# A new type of hydrogen-like charm-pion or charm-kaon matter

Si-Qiang Luo1,2,3,4,5<EMAIL_ADDRESS>Zhan-Wei Liu1,3,4,5
<EMAIL_ADDRESS>Xiang Liu1,3,4,5<EMAIL_ADDRESS>1School of Physical
Science and Technology, Lanzhou University, Lanzhou 730000, China
2School of mathematics and statistics, Lanzhou University, Lanzhou 730000,
China
3Research Center for Hadron and CSR Physics, Lanzhou University and Institute
of Modern Physics of CAS, Lanzhou 730000, China
4Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics
of Gansu Province, Lanzhou University, Lanzhou 730000, China
5Frontiers Science Center for Rare Isotopes, Lanzhou University, Lanzhou
730000, China

###### Abstract

Borrowing the structures of the hydrogen atom, molecular ion, and diatomic
molecule, we predict the nature of a new type of hydrogen-like charm-pion or
charm-kaon matter that could be obtained by replacing the proton and electron
in hydrogen matter with a charmed meson and a pion or a kaon, respectively. We
find that the spectra of the atom, molecular ion, and diatomic molecule can be
obtained simultaneously with the Coulomb potential for the hydrogen, the
charm-pion, and the charm-kaon systems. The predicted charm-pion matter also
allows us to explore the mass shift mediated by the strong interaction. For
the charm-pion and charm-kaon systems, the strong interactions could lead to
binding energy shifts. Our calculations suggests that the binding energy
shifts in charm-pion systems are in the order of several to tens of eV. For
the charm-kaon systems, the results are in the order of tens to hundreds of
eV. Exploring hydrogen-like charm-pion matter must lead to new demands for
high-precision experiments.

Introduction.—As a representative of the precision frontier of particle
physics, the exploration of exotic hadronic matter is always full of
challenges and opportunities, which is as an effective approach to deepen our
understanding of nonperturbative behaviour of quantum chromodynamics (QCD).
With the experimental progress in observing a series of charmoniumlike $XYZ$
states and several $P_{c}$ and $P_{cs}$ states in the last two decades
Chen:2016qju ; Liu:2013waa ; Yuan:2018inv ; Olsen:2017bmm ; Guo:2017jvc ;
Hosaka:2016pey ; Brambilla:2019esw ; Chen:2022asf , our knowledge of hadron
spectroscopy has become rich. Especially recently, the LHCb, CMS and ATLAS
collaborations continue to surprise us with the observation of $P_{\psi
s}^{\Lambda}(4338)$ in $B^{-}\to J/\psi\Lambda\bar{p}$ LHCb:2022jad ,
$T_{c\bar{s}0}^{a}(2900)^{0}$ and $T_{c\bar{s}0}^{a}(2900)^{++}$ in the
$B^{0}\to\bar{D}^{0}D_{s}^{+}\pi^{-}$ and $B^{+}\to{D}^{-}D_{s}^{+}\pi^{+}$
weak decays LHCb:2022xob ; LHCb:2022bkt , and several enhancement structures
in the di-$J/\psi$ invariant mass spectrum ATLAS:2022diJpsi ; CMS:2022diJpsi .
Currently, we are entering new phase of construction of “Particle Zoo 2.0”.

Since the birth of the quark model Gell-Mann:1964ewy ; Zweig:1964ruk ;
Zweig:1964jf , various types of hadronic configurations have been proposed,
including multi-quark state, hadronic molecular state, hybrid and glueball.
Among these exotic hadronic matters, the hadronic molecules, as loosely bound
states composed of color singlet hadrons, have been extensively used to
unravel the properties of these observed new hadronic states Chen:2016qju ;
Liu:2013waa ; Yuan:2018inv ; Olsen:2017bmm ; Guo:2017jvc ; Hosaka:2016pey ;
Brambilla:2019esw ; Chen:2022asf , where these hadronic molecular states are
formed by strong interaction. In fact, a positive charged hadron and a
negative charged hadron can be attracted to each other via the Coulomb
potential, which is similar to that of the atom. These hadronic matters are
expected to have much smaller binding energies and larger sizes than the
corresponding bound states induced directly by the strong interaction. Because
of their special properties, the search for such bound states would be an
interesting task. The typical examples are pion atoms, which are composed of a
pion and a nucleus Chatellard:1997nw ; Sigg:1996qd ; Thomas:1979xu . In 2011,
the SIDDHARTA Collaboration SIDDHARTA:2011dsy ; Bazzi:2012eq  observed a
$N\bar{K}$ system. In addition, kaonic deuterium has been of great concern to
both experimentalists and theorists Curceanu:2013bxa ; Zmeskal:2019ksw ;
Zmeskal:2015efj ; Curceanu:2019uph ; Liu:2020foc ; Meissner:2006gx ;
Barrett:1999cw ; Doleschall:2014wza . Although great progress has been made on
both the theoretical and experimental sides, we should introduce new insights
to reveal new aspects of exotic hadronic matter as we stand at a new stage in
the study of exotic hadronic matter.

Figure 1: The atom, molecular ion, and diatomic molecule in hydrogenous and
charm-pion matters. Here, by replacing $\pi^{-}$ with $K^{-}$, we can obtain
the corresponding charm-kaon matter.

In this Letter, we predict the existence of hydrogen-like charm-pion matters,
where we can borrow some ideas by examining the hydrogen atom, the molecular
ion, and the diatomic molecule, which are typical matter composed of the
positive charged proton and the negative charged electron via the Coulomb
potential, as shown in Fig. 1. Taking the charmed meson $D^{+}$ and $\pi^{-}$
as the counterparts of $p$ and $e^{-}$, one gets a new kind of hydrogen-like
charm-pion matter. As shown in Fig. 1, there is a hydrogen-like atom
($H_{D^{+}}^{\pi^{-}}$), a molecular ion ($H_{D^{+}D^{+}}^{\pi^{-}}$), and a
diatomic molecule ($H_{D^{+}D^{+}}^{\pi^{-}\pi^{-}}$) in charm-pion matter,
which correspond to the known hydrogenous matter. However, the proposed
hydrogen-like charm-pion matter has some unique properties that are different
from those of hydrogenous matter, which is the reason for attracting our
attention. First of all, the wave functions of these hydrogen-like charm-pion
matters are sufficiently simple because the $D^{+}$ and $\pi^{-}$ are spin-0
particles, which can give the explicitly quantitative analysis of them.
Second, because of the Pauli principle, the wave functions of two protons or
two electrons are antisymmetric. The situation is quite different in charm-
pion matter, where the wave functions of two charmed mesons or two pions must
be symmetric. In this letter, we perform a cross-calculation of the two
systems to study how such symmetry affects the solution. In addition, for the
charm-pion matter, the strong interaction as a perturbation contribution can
lead to the energy shift of the discussed system. Thus, the predicted charm-
pion matter can be used to study the nonperturbative behaviour of strong
interaction, which could be tested by the experimental measurement. Of course,
there is also charm-kaon matter as an extension of the focused hydrogeon-like
charm-pion matter, which will also be investigated in this letter.

Table 1: The binding energy $E$, root-mean-square radius $R$ of the atom, molecular ion, and diatomic molecule type systems. For the charm-pion and charm-kaon atoms, the decay widths are also given. | Hydrogen | Charm-pion | Charm-kaon
---|---|---|---
4 |  |  |  |  |  |  |
6 |  |  |  |  |  |  |
8 |  |  |  |  |  |  |
|  | Expt. | Theo. |  | Theo. |  | Theo.
Atom |  $E$ (eV) | $-13.6$ | $-13.6$ |  $E^{\rm QED}$ (keV) | $-3.458$ |  $E^{\rm QED}$ (keV) | $-10.421$
$R$ (nm) |  | 0.09 |  $R$ (fm) | 360.6 |  $R$ (fm) | 119.6
|  |  |  $\Delta E^{\rm OBE}$ (eV) | $-4.4\sim-11.5$ |  $\Delta E^{\rm OBE}$ (eV) | $-44.7\sim-88.0$
|  |  |  $\Gamma$ (eV) | 0.03$\sim$0.47 |  $\Gamma$ (eV) | 0.7$\sim$10.0
Molecular ion |  $B$ (eV) | $-16.25$ | $-16.20$ |  $E^{\rm QED}$ (keV) | $-3.848$ |  $E^{\rm QED}$ (keV) | $-11.182$
$R^{pp}$ (nm) |  | 0.11 |  $R^{D^{+}D^{+}}$ (fm) | 613.0 |  $R^{D^{+}D^{+}}$ (fm) | 259.2
$R^{pe^{-}}$ (nm) |  | 0.10 |  $R^{D^{+}\pi^{-}}$ (fm) | 496.0 |  $R^{D^{+}K^{-}}$ (fm) | 197.2
|  |  |  $\Delta E^{\rm OBE}$ (eV) | $-5.3\sim-14.6$ |  $\Delta E^{\rm OBE}$ (eV) | $-42.8\sim-87.2$
Diatomic molecule |  $E$ (eV) | $-31.65$ | $-31.60$ |  $E^{\rm QED}$ (keV) | $-7.517$ |  $E^{\rm QED}$ (keV) | $-21.889$
$R^{pp}$ (nm) |  | 0.127 |  $R^{D^{+}D^{+}}$ (fm) | 574.3 |  $R^{D^{+}D^{+}}$ (fm) | 187.3
$R^{e^{-}e^{-}}$ (nm) |  | 0.076 |  $R^{\pi^{-}\pi^{-}}$ (fm) | 435.4 |  $R^{K^{-}K^{-}}$ (fm) | 214.3
$R^{pe^{-}}$ (nm) |  | 0.094 |  $R^{D^{+}\pi^{-}}$ (fm) | 433.8 |  $R^{D^{+}K^{-}}$ (fm) | 164.7
|  |  |  $\Delta E^{\rm OBE}$ (eV) | $-13.1\sim-32.0$ |  $\Delta E^{\rm OBE}$ (eV) | $-103.9\sim-209.7$

The hydrogen-like charm-pion/kaon matter produced by the Coulomb
interaction.—In the charm-pion and charm-kaon systems, the electromagnetic
interaction is well established, which is a simple Coulomb potential. For the
systems of atom-type, the Schrödinger equation could be solved analytically.
Explicitly, the binding energy and the root-mean-square radius of the ground
state are

$E=-\frac{1}{2}\mu e^{4}\alpha^{2},~{}~{}~{}R=\frac{\sqrt{3}}{\mu
e^{2}\alpha}.$ (1)

In Eq. (1), the $\mu$, $\alpha$, and $e$ are the reduced mass, the fine
structure constant, and the charge, respectively. In addition, the wave
functions in coordination and momentum spaces are given by

$\psi({\bf r})=\frac{2(\mu e^{2}\alpha)^{3/2}}{\sqrt{4\pi}}{\rm e}^{-\mu
e^{2}\alpha r},~{}~{}~{}\psi({\bf p})=\frac{2\sqrt{2}}{\pi}\frac{(\mu
e^{2}\alpha)^{5/2}}{((\mu e^{2}\alpha)^{2}+p^{2})^{2}}.$ (2)

With the above preparations, one could easily obtain the binding energies of
$D^{+}\pi^{-}$ and $D^{+}K^{-}$ with $-3.456$ and $-10.421$ keV, respectively
(see Table 1). The root-mean-square radius of the $D^{+}\pi^{-}$ and
$D^{+}K^{-}$ could also be obtained by Eq. (1) with 360.6 fm and 119.6 fm,
respectively.

Figure 2: The Jacobi coordinates of the three-body and four-body systems.

However, for the molecular ion and diatomic molecule systems, the calculations
are much difficult. In this work, we use the Gaussian-Expansion-Method (GEM)
Hiyama:2003cu  to solve the three- and four-body Schrödinger equations for the
molecular ion and diatomic molecule systems, respectively. The corresponding
Jacobi coordinates of the three- and four-body systems are shown in Fig. 2. By
considering the symmetries in the two systems, the number of the Jacobi
coordinates of a molecular ion in Ref. Hiyama:2003cu  is reduced from 3 to 2.
The 18 Jacobi coordinates of a diatomic molecule in Ref. Hiyama:2003cu  are
reduced to 6 groups.

To make the calculations more reliable, we first use the hydrogen molecular
ion (${\rm H}_{2}^{+}$) and the hydrogen diatomic molecule (${\rm H}_{2}$) to
test the solutions of the three- and four-body Schrödinger equations,
respectively. As shown in Table 1, the experimental binding energies of ${\rm
H}_{2}^{+}$ and ${\rm H}_{2}$ are $-16.25$ and $-31.65$ eV, respectively,
which are very close to our calculated results of $-16.20$ and $-31.60$ eV,
respectively. This gives us confidence for the following calculations. The
calculated binding energies of $D^{+}D^{+}\pi^{-}$, $D^{+}D^{+}K^{-}$,
$D^{+}D^{+}\pi^{-}\pi^{-}$, and $D^{+}D^{+}K^{-}K^{-}$ are $-3.848$,
$-11.182$, $-7.517$, and $-21.889$ keV, respectively. The root-mean-square
radii of the charm-pion molecular ion and the diatomic molecule are roughly in
the range of $430\sim 610$ fm. For charm-kaon systems, the results are about
in the range of $160\sim 250$ fm.

For two identical particles, the wave functions are symmetric for spin-$0$,
but anti-symmetrical for spin-$\frac{1}{2}$. This leads to different
treatments for hydrogen and charm-pion matters. In order to obtain the details
of the bound state solutions more quickly, we use the following scaling trick.
The Schrödinger equation with electromagnetic interaction could be written as

$\left[\sum\limits_{i}\frac{p_{i}^{2}}{2m_{i}}+\sum\limits_{i<j}e_{i}e_{j}\frac{\alpha}{r_{ij}}\right]\psi=E\psi.$
(3)

If we normalize the mass $m_{i}$ with $m_{i}/{m_{\rm min}}$ and remove the
$\alpha$ in the potential, the Schrödinger equation could be simplified as

$\left[\sum\limits_{i}\frac{p_{i}^{\prime 2}}{2m_{i}/(m_{\rm
min})}+\sum\limits_{i<j}e_{i}e_{j}\frac{1}{r_{ij}^{\prime}}\right]\psi^{\prime}=E_{s}^{\prime}\psi^{\prime}.$
(4)

In the scheme, the solution depends only on the mass ratios. Since the
dimensions of the masses are omitted by dividing $m_{\rm min}$, the ${\bf
p}_{i}^{\prime}$, ${\bf r}_{ij}^{\prime}$, $E_{s}^{\prime}$, and
$\psi^{\prime}$ also could be treated as dimensionless terms. And the
practical solution could be obtained by the relationships

$\begin{split}&E=m_{\rm min}\alpha^{2}E_{s}^{\prime},\\\ &\psi\left(\\{{\bf
r}_{12},{\bf r}_{13},\cdots\\}\right)=(m_{\rm
min}\alpha)^{\frac{3(N-1)}{2}}\psi^{\prime}\left(m_{\rm
min}\alpha\times\\{{\bf r}_{12}^{\prime},{\bf
r}_{13}^{\prime},\cdots\\}\right),\\\ &\psi\left(\\{{\bf p}_{1},{\bf
p}_{2},\cdots\\}\right)=(m_{\rm
min}\alpha)^{-\frac{3(N-1)}{2}}\psi^{\prime}\left((m_{\rm
min}\alpha)^{-1}\times\\{{\bf p}_{1}^{\prime},{\bf
p}_{2}^{\prime},\cdots\\}\right),\end{split}$ (5)

where $N$ is the number of the particles. In the following, we fix $m_{\rm
min}$ and only change the mass ratios to get $E_{s}^{\prime}$.

Figure 3: The normalized binding energy $E_{s}^{\prime}$. Here, we define
$m^{-}$, i.e., the mass of the negative charged particle, as $m_{\rm min}$ in
Eq. (4).

The numerical results are shown in Fig. 3. In the molecular and diatomic
molecule systems, the symmetries of the hydrogen and charm-pion (kaon) the
wave functions follow different rules and lead to different binding energies.
However, this difference could be neglected. One possible reason is that the
Coulomb potential is spin-independent. Thus whether spin-$\frac{1}{2}$ or -$0$
systems, one could refer the binding energy from Fig. 3. Although it is
difficult to find the differences between the two types of the systems, the
symmetrization or anti-symmetrization is still indispensable, otherwise we
might get a non-physical or singular solution.

The mass shift mediated by the strong interaction.—The mass shifts are
important properties for a full understanding of the strong interactions. For
example, we need the hadron-hadron potentials of the strong interactions for
$D^{+}$-$\pi^{-}$, $D^{+}$-$K^{-}$, $D^{+}$-$D^{+}$, $\pi^{-}$-$\pi^{-}$, and
$K^{-}$-$K^{-}$. Here, we use one-boson-exchange (OBE) model to represent
them. According to their symmetries, the strong interactions could occur
through the exchange of $\sigma$ and $\rho$ mesons. For the $V^{D^{+}K^{-}}$,
$V^{D^{+}D^{-}}$, and $V^{K^{-}K^{-}}$, we also consider the potential of
$\omega$ exchange. With the effective Lagrangians in Refs. Chen:2016ypj ;
Lu:2016nlp ; Chen:2011cj , we present the effective potentials explicitly,
i.e.,

$\displaystyle V^{D^{+}\pi^{-}}$ $\displaystyle=$
$\displaystyle\frac{g_{DD\sigma}g_{\pi\pi\sigma}}{2m_{\pi}}Y_{\sigma}+\frac{1}{2}\beta
g_{V}g_{\rho\pi\pi}Y_{\rho},$ (6) $\displaystyle V^{D^{+}K^{-}}$
$\displaystyle=$
$\displaystyle\frac{g_{DD\sigma}g_{KK\sigma}}{2m_{K}}Y_{\sigma}+\frac{1}{2}\beta
g_{V}g_{KK\rho}(Y_{\rho}-Y_{\omega}),$ (7) $\displaystyle V^{D^{+}D^{+}}$
$\displaystyle=$ $\displaystyle-
g_{DD\sigma}g_{DD\sigma}Y_{\sigma}+\frac{1}{4}\beta^{2}g_{V}^{2}(Y_{\rho}+Y_{\omega}),$
(8) $\displaystyle V^{\pi^{-}\pi^{-}}$ $\displaystyle=$
$\displaystyle-\frac{g_{\pi\pi\sigma}^{2}}{4m_{\pi}^{2}}Y_{\sigma}+g_{\pi\pi\rho}^{2}Y_{\rho},$
(9) $\displaystyle V^{K^{-}K^{-}}$ $\displaystyle=$
$\displaystyle-\frac{g_{KK\sigma}^{4}}{4m_{K}^{2}}Y_{\sigma}+g_{KK\rho}^{2}(Y_{\rho}+Y_{\omega})$
(10)

with

$Y_{i}=\frac{{\rm e}^{-m_{i}r}}{4\pi r}-\frac{{\rm e}^{-\Lambda r}}{4\pi
r}-\frac{\Lambda^{2}{\rm e}^{-\Lambda r}}{8\pi\Lambda}+\frac{m_{i}^{2}{\rm
e}^{-\Lambda r}}{8\pi\Lambda}.$ (11)

In Eq. (11), the $m_{i}$ is the mass of the exchanged light flavor boson. The
$\Lambda$ is the cutoff parameter, defined in a monopole form factor ${\cal
F}(q^{2},m_{i}^{2})=({\Lambda^{2}-m_{i}^{2}})/({\Lambda^{2}-q^{2}})$, where
the $q$ is the momentum of the exchanged boson. In previous work, the values
of $\Lambda$ are around 1 GeV Chen:2016ypj ; Chen:2017jjn . In this work, we
extend the range of the $\Lambda$ with $1\sim 2$ GeV to discuss how large the
binding energy shifts are. In addition, we also need the coupling constants in
Eqs. (6)-(10). The coupling constants for the interactions between the $D$
meson and the light flavor mesons are taken to be $\lambda=0.9$, $g_{V}=5.8$,
and $g_{DD\sigma}=0.76$ Chen:2017jjn . Using the central experimental widths
of the $\rho$ and $\sigma$, we obtain $g_{\pi\pi\rho}=-6.01$ and
$g_{\pi\pi\sigma}=-3.52$ GeV, where the signs are determined by the quark
model. In addition, using the quark model relations, we have
$g_{KK\rho}\approx g_{KK\omega}\approx\frac{1}{2}g_{\pi\pi\rho}$ and
$g_{KK\sigma}\approx\frac{m_{K}}{2m_{\pi}}g_{\pi\pi\sigma}$.

Figure 4: The energy shift of the atom (left), the molecular ion (top right),
and the diatomic molecule (bottom right) in th charm-pion and charm-kaon
systems. The dashed dotted and solid dotted lines correspond to the results
for the charm-pion and charm-kaon systems. For the the charm-pion and charm-
kaon atoms, we also show their decay widths.

We first study the atomic type of charm-pion and charm-kaon systems. The
numerical results are presented in Table 1. The details of the binding energy
shifts as a function of the cutoff $\Lambda$ are shown in Fig. 4. The energy
shift of the $D^{+}\pi^{-}$ system is in the range of $-4.4\sim-11.5$ eV,
which is approximately the same as in the $p\pi^{-}$ system. For the
$D^{+}K^{-}$ system, the energy shift is in the range $-44.7\sim-88.0$ eV,
which is smaller than that of the $pK^{-}$ system SIDDHARTA:2011dsy , but
still comparable.

On the other hand, the strong decay behavior is also an important property.
The decay widths could be calculated using the following approach. If two
particles $AB$ form a bound state and are written as $[AB]$, the decay
amplitude could also be calculated by the scattering process with

$\begin{split}{\cal M}_{[AB]\to
CD}=\int&\frac{1}{(2\pi)^{3/2}}\frac{\sqrt{2m_{[AB]}}}{\sqrt{2m_{A}}\sqrt{2m_{B}}}\psi_{AB}({\bf
p}^{\prime})\\\ &\times{\cal M}_{AB\to CD}({\bf p},{\bf p}^{\prime}){\rm
d}^{3}{\bf p}^{\prime},\end{split}$ (12)

where ${\cal M}_{AB\to CD}({\bf p},{\bf p}^{\prime})$ is the scattering
amplitude. Then the decay width is

$\Gamma_{[AB]\to CD}=\frac{p}{32\pi^{2}m_{[AB]}^{2}}\int|{\cal M}_{[AB]\to
CD}|^{2}{\rm d}\Omega.$ (13)

In Eqs. (12)-(13), the ${\bf p}^{\prime}$ and ${\bf p}$ is the momentum of $A$
and $C$, respectively. $\psi_{AB}({\bf p}^{\prime})$ is the wave function of
the bound state $[AB]$ in momentum space. The $D^{+}\pi^{-}$ and $D^{+}K^{-}$
systems have sufficiently simple decay modes. The $D^{0}\pi^{0}$ and
$D^{0}\bar{K}^{0}$ are the main strong decay channels for $D^{+}\pi^{-}$ and
$D^{+}K^{-}$, respectively. So we need the amplitudes of $D^{+}\pi^{-}\to
D^{0}\pi^{0}$ and $D^{+}K^{-}\to D^{0}\bar{K}^{0}$, i.e.,

$\begin{split}i{\cal M}_{D^{+}\pi^{-}\to
D^{0}\pi^{0}}=&[i^{3}g_{\pi\pi\rho}(p_{\pi^{-}}^{0}+p_{\pi^{0}}^{0})]\frac{-ig_{00}+i\frac{q_{0}q_{0}}{m_{\rho}^{2}}}{q^{2}-m_{\rho}^{2}}\\\
&\times[-i\sqrt{2}\beta g_{V}m_{D}]{\cal
F}^{2}(q^{2},\Lambda^{2},m_{\rho}^{2}),\\\ i{\cal M}_{D^{+}K^{-}\to
D^{0}\bar{K}^{0}}=&[-i^{3}\sqrt{2}g_{KK\rho}(p_{K^{-}}^{0}+p_{\bar{K}^{0}}^{0})]\frac{-ig_{00}+i\frac{q_{0}q_{0}}{m_{\rho}^{2}}}{q^{2}-m_{\rho}^{2}}\\\
&\times[-i\sqrt{2}\beta g_{V}m_{D}]{\cal
F}^{2}(q^{2},\Lambda^{2},m_{\rho}^{2}).\end{split}$ (14)

We find that the phase spaces are extremely small, with the initial masses
only a few MeV above the thresholds. Thus, the two states are expected to be
very narrow. As shown in Table 4 and Fig. 4, the widths of $D^{+}\pi^{-}$ and
$D^{+}K^{-}$ are predicted to be of the order of $\sim$0.1 eV and $\sim$1 eV.
Such widths are so narrow when we compare the results with the commonly strong
decays. With the binding energy shifts and widths, we could calculate the
scattering lengths with Meissner:2004jr

$\begin{split}a_{D^{+}\pi^{-}}=&(0.067\sim 0.175)+(0.0002\sim 0.0036)i~{}{\rm
fm},\\\ a_{D^{+}K^{-}}=&(0.074\sim 0.145)+(0.0006\sim 0.0083)i~{}{\rm
fm}.\end{split}$ (15)

For the energy shifts or widths, the numerical results of $D^{+}K^{-}$ are
much larger than $D^{+}\pi^{-}$ using the same cutoff $\Lambda$. This can be
explained qualitatively by the root-mean-square radii of the two systems.
According to Eq. (1), the root-mean-square radii are inversely proportional to
the reduced mass. So we could get the root-mean-square radius relation with
$R_{D^{+}K^{-}}<R_{D^{+}\pi^{-}}$. Since the strong interaction is the typical
short-range force, the system with a small root-mean-square radius can have
larger strong interactions. In this way, it is not difficult to understand the
large numerical differences between charm-pion and charm-kaon matters as shown
in Fig. 4 and Table 1.

For the molecular ion and the diatomic molecule, the strong interactions also
increase the binding energy shifts. In the $H_{D^{+}D^{+}}^{\pi^{-}}$ and
$H_{D^{+}D^{+}}^{K^{-}}$ systems, the shifts are $-5.3\sim-14.6$ and
$-42.8\sim-87.2$ eV, respectively. Such results are similar to the atomic
scheme. For $H_{D^{+}D^{+}}^{\pi^{-}\pi^{-}}$ and
$H_{D^{+}D^{+}}^{K^{-}K^{-}}$, the obtained energy shifts are $-13.1\sim-32.0$
and $-103.9\sim-209.7$ eV, respectively, which are about as twice large as the
results in the atomic or molecular ion situation.

Summary.—It is fascinating to explore new forms of matter, which never cease
to surprises us. Ordinary atoms make up our visible world, but we need to
explore other kinds of atoms to reveal the essence of nature more deeply. The
charm-pion and charm-kaon atoms or ions have many special properties that the
ordinary atoms do not have. The ratios of the constituent masses of such
structures are very different from those of ordinary atoms, and the systems
composed of bosons and fermions would behave differently because of the
statistics. It is also very interesting to study the couplings of
electromagnetic and strong interactions in the formation of this new type of
matter.

Following the conventional hydrogen matter, we study charm-pion and charm-kaon
states with configurations of atom, molecular ion, and diatomic molecule. We
discuss the binding energies and present the strong decay widths. The energy
shifts due to the strong interaction are in the range of several to tens of eV
for charm-pion systems, and are tens to hundreds of eV for the charm-kaon
systems. In this work, the effect of the strong interaction is small, but
important for us to understand its behavior.

In recent years, a number of high energy physics collaborations have produced
many exciting results in hadrons physics. Such observations greatly advance
our understanding of hadron structures and strong interactions. Although we
predict the existence of a new types of hydrogen-like charm-pion or charm-kaon
matter, it is still a challenge to search for such exotic matter
experimentally. However, we believe that the great experimental challenges
would inspire the experimentalists to improve the precision and break the
detection limit in the future.

Acknowledgement.—We would like to thank Zhi-Feng Sun, Fu-Lai Wang and Li-Sheng
Geng for useful discussions. This work is supported by the China National
Funds for Distinguished Young Scientists under Grant No. 11825503, the
National Key Research and Development Program of China under Contract No.
2020YFA0406400, the 111 Project under Grant No. B20063, the National Natural
Science Foundation of China under Grant No. 12175091, 11965016, 12247101, and
the Project for Top-notch Innovative Talents of Gansu Province.

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# Magnetic dipole moments of the hidden-charm pentaquark states:
$P_{c}(4440)$, $P_{c}(4457)$ and $P_{cs}(4459)$

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

###### Abstract

In this work, we employ the light-cone QCD sum rule to calculate the magnetic
dipole moments of the $P_{c}(4440)$, $P_{c}(4457)$ and $P_{cs}(4459)$
pentaquark states by considering them as the diquark-diquark-antiquark and
molecular pictures with quantum numbers $J^{P}=\frac{3}{2}^{-}$,
$J^{P}=\frac{1}{2}^{-}$ and $J^{P}=\frac{1}{2}^{-}$, respectively. In the
analyses, we use the diquark-diquark-antiquark and molecular form of
interpolating currents, and photon distribution amplitudes to obtain the
magnetic dipole moment of pentaquark states. Theoretical examinations on
magnetic dipole moments of the hidden-charm pentaquark states, are essential
as their results can help us better figure out their substructure and the
dynamics of the QCD as the theory of the strong interaction. As a by product,
we extract the electric quadrupole and magnetic octupole moments of the
$P_{c}(4440)$ pentaquark. These values show a non-spherical charge
distribution.

Pentaquarks, magnetic moment, $P_{c}(4440)$, $P_{c}(4457)$, $P_{cs}(4459)$,
diquark-diquark-antiquark picture, molecular picture

##  I Motivation

With the experimental discovery of the exotic X(3872) state, i.e., state that
cannot be interpreted by the conventional meson or baryon picture, a new era
began in high energy physics. Since the discovery of this state, numerous
exotic states have been added to the family of particles that have been
experimentally reported. Study of exotic particles that are composed of
tetraquark, pentaquark or hybrid states is among the most attractive subjects
of the hadron physics. Experimental data collected with various collaborations
in recent years and theoretical developments obtained in different theoretical
models form the field of rapidly growing exotic studies Chen:2016qju ;
Ali:2017jda ; Esposito:2016noz ; Olsen:2017bmm ; Lebed:2016hpi ; Guo:2017jvc ;
Albuquerque:2018jkn ; Brambilla:2019esw ; Liu:2019zoy ; Agaev:2020zad ;
Dong:2021juy .

In 2019, LHCb collaboration reported three new narrow pentaquark states
Aaij:2019vzc  as

$\displaystyle P_{c}(4312):\mbox{M}$ $\displaystyle=4311.9\pm
0.7^{+6.8}_{-0.6}~{}\mbox{MeV},~{}~{}~{}\Gamma=9.80\pm
2.7^{+3.7}_{-4.5}~{}\mbox{MeV},$ $\displaystyle P_{c}(4440):\mbox{M}$
$\displaystyle=4440.3\pm
1.3^{+4.1}_{-4.7}~{}\mbox{MeV},~{}~{}~{}\Gamma=20.6\pm
4.9^{+8.7}_{-10.1}~{}\mbox{MeV},$ $\displaystyle P_{c}(4457):\mbox{M}$
$\displaystyle=4457.3\pm
0.6^{+4.1}_{-1.7}~{}\mbox{MeV},~{}~{}~{}\Gamma=6.40\pm
2.0^{+5.7}_{-1.9}~{}\mbox{MeV}.$

Very recently, the LHCb Collaboration reported a pentaquark state with
strangeness, $P_{cs}(4459)$, in the invariant mass spectrum of $J/\psi\Lambda$
in the $\Xi_{b}^{0}\rightarrow J/\psi\,\Lambda\,K^{-}$ decay Aaij:2020gdg :

$\displaystyle P_{cs}(4459):\mbox{M}$ $\displaystyle=4458.8\pm
2.7^{+4.7}_{-1.1}~{}\mbox{MeV},~{}~{}~{}\Gamma=17.3\pm
6.5^{+8.0}_{-5.7}~{}\mbox{MeV}.$ (1)

But, spin and parity of the $P_{cs}(4459)$ state have not been determined yet.
As regards to their decay products, one can easily conclude that these newly
discovered four states consist of at least five quarks, $\bar{c}cuud$ or
$\bar{c}cuds$, therefore they are perfect candidates of hidden-charm
pentaquark states. After the experimental discovery several phenomenological
models have been adopted to calculate the spectroscopic parameters, decays and
production mechanisms of the pentaquarks, like the QCD sum rule, the meson-
exchange model, the quark delocalization model, and so on Chen:2019asm ;
Chen:2019bip ; Liu:2019tjn ; He:2019ify ; Xiao:2019mvs ; Guo:2019kdc ;
Xiao:2019aya ; Zhang:2019xtu ; Wu:2019rog ; Wang:2019ato ; Xu:2020gjl ;
Peng:2020gwk ; Shimizu:2019ptd ; Zhu:2019iwm ; Wang:2019got ; Wang:2018waa ;
Cheng:2019obk ; Yamaguchi:2019seo ; Pan:2019skd ; Liu:2019zvb ; Fernandez-
Ramirez:2019koa ; Eides:2019tgv ; Guo:2019fdo ; Cao:2019kst ; Wang:2019dsi ;
Wang:2019krd ; Xiao:2020frg ; Mutuk:2019snd ; Yang:2020twg ; Dong:2020nwk ;
Wang:2020rdh ; Du:2019pij ; Wang:2019spc ; Azizi:2020ogm ; Chen:2020opr ;
Chen:2020uif ; Liu:2020hcv ; Peng:2020hql ; Chen:2020kco ; Wang:2020eep ;
Azizi:2021utt ; Zhu:2021lhd . However, the substructure of these states are
not determined yet. In other words, in order to understand the internal
structure and nature of these particles, different properties should be
studied besides their decay channels and spectroscopic properties. For
instance, investigating their electromagnetic form factors may provide
important insights on this point.

Electromagnetic form factors or multi-pole moments of hadrons are important
parameters in study of their electromagnetic structure, and also they can
ensure important knowledge about the dynamics of the QCD at low energy region.
Electromagnetic multi-pole moments, especially magnetic dipole moment, are
also a crucial part in the calculation of J$/\psi$ photo-production cross
sections, which can provide an independent analysis of the hidden-charm
pentaquark states. Investigating electromagnetic features of exotic resonances
is relatively new topic. However, in the literature there are a few studies
where the electromagnetic properties of the hidden-charm pentaquark states are
investigated Wang:2016dzu ; Ozdem:2018qeh ; Ortiz-Pacheco:2018ccl ;
Ozdem:2021btf ; Xu:2020flp . In Ref. Wang:2016dzu , the magnetic dipole moment
of the hidden-charm pentaquark states have been extracted in the diquark-
diquark-antiquark, diquark-triquark and molecular configuration with
$J^{P}=\frac{1}{2}^{\pm},\frac{3}{2}^{\pm},\frac{5}{2}^{\pm}\mbox{and}~{}\frac{7}{2}^{+}$
quantum numbers in the different color-flavor structure. In Ref. Ozdem:2018qeh
, the magnetic dipole, electric quadrupole and magnetic octupole moments of
the $P_{c}(4380)$ pentaquark with $J^{P}=\frac{3}{2}^{-}$ quantum numbers have
been obtained in the diquark-diquark-antiquark and molecular pictures in the
framework of the light-cone QCD sum rule (LCSR). In Ref. Ortiz-Pacheco:2018ccl
, they acquired the the ground state of hidden-charm pentaquarks with
$J^{P}=\frac{3}{2}^{-}$ quantum numbers and their associated magnetic dipole
moments and electromagnetic couplings, of interest to pentaquark
photoproduction experiments in the framework of the constituent quark model.
In Ref. Ozdem:2021btf , the magnetic dipole moment of the $P_{c}(4312)$
pentaquark state have been extracted in the molecular and diquark-diquark-
antiquark pictures via the LCSR with $J^{P}=\frac{1}{2}^{-}$ quantum numbers.
In Ref. Xu:2020flp , they achieved the magnetic dipole moment of the
$P_{c}(4312)$ pentaquark state in the $\Sigma_{c}\bar{D}$ molecular picture by
means of the QCD sum rule (QCDSR) in the external weak electromagnetic field.

In this study, the magnetic dipole moments of the pentaquark states
$P_{c}(4440)$, $P_{c}(4457)$ and $P_{cs}(4459)$ (hereafter we will show these
states as $P_{c1}$, $P_{c2}$ and $P_{cs}$, respectively) are obtained by
employing the diquark-diquark-antiquark and molecular interpolating currents
within the LCSR. The LCSR is a powerful tool to obtain qualitative and
quantitative information about hadron properties. In this method, hadrons are
analytically studied by matching their behaviors at high and low energies.
This matching provides characteristic properties of hadrons from QCD. The
information obtained by LCSR has been used as an input to other theoretical
and experimental approaches for a long time Chernyak:1990ag ; Braun:1988qv ;
Balitsky:1989ry .

The rest of paper is structured as follows. In Sect. II, the method used for
the calculations are described. In Sect. III, we carry out numerical analysis
of the acquired LCSR for the magnetic dipole moment of pentaquark states. The
explicit expressions of the magnetic moment of the $P_{c1}$ pentaquark in the
diquark-diquark-antiquark picture is presented in Appendix A.

##  II Magnetic dipole moments of the pentaquark states via LCSR

In this section, we briefly introduce the LCSR method used to extract the
magnetic dipole moments of the $P_{c1}$, $P_{c2}$ and $P_{cs}$ pentaquark
states. In order to obtain the magnetic dipole moment of the corresponding
states with the QCD sum rule method, we begin by writing the correlation
function suitable for the calculations. This correlation function is obtained
in two representations which are called hadronic and QCD representations. The
LCSR for the physical quantities are obtained from the matches of the
coefficients of the same Lorentz structures achieved on both representations
with the help of the quark-duality ansatz.

###  II.1 Formalism of the $P_{c2}$ and $P_{cs}$ states

The current correlation function for the $P_{c2}$ and $P_{cs}$ pentaquark
states is written as

$\Pi(p,q)=i\int d^{4}xe^{ip\cdot x}\langle
0|\mathcal{T}\\{J_{P_{c2(cs)}}(x)\bar{J}_{P_{c2(cs)}}(0)\\}|0\rangle_{\gamma},$
(2)

where $J_{P_{c2(cs)}}(x)$ is the interpolating current of $P_{c2}$ or $P_{cs}$
pentaquark state. In the diquark-diquark-antiquark and molecular pictures with
quantum number $J^{P}=\frac{1}{2}^{-}$, they are written as

$\displaystyle J_{P_{c2}}^{Di}(x)$ $\displaystyle=$
$\displaystyle\varepsilon^{abc}\varepsilon^{ade}\varepsilon^{bfg}\big{[}u^{T}_{d}(x)C\gamma_{5}d_{e}(x)\,u^{T}_{f}(x)C\gamma_{\mu}c_{g}(x)\,\gamma_{5}\gamma^{\mu}C\bar{c}^{T}_{c}(x)\big{]},$
$\displaystyle J_{P_{c2}}^{Mol}(x)$ $\displaystyle=$
$\displaystyle\big{[}\bar{c}_{d}(x)\gamma_{\mu}u_{d}(x)\big{]}\gamma^{\mu}\gamma_{5}\big{[}\varepsilon^{abc}u^{T}_{a}(x)C\gamma_{\nu}d_{b}(x)\gamma^{\nu}\gamma_{5}c_{c}(x)\big{]},$
(3) $\displaystyle J_{P_{cs}}^{Di}(x)$ $\displaystyle=$
$\displaystyle\varepsilon^{abc}\varepsilon^{ade}\varepsilon^{bfg}\big{[}u^{T}_{d}(x)C\gamma_{5}d_{e}(x)\,s^{T}_{f}(x)C\gamma_{5}c_{g}(x)\,C\bar{c}^{T}_{c}(x)\big{]},$
$\displaystyle J_{P_{cs}}^{Mol}(x)$ $\displaystyle=$
$\displaystyle\big{[}\bar{c}_{d}(x)\gamma_{\mu}u_{d}(x)\big{]}\gamma^{\mu}\gamma_{5}\big{[}\varepsilon^{abc}d^{T}_{a}(x)C\gamma_{5}s_{b}(x)c_{c}(x)\big{]},$
(4)

where $C$ is the charge conjugation matrix; and $a$, $b$… are color indices.

As we mentioned at the beginning of this section, in LCSR studies we need to
evaluate the correlation function at both hadron and quark-gluon degrees of
freedom. At the hadron level, we embedding a complete set of intermediate
pentaquark states into the correlation function to acquire the hadronic
representation, and isolate the ground state pentaquark states, and get the
results:

$\displaystyle\Pi^{Had}(p,q)$ $\displaystyle=\frac{\langle 0\mid
J_{P_{c2(cs)}}\mid{P_{c2(cs)}}(p,s)\rangle}{[p^{2}-m_{P_{c2(cs)}}^{2}]}\langle{P_{c2(cs)}}(p,s)\mid{P_{c2(cs)}}(p+q,s)\rangle_{\gamma}\frac{\langle{P_{c2(cs)}}(p+q,s)\mid\bar{J}_{P_{c2(cs)}}\mid
0\rangle}{[(p+q)^{2}-m_{{P_{c2(cs)}}}^{2}]}+...,$ (5)

The matrix element
$\langle{P_{c2(cs)}}(p,s)\mid{P_{c2(cs)}}(p+q,s)\rangle_{\gamma}$ entering Eq.
(5) can be parameterized in terms of Lorentz invariant form factors as
follows:

$\displaystyle\langle{P_{c2(cs)}}(p,s)\mid{P_{c2(cs)}}(p+q,s)\rangle_{\gamma}$
$\displaystyle=\varepsilon^{\mu}\,\bar{u}(p,s)\Big{[}\big{[}f_{1}(q^{2})+f_{2}(q^{2})\big{]}\gamma_{\mu}+f_{2}(q^{2})\frac{(2p+q)_{\mu}}{2m_{P_{c2(cs)}}}\Big{]}\,u(p+q,s),$
(6)

where $\varepsilon$ and q are the polarization vector and momentum of the
photon, respectively.

Substituting Eq. (6) in Eq. (5) for hadronic side we get

$\displaystyle\Pi^{Had}(p,q)=$
$\displaystyle\lambda^{2}_{P_{c2(cs)}}\gamma_{5}\frac{\Big{(}p\\!\\!\\!/+m_{P_{c2(cs)}}\Big{)}}{[p^{2}-m_{{P_{c2(cs)}}}^{2}]}\varepsilon^{\mu}\Bigg{[}\big{[}f_{1}(q^{2})+f_{2}(q^{2})\big{]}\gamma_{\mu}+f_{2}(q^{2})\,\frac{(2p+q)_{\mu}}{2m_{P_{c2(cs)}}}\Bigg{]}\gamma_{5}\frac{\Big{(}p\\!\\!\\!/+q\\!\\!\\!/+m_{P_{c2(cs)}}\Big{)}}{[(p+q)^{2}-m_{{P_{c2(cs)}}}^{2}]},$
(7)

with $\langle 0\mid J_{P_{c2(cs)}}\mid
P_{c2(cs)}(p,s)\rangle=\lambda_{P_{c2(cs)}}\gamma_{5}\,u(p,s)$. The Lorentz
invariant form factors $f_{1}(q^{2})$ and $f_{2}(q^{2})$ are related to the
magnetic form factor by the relation,

$\displaystyle G_{M}(q^{2})=f_{1}(q^{2})+f_{2}(q^{2}).$ (8)

At $q^{2}=0$, the magnetic form factor is acquired associated with the
functions $f_{1}(q^{2})$ and $f_{2}(q^{2})$ as:

$\displaystyle G_{M}(0)=f_{1}(0)+f_{2}(0).$ (9)

The magnetic dipole moment, $\mu_{P_{c2(cs)}}$, is defined in the following
way:

$\displaystyle\mu_{P_{c2(cs)}}=G_{M}(0)\,\frac{e}{2\,m_{P_{c2(cs)}}}.$ (10)

We observe from Eq.(7) that the correlation function contains many structures,
any of them can be selected in obtaining magnetic dipole moment of the
$P_{c2(cs)}$ pentaquark, and thus we decide on the structure
$\varepsilon\\!\\!\\!/q\\!\\!\\!/$. As a result, the correlation function can
be expressed with regards to the magnetic dipole moment of $P_{c2(cs)}$
pentaquark as,

$\displaystyle\Pi^{Had}(p,q)$
$\displaystyle=\frac{\lambda^{2}_{P_{c2(cs)}}\,m_{P_{c2(cs)}}}{[(p+q)^{2}-m^{2}_{P_{c2(cs)}}]}\,\mu_{P_{c2(cs)}}\,\frac{1}{[p^{2}-m^{2}_{P_{c2(cs)}}]}.$
(11)

At the QCD level, we contract the relevant quark fields in the correlation
function with the help of the Wick’s theorem,

$\displaystyle\Pi^{QCD-Di}_{P_{c2}}(p,q)$
$\displaystyle=i\,\varepsilon^{abc}\varepsilon^{a^{\prime}b^{\prime}c^{\prime}}\varepsilon^{ade}\varepsilon^{a^{\prime}d^{\prime}e^{\prime}}\varepsilon^{bfg}\varepsilon^{b^{\prime}f^{\prime}g^{\prime}}\int
d^{4}xe^{ip\cdot x}\langle 0|\bigg{\\{}$ $\displaystyle
Tr\Big{[}\gamma_{5}S_{d}^{ee^{\prime}}(x)\gamma_{5}\tilde{S}_{u}^{dd^{\prime}}(x)\Big{]}Tr\Big{[}\gamma_{\mu}S_{c}^{gg^{\prime}}(x)\gamma_{\nu}\tilde{S}_{u}^{ff^{\prime}}(x)\Big{]}$
$\displaystyle-
Tr\Big{[}\gamma_{5}S_{d}^{ee^{\prime}}(x)\gamma_{5}\tilde{S}_{u}^{fd^{\prime}}(x)\gamma\mu
S_{c}^{gg^{\prime}}(x)\gamma_{\nu}\tilde{S}_{u}^{df^{\prime}}(x)\Big{]}\bigg{\\}}\Big{(}\gamma_{5}\gamma^{\mu}S_{c}^{c^{\prime}c}(-x)\gamma^{\nu}\gamma_{5}\Big{)}|0\rangle_{\gamma},$
(12) $\displaystyle\Pi^{QCD-Mol}_{P_{c2}}(p,q)$
$\displaystyle=i\,\epsilon^{abc}\epsilon^{a^{\prime}b^{\prime}c^{\prime}}\,\int
d^{4}xe^{ip\cdot x}\langle
0|\Bigg{\\{}\mathrm{Tr}\Big{[}\gamma_{\mu}S_{u}^{dd^{\prime}}(x)\gamma_{\alpha}S_{c}^{d^{\prime}d}(-x)\Big{]}\mathrm{Tr}\Big{[}\gamma_{\nu}S_{d}^{bb^{\prime}}(x)\gamma_{\beta}\widetilde{S}_{u}^{aa^{\prime}}(x)]$
$\displaystyle-\mathrm{Tr}\Big{[}\gamma_{\mu}S_{u}^{da^{\prime}}(x)\gamma_{\beta}\widetilde{S}_{d}^{bb^{\prime}}(x)\Big{]}\mathrm{Tr}\Big{[}\gamma_{\nu}{S}_{u}^{ad^{\prime}}(x)\gamma_{\alpha}S_{c}^{d^{\prime}d}(-x)\Big{]}\Bigg{\\}}\Big{(}\gamma^{\mu}\gamma^{\nu}S_{c}^{cc^{\prime}}(x)\gamma^{\alpha}\gamma^{\beta}\Big{)}\Bigg{|}0\rangle_{\gamma},$
(13)

$\displaystyle\Pi^{QCD-Di}_{P_{cs}}(p,q)=$
$\displaystyle-i\,\varepsilon^{abc}\varepsilon^{a^{\prime}b^{\prime}c^{\prime}}\varepsilon^{ade}\varepsilon^{a^{\prime}d^{\prime}e^{\prime}}\varepsilon^{bfg}\varepsilon^{b^{\prime}f^{\prime}g^{\prime}}\int
d^{4}xe^{ip\cdot x}$ $\displaystyle\langle
0|Tr\Big{[}\gamma_{5}S_{d}^{ee^{\prime}}(x)\gamma_{5}\tilde{S}_{u}^{dd^{\prime}}(x)\Big{]}Tr\Big{[}\gamma_{5}S_{c}^{gg^{\prime}}(x)\gamma_{5}\tilde{S}_{s}^{ff^{\prime}}(x)\Big{]}\tilde{S}_{c}^{c^{\prime}c}(-x)|0\rangle_{\gamma},$
(14) $\displaystyle\Pi_{P_{cs}}^{\mathrm{QCD-Mol}}(p,q)=$
$\displaystyle-i\,\epsilon^{abc}\epsilon^{a^{\prime}b^{\prime}c^{\prime}}\,\int
d^{4}xe^{ip\cdot x}$ $\displaystyle\langle
0|\mathrm{Tr}\Big{[}\gamma_{\nu}S_{u}^{dd^{\prime}}(x)\gamma_{\mu}S_{c}^{d^{\prime}d}(-x)\Big{]}\mathrm{Tr}\Big{[}\gamma^{\nu}\widetilde{S}_{s}^{bb^{\prime}}(x)\gamma^{\mu}S_{d}^{aa^{\prime}}(x)\Big{]}S_{c}^{cc^{\prime}}(x)|0\rangle_{\gamma},$
(15)

where

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

with $S_{q(c)}(x)$ being the full light and charm quark propagators. The
relevant propagators are given as Balitsky:1987bk

$\displaystyle S_{q}(x)$
$\displaystyle=i\frac{{x\\!\\!\\!/}}{2\pi^{2}x^{4}}-\frac{\bar{q}q}{12}\Big{(}1-i\frac{m_{q}x\\!\\!\\!/}{4}\Big{)}-\frac{\bar{q}q}{192}m_{0}^{2}x^{2}\Big{(}1-i\frac{m_{q}x\\!\\!\\!/}{6}\Big{)}-\frac{ig_{s}}{32\pi^{2}x^{2}}~{}G^{\mu\nu}(x)\Big{[}\hbox
to0.0pt{/\hss}{x}\sigma_{\mu\nu}+\sigma_{\mu\nu}\hbox
to0.0pt{/\hss}{x}\Big{]},$ (16) $\displaystyle S_{c}(x)$
$\displaystyle=\frac{m_{c}^{2}}{4\pi^{2}}\Bigg{[}\frac{K_{1}\Big{(}m_{c}\sqrt{-x^{2}}\Big{)}}{\sqrt{-x^{2}}}+i\frac{{x\\!\\!\\!/}~{}K_{2}\Big{(}m_{c}\sqrt{-x^{2}}\Big{)}}{(\sqrt{-x^{2}})^{2}}\Bigg{]}-\frac{g_{s}m_{c}}{16\pi^{2}}\int_{0}^{1}dv\,G^{\mu\nu}(vx)\Bigg{[}(\sigma_{\mu\nu}{x\\!\\!\\!/}+{x\\!\\!\\!/}\sigma_{\mu\nu})$
$\displaystyle\times\frac{K_{1}\Big{(}m_{c}\sqrt{-x^{2}}\Big{)}}{\sqrt{-x^{2}}}+2\sigma_{\mu\nu}K_{0}\Big{(}m_{c}\sqrt{-x^{2}}\Big{)}\Bigg{]},$
(17)

where $K_{i}$ are modified the second kind Bessel functions and $G^{\mu\nu}$
is the gluon field strength tensor.

The QCD representation of the correlation function can be obtained with the
help of photon distribution amplitudes (DAs) according to quark-gluon
properties and after performing the Fourier transform to transfer the
calculations to the momentum space.

As a final step, by applying the double Borel transform on the variables
$-p^{2}$ and $(p+q)^{2}$ and choosing the coefficients of the same Lorentz
structures in both QCD and hadronic representations and matching them
employing the quark-hadron duality approach, we obtain the desired LCSR for
magnetic dipole moments of $P_{c2}$ and $P_{cs}$ states:

$\displaystyle\mu_{P_{c2}}^{Di}\,\lambda^{2-Di}_{P_{c2}}\,m_{P_{c2}}=e^{\frac{m^{2}_{P_{c2}}}{M^{2}}}\,\Delta_{1}^{QCD},$
(18)
$\displaystyle\mu_{P_{c2}}^{Mol}\,\lambda^{2-Mol}_{P_{c2}}\,m_{P_{c2}}=e^{\frac{m^{2}_{P_{c2}}}{M^{2}}}\,\Delta_{2}^{QCD},$
(19)
$\displaystyle\mu_{P_{cs}}^{Di}\,\lambda^{2-Di}_{P_{cs}}\,m_{P_{cs}}=e^{\frac{m^{2}_{P_{cs}}}{M^{2}}}\,\Delta_{3}^{QCD}.$
(20)
$\displaystyle\mu_{P_{cs}}^{Di}\,\lambda^{2-Mol}_{P_{cs}}\,m_{P_{cs}}=e^{\frac{m^{2}_{P_{cs}}}{M^{2}}}\,\Delta_{4}^{QCD}.$
(21)

The $\Delta_{1}^{QCD}$, $\Delta_{2}^{QCD}$, $\Delta_{3}^{QCD}$ and
$\Delta_{4}^{QCD}$ functions are quite lengthy, therefore the explicit
expression of this function are not presented here.

###  II.2 Formalism of the $P_{c1}$ state

In this subsection we derive the LCSR for the magnetic dipole moment of the
$P_{c1}$ pentaquark state. For this purpose, we consider following correlation
function,

$\Pi_{\mu\nu}(p,q)=i\int d^{4}xe^{ip\cdot x}\langle
0|\mathcal{T}\\{J_{\mu}^{P_{c1}}(x)\bar{J}_{\nu}^{P_{c1}}(0)\\}|0\rangle_{\gamma},$
(22)

where $J_{\mu(\nu)}^{P_{c1}}$ is the interpolating current of $P_{c1}$
pentaquark with $J^{P}=\frac{3}{2}^{-}$ quantum numbers. In the diquark-
diquark-antiquark and molecular pictures, it is given as

$\displaystyle J_{\mu}^{P_{c1}-Di}(x)$ $\displaystyle=$
$\displaystyle\varepsilon^{abc}\varepsilon^{ade}\varepsilon^{bfg}\big{[}u^{T}_{d}(x)C\gamma_{5}d_{e}(x)\,u^{T}_{f}(x)C\gamma_{\mu}c_{g}(x)\,C\bar{c}^{T}_{c}(x)\big{]},$
$\displaystyle J_{\mu}^{P_{c1}-Mol}(x)$ $\displaystyle=$
$\displaystyle\big{[}\bar{c}_{d}(x)\gamma_{\mu}u_{d}(x)\big{]}[\varepsilon^{abc}u^{T}_{a}(x)C\gamma_{\nu}d_{b}(x)\gamma^{\nu}\gamma_{5}c_{c}(x)\big{]}$
(23)

The hadronic side of the correlation function is written as,

$\displaystyle\Pi^{Had}_{\mu\nu}(p,q)$ $\displaystyle=$
$\displaystyle\frac{\langle 0\mid
J_{\mu}^{P_{c1}}\mid{P_{c1}}(p)\rangle}{[p^{2}-m_{{P_{c1}}}^{2}]}\langle{P_{c1}}(p)\mid{P_{c1}}(p+q)\rangle_{\gamma}\frac{\langle{P_{c1}}(p+q)\mid\bar{J}_{\nu}^{P_{c1}}\mid
0\rangle}{[(p+q)^{2}-m_{{P_{c1}}}^{2}]}+...$ (24)

The matrix element of the interpolating current between the vacuum and the
$P_{c1}$ pentaquark is defined as

$\langle 0\mid
J_{\mu}^{P_{c1}}(0)\mid{P_{c1}}(p,s)\rangle=\lambda_{{P_{c1}}}u_{\mu}(p,s),$
(25)

where $\lambda_{{P_{c1}}}$ is the residue $P_{c1}$ pentaquark and
$u_{\mu}(p,s)$ is the Rarita-Schwinger spinor.

The transition matrix element
$\langle{P_{c1}}(p)\mid{P_{c1}}(p+q)\rangle_{\gamma}$ entering Eq. (24) can be
parameterized in terms of four Lorentz invariant form factors as follows
Weber:1978dh ; Nozawa:1990gt ; Pascalutsa:2006up ; Ramalho:2009vc :

$\displaystyle\langle{P_{c1}}(p)\mid{P_{c1}}(p+q)\rangle_{\gamma}$
$\displaystyle=-e\bar{u}_{\mu}(p)\Bigg{[}F_{1}(q^{2})g_{\mu\nu}\varepsilon\\!\\!\\!/-\frac{1}{2m_{{P_{c1}}}}\Big{[}F_{2}(q^{2})g_{\mu\nu}+F_{4}(q^{2})\frac{q_{\mu}q_{\nu}}{(2m_{{P_{c1}}})^{2}}\Big{]}\varepsilon\\!\\!\\!/q\\!\\!\\!/$
$\displaystyle+F_{3}(q^{2})\frac{1}{(2m_{{P_{c1}}})^{2}}q_{\mu}q_{\nu}\varepsilon\\!\\!\\!/\Bigg{]}u_{\nu}(p+q).$
(26)

In principle, we can obtain the final expression of the hadronic side of the
correlation function using the above equations, but we encounter two
difficulties: not all Lorentz structures are independent and the correlation
function also includes contributions of spin- 1/2 and these undesirable
contributions must be eliminated. To remove undesirable contributions coming
from the spin-1/2 particles and obtain only independent structures in the
correlation function, we apply the ordering for Dirac matrices as
$\gamma_{\mu}p\\!\\!\\!/\varepsilon\\!\\!\\!/q\\!\\!\\!/\gamma_{\nu}$ and
remove terms with $\gamma_{\mu}$ at the beginning, $\gamma_{\nu}$ at the end
and those proportional to $p_{\mu}$ and $p_{\nu}$ Belyaev:1982cd . As a
result, using Eqs. (22)-(II.2) the hadronic side take the form,

$\displaystyle\Pi^{Had}_{\mu\nu}(p,q)=$
$\displaystyle\frac{\lambda_{{}_{{P_{c1}}}}^{2}}{[(p+q)^{2}-m_{{}_{{P_{c1}}}}^{2}][p^{2}-m_{{}_{{P_{c1}}}}^{2}]}\Bigg{[}g_{\mu\nu}p\\!\\!\\!/\varepsilon\\!\\!\\!/q\\!\\!\\!/\,F_{1}(q^{2})-m_{{P_{c1}}}g_{\mu\nu}\varepsilon\\!\\!\\!/q\\!\\!\\!/\,F_{2}(q^{2})-\frac{F_{3}(q^{2})}{4m_{{P_{c1}}}}q_{\mu}q_{\nu}\varepsilon\\!\\!\\!/q\\!\\!\\!/\,-\frac{F_{4}(q^{2})}{4m_{{P_{c1}}}^{3}}(\varepsilon.p)q_{\mu}q_{\nu}p\\!\\!\\!/q\\!\\!\\!/$
$\displaystyle+\mbox{other independent structures}\Bigg{]}.$ (27)

The final form of the hadronic side in terms of the selected structures in
momentum space is:

$\displaystyle\Pi^{Had}_{\mu\nu}(p,q)$ $\displaystyle=$
$\displaystyle\Pi_{1}^{Had}g_{\mu\nu}p\\!\\!\\!/\varepsilon\\!\\!\\!/q\\!\\!\\!/\,+\Pi_{2}^{Had}g_{\mu\nu}\varepsilon\\!\\!\\!/q\\!\\!\\!/\,+...,$
(28)

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 is represented by dots.

The magnetic, $G_{M}(q^{2})$, form factor is defined in terms of the form
factors $F_{i}(q^{2})$ in the following way Weber:1978dh ; Nozawa:1990gt ;
Pascalutsa:2006up ; Ramalho:2009vc :

$\displaystyle G_{M}(q^{2})$
$\displaystyle=[F_{1}(q^{2})+F_{2}(q^{2})](1+\frac{4}{5}\tau)-\frac{2}{5}[F_{3}(q^{2})+F_{4}(q^{2})]\tau(1+\tau),$
(29)

where $\tau=-\frac{q^{2}}{4m^{2}_{{P_{c1}}}}$. At $q^{2}=0$, the magnetic
dipole moment is obtained in terms of the functions $F_{1}(0)$ and $F_{2}(0)$
as:

$\displaystyle G_{M}(0)$ $\displaystyle=$ $\displaystyle F_{1}(0)+F_{2}(0).$
(30)

The magnetic dipole moment, ($\mu_{{P_{c1}}}$), is defined in the following
way:

$\displaystyle\mu_{{P_{c1}}}$ $\displaystyle=$
$\displaystyle\frac{e}{2m_{{P_{c1}}}}G_{M}(0).$ (31)

The next step is to calculate the correlation function in Eq. (22) in terms of
quark-gluon parameters. When we apply the same procedures as in the previous
subsection, we get the following result:

$\displaystyle\Pi^{QCD-Di}_{\mu\nu}(p,q)$ $\displaystyle=$ $\displaystyle
i\,\varepsilon^{abc}\varepsilon^{a^{\prime}b^{\prime}c^{\prime}}\varepsilon^{ade}\varepsilon^{a^{\prime}d^{\prime}e^{\prime}}\varepsilon^{bfg}\varepsilon^{b^{\prime}f^{\prime}g^{\prime}}\int
d^{4}xe^{ip\cdot x}\langle 0|\Bigg{\\{}$ (32) $\displaystyle
Tr\Big{[}\gamma_{5}S_{d}^{ee^{\prime}}(x)\gamma_{5}\tilde{S}_{u}^{dd^{\prime}}(x)\Big{]}Tr\Big{[}\gamma_{\mu}S_{c}^{gg^{\prime}}(x)\gamma_{\nu}\tilde{S}_{u}^{ff^{\prime}}(x)\Big{]}\tilde{S}_{c}^{c^{\prime}c}(-x)$
$\displaystyle-
Tr\Big{[}\gamma_{5}S_{d}^{ee^{\prime}}(x)\gamma_{5}\tilde{S}_{u}^{fd^{\prime}}(x)\gamma_{\mu}S_{c}^{gg^{\prime}}(x)\gamma_{\nu}\tilde{S}_{u}^{df^{\prime}}(x)\Big{]}\tilde{S}_{c}^{c^{\prime}c}(-x)\Bigg{\\}}|0\rangle_{\gamma},$
$\displaystyle\Pi_{\mu\nu}^{\mathrm{QCD-Mol}}(p,q)$ $\displaystyle=$
$\displaystyle-i\,\epsilon^{abc}\epsilon^{a^{\prime}b^{\prime}c^{\prime}}\,\int
d^{4}xe^{ip\cdot x}\langle
0|\Bigg{\\{}\mathrm{Tr}\Big{[}\gamma_{\mu}S_{u}^{dd^{\prime}}(x)\gamma_{\nu}S_{c}^{d^{\prime}d}(-x)\Big{]}\mathrm{Tr}\Big{[}\gamma_{\beta}\widetilde{S}_{u}^{aa^{\prime}}(x)\gamma_{\alpha}S_{d}^{bb^{\prime}}(x)]$
(33)
$\displaystyle-\mathrm{Tr}\Big{[}\gamma_{\mu}S_{u}^{dd^{\prime}}(x)\gamma_{\nu}S_{c}^{d^{\prime}d}(-x)\Big{]}\mathrm{Tr}\Big{[}\gamma_{\beta}\widetilde{S}_{u}^{ba^{\prime}}(x)\gamma_{\alpha}S_{d}^{ab^{\prime}}(x)\Big{]}\Bigg{\\}}\Big{(}\gamma^{\alpha}\gamma_{5}S_{c}^{cc^{\prime}}(x)\gamma_{5}\gamma^{\beta}\Big{)}\Bigg{|}0\rangle_{\gamma}.$

As a result, the QCD side of the correlation function in terms of the selected
structures is obtained 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)

The following processes are applied as described in the previous subsection
and magnetic dipole moment results are obtained in LCSR. The QCD and hadronic
representations of the correlation function are then matched employing quark-
duality assumption. By equating 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 obtain sum rules for these two form factors. As a result, we get,

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

The explicit expressions of the LCSR for the $F_{1}^{Di}$ and $F_{2}^{Di}$ are
presented in the Appendix A. We are now ready to move on to numerical
analysis.

##  III Numerical analysis and discussions

This section is devoted to the numerical computations for the magnetic dipole
moments of the $P_{c1}$, $P_{c2}$ and $P_{cs}$ pentaquark states. We use
$m_{u}=m_{d}=0$, $m_{s}=96^{+8}_{-4}\,\mbox{MeV}$, $m_{c}=1.275\pm
0.02\,\mbox{GeV}$ Patrignani:2016xqp , $m_{P_{c1}}=4440.3\pm
1.3^{+4.1}_{-4.7}~{}\mbox{MeV}$, $m_{P_{c2}}=4457.3\pm
0.6^{+4.1}_{-1.7}~{}\mbox{MeV}$ Aaij:2019vzc , $m_{P_{cs}}=4458.8\pm
2.7^{+4.7}_{-1.1}~{}\mbox{MeV}$ Aaij:2020gdg ,
$f_{3\gamma}=-0.0039~{}\mbox{GeV}^{2}$ Ball:2002ps ,
$\langle\bar{u}u\rangle=\langle\bar{d}d\rangle=(-0.24\pm
0.01)^{3}\,\mbox{GeV}^{3}$,
$\langle\bar{s}s\rangle=0.8\,\langle\bar{u}u\rangle$ $\,\mbox{GeV}^{3}$
Ioffe:2005ym , $m_{0}^{2}=0.8\pm 0.1\,\mbox{GeV}^{2}$ Ioffe:2005ym , $\langle
g_{s}^{2}G^{2}\rangle=0.88~{}\mbox{GeV}^{4}$ Nielsen:2009uh ,
$\lambda_{P_{c1}}^{Di}=(1.44\pm 0.23)\times 10^{-3}~{}\mbox{GeV}^{6}$,
$\lambda_{P_{c2}}^{Di}=(3.02\pm 0.48)\times 10^{-3}~{}\mbox{GeV}^{6}$
Wang:2019got , $\lambda_{P_{cs}}^{Di}=(1.86\pm 0.31)\times
10^{-3}~{}\mbox{GeV}^{6}$ Wang:2020eep ,
$\lambda_{P_{c1}}^{Mol}=(1.15^{+0.16}_{-0.18})\times
10^{-3}~{}\mbox{GeV}^{6}$,
$\lambda_{P_{c2}}^{Mol}=(2.24^{+0.30}_{-0.34})\times 10^{-3}~{}\mbox{GeV}^{6}$
Chen:2020opr  and $\lambda_{P_{cs}}^{Mol}=1.13\times
10^{-3}~{}\mbox{GeV}^{6}$. Another set of main input parameters are the photon
wavefunctions of different twists, entering the DAs. These wavefunctions are
given in Ref. Ball:2002ps .

Except the above mentioned input parameters, the estimations for the magnetic
dipole moments of pentaquark states depend on two auxiliary parameters: Borel
mass parameter $M^{2}$ and continuum threshold $s_{0}$. According to the
philosophy of the method used, the observables under examination should be
weakly dependent on the variations of these auxiliary parameters. The
continuum threshold is considered to be the point where the excited states and
continuum begin to contribute to the correlation function. The upper and lower
bound of the Borel parameter is decided by demanding that both the
contributions of the higher states and continuum are adequately suppressed and
the contributions coming from higher dimensional terms are small. Our
numerical analysis leads to the conclusion that these requirements are
fulfilled in the regions shown below for the considered pentaquark states.

$\displaystyle 22.0~{}\mbox{GeV}^{2}\leq s_{0}\leq
24.0~{}\mbox{GeV}^{2}~{}P_{c1}~{}\mbox{and}~{}P_{c2}~{}\mbox{states},$
$\displaystyle 23.0~{}\mbox{GeV}^{2}\leq s_{0}\leq
25.0~{}\mbox{GeV}^{2}~{}\mbox{for}~{}P_{cs}~{}\mbox{state},$ $\displaystyle
5.0~{}\mbox{GeV}^{2}\leq M^{2}\leq
7.0~{}\mbox{GeV}^{2}~{}\mbox{for}~{}P_{c1}~{}\mbox{and}~{}P_{c2}~{}\mbox{states},$
$\displaystyle 5.5~{}\mbox{GeV}^{2}\leq M^{2}\leq
7.5~{}\mbox{GeV}^{2}~{}\mbox{for}~{}P_{cs}~{}\mbox{state}.$

By having the values of all input parameters, we can start carrying out
numerical computations. In Fig. 1, as an example, we depict the dependencies
of the magnetic dipole moments in diquark-diquark-antiquark picture on $M^{2}$
and $s_{0}$. As is seen, the deviation of the results in connection with the
$s_{0}$ is remarkable however there is much less dependence of the physical
observables under consideration on the $M^{2}$ in its working window.

Our final results for the magnetic dipole moments are presented in Table. 1.

Picture |  $\mu_{P_{c1}}$ |  $\mu_{P_{c2}}$ |  $\mu_{P_{cs}}$
---|---|---|---
Diquark |  $1.62^{+0.65}_{-0.57}$ |  $0.88^{+0.32}_{-0.29}$ |  $0.34^{+0.13}_{-0.11}$
Molecule |  $3.49^{+1.49}_{-1.31}$ |  $2.78^{+0.94}_{-0.83}$ |  $1.75^{+0.64}_{-0.58}$
Table 1: Results of the magnetic dipole moments (in units of $\mu_{N}$) for
the pentaquark states.

Our results cover errors originated from the uncertainty of the determinations
of auxiliary parameters ($M^{2}$ and $s_{0}$) and other input parameters used
in the analyses. The numerical values of the magnetic dipole moments acquired
by means of two configurations differ considerably from each other, which can
be used to determine the fundamental structure of these pentaquark states.
That is to say, as many theoretical models give compatible results on the
decay channels and spectroscopic parameters with the experimental data
preventing us assigning precise substructure for pentaquarks, the experimental
measurement of the magnetic dipole moments of these pentaquarks indeed can
help us exactly distinguish their substructures. Since there are no
theoretical result and experimental data for the results we obtained for these
pentaquark states, we cannot compare our results. However, we may compare the
result of the $P_{cs}$ state with the $P_{c}(4312)$ state magnetic dipole
moments. It can be seen that the quark configurations of the $P_{cs}$ and
$P_{c}(4312)$ particles are similar and therefore the magnetic dipole moment
results of these particles can be expected to be close to each other. In Ref.
Ozdem:2021btf , the magnetic dipole moment of $P_{c}(4312)$ in the diquark-
diquark-antiquark and molecular pictures are extracted as $\mu_{P_{c}}=0.40\pm
0.15\,\mu_{N}$ and $\mu_{P_{c}}=1.98\pm 0.75\,\mu_{N}$, respectively. The
magnetic dipole moment of the $P_{c}(4312)$ pantaquark state has also been
extracted via the QCD sum rule and its extension in the weak electromagnetic
field by employing a molecular type interpolating current in Ref. Xu:2020flp .
The numerical value was obtained as $\mu_{P_{c}}=0.59^{+10}_{-0.20}$. As one
can see from these estimations, the numerical values for the magnetic dipole
moments of $P_{c}(4312)$ and $P_{cs}$ states acquired in the present study are
close to each other and we see a reasonable SU(3) flavor violation which is
roughly $\%15$. The obtained results in both pictures are considerably
different compared to the result of Ref. Xu:2020flp . Comparing the results
obtained using different theoretical models with our results can give an idea
about the consistency of our predictions.

As a by product, we also obtain the electric quadrupole ($Q_{P_{c1}}$) and
magnetic octupole ($O_{P_{c1}}$) moments of the $P_{c1}$ pentaquark as

$\displaystyle Q_{P_{c1}}^{Di}=(2.0^{+0.6}_{-0.5})\times
10^{-2}~{}\mbox{fm${}^{2}$},~{}~{}~{}~{}~{}~{}~{}~{}Q_{P_{c1}}^{Mol}=0.28^{+0.09}_{-0.07}~{}\mbox{fm${}^{2}$},$
(36) $\displaystyle O_{P_{c1}}^{Di}=(0.24^{+0.07}_{-0.06})\times
10^{-3}~{}\mbox{fm${}^{3}$},~{}~{}~{}~{}~{}~{}O_{P_{c1}}^{Mol}=(4.83^{+1.36}_{-1.13})\times
10^{-3}~{}\mbox{fm${}^{3}$}.$ (37)

Just like the magnetic dipole moment results, we can see that the electric
quadrupole and magnetic octupole moments results obtained by using two
different pictures are different from each other. The $Q_{P_{c1}}$ and
$O_{P_{c1}}$ of the $P_{c1}$ pentaquark state showing a non-spherical charge
distribution.

In summary, the observation of new pentaquark states such as $P_{c}(4312)$,
$P_{c}(4440)$, $P_{c}(4457)$ and $P_{cs}(4459)$, ensures a new platform to
investigate the exotic states in QCD. There are different interpretations of
their inner structures and quantum numbers, and these should be shedded light
on with further research. In the present work, stimulated by the observation
of the hidden-charm pentaquark states we have achieved the magnetic dipole
moments of the $P_{c}(4440)$, $P_{c}(4457)$ and $P_{cs}(4459)$ by considering
them as diquark-diquark-antiquark and molecular pictures with quantum numbers
$J^{P}=\frac{3}{2}^{-}$, $J^{P}=\frac{1}{2}^{-}$ and $J^{P}=\frac{1}{2}^{-}$
by means of the light-cone QCD sum rule, respectively. As a by product, the
electric quadrupole and magnetic octupole moments of the $P_{c}(4440)$
pentaquark have also extracted. Our predictions for electromagnetic multi-pole
moments of pentaquark states may be checked via different theoretical models
such as Lattice QCD, chiral perturbation theory etc. With the latest
developments in the experimental side, we hope that we will be able to measure
the magnetic dipole moment of newly formed multi-quark states, especially
pantaquark states in the future. Any experimental measurements of the
electromagnetic multi-pole moments of the hidden-charm pentaquark states and
comparison of the obtained results with the predictions of the this study may
provide as helpful knowledge on the internal structure of the these states as
well as the non-perturbative behaviors of the strong interaction at the low-
energy region.

##  IV Acknowledgments

We thank Hua-Xing Chen for providing us the numeric value of the residue of
the $P{cs}(4459)$ state.

Figure 1: Variations of the magnetic dipole moments of $\mu_{P_{c1}}$,
$\mu_{P_{c2}}$ and $\mu_{P_{cs}}$in diquark-diquark-antiquark picture with
$M^{2}$ and $s_{0}$.

## Appendix A: Explicit forms of the $F_{1}^{Di}$ and $F_{2}^{Di}$ functions

In this appendix, we give the explicit expressions for the $F_{1}^{Di}$ and
$F_{2}^{Di}$ functions:

$\displaystyle F_{1}^{Di}$
$\displaystyle=\frac{e^{\frac{m^{2}_{P_{c1}}}{M^{2}}}}{\lambda^{2-Di}_{P_{c1}}}\Bigg{\\{}\frac{\langle
g_{s}^{2}G^{2}\rangle}{108716359680\,\pi^{7}}\Bigg{[}20\,\pi^{2}\,f_{3\gamma}\Big{(}4\,(32e_{d}+41e_{u})\,I[0,4,4,0]+3\,(9e_{d}+20e_{u})\,I[0,4,5,0]\Big{)}I_{2}[\mathcal{V}]$
$\displaystyle+(72e_{d}+720e_{u}-702e_{c})\,I[0,5,2,1]-(324e_{d}+2184e_{u}-2673e_{c})\,I[0,5,2,2]+(492e_{d}+2456e_{u}-3511e_{c})$
$\displaystyle\times
I[0,5,2,4]-(300e_{d}+1240e_{u}-1811e_{c})\,I[0,5,2,4]+(60e_{d}+248e_{u}-271e_{c})\,I[0,5,2,5]+(216e_{d}+2160e_{u}$
$\displaystyle-2106)\,I[0,5,3,1]+(684e_{d}+4728e_{d}-5693e_{c})\,I[0,5,3,2]-(624e_{d}+3424e_{u}-4483e_{c})\,I[0,5,3,3]+(156e_{d}$
$\displaystyle+856e_{u}-893e_{c})\,I[0,5,3,4]+(216e_{d}+2160e_{u}-2106e_{c})\,I[0,5,4,1]-(396e_{d}+2904e_{u}-3375e_{c})$
$\displaystyle\times
I[0,5,4,2]+(132e_{d}+968e_{u}-973e_{c})\,I[0,5,4,3]-(72e_{d}+720e_{u}-702e_{c})\,I[0,5,5,1]+(36e_{d}+360e_{u}$
$\displaystyle-351e_{c})\,I[0,5,5,2]\Bigg{]}$
$\displaystyle-\frac{m_{c}\,\langle\bar{q}q\rangle}{3774873\,\pi^{5}}\Bigg{[}192\,e_{c}\,\Big{(}I[0,5,2,2]-2\,I[0,5,2,3]+I[0,5,2,4]-2\,I[0,5,3,2]+2\,I[0,5,3,3]+I[0,5,4,2]\Big{)}$
$\displaystyle-40\,\pi^{2}\,f_{3\gamma}(e_{d}+10e_{u})\,I_{2}[\mathcal{V}]\,I[0,4,4,0]+3(e_{d}+14e_{u})\,I_{4}[\tilde{S}]\,I[0,5,4,0]\Bigg{]}$
$\displaystyle+\frac{f_{3\gamma}}{3019898880\,\pi^{5}}(13e_{d}+58e_{u})\,I_{2}[\mathcal{V}]\,I[0,6,5,0]$
$\displaystyle-\frac{e_{c}}{880803840\,\pi^{7}}\Bigg{[}4\,I[0,7,2,3]-13\,I[0,7,2,4]+15\,I[0,7,2,5]-7\,I[0,7,2,6]+I[0,7,2,7]-12\,I[0,7,3,3]$
$\displaystyle+27\,I[0,7,3,4]-18\,I[0,7,3,5]+3\,I[0,7,3,6]+12\,I[0,7,4,3]+3\,I[0,7,4,5]-4\,I[0,7,5,3]+I[0,7,5,4]\Bigg{]}\Bigg{\\}},$
(38) $\displaystyle F_{2}^{Di}$
$\displaystyle=\frac{m_{P_{c1}}\,e^{\frac{m^{2}_{P_{c1}}}{M^{2}}}}{\lambda^{2-Di}_{P_{c1}}}\Bigg{\\{}\frac{\langle
g_{s}^{2}G^{2}\rangle}{108716359680\,\pi^{7}}\Bigg{[}20\,f_{3\gamma}\pi^{2}\,\Big{\\{}-4\Big{(}(52e_{d}-19e_{u})I_{2}[\mathcal{A}]-9(32e_{d}+41e_{u})I_{2}[\mathcal{V}]+4(3e_{d}+2e_{u})$
$\displaystyle\times
I_{6}[\psi^{\nu}]\Big{)}\,I[0,4,4,0]+3\Big{(}8(-34e_{d}+e_{u})I_{2}[\mathcal{A}]+9(9e_{d}+220e_{u})\,I_{2}[\mathcal{V}]\Big{)}\,I[0,4,5,0]\Big{\\}}-\Big{\\{}(702e_{c}-72e_{d}-720e_{u})$
$\displaystyle\times
I[0,5,2,1]+(324e_{d}-2673e_{c}+2184e_{u})\,I[0,5,2,2]+(-492e_{d}+3511e_{c}-2456e_{u})\,I[0,5,2,3]+(300e_{d}$
$\displaystyle-1811e_{c}+1240e_{u})\,I[0,5,2,4]+(-60e_{d}+271e_{c}-248e_{u})\,I[0,5,2,5]+(216e_{d}-2106e_{c}+2160e_{u})\,I[0,5,3,1]$
$\displaystyle+(-684e_{d}+5697e_{c}-4728eu)I[0,5,3,2]+(624e_{d}-4484e_{c}+3424e_{u})\,I[0,5,3,3]+(893e_{c}-156e_{d}-856e_{u})$
$\displaystyle\times
I[0,5,3,4]+(-216e_{d}+2106e_{c}-2160e_{u})\,I[0,5,4,1]+(396e_{d}-3375e_{c}+2904e_{u})\,I[0,5,4,2]+(-132e_{d}$
$\displaystyle+973e_{c}-968e_{u})I[0,5,4,3]+(72e_{d}-702e_{c}+720e_{u})\,I[0,5,5,1]+(-36e_{d}+351e_{c}-360e_{u})\,I[0,5,5,2]\Big{\\}}\Bigg{]}$
$\displaystyle-\frac{m_{c}\,\langle\bar{q}q\rangle}{125829120\,\pi^{5}}\Bigg{[}-120\,(e_{d}+10e_{u})\,f_{3\gamma}\,\pi^{2}\,I_{2}[\mathcal{V}]\,I[0,4,4,0]+(3e_{d}+14e_{u})I_{4}[\mathcal{S}]+2e_{d}\,I_{4}[\mathcal{T}_{1}]I[0,5,4,0]$
$\displaystyle+64\,e_{c}\,\Big{(}I[0,5,2,2]-2\,I[0,5,2,3]+I[0,5,2,4]-2\,I[0,5,3,2]+2\,I[0,5,3,3]+I[0,5,4,2]\Big{)}\Bigg{]}$
$\displaystyle+\frac{f_{3\gamma}}{3019898880\,\pi^{5}}(13e_{d}+58e_{u})I_{2}[\mathcal{V}]\,I[0,6,5,0]$
$\displaystyle-\frac{e_{c}}{880803840\,\pi^{7}}\Bigg{[}4\,I[0,7,2,3]-13\,I[0,7,2,4]+15\,I[0,7,2,5]-7\,I[0,7,2,6]+I[0,7,2,7]-12\,I[0,7,3,3]$
$\displaystyle+27\,I[0,7,3,4]-18\,I[0,7,3,5]+3\,I[0,7,3,6]+12\,I[0,7,4,3]-15\,I[0,7,4,4]+3\,I[0,7,4,5]-4\,I[0,7,5,3]$
$\displaystyle+I[0,7,5,4]\Bigg{]}\Bigg{\\}}.$ (39)

We should also point out that in the above expressions, for simplicity we have
only given the terms that give important contributions to the numerical values
of the magnetic moments, and have not presented many higher dimensional
contributions, although they have been considered in the numerical
calculations.

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

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

where $\mathcal{A}$ represents the corresponding photon distribution
amplitudes.

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