# Electromagnetic properties of doubly heavy pentaquark states

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

###### Abstract

Motivated by the latest discovery of doubly-charmed tetraquark $T^{+}_{cc}$ by
LHCb collaboration, we have studied the magnetic moments of the possible
doubly-heavy pentaquark states with quantum numbers $J^{P}=1/2^{-}$ and
$J^{P}=3/2^{-}$ within the light-cone sum rules method. In the analysis, these
possible pentaquark states are considered in diquark-diquark-antiquark
structure. The magnetic moments of hadrons encode helpful details about the
distributions of the charge and magnetization inside the hadrons, which help
us to figure out their geometric configurations. As a by product, the electric
quadrupole and magnetic octupole moments of the spin-3/2 doubly-heavy
pentaquark states are also extracted. These values show a non-spherical charge
distribution. It will be interesting and useful to examine the magnetic
moments of these possible doubly-heavy pentaquark states with different
theoretical approaches.

Doubly-heavy pentaquarks, magnetic moment, diquark-diquark-antiquark picture,
light-cone sum rule

##  I Introduction

The first experimental discovery came in 2003 with the discovery of the
X(3872) state [1], although they were theoretically predicted long ago that
states other than conventional hadrons could exist. Since then, scientist have
paid more and more attention to the study of exotic states that are very
different from conventional hadrons. The investigation of exotic states and
how the quarks are got together inside plays a important role for
comprehension the low energy QCD, and it is very crucial to search for them in
experiments. To date, many of multi-quark states have been observed by
different experimental facilities. Recent progress regarding these multi-quark
states can be seen in Refs. [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,
16].

In 2015, the LHCb collaboration explored the $\Lambda_{b}^{0}$ $\rightarrow$
$J/\psi\,K^{-}\,p$ process and announced the observation of the two candidates
for pentaquark states, $P_{c}(4380)$ and $P_{c}(4450)$, in the $J/\psi\,p$
invariant mass distribution [17]. In 2019, three new pentaquark states,
$P_{c}(4312)$, $P_{c}(4440)$, and $P_{c}(4457)$, were reported in the updated
analyses of the LHCb collaboration [18]. These pentaquark states were
discovered in the $J/\psi\,p$ invariant mass spectrum, indicating that all the
states include a combination of the $uudc\bar{c}$ quark flavors. In 2020, the
LHCb Collaboration announced a pentaquark state, $P_{cs}(4459)$, in the
invariant mass spectrum of $J/\psi\Lambda$ in the $\Xi_{b}^{0}\rightarrow
J/\psi\,\Lambda\,K^{-}$ decay [19]. Since this pentaquark state is observed in
the invariant mass distribution $J/\psi\Lambda$, the quark content is
considered to be $udsc\bar{c}$. It should be noted here that most of the heavy
multi-quark states discovered in the experiment so far have a hidden-
charm/bottom quark structure. Recently, the first two charm quark-containing
exotic state, $T_{cc}^{+}$, was observed by the LHCb collaboration in the
$D^{0}D^{0}\pi^{+}$ mass spectrum [20, 21]. Its mass, with respect to the
$D^{0}D^{\ast+}$ threshold, and width have been measured to be $\delta
m=m_{T_{cc}^{+}}-(m_{D^{0}}+m_{D^{\ast+}})=-273\pm 61\pm
5_{-14}^{+11}~{}\mathrm{KeV}$, $\Gamma=410\pm 165\pm
43_{-38}^{+18}~{}\mathrm{KeV}$. The spin-parity of $T_{cc}^{+}$ tetraquark
state was estimated by the experiment as $J^{P}=1^{+}$. Clearly, the discovery
of $T^{+}_{cc}$ tetraquark state will open a new window to search for the new
states beyond the standard hadrons, both experimentally and theoretically. If
the $T^{+}_{cc}$ is the doubly-charmed tetraquark state exist, there may also
exist the doubly-charmed pentaquarks state. This argument is very similar to
the underlying reasoning of predicting the hidden-charm pentaquarks from the
existence of the hidden-charm tetraquark states. The only difference is that
now we have two charm quarks instead of a $\bar{c}c$ pair. If these doubly-
charmed pentaquark states do not exist, it also is important, in our opinion,
to explore the reasons why they are not. Inspired by this, it is well-
motivated and very interesting to search for the possible doubly-charmed
pentaquark states. We would also like to point out that in Refs. [22, 23, 24],
mass and possible decay channels of possible doubly-charmed pentaquark states
have been investigated within molecular pictures. In these studies, it has
been shown that these possible pentaquark states are below the threshold of
the possible meson-baryon state. In the study conducted in Ref. [25], possible
pentaquark states are considered as compact pentaquarks and the results
obtained seem to be consistent with the results obtained with the molecular
pictures. Based on results of these studies, it can be seen that these
possible pentaquark states might be below threshold and hence stable.

Compared to the doubly-heavy baryons, the doubly-heavy pentaquark states
should be expected to be heavier. But, the sophisticated interactions within
multi-quark states may lower the mass, which likely makes it hard to separate
experimentally a traditional baryon from a pentaquark state just from the mass
consideration. Examining other properties of these states along with mass,
such as electromagnetic properties, can help shed light on the inner nature of
these states. The magnetic dipole and higher moments of hadrons can help us to
obtain helpful details on the charge and magnetization distributions as well
as their geometric shape. In this work, we study the magnetic moments of
spin-1/2 and spin-3/2 doubly-heavy pentaquark (For short $P^{1/2}_{QQ}$ and
$P^{3/2}_{QQ}$, respectively) states using the light-cone sum rule formalism
[26, 27, 28] in the compact pentaquark picture. The attractive interaction
induced by one-gluon exchange supports formation of the diquarks in color
antitriplet and the supported configurations are the scalar and axialvector
diquark states from the QCD sum rules [29, 30, 31, 32]. In case of the heavy
diquark systems, only the tensor and axialvector diquarks remain because of
the Fermi-Dirac statistics, the axialvector diquarks are more stable than the
tensor diquarks. Therefore, in this study, we choose the axialvector type
heavy diquark interpolating currents. There are some theoretical estimations
on the internal structure of $P^{1/2}_{QQ}$ and $P^{3/2}_{QQ}$ states, their
masses, production mechanisms and decay channels using different
configurations and models [22, 23, 24, 25, 33, 34, 35, 36, 37]. It should be
noted here that, magnetic moments of hidden-charm pentaquark states have been
obtained via different models and substructures [38, 39, 40, 41, 42, 43, 44,
45]. Though the short lifetimes of the pentaquark states make the magnetic
moment difficult to be measured at present, more data accumulation in
different experiments in the future may make this possible. The
$\Delta^{+}(1232)$ baryon has also a very short lifetime, however, its
magnetic moment have been obtained by means of $\gamma N$ $\rightarrow$
$\Delta$ $\rightarrow$ $\Delta\gamma$ $\rightarrow$ $\pi N\gamma$ process [46,
47, 48]. The electromagnetic properties of baryons containing two charm quarks
have been calculated in the framework of the lattice QCD method [49, 50], and
it may be possible to generalize these analyzes to exotic states in the near
future.

This work has the following structure. In Sect. II, we briefly discuss the
formalism and calculate the light-cone sum rule for the magnetic moments under
investigation. In Sect. III, the numerical analysis and discussions for the
magnetic moments are presented. A brief summary of the article is presented in
Sect. IV. The explicit expressions of the photon distribution amplitudes and
magnetic moments for spin-1/2 doubly-heavy pentaquark states are given in
Appendixes A and B.

##  II Formalism

For magnetic moment analysis, calculations are started by writing the
appropriate correlation function in the light-cone sum rules. This correlation
function allows us to calculate the physical quantity to be calculated, in our
case, the magnetic moment, in terms of both QCD and hadron parameters. Then,
the correlation function calculated in two different ways is equalized to each
other using the quark-hadron duality. As a final step, Borel transform and
continuum subtraction are performed to reduce continuum and higher states
effects.

###  II.1 Formalism of the $P^{1/2}_{QQ}$ states

The correlation function required for the computations magnetic moment has the
following form:

$\displaystyle\Pi(p,q)$ $\displaystyle=$ $\displaystyle i\int d^{4}xe^{ip\cdot
x}\langle
0|T\left\\{J^{P^{1/2}_{QQ}}(x)\bar{J}^{P^{1/2}_{QQ}}(0)\right\\}|0\rangle_{\gamma}\,,$
(1)

where $\gamma$ is the external electromagnetic field and the $J(x)$ stands for
interpolating currents of the considered spin-1/2 doubly-heavy pentaquark
states. We would like to point out that many possible interpolating currents
can be written for exotic states, but the number of possible interpolating
currents that can be written can be slightly reduced when the QCD sum rules
and the states to be examined are taken into account. As we mentioned in the
introduction of the text, only axialvector or tensor structures make
significant contributions in case of heavy-heavy diquark configurations. Among
these two structures, axialvector structures were preferred because they are
more stable (Considering the heavy quark limit, it can be seen that the tensor
structure is proportional to zero). Apart from the heavy-heavy diquark case,
heavy-light diquarks may also be in question. However, due to the lack of
spectroscopic parameters in the QCD sum rules, they are excluded from this
work for now, considering that they can be examined in future studies.
Therefore, in this work, we choose the axialvector type heavy-heavy diquark
interpolating current and its given as follows [25]:

$\displaystyle J^{P^{1/2}_{QQ}}(x)$ $\displaystyle=$
$\displaystyle\varepsilon^{abc}\varepsilon^{ade}\varepsilon^{bfg}\big{[}Q^{T}_{d}(x)C\gamma_{\mu}Q_{e}(x)\big{]}\big{[}u^{T}_{f}(x)C\gamma_{5}d_{g}(x)\big{]}\gamma_{5}\gamma^{\mu}C\bar{q}^{T}_{c}(x)\,,$
(2)

where $Q$ is c or b-quark and $q$ is u or d-quark.

In order to get the hadronic degrees of freedom of the correlation function,
we insert a complete set of intermediate $P^{1/2}_{QQ}$ states with the same
quantum numbers as the interpolating currents into the correlation function.
As a result, we get

$\displaystyle\Pi^{Had}(p,q)$ $\displaystyle=\frac{\langle 0\mid
J(x)\mid{P^{1/2}_{QQ}}(p,s)\rangle}{[p^{2}-m_{P^{1/2}_{QQ}}^{2}]}\langle{P^{1/2}_{QQ}}(p,s)\mid{P^{1/2}_{QQ}}(p+q,s)\rangle_{\gamma}\frac{\langle{P^{1/2}_{QQ}}(p+q,s)\mid\bar{J}(0)\mid
0\rangle}{[(p+q)^{2}-m_{{P^{1/2}_{QQ}}}^{2}]}+\cdots$ (3)

The matrix element
$\langle{P^{1/2}_{QQ}}(p,s)\mid{P^{1/2}_{QQ}}(p+q,s)\rangle_{\gamma}$
inserting Eq. (3) can be described in connection with form factors as follows:

$\displaystyle\langle{P^{1/2}_{QQ}}(p,s)\mid{P^{1/2}_{QQ}}(p+q,s)\rangle_{\gamma}$
$\displaystyle=\varepsilon^{\mu}\,\bar{u}(p,s)\Big{[}\big{[}F_{1}(q^{2})+F_{2}(q^{2})\big{]}\gamma_{\mu}+F_{2}(q^{2})\frac{(2p+q)_{\mu}}{2m_{P^{1/2}_{QQ}}}\Big{]}\,u(p+q,s).$
(4)

We insert Eq. (4) in Eq. (3). Then, after some calculations are made, we get
the result for the hadronic side as follows

$\displaystyle\Pi^{Had}(p,q)=$
$\displaystyle\lambda^{2}_{P^{1/2}_{QQ}}\gamma_{5}\frac{\Big{(}p\\!\\!\\!/+m_{P^{1/2}_{QQ}}\Big{)}}{[p^{2}-m_{{P^{1/2}_{QQ}}}^{2}]}\varepsilon^{\mu}\Bigg{[}\big{[}F_{1}(q^{2})+F_{2}(q^{2})\big{]}\gamma_{\mu}+F_{2}(q^{2})\,\frac{(2p+q)_{\mu}}{2m_{P^{1/2}_{QQ}}}\Bigg{]}\gamma_{5}\frac{\Big{(}p\\!\\!\\!/+q\\!\\!\\!/+m_{P^{1/2}_{QQ}}\Big{)}}{[(p+q)^{2}-m_{{P^{1/2}_{QQ}}}^{2}]}.$
(5)

The value of form factors $F_{1}(q^{2})$ and $F_{2}(q^{2})$ give us the
magnetic form factor $F_{M}(q^{2})$ at different $q^{2}$ :

$\displaystyle F_{M}(q^{2})=F_{1}(q^{2})+F_{2}(q^{2}).$ (6)

At static limit, i.e. $q^{2}=0$, magnetic form factor $F_{M}(q^{2}=0)$ is
proportional to the magnetic moment $\mu_{P_{QQ}}$ for real photon :

$\displaystyle\mu_{P_{QQ}}=\frac{e}{2\,m_{P_{QQ}}}\,F_{M}(q^{2}=0).$ (7)

The next step is to obtain the correlation function in terms of QCD degrees of
freedom. When calculating the correlation function in terms of QCD degrees of
freedom, the explicit forms of the interpolating currents are substituted into
the correlation function. Then, the relevant light and heavy quark fields are
contracted via the Wick’s theorem, and the desired results are obtained.
Consequently, we get

$\displaystyle\Pi^{QCD}(p,q)$ $\displaystyle=$
$\displaystyle-i\,\varepsilon_{abc}\varepsilon_{ade}\varepsilon_{bfg}\varepsilon_{a^{\prime}d^{\prime}e^{\prime}}\varepsilon_{a^{\prime}b^{\prime}c^{\prime}}\varepsilon_{b^{\prime}f^{\prime}g^{\prime}}\int
d^{4}x\,e^{ip\cdot x}\langle
0\mid\gamma_{5}\gamma^{\mu}\widetilde{S}_{q}^{c^{\prime}c}(-x)\gamma^{\nu}\gamma_{5}$
(8)
$\displaystyle\Big{\\{}\mbox{Tr}\Big{[}\gamma_{\mu}S_{Q}^{ee^{\prime}}(x)\gamma_{\nu}\widetilde{S}_{Q}^{dd^{\prime}}(x)\Big{]}\,\mbox{Tr}\Big{[}\gamma_{5}S_{d}^{gg^{\prime}}(x)\gamma_{5}\widetilde{S}_{u}^{ff^{\prime}}(x)\Big{]}$
$\displaystyle-\mbox{Tr}\Big{[}\gamma_{\mu}S_{Q}^{ed^{\prime}}(x)\gamma_{\nu}\widetilde{S}_{Q}^{de^{\prime}}(x)\Big{]}\,\mbox{Tr}\Big{[}\gamma_{5}S_{d}^{gg^{\prime}}(x)\gamma_{5}\widetilde{S}_{u}^{ff^{\prime}}(x)\Big{]}\Big{\\}}\mid
0\rangle_{\gamma}\,,$

where $\widetilde{S}_{Q(q)}^{ij}(x)=CS_{Q(q)}^{ij\mathrm{T}}(x)C$ and ,
$S_{q}(x)$ and $S_{Q}(x)$ are the light and heavy quark propagators,
respectively. Their explicit expressions in the x-space are presented as

$\displaystyle S_{q}(x)$
$\displaystyle=i\frac{{x\\!\\!\\!/}}{2\pi^{2}x^{4}}-\frac{\langle\bar{q}q\rangle}{12}\Big{(}1-i\frac{m_{q}x\\!\\!\\!/}{4}\Big{)}-\frac{\langle\bar{q}q\rangle}{192}m_{0}^{2}x^{2}\Big{(}1-i\frac{m_{q}x\\!\\!\\!/}{6}\Big{)}-\frac{ig_{s}}{32\pi^{2}x^{2}}~{}G^{\mu\nu}(x)\Big{[}\hbox
to0.0pt{/\hss}{x}\sigma_{\mu\nu}+\sigma_{\mu\nu}\hbox
to0.0pt{/\hss}{x}\Big{]},$ (9) $\displaystyle S_{Q}(x)$
$\displaystyle=\frac{m_{Q}^{2}}{4\pi^{2}}\Bigg{[}\frac{K_{1}\Big{(}m_{Q}\sqrt{-x^{2}}\Big{)}}{\sqrt{-x^{2}}}+i\frac{{x\\!\\!\\!/}~{}K_{2}\Big{(}m_{Q}\sqrt{-x^{2}}\Big{)}}{(\sqrt{-x^{2}})^{2}}\Bigg{]}-\frac{g_{s}m_{Q}}{16\pi^{2}}\int_{0}^{1}dv\,G^{\mu\nu}(vx)\Bigg{[}(\sigma_{\mu\nu}{x\\!\\!\\!/}+{x\\!\\!\\!/}\sigma_{\mu\nu})$
$\displaystyle\times\frac{K_{1}\Big{(}m_{Q}\sqrt{-x^{2}}\Big{)}}{\sqrt{-x^{2}}}+2\sigma_{\mu\nu}K_{0}\Big{(}m_{Q}\sqrt{-x^{2}}\Big{)}\Bigg{]}.$
(10)

where $\langle\bar{q}q\rangle$ is quark condensate, $m_{0}$ is characterized
via the quark-gluon mixed condensate $\langle
0\mid\bar{q}\,g_{s}\,\sigma_{\mu\nu}\,G^{\mu\nu}\,q\mid
0\rangle=m_{0}^{2}\,\langle\bar{q}q\rangle$, $K_{0}$, $K_{1}$ and $K_{2}$ are
modified the second kind Bessel functions, $v$ is line variable and
$G^{\mu\nu}$ is the gluon field strength tensor. The first term of the light
and massive quark propagators corresponds to the free or perturbative part,
and the remaining part related to the interacting parts.

The correlation function in Eq. (8) includes different types of contributions:
The photon can be emitted both perturbatively and non-perturbatively. In the
first case, one of the free light or massive quark propagators in Eq. (8) is
replaced by

$\displaystyle S^{free}\rightarrow\int d^{4}y\,S^{free}(x-y)\,\hbox
to0.0pt{/\hss}{\\!A}(y)\,S^{free}(y)\,,$ (11)

the remaining four propagators are replaced with the full quark propagators.
The light-cone sum rule calculations are most conveniently done in the fixed-
point gauge. For electromagnetic field, it is defined by $x_{\mu}A^{\mu}=0$.
In this gauge, the electromagnetic potential is given by

$\displaystyle
A_{\alpha}=-\frac{1}{2}F_{\alpha\beta}y^{\beta}=-\frac{1}{2}(\varepsilon_{\alpha}q_{\beta}-\varepsilon_{\beta}q_{\alpha})\,y^{\beta}.$
(12)

Equation (12) is plugged into Eq. (11), as a result of which we obtain

$\displaystyle
S^{free}\rightarrow-\frac{1}{2}(\varepsilon_{\alpha}q_{\beta}-\varepsilon_{\beta}q_{\alpha})\int\,d^{4}y\,y^{\beta}\,S^{free}(x-y)\,\gamma_{\alpha}\,S^{free}(y)\,,$
(13)

After some calculations for $S_{q}^{free}$ and $S_{Q}^{free}$ we get

$\displaystyle
S_{q}^{free}=\frac{e_{q}}{32\pi^{2}x^{2}}\Big{(}\varepsilon_{\alpha}q_{\beta}-\varepsilon_{\beta}q_{\alpha}\Big{)}\Big{(}x\\!\\!\\!/\sigma_{\alpha\beta}+\sigma_{\alpha\beta}x\\!\\!\\!/\Big{)},$
$\displaystyle
S_{Q}^{free}=-i\frac{e_{Q}m_{Q}}{32\pi^{2}}\Big{(}\varepsilon_{\alpha}q_{\beta}-\varepsilon_{\beta}q_{\alpha}\Big{)}\Big{[}2\sigma_{\alpha\beta}K_{0}\Big{(}m_{Q}\sqrt{-x^{2}}\Big{)}+\frac{K_{1}\Big{(}m_{Q}\sqrt{-x^{2}}\Big{)}}{\sqrt{-x^{2}}}\Big{(}x\\!\\!\\!/\sigma_{\alpha\beta}+\sigma_{\alpha\beta}x\\!\\!\\!/\Big{)}\Big{]}.$
(14)

In the second case one of the light quark propagators in Eq. (8) are replaced
by

$\displaystyle
S_{\alpha\beta}^{ab}\rightarrow-\frac{1}{4}(\bar{q}^{a}\Gamma_{i}q^{b})(\Gamma_{i})_{\alpha\beta},$
(15)

and the remaining propagators are full quark propagators including the
perturbative as well as the nonperturbative contributions. Here as an example,
we give a short detail of the calculations of the QCD representations. In
second case for simplicity, we only consider the first trace in Eq. (8),

$\displaystyle\Pi^{QCD}(p,q)$
$\displaystyle=-i\,\varepsilon_{abc}\varepsilon_{ade}\varepsilon_{bfg}\varepsilon_{a^{\prime}d^{\prime}e^{\prime}}\varepsilon_{a^{\prime}b^{\prime}c^{\prime}}\varepsilon_{b^{\prime}f^{\prime}g^{\prime}}\int
d^{4}x\,e^{ip\cdot x}\langle
0\mid\gamma_{5}\gamma^{\mu}\widetilde{S}_{q}^{c^{\prime}c}(-x)\gamma^{\nu}\gamma_{5}$
$\displaystyle\Big{\\{}\mbox{Tr}\Big{[}\gamma_{\mu}S_{Q}^{ee^{\prime}}(x)\gamma_{\nu}\widetilde{S}_{Q}^{dd^{\prime}}(x)\Big{]}\,\mbox{Tr}\Big{[}\gamma_{5}S_{d}^{gg^{\prime}}(x)\gamma_{5}\widetilde{S}_{u}^{ff^{\prime}}(x)\Big{]}\Big{\\}}\mid
0\rangle_{\gamma}\,+...$ (16)

By replacing one of light propagators with the expressions in Eq. (9) and
making use of

$\displaystyle\bar{q}^{a}(x)\Gamma_{i}q^{a^{\prime}}(0)\rightarrow\frac{1}{3}\delta^{aa^{\prime}}\bar{q}(x)\Gamma_{i}q(0),$
(17)

the Eq. (II.1) takes the form

$\displaystyle\Pi^{QCD}(p,q)$
$\displaystyle=i\,\varepsilon_{abc}\varepsilon_{ade}\varepsilon_{bfg}\varepsilon_{a^{\prime}d^{\prime}e^{\prime}}\varepsilon_{a^{\prime}b^{\prime}c^{\prime}}\varepsilon_{b^{\prime}f^{\prime}g^{\prime}}\int
d^{4}x\,e^{ip\cdot x}$
$\displaystyle\Bigg{\\{}\gamma_{5}\gamma^{\mu}\Gamma_{i}\gamma^{\nu}\gamma_{5}\mbox{Tr}\Big{[}\gamma_{\mu}S_{Q}^{ee^{\prime}}(x)\gamma_{\nu}\widetilde{S}_{Q}^{dd^{\prime}}(x)\Big{]}\,\mbox{Tr}\Big{[}\gamma_{5}S_{d}^{gg^{\prime}}(x)\gamma_{5}\widetilde{S}_{u}^{ff^{\prime}}(x)\Big{]}\delta^{c^{\prime}c}$
$\displaystyle+\gamma_{5}\gamma^{\mu}\widetilde{S}_{q}^{c^{\prime}c}(-x)\gamma^{\nu}\gamma_{5}\mbox{Tr}\Big{[}\gamma_{\mu}S_{Q}^{ee^{\prime}}(x)\gamma_{\nu}\widetilde{S}_{Q}^{dd^{\prime}}(x)\Big{]}\,\mbox{Tr}\Big{[}\gamma_{5}\Gamma_{i}\gamma_{5}\widetilde{S}_{u}^{ff^{\prime}}(x)\Big{]}\delta^{gg^{\prime}}$
$\displaystyle+\gamma_{5}\gamma^{\mu}\widetilde{S}_{q}^{c^{\prime}c}(-x)\gamma^{\nu}\gamma_{5}\mbox{Tr}\Big{[}\gamma_{\mu}S_{Q}^{ee^{\prime}}(x)\gamma_{\nu}\widetilde{S}_{Q}^{dd^{\prime}}(x)\Big{]}\,\mbox{Tr}\Big{[}\gamma_{5}S_{d}^{gg^{\prime}}(x)\gamma_{5}\Gamma_{i}\Big{]}\delta^{ff^{\prime}}\Bigg{\\}}\frac{1}{12}\langle\gamma(q)|\bar{q}(x)\Gamma_{i}q(0)|0\rangle+...,$
(18)

where
$\Gamma_{i}=I,\gamma_{5},\gamma_{\mu},i\gamma_{5}\gamma_{\mu},\sigma_{\mu\nu}/2$.
Similarly, when a light propagator interacts with the photon, a gluon may be
released from one of the remaining four propagators. The expression obtained
in this case is as follows:

$\displaystyle\Pi_{\mu\nu}^{\mathrm{QCD}}(p,q)$
$\displaystyle=i\,\varepsilon_{abc}\varepsilon_{ade}\varepsilon_{bfg}\varepsilon_{a^{\prime}d^{\prime}e^{\prime}}\varepsilon_{a^{\prime}b^{\prime}c^{\prime}}\varepsilon_{b^{\prime}f^{\prime}g^{\prime}}\int
d^{4}x\,e^{ip\cdot x}$
$\displaystyle\Bigg{\\{}\gamma_{5}\gamma^{\mu}\Gamma_{i}\gamma^{\nu}\gamma_{5}\mbox{Tr}\Big{[}\gamma_{\mu}S_{Q}^{ee^{\prime}}(x)\gamma_{\nu}\widetilde{S}_{Q}^{dd^{\prime}}(x)\Big{]}\,\mbox{Tr}\Big{[}\gamma_{5}S_{d}^{gg^{\prime}}(x)\gamma_{5}\widetilde{S}_{u}^{ff^{\prime}}(x)\Big{]}\Big{[}\Big{(}\delta^{ce}\delta^{c^{\prime}e^{\prime}}-\frac{1}{3}\delta^{cc^{\prime}}\delta^{ee^{\prime}}\Big{)}$
$\displaystyle+\Big{(}\delta^{cd}\delta^{c^{\prime}d^{\prime}}-\frac{1}{3}\delta^{cc^{\prime}}\delta^{dd^{\prime}}\Big{)}+\Big{(}\delta^{cg}\delta^{c^{\prime}g^{\prime}}-\frac{1}{3}\delta^{cc^{\prime}}\delta^{gg^{\prime}}\Big{)}+\Big{(}\delta^{cf}\delta^{c^{\prime}f^{\prime}}-\frac{1}{3}\delta^{cc^{\prime}}\delta^{ff^{\prime}}\Big{)}\Big{]}$
$\displaystyle+\gamma_{5}\gamma^{\mu}\widetilde{S}_{q}^{c^{\prime}c}(-x)\gamma^{\nu}\gamma_{5}\mbox{Tr}\Big{[}\gamma_{\mu}S_{Q}^{ee^{\prime}}(x)\gamma_{\nu}\widetilde{S}_{Q}^{dd^{\prime}}(x)\Big{]}\,\mbox{Tr}\Big{[}\gamma_{5}\Gamma_{i}\gamma_{5}\widetilde{S}_{u}^{ff^{\prime}}(x)\Big{]}\Big{[}\Big{(}\delta^{gc}\delta^{g^{\prime}c^{\prime}}-\frac{1}{3}\delta^{gg^{\prime}}\delta^{cc^{\prime}}\Big{)}$
$\displaystyle+\Big{(}\delta^{ge}\delta^{g^{\prime}e^{\prime}}-\frac{1}{3}\delta^{ee^{\prime}}\delta^{gg^{\prime}}\Big{)}+\Big{(}\delta^{gd}\delta^{g^{\prime}d^{\prime}}-\frac{1}{3}\delta^{gg^{\prime}}\delta^{dd^{\prime}}\Big{)}+\Big{(}\delta^{gf}\delta^{g^{\prime}f^{\prime}}-\frac{1}{3}\delta^{gg^{\prime}}\delta^{ff^{\prime}}\Big{)}\Big{]}$
$\displaystyle+\gamma_{5}\gamma^{\mu}\widetilde{S}_{q}^{c^{\prime}c}(-x)\gamma^{\nu}\gamma_{5}\mbox{Tr}\Big{[}\gamma_{\mu}S_{Q}^{ee^{\prime}}(x)\gamma_{\nu}\widetilde{S}_{Q}^{dd^{\prime}}(x)\Big{]}\,\mbox{Tr}\Big{[}\gamma_{5}S_{d}^{gg^{\prime}}(x)\gamma_{5}\Gamma_{i}\Big{]}\Big{[}\Big{(}\delta^{fc}\delta^{f^{\prime}c^{\prime}}-\frac{1}{3}\delta^{ff^{\prime}}\delta^{cc^{\prime}}\Big{)}$
$\displaystyle+\Big{(}\delta^{fe}\delta^{f^{\prime}e^{\prime}}-\frac{1}{3}\delta^{ff^{\prime}}\delta^{ee^{\prime}}\Big{)}+\Big{(}\delta^{fd}\delta^{f^{\prime}d^{\prime}}-\frac{1}{3}\delta^{ff^{\prime}}\delta^{dd^{\prime}}\Big{)}+\Big{(}\delta^{fg}\delta^{f^{\prime}g^{\prime}}-\frac{1}{3}\delta^{ff^{\prime}}\delta^{gg^{\prime}}\Big{)}\Big{]}\Bigg{\\}}$
$\displaystyle\times\frac{1}{32}\langle\gamma(q)|\bar{q}(x)\Gamma_{i}G_{\mu\nu}(vx)q(0)|0\rangle+...,$
(19)

where we inserted

$\displaystyle\bar{q}^{a}(x)\Gamma_{i}G_{\mu\nu}^{bb^{\prime}}(vx)q^{a^{\prime}}(0)\rightarrow\frac{1}{8}\Big{(}\delta^{ab}\delta^{a^{\prime}b^{\prime}}-\frac{1}{3}\delta^{aa^{\prime}}\delta^{bb^{\prime}}\Big{)}\bar{q}(x)\Gamma_{i}G_{\mu\nu}(vx)q(0).$
(20)

As is seen, there appear matrix elements such as
$\langle\gamma(q)\left|\bar{q}(x)\Gamma_{i}q(0)\right|0\rangle$ and
$\langle\gamma(q)\left|\bar{q}(x)\Gamma_{i}G_{\mu\nu}(vx)q(0)\right|0\rangle$,
representing the non-perturbative contributions. These matrix elements can be
expressed in terms of photon distribution amplitudes (DAs) and wave functions
with definite twists, whose expressions are given in Appendix A. Besides these
matrix elements, non-local operators such as two gluons ($\bar{q}GGq$) and
four quarks ($\bar{q}q\bar{q}q$) are expected to seem. However, it is known
that the effects of such operators are small, which is justified by the
conformal spin expansion [51, 52], and thus we shall ignore them. The QCD
representation of the correlation function is obtained by using Eqs. (8-20).
Then, the Fourier transformation is applied to transfer expressions in x-space
to the momentum space.

To find the desired sum rules, we obtain the invariant amplitude
$\Pi^{QCD}(p,q)$ corresponding to the structure
$\varepsilon\\!\\!\\!/q\\!\\!\\!/$, and match it to $\Pi^{Had}(p,q)$. We
perform the double Borel transformation to both representations of the
acquired equality, which is needed to suppress contributions of the higher
resonances and continuum states. The last operation to be applied is continuum
subtraction, which is obtained by invoking assumption on quark-hadron duality.
After these steps, we acquire the required sum rules for the magnetic moments:

$\displaystyle\mu_{P^{1/2}_{QQ}}\,\lambda^{2}_{P^{1/2}_{QQ}}\,m_{P^{1/2}_{QQ}}=e^{\frac{m^{2}_{P^{1/2}_{QQ}}}{M^{2}}}\,\Delta^{QCD}.$
(21)

Explicit forms of the analytical expressions obtained for the $\Delta^{QCD}$
function are given in Appendix B.

###  II.2 Formalism of the $P^{3/2}_{QQ}$ states

In the present subsection, we derive the light-cone sum rule for the magnetic
moments of the $P^{3/2}_{QQ}$ pentaquark states. To do this, we begin with
subsequent correlation function,

$\Pi_{\mu\nu}(p,q)=i\int d^{4}xe^{ip\cdot x}\langle
0|\mathcal{T}\\{J_{\mu}^{P^{3/2}_{QQ}}(x)\bar{J}_{\nu}^{P^{3/2}_{QQ}}(0)\\}|0\rangle_{\gamma},$
(22)

where the interpolating current of doubly-heavy pentaquark states with
$J^{P}=\frac{3}{2}^{-}$ is denoted $J_{\mu(\nu)}^{P^{3/2}_{QQ}}$. In the
compact pentaquark picture, it is given as [25]

$\displaystyle J_{\mu}^{P^{3/2}_{QQ}}(x)$ $\displaystyle=$
$\displaystyle\varepsilon^{abc}\varepsilon^{ade}\varepsilon^{bfg}\big{[}Q^{T}_{d}(x)C\gamma_{\mu}Q_{e}(x)\big{]}\big{[}u^{T}_{f}(x)C\gamma_{5}d_{g}(x)\big{]}C\bar{q}^{T}_{c}(x)\,,$
(23)

The correlation function obtained depending on the hadron parameters is
written as,

$\displaystyle\Pi^{Had}_{\mu\nu}(p,q)$ $\displaystyle=$
$\displaystyle\frac{\langle 0\mid
J_{\mu}^{P^{3/2}_{QQ}}(x)\mid{P^{3/2}_{QQ}}(p,s)\rangle}{[p^{2}-m_{{P^{3/2}_{QQ}}}^{2}]}\langle{P^{3/2}_{QQ}}(p,s)\mid{P^{3/2}_{QQ}}(p+q,s)\rangle_{\gamma}\frac{\langle{P^{3/2}_{QQ}}(p+q,s)\mid\bar{J}_{\nu}^{P^{3/2}_{QQ}}(0)\mid
0\rangle}{[(p+q)^{2}-m_{{P^{3/2}_{QQ}}}^{2}]}+...$ (24)

The matrix element of the interpolating current between the vacuum and the
$P^{3/2}_{QQ}$ pentaquark is defined as

$\displaystyle\langle 0\mid
J_{\mu}^{P^{3/2}_{QQ}}(x)\mid{P^{3/2}_{QQ}}(p,s)\rangle$
$\displaystyle=\lambda_{{P^{3/2}_{QQ}}}u_{\mu}(p,s),$
$\displaystyle\langle{P^{3/2}_{QQ}}(p+q,s)\mid\bar{J}_{\nu}^{P^{3/2}_{QQ}}(0)\mid
0\rangle$ $\displaystyle=\lambda_{{P^{3/2}_{QQ}}}\bar{u}_{\nu}(p+q,s),$ (25)

where the $u_{\mu}(p,s)$, $u_{\nu}(p+q,s)$ and $\lambda_{{P^{3/2}_{QQ}}}$ are
the spinors and residue doubly-heavy $P^{3/2}_{QQ}$ pentaquark states,
respectively.

The transition matrix element
$\langle{P^{3/2}_{QQ}}(p)\mid{P^{3/2}_{QQ}}(p+q)\rangle_{\gamma}$ entering Eq.
(24) can be written as follows [53, 54, 55, 56]:

$\displaystyle\langle{P^{3/2}_{QQ}}(p,s)\mid{P^{3/2}_{QQ}}(p+q,s)\rangle_{\gamma}$
$\displaystyle=-e\bar{u}_{\mu}(p,s)\Bigg{[}F_{1}(q^{2})g_{\mu\nu}\varepsilon\\!\\!\\!/-\frac{1}{2m_{{P^{3/2}_{QQ}}}}\Big{[}F_{2}(q^{2})g_{\mu\nu}+F_{4}(q^{2})\frac{q_{\mu}q_{\nu}}{(2m_{{P^{3/2}_{QQ}}})^{2}}\Big{]}\varepsilon\\!\\!\\!/q\\!\\!\\!/$
$\displaystyle+F_{3}(q^{2})\frac{1}{(2m_{{P^{3/2}_{QQ}}})^{2}}q_{\mu}q_{\nu}\varepsilon\\!\\!\\!/\Bigg{]}u_{\nu}(p+q,s).$
(26)

where $F_{i}$’s are the Lorentz invariant form factors.

In principle, we can derive the hadronic representation of the correlation
function employing Eqs. (22)-(II.2), but in this case we run into two
undesirable problems. The first of these problems is that the Lorentz
structures in the correlation function are not independent, and the second is
that the correlation function also contains spin-1/2 contributions. Indeed,
the matrix element of the current $J_{\mu}$ between vacuum and spin-1/2
doubly-heavy pentaquarks is non-zero and is determined as

$\langle 0\mid J_{\mu}(0)\mid
B(p,s=1/2)\rangle=(Ap_{\mu}+B\gamma_{\mu})u(p,s=1/2).$ (27)

As is seen the undesired spin-1/2 effects are proportional to $\gamma_{\mu}$
and $p_{\mu}$. By multiplying both sides with $\gamma^{\mu}$ and using the
condition $\gamma^{\mu}J_{\mu}=0$ one can specify the constant A in terms of
B. To eliminate unwanted effects of the spin-1/2 states and acquire only
independent Lorentz structures in the correlation function, we carry out the
ordering for Dirac matrices as
$\gamma_{\mu}p\\!\\!\\!/\varepsilon\\!\\!\\!/q\\!\\!\\!/\gamma_{\nu}$ and
eliminate expressions with $\gamma_{\mu}$ at the beginning, $\gamma_{\nu}$ at
the end and those proportional to $p_{\mu}$ and $p_{\nu}$ [57]. Consequently,
employing Eqs. (22)-(II.2) the hadronic side takes the form,

$\displaystyle\Pi^{Had}_{\mu\nu}(p,q)$
$\displaystyle=\frac{\lambda_{{}_{{P^{3/2}_{QQ}}}}^{2}}{[(p+q)^{2}-m_{{}_{{P^{3/2}_{QQ}}}}^{2}][p^{2}-m_{{}_{{P^{3/2}_{QQ}}}}^{2}]}\Bigg{[}g_{\mu\nu}p\\!\\!\\!/\varepsilon\\!\\!\\!/q\\!\\!\\!/\,F_{1}(q^{2})-m_{{P^{3/2}_{QQ}}}g_{\mu\nu}\varepsilon\\!\\!\\!/q\\!\\!\\!/\,F_{2}(q^{2})-\frac{F_{3}(q^{2})}{4m_{{P^{3/2}_{QQ}}}}q_{\mu}q_{\nu}\varepsilon\\!\\!\\!/q\\!\\!\\!/\,$
$\displaystyle-\frac{F_{4}(q^{2})}{4m_{{P^{3/2}_{QQ}}}^{3}}(\varepsilon.p)q_{\mu}q_{\nu}p\\!\\!\\!/q\\!\\!\\!/+...\Bigg{]}.$
(28)

The final form of the hadronic description associated with the chosen
structures as follows:

$\displaystyle\Pi^{Had}_{\mu\nu}(p,q)$ $\displaystyle=$
$\displaystyle\Pi_{1}^{Had}g_{\mu\nu}p\\!\\!\\!/\varepsilon\\!\\!\\!/q\\!\\!\\!/\,+\Pi_{2}^{Had}g_{\mu\nu}\varepsilon\\!\\!\\!/q\\!\\!\\!/\,+...,$
(29)

where $\Pi_{1}^{Had}$ and $\Pi_{2}^{Had}$ are functions of the form factors
$F_{1}(q^{2})$ and $F_{2}(q^{2})$, respectively; and other independent
structures and form factors are denoted by dots.

The magnetic form factor, $G_{M}(q^{2})$, is characterized with respect to the
form factors $F_{i}(q^{2})$ as follows [53, 54, 55, 56]:

$\displaystyle G_{M}(q^{2})$
$\displaystyle=[F_{1}(q^{2})+F_{2}(q^{2})](1+\frac{4}{5}\tau)-\frac{2}{5}[F_{3}(q^{2})+F_{4}(q^{2})]\tau(1+\tau),$
(30)

where $\tau=-\frac{q^{2}}{4m^{2}_{{P^{3/2}_{QQ}}}}$. At $q^{2}=0$, the
magnetic moment is obtained with respect to the functions $F_{1}(0)$ and
$F_{2}(0)$ form factors as:

$\displaystyle G_{M}(0)$ $\displaystyle=$ $\displaystyle F_{1}(0)+F_{2}(0).$
(31)

The magnetic moment, ($\mu_{{P^{3/2}_{QQ}}}$), is described as follows,

$\displaystyle\mu_{{P^{3/2}_{QQ}}}$ $\displaystyle=$
$\displaystyle\frac{e}{2m_{{P^{3/2}_{QQ}}}}G_{M}(0).$ (32)

When we perform the above processes, the calculations in terms of hadronic
parameters, which are the first step of light-cone sum rule calculations, are
completed.

The second step in light-cone sum rule calculations is to evaluate the
correlation function in Eq. (22) in connection with quark-gluon parameters as
well as photon DAs. Repeating the processes in the previous subsection gives
the subsequent result:

$\displaystyle\Pi^{QCD}(p,q)$ $\displaystyle=$ $\displaystyle
i\,\varepsilon_{abc}\varepsilon_{ade}\varepsilon_{bfg}\varepsilon_{a^{\prime}d^{\prime}e^{\prime}}\varepsilon_{a^{\prime}b^{\prime}c^{\prime}}\varepsilon_{b^{\prime}f^{\prime}g^{\prime}}\int
d^{4}x\,e^{ip\cdot x}\langle 0\mid\widetilde{S}_{q}^{c^{\prime}c}(-x)$ (33)
$\displaystyle\Big{\\{}\mbox{Tr}\Big{[}\gamma_{\mu}S_{Q}^{ee^{\prime}}(x)\gamma_{\nu}\widetilde{S}_{Q}^{dd^{\prime}}(x)\Big{]}\,\mbox{Tr}\Big{[}\gamma_{5}S_{d}^{gg^{\prime}}(x)\gamma_{5}\widetilde{S}_{u}^{ff^{\prime}}(x)\Big{]}$
$\displaystyle-\mbox{Tr}\Big{[}\gamma_{\mu}S_{Q}^{ed^{\prime}}(x)\gamma_{\nu}\widetilde{S}_{Q}^{de^{\prime}}(x)\Big{]}\,\mbox{Tr}\Big{[}\gamma_{5}S_{d}^{gg^{\prime}}(x)\gamma_{5}\widetilde{S}_{u}^{ff^{\prime}}(x)\Big{]}\Big{\\}}\mid
0\rangle_{\gamma}\,.$

Consequently, the QCD representation of the correlation function in connection
with the chosen structures is computed as

$\displaystyle\Pi^{QCD}_{\mu\nu}(p,q)$ $\displaystyle=$
$\displaystyle\Pi_{1}^{QCD}g_{\mu\nu}p\\!\\!\\!/\varepsilon\\!\\!\\!/q\\!\\!\\!/\,+\Pi_{2}^{QCD}g_{\mu\nu}\varepsilon\\!\\!\\!/q\\!\\!\\!/\,+....$
(34)

Since the $\Pi_{i}^{QCD}$ functions are very lengthy, their explicit forms are
not given here.

We have obtained the correlation function in terms of both QCD and hadronic
parameters. For the magnetic moment calculations, the QCD and hadronic
descriptions of the correlation function are equalized using the quark-hadron
duality ansatz. By matching the coefficients of the structures
$g_{\mu\nu}p\\!\\!\\!/\varepsilon\\!\\!\\!/q\\!\\!\\!/$ and
$g_{\mu\nu}\varepsilon\\!\\!\\!/q\\!\\!\\!/$, respectively for the $F_{1}$ and
$F_{2}$ we get light-cone sum rules for these two form factors. Consequently,
we acquire,

$\displaystyle\Pi^{Had}_{\mu\nu}(p,q)=\Pi^{QCD}_{\mu\nu}(p,q).$ (35)

Analytical expressions have also been obtained for the $P_{QQ}^{3/2}$ doubly-
heavy pentaquarks. The next step will be to perform numerical calculations for
both $P_{QQ}^{1/2}$ and $P_{QQ}^{3/2}$ doubly-heavy pentaquarks.

##  III Numerical analysis

The light-cone sum rule for magnetic moments of the $P^{1/2}_{QQ}$ and
$P^{3/2}_{QQ}$ states contains many input parameters that we need their
numerical values. We use $m_{u}=m_{d}=0$, $m_{c}=1.275\pm 0.02\,\mbox{GeV}$,
$m_{b}=4.18^{+0.03}_{-0.02}\,\mbox{GeV}$ [58],
$f_{3\gamma}=-0.0039~{}\mbox{GeV}^{2}$ [59],
$\langle\bar{u}u\rangle=\langle\bar{d}d\rangle=(-0.24\pm
0.01)^{3}\,\mbox{GeV}^{3}$ [60], $m_{0}^{2}=0.8\pm 0.1\,\mbox{GeV}^{2}$ [60],
$\langle g_{s}^{2}G^{2}\rangle=0.88~{}\mbox{GeV}^{4}$ [61]. To further the
numerical analysis we also need the numerical values of the mass and residues
of these states. In Ref. [25], these values were obtained within the framework
of the mass sum rules. The obtained results for masses and residues are given
as $m_{P_{cc}^{1/2}}=4.21^{+0.10}_{-0.11}~{}\mbox{GeV}$,
$m_{P_{cc}^{3/2}}=4.27^{+0.11}_{-0.10}~{}\mbox{GeV}$,
$m_{P_{bb}^{1/2}}=10.75^{+0.12}_{-0.12}~{}\mbox{GeV}$,
$m_{P_{bb}^{3/2}}=10.76^{+0.11}_{-0.13}~{}\mbox{GeV}$,
$\lambda_{P_{cc}^{1/2}}=(2.51^{+0.46}_{-0.39})\times
10^{-3}~{}\mbox{GeV}^{6}$,
$\lambda_{P_{cc}^{3/2}}=(1.65^{+0.30}_{-0.25})\times
10^{-3}~{}\mbox{GeV}^{6}$,
$\lambda_{P_{bb}^{1/2}}=(7.53^{+1.52}_{-1.39})\times 10^{-3}~{}\mbox{GeV}^{6}$
and $\lambda_{P_{bb}^{3/2}}=(4.27^{+0.85}_{-0.78})\times
10^{-3}~{}\mbox{GeV}^{6}$. One of the fundamental components of the light-cone
sum rules for the magnetic moment is the photon DAs. The values of DAs are
obtained in Ref. [59], which we will employ in our numerical computations. The
explicit expressions of the photon DAs are given in Appendix A.

The light-cone sum rules calculation for magnetic moments of doubly-heavy
pentaquarks also contains two arbitrary parameters, the Borel mass $M^{2}$ and
the continuum threshold $s_{0}$. According to the philosophy of the method
used, we should find the working intervals in which the magnetic moments are
practically insensitive to variations of these parameters. To do this, two
constraints are applied, such as pole contribution (PC) and convergence of
operator product expansion (OPE). Our numerical analysis indicates that the
requirements of the method are fulfilled in the regions of arbitrary
parameters presented as

$\displaystyle 23.0~{}\mbox{GeV}^{2}\leq s_{0}\leq 25.0~{}\mbox{GeV}^{2},$
$\displaystyle 5.0~{}\mbox{GeV}^{2}\leq M^{2}\leq 7.0~{}\mbox{GeV}^{2},$ (36)

for doubly-charmed pentaquark states and

$\displaystyle 121.0~{}\mbox{GeV}^{2}\leq s_{0}\leq 125.0~{}\mbox{GeV}^{2},$
$\displaystyle 11.0~{}\mbox{GeV}^{2}\leq M^{2}\leq 15.0~{}\mbox{GeV}^{2},$
(37)

for doubly-bottom pentaquark states. In our calculations, PC changes on
average within limits $0.33\leq$ PC $\leq 0.58$, which is acceptable for
multiquark states. When we examine the OPE convergence, we have obtained that
the contribution of the higher dimensional term in OPE is less than $\sim
1\%$. As can be seen from these results, the chosen working regions for
$M^{2}$ and $s_{0}$ meet the requirements of the method.

In Fig. 1, as an example, we plot the dependencies of the magnetic moments of
doubly-charmed pentaquark states on $M^{2}$ at several fixed values of the
$s_{0}$. As can be seen from the figure, though being not completely
insensitive, the magnetic moments show reasonable dependency on the arbitrary
parameters, $s_{0}$ and $M^{2}$ which is acceptable in the error limits of the
light-cone sum rule method.

(a)  (b)

(c)  (d)

Figure 1: Dependencies of magnetic moments of $P^{1/2}_{cc}$ and
$P^{3/2}_{cc}$ states on $M^{2}$ at three different values of $s_{0}$; (a) and
(b) for $P^{1/2}_{cc}$ states; and (c) and (d) for $P^{3/2}_{cc}$ states.

We have determined all the parameters we need to complete the numerical
analysis of magnetic moments. Using the values of the parameters we
determined, we give the numerical results we obtained for the magnetic moment
as follows

$\displaystyle\mu_{P_{ccud\bar{u}}}$
$\displaystyle=1.08^{+0.38}_{-0.34}\,\mu_{N},~{}~{}~{}~{}~{}~{}~{}\mu_{P_{bbud\bar{u}}}=-2.48^{+0.74}_{-0.67}\,\mu_{N},$
(38) $\displaystyle\mu_{P_{ccud\bar{d}}}$
$\displaystyle=0.93^{+0.32}_{-0.29}\,\mu_{N},~{}~{}~{}~{}~{}~{}~{}\mu_{P_{bbud\bar{d}}}=-3.09^{+0.93}_{-0.84}\,\mu_{N},$
(39)

for spin-1/2 doubly-heavy pentaquark states,

$\displaystyle\mu_{P_{ccud\bar{u}}}$
$\displaystyle=4.23^{+2.00}_{-1.58}\,\mu_{N},~{}~{}~{}~{}~{}~{}~{}\mu_{P_{bbud\bar{u}}}=5.46^{+1.04}_{-1.00}\,\mu_{N},$
(40) $\displaystyle\mu_{P_{ccud\bar{d}}}$
$\displaystyle=2.01^{+0.86}_{-0.75}\,\mu_{N},~{}~{}~{}~{}~{}~{}~{}\mu_{P_{bbud\bar{d}}}=3.14^{+0.64}_{-0.61}\,\mu_{N},$
(41)

for spin-3/2 doubly-heavy pentaquark states. The errors in the results given
in Eqs. (38) and (41) are due to all input parameters, extra parameters such
as $s_{0}$ and $M^{2}$, as well as the parameters on which the wave functions
used in the photon distribution amplitudes depend. The magnetic moments of the
doubly-heavy pentaquark states have been extracted from the light-cone sum
rules employing for their physical representations a simple-pole approximation
[see, Eqs. (3) and (24)]. In the case of the multiquark hadrons such
approximation should be verified by supplementary arguments, because a
physical side of relevant sum rules receives contributions from two-hadron
reducible terms as well. This problem was first proposed during theoretical
studies of the pentaquarks [62, 63]. Two-hadron contaminating terms have to be
considered when extracting parameters of multiquark hadrons. In the case of
the multiquark hadrons they lead to modification in the quark propagator

$\frac{1}{m^{2}-p^{2}}\rightarrow\frac{1}{m^{2}-p^{2}-i\sqrt{p^{2}}\Gamma(p)},$
(42)

where $\Gamma(p)$ is the finite width of the multiquark hadrons generated by
two-hadron scattering states. When these effects are properly considered in
the sum rules, they rescale the residue of the multiquark hadrons under
investigations leaving its mass unchanged. Detailed investigations show that
two-hadron scattering effects are small for multiquark hadrons (see Refs. [63,
64, 65, 66, 67, 68, 69, 70, 71, 72, 73]). Therefore, in the present study the
zero-width single-pole approximation has been used.

When the results in Eqs. (38)-(41) are examined, it can be seen that the
results acquired for the magnetic moments are of measurable size in the
experiments. It is seen that the spin-1/2 doubly-heavy pentaquarks results are
close to each other, but the difference between the results of the spin-3/2
doubly-heavy pentaquark states is on the order of two. To understand the
reason for this difference, we extracted the individual quark contributions to
the magnetic moments. This can be done by dialing the corresponding charge
factors $e_{q}$ and $e_{Q}$. In case of spin-1/2 doubly-heavy pentaquarks, we
obtained that these magnetic moments are dominantly determined by the heavy-
quarks. The situation is the opposite in the spin-3/2 doubly-heavy
pentaquarks. In this case, the dominant contribution comes from light-quarks
and the contribution of heavy-quarks is negligible. A more detail
investigation shows that the smallness of the heavy-quarks contributions are
due to an almost exact cancellation of the expressions involving the heavy-
quarks, though these expressions are not small themselves. It will be
interesting and useful to examine the magnetic moments of these doubly-heavy
pentaquark states with different theoretical approaches.

In Refs. [41, 42], the magnetic moments of the $P_{c}(4312)$ and $P_{c}(4380)$
hidden-charm pentaquark states have been acquired within light-cone sum rules
by assuming them as diquark-diquark-antiquark and molecular configurations. In
these studies, the quark content of both $P_{c}(4312)$ and $P_{c}(4380)$
hidden-charm pentaquark states are considered as $c\bar{c}udu$, and the
obtained magnetic moment results for the diquark-diquark-antiquark picture
have been given as $\mu_{P_{c}(4312)}=0.40\pm 0.15~{}\mu_{N}$ and
$\mu_{P_{c}(4380)}=1.30\pm 0.50~{}\mu_{N}$. When the comparison is made for
the states with the same quark content, it is seen that there is a significant
difference between the magnetic moment results obtained for the hidden-charm
and doubly-charm pentaquarks. Whether the results obtained in this study are
consistent or not can be seen by examining the magnetic moments of these
possible doubly-heavy pentaquark states with other theoretical models.

For completeness, we have also acquired higher multipole moments, electric
quadrupole ($\mathcal{Q}$) and magnetic octupole ($\mathcal{O}$), of the
$P^{3/2}_{QQ}$ pentaquark states as

$\displaystyle\mbox{$P_{ccud\bar{u}}$
state}:\mathcal{Q}=-0.048^{+0.016}_{-0.014}~{}\mbox{fm${}^{2}$},~{}~{}~{}~{}~{}~{}~{}\mathcal{O}=-0.022^{+0.008}_{-0.006}~{}\mbox{fm${}^{3}$},$
(43) $\displaystyle\mbox{$P_{bbud\bar{u}}$
state}:\mathcal{Q}=-0.22^{+0.07}_{-0.07}~{}\mbox{fm${}^{2}$},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathcal{O}=-0.10^{+0.04}_{-0.03}~{}\mbox{fm${}^{3}$},$
(44) $\displaystyle\mbox{$P_{ccud\bar{d}}$
state}:\mathcal{Q}=0.024^{+0.008}_{-0.008}~{}\mbox{fm${}^{2}$},~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathcal{O}=0.011^{+0.005}_{-0.004}~{}\mbox{fm${}^{3}$},$
(45) $\displaystyle\mbox{$P_{bbud\bar{d}}$
state}:\mathcal{Q}=0.11^{+0.04}_{-0.03}~{}\mbox{fm${}^{2}$},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathcal{O}=0.06^{+0.002}_{-0.002}~{}\mbox{fm${}^{3}$}.$
(46)

As can be seen from these results, the charge distribution of the
$P^{3/2}_{QQ}$ pentaquark states is non-spherical. The sign of electric
quadrupole moment is positive for $P_{QQud\bar{d}}$ pentaquark states and
negative for $P_{QQud\bar{u}}$ pentaquark states, which correspond to the
prolate and oblate charge distributions, respectively.

##  IV Summary

Motivated by the latest discovery of doubly-charmed tetraquark $T^{+}_{cc}$ by
LHCb collaboration, we have studied the magnetic moments of the possible
doubly-heavy pentaquark states with quantum numbers $J^{P}=1/2^{-}$ and
$J^{P}=3/2^{-}$ in the framework of the light-cone sum rules method. In the
analysis, these possible pentaquark states are considered in diquark-diquark-
antiquark structure. We have also acquired non-vanishing values for the
electric quadrupole and magnetic octupole moments of spin-3/2 doubly-heavy
pentaquark states which mean non-spherical charge distribution. The magnetic
dipole and higher multipole moments of hadrons encode helpful details about
the distributions of the charge and magnetization inside the hadrons, which
help us to figure out their geometric configurations. The discovery of the
first doubly-charmed tetraquark state gave a new platform for hadron physics.
More theoretical and experimental attempts are required for to figure out its
fundamental structure and non-perturbative nature of QCD dynamics in this
region. It would be exciting to predict future experimental attempts that will
search possible doubly-heavy pentaquark states and test the obtainment from
the present analysis.

Data Availability Statement: This manuscript has no associated data or the
data will not be deposited. [Authors’ comment: This is a theoretical research
work, so no additional data are associated with this work.]

## Appendix A: Distribution Amplitudes of the photon

In the present appendix, 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 [59],

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

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 entering into the matrix elements above are
described as follows:

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

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

## Appendix B: The explicit expression of $\Delta^{QCD}$ function

In here, we present the explicit expression for the function $\Delta^{QCD}$
acquired from the light-cone sum rule in subsection II.1. It is acquired by
selecting the $\varepsilon\\!\\!\\!/p\\!\\!\\!/$ structure as follows

$\displaystyle\Delta^{QCD}$
$\displaystyle=\frac{P_{2}}{849346560\,\pi^{5}}\Bigg{\\{}10P_{1}(e_{d}+e_{u})\Bigg{[}4m_{Q}^{2}\Bigg{(}3m_{0}^{2}\Big{(}I[0,2,1,0]-2I[0,2,1,1]+I[0,2,1,2]-2I[0,2,2,0]$
$\displaystyle+2I[0,2,2,1]+I[0,2,3,0]-2\big{(}I[1,1,1,0]-2I[1,1,1,1]+I[1,1,1,2]-2I[1,1,2,0]+2I[1,1,2,1]$
$\displaystyle+I[1,1,3,0]\big{)}\Big{)}+4\Big{(}I[0,3,1,0]-3I[0,3,1,1]+3I[0,3,1,2]-I[0,3,1,3]-2I[0,3,2,0]+4I[0,3,2,1]$
$\displaystyle+I[0,3,3,0]-I[0,3,3,1]+3\big{(}I[1,2,1,1]-2I[1,2,1,2]+I[1,2,1,3]-2I[1,2,2,1]+2I[1,2,2,2]$
$\displaystyle-2I[0,3,2,2]+I[1,2,3,1]\big{)}\Big{)}\Bigg{)}+3\Big{(}I[0,4,2,0]-3I[0,4,2,1]+3I[0,4,2,2]-I[0,4,2,3]-3I[0,4,3,0]$
$\displaystyle+6I[0,4,3,1]-3I[0,4,3,2]+3I[0,4,4,0]-3I[0,4,4,1]-I[0,4,5,0]+4\big{(}3I[1,3,2,1]-3I[1,3,2,2]$
$\displaystyle+I[1,3,2,3]+3\big{(}-2I[1,3,3,1]+I[1,3,3,2]+I[1,3,4,1]\big{)}\big{)}\Big{)}\Bigg{]}$
$\displaystyle+9e_{Q}\Bigg{[}16m_{Q}^{2}\Bigg{(}5P_{1}\Big{(}I[0,3,1,0]-2I[0,3,1,1]+I[0,3,1,2]-2I[0,3,2,0]+2I[0,3,2,1]+I[0,3,3,0]\Big{)}$
$\displaystyle-45m_{0}^{2}\Big{(}I[0,4,1,1]-2I[0,4,1,2]+I[0,4,1,3]-2I[0,4,2,1]+2I[0,4,2,2]+I[0,4,3,1]\Big{)}+18\Big{(}I[0,5,1,2]$
$\displaystyle-2I[0,5,1,3]+I[0,5,1,4]-2I[0,5,2,2]+2I[0,5,2,3]+I[0,5,3,2]\Big{)}\Bigg{)}+3\Bigg{(}5P_{1}\Big{(}I[0,4,2,0]-3I[0,4,2,1]$
$\displaystyle+3I[0,4,2,2]-I[0,4,2,3]-3I[0,4,3,0]+6I[0,4,3,1]-3I[0,4,3,2]+3I[0,4,4,0]-3I[0,4,4,1]-I[0,4,5,0]$
$\displaystyle+4\big{(}3I[1,3,2,1]-3I[1,3,2,2]+I[1,3,2,3]+3\big{(}-2I[1,3,3,1]+I[1,3,3,2]+I[1,3,4,1]\big{)}\big{)}\Big{)}$
$\displaystyle+36m_{0}^{2}\Big{(}I[0,5,2,0]-4I[0,5,2,1]+6I[0,5,2,2]-4I[0,5,2,3]+I[0,5,2,4]-3I[0,5,3,0]+9I[0,5,3,1]$
$\displaystyle-9I[0,5,3,2]+3I[0,5,3,3]+3I[0,5,4,0]-6I[0,5,4,1]+3I[0,5,4,2]-I[0,5,5,0]+I[0,5,5,1]$
$\displaystyle+5\big{(}I[1,4,2,1]-3I[1,4,2,2]+3I[1,4,2,3]-I[1,4,2,4]-3\big{(}I[1,4,3,1]-2I[1,4,3,2]+I[1,4,3,3]$
$\displaystyle-I[1,4,4,1]+I[1,4,4,2]\big{)}-I[1,4,5,1]\big{)}\Big{)}\Bigg{)}-36\Bigg{(}2I[0,6,2,1]-7I[0,6,2,2]+9I[0,6,2,3]-5I[0,6,2,4]$
$\displaystyle+I[0,6,2,5]-6I[0,6,3,1]+15I[0,6,3,2]-12I[0,6,3,3]+3I[0,6,3,4]+6I[0,6,4,1]-9I[0,6,4,2]$
$\displaystyle+3I[0,6,4,3]-2I[0,6,5,1]+I[0,6,5,2]+6\Big{(}I[1,5,2,2]-3I[1,5,2,3]+3I[1,5,2,4]-I[1,5,2,5]$
$\displaystyle-3\big{(}I[1,5,3,2]-2I[1,5,3,3]+I[1,5,3,4]-I[1,5,4,2]+I[1,5,4,3]\big{)}-I[1,5,5,2]\Big{)}\Bigg{)}\Bigg{]}\Bigg{\\}}$
$\displaystyle+\frac{P_{2}}{(13589544960\,\pi^{5}}\Bigg{\\{}2560(e_{d}-e_{u})f_{3\gamma}m_{Q}^{2}P_{1}\pi^{2}(2I_{2}[\mathcal{A}]I[0,2,2,0]-I_{5}[\psi_{a}]I[0,2,3,0])$
$\displaystyle-80m_{Q}^{2}\Big{(}144(2e_{d}+e_{u})f_{3\gamma}m_{0}^{2}\pi^{2}I_{2}[\mathcal{V}]+e_{q}P_{1}\big{(}-44I_{1}[\mathcal{S}]-33I_{1}[\mathcal{T}_{2}]-184I_{3}[\mathcal{S}]+138I_{3}[\mathcal{T}_{1}]+193I_{3}[\mathcal{T}_{2}]$
$\displaystyle+11I_{3}[\mathcal{T}_{4}]\big{)}\Big{)}I[0,3,3,0]-345e_{q}P_{1}\Big{(}4I_{1}[\mathcal{S}]-3I_{1}[\mathcal{T}_{2}]+6I_{3}[\mathcal{T}_{2}]\Big{)}I[0,4,5,0]+360(2e_{d}+e_{u})f_{3\gamma}\pi^{2}I_{2}[\mathcal{V}]$
$\displaystyle\times\Big{(}32m_{Q}^{2}I[0,4,3,0]+3m0^{2}I[0,4,5,0]\Big{)}+864\Big{(}8e_{q}m_{Q}^{2}\big{(}4I_{3}[\mathcal{S}]-3I_{3}[\mathcal{T}_{1}]-3I_{3}[\mathcal{T}_{2}]\big{)}I[0,5,3,0]$
$\displaystyle+(2e_{d}+e_{u})f_{3\gamma}\pi^{2}I_{2}[\mathcal{V}]I[0,5,5,0]\Big{)}-432e_{q}\Big{(}4I_{1}[\mathcal{S}]-3I_{1}[\mathcal{T}_{2}]+2(-8I_{3}[\mathcal{S}]+6I_{3}[\mathcal{T}_{1}]+8I_{3}[\mathcal{T}_{2}]+I_{3}[\mathcal{T}_{4}])\Big{)}$
$\displaystyle\times I[0,6,5,0]$
$\displaystyle+2560(e_{d}-e_{u})f_{3\gamma}m_{Q}^{2}P_{1}\pi^{2}\Big{(}I[0,2,1,0]-2I[0,2,1,1]+I[0,2,1,2]-2I[0,2,2,0]+2I[0,2,2,1]+I[0,2,3,0]\Big{)}$
$\displaystyle\times\psi_{a}[u_{0}]\Bigg{\\}},$ (47)

where $P_{1}=\langle g_{s}^{2}G^{2}\rangle$ is gluon condensate,
$P_{2}=\langle\bar{q}q\rangle$ stands for u/d quark condensate. We should also
remark that in the Eq.(47), for simplicity we have only given the terms that
give significant contributions to the numerical values of the magnetic moments
and neglected to give many higher dimensional operators though they have been
taken into account in the numerical analyses. The $I[n,m,l,k]$,
$I_{1}[\mathcal{F}]$, $I_{2}[\mathcal{F}]$, $I_{3}[\mathcal{F}]$,
$I_{4}[\mathcal{F}]$, $I_{5}[\mathcal{F}]$, and $I_{6}[\mathcal{F}]$ functions
are defined as:

$\displaystyle I[n,m,l,k]$
$\displaystyle=\int_{4m_{Q}^{2}}^{s_{0}}ds\int_{0}^{1}dt\int_{0}^{1}dw~{}e^{-s/M^{2}}~{}s^{n}\,(s-4\,m_{Q}^{2})^{m}\,t^{l}\,w^{k},$
$\displaystyle I_{1}[\mathcal{F}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{A}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta^{\prime}(\alpha_{q}+\bar{v}\alpha_{g}-u_{0}),$
$\displaystyle I_{2}[\mathcal{F}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{A}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta^{\prime}(\alpha_{\bar{q}}+v\alpha_{g}-u_{0}),$
$\displaystyle I_{3}[\mathcal{F}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{A}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta(\alpha_{q}+\bar{v}\alpha_{g}-u_{0}),$
$\displaystyle I_{4}[\mathcal{F}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{A}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta(\alpha_{\bar{q}}+v\alpha_{g}-u_{0}),$
$\displaystyle I_{5}[\mathcal{F}]$
$\displaystyle=\int_{0}^{1}du~{}A(u)\delta^{\prime}(u-u_{0}),$ $\displaystyle
I_{6}[\mathcal{F}]$ $\displaystyle=\int_{0}^{1}du~{}A(u),$

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

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