11institutetext:  School of Physics, Southeast University, Nanjing 210094,
China 22institutetext: School of Physics, Sun Yat-Sen University, Guangzhou
510275, China 33institutetext: School of Physical Science and Technology,
Lanzhou University, Lanzhou 730000, China 44institutetext: Research Center for
Hadron and CSR Physics, Lanzhou University and Institute of Modern Physics of
CAS, Lanzhou 730000, China 55institutetext: Center for Joint Quantum Studies
and Department of Physics, School of Science, Tianjin University, Tianjin
300350, China

# Establishing the first hidden-charm pentaquark with strangeness

Hua-Xing Chen<EMAIL_ADDRESS>Wei Chen<EMAIL_ADDRESS>Xiang
Liu<EMAIL_ADDRESS>Xiao-Hai Liu<EMAIL_ADDRESS>
(Received: date / Revised version: date)

###### Abstract

We study the $P_{cs}(4459)^{0}$ recently observed by LHCb using the method of
QCD sum rules. Our results support its interpretation as the
$\bar{D}^{*}\Xi_{c}$ hadronic molecular state of either $J^{P}=1/2^{-}$ or
$3/2^{-}$. Within the hadronic molecular picture, the three LHCb experiments
observing $P_{c}$ and $P_{cs}$ states lhcb ; Aaij:2015tga ; Aaij:2019vzc  can
be well understood as a whole. This strongly supports the existence of
hadronic molecules, whose studies can significantly improve our understanding
on the construction of the subatomic world. To verify this picture, we propose
to further investigate the $P_{cs}(4459)^{0}$ to examine whether it can be
separated into two states, and to search for the $\bar{D}\Xi_{c}$ molecular
state of $J^{P}=1/2^{-}$.

###### pacs:

12.39.MkGlueball and nonstandard multi-quark/gluon states and 12.38.LgOther
nonperturbative calculations

††offprints:

Introduction.—– Atomic nuclei are made of protons and neutrons, which are
themselves composed of quarks and gluons. In the past century a huge number of
subatomic particles, called hadrons, were discovered in particle experiments,
whose properties are similar to the proton and neutron pdg . One naturally
raises an interesting question: are there subatomic particles corresponding to
the nucleus? Nowadays we call them “hadronic molecules”, whose studies can
significantly improve our understanding on the construction of the subatomic
world, as illustrated in Fig. 1.

Figure 1: A possible way to construct the subatomic world.

It is not so easy to answer the above question. Most of the experimentally
observed hadrons can be described as $q\bar{q}$ mesons or $qqq$ baryons in the
conventional quark model, while there can also exist (compact)
$q\bar{q}q\bar{q}$ tetraquarks and $qqqq\bar{q}$ pentaquarks GellMann:1964nj ;
Zweig:1981pd , etc. With the experimental progress on this issue over the past
decade, dozens of $XYZ$ charmonium-like states were reported, providing us
good opportunities to identify exotic hidden-charm tetraquarks pdg . However,
these states may be explained as hadronic molecules, while they may also be
explained as compact tetraquarks that are still hadrons.

Fortunately, in the LHCb experiments performed in 2015 and 2019 Aaij:2015tga ;
Aaij:2019vzc , the famous hidden-charm pentaquark states, $P_{c}(4312)$,
$P_{c}(4380)$, $P_{c}(4440)$, and $P_{c}(4457)$, were discovered. These four
$P_{c}$ states contain at least five quarks $\bar{c}cuud$, so they are perfect
candidates of hidden-charm pentaquark states. Among them, the three narrow
states $P_{c}(4312)$, $P_{c}(4440)$, and $P_{c}(4457)$ are just below the
$\bar{D}\Sigma_{c}$ and $\bar{D}^{*}\Sigma_{c}$ thresholds, so their natural
interpretations are the $\bar{D}\Sigma_{c}$ and $\bar{D}^{*}\Sigma_{c}$
hadronic molecular states Chen:2015loa ; Chen:2015moa ; Roca:2015dva ;
Meissner:2015mza ; Liu:2019tjn , whose existence had been predicted in Refs.
Wu:2010jy ; Wang:2011rga ; Yang:2011wz ; Li:2014gra ; Karliner:2015ina  before
the LHCb experiment performed in 2015 Aaij:2015tga . However, there still
exist other possible explanations Maiani:2015vwa ; Lebed:2015tna ;
Wang:2015epa ; Guo:2015umn ; Liu:2015fea ; Bayar:2016ftu , and we refer to the
reviews Chen:2016qju ; Liu:2019zoy ; Guo:2017jvc ; Ali:2017jda ; Olsen:2017bmm
; Brambilla:2019esw  for detailed discussions.

It is natural to conjecture whether the hidden-charm pentaquark state with
strangeness exists or not. Such a state is usually denoted as “$P_{cs}$”,
whose quark content is $\bar{c}csqq$ ($q=u/d$). There have been some but not
many theoretical studies on it Cheng:2015cca ; Anisovich:2015zqa ;
Wang:2015wsa ; Feijoo:2015kts ; Lu:2016roh ; Chen:2016ryt ; Xiao:2019gjd ;
Zhang:2020cdi ; Shen:2020gpw . Especially, we proposed in Ref. Chen:2015sxa
to search for the $P_{cs}$ in the $J/\psi\Lambda$ invariant mass spectrum of
the $\Xi_{b}^{-}\to J/\psi K^{-}\Lambda$ decays. Besides, in Ref. Wang:2019nvm
the authors studied the $P_{cs}$ using the chiral effective field theory, and
calculated the mass of the $J^{P}=1/2^{-}$ $\bar{D}^{*}\Xi_{c}$ molecular
state to be $4456.9^{+3.2}_{-3.3}$ MeV. Note that both of these two references
are based on the hadronic molecular picture.

Very recently, the LHCb Collaboration reported the evidence of a hidden-charm
pentaquark state with strangeness, $P_{cs}(4459)^{0}$, in the $J/\psi\Lambda$
invariant mass spectrum of the $\Xi_{b}^{-}\to J/\psi K^{-}\Lambda$ decays
lhcb . Its mass and width were measured to be:

$\displaystyle P_{cs}(4459)^{0}:$ $\displaystyle M=$ $\displaystyle 4458.8\pm
2.9^{+4.7}_{-1.1}\mbox{ MeV}\,,$ $\displaystyle\Gamma=$ $\displaystyle 17.3\pm
6.5^{+8.0}_{-5.7}\mbox{ MeV}\,,$

while its spin-parity quantum number was not determined since the statistic is
not enough. Note that the channel observing the $P_{cs}(4459)^{0}$ is just the
one proposed by us in Ref. Chen:2015sxa , and the above mass value is almost
identical to the mass of the $J^{P}=1/2^{-}$ $\bar{D}^{*}\Xi_{c}$ molecular
state predicted in Ref. Wang:2019nvm . Therefore, the present LHCb experiment
lhcb  strongly supports the hadronic molecular picture once more.

Actually, as indicated by LHCb, the $P_{cs}(4459)^{0}$ is about 19 MeV below
the $\bar{D}^{*0}\Xi_{c}^{0}$ threshold lhcb , so it is natural to interpret
it as the $\bar{D}^{*}\Xi_{c}$ molecular state, with the spin-parity quantum
number $J^{P}=1/2^{-}$ or $3/2^{-}$. Accordingly, in this letter we shall
study the $\bar{D}^{*}\Xi_{c}$ molecular states of $J^{P}=1/2^{-}$ and
$3/2^{-}$ using the method of QCD sum rules, and at the same time we shall
also study the $\bar{D}\Xi_{c}$ molecular state of $J^{P}=1/2^{-}$. We
calculate masses of $\bar{D}^{*}\Xi_{c}$ molecular states to be
$4.46^{+0.16}_{-0.14}$ GeV for the $J^{P}=1/2^{-}$ one and
$4.47^{+0.19}_{-0.15}$ GeV for the $J^{P}=3/2^{-}$ one. These two values are
both consistent with the experimental mass of the $P_{cs}(4459)^{0}$,
supporting its interpretation as the $\bar{D}^{*}\Xi_{c}$ molecular state of
either $J^{P}=1/2^{-}$ or $3/2^{-}$. We also calculate the mass of the
$J^{P}=1/2^{-}$ $\bar{D}\Xi_{c}$ molecular state to be $4.29^{+0.13}_{-0.12}$
GeV.

To verify the hadronic molecular picture, we propose to further investigate
the $P_{cs}(4459)^{0}$ state in future experiments to examine whether it can
be separated into two states, and to search for the $J^{P}=1/2^{-}$
$\bar{D}\Xi_{c}$ molecular state. If the hadronic molecular picture turns out
to be correct, our understanding on the construction of the subatomic world
would be significantly improved. Besides, these studies are helpful to improve
our understanding on the non-perturbative behaviors of the strong interaction
at the low energy region.

Hidden-charm pentaquark currents.—– We use the $\bar{c}$, $c$, $s$, $u$, and
$d$ quarks to construct hidden-charm pentaquark interpolating currents with
strangeness. To study $\bar{D}^{(*)}\Xi_{c}$ molecular states, we consider the
following type of currents:

$\eta(x)=[\bar{c}_{a}(x)\Gamma_{1}u_{b}(x)]~{}\Big{[}[d^{T}_{c}(x)\mathbb{C}\Gamma_{2}s_{d}(x)]~{}\Gamma_{3}c_{e}(x)\Big{]}\,,$
(2)

where $a\cdots e$ are color indices, $\Gamma_{1/2/3}$ are Dirac matrices, and
$\mathbb{C}=i\gamma_{2}\gamma_{0}$ is the charge-conjugation operator. The
other type of currents:

$\eta^{\prime}(x)=[\bar{c}_{a}(x)\Gamma_{1}d_{b}(x)]~{}\Big{[}[u^{T}_{c}(x)\mathbb{C}\Gamma_{2}s_{d}(x)]~{}\Gamma_{3}c_{e}(x)\Big{]}\,,$
(3)

can be similarly studied, but they just lead to the same QCD sum rule results
as the $\eta(x)$ currents. Hence, the present study can not distinguish the
isospin of $P_{cs}$ states.

The $\eta(x)$ currents can be constructed by combining charmed meson operators
and charmed baryon fields. We need the charmed meson operators
$J_{\mathcal{D}}$, which couple to the ground-state charmed mesons
$\mathcal{D}=\bar{D}^{0}/\bar{D}^{*0}$:

$\displaystyle J_{\bar{D}^{0}}$ $\displaystyle=$
$\displaystyle\bar{c}_{d}\gamma_{5}u_{d}\,,$ (4) $\displaystyle
J_{\bar{D}^{*0}}$ $\displaystyle=$
$\displaystyle\bar{c}_{d}\gamma_{\mu}u_{d}\,.$

We also need the charmed baryon field $J_{\mathcal{B}}$, which couples to the
ground-state charmed baryon $\mathcal{B}=\Xi_{c}^{0}$:

$\displaystyle J_{\Xi_{c}^{0}}$ $\displaystyle=$
$\displaystyle\epsilon^{abc}[d_{a}^{T}\mathbb{C}\gamma_{5}s_{b}]c_{c}\,.$ (5)

There can be altogether three $\bar{D}^{(*)}\Xi_{c}$ hadronic molecular
states, that are $\bar{D}\Xi_{c}$ of $J^{P}={1/2}^{-}$, $\bar{D}^{*}\Xi_{c}$
of $J^{P}={1/2}^{-}$, and $\bar{D}^{*}\Xi_{c}$ of $J^{P}={3/2}^{-}$. Their
relevant interpolating currents are:

$\displaystyle\eta_{1}$ $\displaystyle=$
$\displaystyle[\delta^{ab}\bar{c}_{a}\gamma_{5}u_{b}]~{}[\epsilon^{cde}d_{c}^{T}\mathbb{C}\gamma_{5}s_{d}c_{e}]$
(6) $\displaystyle=$ $\displaystyle\bar{D}^{0}~{}\Xi_{c}^{0}\,,$
$\displaystyle\eta_{2}$ $\displaystyle=$
$\displaystyle[\delta^{ab}\bar{c}_{a}\gamma_{\nu}u_{b}]~{}\gamma^{\nu}\gamma_{5}~{}[\epsilon^{cde}d_{c}^{T}\mathbb{C}\gamma_{5}s_{d}c_{e}]$
(7) $\displaystyle=$
$\displaystyle\bar{D}^{*0}_{\nu}~{}\gamma^{\nu}\gamma_{5}~{}\Xi_{c}^{0}\,,$
$\displaystyle\eta_{3}^{\alpha}$ $\displaystyle=$ $\displaystyle
P_{3/2}^{\alpha\nu}~{}[\delta^{ab}\bar{c}_{a}\gamma_{\nu}u_{b}]~{}[\epsilon^{cde}d_{c}^{T}\mathbb{C}\gamma_{5}s_{d}c_{e}]$
(8) $\displaystyle=$ $\displaystyle
P_{3/2}^{\alpha\nu}~{}\bar{D}^{*0}_{\nu}~{}\Xi_{c}^{0}\,.$

In the above expressions, we have used $\mathcal{D}$ and $\mathcal{B}$ to
denote the charmed meson operators $J_{\mathcal{D}}$ and the charmed baryon
field $J_{\mathcal{B}}$; $P_{3/2}^{\mu\nu}$ is the spin-3/2 projection
operator

$P_{3/2}^{\mu\nu}=g^{\mu\nu}-{1\over 4}\gamma^{\mu}\gamma^{\nu}\,.$ (9)

QCD sum rule studies.—– We use the method of QCD sum rules Shifman:1978bx ;
Reinders:1984sr  to study $\bar{D}^{(*)}\Xi_{c}$ molecular states through the
currents $\eta_{1,2}$ of $J^{P}=1/2^{-}$ and $\eta^{\alpha}_{3}$ of
$J^{P}=3/2^{-}$. Taking the current $\eta_{2}$ as an example, we assume it
couples to the $\bar{D}^{*}\Xi_{c}$ molecular state of $J^{P}=1/2^{-}$ through

$\displaystyle\langle 0|\eta_{2}|X;1/2^{-}\rangle$ $\displaystyle=$
$\displaystyle f_{X}u(p)\,,$

where $u(p)$ is the Dirac spinor of this state, denoted as $X$ for simplicity.
The two-point correlation function extracted from $\eta_{2}$ can be written
as:

$\displaystyle\Pi\left(q^{2}\right)$ $\displaystyle=$ $\displaystyle i\int
d^{4}xe^{iq\cdot x}\langle
0|T\left[\eta_{2}(x)\bar{\eta}_{2}(0)\right]|0\rangle$ (10) $\displaystyle=$
$\displaystyle(q\\!\\!\\!/~{}+M_{X})~{}\Pi_{0}\left(q^{2}\right)\,.$

In QCD sum rule studies we calculate the two-point correlation function
$\Pi_{0}\left(q^{2}\right)$ at both hadron and quark-gluon levels. At the
hadron level, we use the dispersion relation to write it as

$\Pi_{0}(q^{2})={\frac{1}{\pi}}\int^{\infty}_{s_{<}}\frac{{\rm
Im}\Pi_{0}(s)}{s-q^{2}-i\varepsilon}ds\,,$ (11)

where $s_{<}$ is the physical threshold. We further define the imaginary part
of the correlation function as the spectral density $\rho(s)$, which is
usually evaluated at the hadron level by inserting intermediate hadron states
$\sum_{n}|n\rangle\langle n|$:

$\displaystyle\rho_{\rm phen}(s)$ $\displaystyle\equiv$ $\displaystyle{\rm
Im}\Pi_{0}(s)/\pi$ $\displaystyle=$
$\displaystyle\sum_{n}\delta(s-M^{2}_{n})\langle 0|\eta|n\rangle\langle
n|{\eta^{\dagger}}|0\rangle$ $\displaystyle=$ $\displaystyle
f_{X}^{2}\delta(s-m_{X}^{2})+\mbox{continuum}.$

In the last step we have adopted the usual parametrization of one-pole
dominance for the ground state $X$ and a continuum contribution.

At the quark-gluon level we calculate $\Pi_{0}\left(q^{2}\right)$ using the
method of operator product expansion (OPE), and derive its corresponding
spectral density $\rho_{\rm OPE}(s)$. After performing the Borel
transformation to Eq. (11) at both hadron and quark-gluon levels, we can
approximate the continuum using the spectral density above a threshold value
$s_{0}$ (quark-hadron duality), and obtain the sum rule equation

$\Pi_{0}(s_{0},M_{B}^{2})\equiv
f^{2}_{X}e^{-M_{X}^{2}/M_{B}^{2}}=\int^{s_{0}}_{s_{<}}e^{-s/M_{B}^{2}}\rho_{\rm
OPE}(s)ds\,.$ (13)

It can be used to further calculate $M_{X}$ through

$\displaystyle M^{2}_{X}(s_{0},M_{B})$ $\displaystyle=$
$\displaystyle\frac{\int^{s_{0}}_{s_{<}}e^{-s/M_{B}^{2}}s\rho_{\rm
OPE}(s)ds}{\int^{s_{0}}_{s_{<}}e^{-s/M_{B}^{2}}\rho_{\rm OPE}(s)ds}\,.$ (14)

In the present study we calculate OPEs at the leading order of $\alpha_{s}$
and up to the $D({\rm imension})=10$ terms, including the perturbative term,
the charm quark mass, the quark condensates
$\langle\bar{q}q\rangle/\langle\bar{s}s\rangle$, the gluon condensate $\langle
g_{s}^{2}GG\rangle$, the quark-gluon mixed condensates $\langle
g_{s}\bar{q}\sigma Gq\rangle/\langle g_{s}\bar{s}\sigma Gs\rangle$, and their
combinations. We summarized the obtained spectral densities $\rho_{1\cdots
3}(s)$ in Appendix A, which are extracted from the currents $\eta_{1\cdots
3}$, respectively. In the calculations we ignore the chirally suppressed terms
with the up and down quark masses, and adopt the factorization assumption of
vacuum saturation for higher dimensional condensates. We find that the $D=3$
quark condensates $\langle\bar{q}q\rangle/\langle\bar{s}s\rangle$ and the
$D=5$ mixed condensates $\langle g_{s}\bar{q}\sigma Gq\rangle/\langle
g_{s}\bar{s}\sigma Gs\rangle$ are multiplied by the charm quark mass, which
are thus important power corrections.

Numerical analyses.——– We still use the current $\eta_{2}$ as an example to
perform numerical analyses, where we use the following values for various QCD
sum rule parameters pdg ; Yang:1993bp ; Eidemuller:2000rc ; Narison:2002pw ;
Gimenez:2005nt ; Jamin:2002ev ; Ioffe:2002be ; Ovchinnikov:1988gk ;
Ellis:1996xc ; colangelo :

$\displaystyle m_{s}$ $\displaystyle=$ $\displaystyle 96^{+8}_{-4}\mbox{
MeV}\,,$ $\displaystyle m_{c}$ $\displaystyle=$ $\displaystyle
1.275^{+0.025}_{-0.035}\mbox{ GeV}\,,$ $\displaystyle\langle\bar{q}q\rangle$
$\displaystyle=$ $\displaystyle-(0.240\pm 0.010)^{3}\mbox{ GeV}^{3}\,,$
$\displaystyle\langle\bar{s}s\rangle$ $\displaystyle=$ $\displaystyle(0.8\pm
0.1)\times\langle\bar{q}q\rangle\,,$ (15) $\displaystyle\langle
g_{s}^{2}GG\rangle$ $\displaystyle=$ $\displaystyle(0.48\pm 0.14)\mbox{
GeV}^{4}\,,$ $\displaystyle\langle g_{s}\bar{q}\sigma Gq\rangle$
$\displaystyle=$ $\displaystyle-M_{0}^{2}\times\langle\bar{q}q\rangle\,,$
$\displaystyle\langle g_{s}\bar{s}\sigma Gs\rangle$ $\displaystyle=$
$\displaystyle-M_{0}^{2}\times\langle\bar{s}s\rangle\,,$ $\displaystyle
M_{0}^{2}$ $\displaystyle=$ $\displaystyle(0.8\pm 0.2)\mbox{ GeV}^{2}\,.$

Here the running mass in the $\overline{MS}$ scheme is used for the charm
quark.

There are two free parameters in Eqs. (14): the Borel mass $M_{B}$ and the
threshold value $s_{0}$. We use two criteria to constrain the Borel mass
$M_{B}$ for a fixed $s_{0}$. The first criterion is used to insure the
convergence of the OPE series. It is done by requiring the $D=10$ terms
($m_{c}\langle\bar{q}q\rangle^{3}$ and $\langle g_{s}\bar{q}\sigma
Gq\rangle^{2}$) to be less than 20%, which can determine the lower limit
$M_{B}^{min}$:

$\mbox{Convergence
(CVG)}\equiv\left|\frac{\Pi^{D=10}(\infty,M_{B})}{\Pi(\infty,M_{B})}\right|\leq
20\%\,.$ (16)

This criterion leads to $\left(M_{B}^{min}\right)^{2}=2.93$ GeV2, when setting
$s_{0}=25.8$ GeV2.

The second criterion is used to insure the validity of one-pole
parametrization. It is done by requiring the pole contribution to be larger
than 50%, which can determine the upper limit $M_{B}^{max}$:

$\mbox{Pole Contribution
(PC)}\equiv\frac{\Pi(s_{0},M_{B})}{\Pi(\infty,M_{B})}\geq 50\%\,.$ (17)

This criterion leads to $\left(M_{B}^{max}\right)^{2}=3.07$ GeV2, when setting
$s_{0}=25.8$ GeV2.

Figure 2: The variation of $M_{X}$ with respect to the Borel mass $M_{B}$,
calculated using the current $\eta_{2}$. The short-dashed, solid, and long-
dashed curves are obtained by setting $s_{0}=24.8$, $25.8$, and $26.8$ GeV2,
respectively.

Altogether we extract the working region of Borel mass to be $2.93$
GeV${}^{2}<M_{B}^{2}<3.07$ GeV2 for the current $\eta_{2}$ with the threshold
value $s_{0}=25.8$ GeV2. We show the variation of $M_{X}$ with respect to the
Borel mass $M_{B}$ in Fig. 2 in a broader region $2.7$ GeV${}^{2}\leq
M_{B}^{2}\leq 3.2$ GeV2, and find it more stable inside the above Borel
window.

Redoing the same procedures by changing $s_{0}$, we find that there are non-
vanishing Borel windows as long as $s_{0}\geq s_{0}^{min}=24.8$ GeV2.
Accordingly, we choose the threshold value $s_{0}$ to be about 1.0 GeV larger
with the uncertainty $\pm 1.0$ GeV, i.e., $24.8$ GeV${}^{2}\leq s_{0}\leq
26.8$ GeV2. Altogether, our working regions for the current $\eta_{2}$ are
determined to be $24.8$ GeV${}^{2}\leq s_{0}\leq 26.8$ GeV2 and $2.93$
GeV${}^{2}<M_{B}^{2}<3.07$ GeV2, where the mass is extracted to be:

$M_{\bar{D}^{*}\Xi_{c};1/2^{-}}=4.46^{+0.16}_{-0.14}\mbox{ GeV}\,.$ (18)

Here the central value corresponds to $M_{B}^{2}=3.00$ GeV2 and $s_{0}=25.8$
GeV2. Its uncertainty comes from the Borel mass $M_{B}$, the threshold value
$s_{0}$, the charm quark mass $m_{c}$, and various QCD sum rule parameters
listed in Eqs. (15). This mass value is consistent with the experimental mass
of the $P_{cs}(4459)^{0}$, supporting its interpretation as the
$\bar{D}^{*}\Xi_{c}$ molecular state of $J^{P}=1/2^{-}$.

Similarly, we use the currents $\eta_{1}$ and $\eta^{\alpha}_{3}$ to perform
numerical analyses, and extract masses of the $J^{P}=1/2^{-}$ $\bar{D}\Xi_{c}$
molecular state and the $J^{P}=3/2^{-}$ $\bar{D}^{*}\Xi_{c}$ molecular state
to be:

$\displaystyle M_{\bar{D}\Xi_{c};1/2^{-}}$ $\displaystyle=$ $\displaystyle
4.29^{+0.13}_{-0.12}\mbox{ GeV}\,,$ (19) $\displaystyle
M_{\bar{D}^{*}\Xi_{c};3/2^{-}}$ $\displaystyle=$ $\displaystyle
4.47^{+0.19}_{-0.15}\mbox{ GeV}\,.$ (20)

Hence, our results also support the interpretation of the $P_{cs}(4459)^{0}$
as the $\bar{D}^{*}\Xi_{c}$ molecular state of $J^{P}=3/2^{-}$. We summarize
all the above results in Table 1.

Generally speaking, understanding the nature of exotic hadrons is a
complicated topic, since different structures with the same quantum numbers
can contribute to the same state, and different structures may have similar
masses. However, these different structures may lead to different decay
processes. Therefore, to determine whether the $P_{cs}(4459)^{0}$ is the
$\bar{D}^{*}\Xi_{c}$ molecular state of $J^{P}=1/2^{-}$ or the one of
$J^{P}=3/2^{-}$ in our framework, we shall further calculate its width in our
future study, to be compared with its experimental value
$\Gamma_{P_{cs}(4459)^{0}}=17.3\pm 6.5^{+8.0}_{-5.7}$ MeV lhcb .

Table 1: Masses of the $\bar{D}^{(*)}\Xi_{c}$ hadronic molecular states, extracted from the currents $\eta^{(\alpha)}_{1\cdots 3}$. Currents |  Configuration |  $s_{0}^{min}~{}[{\rm GeV}^{2}]$ | Working Regions |  Pole [%] |  Mass [GeV]
---|---|---|---|---|---
$s_{0}~{}[{\rm GeV}^{2}]$ |  $M_{B}^{2}~{}[{\rm GeV}^{2}]$
$\eta_{1}$ | $|\bar{D}\Xi_{c};1/2^{-}\rangle$ | $22.3$ | $23.3\pm 1.0$ |  $2.61$–$2.77$ |  $50$–$56$ | $4.29^{+0.13}_{-0.12}$
$\eta_{2}$ | $|\bar{D}^{*}\Xi_{c};1/2^{-}\rangle$ | $24.8$ | $25.8\pm 1.0$ |  $2.93$–$3.07$ |  $50$–$55$ | $4.46^{+0.16}_{-0.14}$
$\eta_{3}^{\alpha}$ | $|\bar{D}^{*}\Xi_{c};3/2^{-}\rangle$ | $24.4$ | $25.4\pm 1.0$ |  $2.81$–$2.95$ |  $50$–$55$ | $4.47^{+0.19}_{-0.15}$

Summary and Discussions.—– Very recently, the LHCb Collaboration reported the
evidence of a hidden-charm pentaquark state with strangeness,
$P_{cs}(4459)^{0}$, in the $J/\psi\Lambda$ invariant mass spectrum of the
$\Xi_{b}^{-}\to J/\psi K^{-}\Lambda$ decays lhcb . This state contains at
least five quarks $\bar{c}csqq$ ($q=u/d$), with one of them the $strange$
quark. This LHCb experiment indicates that there probably exist many more
exotic hadrons with strangeness to be discovered in the near future, so a new
hadron spectrum is waiting to be constructed.

The $P_{cs}(4459)^{0}$ is about 19 MeV below the $\bar{D}^{*0}\Xi_{c}^{0}$
threshold, so it is natural to interpret it as the $\bar{D}^{*}\Xi_{c}$
molecular state. Accordingly, in this letter we use the method of QCD sum
rules to study the $\bar{D}^{*}\Xi_{c}$ molecular states of $J^{P}=1/2^{-}$
and $3/2^{-}$, and at the same time we also study the $\bar{D}\Xi_{c}$
molecular state of $J^{P}=1/2^{-}$. We evaluate their masses to be

$\displaystyle M_{\bar{D}\Xi_{c};1/2^{-}}$ $\displaystyle=$ $\displaystyle
4.29^{+0.13}_{-0.12}\mbox{ GeV}\,,$ $\displaystyle
M_{\bar{D}^{*}\Xi_{c};1/2^{-}}$ $\displaystyle=$ $\displaystyle
4.46^{+0.16}_{-0.14}\mbox{ GeV}\,,$ $\displaystyle
M_{\bar{D}^{*}\Xi_{c};3/2^{-}}$ $\displaystyle=$ $\displaystyle
4.47^{+0.19}_{-0.15}\mbox{ GeV}\,.$

Hence, our QCD sum rule results support the interpretation of the
$P_{cs}(4459)^{0}$ as the $\bar{D}^{*}\Xi_{c}$ molecular state of either
$J^{P}=1/2^{-}$ or $3/2^{-}$.

(a) $\Xi_{b}^{-}\to\bar{D}^{(*)0}\Xi_{c}^{0}K^{-}$ (b)
$\Lambda_{b}^{0}\to\bar{D}^{(*)0}\Lambda_{c}^{+}K^{-}$

(c) $\Lambda_{b}^{0}\to\bar{D}^{(*)0}\Sigma_{c}^{(*)+}K^{-}$

Figure 3: Possible production mechanisms of the $P_{c}^{+}$ and $P_{cs}^{0}$
states in $\Lambda_{b}^{0}/\Xi_{b}^{-}$ decays.

Within the hadronic molecular picture, the three LHCb experiments observing
$P_{c}$ and $P_{cs}$ states lhcb ; Aaij:2015tga ; Aaij:2019vzc  can be well
understood as a whole:

  * •

The transition of $\Xi_{b}^{-}\to J/\psi K^{-}\Lambda$ is dominated by the
Cabibbo-favored weak decay of $b\to c+\bar{c}s$ via the $V$-$A$ current. This
leads to an intuitive expectation that the $s$ and $d$ pair in $\Xi_{b}^{-}$
may exist as a spectator, so that their total spin $S=0$ is conserved. As a
consequence, if this $sd$ pair is to be combined with a $c$ quark to form a
charmed baryon, it will favor the $\Xi_{c}$ instead of the $\Xi_{c}^{\prime}$
and $\Xi_{c}^{*}$, of which the total spin of the $sd$ pair is $S=1$.
Accordingly, the $P_{cs}(4459)^{0}$ possibly as the $\bar{D}^{*}\Xi_{c}$
molecular state was observed in this channel by LHCb lhcb , other than those
possibly existing $\bar{D}^{(*)}\Xi_{c}^{\prime}$ and
$\bar{D}^{(*)}\Xi_{c}^{*}$ molecular states. We illustrate this process in
Fig. 3(a).

  * •

The transition of $\Lambda_{b}^{0}\to J/\psi K^{-}p$ can be similarly
analysed, which may favor the $\Lambda_{c}$ instead of the $\Sigma_{c}$ and
$\Sigma_{c}^{*}$. However, in this case the $\Lambda_{c}$ is probably not
bounded with the charmed mesons due to the lack of $\pi$ exchanges. It is only
because of the significantly larger data sample (680k for the
$\Lambda_{b}^{0}\to J/\psi K^{-}p$ decays and only 4k for the $\Xi_{b}^{-}\to
J/\psi K^{-}\Lambda$ decays lhcb ), that the $P_{c}(4312)$, $P_{c}(4440)$, and
$P_{c}(4457)$ possibly as the $\bar{D}\Sigma_{c}$ and $\bar{D}^{*}\Sigma_{c}$
molecular states were observed in this channel by LHCb Aaij:2015tga ;
Aaij:2019vzc . We illustrate the relevant two processes in Fig. 3(b,c).

Therefore, the three LHCb experiments observing $P_{c}$ and $P_{cs}$ states
lhcb ; Aaij:2015tga ; Aaij:2019vzc  strongly support the hadronic molecular
picture and the existence of “hadronic molecules”.

To further verify the above hadronic molecular picture, we propose to
investigate the $P_{cs}(4459)^{0}$ in future experiments to examine whether it
can be separated into two states. We also propose to search for the
$\bar{D}\Xi_{c}$ molecular state of $J^{P}=1/2^{-}$, whose mass is predicted
to be $4.29^{+0.13}_{-0.12}$ GeV. If the hadronic molecular picture turns out
to be correct, our understanding on the construction of the subatomic world be
significantly improved, and our understanding on the non-perturbative
behaviors of the strong interaction at the low energy region would also be
significantly improved.

## Acknowledgments

This project is supported by the National Natural Science Foundation of China
under Grants No. 11722540, No. 11975165, and No. 12075019, the China National
Funds for Distinguished Young Scientists under Grant No. 11825503, the
National Program for Support of Top-notch Young Professionals, and the
Fundamental Research Funds for the Central Universities.

##  Appendix A Spectral densities

In this appendix we list the spectral densities $\rho_{1\cdots 3}(s)$
extracted for the currents $\eta_{1\cdots 3}$. In the following expressions,
$\mathcal{F}(s)=\left[(\alpha+\beta)m_{c}^{2}-\alpha\beta s\right]$,
$\mathcal{H}(s)=\left[m_{c}^{2}-\alpha(1-\alpha)s\right]$, and the integration
limits are $\alpha_{min}=\frac{1-\sqrt{1-4m_{c}^{2}/s}}{2}$,
$\alpha_{max}=\frac{1+\sqrt{1-4m_{c}^{2}/s}}{2}$, $\beta_{min}=\frac{\alpha
m_{c}^{2}}{\alpha s-m_{c}^{2}}$, and $\beta_{max}=1-\alpha$. In the
calculations we take into account both the $m_{s}$ and $m_{s}^{2}$ terms, but
here we list only the $m_{s}$ terms.

The spectral density $\rho_{1}(s)$ extracted for the current $\eta_{1}$ is

$\displaystyle\rho_{1}(s)$ $\displaystyle=$
$\displaystyle\rho^{pert}_{1}(s)+\rho^{\langle\bar{q}q\rangle}_{1}(s)+\rho^{\langle
GG\rangle}_{1}(s)+\rho^{\langle\bar{q}Gq\rangle}_{1}(s)+\rho^{\langle\bar{q}q\rangle^{2}}_{1}(s)+\rho^{\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}_{1}(s)+\rho^{\langle\bar{q}Gq\rangle^{2}}_{1}(s)+\rho^{\langle\bar{q}q\rangle^{3}}_{1}(s)\,,$
(21)

where

$\displaystyle\rho^{pert}_{1}(s)$ $\displaystyle=$
$\displaystyle\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{5}\times\frac{(1-\alpha-\beta)^{3}}{163840\pi^{8}\alpha^{5}\beta^{4}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle}_{1}(s)$ $\displaystyle=$
$\displaystyle\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{3}\times\frac{(1-\alpha-\beta)((\alpha+\beta-1)\beta
m_{c}\langle\bar{q}q\rangle+\beta
m_{s}(\langle\bar{s}s\rangle-2\langle\bar{q}q\rangle))}{2048\pi^{6}\alpha^{3}\beta^{3}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle GG\rangle}_{1}(s)$ $\displaystyle=$
$\displaystyle{\langle
g_{s}^{2}GG\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}m_{c}^{2}\mathcal{F}(s)^{2}\times\frac{(1-\alpha-\beta)^{3}\left(\alpha^{3}+\beta^{3}\right)}{196608\pi^{8}\alpha^{5}\beta^{4}}$
$\displaystyle~{}~{}~{}~{}~{}~{}+\mathcal{F}(s)^{3}\times\frac{(\alpha+\beta-1)\left(3\alpha^{3}-\alpha^{2}(\beta+3)-2\alpha(\beta-1)\beta-(\beta-1)^{2}\beta\right)}{196608\pi^{8}\alpha^{5}\beta^{3}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}Gq\rangle}_{1}(s)$ $\displaystyle=$
$\displaystyle\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{2}\times\frac{-3\left(\alpha^{2}+\alpha(3\beta-2)+2\beta^{2}-3\beta+1\right){m_{c}\langle
g_{s}\bar{q}\sigma Gq\rangle}+3\beta^{2}{m_{s}\langle g_{s}\bar{q}\sigma
Gq\rangle}}{4096\pi^{6}\alpha^{2}\beta^{3}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle^{2}}_{1}(s)$ $\displaystyle=$
$\displaystyle\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\frac{-\mathcal{F}(s)^{2}\times\langle\bar{q}q\rangle\langle\bar{s}s\rangle+m_{c}m_{s}\mathcal{F}(s)\times\alpha(2\langle\bar{q}q\rangle^{2}-\langle\bar{q}q\rangle\langle\bar{s}s\rangle)}{256\pi^{4}\alpha^{2}\beta}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}_{1}(s)$
$\displaystyle=$
$\displaystyle\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\Bigg{\\{}\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\frac{m_{c}m_{s}\langle
g_{s}\bar{q}\sigma
Gq\rangle(2\langle\bar{q}q\rangle-\langle\bar{s}s\rangle)}{512\pi^{4}\beta}\Bigg{\\}}$
$\displaystyle~{}~{}~{}~{}~{}~{}+\mathcal{H}(s)\times\frac{\langle\bar{q}q\rangle\langle
g_{s}\bar{s}\sigma Gs\rangle+\langle\bar{s}s\rangle\langle g_{s}\bar{q}\sigma
Gq\rangle}{512\pi^{4}\alpha}-\frac{m_{c}m_{s}\langle g_{s}\bar{q}\sigma
Gq\rangle(4\langle\bar{q}q\rangle-\langle\bar{s}s\rangle)}{1024\pi^{4}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}Gq\rangle^{2}}_{1}(s)$ $\displaystyle=$
$\displaystyle\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\Bigg{\\{}\frac{(\alpha-1)\langle
g_{s}\bar{q}\sigma Gq\rangle\langle g_{s}\bar{s}\sigma
Gs\rangle}{1024\pi^{4}}\Bigg{\\}}+\int^{1}_{0}d\alpha\Bigg{\\{}\delta\left(s-{m_{c}^{2}\over\alpha(1-\alpha)}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}\times\frac{\langle g_{s}\bar{q}\sigma
Gq\rangle\left(m_{c}^{3}m_{s}\langle g_{s}\bar{q}\sigma
Gq\rangle/M_{B}^{2}+(1-\alpha)m_{c}^{2}\langle g_{s}\bar{s}\sigma
Gs\rangle-\alpha(\alpha+1)m_{c}m_{s}\langle g_{s}\bar{q}\sigma
Gq\rangle\right)}{2048\pi^{4}(\alpha-1)\alpha}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle^{3}}_{1}(s)$ $\displaystyle=$
$\displaystyle{m_{c}\langle\bar{q}q\rangle^{2}\langle\bar{s}s\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\Bigg{\\{}\frac{1}{96\pi^{2}}\Bigg{\\}}\,.$

The spectral density $\rho_{2}(s)$ extracted for the current $\eta_{2}$ is

$\displaystyle\rho_{2}(s)$ $\displaystyle=$
$\displaystyle\rho^{pert}_{2}(s)+\rho^{\langle\bar{q}q\rangle}_{2}(s)+\rho^{\langle
GG\rangle}_{2}(s)+\rho^{\langle\bar{q}Gq\rangle}_{2}(s)+\rho^{\langle\bar{q}q\rangle^{2}}_{2}(s)+\rho^{\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}_{2}(s)+\rho^{\langle\bar{q}Gq\rangle^{2}}_{2}(s)+\rho^{\langle\bar{q}q\rangle^{3}}_{2}(s)\,,$
(22)

where

$\displaystyle\rho^{pert}_{2}(s)$ $\displaystyle=$
$\displaystyle\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{5}\times\frac{(1-\alpha-\beta)^{3}}{81920\pi^{8}\alpha^{5}\beta^{4}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle}_{2}(s)$ $\displaystyle=$
$\displaystyle\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{3}\times\frac{(1-\alpha-\beta)(2(\alpha+\beta-1)m_{c}\langle\bar{q}q\rangle+\beta
m_{s}(\langle\bar{s}s\rangle-2\langle\bar{q}q\rangle))}{1024\pi^{6}\alpha^{3}\beta^{3}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle GG\rangle}_{2}(s)$ $\displaystyle=$
$\displaystyle{\langle
g_{s}^{2}GG\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}m_{c}^{2}\mathcal{F}(s)^{2}\times\frac{(1-\alpha-\beta)^{3}\left(\alpha^{3}+\beta^{3}\right)}{98304\pi^{8}\alpha^{5}\beta^{4}}$
$\displaystyle~{}~{}~{}~{}~{}~{}+\mathcal{F}(s)^{3}\times\frac{(1-\alpha-\beta)\left(3\alpha^{3}+\alpha^{2}(7\beta-3)+2\alpha(\beta-1)\beta+(\beta-1)^{2}\beta\right)}{98304\pi^{8}\alpha^{5}\beta^{3}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}Gq\rangle}_{2}(s)$ $\displaystyle=$
$\displaystyle\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{2}\times\frac{-3\langle
g_{s}\bar{q}\sigma Gq\rangle(2(\alpha+\beta-1)m_{c}-\beta
m_{s})}{2048\pi^{6}\alpha^{2}\beta^{2}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle^{2}}_{2}(s)$ $\displaystyle=$
$\displaystyle\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\frac{-\mathcal{F}(s)^{2}\times\langle\bar{q}q\rangle\langle\bar{s}s\rangle-
m_{c}m_{s}\mathcal{F}(s)\times
2\alpha(\langle\bar{s}s\rangle-2\langle\bar{q}q\rangle)}{128\pi^{4}\alpha^{2}\beta}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}_{2}(s)$
$\displaystyle=$
$\displaystyle\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\Bigg{\\{}\mathcal{H}(s)\times\frac{\langle\bar{q}q\rangle\langle
g_{s}\bar{s}\sigma Gs\rangle+\langle\bar{s}s\rangle\langle g_{s}\bar{q}\sigma
Gq\rangle}{256\pi^{4}\alpha}-\frac{m_{c}m_{s}\langle g_{s}\bar{q}\sigma
Gq\rangle(4\langle\bar{q}q\rangle-\langle\bar{s}s\rangle)}{256\pi^{4}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}Gq\rangle^{2}}_{2}(s)$ $\displaystyle=$
$\displaystyle\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\Bigg{\\{}\frac{(\alpha-1)\langle
g_{s}\bar{q}\sigma Gq\rangle\langle g_{s}\bar{s}\sigma
Gs\rangle}{512\pi^{4}}\Bigg{\\}}+\int^{1}_{0}d\alpha\Bigg{\\{}\delta\left(s-{m_{c}^{2}\over\alpha(1-\alpha)}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}\times\frac{\langle g_{s}\bar{q}\sigma
Gq\rangle\left(2m_{c}^{3}m_{s}\langle g_{s}\bar{q}\sigma
Gq\rangle/M_{B}^{2}+(1-\alpha)m_{c}^{2}\langle g_{s}\bar{s}\sigma
Gs\rangle-2(\alpha-1)\alpha m_{s}\langle g_{s}\bar{q}\sigma
Gq\rangle\right)}{1024\pi^{4}(\alpha-1)\alpha}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle^{3}}_{2}(s)$ $\displaystyle=$
$\displaystyle{m_{c}\langle\bar{q}q\rangle^{2}\langle\bar{s}s\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\Bigg{\\{}\frac{1}{24\pi^{2}}\Bigg{\\}}\,.$

The spectral density $\rho_{3}(s)$ extracted for the current
$\eta_{3}^{\alpha}$ is

$\displaystyle\rho_{3}(s)$ $\displaystyle=$
$\displaystyle\rho^{pert}_{3}(s)+\rho^{\langle\bar{q}q\rangle}_{3}(s)+\rho^{\langle
GG\rangle}_{3}(s)+\rho^{\langle\bar{q}Gq\rangle}_{3}(s)+\rho^{\langle\bar{q}q\rangle^{2}}_{3}(s)+\rho^{\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}_{3}(s)+\rho^{\langle\bar{q}Gq\rangle^{2}}_{3}(s)+\rho^{\langle\bar{q}q\rangle^{3}}_{3}(s)\,,$
(23)

where

$\displaystyle\rho^{pert}_{3}(s)$ $\displaystyle=$
$\displaystyle\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{5}\times\frac{(1-\alpha-\beta)^{3}(\alpha+\beta+4)}{1310720\pi^{8}\alpha^{5}\beta^{4}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle}_{3}(s)$ $\displaystyle=$
$\displaystyle\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{3}\times\frac{(\alpha+\beta-1)(6(1-\alpha-\beta)m_{c}\langle\bar{q}q\rangle+\beta(2\alpha+2\beta+3)m_{s}(2\langle\bar{q}q\rangle-\langle\bar{s}s\rangle))}{16384\pi^{6}\alpha^{3}\beta^{3}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle GG\rangle}_{3}(s)$ $\displaystyle=$
$\displaystyle{\langle
g_{s}^{2}GG\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}m_{c}^{2}\mathcal{F}(s)^{2}\times\frac{(1-\alpha-\beta)^{3}(\alpha+\beta+4)\left(\alpha^{3}+\beta^{3}\right)}{1572864\pi^{8}\alpha^{5}\beta^{4}}$
$\displaystyle~{}~{}~{}~{}~{}~{}+\mathcal{F}(s)^{3}\times\Bigg{(}\frac{4\alpha^{5}-3\alpha^{4}(3\beta+11)-6\alpha^{3}\left(6\beta^{2}+13\beta-9\right)-\alpha^{2}\left(32\beta^{3}+51\beta^{2}-108\beta+25\right)}{4718592\pi^{8}\alpha^{5}\beta^{3}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{-3\alpha(\beta-1)^{2}\beta(4\beta+11)-3(\beta-1)^{3}\beta(\beta+4)}{4718592\pi^{8}\alpha^{5}\beta^{3}}\Bigg{)}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}Gq\rangle}_{3}(s)$ $\displaystyle=$
$\displaystyle\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{2}\times\frac{3\langle
g_{s}\bar{q}\sigma
Gq\rangle(-6(\alpha+\beta-1)m_{c}+\beta(4\alpha+4\beta+1)m_{s})}{32768\pi^{6}\alpha^{2}\beta^{2}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle^{2}}_{3}(s)$ $\displaystyle=$
$\displaystyle\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\frac{-\mathcal{F}(s)^{2}\times(4\alpha+4\beta+1)\langle\bar{q}q\rangle\langle\bar{s}s\rangle-
m_{c}m_{s}\mathcal{F}(s)\times
6\alpha(\langle\bar{s}s\rangle-2\langle\bar{q}q\rangle)}{2048\pi^{4}\alpha^{2}\beta}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}_{3}(s)$
$\displaystyle=$
$\displaystyle\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\Bigg{\\{}\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)\times\frac{-\langle\bar{q}q\rangle\langle
g_{s}\bar{s}\sigma Gs\rangle-\langle\bar{s}s\rangle\langle g_{s}\bar{q}\sigma
Gq\rangle}{1024\pi^{4}\alpha}\Bigg{\\}}$
$\displaystyle~{}~{}~{}~{}~{}~{}+\mathcal{H}(s)\times\frac{5\langle\bar{q}q\rangle\langle
g_{s}\bar{s}\sigma Gs\rangle+5\langle\bar{s}s\rangle\langle g_{s}\bar{q}\sigma
Gq\rangle}{4096\pi^{4}\alpha}-\frac{3m_{c}m_{s}\langle g_{s}\bar{q}\sigma
Gq\rangle(4\langle\bar{q}q\rangle-\langle\bar{s}s\rangle)}{4096\pi^{4}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}Gq\rangle^{2}}_{3}(s)$ $\displaystyle=$
$\displaystyle\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\Bigg{\\{}\frac{3(\alpha-1)\langle
g_{s}\bar{q}\sigma Gq\rangle\langle g_{s}\bar{s}\sigma
Gs\rangle}{8192\pi^{4}}\Bigg{\\}}+\int^{1}_{0}d\alpha\Bigg{\\{}\delta\left(s-{m_{c}^{2}\over\alpha(1-\alpha)}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}\times\frac{\langle g_{s}\bar{q}\sigma
Gq\rangle\left(6m_{c}^{3}m_{s}\langle g_{s}\bar{q}\sigma
Gq\rangle/M_{B}^{2}-5(\alpha-1)m_{c}^{2}\langle g_{s}\bar{s}\sigma
Gs\rangle-6(\alpha-1)\alpha m_{c}m_{s}\langle g_{s}\bar{q}\sigma
Gq\rangle\right)}{16384\pi^{4}(\alpha-1)\alpha}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle^{3}}_{3}(s)$ $\displaystyle=$
$\displaystyle{m_{c}\langle\bar{q}q\rangle^{2}\langle\bar{s}s\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\Bigg{\\{}\frac{1}{128\pi^{2}}\Bigg{\\}}\,.$

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