# A new look at the $P_{cs}$ states from a molecular perspective

Albert Feijoo1<EMAIL_ADDRESS>Wen-Fei Wang1,2<EMAIL_ADDRESS>Chu-Wen
Xiao3<EMAIL_ADDRESS>Jia-Jun Wu4<EMAIL_ADDRESS>Eulogio Oset1
<EMAIL_ADDRESS>Juan Nieves1<EMAIL_ADDRESS>Bing-Song Zou4,5
<EMAIL_ADDRESS>1Departamento de Física Teórica and IFIC, Centro Mixto
Universidad de Valencia-CSIC Institutos de Investigación de Paterna, Aptdo.
22085, 46071 Valencia, Spain
2Institute of Theoretical Physics, Shanxi University, Taiyuan, Shanxi 030006,
China
3School of Physics and Electronics, Central South University, Changsha 410083,
China
4School of Physical Sciences, University of Chinese Academy of Sciences
(UCAS), Beijing 100049, China
5Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing 100190, China

###### Abstract

We have a look at the $P_{cs}$ states generated from the interaction of
$\bar{D}^{(*)}\Xi^{(\prime*)}_{c}$ coupled channels. We consider the blocks of
pseudoscalar-baryon $({\frac{1}{2}}^{+},{\frac{3}{2}}^{+})$ and vector-baryon
$({\frac{1}{2}}^{+},{\frac{3}{2}}^{+})$, and find $10$ resonant states
coupling mostly to
$\bar{D}\Xi_{c},\bar{D}^{*}\Xi_{c},\bar{D}\Xi^{\prime}_{c},\bar{D}^{*}\Xi^{\prime}_{c},\bar{D}\Xi^{*}_{c}$
and $\bar{D}^{*}\Xi^{*}_{c}$. A novel aspect of the work is the realization
that the $\bar{D}\Xi_{c},\bar{D}_{s}\Lambda_{c}$ or
$\bar{D}^{*}\Xi_{c},\bar{D}^{*}_{s}\Lambda_{c}$ channels, with a strong
transition potential, collaborate to produce a larger attraction than the
corresponding states $\bar{D}\Sigma_{c},\bar{D}\Lambda_{c}$ or
$\bar{D}^{*}\Sigma_{c},\bar{D}^{*}\Lambda_{c}$ appearing in the generation of
the strangenessless $P_{c}$ states, since in the latter case the transition
potential between those channels is zero. The extra attraction obtained in the
$\bar{D}\Xi_{c},\bar{D}^{*}\Xi_{c}$ pairs preclude the association of these
channels to the $P_{cs}(4338)$ and $P_{cs}(4459)$ states respectively. Then we
find a natural association of the $P_{cs}(4338)$ state coupling mostly to
$\bar{D}^{*}\Xi_{c}$ while the $P_{cs}(4459)$ is associated to the state found
that couples mostly to $\bar{D}\Xi^{\prime}_{c}$. Four more states appear,
like in other molecular pictures, and some of the states are degenerate in
spin. Counting different spin states we find $10$ states, which we hope can be
observed in the near future.

##  I Introduction

The discovery of the two pentaquark states by the LHCb collaboration in $2015$
in the $J/\psi p$ spectrum of the $\Lambda^{0}_{b}\to J/\psi K^{-}p$ decay
Aaij  _et al._ (2015, 2016) was a turning point in hadron physics which
stimulated a tremendous amount of theoretical work aimed at describing the
nature of such states. This work is reflected in a large number of review
papers Chen  _et al._ (2016a); Hosaka  _et al._ (2016); Chen  _et al._
(2017a); Lebed  _et al._ (2017); Esposito  _et al._ (2017); Oset  _et al._
(2016); Guo  _et al._ (2018); Ali  _et al._ (2017a); Olsen  _et al._ (2018);
Karliner  _et al._ (2018); Yuan (2018); Liu  _et al._ (2019) and different
pictures have been proposed to explain these states, in most cases as
molecular states of meson baryon or as compact pentaquark states. In $2019$
the LHCb collaboration updated the former results with additional data from
Run-2, where three narrow structures were observed Aaij  _et al._ (2019), the
$P_{c}(4312)$, $P_{c}(4440)$ and $P_{c}(4457)$. Interestingly, in Refs. Wu
_et al._ (2010, 2011) predictions were done in $2010$ about such states of
molecular nature, which agreed qualitatively with the states found. The work
of Wu  _et al._ (2010, 2011) was revisited from the point of view of heavy
quark spin symmetry Isgur and Wise (1989); Neubert (1994); Manohar and Wise
(2000) in Xiao  _et al._ (2013) and the only parameter of the theoretical
model, the regulator of the meson-baryon loops, was fine tuned to get a good
description of the experimental data. The same work in Refs. Wu  _et al._
(2010, 2011) predicted also states of hidden charm and strange content,
$c\bar{c}sqq$, and work followed in Refs. Santopinto and Giachino (2017); Chen
_et al._ (2017b); Wang  _et al._ (2020). One of such states, named
$P_{cs}(4459)$, was found again by the LHCb collaboration in the
$J/\psi\Lambda$ invariant mass distribution of the $\Xi^{-}_{b}\to J/\psi
K^{-}\Lambda$ Aaij  _et al._ (2021), which had been suggested in Santopinto
and Giachino (2017); Xiao  _et al._ (2019); Chen  _et al._ (2016b) as an ideal
reaction to look for these states. The experimental finding drew much
theoretical attention intended to explain the nature of the state found, while
making further predictions Chen  _et al._ (2021); Wang (2021); Azizi  _et al._
(2021); Özdem (2021); Peng  _et al._ (2021); Chen (2021a, b); Liu  _et al._
(2021a); Zhu  _et al._ (2021); Liu  _et al._ (2021b); Wang and Liu (2022);
Özdem (2023). Furthermore, very recently, a new $P_{cs}$ state, $P_{cs}(4338)$
was found in the $J/\psi\Lambda$ mass distribution of $B^{-}\to
J/\psi\Lambda\bar{p}$ decay Aaij  _et al._ (2022).

After the finding of the $P_{c}$ states, predictions for $P_{cs}$ states were
made in Ref. Xiao  _et al._ (2019) within the heavy quark spin symmetry scheme
used successfully in Ref. Xiao  _et al._ (2013). The predictions made could be
considered in rough qualitative agreement with the experimental states found
later. Yet, with the perspective given by the two states found and the
theoretical work done after the observation of the latest $P_{cs}(4338)$
state, a fresh look at the problem is most opportune and we address this issue
here.

In Wang and Wang (2023), the QCD sum rule method was used and the molecular
structure of $\bar{D}\Xi_{c},\bar{D}^{*}\Xi_{c}$ for the $P_{cs}(4338)$ and
$P_{cs}(4459)$ states was suggested. In Meng  _et al._ (2022) the boson
exchange model was used and, once again the $\bar{D}\Xi_{c}$ is associated
with the $P_{cs}(4338)$ while the $\bar{D}^{*}\Xi_{c}$ , with two structures
as ${\frac{1}{2}}^{-},{\frac{3}{2}}^{-}$, is associated to the $P_{cs}(4459)$.
At the same time predictions are made of states of
$\bar{D}^{(*)}\Xi^{\prime}_{c}$ structure. The lineshape of the
$J/\psi\Lambda$ mass distribution is considered in Meng  _et al._ (2022)
assuming that the $P_{cs}(4338)$ state is made from the
$\bar{D}^{0}\Xi_{c}^{0}$ and $D^{-}\Xi_{c}^{+}$ components. A different view
is presented in Burns and Swanson (2022) suggesting that the $P_{cs}(4338)$
peak, together with another peak, are the result of triangle singularities,
relying on a formalism that contains a few free parameters. A constituent
quark model is investigated in Ortega  _et al._ (2022) looking at the
molecular meson-baryon structures and 9 states are obtained of
$\eta\Lambda,J/\psi\Lambda,\bar{D}^{(*)}_{s}\Lambda_{c},\bar{D}^{(*)}\Xi_{c},\bar{D}^{(*)}\Xi^{\prime}_{c},\bar{D}^{(*)}\Xi_{c}^{(*)}$
type. Once again the $P_{cs}(4338)$ is associated mostly to the
$\bar{D}\Xi_{c}$ state, although the $\bar{D}^{(*)}_{s}\Lambda_{c}$ channel
also has a sizable component, while the $\bar{D}^{*}\Xi_{c}^{*}$ has a small
one. In Wang and Liu (2022) the molecular picture is invoked suggesting that
the $P_{cs}(4459)$ may contain two substructures of $\bar{D}^{*}\Xi_{c}$ with
$J^{P}={1/2}^{-},{3/2}^{-}$ and there is the corresponding $\bar{D}\Xi_{c}$
state with $J^{P}={1/2}^{-}$. A pentaquark structure is assumed for the two
states in Özdem (2023) and magnetic moments are evaluated using QCD light cone
sum rules. In Zhu  _et al._ (2022), a potential kernel constructed from meson
exchange is used in the Bethe-Salpeter equation and association of
$\bar{D}\Xi_{c}$ to the $P_{cs}(4338)$ is made while $\bar{D}^{*}\Xi_{c}$
appears in ${1/2}^{-},{3/2}^{-}$, mostly in $J^{P}={1/2}^{-}$, and is
associated to the $P_{cs}(4459)$. In Nakamura and Wu (2022), the authors
determine the pole of $P_{cs}(4338)$ within a coupled channel framework by
fitting the various distributions of the final states in $B^{-}\to
J/\psi\Lambda\bar{p}$ and also confirm that the peak in the spectrum of
$J/\psi\Lambda$ cannot come just from a cusp. In that model, the
$P_{cs}(4338)$ is also related to the $\bar{D}\Xi_{c}$ channel and there is
another possible pole named as $P_{cs}(4255)$ hinted by one datum around the
$\bar{D}_{s}\Lambda_{c}$ threshold which is, however, compatible with a
statistical fluctuation 111Tim Gerson private communication.. In Chen  _et
al._ (2022), the meson-baryon interaction is parametrized according to
$SU(3)_{f}$ light flavor and heavy quark symmetries, and coupled channels are
considered. Once again the $P_{cs}(4338)$ state is associated to the
$\bar{D}\Xi_{c}$ component, while two states of $\bar{D}^{*}\Xi_{c}$ nature
with $J^{P}={1/2}^{-},{3/2}^{-}$ would be associated to the $P_{cs}(4459)$,
although a state of $\bar{D}\Xi^{\prime}_{c}$ nature appears around $4434$ MeV
with a very small width, which would make it incompatible with the observed
$P_{cs}(4459)$ state. One interesting feature of Chen  _et al._ (2022) is the
important effect of the $\bar{D}\Xi_{c}-\bar{D}^{(*)}_{s}\Lambda_{c}$ coupled
channels that reduces the binding of the $\bar{D}\Xi_{c}$ state considered as
single channel. We shall discuss the effect of coupled channels in detail in
this paper. In Yang  _et al._ (2022), using as the source of interaction
$\sigma,\rho,\omega$ exchange, once again, the association of the
$P_{cs}(4338)$ to $\bar{D}\Xi_{c}$ and $P_{cs}(4459)$ to $\bar{D}^{*}\Xi_{c}$
is favored.

One important issue affecting both the $P_{c}$ and $P_{cs}$ states is the
present status concerning the spin-parity of the states. While all the
theoretical studies suggest $J^{P}$ values, mostly
$J^{P}={1/2}^{-},{3/2}^{-}$, for the different states found, experimentally
these quantum numbers are not yet established. It is fair to assume that
future runs of the LHCb with more statistics will allow the determination of
these important quantities, which will help put some order into the different
theoretical predictions. One promising path of progress in this direction is
given by the recent work Zhang  _et al._ (2023) where the spectrum, analyzed
with the help of machine learning techniques, sheds light on this particular
issue.

The present work offers and explanation of the two $P_{cs}$ pentaquark states
in a molecular picture which successfully reproduced the observed $P_{c}$
states. There are however alternative proposals based on compact pentaquark
states starting with the interaction of the quarks themselves. There are
multiple works on the issue which is not our purpose to discuss here, but it
becomes instructive to discuss some of the main differences between these
pictures and how future experiments can help to find the picture adjusting
more closely to the reality. Reviews on the subject and comparison between
models have been quoted before, and adjusting more to the compact pentaquark
structures one can mention Lebed  _et al._ (2017); Esposito  _et al._ (2017);
Ali  _et al._ (2017a); Liu  _et al._ (2019). One characteristics of these
models is the large freedom to make substructures of quarks. In contrast to
the molecules that have the quarks clustered into a meson and a baryon, the
compact pentaquarks rely on diquark structures. Yet, even then, there is a
large freedom on how to clusterize the five quarks. As an example we choose a
few works that evidence this freedom. In Ali  _et al._ (2019, 2016, 2017b)
compact hidden-charm diquark-diquark-antiquark pentaquarks structures are
considered, while in Giron  _et al._ (2021); Giron and Lebed (2021) the
authors consider diquark-triquark structures. In Shi  _et al._ (2021) two
diquarks and an antiquark structures are taken, as in Ali  _et al._ (2019,
2016, 2017b), but there are differences in the details of the interaction,
which lead to large differences in the obtained spectra. Even between Ali  _et
al._ (2019) and Ali  _et al._ (2016, 2017b) by the same group, differences of
mass of $100$-$300$ MeV are found for the same states. There are also
differences in the assignment of spin and parity to the observed states.
Indeed, while in Shi  _et al._ (2021) states of negative parity are preferred,
in Ali  _et al._ (2019) some states would be of positive parity and others of
negative parity. In Giron  _et al._ (2021); Giron and Lebed (2021) structures
corresponding to orbital excitation of the clusters with $L=1$, are preferred,
leading to positive parity states. One common characteristics of all these
models is the large amount of states predicted, of the order of $30$ in Ali
_et al._ (2019); Shi  _et al._ (2021), but it is discussed in Ali  _et al._
(2019) that if arguments of heavy quark spin symmetry are used, the number is
reduced. Another problem related to these models is the width of the states
which is only discussed qualitatively but no precise numbers are given. The
widths appear naturally large and arguments are given some times to justify
that restrictions imposed by heavy quark symmetry could be responsible for the
small observed widths Ali  _et al._ (2019). Different arguments are given in
Brodsky  _et al._ (2014), suggesting that the small widths originate from the
significant spatial separation between the diquark or triquark quasiparticles,
which hinders the rearrangement of their component quarks into color-singlet
hadrons. On the other hand in Giron  _et al._ (2021) the large widths expected
from states with orbital angular momentum $L=0$ are taken as an argument to
prefer the $L=1$ pentaquarks, that lead automatically to positive parity.

In the molecular picture there is less freedom and one obtains fewer states.
The states obtained are all of negative parity since the S-wave attractive
interaction is applied between negative parity meson and a positive parity
baryon for $P_{c}$ and $P_{cs}$ states. The widths obtained are small and they
result automatically from decay to states of meson baryon type, used in the
same set of the coupled channels which are allowed to interact in a coupled
channels unitary scheme.

Concerning the $P_{cs}$ states, the works discussed above only refer to the
$P_{cs}(4459)$ and the new $P_{cs}(4338)$ is not discussed. The width of the
$P_{cs}(4459)$ state is not addressed either. It would be interesting to
extend the predictions of these models to the new state and its properties. In
any case, the discussion done above leads us to the conclusion that to make
advances in the field and pin down the nature of the $P_{c}$ and $P_{cs}$
states, the experimental determination of the spin and parity of these states
is crucial. On the other hand, both the compact pentaquark models and the
molecular ones predict more states than have been observed. Since all the
models, adjusting their input to present states, make different predictions
for other states, the likely observation of new states in future runs of
present facilities, with more statistics, will also provide very valuable
information to help us get a closer look at the structure of these states.

##  II Formalism

We start from the formalism of Xiao  _et al._ (2019), which used heavy quark
spin symmetry (HQSS) together with the local hidden gauge approach (LHG) Bando
_et al._ (1985, 1988); Meissner (1988); Nagahiro  _et al._ (2009) extrapolated
to the charm sector in order to determine the coefficients of the lowest order
(LO) HQSS effective theory scheme. We divert a bit here from this picture and
evaluate all the matrix elements of the coupled channels potential using the
local hidden gauge approach. There are two reasons for it. The extrapolation
of the local hidden gauge approach to the heavy quark sector satisfies
strictly the rules of HQSS for the dominant terms. Indeed, this approach
relies upon the exchange of vector mesons. The leading order dominant terms
correspond to the exchange of light vectors, where the heavy quarks are
spectators, which automatically makes the matrix elements independent of the
heavy quarks and the rules of heavy quark symmetry are automatically
fulfilled. However the LHG approach also allows for the exchange of heavy
vector mesons which are suppressed by a $(m_{V}/m_{D^{*}})^{2}$ factor, and
hence, vanishing in the infinite heavy quark mass limit, but still non
negligible at the charm quark mass scale. Second, in Xiao  _et al._ (2019) the
HQSS rules were extended to some terms which are subdominant, for instance to
relate some non diagonal pseudoscalar-baryon (PB) and vector-baryon (VB)
components. While this had minor numerical effects, it is not justified from
the fundamental point of view and the use of the strict LHG approach avoids
this. Another novelty in the present approach is the use of the cutoff
regularization of the loop functions $G_{l}$ instead of the dimensional
regularization employed in Xiao  _et al._ (2019). The reason is that in the
charm sector the $G$ functions in dimensional regularization can become
positive below threshold, eventually allowing the appearance of bound states
with a repulsive potential Wu  _et al._ (2012). On the other hand we rely on
the fact that the PB and VB components are related by the exchange of
pseudoscalar mesons and their mixing is small as discussed in detail in
Appendix B of Ref. Dias  _et al._ (2021). This said, we have now four block of
coupled channels that we do not mix:

  * 1)

$PB(\frac{1}{2}^{+})$: Pseudoscalar meson-baryon of $J^{P}=\frac{1}{2}^{+}$,
$\eta_{c}\Lambda(4099.6)$, $\bar{D}_{s}\Lambda_{c}(4259.8)$,
$\bar{D}\Xi_{c}(4336.3)$, $\bar{D}\Xi_{c}^{\prime}(4445.7)$;

  * 2)

$PB(\frac{3}{2}^{+})$: Pseudoscalar meson-baryon of $J^{P}=\frac{3}{2}^{+}$,
$\bar{D}\Xi_{c}^{*}(4512.9)$;

  * 3)

$VB(\frac{1}{2}^{+})$: Vector meson-baryon of $J^{P}=\frac{1}{2}^{+}$,
$J/\psi\Lambda(4212.6)$, $\bar{D}^{*}_{s}\Lambda_{c}(4398.7)$,
$\bar{D}^{*}\Xi_{c}(4477.6)$, $\bar{D}^{*}\Xi_{c}^{\prime}(4587.0)$;

  * 4)

$VB(\frac{3}{2}^{+})$: Vector meson-baryon of $J^{P}=\frac{3}{2}^{+}$,
$\bar{D}^{*}\Xi_{c}^{*}(4654.2)$;

where the numbers in parenthesis are the threshold masses of the coupled
channels in units of MeV. In the nomenclature $PB(1/2^{+})$, etc., the
$1/2^{+}$ or $3/2^{+}$ refers to the $J^{P}$ of the baryon alone.

The wave functions of the elementary baryons in terms of quarks are taken as
in Xiao  _et al._ (2019) (section 2.3), where, following Roberts and Pervin
(2008), we isolate the heavy quark and put the spin-flavor symmetry of the
wave function in the light quarks. This allows for an easy evaluation of the
$VBB$ vertices in terms quarks, while the $PPV$ of $VVV$ vertices are trivial
since mesons are simple $q\bar{q}$ structures. Then we obtain the transition
potentials between channels as

$\displaystyle V_{ij}=C_{ij}\frac{1}{4f_{\pi}^{2}}(p^{0}+p^{\prime 0})\,,$ (1)

where $f_{\pi}=93$ MeV, and $p^{0},p^{\prime 0}$ are the energies of the
incoming and outgoing mesons in the $PB\to P^{\prime}B^{\prime}$, etc.
transitions. The $C_{ij}$ coefficients for $I=0$ are easily evaluated and
coincide with those obtained in Xiao  _et al._ (2019) for the dominant terms.
We write the results in Tables 1-2 for the $PB(1/2^{+})$ and $VB(1/2^{+})$,
respectively. The coefficient $\lambda$ corresponds to the exchange of $D^{*}$
or $D^{*}_{s}$, which is suppressed by approximately the ratio
$(m_{V}/m_{D^{*}})^{2}$ (with $m_{V}=800$ MeV for the light vectors). As in
Mizutani and Ramos (2006) we take the value $\lambda=0.25$. For the sectors
$PB(3/2^{+})$ and $VB(3/2^{+})$, we have only one channel for each of them,
$\bar{D}\Xi_{c}^{*}\to\bar{D}\Xi_{c}^{*}$ and
$\bar{D}^{*}\Xi_{c}^{*}\to\bar{D}^{*}\Xi_{c}^{*}$, respectively, with the same
value of $-1$ for the coefficient $C_{11}$.

As mentioned above, the dominant terms, this is to say those which remain
finite in the heavy quark limit, compiled in Tables 1 and 2, agree with those
found in Ref. Xiao  _et al._ (2019). In the later work, the subdominant terms,
driven by the exchanged of a heavy meson, were fixed from the predictions of
the LHG model for the $PB(1/2^{+})$ block, and used, assuming HQSS, to all the
rest of channels. This is inexact because the HQSS matrices derived in Xiao
_et al._ (2019) (Eqs. (5), (6) and (7)) are only correct in the strict heavy
quark limit. Indeed, there exist corrections when the interaction is due to
the exchanges of heavy mesons which break the invariance under spin rotations
of the heavy quarks. Here we evaluate this HQSS terms within the LHG scheme.

Table 1: Coefficients $C_{ij}$ for the PB sector with $J^{P}=\frac{1}{2}^{-}$ and $I=0$. | $\eta_{c}\Lambda\;$ | $\bar{D}_{s}\Lambda_{c}$ | $\bar{D}\Xi_{c}$ | $\bar{D}\Xi_{c}^{\prime}$
---|---|---|---|---
$\eta_{c}\Lambda$ | $0$ | $-\frac{1}{\sqrt{3}}\lambda$ | $\frac{1}{\sqrt{6}}\lambda$ | $-\frac{1}{\sqrt{2}}\lambda$
$\bar{D}_{s}\Lambda_{c}$ |  | $0$ | $-\sqrt{2}$ | $0$
$\bar{D}\Xi_{c}$ |  |  | $-1$ | $0$
$\bar{D}\Xi_{c}^{\prime}$ |  |  |  | $-1$
Table 2: Coefficients $C_{ij}$ for the VB sector with $J^{P}=\frac{1}{2}^{-},\frac{3}{2}^{-}$ and $I=0$. | $J/\psi\Lambda\;$ | $\bar{D}^{*}_{s}\Lambda_{c}$ | $\bar{D}^{*}\Xi_{c}$ | $\bar{D}^{*}\Xi_{c}^{\prime}$
---|---|---|---|---
$J/\psi\Lambda$ | $0$ | $-\frac{1}{\sqrt{3}}\lambda$ | $\frac{1}{\sqrt{6}}\lambda$ | $-\frac{1}{\sqrt{2}}\lambda$
$\bar{D}^{*}_{s}\Lambda_{c}$ |  | $0$ | $-\sqrt{2}$ | $0$
$\bar{D}^{*}\Xi_{c}$ |  |  | $-1$ | $0$
$\bar{D}^{*}\Xi_{c}^{\prime}$ |  |  |  | $-1$

Here comes an important observation. As we see, in the matrices of sectors
$1)$ and $3)$ in Tables 1 and 2, respectively, the only non zero diagonal
terms correspond to $\bar{D}\Xi_{c}$, $\bar{D}\Xi_{c}^{\prime}$ and
$\bar{D}^{*}\Xi_{c}$, $\bar{D}^{*}\Xi_{c}^{\prime}$ with $C_{ii}=-1$. We
should not expect bound states for $\eta_{c}\Lambda$ and
$\bar{D}_{s}\Lambda_{c}$ in $PB(1/2^{+})$ sector or $J/\psi\Lambda$ and
$\bar{D}^{*}_{s}\Lambda_{c}$ in $VB(1/2^{+})$ sector. However, the transition
$C_{ij}$ from $\bar{D}_{s}\Lambda_{c}$ to $\bar{D}\Xi_{c}$ or
$\bar{D}^{*}_{s}\Lambda_{c}$ to $\bar{D}^{*}\Xi_{c}$ is $-\sqrt{2}$, strong
compared with the diagonal terms. The relevance of these two strongly coupled
channels was already stressed in Chen  _et al._ (2022), however with a
different interaction.

To understand the effect of the coupled channels, let us recall that the
molecular states are obtained by the solution of the Bethe-Salpeter equation
in coupled channels

$\displaystyle T=[1-VG]^{-1}V\,,$ (2)

with $G$ the loop function of a meson-baryon intermediate state

$\displaystyle G_{l}$ $\displaystyle=$ $\displaystyle\int_{|{\bf
q}|<q_{max}}\frac{d^{3}q}{(2\pi)^{3}}\frac{1}{2\omega_{l}({\bf
q})}\frac{M_{l}}{E_{l}({\bf q})}$ (3) $\displaystyle\cdot$
$\displaystyle\frac{1}{\sqrt{s}-\omega_{l}({\bf q})-E_{l}({\bf
q})+i\epsilon}\,,$

where $\omega_{l}({\bf q})=\sqrt{m^{2}_{l}+{\bf q}^{2}}$ for the meson and
$E_{l}=\sqrt{M^{2}_{l}+{\bf q}^{2}}$ for the baryon and $q_{max}$ is a
regulator tied to the range of the interaction in momentum space Gamermann
_et al._ (2010); Song  _et al._ (2022). It is clear that this function is
always negative below threshold $\sqrt{s}=M_{th}=m_{l}+M_{l}$. At
$\sqrt{s}=M_{th}-\epsilon$, the real part Re$G$ of the loop function has
infinite slope and for $\sqrt{s}>M_{th}$ it grows smoothly, but keeps negative
in a long range of energies where one whishes to apply the theory. In the case
of using dimensional regularization with non relativistic kinematics
Re$G_{l}(\sqrt{s})$ for $\sqrt{s}>M_{th}$ is a constant, equal to the value of
threshold, which we should take negative as it corresponds to the evaluation
of Eq. (3) at threshold with any cutoff Kaplan  _et al._ (1996). In the
present case and taking $q_{max}=600$ MeV suited to reproduce the results of
Refs. Xiao  _et al._ (2013, 2019), for $P_{c}$ and $P_{cs}$, where the
dimensional regularization was used for the loops, the situation is as shown
in Fig. 1. In Fig. 2 we show the same $G$ functions in dimensional
regularization, with relativistic kinematics, having the same value at the
threshold as using the cutoff regularization with $q_{max}=600$ MeV.

Figure 1: Re$G_{l}(\sqrt{s})$ for the $\bar{D}_{s}\Lambda_{c}$ and
$\bar{D}\Xi_{c}$ channels using cut off regularization with $q_{max}=600$ MeV.
Figure 2: Re$G_{l}(\sqrt{s})$ for the $\bar{D}_{s}\Lambda_{c}$ and
$\bar{D}\Xi_{c}$ channels using dimensional regularization.

As we can see, in the region of the $\bar{D}\Xi_{c}$ channel around $4338$
MeV, the real part of the two $G$ functions is negative and in Fig. 2 we
observe that Re$G_{\bar{D}\Xi_{c}}$ can become positive below threshold as
discussed in Wu  _et al._ (2012), when dimensional regularization is used.

Let us take Eq. (2) with the two relevant channels $\bar{D}\Xi_{c}$ and
$\bar{D}_{s}\Lambda_{c}$, which we label now as $1$ and $2$, respectively, for
the discussion that follows. We have a $V$ matrix of the type

$\left(\begin{array}[]{cc}V_{11}&V_{12}\\\ V_{12}&0\\\ \end{array}\right)$ (4)

and Eq. (2) is immediately solved with the solution

$T_{11}=\frac{V_{11}+V_{12}^{2}G_{2}}{1-(V_{11}+V_{12}^{2}G_{2})G_{1}}\,.$ (5)

This is also discussed in section $6$ of Aceti  _et al._ (2014) (see also
Hyodo (2013)). Eq. (5) tells us that we can eliminate channel $2$ and work
with only channel $1$ with an effective potential

$V_{eff}=V_{11}+V_{12}^{2}G_{2}.$ (6)

If the channel $2$ had a higher threshold than channel $1$, Re$G_{2}$ would be
always negative when looking for a bound state of channel $1$. In the opposite
situation, as we have in Fig. 1, Re$G_{2}$ can reach positive values, but, for
$q_{max}=600$ MeV, it is negative around the threshold of the $\bar{D}\Xi_{c}$
channel and below, where the bound states are expected. One should note that
the raise of Re$G_{\bar{D}_{s}\Lambda_{c}}$ (Re$G_{2}$) is tied to the
relatively small cutoff. Indeed if $q_{max}=q_{on}$, where $q_{on}$ is the on
shell momentum for the $\bar{D}_{s}\Lambda_{c}$ channel
($\sqrt{s}=\omega(q_{on})+E(q_{on})$), then Re$G\to+\infty$ as $\sqrt{s}$
increases, because there is only one branch to the left of $q_{max}$ in the
evaluation of the principal value of the integral, and the branch above
$q_{max}$ that would cancel the infinite is not allowed. This is certainly not
realistic and one should not use this $G$ function in that region, but the $G$
function is well behaved in the region where $\bar{D}\Xi_{c}$ bound states
appear. To clarify further the situation we also plot
Re$G_{\bar{D}_{s}\Lambda_{c}}$ in dimensional regularization in Fig. 2
normalized to the same value at threshold, and we see that
Re$G_{\bar{D}_{s}\Lambda_{c}}$ is always negative in the wide region of the
plot. The consequence of this is that the effective potential has incremented
$V_{11}$ (the $\bar{D}\Xi_{c}$ potential) by $V_{12}^{2}G_{2}$, which is
negative in the region of relevance to us and, hence, the addition of the new
$\bar{D}_{s}\Lambda_{c}$ channel has the effect of an extra attraction in the
$\bar{D}\Xi_{c}$ channel. We have extended this discussion because the
conclusion in Ref. Chen  _et al._ (2022) is opposite, and in that work the
inclusion of the $\bar{D}_{s}\Lambda_{c}$ channel produces repulsion. While
this is possible depending on the regularization procedure followed, our
arguments strongly favor the general rule that the elimination of one channel
reverts into extra attraction in the channel kept.

This has strong consequences in the comparison of the results that we obtain
with those obtained for the $P_{c}$ states. Indeed, the diagonal channels
corresponding to $\bar{D}\Xi_{c}$ and $\bar{D}_{s}\Lambda_{c}$ for the $P_{c}$
state in Ref. Xiao  _et al._ (2013) are $\bar{D}\Sigma_{c}$ and
$\bar{D}\Lambda_{c}$, but in Xiao  _et al._ (2013) the $C_{ij}$ coefficient
for the transition between these latter channels (see Eq. (29) of Xiao  _et
al._ (2013)) is zero. Thus, we should expect that the $\bar{D}\Xi_{c}$ state
that we obtain here is more bound than the $\bar{D}\Sigma_{c}$ found in Xiao
_et al._ (2013) at $4262$ MeV, which is bound by about $58$ MeV. Hence we
should expect the binding of the $\bar{D}\Xi_{c}$ state reasonable bigger.
This also means that we cannot by any means associate the new $P_{cs}(4338)$
state to the $\bar{D}\Xi_{c}$ channel which has the mass $4336.3$ MeV, and it
should correspond to a different channel. Even if we make the
$\bar{D}\Sigma_{c}$ less bound to correspond to the $P_{c}(4312)$, the
$\bar{D}\Xi_{c}$ state still will be bound by several tens of MeV and there is
no way to make it correspond to the $P_{cs}(4338)$. On the other hand,
$\bar{D}\Xi_{c}^{\prime}$ is the equivalent to $\bar{D}\Sigma_{c}$ (mixed
symmetric spin wave functions) and one might expect tens of MeV of bounding.

##  III Results

Table 3: Coupling constants and wave functions at the origin of all channels, corresponding to poles of the $T$ matrix in the $PB(1/2^{+})$ sector with $J^{P}=\frac{1}{2}^{-}$. Masses and half width in units of MeV, we stress in bold face the dominant channel. Poles |  | $\eta_{c}\Lambda$ | $\bar{D}_{s}\Lambda_{c}$ | $\bar{D}\Xi_{c}$ | $\bar{D}\Xi_{c}^{\prime}$
---|---|---|---|---|---
$4198.94+i0.11$ | $g_{i}$ | ${0.12-i0.00}$ | $3.01-i0.01$ | $\bm{4.85+i0.01}$ | $0.01-i0.03$
$g_{i}G^{II}_{i}$ | ${-0.35+i1.01}$ | $-19.24+i0.05$ | $\bm{-20.35-i0.07}$ | $-0.03+i0.09$
$4422.79+i7.75$ | $g_{i}$ | $0.71-i0.08$ | $0.05+i0.00$ | ${-0.06+i0.04}$ | $\bm{2.79-i0.35}$
$g_{i}G^{II}_{i}$ | $1.14+i10.71$ | $0.76+i1.82$ | ${-0.43-i1.53}$ | $\bm{-26.54+i0.64}$
Table 4: The same as in Table 3, but for the $VB(1/2^{+})$ sector with $J^{P}=\frac{1}{2}^{-},\frac{3}{2}^{-}$. Poles |  | $J/\psi\Lambda$ | $\bar{D}^{*}_{s}\Lambda_{c}$ | $\bar{D}^{*}\Xi_{c}$ | $\bar{D}^{*}\Xi_{c}^{\prime}$
---|---|---|---|---|---
$4337.98+i0.12$ | $g_{i}$ | ${0.11-i0.00}$ | $3.17-i0.01$ | $\bm{5.07+i0.01}$ | $0.00-i0.04$
$g_{i}G^{II}_{i}$ | ${-0.13+i1.07}$ | $-18.57+i0.06$ | $\bm{-19.94-i0.07}$ | $-0.01+i0.10$
$4565.73+i15.58$ | $g_{i}$ | $0.70-i0.16$ | $0.09-i0.03$ | ${-0.10+i0.09}$ | $\bm{2.84-i0.72}$
$g_{i}G^{II}_{i}$ | $6.24+i21.04$ | $2.07+i3.40$ | ${-1.37-i2.81}$ | $\bm{-26.31+i1.15}$

In Table 3 we show the results that we obtain for the $PB(1/2^{+})$ sector,
showing the masses of the states, the couplings $g_{i}$ to each channel
evaluated with $q_{max}=600$ MeV and the wave functions evaluated at the
origin, $g_{i}G^{II}$ ($G^{II}$ is the loop function in the second Riemann
sheet) Gamermann  _et al._ (2010). In Table 4 we show the equivalent results
in the $VB(1/2^{+})$ sector. In addition we have the single channel for
$PB(3/2^{+})$ and $VB(3/2^{+})$, with their poles and couplings as

$\displaystyle~{}~{}\;\bar{D}\Xi_{c}^{*}:~{}4488.52~{}{\rm
MeV},~{}g=2.81,~{}gG^{II}=-26.47,$ $\displaystyle~{}~{}\quad\;\qquad
J^{P}=\frac{3}{2}^{-};$ (7)
$\displaystyle~{}\bar{D}^{*}\Xi_{c}^{*}:~{}4627.48~{}{\rm
MeV},~{}g=2.96,~{}gG^{II}=-25.93,$ $\displaystyle~{}~{}\quad\;\qquad
J^{P}=\frac{1}{2}^{-},\frac{3}{2}^{-},\frac{5}{2}^{-}.$ (8)

Figure 3: Loop diagram for the $\bar{D}^{*}\Xi_{c}^{(\prime)}$ state to decay
to $\bar{D}_{s}\Lambda_{c}$.

There is a caveat to be taken into account. The $\eta_{c}\Lambda$ channel has
a threshold at $4100$ MeV, $326$ MeV below the state obtained in Table 3
around $4423$ MeV. Channels far away in energy should not be relevant for the
structure of the state obtained, except as being potential decay channels.
This is why in Refs. Dai  _et al._ (2020); Wang  _et al._ (2021) a strategy
was used, and proved successful, which is to neglect the real part of the $G$
function of such channels and consider only their imaginary part. We use the
same strategy here, which is further justified since for $q_{max}=600$ MeV we
find a situation like the one of Fig. 1 for Re$G$ of the
$\bar{D}_{s}\Lambda_{c}$ channel where Re$G_{\eta_{c}\Lambda}$ blows up at
$\sqrt{s}\approx 4305$ MeV. We take here $G_{\eta_{c}\Lambda}\to
i$Im$G_{\eta_{c}\Lambda}(M_{\rm
inv})=-i\frac{1}{4\pi}p_{\eta_{c}}\frac{M_{\Lambda}}{M_{\rm inv}}$, with
$p_{\eta_{c}}$ the on shell momentum for $\eta_{c}$ in the decay of an object
of energy $\sqrt{s}$ to $\eta_{c}\Lambda$.

Inspection of Tables 3-4 shows that the state to be associated to the
$P_{cs}(4338)$ is the state that couples mostly to $\bar{D}^{*}\Xi_{c}$ in
Table 4. The mass coincides with the experimental value. The width is very
small, ($0.24$ MeV), with the state decaying to $J/\psi\Lambda$ which has a
very small coupling. The width would become a bit large if we consider the
transition to the $PB(1/2^{+})$ channels, essentially $\bar{D}_{s}\Lambda_{c}$
given the threshold masses. This requires the triangle loop of Fig. 3.
However, the dominant intermediate state $\bar{D}^{*}\Xi_{c}$ does not
contribute because the $q\bar{q}\bm{\sigma}\cdot\bm{q}$ operator at the quark
level, with the wave functions used in Xiao  _et al._ (2019), does not connect
the $\Xi_{c}$ and $\Lambda_{c}$ states that go both with the spin $\chi_{MA}$
function. The $\bar{D}^{*}\Xi_{c}^{\prime}$ channel can contribute, but we see
in Table 4 that the obtained coupling of the $4338$ MeV state to
$\bar{D}^{*}\Xi_{c}^{\prime}$ is very small. We expect an extra very small
width. Actually, the experimental width is $\Gamma_{P_{cs}(4338)}=7.0\pm
1.2\pm 1.3$ MeV, from where we should also exclude the effects of the
experimental resolution. In this sense our predictions of a very small width
should be considered as positive feature of the calculation. We should note
that the state found also couples strongly to the $\bar{D}^{*}_{s}\Lambda_{c}$
component, although with weaker coupling than to $\bar{D}^{*}\Xi_{c}$.

Furthermore, we should note that we get two degenerate states of
$\frac{1}{2}^{-}$ and $\frac{3}{2}^{-}$ in this case. In practice,
consideration of $\pi$ exchange to couple the $VB$ with $PB$ channels will
break the degeneracy, if the two states are very close, this could justify the
observed experimental results. Actually, this happened with the splitting of
the $P_{c}$ state observed in Aaij  _et al._ (2015) into two states observed
in Aaij  _et al._ (2019) with more statistics. Future measurements with higher
resolution could tell us more about this suggestion.

Then we have to see a candidate for the $P_{cs}(4459)$. Our natural choice is
the $PB(1/2^{+})$ state obtained at $4423$ MeV in Table 3, which couples
mostly to $\bar{D}\Xi_{c}^{\prime}$. In favor of this association we should
mention that we find a width of $15$ MeV which compares favorably with the
experimental one of $\Gamma_{P_{cs}(4459)}=17.3\pm 6.5^{+8.0}_{-5.7}$ MeV.

This reasonable association to the observed states still leaves some states
unobserved. The one we predict at $4198.94$ MeV can decay to $\eta_{c}\Lambda$
according to Table 3 and we should expect a small width, as shown by the
calculation, given the small coupling of this state to $\eta_{c}\Lambda$. The
other states coupling mostly to $\bar{D}^{*}\Xi_{c}^{\prime}$,
$\bar{D}\Xi_{c}^{*}$ and $\bar{D}^{*}\Xi_{c}^{*}$ appear at energies around
$4500$ MeV and above. The $\bar{D}^{*}\Xi_{c}^{\prime}$ state, shown in Table
4, appears degenerate in our approach with
$J^{P}=\frac{1}{2}^{-},\frac{3}{2}^{-}$, while the $\bar{D}\Xi_{c}^{*}$ and
$\bar{D}^{*}\Xi_{c}^{*}$, as shown in Eqs. (7)-(8), appear with
$J^{P}=\frac{3}{2}^{-}$ and
$J^{P}=\frac{1}{2}^{-},\frac{3}{2}^{-},\frac{5}{2}^{-}$, respectively. In
total, and counting the different spins, we have $10$ states. They also show
up at high energy in the molecular papers mentioned above and we are hopeful
that it is a question of time that they are observed.

##  IV Further considerations

Associating the state at $4338$ MeV to the $\bar{D}^{*}\Xi_{c}$ implies a
binding of this channel by $140$ MeV. This is certainly bigger than the
binding of the $\bar{D}\Sigma_{c}$ with respect to the $P_{c}(4312)$ of $13$
MeV. As we have already discussed, we expect a more binding here because of
the effect of the coupled channel $\bar{D}^{*}_{s}\Lambda_{c}$. To make it
more explicit we show in Fig. 4 how the state appears in the case of the
$\bar{D}^{*}\Xi_{c}$ and $\bar{D}^{*}_{s}\Lambda_{c}$ coupled channels.

Figure 4: Re$G$ for the $\bar{D}^{*}\Xi_{c}$ and $\bar{D}^{*}_{s}\Lambda_{c}$
channels. The dashed line corresponds to
$V^{-1}_{\bar{D}^{*}\Xi_{c},\bar{D}^{*}\Xi_{c}}$, the dashed dotted line to
$({\rm Re}V_{eff})^{-1}$ (Eq. (9)).

Should we have only one channel, $\bar{D}^{*}\Xi_{c}$, the pole appears when
$V^{-1}_{\bar{D}^{*}\Xi_{c},\bar{D}^{*}\Xi_{c}}=G_{\bar{D}^{*}\Xi_{c}}$. This
produces a binding of about $20$ MeV, of the order of magnitude of the one of
$\bar{D}\Sigma_{c}$ in the $P_{c}(4312)$ state. But, when we consider the
effect of the $\bar{D}^{*}_{s}\Lambda_{c}$ coupled channel, as shown in Eq.
(6), this gives rise to a $\bar{D}^{*}\Xi_{c}$ effective potential

$\displaystyle
V_{eff}(\bar{D}^{*}\Xi_{c})=V_{\bar{D}^{*}\Xi_{c},\bar{D}^{*}\Xi_{c}}+V^{2}_{\bar{D}^{*}\Xi_{c},\bar{D}^{*}_{s}\Lambda_{c}}\cdot
G_{\bar{D}^{*}_{s}\Lambda_{c}}.$ (9)

We observe that now $V_{eff}^{-1}$ cuts Re$G_{\bar{D}^{*}\Xi_{c}}$ at the
energy of $4338$ MeV as found in Table 4, with a binding much bigger than for
the original single channel. We plot $({\rm Re}V_{eff})^{-1}$, but below the
$\bar{D}^{*}_{s}\Lambda_{c}$ threshold where the pole appears $V_{eff}$ is
real. It is logical to obtain this result. Indeed one can see that the
additional potential in $V_{eff}$ is as large as
$V_{\bar{D}^{*}\Xi_{c},\bar{D}^{*}\Xi_{c}}$ which makes
$V_{eff}^{-1}\approx\frac{1}{2}(V_{\bar{D}^{*}\Xi_{c},\bar{D}^{*}\Xi_{c}})^{-1}$
and this cuts Re$G_{\bar{D}^{*}\Xi_{c}}$ at much lower energies. Should Re$G$
be a linear function, we would obtain a larger binding, but the slope of Re$G$
below the threshold is infinite and then gradually it becomes smaller in
modulus such that a double potential means in practice a much bigger binding
than we would obtain with a linear $G$ function, as one can appreciate in the
figure.

On the other hand, the $\bar{D}\Xi_{c}^{\prime}$ channel does not benefit from
the coupled channel effect since it only couples and weakly to the
$\eta_{c}\Lambda$ channel, so one must rely in this case on the diagonal
$\bar{D}\Xi_{c}^{\prime}$ interaction. Thus, with respect to the
$\bar{D}\Xi_{c}^{\prime}$ threshold, the channel is only bound by about $22$
MeV, in line with the single potential binding of the $\bar{D}^{*}\Xi_{c}$
state. Yet, the experimental state $P_{cs}(4459)$ is about $13$ MeV above the
$\bar{D}\Xi_{c}^{\prime}$ threshold, although with experimental errors it is
closer. It is not easy for us to obtain a state above threshold, although the
energy dependence of the potential allows it. However, should we admit a
slightly different cutoff in the $PB$ than the $VB$ threshold and using
$q_{max}=500$ MeV in the $PB$ sector, one obtains a mass of $4438$ MeV for the
state, closer to the experimental value, and one would obtain the result ever
closer with small value of $q_{max}$ as used to obtain the $T_{cc}$ state
Feijoo  _et al._ (2021) of the order of $450$ MeV. Admitting that a more
complicated picture could emerge for this state, yet, we find it reasonable to
accept that the channel $\bar{D}\Xi_{c}^{\prime}$ with a threshold around the
nominal mass of the $P_{cs}(4459)$ has much to do with that observed state.
Incidentally, the state that appeared before at $4200$ MeV appears now at
$4242$ MeV when $q_{max}=500$ MeV is used. Hence, that state is more sensitive
to the cutoff and one must also accept larger uncertainties in that
prediction. It will also be interesting to pay attention to future runs with
more statistics for the experiment reporting that state, since other possible
structures are hinted in the present spectrum.

##  V Conclusions

We have studied the interaction in coupled channels of pseudoscalar mesons and
vector mesons with baryons of $J^{P}=1/2^{+}$ and $3/2^{+}$, evaluating the
scattering matrix $T$, and looking for poles in the second Riemann sheet, from
where we obtain the couplings looking at the residue of $T_{ij}$ at the pole.
We obtain $10$ states coupling mostly to $\bar{D}\Xi_{c}$,
$\bar{D}^{*}\Xi_{c}$, $\bar{D}\Xi_{c}^{\prime}$,
$\bar{D}^{*}\Xi_{c}^{\prime}$, $\bar{D}\Xi_{c}^{*}$ and
$\bar{D}^{*}\Xi_{c}^{*}$. We associated the $P_{cs}(4338)$ state to the one
obtained with the same energy, coupling mostly to $\bar{D}^{*}\Xi_{c}$. The
$P_{cs}(4459)$ was associated to the state that couples mostly to
$\bar{D}\Xi_{c}^{\prime}$ and the other states, one below the $J/\psi\Lambda$
threshold, which cannot be observed in this channel, and the other states
appearing at higher energies, which are still unobserved.

The key point leading to this association was the realization that the
$\bar{D}_{s}\Lambda_{c}$ and $\bar{D}^{*}_{s}\Lambda_{c}$ channels play an
important role given the strong transition potential from these channels to
the $\bar{D}\Xi_{c}$ and $\bar{D}^{*}\Xi_{c}$ respectively, which has no
counterpart in the corresponding $\bar{D}\Sigma_{c}$, $\bar{D}\Lambda_{c}$ and
$\bar{D}^{*}\Sigma_{c}$, $\bar{D}^{*}\Lambda_{c}$ channels that appear in the
evaluation of the $P_{c}$ states. Indeed, in this latter case the transition
potentials between $\bar{D}\Sigma_{c}$ and $\bar{D}\Lambda_{c}$, or
$\bar{D}^{*}\Sigma_{c}$ and $\bar{D}^{*}\Lambda_{c}$ are zero. We showed that
this transition reverted into extra attraction. Hence, there is more
attraction in the $\bar{D}\Xi_{c}$ or $\bar{D}^{*}\Xi_{c}$ channels than in
the $\bar{D}\Sigma_{c}$ or $\bar{D}^{*}\Sigma_{c}$ ones and hence we should
expect more binding in $\bar{D}\Xi_{c}$ and $\bar{D}^{*}\Xi_{c}$ than was
found for $\bar{D}\Sigma_{c}$ and $\bar{D}^{*}\Sigma_{c}$ which produced the
observed $P_{c}$ pentaquarks. It is then hard to associate the $P_{cs}(4338)$
directly to the $\bar{D}\Xi_{c}$ channel, which has a threshold mass at the
same energy, and still be consistent with the commonly accepted molecular
picture for the $P_{c}$ states.

In view of that, the natural picture that emerges from the present study is
that the $P_{cs}(4338)$ state must be associated to the state coupling mostly
to $\bar{D}^{*}\Xi_{c}$, while the $P_{cs}(4459)$ should be associated with
the state coupling mostly to $\bar{D}\Xi_{c}^{\prime}$. The state coupling
mostly to $\bar{D}\Xi_{c}$ is predicted below the $J/\psi\Lambda$ threshold
and could be observed in $\eta_{c}\Lambda$. The other states appear at higher
energies. These findings should stimulate further experimental research. They
are also predicted in other molecular pictures at different energies and their
observation would be most helpful to shed light on present theoretical
approaches.

###### Acknowledgements.

This work was supported by the Spanish Ministerio de Economia y Competitividad
(MINECO) and European FEDER funds under Contracts No. PID2020-112777GB-I00,
and by Generalitat Valenciana under contract PROMETEO/2020/023. This project
has received funding from the European Union Horizon 2020 research and
innovation programme under the program H2020-INFRAIA-2018-1, grant agreement
No. 824093 of the STRONG-2020 project. The work of A. F. was partially
supported by the Generalitat Valenciana and European Social Fund
APOSTD-2021-112. This work of W. F. W. was supported in part by the National
Natural Science Foundation of China under Grant No. 12147215. This work is
also partly supported by the Fundamental Research Funds for the Central
Universities (JJW), and NSFC under Grant No. 12070131001 (CRC110 cofunded by
DFGand NSFC), Grant No. 11835015, No. 12047503, and by the Chinese Academy of
Sciences (CAS) under Grants No.XDB34030000 (BSZ).

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