# Magnetic dipole moments of $B_{(s)}^{(*)}B_{(s)}^{(*)}$ states

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

###### Abstract

We systematically study the magnetic dipole moments of multiquark states. In
this study, the magnetic dipole moments of possible $B^{-}B^{*-}$,
$B^{0}B^{*-}$, $B^{-}B^{*0}$, $B^{0}B^{*0}$, $B_{s}^{0}B^{*-}$,
$B^{-}B_{s}^{*0}$, $B_{s}^{0}B^{*0}$, $B^{0}B_{s}^{*0}$ and
$B^{0}_{s}B_{s}^{*0}$ states are extracted using light-cone sum rules. We
explore magnetic dipole moments of these states as molecular picture with
spin-parity $J^{P}=1^{+}$. The magnetic dipole moments of hadrons include
useful information on the distributions of internal charge and magnetization,
which can be used to understand their geometrical shapes and quark-gluon
organization. The results of the present study along with the spectroscopic
parameters may help the future theoretical and experimental research on the
characteristics of doubly-bottom tetraquark states.

Magnetic dipole moment, doubly-bottom tetraquarks, $T_{bb}$ states, light-cone
sum rules

##  I Introduction

Recently, the LHCb Collaboration made a great breakthrough in the search for
multiquark states and reported a new state ($T_{cc}^{+}$ for short) below the
$D^{0}D^{\ast+}$ mass threshold in the $D^{0}D^{0}\pi^{+}$ invariant mass
spectrum [1, 2]. The fact that this observed state contains two charm quarks
and has an electrical charge makes it a good candidate for an exotic state
with the quark content $cc\bar{u}\bar{d}$. The significance of the
$T^{+}_{cc}$ state is the same as that of $X(3872$); therefore the newly
discovered state provides an important new platform for both experimental and
theoretical hadron physics. In literature, there are numerous theoretical
studies that have been performed to understand the spectroscopic parameters,
magnetic dipole moments, production mechanisms and decay modes of doubly-
charmed tetraquark states within different models [3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 5].

If $T^{+}_{cc}$ is the doubly-charmed tetraquark state, there may also be
doubly-bottom tetraquark states. Even if these doubly-bottom tetraquark states
do not exist, it is important, in our opinion, to explore the reasons why.
Inspired by this, we are well-motivated to search for the possible doubly-
bottom tetraquark states. Therefore, besides the doubly-charmed states, the
properties of doubly-bottom tetraquark states have also been extracted in
different configurations [24, 25, 5, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48]. In Ref. [24], the mass and decay width of the
$T_{cc}$ and $T_{bb}$ states were investigated in the framework of the one-
boson exchange potential model. The researchers predicted that $T_{bb}$ states
are more stable than $T_{cc}$ states. In Ref. [31], the authors studied the
interaction of $T_{bb}$ states by means of vector meson exchange with
Lagrangians from an extension of the local hidden-gauge approach. They
predicted that only $B^{*}B$, ($B_{s}^{*}B-B^{*}B_{s}$), $B^{*}B^{*}$ and
$B_{s}^{*}B^{*}$ states form bound states with the quantum numbers
$J^{P}=1^{+}$. In Ref. [35], the masses of the $QQ\bar{q}\bar{q}$ tetraquark
states were obtained with the help of heavy diquark-antiquark symmetry and the
chromomagnetic interaction model. They predicted that only the
$bb\bar{q}\bar{q}$ and $bb\bar{q}\bar{s}$ states are stable with respect to
strong decays and discussed the constraints on the masses of the these
tetraquark states. In Ref. [36], the authors predicted the masses of doubly-
heavy tetraquark states in the heavy quark limit. They found that only doubly-
bottom tetraquarks with $\bar{u}\bar{d}$, $\bar{s}\bar{u}$ and
$\bar{s}\bar{d}$ are stable with strong decays. In Ref. [37], the bound states
of doubly-heavy tetraquarks were investigated via the non-relativistic quark
model. The authors obtained several stable states, one of which was a strongly
bound $bb\bar{q}\bar{q}$ with isospin and spin-parity $I(J^{P})=0(1^{+})$. In
Ref. [46], the spectroscopic parameters of $T_{bb}$ states were investigated
within QCD sum rules using molecular pictures with quantum numbers
$J^{P}=1^{+}$. Furthermore, in Ref. [47], the authors attempted to extract
possible $bb\bar{u}\bar{d}$ states within lattice QCD and found that one of
these states form a bound state. In addition to spectroscopic parameters, the
semi leptonic and nonleptonic decays of double-bottom tetraquark states were
extracted in the framework of the QCD sum rule method in Ref. [41, 45].

In Refs. [26, 27], we extracted the magnetic dipole moments of doubly-charmed
tetraquark states in the molecular picture using the light-cone sum rule
method. For the case of $T^{+}_{cc}$, the magnetic dipole moment was also
obtained by considering it as a diquark-antidiquark state. We extend our work
to doubly-bottom tetraquark states. In this study, we evaluate the magnetic
dipole moments of doubly-bottom tetraquark states in the molecular framework
using the light-cone sum rule method [49, 50, 51]. The light-cone sum rules
method has been employed in literature to obtain information about the dynamic
and static parameters of conventional and non-conventional hadrons providing
successful predictions that are consistent with experimental ones. The
magnetic dipole moment of hadrons represents an important tool for
understanding their internal structure in terms of quarks and gluons. Thus, it
is important and also interesting to investigate the magnetic dipole moments
of conventional and non-conventional hadrons.

The paper is organized as follows: In Sec. II, light-cone sum rules for the
magnetic dipole moments of doubly-bottom tetraquark states are calculated.
Sec. III is devoted to the numerical calculations of the magnetic dipole
moment sum rules and discussion. Explicit expressions for the magnetic dipole
moments of the doubly-bottom tetraquark states and distribution amplitudes
(DAs) of photons are presented in the appendices.

##  II Formalism

In the light-cone sum rules method, the correlation function is evaluated in
terms of hadrons (hadronic side) and quark-gluon degrees of freedom (QCD
side). Then, by applying the continuum subtraction and double Borel
transformation to eliminate the effects from higher states and the continuum
and enhance the contributions of the ground state, and matching these results,
we can obtain the desired sum rules.

To identify the magnetic dipole moments of doubly-bottom tetraquark states
within the light-cone sum rules, we introduce the following correlation
function:

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

where $\gamma$ indicates the external electromagnetic field and $J_{\mu}(x)$
is the interpolating current of $T_{bb}$ states with the spin-parity
$J^{P}=1^{+}$. Following the molecular configuration for $T_{bb}$ states, we
consider the interpolating current as

$\displaystyle J_{\mu}(x)$
$\displaystyle=\big{[}\bar{q_{1}}^{a}(x)i\gamma_{5}b^{a}(x)][\bar{q}_{2}^{b}(x)\gamma_{\mu}b^{b}(x)],$
(2)

where $q_{1}$ and $q_{2}$ denote the $u$, $d$ and $s$-quarks.

To acquire the hadronic side of the correlation function we insert a complete
set of hadronic states with quantum number $J^{P}=1^{+}$ into the correlation
function and isolate the contributions of the ground state of doubly-bottom
tetraquark states. As a result of these calculations, we obtain the following:

$\displaystyle\Pi_{\mu\nu}^{Had}(p,q)$ $\displaystyle={\frac{\langle 0\mid
J_{\mu}(x)\mid T_{bb}(p,\varepsilon^{\theta})\rangle}{p^{2}-m_{T_{bb}}^{2}}}$
$\displaystyle\langle T_{bb}(p,\varepsilon^{\theta})\mid
T_{bb}(p+q,\varepsilon^{\delta})\rangle_{\gamma}$ $\displaystyle\frac{\langle
T_{bb}(p+q,\varepsilon^{\delta})\mid{J^{\dagger}}_{\nu}(0)\mid
0\rangle}{(p+q)^{2}-m_{T_{bb}}^{2}}$ $\displaystyle+\mbox{higher states}\,.$
(3)

The amplitude $\langle 0\mid J_{\mu}(x)\mid
T_{bb}(p,\varepsilon^{\theta})\rangle$ can be parameterized in terms of the
residue $\lambda_{T_{bb}}$ and polarization vector
$\varepsilon_{\mu}^{\theta}$ of $T_{bb}$ states as

$\displaystyle\langle 0\mid J_{\mu}(x)\mid
T_{bb}(p,\varepsilon^{\theta})\rangle$
$\displaystyle=\lambda_{T_{bb}}\varepsilon_{\mu}^{\theta}\,,$ (4)

whereas the matrix element $\langle T_{bb}(p,\varepsilon^{\theta})\mid
T_{bb}(p+q,\varepsilon^{\delta})\rangle_{\gamma}$ is given by

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

where $\varepsilon^{\tau}$ is the polarization of the photon, and
$G_{1}(Q^{2})$, $G_{2}(Q^{2})$, and $G_{3}(Q^{2})$ are electromagnetic form
factors, with $Q^{2}=-q^{2}$.

Using Eqs. (II)-(5) and after doing some necessary calculations, we obtain the
hadronic side of the correlation function as follows:

$\displaystyle\Pi_{\mu\nu}^{Had}(p,q)$
$\displaystyle=\frac{\varepsilon_{\rho}\,\lambda_{T_{bb}}^{2}}{[m_{T_{bb}}^{2}-(p+q)^{2}][m_{T_{bb}}^{2}-p^{2}]}\Big{\\{}G_{2}(Q^{2})$
$\displaystyle\times\Big{(}q_{\mu}g_{\rho\nu}-q_{\nu}g_{\rho\mu}-\frac{p_{\nu}}{m_{T_{bb}}^{2}}\big{(}q_{\mu}p_{\rho}-\frac{1}{2}Q^{2}g_{\mu\rho}\big{)}+$
$\displaystyle+\frac{(p+q)_{\mu}}{m_{T_{bb}}^{2}}\big{(}q_{\nu}(p+q)_{\rho}+\frac{1}{2}Q^{2}g_{\nu\rho}\big{)}$
$\displaystyle-\frac{(p+q)_{\mu}p_{\nu}p_{\rho}}{m_{T_{bb}}^{4}}\,Q^{2}\Big{)}$
$\displaystyle+\mbox{other independent structures}\Big{\\}}.$ (6)

To characterize the magnetic dipole moment, we demand the value of the
$G_{2}(Q^{2})$ form factor to be at $Q^{2}=0$. The magnetic form factor
$F_{M}(Q^{2})$ is determined as:

$\displaystyle F_{M}(Q^{2})=G_{2}(Q^{2})\,,$ (7)

and the magnetic dipole moment $\mu_{T_{bb}}$ is described in terms of
$F_{M}(Q^{2}=0)$ as follows:

$\displaystyle\mu_{T_{bb}}=\frac{e}{2\,m_{T_{bb}}}\,F_{M}(Q^{2}=0).$ (8)

The next step in obtaining the analytical expressions of the magnetic dipole
moment calculations is used to calculate the QCD side of the correlation
function,which can be obtained by inserting the expression of the
interpolating current given in Eq. (2) into Eq. (1) and using the Wick
theorem. As a result, we get

$\displaystyle\Pi_{\mu\nu}^{\mathrm{QCD}}(p,q)$ $\displaystyle=$
$\displaystyle-i\int d^{4}xe^{ip\cdot x}\,\langle 0\mid$ (9)
$\displaystyle\Big{\\{}\mathrm{Tr}\Big{[}\gamma_{5}S_{b}^{aa^{\prime}}(x)\gamma_{5}S_{q_{1}}^{a^{\prime}a}(-x)\Big{]}\mathrm{Tr}\Big{[}\gamma_{\mu}S_{b}^{bb^{\prime}}(x)$
$\displaystyle\times\gamma_{\nu}S_{q_{2}}^{bb^{\prime}}(-x)\Big{]}-\mathrm{Tr}\Big{[}\gamma_{5}S_{b}^{ab^{\prime}}(x)\gamma_{\nu}S_{q_{2}}^{b^{\prime}b}(-x)$
$\displaystyle\times\gamma_{\mu}S_{b}^{ba^{\prime}}(x)\gamma_{5}S_{q_{1}}^{a^{\prime}a}(-x)\Big{]}\Big{\\}}\mid
0\rangle_{\gamma},$

where $S_{q}(x)$ and $S_{b}(x)$ denote the light and bottom-quark propagators.
During our calculations, we utilize the x-space expressions for the light and
bottom-quark propagators,

$\displaystyle S_{q}(x)$
$\displaystyle=i\frac{{x\\!\\!\\!/}}{2\pi^{2}x^{4}}-\frac{\langle\bar{q}q\rangle}{12}\Big{(}1-i\frac{m_{q}x\\!\\!\\!/}{4}\Big{)}-\frac{\langle\bar{q}q\rangle}{192}m_{0}^{2}x^{2}$
$\displaystyle\times\Big{(}1-i\frac{m_{q}x\\!\\!\\!/}{6}\Big{)}-\frac{ig_{s}}{32\pi^{2}x^{2}}~{}G^{\mu\nu}(x)\Big{[}\hbox
to0.0pt{/\hss}{x}\sigma_{\mu\nu}+\sigma_{\mu\nu}\hbox
to0.0pt{/\hss}{x}\Big{]},$ (10) $\displaystyle S_{b}(x)$
$\displaystyle=\frac{m_{b}^{2}}{4\pi^{2}}\Bigg{[}\frac{K_{1}\Big{(}m_{b}\sqrt{-x^{2}}\Big{)}}{\sqrt{-x^{2}}}+i\frac{{x\\!\\!\\!/}~{}K_{2}\Big{(}m_{b}\sqrt{-x^{2}}\Big{)}}{(\sqrt{-x^{2}})^{2}}\Bigg{]}$
$\displaystyle-\frac{g_{s}m_{b}}{16\pi^{2}}\int_{0}^{1}dv\,G^{\mu\nu}(vx)\Bigg{[}\big{(}\sigma_{\mu\nu}{x\\!\\!\\!/}+{x\\!\\!\\!/}\sigma_{\mu\nu}\big{)}$
$\displaystyle\times\frac{K_{1}\Big{(}m_{b}\sqrt{-x^{2}}\Big{)}}{\sqrt{-x^{2}}}+2\sigma_{\mu\nu}K_{0}\Big{(}m_{b}\sqrt{-x^{2}}\Big{)}\Bigg{]},$
(11)

where $\langle\bar{q}q\rangle$ is the light-quark condensate, $m_{0}$ is
characterized via the quark-gluon mixed condensate $\langle
0\mid\bar{q}\,g_{s}\,\sigma_{\alpha\beta}\,G^{\alpha\beta}\,q\mid
0\rangle=m_{0}^{2}\,\langle\bar{q}q\rangle$, $v$ is the line variable,
$G^{\mu\nu}$ is the gluon field strength tensor, and $K_{1}$, $K_{2}$, and
$K_{3}$ are modified Bessel functions of the second kind.

(a)

(b)

Figure 1:  Feynman diagrams of the magnetic dipole moments of doubly-bottom
tetraquark states. The thick, thin, wavy and curly lines represent the heavy
quark, light quark, photon and gluon propagators, respectively. Diagrams (a)
correspond to the perturbative photon vertex and, diagrams (b) represent the
contributions originating from the DAs of the photon.

In Fig. 1, we present some of the possible Feynman diagrams that contribute to
the QCD side of the analysis. For simplicity, we do not presented the Feynman
diagrams of terms proportional to higher-dimensional operators, but they are
considered in numerical analysis.

The correlation function in Eq. (9) includes different contributions,that is,
the photon can be emitted both perturbatively (contributions of short-
distance) and non-perturbatively (contributions of long-distance). In the
first case, the photon perturbatively interacts with one of the light or heavy
quarks; To obtain this contribution, the propagator of the quark interacting
with the photon perturbatively is modified via

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

where $S^{free}(x)$ is the first term of the light and bottom quark
propagators, and the remaining three propagators in Eq. (9) are replaced with
full quark propagators involving the perturbative and non-perturbative
contributions. In the second case, one of the light quark propagators in Eq.
(9), defined as the photon emission at large distances, is replaced via

$\displaystyle
S_{\mu\nu}^{ab}(x)\rightarrow-\frac{1}{4}\big{[}\bar{q}^{a}(x)\Gamma_{i}q^{b}(x)\big{]}\big{(}\Gamma_{i}\big{)}_{\mu\nu},$
(13)

where $S_{\mu\nu}^{ab}(x)$ is one of the light quark propagators given in Eq.
(II),
$\Gamma_{i}=I,\gamma_{5},\gamma_{\mu},i\gamma_{5}\gamma_{\mu},\sigma_{\mu\nu}/2$,
and the remaining light and heavy quark propagators are replaced with the full
quark propagators. Because a photon interacts with light-quark fields at long
distances, there are the matrix elements of nonlocal operators between the
vacuum and photon state such as
$\langle\gamma(q)\left|\bar{q}(x)\Gamma_{i}q(0)\right|0\rangle$ and
$\langle\gamma(q)\left|\bar{q}(x)\Gamma_{i}G_{\mu\nu}q(0)\right|0\rangle$.
These matrix elements are described with respect to the DAs of the photon,
which are defined in Ref. [52]. After these calculations, the QCD side of the
correlation function is obtained.

In the final step, by applying a double Borel transformation over the
variables $-p^{2}$ and $-(p+q)^{2}$, choosing the coefficients of the same
Lorentz structures ($\varepsilon.p\,(p_{\mu}q_{\nu}-p_{\nu}q_{\mu})$) in both
the QCD and hadronic sides and equating them, and performing quark-hadron
duality approximation, we obtain the required light-cone sum rules for these
magnetic dipole moments:

$\displaystyle\mu_{T_{bb}}=m_{T_{bb}}^{2}\,\frac{e^{\frac{m_{T_{bb}}^{2}}{M^{2}}}}{\lambda_{T_{bb}}^{2}}\,\,\Delta(M^{2},s_{0}).$
(14)

The explicit expression of the $\Delta(M^{2},s_{0})$ function is given in
Appendix A.

##  III Numerical analysis and conclusions

In this section, numerical computations are performed for the magnetic dipole
moments of doubly-bottom tetraquark states. The light-cone sum rules contain
various input parameters, such as the light and heavy quark masses; and light-
quark and gluon condensates. These parameters are given as: $m_{u}=m_{d}=0$,
$m_{s}=96^{+8}_{-4}\,\mbox{MeV}$, $m_{b}=(4.78\pm 0.06)\,$GeV,
$f_{3\gamma}=-0.0039~{}GeV^{2}$ [52], $\langle\bar{s}s\rangle$=
$0.8\langle\bar{u}u\rangle$ with $\langle\bar{u}u\rangle$=$(-0.24\pm
0.01)^{3}\,$GeV3 [53], $m_{0}^{2}=0.8\pm 0.1$ GeV2 [53], $\langle
g_{s}^{2}G^{2}\rangle=0.88~{}$GeV4 [54] and $\chi=-2.85\pm 0.5$ GeV-2 [55]. To
obtain a numerical value for the magnetic dipole moments, we must define the
values of the mass and residue of the doubly-bottom tetraquark states. These
parameters are borrowed from Ref. [46]. Another set of crucial input
parameters are the photon DAs of different twists. Explicit expression of
these DAs is given in Appendix B.

It follows from the explicit expressions of the light-cone sum rules for the
magnetic dipole moments of the doubly-bottom tetraquark states that; in
addition to DAs, they contain two arbitrary parameters, namely, the Borel mass
parameter $M^{2}$ and continuum threshold $s_{0}$. According to the light-cone
sum rules methodology, we must find working intervals of these arbitrary
parameters, where the magnetic dipole moments are insensitive to variations in
these arbitrary parameters in their working intervals. However, in practice,
it is necessary to find the working region where the variation in the
calculations of the magnetic dipole moments of doubly-bottom tetraquark states
according to these parameters is minimum. The continuum threshold $s_{0}$ is
not arbitrary and is related to the energy of the first excited state in the
initial channel. However, because we have very limited information on the
energy of excited states, we must decide on the method of choosing the working
interval of $s_{0}$. There are various proposals on how to determine this
parameter. The analysis of various sum rules predicts that
$s_{0}\simeq(m_{ground}+0.5^{+0.1}_{-0.1})^{2}$ GeV2. We use two criteria to
determine the working region of $M^{2}$. The lower bound of $M^{2}$ is
constrained by operator product expansion (OPE) convergence, demanding the
higher twist and higher condensate terms to less than $10\%$ of the total. The
upper bound of $M^{2}$ is constrained by the pole contribution (PC)

$\displaystyle\mbox{PC}=\frac{\Delta(M^{2},s_{0})}{\Delta(M^{2},\infty)},$
(15)

which represents the lowest-lying state contribution to the correlation
function. Our numerical calculations indicate that the requirements of the
light-cone sum rules method are satisfied in the working intervals of the
arbitrary parameters presented in Table 1. In our computations, for the
magnetic dipole moments of doubly-bottom tetraquark states, the PC varies on
average within the limits $0.35-0.61$. In the standard analysis of sum rules,
the PC is expected to be larger than $0.5$ for conventional hadrons. However,
in the case of tetraquark states, PC $>0.2$. When we analyze the OPE
convergence, we have obtained that the contribution of the higher dimensional
term in OPE is less than $\sim 1\%$. As these results suggest, the chosen
working intervals for $M^{2}$ and $s_{0}$ satisfy the requirements of the
light-cone sum rules method. In Fig. 2, we present the dependence of the
magnetic dipole moments of doubly-bottom tetraquark states on $M^{2}$ at
various $s_{0}$ values. As shown, the variation in magnetic dipole moment with
respect to $M^{2}$ is roughly $10-15\%$. Although the magnetic dipole moments
indicate some dependence on $s_{0}$, it remains inside the limits allowed by
the light-cone sum rules and generates most of the uncertainties.

Table 1: Working intervals of $s_{0}$, $M^{2}$ and the PC for magnetic dipole moments. $T_{bb}$ States |  $s_{0}$ [GeV2] |  $M^{2}$ [GeV2] | PC
---|---|---|---
$B^{-}B^{*-}$ | $115-119$ | $11-15$ | $0.38-0.61$
$B^{0}B^{*-}$ | $115-119$ | $11-15$ | $0.35-0.58$
$B^{-}B^{*0}$ | $115-119$ | $11-15$ | $0.35-0.61$
$B^{0}B^{*0}$ | $115-119$ | $11-15$ | $0.37-0.59$
$B_{s}^{0}B^{*-}$ | $117-121$ | $11-15$ | $0.36-0.58$
$B^{-}B_{s}^{*0}$ | $117-121$ | $11-15$ | $0.37-0.60$
$B_{s}^{0}B^{*0}$ | $117-121$ | $11-15$ | $0.36-0.59$
$B^{0}B_{s}^{*0}$ | $117-121$ | $11-15$ | $0.35-0.61$
$B^{0}_{s}B_{s}^{*0}$ | $121-125$ | $11-15$ | $0.35-0.60$

After performing the numerical calculations, the acquired values of the
magnetic dipole moments of doubly-bottom tetraquark states are collected in
Table 2. The uncertainties on the results are due to the variation in the
$s_{0}$, $M^{2}$ and, errors in the values of the input parameters.

Table 2: Numerical values of the magnetic dipole moments (in units of nuclear magneton $\mu_{N}$). $T_{bb}$ States |  Magnetic dipole moment
---|---
$B^{-}B^{*-}$ |  $~{}1.72\pm 0.67$
$B^{0}B^{*-}$ |  $~{}1.38\pm 0.56$
$B^{-}B^{*0}$ |  $-0.44\pm 0.17$
$B^{0}B^{*0}$ |  $-0.77\pm 0.28$
$B_{s}^{0}B^{*-}$ |  $~{}1.47\pm 0.42$
$B^{-}B_{s}^{*0}$ |  $-0.42\pm 0.15$
$B_{s}^{0}B^{*0}$ |  $-0.77\pm 0.25$
$B^{0}B_{s}^{*0}$ |  $-0.73\pm 0.24$
$B^{0}_{s}B_{s}^{*0}$ |  $-1.11\pm 0.31$

The magnitude of the magnetic dipole moments indicates their measurability in
experiments. From this perspective, it can be said that it is possible to
measure the magnetic dipole moments acquired experimentally. $SU(3)_{f}$
breaking effects are considered through a nonzero s-quark mass and s-quark
condensate, and we predict that $SU(3)_{f}$ symmetry violation in the magnetic
dipole moments is small, except for the relation between $B^{0}B^{*0}$ and
$B_{s}B_{s}^{*}$, where $SU(3)_{f}$ symmetry violation is large. In Ref. [5],
the authors performed a systematic investigation of the magnetic dipole
moments of doubly heavy tetraquark states with the molecule configuration
within the framework of the non-relativistic quark model with the help of the
Gaussian expansion method. The obtained magnetic dipole moments depending on
their spatial configurations are given as
$\mu_{T_{bb}^{-}}=0.49-0.98~{}\mu_{N}$ and
$\mu_{T_{bbs}^{-}}=1.29-1.40~{}\mu_{N}$.

As shown by the analytical results given in the appendix, we choose to keep
the quark charge factors explicit. The advantage of this is that it can make
it possible to investigate individual quark contribution to the magnetic
dipole moment. Via a profound analysis, it is observed that the sign of the
magnetic dipole moment is determined by the $q_{2}$-quark in the interpolating
current. If $q_{2}$ is a $u$-quark, the sign of the magnetic dipole moment is
positive ($B^{-}B^{*-}$, $B^{0}B^{*-}$ and $B_{s}^{0}B^{*-}$), and if $q_{2}$
is one of the $d$ or $s$-quarks, the sign of the magnetic dipole moment is
negative ($B^{-}B^{*0}$, $B^{0}B^{*0}$, $B^{-}B_{s}^{*0}$, $B_{s}^{0}B^{*0}$,
$B^{0}B_{s}^{*0}$, and $B^{0}_{s}B_{s}^{*0}$). A detailed examination
indicates that the smallness of the $e_{b}$ and $e_{q_{1}}$ contributions are
due to an almost exact cancellation of terms involving $e_{b}$ and
$e_{q_{1}}$. We would also like to discuss the amount of perturbative and non-
perturbative contributions (for example,quark-gluon condensates; and the DAs
of the photon) to the total results. Our numerical computations indicate that
roughly $85\%$ of the total contribution belongs to the perturbative part, and
the surviving $15\%$ corresponds to the non-perturbative contributions.

In conclusion, we extracted the magnetic dipole moments within the light-cone
sum rules method by employing molecular type interpolating currents for doubly
bottom tetraquark states with spin-parity $J^{P}=1^{+}$. The acquired results
along with the spectroscopic parameters may help the future theoretical and
experimental research on the characteristics of doubly-bottom tetraquark
states. It would be exciting to predict future experimental attempts at
exploring possible doubly-bottom tetraquark states and test the results of the
present analysis.

##  IV Acknowledgements

We are grateful to A. Ozpineci for useful discussions, comments and remarks.

(a)  (b)  (c)

(d)  (e)  (f)

(g)  (h)  (i)

Figure 2: Dependence of the magnetic dipole moments of doubly-bottom
tetraquark states on $M^{2}$ at three different values of $s_{0}$; (a), (b),
(c), (d), (e), (f), (g), (h) and (i) represent $B^{-}B^{*-}$, $B^{0}B^{*-}$,
$B^{-}B^{*0}$, $B^{0}B^{*0}$, $B_{s}^{0}B^{*-}$, $B^{-}B_{s}^{*0}$,
$B_{s}^{0}B^{*0}$, $B^{0}B_{s}^{*0}$, and $B^{0}_{s}B_{s}^{*0}$ states,
respectively.

### Appendix A: Explicit expression for $\Delta(M^{2},s_{0})$

In Appendix A, we present the explicit expressions of the function
$\Delta(M^{2},s_{0})$ for the magnetic dipole moments of doubly-bottom
tetraquark states entering the sum rule.

$\displaystyle\Delta(M^{2},s_{0})$ $\displaystyle=\frac{e_{q_{1}}m_{b}\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}_{1}q_{1}\rangle}{3538944\pi^{3}}\Bigg{[}8\Big{(}I[0,1,1,0]-2I[0,1,1,1]+I[0,1,1,2]-2I[0,1,2,0]+2I[0,1,2,1]\Big{)}\mathcal{A}[u_{0}]$
$\displaystyle+(8A[u_{0}]+3I_{3}[\mathcal{S}]-8I_{3}[\mathcal{\tilde{S}}])I[0,1,3,0]-4\chi\Big{(}9I[0,2,1,0]-23I[0,2,1,1]+19I[0,2,1,2]-5I[0,2,1,3]$
$\displaystyle-26I[0,2,2,0]+44I[0,2,2,1]-18I[0,2,2,2]+25I[0,2,3,0]-21I[0,2,3,1]-8I[0,2,4,0]\Big{)}\varphi_{\gamma}[u_{0}]\Bigg{]}$
$\displaystyle-\frac{m_{b}\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}_{2}q_{2}\rangle}{3538944\pi^{3}}\Bigg{[}-13e_{q_{2}}\big{(}I_{3}[\mathcal{S}]+I_{3}[\mathcal{\tilde{S}}]\big{)}I[0,1,3,0]+24e_{b}\Big{(}I[0,1,1,0]-2I[0,1,1,1]+I[0,1,1,2]$
$\displaystyle-2I[0,1,2,0]+2I[0,1,2,1]+I[0,1,3,0]\Big{)}\Bigg{]}$
$\displaystyle-\frac{\langle g_{s}^{2}G^{2}\rangle
f_{3\gamma}}{28311552\pi^{3}}\Bigg{[}9e_{q_{1}}I_{1}[\mathcal{V}]I[0,2,4,0]+48e_{q_{2}}\Big{(}4m_{b}(m_{b}+2m_{q_{1}})\big{(}I[0,1,1,0]-2I[0,1,1,1]+I[0,1,1,2]$
$\displaystyle-2I[0,1,2,0]+2I[0,1,2,1]+I[0,1,3,0]\big{)}+I[0,2,2,0]-2I[0,2,2,1]+I[0,2,2,2]-2I[0,2,3,0]$
$\displaystyle+2I[0,2,3,1]+I[0,2,4,0]\Big{)}+16e_{q_{1}}\Big{(}4m_{b}(m_{b}+2m_{q_{2}})\big{(}I[0,1,1,0]-2I[0,1,1,1]+I[0,1,1,2]$
$\displaystyle-2I[0,1,2,0]+2I[0,1,2,1]+I[0,1,3,0]\big{)}+I[0,2,2,0]-2I[0,2,2,1]+I[0,2,2,2]-2I[0,2,3,0]$
$\displaystyle+2I[0,2,3,1]+I[0,2,4,0]\Big{)}\psi^{a}[u_{0}]\Bigg{]}$
$\displaystyle-\frac{\langle
g_{s}^{2}G^{2}\rangle}{28311552\pi^{3}}\Bigg{[}216e_{q_{2}}\Big{(}I[0,3,2,0]-3I[0,3,2,1]+3I[0,3,2,2]-I[0,3,2,3]-3I[0,3,3,0]-6I[0,3,3,1]$
$\displaystyle+3I[0,3,3,2]-3I[0,3,4,0]+3I[0,3,4,1]-I[0,3,5,0]\Big{)}+e_{b}\Bigg{(}-12m_{b}\Big{(}6m_{q_{2}}\big{(}I[0,2,1,1]-2I[0,2,1,2]$
$\displaystyle+I[0,2,1,3]-2I[0,2,2,1]+2I[0,2,2,2]+I[0,2,3,1]\big{)}+m_{b}\big{(}I[0,2,1,1]-10I[0,2,1,2]+9I[0,2,1,3]$
$\displaystyle-2I[0,2,2,1]+10I[0,2,2,2]+I[0,2,3,1]\big{)}\Big{)}-3I[0,3,2,0]-127I[0,3,2,1]+263I[0,3,2,2]$
$\displaystyle-133I[0,3,2,3]+9I[0,3,3,0]+254I[0,3,3,1]-263I[0,3,3,2]-9I[0,3,4,0]-127I[0,3,4,1]$
$\displaystyle+3I[0,3,5,0]\Bigg{)}\Bigg{]}$
$\displaystyle+\frac{m_{b}\langle\bar{q}_{1}q_{1}\rangle}{196608\pi^{3}}\Bigg{[}72e_{q_{2}}I[0,3,2,0]+e_{b}\Big{(}3m_{0}^{2}\big{(}3I[0,2,1,0]-9I[0,2,1,1]+9I[0,2,1,2]-3I[0,2,1,3]$
$\displaystyle+2I[0,2,2,0]-4I[0,2,2,1]+2I[0,2,2,2]-13I[0,2,3,0]-13I[0,2,3,1]+8I[0,2,4,0]\big{)}-4\big{(}3I[0,3,1,1]$
$\displaystyle-9I[0,3,1,2]+9I[0,3,1,3]-3I[0,3,1,4]+2I[0,3,2,1]-4I[0,3,2,2]+2I[0,3,2,3]-13(I[0,3,3,1]$
$\displaystyle-I[0,3,3,2])+8I[0,3,4,1]\big{)}\Big{)}-72e_{q_{2}}\Big{(}3I[0,3,2,1]-3I[0,3,2,2]+I[0,3,2,3]+3I[0,3,3,0]-6I[0,3,3,1]$
$\displaystyle+3I[0,3,3,2]-3I[0,3,4,0]+3I[0,3,4,1]+I[0,3,5,0]\Big{)}+e_{q_{1}}\Big{(}-4(4I_{3}[\mathcal{S}]+3I_{3}[\mathcal{\tilde{S}}])I[0,3,4,0]$
$\displaystyle+3\big{(}I[0,3,2,0]-3I[0,3,2,1]+3I[0,3,2,2]-I[0,3,2,3]-3I[0,3,3,0]-6I[0,3,3,1]+3I[0,3,3,2]$
$\displaystyle-3I[0,3,4,0]+3I[0,3,4,1]-I[0,3,5,0]\big{)}\mathcal{A}[u_{0}]-3I_{3}[\mathcal{S}]I[0,3,5,0]+3\chi\big{(}-I[0,4,2,1]+3I[0,4,2,2]$
$\displaystyle-3I[0,4,2,3]+I[0,4,2,4]+3I[0,4,3,1]-6I[0,4,3,2]+3I[0,4,3,3]-3I[0,4,4,1]+3I[0,4,4,2]$
$\displaystyle+I[0,4,5,1]\big{)}\varphi_{\gamma}[u_{0}]\Big{)}\Bigg{]}$ (16)
$\displaystyle+\frac{m_{b}\langle\bar{q}_{2}q_{2}\rangle}{196608\pi^{3}}\Bigg{[}-3e_{b}m_{0}^{2}(I[0,2,2,0]-2I[0,2,2,1]+I[0,2,2,2]-2I[0,2,3,0]+2I[0,2,3,1]+I[0,2,4,0])$
$\displaystyle+4e_{b}\Big{(}I[0,3,2,1]-2I[0,3,2,2]+I[0,3,2,3]-2I[0,3,3,1]+2I[0,3,3,2]+I[0,3,4,1]\Big{)}$
$\displaystyle+e_{q_{2}}\big{(}I_{3}[\mathcal{S}]+I_{3}[\mathcal{\tilde{S}}]\big{)}\big{(}-4I[0,3,4,0]+3I[0,3,5,0]\big{)}\Bigg{]}$
$\displaystyle+\frac{m_{b}^{2}}{786432\pi^{5}}\Bigg{[}e_{b}\Big{(}I[0,4,1,3]-2I[0,4,1,4]+I[0,4,1,5]-2I[0,4,2,3]+2I[0,4,2,4]+I[0,4,3,3]\Big{)}$
$\displaystyle-8(e_{q_{1}}-e_{q_{2}})f_{3\gamma}\pi^{2}\Big{(}I[0,3,2,1]-2I[0,3,2,2]+I[0,3,2,3]-2I[0,3,3,1]+2I[0,3,3,2]+I[0,3,4,1]\Big{)}\psi^{a}[u_{0}]\Bigg{]}$
$\displaystyle+\frac{f_{3\gamma}}{1048576\pi^{3}}\Bigg{[}e_{q_{1}}\
I_{1}[\mathcal{V}](-4I[0,4,5,0]+3I[0,4,6,0])-6(e_{q_{1}}+11e_{q_{2}})\Big{(}I[0,4,3,0]-3I[0,4,3,1]+3I[0,4,3,2]$
$\displaystyle-I[0,4,3,3]-3I[0,4,4,0]-6I[0,4,4,1]+3I[0,4,4,2]-3I[0,4,5,0]+3I[0,4,5,1]-I[0,4,6,0]\Big{)}\psi^{a}[u_{0}]\Bigg{]}$
$\displaystyle+\frac{m_{b}}{524288\pi^{5}}\Bigg{[}e_{b}\Bigg{(}-m_{q_{2}}\Big{(}I[0,4,2,2]-2I[0,4,2,3]+I[0,4,2,4]-2I[0,4,3,2]+2I[0,4,3,3]+I[0,4,4,2]\Big{)}$
$\displaystyle+m_{q_{1}}\Big{(}3I[0,4,1,2]-9I[0,4,1,3]+9I[0,4,1,4]-3I[0,4,1,5]+2I[0,4,2,2]-4I[0,4,2,3]+2I[0,4,2,4]$
$\displaystyle-13(I[0,4,3,2]-I[0,4,3,3])+8I[0,4,4,2]\Big{)}\Bigg{)}+36e_{q_{1}}m_{q_{2}}\Big{(}-I[0,4,2,1]+3I[0,4,2,2]-3I[0,4,2,3]+I[0,4,2,4]$
$\displaystyle+3(I[0,4,3,1]-2I[0,4,3,2]+I[0,4,3,3]-I[0,4,4,1]+I[0,4,4,2])+I[0,4,5,1]\Big{)}$
$\displaystyle-16(5e_{q_{2}}m_{q_{1}}+e_{q_{1}}m_{q_{2}})f_{3\gamma}\pi^{2}\Big{(}I[0,3,2,0]-3I[0,3,2,1]+3I[0,3,2,2]-I[0,3,2,3]-3(I[0,3,3,0]-2I[0,3,3,1]$
$\displaystyle+I[0,3,3,2]-I[0,3,4,0]+I[0,3,4,1])-I[0,3,5,0]\Big{)}\psi^{a}[u_{0}]\Bigg{]}$
$\displaystyle+\frac{3}{5242880\pi^{5}}\Bigg{[}-36e_{q_{2}}I[0,5,3,1]+11e_{b}\Big{(}I[0,5,2,2]-3I[0,5,2,3]+3I[0,5,2,4]-I[0,5,2,5]-3(I[0,5,3,2]$
$\displaystyle-2I[0,5,3,3]+I[0,5,3,4]-I[0,5,4,2]+I[0,5,4,3])-I[0,5,5,2]\Big{)}+36e_{q_{2}}\Big{(}3I[0,5,3,2]-3I[0,5,3,3]+I[0,5,3,4]$
$\displaystyle+3(I[0,5,4,1]-2I[0,5,4,2]+I[0,5,4,3]-I[0,5,5,1]+I[0,5,5,2])+I[0,5,6,1]\Big{)}\Bigg{]},$
(17)

where $u_{0}=\frac{M_{1}^{2}}{M_{1}^{2}+M_{2}^{2}}$, and
$\frac{1}{M^{2}}=\frac{1}{M_{1}^{2}}+\frac{1}{M_{2}^{2}}$ with $M_{1}^{2}$ and
$M_{2}^{2}$ as the Borel parameters in the initial and final states,
respectively. Here $e_{q_{1}(q_{2})}$, $m_{q_{1}(q_{2})}$, and
$\langle\bar{q}_{1(2)}q_{1(2)}\rangle$ are the electric charge, mass, and
condensates of the corresponding light-quark, respectively. For simplicity we
do not present the terms proportional to many higher dimensional operators;
however in the numerical computations we take these terms into account.

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:

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

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

## Appendix B: Distribution Amplitudes of the photon

In Appendix B, the matrix elements
$\langle\gamma(q)\left|\bar{q}(x)\Gamma_{i}q(0)\right|0\rangle$ and
$\langle\gamma(q)\left|\bar{q}(x)\Gamma_{i}G_{\mu\nu}q(0)\right|0\rangle$
associated with the photon DAs are presented as follows [52]:

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

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

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

The expressions of the DAs that are entered into the matrix elements above are
as follows:

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

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

## References

  * [1] R. Aaij, et al., Observation of an exotic narrow doubly charmed tetraquark, Nature Phys. 18 (7) (2022) 751–754.  arXiv:2109.01038, doi:10.1038/s41567-022-01614-y.
  * [2] R. Aaij, et al., Study of the doubly charmed tetraquark $T_{cc}^{+}$, Nature Commun. 13 (1) (2022) 3351.  arXiv:2109.01056, doi:10.1038/s41467-022-30206-w.
  * [3] Q. Qin, Y.-F. Shen, F.-S. Yu, Discovery potentials of double-charm tetraquarks, Chin. Phys. C 45 (10) (2021) 103106.  arXiv:2008.08026, doi:10.1088/1674-1137/ac1b97.
  * [4] A. Feijoo, W. H. Liang, E. Oset, D0D0$\pi$\+ mass distribution in the production of the Tcc exotic state, Phys. Rev. D 104 (11) (2021) 114015.  arXiv:2108.02730, doi:10.1103/PhysRevD.104.114015.
  * [5] C. Deng, S.-L. Zhu, Tcc+ and its partners, Phys. Rev. D 105 (5) (2022) 054015\.  arXiv:2112.12472, doi:10.1103/PhysRevD.105.054015.
  * [6] 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.
  * [7] M.-J. Yan, M. P. Valderrama, Subleading contributions to the decay width of the Tcc+ tetraquark, Phys. Rev. D 105 (1) (2022) 014007.  arXiv:2108.04785, doi:10.1103/PhysRevD.105.014007.
  * [8] F.-L. Wang, X. Liu, Investigating new type of doubly charmed molecular tetraquarks composed of charmed mesons in the H and T doublets, Phys. Rev. D 104 (9) (2021) 094030.  arXiv:2108.09925, doi:10.1103/PhysRevD.104.094030.
  * [9] Y. Huang, H. Q. Zhu, L.-S. Geng, R. Wang, Production of Tcc+ exotic state in the $\gamma$p→D+T¯cc-$\Lambda$c+ reaction, Phys. Rev. D 104 (11) (2021) 116008.  arXiv:2108.13028, doi:10.1103/PhysRevD.104.116008.
  * [10] S. S. Agaev, K. Azizi, H. Sundu, Newly observed exotic doubly charmed meson Tcc+, Nucl. Phys. B 975 (2022) 115650.  arXiv:2108.00188, doi:10.1016/j.nuclphysb.2022.115650.
  * [11] X. Chen, Y. Yang, Doubly-heavy tetraquark states and *, Chin. Phys. C 46 (5) (2022) 054103.  arXiv:2109.02828, doi:10.1088/1674-1137/ac4ee8.
  * [12] Y. Jin, S.-Y. Li, Y.-R. Liu, Q. Qin, Z.-G. Si, F.-S. Yu, Color and baryon number fluctuation of preconfinement system in production process and Tcc structure, Phys. Rev. D 104 (11) (2021) 114009.  arXiv:2109.05678, doi:10.1103/PhysRevD.104.114009.
  * [13] X.-Z. Ling, M.-Z. Liu, L.-S. Geng, E. Wang, J.-J. Xie, Can we understand the decay width of the Tcc+ state?, Phys. Lett. B 826 (2022) 136897.  arXiv:2108.00947, doi:10.1016/j.physletb.2022.136897.
  * [14] Y. Hu, J. Liao, E. Wang, Q. Wang, H. Xing, H. Zhang, Production of doubly charmed exotic hadrons in heavy ion collisions, Phys. Rev. D 104 (11) (2021) L111502.  arXiv:2109.07733, doi:10.1103/PhysRevD.104.L111502.
  * [15] K. Chen, R. Chen, L. Meng, B. Wang, S.-L. Zhu, Systematics of the heavy flavor hadronic molecules, Eur. Phys. J. C 82 (7) (2022) 581.  arXiv:2109.13057, doi:10.1140/epjc/s10052-022-10540-5.
  * [16] M. Albaladejo, Tcc+ coupled channel analysis and predictions, Phys. Lett. B 829 (2022) 137052.  arXiv:2110.02944, doi:10.1016/j.physletb.2022.137052.
  * [17] L. M. Abreu, F. S. Navarra, M. Nielsen, H. P. L. Vieira, Interactions of the doubly charmed state $T_{cc}^{+}$ with a hadronic medium, Eur. Phys. J. C 82 (4) (2022) 296.  arXiv:2110.11145, doi:10.1140/epjc/s10052-022-10238-8.
  * [18] M.-L. Du, V. Baru, X.-K. Dong, A. Filin, F.-K. Guo, C. Hanhart, A. Nefediev, J. Nieves, Q. Wang, Coupled-channel approach to Tcc+ including three-body effects, Phys. Rev. D 105 (1) (2022) 014024.  arXiv:2110.13765, doi:10.1103/PhysRevD.105.014024.
  * [19] L. R. Dai, R. Molina, E. Oset, Prediction of new Tcc states of D*D* and Ds*D* molecular nature, Phys. Rev. D 105 (1) (2022) 016029.  arXiv:2110.15270, doi:10.1103/PhysRevD.105.016029.
  * [20] F.-L. Wang, R. Chen, X. Liu, A new group of doubly charmed molecule with $T$-doublet charmed meson pair (10 2021).  arXiv:2111.00208.
  * [21] L. Meng, G.-J. Wang, B. Wang, S.-L. Zhu, Probing the long-range structure of the Tcc+ with the strong and electromagnetic decays, Phys. Rev. D 104 (5) (2021) 051502.  arXiv:2107.14784, doi:10.1103/PhysRevD.104.L051502.
  * [22] S. Fleming, R. Hodges, T. Mehen, Tcc+ decays: Differential spectra and two-body final states, Phys. Rev. D 104 (11) (2021) 116010.  arXiv:2109.02188, doi:10.1103/PhysRevD.104.116010.
  * [23] Q. Xin, Z.-G. Wang, Analysis of the doubly-charmed tetraquark molecular states with the QCD sum rules, Eur. Phys. J. A 58 (6) (2022) 110.  arXiv:2108.12597, doi:10.1140/epja/s10050-022-00752-4.
  * [24] H. Ren, F. Wu, R. Zhu, Hadronic Molecule Interpretation of Tcc+ and Its Beauty Partners, Adv. High Energy Phys. 2022 (2022) 9103031.  arXiv:2109.02531, doi:10.1155/2022/9103031.
  * [25] R. Albuquerque, S. Narison, D. Rabetiarivony, Improved XTZ masses and mass ratios from Laplace sum rules at NLO, Nucl. Phys. A 1023 (2022) 122451.  arXiv:2201.13449, doi:10.1016/j.nuclphysa.2022.122451.
  * [26] K. Azizi, U. Özdem, Magnetic dipole moments of the Tcc+ and ZV++ tetraquark states, Phys. Rev. D 104 (11) (2021) 114002.  arXiv:2109.02390, doi:10.1103/PhysRevD.104.114002.
  * [27] U. Özdem, Magnetic moments of the doubly charged axial-vector Tcc++ states, Phys. Rev. D 105 (5) (2022) 054019.  arXiv:2112.10402, doi:10.1103/PhysRevD.105.054019.
  * [28] Y. Kim, M. Oka, K. Suzuki, Doubly heavy tetraquarks in a chiral-diquark picture, Phys. Rev. D 105 (7) (2022) 074021.  arXiv:2202.06520, doi:10.1103/PhysRevD.105.074021.
  * [29] L. M. Abreu, F. S. Navarra, H. P. L. Vieira, Multiplicity of the doubly charmed state Tcc+ in heavy-ion collisions, Phys. Rev. D 105 (11) (2022) 116029\.  arXiv:2202.10882, doi:10.1103/PhysRevD.105.116029.
  * [30] S. S. Agaev, K. Azizi, H. Sundu, Hadronic molecule model for the doubly charmed state ${T}_{cc}^{+}$, JHEP 06 (2022) 057.  arXiv:2201.02788, doi:10.1007/JHEP06(2022)057.
  * [31] L. R. Dai, E. Oset, A. Feijoo, R. Molina, L. Roca, A. M. Torres, K. P. Khemchandani, Masses and widths of the exotic molecular B(s)(*)B(s)(*) states, Phys. Rev. D 105 (7) (2022) 074017.  arXiv:2201.04840, doi:10.1103/PhysRevD.105.074017.
  * [32] L.-Y. Dai, X. Sun, X.-W. Kang, A. P. Szczepaniak, J.-S. Yu, Pole analysis on the doubly charmed meson in D0D0$\pi$\+ mass spectrum, Phys. Rev. D 105 (5) (2022) L051507.  arXiv:2108.06002, doi:10.1103/PhysRevD.105.L051507.
  * [33] M. Karliner, J. L. Rosner, Discovery of doubly-charmed $\Xi_{cc}$ baryon implies a stable ($bb\bar{u}\bar{d}$) tetraquark, Phys. Rev. Lett. 119 (20) (2017) 202001.  arXiv:1707.07666, doi:10.1103/PhysRevLett.119.202001.
  * [34] E. J. Eichten, C. Quigg, Heavy-quark symmetry implies stable heavy tetraquark mesons $Q_{i}Q_{j}\bar{q}_{k}\bar{q}_{l}$, Phys. Rev. Lett. 119 (20) (2017) 202002.  arXiv:1707.09575, doi:10.1103/PhysRevLett.119.202002.
  * [35] J.-B. Cheng, S.-Y. Li, Y.-R. Liu, Z.-G. Si, T. Yao, Double-heavy tetraquark states with heavy diquark-antiquark symmetry, Chin. Phys. C 45 (4) (2021) 043102\.  arXiv:2008.00737, doi:10.1088/1674-1137/abde2f.
  * [36] E. Braaten, L.-P. He, A. Mohapatra, Masses of doubly heavy tetraquarks with error bars, Phys. Rev. D 103 (1) (2021) 016001.  arXiv:2006.08650, doi:10.1103/PhysRevD.103.016001.
  * [37] Q. Meng, E. Hiyama, A. Hosaka, M. Oka, P. Gubler, K. U. Can, T. T. Takahashi, H. S. Zong, Stable double-heavy tetraquarks: spectrum and structure, Phys. Lett. B 814 (2021) 136095.  arXiv:2009.14493, doi:10.1016/j.physletb.2021.136095.
  * [38] J. M. Dias, S. Narison, F. S. Navarra, M. Nielsen, J. M. Richard, Relation between $T_{cc,bb}$ and $X_{c,b}$ from QCD, Phys. Lett. B 703 (2011) 274–280.  arXiv:1105.5630, doi:10.1016/j.physletb.2011.07.082.
  * [39] F. S. Navarra, M. Nielsen, S. H. Lee, QCD sum rules study of QQ - anti-u anti-d mesons, Phys. Lett. B 649 (2007) 166–172.  arXiv:hep-ph/0703071, doi:10.1016/j.physletb.2007.04.010.
  * [40] D. Gao, D. Jia, Y.-J. Sun, Z. Zhang, W.-N. Liu, Q. Mei, Masses of doubly heavy tetraquark states with isospin = $\frac{1}{2}$ and 1 and spin-parity $1^{+\pm}$ (7 2020).  arXiv:2007.15213.
  * [41] S. S. Agaev, K. Azizi, B. Barsbay, H. Sundu, Semileptonic and nonleptonic decays of the axial-vector tetraquark $T_{bb;\overline{u}\overline{d}}^{-}$, Eur. Phys. J. A 57 (3) (2021) 106.  arXiv:2008.02049, doi:10.1140/epja/s10050-021-00428-5.
  * [42] S. S. Agaev, K. Azizi, B. Barsbay, H. Sundu, A family of double-beauty tetraquarks: Axial-vector state $T_{bb;\bar{u}\bar{s}}^{-}$, Chin. Phys. C 45 (1) (2021) 013105.  arXiv:2002.04553, doi:10.1088/1674-1137/abc16d.
  * [43] S. S. Agaev, K. Azizi, B. Barsbay, H. Sundu, Stable scalar tetraquark $T_{bb;\bar{u}\bar{d}}^{-}$ (1 2020).  arXiv:2001.01446, doi:10.1140/epja/s10050-020-00187-9.
  * [44] S. S. Agaev, K. Azizi, B. Barsbay, H. Sundu, Heavy exotic scalar meson $T_{bb;\bar{u}\bar{s}}^{-}$, Phys. Rev. D 101 (9) (2020) 094026.  arXiv:1912.07656, doi:10.1103/PhysRevD.101.094026.
  * [45] S. S. Agaev, K. Azizi, B. Barsbay, H. Sundu, Weak decays of the axial-vector tetraquark $T_{bb;\bar{u}\bar{d}}^{-}$, Phys. Rev. D 99 (3) (2019) 033002.  arXiv:1809.07791, doi:10.1103/PhysRevD.99.033002.
  * [46] T. M. Aliev, S. Bilmis, M. Savci, Determination of the spectroscopic parameters of beauty-partners of Tcc from QCD, Phys. Rev. D 105 (7) (2022) 074038\.  arXiv:2111.01081, doi:10.1103/PhysRevD.105.074038.
  * [47] P. Mohanta, S. Basak, Construction of $bb\bar{u}\bar{d}$ tetraquark states on lattice with NRQCD bottom and HISQ up and down quarks, Phys. Rev. D 102 (9) (2020) 094516.  arXiv:2008.11146, doi:10.1103/PhysRevD.102.094516.
  * [48] H.-W. Ke, Y.-L. Shi, Study of possible molecular states of Ds(*)Ds(*) and Bs(*)Bs(*), Phys. Rev. D 105 (11) (2022) 114019.  arXiv:2202.13380, doi:10.1103/PhysRevD.105.114019.
  * [49] 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.
  * [50] 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.
  * [51] 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.
  * [52] 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.
  * [53] 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.
  * [54] R. D. Matheus, S. Narison, M. Nielsen, J. M. Richard, Can the X(3872) be a 1++ four-quark state?, Phys. Rev. D 75 (2007) 014005.  arXiv:hep-ph/0608297, doi:10.1103/PhysRevD.75.014005.
  * [55] 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.