcal{S}]\,I[0,5,3,0]+2e_{q}\,I_{4}[\mathcal{T}_{1}]I[0,5,4,0]+476\,e_{q}\,f_{3\gamma}\,\pi^{2}\,I_{2}[\mathcal{V}]\,I[0,4,4,0]\Big{)}+\chi\,m_{0}^{2}\Big{(}238\,e_{q}$
$\displaystyle\times
I_{1}[\mathcal{V}]\,I[0,6,4,0]+13e_{q}I_{3}[\mathcal{S}]\,I[0,6,3,0]+8e_{q}\,I_{3}[\mathcal{T}_{1}]I[0,6,2,0]\Big{)},$
(30) $\displaystyle\Delta^{QCD}_{2}$
$\displaystyle=\frac{e_{b}}{11796480\pi^{5}}\Bigg{[}960\,m_{0}^{2}\,m_{c}\,P_{2}\pi^{2}\Big{(}I[0,3,2,0]-2I[0,3,2,1]+I[0,3,2,2]-2I[0,3,3,0]+2I[0,3,3,1]$
$\displaystyle+I[0,3,4,0]+6I[1,2,2,1]-3I[1,2,2,2]-6I[1,2,3,1]\Big{)}+5\Bigg{(}576m_{b}P_{2}\pi^{2}\Big{(}I[0,4,1,0]-3I[0,4,1,1]$
$\displaystyle+3I[0,4,1,2]-I[0,4,1,3]-3(I[0,4,2,0]-2I[0,4,2,1]+I[0,4,2,2]-I[0,4,3,0]+I[0,4,3,1])-I[0,4,4,0]\Big{)}$
$\displaystyle+(P_{1}-192m_{c}P_{2}\pi^{2})\Big{(}I[0,4,2,0]-3I[0,4,2,1]+3I[0,4,2,2]-I[0,4,2,3]-2I[0,4,3,0]+4I[0,4,3,1]$
$\displaystyle-2I[0,4,3,2]+I[0,4,4,0]-I[0,4,4,1]+4\big{(}I[1,3,2,1]-2I[1,3,2,2]+I[1,3,2,3]-2I[1,3,3,1]+2I[1,3,3,2]$
$\displaystyle+I[1,3,4,1]\big{)}\Big{)}\Bigg{)}-18\Big{(}2I[0,6,2,1]-7I[0,6,2,2]+9I[0,6,2,3]-5I[0,6,2,4]+I[0,6,2,5]-6I[0,6,3,1]$
$\displaystyle+15I[0,6,3,2]-12I[0,6,3,3]+3I[0,6,3,4]+6I[0,6,4,1]-9I[0,6,4,2]+3I[0,6,4,3]-2I[0,6,5,1]$
$\displaystyle+I[0,6,5,2]+6\big{(}I[1,5,2,2]-3I[1,5,2,3]+3I[1,5,2,4]-I[1,5,2,5]-3(I[1,5,3,2]-2I[1,5,3,3]$
$\displaystyle+I[1,5,3,4]-I[1,5,4,2]+I[1,5,4,3])-I[1,5,5,2]\big{)}\Big{)}\Bigg{]}$
$\displaystyle+\frac{5e_{c}P_{1}}{56623104\pi^{5}}\Bigg{[}12I[0,4,2,1]-4I[0,4,2,0]-12I[0,4,2,2]+4I[0,4,2,3]+5I[0,4,3,0]-10I[0,4,3,1]$
$\displaystyle+5I[0,4,3,2]+2I[0,4,4,0]-2I[0,4,4,1]-3I[0,4,5,0]-64m_{b}\Bigg{(}P_{2}\pi^{2}\Big{(}I[0,2,1,0]-2I[0,2,1,1]$
$\displaystyle+I[0,2,1,2]-2I[0,2,2,0]+2I[0,2,2,1]+I[0,2,3,0]-2I[1,1,1,0]-4I[1,1,1,1]+2I[1,1,1,2]-4I[1,1,2,0]$
$\displaystyle+4I[1,1,2,1]+2I[1,1,3,0]\Big{)}\Bigg{)}-16I[1,3,2,1]-16I[1,3,2,2]+32I[1,3,2,3]-8I[1,3,3,1]+24I[1,3,3,2]$
$\displaystyle-16I[1,3,4,1]\Bigg{]}$
$\displaystyle+\frac{e_{q}}{141557760\pi^{5}}\Bigg{[}-800m_{b}P_{1}\Bigg{(}4P_{2}\
\pi^{2}\Big{(}I[0,2,1,0]-2I[0,2,1,1]+I[0,2,1,2]-2I[0,2,2,0]+2I[0,2,2,1]$
$\displaystyle+I[0,2,3,0]-2\big{(}I[1,1,1,0]-2I[1,1,1,1]+I[1,1,1,2]-2I[1,1,2,0]+2I[1,1,2,1]+I[1,1,3,0]\big{)}\Big{)}\Bigg{)}$
$\displaystyle-5P1\Big{(}20I[0,4,2,0]-60I[0,4,2,1]+60I[0,4,2,2]-20I[0,4,2,3]-7I[0,4,3,0]+14I[0,4,3,1]-7I[0,4,3,2]$
$\displaystyle-46I[0,4,4,0]+46I[0,4,4,1]+33I[0,4,5,0]+4\big{(}20I[1,3,2,1]-40I[1,3,2,2]+20I[1,3,2,3]+26I[1,3,3,1]$
$\displaystyle+7I[1,3,3,2]-46I[1,3,4,1]\big{)}\Big{)}-216\Bigg{(}4m_{c}\Big{(}20P_{2}\pi^{2}\big{(}I[0,4,3,0]-2I[0,4,3,1]+I[0,4,3,2]-2I[0,4,4,0]$
$\displaystyle+2I[0,4,4,1]+I[0,4,5,0]+8I[1,3,3,1]-4I[1,3,3,2]-8I[1,3,4,1]\big{)}\Big{)}+3\Big{(}I[0,6,3,0]-4I[0,6,3,1]$
$\displaystyle+6I[0,6,3,2]+I[0,6,3,4]-3I[0,6,4,0]+9I[0,6,4,1]-9I[0,6,4,2]+3I[0,6,4,3]+3I[0,6,5,0]$
$\displaystyle-6I[0,6,5,1]+3I[0,6,5,2]-I[0,6,6,0]+I[0,6,6,1]+6\big{(}I[1,5,3,1]-3I[1,5,3,2]+3I[1,5,3,3]$
$\displaystyle-I[1,5,3,4]-3(I[1,5,4,1]-2I[1,5,4,2]+I[1,5,4,3]-I[1,5,5,1]+I[1,5,5,2])-I[1,5,6,1]\Big{)}\Big{)}\Bigg{)}\Bigg{]}$
$\displaystyle+32m_{0}^{2}\Bigg{[}\Big{(}26e_{q}I_{4}[\mathcal{S}]\,I[0,5,3,0]+12e_{q}\,I_{4}[\mathcal{T}_{1}]I[0,5,4,0]+238\,e_{q}\,f_{3\gamma}\,\pi^{2}\,I_{2}[\mathcal{V}]\,I[0,4,4,0]\Big{)}+\chi\,\Big{(}476\,e_{q}\,I_{1}[\mathcal{V}]$
$\displaystyle\times\,I[0,6,5,0]+26e_{q}I_{3}[\mathcal{S}]\,I[0,6,4,0]+12e_{q}\,I_{3}[\mathcal{T}_{1}]I[0,6,3,0]\Big{)}\Bigg{]},$
(31)

where $P_{1}=\langle g_{s}^{2}G^{2}\rangle$, $P_{2}=\langle\bar{q}q\rangle$
are gluon and u/d-quark condensates, respectively. We should also mention
that, in Eqs. (30) and (31), for the sake of brevity we have only presented
the terms that give substantial contributions to the numerical values of the
observables under investigation and ignored to present many higher dimensional
operators though they have been considered in the numerical calculations.

The functions $I[n,m,l,k]$, $I_{1}[\mathcal{F}]$, $I_{2}[\mathcal{F}]$,
$I_{3}[\mathcal{F}]$ and $I_{4}[\mathcal{F}]$ are defined as:

$\displaystyle I[n,m,l,k]$
$\displaystyle=\int_{(m_{c}+m_{b})^{2}}^{s_{0}}ds\int_{0}^{1}dt\int_{0}^{1}dw~{}e^{-s/M^{2}}~{}s^{n}\,(s-(m_{c}+m_{b})^{2})^{m}\,t^{l}\,w^{k},$
$\displaystyle I_{1}[\mathcal{F}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{F}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta^{\prime}(\alpha_{q}+\bar{v}\alpha_{g}-u_{0}),$
$\displaystyle I_{2}[\mathcal{F}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{F}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta^{\prime}(\alpha_{\bar{q}}+v\alpha_{g}-u_{0}),$
$\displaystyle I_{3}[\mathcal{F}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{F}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta(\alpha_{q}+\bar{v}\alpha_{g}-u_{0}),$
$\displaystyle I_{4}[\mathcal{F}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{F}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta(\alpha_{\bar{q}}+v\alpha_{g}-u_{0}),$

where $\mathcal{F}$ stands for the corresponding photon DAs.

## References

  * [1] S. K. Choi, et al., Observation of a narrow charmonium-like state in exclusive $B^{\pm}\to K^{\pm}\pi^{+}\pi^{-}J/\psi$ decays, Phys. Rev. Lett. 91 (2003) 262001\.  arXiv:hep-ex/0309032, doi:10.1103/PhysRevLett.91.262001.
  * [2] A. Esposito, A. L. Guerrieri, F. Piccinini, A. Pilloni, A. D. Polosa, Four-Quark Hadrons: an Updated Review, Int. J. Mod. Phys. A 30 (2015) 1530002\.  arXiv:1411.5997, doi:10.1142/S0217751X15300021.
  * [3] A. Esposito, A. Pilloni, A. D. Polosa, Multiquark Resonances, Phys. Rept. 668 (2017) 1–97.  arXiv:1611.07920, doi:10.1016/j.physrep.2016.11.002.
  * [4] S. L. Olsen, T. Skwarnicki, D. Zieminska, Nonstandard heavy mesons and baryons: Experimental evidence, Rev. Mod. Phys. 90 (1) (2018) 015003.  arXiv:1708.04012, doi:10.1103/RevModPhys.90.015003.
  * [5] R. F. Lebed, R. E. Mitchell, E. S. Swanson, Heavy-Quark QCD Exotica, Prog. Part. Nucl. Phys. 93 (2017) 143–194.  arXiv:1610.04528, doi:10.1016/j.ppnp.2016.11.003.
  * [6] M. Nielsen, F. S. Navarra, S. H. Lee, New Charmonium States in QCD Sum Rules: A Concise Review, Phys. Rept. 497 (2010) 41–83.  arXiv:0911.1958, doi:10.1016/j.physrep.2010.07.005.
  * [7] N. Brambilla, S. Eidelman, C. Hanhart, A. Nefediev, C.-P. Shen, C. E. Thomas, A. Vairo, C.-Z. Yuan, The $XYZ$ states: experimental and theoretical status and perspectives, Phys. Rept. 873 (2020) 1–154.  arXiv:1907.07583, doi:10.1016/j.physrep.2020.05.001.
  * [8] S. Agaev, K. Azizi, H. Sundu, Four-quark exotic mesons, Turk. J. Phys. 44 (2) (2020) 95–173.  arXiv:2004.12079, doi:10.3906/fiz-2003-15.
  * [9] H.-X. Chen, W. Chen, X. Liu, S.-L. Zhu, The hidden-charm pentaquark and tetraquark states, Phys. Rept. 639 (2016) 1–121.  arXiv:1601.02092, doi:10.1016/j.physrep.2016.05.004.
  * [10] A. Ali, J. S. Lange, S. Stone, Exotics: Heavy Pentaquarks and Tetraquarks, Prog. Part. Nucl. Phys. 97 (2017) 123–198.  arXiv:1706.00610, doi:10.1016/j.ppnp.2017.08.003.
  * [11] F.-K. Guo, C. Hanhart, U.-G. Meißner, Q. Wang, Q. Zhao, B.-S. Zou, Hadronic molecules, Rev. Mod. Phys. 90 (1) (2018) 015004, [Erratum: Rev.Mod.Phys. 94, 029901 (2022)].  arXiv:1705.00141, doi:10.1103/RevModPhys.90.015004.
  * [12] Y.-R. Liu, H.-X. Chen, W. Chen, X. Liu, S.-L. Zhu, Pentaquark and Tetraquark states, Prog. Part. Nucl. Phys. 107 (2019) 237–320.  arXiv:1903.11976, doi:10.1016/j.ppnp.2019.04.003.
  * [13] G. Yang, J. Ping, J. Segovia, Tetra- and penta-quark structures in the constituent quark model, Symmetry 12 (11) (2020) 1869.  arXiv:2009.00238, doi:10.3390/sym12111869.
  * [14] X.-K. Dong, F.-K. Guo, B.-S. Zou, A survey of heavy-antiheavy hadronic molecules, Progr. Phys. 41 (2021) 65–93.  arXiv:2101.01021, doi:10.13725/j.cnki.pip.2021.02.001.
  * [15] X.-K. Dong, F.-K. Guo, B.-S. Zou, A survey of heavy–heavy hadronic molecules, Commun. Theor. Phys. 73 (12) (2021) 125201.  arXiv:2108.02673, doi:10.1088/1572-9494/ac27a2.
  * [16] L. Meng, B. Wang, G.-J. Wang, S.-L. Zhu, Chiral perturbation theory for heavy hadrons and chiral effective field theory for heavy hadronic molecules (4 2022).  arXiv:2204.08716.
  * [17] H.-X. Chen, W. Chen, X. Liu, Y.-R. Liu, S.-L. Zhu, An updated review of the new hadron states, Rept. Prog. Phys. 86 (2) (2023) 026201.  arXiv:2204.02649, doi:10.1088/1361-6633/aca3b6.
  * [18] J.-R. Zhang, M.-Q. Huang, Q anti-qanti-Q-(prime)q molecular states, Phys. Rev. D 80 (2009) 056004.  arXiv:0906.0090, doi:10.1103/PhysRevD.80.056004.
  * [19] J.-R. Zhang, M.-Q. Huang, Q anti-santi-Q-(prime)s molecular states in QCD sum rules, Commun. Theor. Phys. 54 (2010) 1075–1090.  arXiv:0905.4672, doi:10.1088/0253-6102/54/6/22.
  * [20] Z.-F. Sun, X. Liu, M. Nielsen, S.-L. Zhu, Hadronic molecules with both open charm and bottom, Phys. Rev. D 85 (2012) 094008.  arXiv:1203.1090, doi:10.1103/PhysRevD.85.094008.
  * [21] R. M. Albuquerque, X. Liu, M. Nielsen, Exotic $B_{c}$-like molecules in QCD Sum Rules, Phys. Lett. B 718 (2012) 492–498.  arXiv:1203.6569, doi:10.1016/j.physletb.2012.10.063.
  * [22] W. Chen, T. G. Steele, S.-L. Zhu, Exotic open-flavor $bc\bar{q}\bar{q}$, $bc\bar{s}\bar{s}$ and $qc\bar{q}\bar{b}$, $sc\bar{s}\bar{b}$ tetraquark states, Phys. Rev. D 89 (5) (2014) 054037.  arXiv:1310.8337, doi:10.1103/PhysRevD.89.054037.
  * [23] S. S. Agaev, K. Azizi, H. Sundu, Open charm-bottom scalar tetraquarks and their strong decays, Phys. Rev. D 95 (3) (2017) 034008.  arXiv:1611.00293, doi:10.1103/PhysRevD.95.034008.
  * [24] S. S. Agaev, K. Azizi, H. Sundu, Open charm-bottom axial-vector tetraquarks and their properties, Eur. Phys. J. C 77 (5) (2017) 321.  arXiv:1702.08230, doi:10.1140/epjc/s10052-017-4892-8.
  * [25] Q.-N. Wang, W. Chen, Fully open-flavor tetraquark states $bc\bar{q}\bar{s}$ and $sc\bar{q}\bar{b}$ with $J^{P}=0^{+},1^{+}$, Eur. Phys. J. C 80 (5) (2020) 389.  arXiv:2002.04243, doi:10.1140/epjc/s10052-020-7938-2.
  * [26] Z.-G. Wang, Analysis of the axialvector $B_{c}$-like tetraquark states with the QCD sum rules, EPL 128 (1) (2019) 11001.  arXiv:1907.10921, doi:10.1209/0295-5075/128/11001.
  * [27] J. Wu, X. Liu, Y.-R. Liu, S.-L. Zhu, Systematic studies of charmonium-, bottomonium-, and $B_{c}$-like tetraquark states, Phys. Rev. D 99 (1) (2019) 014037\.  arXiv:1810.06886, doi:10.1103/PhysRevD.99.014037.
  * [28] P. G. Ortega, J. Segovia, D. R. Entem, F. Fernandez, Spectroscopy of $\mathbf{B_{c}}$ mesons and the possibility of finding exotic $\mathbf{B_{c}}$-like structures, Eur. Phys. J. C 80 (3) (2020) 223.  arXiv:2001.08093, doi:10.1140/epjc/s10052-020-7764-6.
  * [29] V. L. Chernyak, I. R. Zhitnitsky, B meson exclusive decays into baryons, Nucl. Phys. B 345 (1990) 137–172.  doi:10.1016/0550-3213(90)90612-H.
  * [30] V. M. Braun, I. E. Filyanov, QCD Sum Rules in Exclusive Kinematics and Pion Wave Function, Z. Phys. C 44 (1989) 157.  doi:10.1007/BF01548594.
  * [31] I. I. Balitsky, V. M. Braun, A. V. Kolesnichenko, Radiative Decay Sigma+ —$>$ p gamma in Quantum Chromodynamics, Nucl. Phys. B 312 (1989) 509–550.  doi:10.1016/0550-3213(89)90570-1.
  * [32] U. Özdem, Magnetic moments of the vector hidden-charm tetraquark states, Phys. Rev. D 105 (11) (2022) 114030.  arXiv:2206.05196, doi:10.1103/PhysRevD.105.114030.
  * [33] Q. Xin, Z.-G. Wang, X.-S. Yang, Analysis of the X(3960) and related tetraquark molecular states via the QCD sum rules, AAPPS Bull. 32 (1) (2022) 37.  arXiv:2207.09910, doi:10.1007/s43673-022-00070-3.
  * [34] S. S. Agaev, K. Azizi, H. Sundu, Resonance X(4630), Phys. Rev. D 106 (1) (2022) 014025.  arXiv:2203.02542, doi:10.1103/PhysRevD.106.014025.
  * [35] Z.-G. Wang, Analysis of the X(6600), X(6900), X(7300) and related tetraquark states with the QCD sum rules, Nucl. Phys. B 985 (2022) 115983.  arXiv:2207.08059, doi:10.1016/j.nuclphysb.2022.115983.
  * [36] U. Özdem, A. K. Yıldırım, Magnetic dipole moments of the Zc(4020)+, Zc(4200)+, Zcs(4000)+ and Zcs(4220)+ states in light-cone QCD, Phys. Rev. D 104 (5) (2021) 054017.  arXiv:2104.13074, doi:10.1103/PhysRevD.104.054017.
  * [37] S. S. Agaev, K. Azizi, H. Sundu, Strong $Z_{c}^{+}(3900)\rightarrow J/\psi\pi^{+};\eta_{c}\rho^{+}$ decays in QCD, Phys. Rev. D 93 (7) (2016) 074002\.  arXiv:1601.03847, doi:10.1103/PhysRevD.93.074002.
  * [38] W. Chen, T. G. Steele, H.-X. Chen, S.-L. Zhu, Mass spectra of Zc and Zb exotic states as hadron molecules, Phys. Rev. D 92 (5) (2015) 054002.  arXiv:1505.05619, doi:10.1103/PhysRevD.92.054002.
  * [39] Y.-J. Xu, Y.-L. Liu, C.-Y. Cui, M.-Q. Huang, $\bar{D}^{(*)}_{s}D^{(*)}$ molecular state with $J^{P}$= $1^{+}$, Phys. Rev. D 104 (9) (2021) 094028.  arXiv:2011.14313, doi:10.1103/PhysRevD.104.094028.
  * [40] U. Özdem, K. Azizi, Magnetic dipole moment of the $Z_{cs}(3985)$ state: diquark–antidiquark and molecular pictures, Eur. Phys. J. Plus 136 (9) (2021) 968.  arXiv:2102.09231, doi:10.1140/epjp/s13360-021-01977-w.
  * [41] Y.-J. Xu, Y.-L. Liu, M.-Q. Huang, The magnetic moment of $Z_{c}(3900)$ as an axial-vector molecular state, Eur. Phys. J. C 80 (10) (2020) 953.  arXiv:2007.15214, doi:10.1140/epjc/s10052-020-08515-5.
  * [42] Z.-G. Wang, Analysis of Zcs (3985) as the axialvector tetraquark state, Chin. Phys. C 45 (7) (2021) 073107.  arXiv:2011.10959, doi:10.1088/1674-1137/abfa83.
  * [43] U. Ozdem, K. Azizi, Magnetic dipole moment of $Z_{b}(10610)$ in light-cone QCD, Phys. Rev. D 97 (1) (2018) 014010.  arXiv:1709.09714, doi:10.1103/PhysRevD.97.014010.
  * [44] Z.-G. Wang, Vector tetraquark state candidates: $Y(4260/4220)$, $Y(4360/4320)$, $Y(4390)$ and $Y(4660/4630)$, Eur. Phys. J. C 78 (6) (2018) 518\.  arXiv:1803.05749, doi:10.1140/epjc/s10052-018-5996-5.
  * [45] G. L. Yu, Z. G. Wang, Z. Y. Li, The analysis of the charmonium-like states $X^{*}(3860)$,$X(3872)$, $X(3915)$, $X(3930)$ and $X(3940)$ according to its strong decay behaviors, Chin. Phys. C 42 (4) (2018) 043107.  arXiv:1704.06763, doi:10.1088/1674-1137/42/4/043107.
  * [46] U. Ozdem, K. Azizi, Magnetic and quadrupole moments of the $Z_{c}(3900)$, Phys. Rev. D 96 (7) (2017) 074030.  arXiv:1707.09612, doi:10.1103/PhysRevD.96.074030.
  * [47] H. Sundu, S. S. Agaev, K. Azizi, New charged resonance $Z_{c}^{-}(4100)$: the spectroscopic parameters and width, Eur. Phys. J. C 79 (3) (2019) 215.  arXiv:1812.10094, doi:10.1140/epjc/s10052-019-6737-0.
  * [48] S. S. Agaev, K. Azizi, H. Sundu, Spectroscopic parameters and decays of the resonance $Z_{b}(10610)$, Eur. Phys. J. C 77 (12) (2017) 836.  arXiv:1709.03148, doi:10.1140/epjc/s10052-017-5421-5.
  * [49] Z.-G. Wang, Vector hidden-bottom tetraquark candidate: $Y(10750)$, Chin. Phys. C 43 (12) (2019) 123102.  arXiv:1905.06610, doi:10.1088/1674-1137/43/12/123102.
  * [50] Z.-G. Wang, Axialvector tetraquark candidates for $Z_{c}(3900)$, $Z_{c}(4020)$, $Z_{c}(4430)$, $Z_{c}(4600)$, Chin. Phys. C 44 (6) (2020) 063105.  arXiv:1901.10741, doi:10.1088/1674-1137/44/6/063105.
  * [51] S. S. Agaev, K. Azizi, H. Sundu, Treating $Z_{c}(3900)$ and $Z(4430)$ as the ground-state and first radially excited tetraquarks, Phys. Rev. D 96 (3) (2017) 034026.  arXiv:1706.01216, doi:10.1103/PhysRevD.96.034026.
  * [52] S. S. Agaev, K. Azizi, H. Sundu, Exploring the resonances $X(4140)$ and $X(4274)$ through their decay channels, Phys. Rev. D 95 (11) (2017) 114003.  arXiv:1703.10323, doi:10.1103/PhysRevD.95.114003.
  * [53] Z.-G. Wang, Tetraquark state candidates: $Y(4260)$, $Y(4360)$, $Y(4660)$ and $Z_{c}(4020/4025)$, Eur. Phys. J. C 76 (7) (2016) 387.  arXiv:1601.05541, doi:10.1140/epjc/s10052-016-4238-y.
  * [54] H.-X. Chen, E.-L. Cui, W. Chen, X. Liu, S.-L. Zhu, Understanding the internal structures of the $X(4140)$, $X(4274)$, $X(4500)$ and $X(4700)$, Eur. Phys. J. C 77 (3) (2017) 160.  arXiv:1606.03179, doi:10.1140/epjc/s10052-017-4737-5.
  * [55] W. Chen, T. G. Steele, H.-X. Chen, S.-L. Zhu, $Z_{c}(4200)^{+}$ decay width as a charmonium-like tetraquark state, Eur. Phys. J. C 75 (8) (2015) 358.  arXiv:1501.03863, doi:10.1140/epjc/s10052-015-3578-3.
  * [56] Z.-G. Wang, The magnetic moment of the $Z_{c}(3900)$ as an axialvector tetraquark state with QCD sum rules, Eur. Phys. J. C 78 (4) (2018) 297.  arXiv:1712.05664, doi:10.1140/epjc/s10052-018-5794-0.
  * [57] S. J. Brodsky, J. R. Hiller, Universal properties of the electromagnetic interactions of spin one systems, Phys. Rev. D 46 (1992) 2141–2149.  doi:10.1103/PhysRevD.46.2141.
  * [58] K.-C. Yang, W. Y. P. Hwang, E. M. Henley, L. S. Kisslinger, QCD sum rules and neutron proton mass difference, Phys. Rev. D 47 (1993) 3001–3012.  doi:10.1103/PhysRevD.47.3001.
  * [59] V. M. Belyaev, B. Y. Blok, CHARMED BARYONS IN QUANTUM CHROMODYNAMICS, Z. Phys. C 30 (1986) 151.  doi:10.1007/BF01560689.
  * [60] B. L. Ioffe, QCD at low energies, Prog. Part. Nucl. Phys. 56 (2006) 232–277.  arXiv:hep-ph/0502148, doi:10.1016/j.ppnp.2005.05.001.
  * [61] P. Ball, V. M. Braun, N. Kivel, Photon distribution amplitudes in QCD, Nucl. Phys. B 649 (2003) 263–296.  arXiv:hep-ph/0207307, doi:10.1016/S0550-3213(02)01017-9.
  * [62] J. Rohrwild, Determination of the magnetic susceptibility of the quark condensate using radiative heavy meson decays, JHEP 09 (2007) 073.  arXiv:0708.1405, doi:10.1088/1126-6708/2007/09/073.
  * [63] S. K. Bose, L. P. Singh, Magnetic Moments of Charmed and $B$ Flavored Hadrons in MIT Bag Model, Phys. Rev. D 22 (1980) 773.  doi:10.1103/PhysRevD.22.773.
  * [64] V. Simonis, Improved predictions for magnetic moments and M1 decay widths of heavy hadrons (3 2018).  arXiv:1803.01809.
  * [65] T. A. Lahde, Exchange current operators and electromagnetic dipole transitions in heavy quarkonia, Nucl. Phys. A 714 (2003) 183–212.  arXiv:hep-ph/0208110, doi:10.1016/S0375-9474(02)01362-3.
  * [66] I. I. Balitsky, V. M. Braun, Evolution Equations for QCD String Operators, Nucl. Phys. B 311 (1989) 541–584.  doi:10.1016/0550-3213(89)90168-5.
  * [67] V. M. Braun, I. E. Filyanov, Conformal Invariance and Pion Wave Functions of Nonleading Twist, Z. Phys. C 48 (1990) 239–248.  doi:10.1007/BF01554472.