# Insights into the nature of the $P_{cs}(4459)$

Meng-Lin Dua111<EMAIL_ADDRESS>, Zhi-Hui Guob222<EMAIL_ADDRESS>, J.
A. Ollerc333<EMAIL_ADDRESS>
a Instituto de Física Corpuscular (centro mixto CSIC-UV),
Institutos de Investigación de Paterna, Apartado 22085, 46071, Valencia, Spain
b School of Physics, Southeast University, Nanjing 211189, China
c Departamento de Física. Universidad de Murcia. E-30071 Murcia. Spain

###### Abstract

We study the nature of the recently observed $P_{cs}(4459)$ by the LHCb
collaboration by employing three methods based on the elastic effective-range
expansion and the resulting size of the effective-range, the saturation of the
compositeness relation and width of the resonance, and a direct fit to data
involving the channels $J/\psi\Lambda$, $\Xi^{\prime}_{c}\bar{D}$ and
$\Xi_{c}\bar{D}^{*}$. We have also considered the addition of a CDD pole but
this scenario can be discarded. Our different analyses clearly indicate the
molecular nature of the $P_{cs}(4459)$ with a clear $\Xi_{c}\bar{D}^{*}$
dominant component. In relation with heavy-quark-spin symmetry our results
also favor the actual existence of two resonances with $J=1/2$ (the lighter
one) and $3/2$ (the heavier one) in the energy region of the $P_{cs}(4459)$.
In the scenario of two-resonance for the $P_{cs}(4459)$, the inclusion of the
$\Xi_{c}^{\prime}\bar{D}$ channel is required for the their mass splitting and
it allows one to determine the spin structures of the two resonances.

##  1 Introduction

The study of exotic hadrons beyond the conventional quark-antiquark
($q\bar{q}$) configurations for mesons and the three-quark ($3q$) ones for
baryons has been an interesting topic in strong interactions since long time
ago, e.g. one could mention the discussions on the lightest scalar resonances
$f_{0}(500)$, $f_{0}(980)$, $a_{0}(980)$ and $\kappa(900)$ [1, 2, 3, 4, 5],
the case of the $\Lambda(1405)$ resonances [6, 7, 8], the Roper [9], etc. The
interest on this topic has received a great boost since the discoveries in
2003 of the $D_{s0}^{*}(2317)$ [10] and the $\chi_{c1}(3872)$ [11], aka
$X(3872)$, whose properties cannot be accommodated within the conventional
quark model expectations. As a result a new era has developed in hadron
spectroscopy based on the detection of numerous multiquark states with heavy
quarks, particularly the charm quark, and more recently the bottom quark as
well.

An important outcome in this fruitful field of research was the discovery by
the LHCb in 2015 of the charmonium pentaquarks $P_{c}(4450)$ and $P_{c}(4380)$
decaying into $J/\psi p$ [12]. With higher statistics in 2019 [13] the same
collaboration in the same decay process determined the existence of the
pentaquarks $P_{c}(4457)$, $P_{c}(4440)$ (instead of just one resonance
$P_{c}(4450)$) and the lighter and narrow $P_{c}(4312)$. In Refs. [14, 15] a
through analysis of the experimental data employing coupled-channel dynamics,
heavy-quark-spin symmetry (HQSS) and one-pion exchange provided a clear hint
for the existence of a narrow $P_{c}(4380)$.

Very recently the LHCb collaboration [16] has reported the first evidence of a
charmonium pentaquark resonance with strangeness, the so called $P_{cs}(4459)$
appearing as a peak in the $J/\psi\Lambda$ event distribution from an
amplitude analysis of the $\Xi_{b}^{-}\to J/\psi\Lambda K^{-}$ decay. Ref.
[16] also concludes that this structure is consistent with two resonances.
Indeed strange pentaquark resonances were already predicted theoretically in
several references [17, 18, 19, 20, 21, 22, 23, 24], being also suggested to
look for them in the decay process $\Xi_{b}^{-}\to J/\psi\Lambda K^{-}$ by the
Refs. [25, 19].

The discovery of the $P_{cs}(4459)$ by the LHCb collaboration has prompted
numerous studies of this resonance [26, 27, 28, 29, 30, 31]. These studies
typically favor a molecular-type picture with a dominant component of the
channel $\Xi_{c}\bar{D}^{*}$, whose threshold lies only about 19 MeV above the
nominal mass of the $P_{cs}(4459)$. However, there is no such a general
consensus regarding the spin of the resonance between the two values $J=1/2$
and $3/2$ and, indeed, some studies point out towards the existence of two
resonances around 4.45 GeV in the $J/\psi\Lambda$ event distributions.

In this work we intent to scrutinize the nature of the $P_{cs}(4459)$ without
ascribing to a specific dynamical model, by making use of general arguments
based on unitarity and analyticity in $S$-matrix theory. We exploit three
general methods, recently applied in a similar way by two of us in the study
of the $X(6900)$ [32], all of which come to the conclusion of the molecular
nature of the $P_{cs}(4459)$ with a clearly dominant $\Xi_{c}\bar{D}^{*}$
component.

The first method is based on the fact that a CDD pole [33] in the near-
threshold region gives rise to a strong impact [34] in the values of the
scattering length $a$ and effective range $r$. Particularly relevant is to
focus on $r$ because a priori its absolute value is given by the typical range
of strong interactions around 1 fm [35], while the presence of a nearby CDD
pole would typically imply much larger values [34]. This signature could then
be taken as favoring the presence in the state of interest of a large bare
component, or of large components from other channels, to which the CDD pole
would be associated [36]. However, we find here that $r$ has a natural value
around $-1$ fm by applying the elastic effective-range-expansion (ERE) study
to $\Xi_{c}\bar{D}^{*}$, suggesting the $\Xi_{c}\bar{D}^{*}$ molecular nature
of this resonance. We also observe that when this method is applied to the
$\Xi_{c}^{\prime}\bar{D}$ channel (whose threshold is around 13 MeV below the
resonance mass) the resulting $r$ has instead a much larger modulus indicating
that the $P_{cs}(4459)$ is not dynamically generated by this channel.

The second method is a coupled-channel generalization that relies on the
saturation of the width $\Gamma$ and compositeness $X$ of the resonance in
terms of the $J/\psi\Lambda$ and
$\Xi_{c}\bar{D}^{*}$($\Xi_{c}^{\prime}\bar{D})$ channels. The latter concept
refers to the weight of the baryon-meson components in the $P_{cs}(4459)$. For
its calculation we follow Ref. [37] that provides a probabilistic
interpretation for the compositeness of a resonance. The optimal way to
calculate the compositeness is still under debate [38, 39, 40, 37, 41, 41]. By
following this method we have at our disposal two equations, though the
outcome depends on the value assumed for the total compositeness $X$, while
$\Gamma$ is taken from the LHCb collaboration Ref. [16]. Nonetheless, given a
value of $X$ the method provides us with the partial decay widths and the
partial compositeness coefficients for each channel.

The third method allows one to also include coupled channels and avoid having
to take for granted a value for $X$. We proceed with a direct fit to the
$J/\psi\Lambda$ event distribution around the $P_{cs}(4459)$ resonance energy
region explicitly taking into account the three coupled channels
$J/\psi\Lambda$, $\Xi_{c}^{\prime}\bar{D}$ and $\Xi_{c}\bar{D}^{*}$, together
with constraints from HQSS. The lightest channel accounts in an effective way
of those channels whose thresholds are relatively far from the resonance mass,
in particular this is the case for the $\eta_{c}\Lambda$ which is degenerate
to the $J/\psi\Lambda$ in the heavy-quark limit. Our analysis also favors the
presence of two resonances (with $J=\frac{1}{2}$ and $J=\frac{3}{2}$
respectively) in the peak of the $J/\psi\Lambda$ event distribution associated
to the $P_{cs}(4459)$. Since the $S$-wave $\Xi_{c}^{\prime}\bar{D}$ only
couples to $\Xi_{c}\bar{D}^{*}$ in $J=\frac{1}{2}$, it could affect the
dynamics of this channel and leave significant impact on the line shape. As a
result, it leads to the mass split of the two resonances and allows one to
determine their spins. We stress that the HQSS breaking of the potential,
which is of order $\mathcal{O}(\frac{\Lambda_{\text{QCD}}}{m_{c}})$, cannot
account for the mass splitting of the two resonances, cf. Sec. 4.1. In other
words, the inclusion of the $\Xi_{c}^{\prime}\bar{D}$ channel provides an
extra mechanism which is responsible for the mass splitting of the two
resonances while keeping the HQSS constraints on the interacting potential. In
addition, we have also allowed the presence of a CDD pole in the
$\Xi_{c}\bar{D}^{*}$ or $\Xi_{c}^{\prime}\bar{D}$ channel but no acceptable
fits result.

We find that the methods 1 and 3 agree in the basic fact that the
$P_{cs}(4459)$ is mainly a $\Xi_{c}\bar{D}^{*}$ molecular resonance, while the
methods 2 and 3 provide also very compatible values for the couplings of the
resonance to the different channels when the same value for $X$ is used in the
second method.

The contents of the manuscript are structured as follows. After this
introduction we describe and apply the methods 1, 2 and 3 to the study the
$P_{cs}(4459)$ in the Secs. 2, 3 and 4, respectively. In the latter section we
also compare between several types of fits with different number of channels
and partial waves involved, as well as with different degree of imposition of
the HQSS constraints. The conclusions are gathered in Sec. 5.

##  2 Effective range expansion for the elastic scattering

First we rely on the ERE method to perform an exploratory study of the
$P_{cs}(4459)$ near the $\Xi_{c}\bar{D}^{*}$ and ${\Xi^{\prime}}_{c}\bar{D}$
thresholds. The elastic $S$-wave ERE formula reads

$\displaystyle T(E)=\frac{1}{-\frac{1}{a}+\frac{1}{2}r\,k^{2}-i\,k}\,,$ (1)

where $a$ and $r$ are the scattering length and the effective range in order,
and the non-relativistic relation between the three-momentum $k$ and the
energy $E$ in the center of mass (CM) frame is

$\displaystyle k=\sqrt{2\mu_{m}(E-m_{\rm th})}\,,$ (2)

being $\mu_{m}=m_{1}m_{2}/(m_{1}+m_{2})$ the reduced mass and $m_{\rm
th}=m_{1}+m_{2}$ the threshold of the two states in the scattering. Above the
threshold, the unitarity relation

${\rm Im}\,T(E)^{-1}=-k\,,\quad(E>m_{\rm th})\,,$ (3)

is automatically satisfied by the amplitude in Eq. (1).

There are two caveats when applying the ERE formalism. First the left hand
cuts originated from the particle exchanges in the crossed channels are
neglected. This assumption is consistent with the pionless effective filed
theory description, which turns out to be able to explain the various
pentaquark candidates [42, 43]. Nevertheless, further studies on the inclusion
of the light-meson exchanges reveal possibly noticeable effects on the
assignments of the spin for the different pentaquark states [44, 45, 15, 14].
The other caveat is about the possible underlying CDD pole near the threshold.
It is rather challenging to infer the existence of the near-threshold CDD pole
a priori. In Refs. [46, 34], a practicable criteria is provided to discern the
appearance of a near-threshold CDD pole based on the large magnitude of the
effective range $r$. This is because the effective range is found to scale as

$\displaystyle r\propto\frac{1}{(M_{\text{CDD}}-m_{\rm th})^{2}}\,,$ (4)

when the CDD position in energy, $M_{\text{CDD}}$, approaches to the
threshold. To be more specific, if the resulting magnitude of $r$ in the ERE
study is large, as compared to the typical range around 1 fm of strong
interactions, it would be a clear hint for the existence of a CDD pole near
threshold, indicating as well that the ERE expansion is not suitable to handle
such systems. On the contrary, if the resulting $|r|$ has a value less than
$1\sim 2$ fm, a near-threshold CDD pole is not necessary and the ERE would not
be spoiled by its presence, and it could be used to describe such scattering
processes. 444Note that a large value of $|r|$ can also be caused by a near
coupled-channel effect. However, this effect could also be related to the
presence of a CDD pole in the elastic scattering of the open channel.

Regarding the previous discussion, by identifying the observed pentaquark
candidate $P_{cs}(4459)$ as the resonance pole in the elastic scattering
amplitude of $\Xi_{c}\bar{D}^{*}$ or ${\Xi^{\prime}}_{c}\bar{D}$, one can then
fix in every case the corresponding scattering length and effective range via
the mass and width of the $P_{cs}(4459)$. The size obtained for the effective
range would be an indication of whether the resonance can be considered as
“molecular” in terms of the channel considered or it better corresponds to a
bare state (preexisting resonance). The resonance pole lies in the second
Riemann sheet (RS) in the elastic scattering case and the amplitude
$T_{II}(E)$ in this RS has the form

$\displaystyle T_{II}(E)=\frac{1}{-\frac{1}{a}+\frac{1}{2}r\,k^{2}\,+i\,k}\,.$
(5)

In this convention, the imaginary part of $k$ is understood to be positively
taken, namely ${\rm Im}k>0$, in Eqs. (1) and (5). The solution of
$T_{II}(E_{R})^{-1}=0$ gives the resonance pole position
$E_{R}=M_{R}-i\Gamma_{R}/2$, being $M_{R}$ and $\Gamma_{R}$ the resonance mass
and width, respectively. Alternatively, one could also determine $a$ and $r$
in terms of the resonance pole position $E_{R}$ and their explicit relations
turn out to be [46]

$\displaystyle a=-\frac{2k_{i}}{|k_{r}|^{2}+|k_{i}|^{2}}\,,\quad
r=-\frac{1}{k_{i}}\,,$ (6)

where $k_{r}$ and $k_{i}$ stand for the real and imaginary parts of the three-
momentum $k_{R}$ evaluated at the pole, namely,
$k_{R}=\sqrt{2\mu_{m}(E_{R}-m_{\rm th})}$. For clarity, let us point out again
that the convention $k_{i}>0$ is taken.

The ERE scattering amplitude is completely fixed after $a$ and $r$ are known,
and with it an important quantity, i.e. the residue, which can then be
inferred by taking the Laurent expansion of $T_{II}(E(k))$ around the pole
position

$\displaystyle T_{II}(k)=\frac{-k_{i}/k_{r}}{k-k_{R}}+{\cal
O}[(k-k_{R})^{0}]\,.$ (7)

In our previous studies [37, 34], this residue $-k_{i}/k_{r}$ has been
demonstrated to be equal to the compositeness coefficient $X$ [37], which
corresponds to the probability of the two-body component in the resonance,
i.e.,

$\displaystyle X=-\frac{k_{i}}{k_{r}}\,.$ (8)

As proven in Ref. [34] when $M_{R}-m_{\rm th}>0$ it follows that the
compositeness $X=-k_{i}/k_{r}$ lies within the range $[0,1]$, a result that is
also consistent with the working condition of the probabilistic interpretation
revealed in Ref. [37]. It should be stressed that the expression of the
compositeness coefficient $X$ in Eq. (8) is only valid in the context of the
single-channel ERE when there is a resonance pole in the amplitude. General
discussions on the compositeness relations for resonances will be given in the
next section.

The mass and width of the $P_{cs}(4459)$ from the single-resonance fit of the
LHCb collaboration [16] read

$\displaystyle M_{R}=4458.8\pm
2.9^{+4.7}_{-1.1}~{}\text{MeV}\,,\qquad\Gamma_{R}=17.3\pm
6.5^{+8.0}_{-5.7}~{}\text{MeV}\,,$ (9)

where the first errors are statistics and the second ones are systematic. We
will take a conservative estimation and combine the two types of uncertainties
by adding in quadrature the statistic error and the largest of the two
systematic errors. The explicit values of the mass and width used in our
analyses are given in the second and third columns of Table 1, respectively,
together with the thresholds of ${\Xi^{\prime}}_{c}\bar{D}$ and
$\Xi_{c}\bar{D}^{*}$ in the next column.

Resonance | Mass | Width | Threshold | $a$ | $r$ | $X$
---|---|---|---|---|---|---
| (MeV) | (MeV) | (MeV) | (fm) | (fm) |
$P_{cs}$ | $4458.8\pm 5.5$ | $17.3\pm 10.3$ |  ${\Xi^{\prime}}_{c}D$ (4446.0) | $-0.63\pm 0.38$ | $-3.68\pm 2.11$ | $0.31\pm 0.19$
$P_{cs}$ | $4458.8\pm 5.5$ | $17.3\pm 10.3$ |  $\Xi_{c}D^{*}$ (4478.0) | $-1.79\pm 0.23$ | $-0.94\pm 0.13$ | $--$
Table 1: Results for the scattering length, effective range and the
compositeness coefficient from the elastic ERE studies given in the last three
columns in this order. The channel involved in the scattering is indicated in
the fourth column.

As a tentative study, let us now focus on the two elastic scattering cases to
address the $P_{cs}(4459)$, i.e. those stemming from taking separately the
${\Xi^{\prime}}_{c}\bar{D}$ and $\Xi_{c}\bar{D}^{*}$, whose thresholds lie
near the $P_{cs}(4459)$ peak observed by the LHCb collaboration [16].
Therefore, we separately consider in the ERE analysis that the $P_{cs}(4459)$
corresponds to the poles in the elastic ${\Xi^{\prime}}_{c}\bar{D}$ and
$\Xi_{c}\bar{D}^{*}$ scattering processes. The coupled-channel studies will be
postponed until the next section. The determination of the scattering length
and effective range for each case is summarized in the columns fifth and sixth
of Table 1.

In the elastic ${\Xi^{\prime}}_{c}\bar{D}$ scattering, the $P_{cs}(4459)$ pole
meets the working condition raised in Ref. [37], i.e., its mass is above the
${\Xi^{\prime}}_{c}\bar{D}$ threshold, so that the prescription in Eq. (8) is
eligible for the probabilistic interpretation. The small value of the
compositeness $X$ of the ${\Xi^{\prime}}_{c}\bar{D}$ component, shown in the
last column, indicates that other components than the
${\Xi^{\prime}}_{c}\bar{D}$, play more important roles in the constitution of
the $P_{cs}(4459)$. The small value of $X$, whose central value is $31\%$, is
also consistent with the rather large magnitude of $r$, as the latter hints
towards the appearance (within this simplistic elastic picture of
$\Xi^{\prime}_{c}\bar{D}$ scattering) of an underlying CDD pole near
threshold, see Eq. (4). For the case of the elastic $\Xi_{c}\bar{D}^{*}$
scattering, the $P_{cs}(4459)$ pole does not meet the requirement of the
working condition in Ref. [37], since the mass of the pole is below threshold.
As a result, the prescription of Eq. (8) loses the probabilistic
interpretation in the $\Xi_{c}\bar{D}^{*}$ case. Nevertheless, the magnitudes
of $r$ and $a$ all correspond to the conventional strong interactions, which
hint the natural explanation of the $P_{cs}(4459)$ as a molecular of the
$\Xi_{c}\bar{D}^{*}$ system. This conclusion is further corroborated when
discussing the coupled-channel case.

Note that the total width of the $P_{cs}(4459)$ is assumed to be saturated by
only one channel, either the $\Xi_{c}\bar{D}^{*}$ or
${\Xi^{\prime}}_{c}\bar{D}$, in the elastic ERE framework. This assumption
should be revisited since the partial decay width to the $J/\psi\Lambda$
channel is likely non-negligible, which is indeed the channel that is used to
measure the $P_{cs}(4459)$ in experiment [16]. As a result, a more realistic
study would require the information of the partial decay width to
$\Xi_{c}\bar{D}^{*}$ and ${\Xi^{\prime}}_{c}\bar{D}$, which will be
investigated in the next section within a coupled-channel analysis. Later on
various coupled-channel fits to the experimental event distributions of
$J/\psi\Lambda$ will be also considered.

##  3 Coupled-channel study of $J/\psi\Lambda$ and $\Xi_{c}\bar{D}^{*}$ by
saturating the compositeness and the decay width

There are intensive discussions in the literature to extend the Weinberg’s
compositeness relation for the bound states [47, 48] to the resonance cases
[49, 50, 38, 39, 40, 37, 51, 41]. The key obstacle in the latter situation is
that the compositeness coefficient $X$ usually becomes complex [52]. Different
proposals have been given in Refs. [38, 39, 40, 37, 41] to cure this problem.
In Ref. [37] the Weinberg’s compositeness relation for bound sates is
generalized to resonances by performing a phase transformation of the $S$
matrix, so that the compositeness coefficient $X$ corresponding to the
resonance state turns to be a positive number between 0 and 1, which can be
interpreted as the probability to find the two-particle component in a
resonance. It is demonstrated in Ref. [37] that the compositeness coefficient
$X_{j}$ of a specific channel $j$ reduces to taking the absolute value of the
product of the complex residue/coupling squared $g_{j}^{2}$ and the derivative
of the two-point Green function at the resonance pole position
$s_{R}=(M_{R}-i\Gamma_{R}/2)^{2}$ [37], i.e.

$\displaystyle X_{j}$ $\displaystyle=$
$\displaystyle|g_{j}|^{2}\left|\frac{dG_{j}(s_{R})}{ds}\right|\,,$ (10)

where $G_{j}(s)$ is the one-loop two-point Green function of the $j$th
channel. We will take the explicit formula of $G_{j}(s)$ from the dimensional
regularization method

$\displaystyle G_{j}(s)$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi^{2}}\left[a(\mu^{2})+\log\frac{m_{2}^{2}}{\mu^{2}}-x_{+}\log\frac{x_{+}-1}{x_{+}}-x_{-}\log\frac{x_{-}-1}{x_{-}}\right]\,,$
$\displaystyle x_{\pm}$ $\displaystyle=$
$\displaystyle\frac{s+m_{1}^{2}-m_{2}^{2}}{2s}\pm\frac{q_{j}(s)}{\sqrt{s}}\,,$
(11)

where the subtraction constant $a(\mu^{2})$ is introduced to replace the
divergent term. The expression given in Eq. (3) represents $G_{j}(s)$ in the
first or physical RS, and its corresponding formula in the unphysical RS reads
[4]

$\displaystyle G_{j}(s)^{\rm II}=G_{j}(s)-i\frac{q_{j}(s)}{4\pi\sqrt{s}}\,,$
(12)

where $q_{j}(s)$ denotes the relativistic CM three-momentum of the $j$th
channel

$\displaystyle
q_{j}(s)=\frac{\sqrt{[s-(m_{1}+m_{2})^{2}][s-(m_{1}-m_{2})^{2}]}}{2\sqrt{s}}\,,$
(13)

with $m_{1}$ and $m_{2}$ the masses of the two particles in that channel. The
imaginary part of $G_{j}(s)^{\rm II}$ is opposite with the one of $G_{j}(s)$
along the real $s$ axis above the threshold. The explicit $\mu$ dependences in
the first two terms in Eq. (3) cancel each other, which implies that the
$G_{j}(s)$ function is actually independent on the scale $\mu$. For the
practical purpose, we will set $\mu=770$ MeV throughout. The natural value of
the subtraction constant $a(\mu)$ can be estimated by matching the functions
$G_{j}(s)$ calculated in dimensional regularization and with a three-momentum
cut-off $q_{\text{max}}$ at threshold, as explained in Refs. [7, 53], which
leads to

$\displaystyle
a=-\frac{2}{m_{1}+m_{2}}\left[m_{1}\log\left(1+\sqrt{1+\frac{m_{1}^{2}}{q_{\rm
max}^{2}}}\right)+m_{2}\log\left(1+\sqrt{1+\frac{m_{2}^{2}}{q_{\rm
max}^{2}}}\right)\right]\simeq-2.5\,,$ (14)

by taking $q_{\rm max}=1.0$ GeV, $m_{1}=m_{\Xi_{c}}$ and $m_{2}=m_{D^{*}}$. We
will take a universal value for the subtraction constants in the two channels
(since the variation in $a$ from Eq. (14) by taking the masses of the
particles in the first channel is small) and fix it to the one given in Eq.
(14). It is worth noticing that the derivatives of the $G_{j}(s)$ and
$G_{j}(s)^{\rm II}$ functions appearing in the partial compositeness
coefficient in Eq. (10) do not depend on the subtraction constant $a(\mu)$ nor
on $\mu$.

We introduce the subscript $j=1$ and 2 to distinguish between the channels
$J/\psi\Lambda$ and $\Xi_{c}D^{*}$, respectively. The saturation of the
compositeness relation can be written as

$X=X_{1}+X_{2}=|g_{1}|^{2}\left|\frac{dG_{1}^{\rm
II}(s_{R})}{ds}\right|+|g_{2}|^{2}\left|\frac{dG_{2}(s_{R})}{ds}\right|\,,$
(15)

where $X$ in the left-hand side is the total compositeness coefficient
contributed by the $J/\psi\Lambda$ and $\Xi_{c}\bar{D}^{*}$ channels.
Regarding the saturation of the decay width of $P_{cs}(4459)$, we also assume
that it is exclusively given by these two channels. For the lighter case with
$J/\psi\Lambda$, since its threshold is clearly below the resonance mass, one
can safely use the standard two-body decay width formula [54]

$\displaystyle\Gamma_{1}=|g_{1}|^{2}\frac{q_{1}(M_{R}^{2})}{8\pi
M_{R}^{2}}\,.$ (16)

While for the heavier channel $\Xi_{c}\bar{D}^{*}$, its threshold is slightly
above the resonance mass and the standard decay width formula can not be
applied. One way to estimate its partial decay width is to introduce the
spectral distribution of the resonance taking into account its finite-width
[4, 55, 34]

$\displaystyle\Gamma_{2}=|g_{2}|^{2}\int_{m_{\rm
th,2}}^{M_{R}+n\Gamma_{R}}dw\,\frac{q_{2}(w^{2})}{16\pi^{2}\,w^{2}}\frac{\Gamma_{R}}{(M_{R}-w)^{2}+\Gamma_{R}^{2}/4}\,.$
(17)

Due to the long tail of the Lorentzian spectral distribution in the integrand
of the above equation, the upper integration limit should be taken with care
to guarantee the coverage of the resonance peak in Eq. (17). For the partial
decay width of the $P_{cs}(4459)$ into the $\Xi_{c}\bar{D}^{*}$ channel, the
distance of its threshold to the $P_{cs}(4459)$ mass is slightly larger than
the somewhat narrow width of the $P_{cs}(4459)$. Therefore, somewhat large
values of $n$ in the upper integration limit of Eq. (17) turn out to be
required. For example, it is found that when taking $n=8$, as addressed in
detail in the next section, the decay width formulas for the near-threshold
channels $\Xi_{c}\bar{D}^{*}$ and ${\Xi^{\prime}}_{c}\bar{D}$, calculated
using Eq. (17), together with the standard formula of Eq. (16) for
$J/\psi\Lambda$, lead to consistent results from the pole determinations of
both the total width and the couplings fixed from the residues of the pole
position. Consequently, we will take this value of $n=8$ in the following
analyses. The width saturation is then given by

$\displaystyle\Gamma=\Gamma_{1}+\Gamma_{2}=|g_{1}|^{2}\frac{q_{1}(M_{R}^{2})}{8\pi
M_{R}^{2}}+|g_{2}|^{2}\int_{m_{\rm
th,2}}^{M_{R}+n\Gamma_{R}}dw\,\frac{q_{2}(w^{2})}{16\pi^{2}\,w^{2}}\frac{\Gamma_{R}}{(M_{R}-w)^{2}+\Gamma_{R}^{2}/4}\,.$
(18)

Let us stress that we have also explicitly tried to set $n$ at larger values,
say up to 50, and the results are found to be quantitatively similar to $n=8$.
Therefore we conclude that the values presented in Tables 2 and 3 are robust
under the mild changes of $n$.

The equations that represent the saturation of the compositeness coefficient
$X$ (15) and the decay width $\Gamma$ (18) can be used to determine the
couplings $g_{1}$ and $g_{2}$, which in turn allow us to further analyze the
partial compositeness factors and the partial decay widths. This approach
based on saturating given values of $X$ and $\Gamma$ is not suitable to
distinguish between the different total angular momenta $J$. The point is that
the coupling strengths $|g_{j}|^{2}$ would appear in the same way for any
total angular momentum $J$. Possible contributions from lighter channels, such
as the $\eta_{c}\Lambda$ channel, which threshold is rather distant from the
interested energy region around 4.46 GeV, is effectively reabsorbed in the
$J/\psi\Lambda$ channel.

To solve Eqs. (15) and (18), one must first provide the value of the total
compositeness $X$, which is however typically unknown a priori. Here we carry
out exploring studies by giving some tentative values to $X$, and the results
are summarized in Table 2 for the situation in which the $J/\psi\Lambda$ and
$\Xi_{c}\bar{D}^{*}$ coupled channels are included, and in Table 3 for the
$J/\psi\Lambda$ and ${\Xi^{\prime}}_{c}\bar{D}$ case.

Three different values of $X$, namely $X=0.1,0.5$ and 1.0, are used to analyze
the $J/\psi\Lambda$ and $\Xi_{c}\bar{D}^{*}$ system. It turns out that the
coupling strengths of $|g_{1}|$ and $|g_{2}|$ tends to decrease and increase,
respectively, when increasing the values of $X$, so do the partial decay
widths of $\Gamma_{1}$ and $\Gamma_{2}$. This trend is typical in this type of
studies as previously observed [56, 32]. In all the cases, the partial
compositeness factor $X_{2}$ of the $\Xi_{c}\bar{D}^{*}$ is always much larger
than the compositeness coefficient $X_{1}$ contributed by the $J/\psi\Lambda$
channel. Even when taking $X=1.0$, the magnitudes of the two partial widths of
the $J/\psi\Lambda$ and $\Xi_{c}\bar{D}^{*}$ channels are similar because
$|g_{2}|\gg|g_{1}|$. While, when taking smaller values for the total
compositeness coefficient $X$, the widths of the $P_{cs}(4459)$ are found to
be dominated by the $J/\psi\Lambda$ channel.

For the $J/\psi\Lambda$ and ${\Xi^{\prime}}_{c}\bar{D}$ coupled-channel system
the maximum value of $X$ that results by solving Eqs. (15) and (18) turns out
to be around 0.3. According to the results shown in Table 3, the partial
compositeness factor $X_{2}$ is always much larger than the value of $X_{1}$,
regardless of the total compositeness coefficient $X$ taken in the study. In
contrast, the magnitudes of the two partial widths are rather sensitive to the
values of $X$. Larger values of $X$ lead to bigger partial widths $\Gamma_{2}$
of the ${\Xi^{\prime}}_{c}\bar{D}$ channel, and smaller $\Gamma_{1}$ for
$J/\psi\Lambda$.

It is clear that definite values of $X$ will be obviously helpful to give more
concrete conclusions of the properties of $P_{cs}(4459)$. In the following
section, we perform the fits within dynamical unitarized models to the
experimental event distributions of the $J/\psi\Lambda$ from the LHCb
collaboration [16]. In this way, we are able to obtain more precise values of
the coupling strengths $g_{i}$, which in turn allow us to perform predictions
for the partial decay widths and compositeness coefficients contributed from
different channels.

| $|g_{1}|$ | $|g_{2}|$ | $\Gamma_{1}$ | $\Gamma_{2}$ | $X_{1}$ | $X_{2}$
---|---|---|---|---|---|---
| (GeV) | (GeV) | (MeV) | (MeV) |  |
$X=0.1$ | $3.5_{-0.8}^{+1.0}$ | $4.3_{-0.4}^{+0.2}$ | $16.4_{-6.7}^{+9.5}$ | $0.9_{-0.5}^{+0.5}$ | $0.02_{-0.01}^{+0.01}$ | $0.08_{-0.01}^{+0.01}$
$X=0.5$ | $3.1_{-0.7}^{+0.7}$ | $10.4_{-0.8}^{+0.6}$ | $12.3_{-4.9}^{+5.9}$ | $5.0_{-2.9}^{+4.7}$ | $0.01_{-0.01}^{+0.01}$ | $0.49_{-0.01}^{+0.01}$
$X=1.0$ | $2.3_{-0.4}^{+0.4}$ | $14.8_{-1.0}^{+1.0}$ | $7.1_{-2.3}^{+1.7}$ | $10.2_{-5.5}^{+9.5}$ | $0.0_{-0.0}^{+0.0}$ | $1.0_{-0.0}^{+0.0}$
Table 2: Results from the $J/\psi\Lambda$ and $\Xi_{c}\bar{D}^{*}$ coupled-channel studies by assuming the saturation of the compositeness and decay width of the $P_{cs}(4459)$.  | $|g_{1}|$ | $|g_{2}|$ | $\Gamma_{1}$ | $\Gamma_{2}$ | $X_{1}$ | $X_{2}$
---|---|---|---|---|---|---
| (GeV) | (GeV) | (MeV) | (MeV) |  |
$X=0.1$ | $3.2_{-1.0}^{+1.2}$ | $3.8_{-0.4}^{+0.2}$ | $13.0_{-6.4}^{+12.2}$ | $4.3_{-1.4}^{+1.8}$ | $0.01_{-0.00}^{+0.02}$ | $0.09_{-0.02}^{+0.00}$
$X=0.3$ | $1.4_{-0.0}^{+2.0}$ | $7.0_{-0.8}^{+0.4}$ | $2.5_{-0.0}^{+12.2}$ | $14.8_{-6.1}^{+4.3}$ | $0.00_{-0.00}^{+0.02}$ | $0.30_{-0.02}^{+0.00}$
Table 3: Results from the $J/\psi\Lambda$ and ${\Xi^{\prime}}_{c}\bar{D}$
coupled-channel studies by assuming the saturation of the compositeness and
decay width of the $P_{cs}(4459)$. Notice that the maximum value of $X$
admitting solutions of the two equations (15) and (18) is 0.3.

##  4 Fits to the experimental event distributions

In the following, we perform fits to the $J/\psi\Lambda$ invariant-mass
distributions from the LHCb [16] data on the decay $\Xi^{-}_{b}\to
J/\psi\Lambda K^{-}$, in order to obtain the couplings and the resonance pole
position. In this way one can further constrain the coupling strengths which
will in turn give more definite predictions for $X$.

We consider first the $\Xi_{c}\bar{D}^{*}$ channel, and later also include the
$\Xi^{\prime}_{c}\bar{D}$ one, as these are the two-body channels whose
thresholds, lying around 4478 and 4446 MeV, respectively, are very close to
the resonance region of the $P_{cs}(4459)$. Because of this fact we study
their interaction in $S$ wave only. In addition, we also include the much
lighter channel $J/\psi\Lambda$, whose event distribution will be fitted. In
principle one could also include the $\eta_{c}\Lambda$ since it is related to
$J/\psi\Lambda$ by the HQSS, in which limit the mass of the $J/\psi$ and
$\eta_{c}$ are equal. As we are interested only in the energy region around
the $P_{cs}(4459)$ resonance it is then legitimate to characterize both in
terms of just one effective channel, namely the $J/\psi\Lambda$ (more detailed
discussions on this point are given below).

It is illustrative to consider the decomposition of the two-body states in the
basis of HQSS eigenstates $|s_{H}\otimes j_{\ell}\rangle$, where $s_{H}$ and
$j_{\ell}$ are the total spin and angular momentum of the heavy and light
quarks, respectively. The $|\Xi_{c}\rangle$ and $|\bar{D}^{*}\rangle$ spin
multiplets are $|\frac{1}{2}\otimes 0\rangle_{1/2}$ and
$|\frac{1}{2}\otimes\frac{1}{2}\rangle_{1}$ in this basis, where the subscript
indicates the spin $j_{i}$ of each state. The latter arises from the angular
momentum combination of $s_{H_{i}}\otimes j_{\ell_{i}}$, with the index $i=1$
referring to $\Xi_{c}$ and $i=2$ to $\bar{D}^{*}$. One can then combine
$j_{1}\otimes j_{2}$ in states of definite total angular momentum $J$,
$|s_{H_{1}}j_{\ell_{1}}j_{1};s_{H_{2}}j_{\ell_{2}}j_{2};J\rangle$, which we
simply designate in the following by $|\Xi_{c}\bar{D}^{*}\rangle_{J}$. Another
way is to combine first the heavy spins $s_{H_{1}}\otimes s_{H_{2}}$ and the
light angular momenta $j_{\ell_{1}}\otimes j_{\ell_{2}}$ separately into
states with well defined total heavy spin $s_{H}$ and total light angular
momentum $j_{L}$, respectively. Then, they are composed to give states of the
form $|s_{H_{1}}s_{H_{2}}s_{H};j_{\ell_{1}}j_{\ell_{2}}j_{L};J\rangle$, that
we simply designate in the following by $|s_{H}\otimes j_{L}\rangle_{J}$, with
definite total angular momentum $J$. The relation between the two sets of
basis vectors with total angular momentum well defined is through the Wigner
$9j$ symbols. See e.g. Appendix A of Ref. [57] for further details in this
formalism. In this way, it follows that

$\displaystyle|\Xi_{c}\bar{D}^{*}\rangle_{\frac{1}{2}}$
$\displaystyle=\frac{\sqrt{3}}{2}|0\otimes\frac{1}{2}\rangle_{\frac{1}{2}}-\frac{1}{2}|1\otimes\frac{1}{2}\rangle_{\frac{1}{2}}~{},$
(19) $\displaystyle|\Xi_{c}\bar{D}^{*}\rangle_{\frac{3}{2}}$
$\displaystyle=|1\otimes\frac{1}{2}\rangle_{\frac{3}{2}}~{}.$

The states $|J/\psi\rangle$, $|\eta_{c}\rangle$ and $|\Lambda\rangle$ are
$|1\otimes 0\rangle$, $|0\otimes 0\rangle$ and $|0\otimes\frac{1}{2}\rangle$,
in this order. Therefore,
$|\eta_{c}\Lambda\rangle=|0\otimes\frac{1}{2}\rangle_{\frac{1}{2}}$, while
$|J/\psi\Lambda\rangle_{J}=|1\otimes\frac{1}{2}\rangle_{J}$ has both $J=1/2$
and $3/2$. However, the transformation matrix for the
$|\Xi_{c}\bar{D}^{*}\rangle_{3/2}$ states is just the identity, as it is clear
from Eq. (19).

Here we only consider $S$-wave scattering because: i) The
$|\Xi_{c}\bar{D}^{*}\rangle$ and $|\Xi_{c}^{\prime}\bar{D}\rangle$ channels
are very close to threshold in the region of interest. ii) The
$|J/\psi\Lambda\rangle$ ($|\eta_{c}\Lambda\rangle$) does not couple to
$|1\otimes\frac{1}{2}\rangle$($|0\otimes\frac{1}{2}\rangle$) in $D$ and higher
partial waves in the heavy-quark limit.

The coupled-channel scattering amplitudes take the form [58, 59]

$\displaystyle\mathcal{T}_{J}(s)=[\mathbb{I}-\mathcal{V}_{J}\cdot
G(s)]^{-1}\cdot\mathcal{V}_{J}(s)\,.$ (20)

The Eq. (20) satisfies unitarity [58, 7] as the right-hand cut is generated by
the diagonal matrix $G(s)$, whose elements are $G_{j}(s)\equiv
G(s,m_{j1}^{2},m_{j2}^{2})$, as defined in Eq. (3).

In all cases analyzed below, the matrix elements of $\mathcal{V}_{J}$
involving the $J/\psi$ or $\eta_{c}$ are taken constant because the energy
region studied is narrow as compared with the energy distance to the
thresholds of these channels. For the transitions involving only the
$|\Xi_{c}\bar{D}^{*}\rangle$ or $|\Xi_{c}^{\prime}\bar{D}\rangle$ this will be
also typically the case, except when we consider for comparison a possible
linear dependence in $s$. Taking into account the conservation of the heavy-
quark spin $s_{H}$ in the heavy-quark limit, and that $j_{\ell}=1/2$ always,
we can write $V_{J}$ for $J=1/2$ and $3/2$ when explicitly including the
$J/\psi\Lambda,\eta_{c}\Lambda$ and $\Xi_{c}\bar{D}^{*}$ channels as

$\displaystyle\begin{array}[]{ll}V_{\frac{1}{2}}=\left(\begin{array}[]{lll}0&0&-\frac{1}{2}g\\\
0&0&\frac{\sqrt{3}}{2}g\\\
-\frac{1}{2}g&\frac{\sqrt{3}}{2}g&C_{\frac{1}{2}}\end{array}\right)\,,{}&V_{\frac{3}{2}}=\left(\begin{array}[]{lll}0&0&g\\\
0&0&0\\\ g&0&C_{\frac{3}{2}}\end{array}\right)~{}.\end{array}$ (28)

Though HQSS requires that $C_{\frac{1}{2}}=C_{\frac{3}{2}}$ we treat them
separately because if significantly different values emerge when fitting data
this could probably indicate that the model is incomplete. We have set to zero
the direct transitions between the lighter channels $|J/\psi\Lambda\rangle$
and $|\eta_{c}\Lambda\rangle$ because they are Okubo-Zweig-Izuka suppressed.
Furthermore, a recent lattice QCD study of the related $|J/\psi N\rangle$ and
$|\eta_{c}N\rangle$ scattering [60] found that these interactions are very
weak. We also mention that the coupled-channel model estimate of Ref. [61]
gives a $J/\psi p$ scattering length of the order of $10^{-3}$ fm.

The typical formalism [59] for implementing final-state interactions (FSI) and
obtaining the needed amplitudes for the calculation of the actual event
distribution of $J/\psi\Lambda$, that we call $F_{J}$, can be constructed by
making use of the matrix $D_{J}(s)$,

$\displaystyle D_{J}(s)$ $\displaystyle=\mathbb{I}-\mathcal{V}_{J}\cdot
G(s)~{}.$ (29)

The matrix $D_{J}(s)$ appears in Eq. (20) because its inverse resums the re-
scattering effects, there starting from a basic interaction $V_{J}(s)$. But
now the rescattering effects stem from the elementary production vertices
$P_{J}$’s,

$\displaystyle P_{J}=\left(\begin{array}[]{l}0\\\ 0\\\
d_{J}\end{array}\right)\,,$ (33)

such that

$\displaystyle F_{J}(s)$ $\displaystyle=D_{J}(s)^{-1}P_{J}\,.$ (34)

To be consistent with the assumptions of Eq. (28), the direct transition
vertices with the $J/\psi\Lambda$ and $\eta_{c}\Lambda$ are also set to zero
in Eq. (33).

However, the direct application of Eq. (34) for calculating $F_{J}(s)$ with a
constant $P_{J}$ is not adequate because the production vertex $F_{J}(s)$ is
proportional to $G(s,M_{\Xi_{c}}^{2},m_{D^{*}}^{2})$, as it follows from Eqs.
(29)–(34). But, this loop function has a zero at an energy $E_{z}\simeq
4444~{}$MeV (below the $\Xi_{c}\bar{D}^{*}$ channel), just at the region of
interest, so that $F_{J}(s)$ would also vanish at the same energy (something
not seen in the data) unless $P_{J}(s)$ has a pole there to cancel the zero.

The simplest way to remedy this situation is to realize that the resonance
signal is actually contained in the determinant of $D_{J}(s)$, that we call
$\Delta_{J}(s)$, since a resonance stems from a zero in it. The function
$\Delta_{J}(s)$ also includes, of course, the cusp effects associated to the
relevant nearby $\Xi_{c}\bar{D}^{*}$ threshold (and to the
$\Xi_{c}^{\prime}\bar{D}$ one when later included).555Another way would be to
replace $d_{J}$ in $P_{J}$, Eq. (33), by $d_{J}/(s-E_{z}^{2})$. Nonetheless,
we prefer to keep a constant $P_{J}(s)$, as in the original Eq. (33). As a
result, instead of Eq. (34), the final expression that we use for the
calculation of $F_{J}(s)$ is

$\displaystyle F_{J}(s)$ $\displaystyle=\frac{d_{J}}{\Delta_{J}(s)}~{}.$ (35)

Taking as in the HQSS limit that
$G(s,M_{J/\psi}^{2},M_{\Lambda}^{2})=G(s,M_{\eta_{c}}^{2},M_{\Lambda}^{2})$,
which is actually a good approximation because the region under study is much
higher than the $J/\psi\Lambda$ and $\eta_{c}\Lambda$ thresholds, we have that
both $\Delta_{1/2}$ and $\Delta_{3/2}$ become equal to

$\displaystyle\Delta_{J}=1-G_{3}(C_{J}+G_{1}g^{2})~{}.$ (36)

Because of these reasons in the following we do not take explicitly into
consideration the $\eta_{c}\Lambda$ channel and only keep the $J/\psi\Lambda$
one. As a result, the number of coupled channels is reduced to two.

The invariant-mass distribution of the $J/\psi\Lambda$, $dN(s)/d\sqrt{s}$, is
calculated from $F_{J}(s)$ by implementing the three-body phase space as

$\displaystyle\frac{dN(s)}{d\sqrt{s}}$
$\displaystyle=\frac{1}{128\pi^{3}M^{3}_{\Xi_{b}}}\frac{\sqrt{\lambda(M^{2}_{\Xi_{b}},s,M_{K}^{2})\lambda(s,M^{2}_{J/\psi},M_{\Lambda}^{2})}}{\sqrt{s}}\sum_{J}|F_{J}|^{2}~{}.$
(37)

In addition, we take into account the width of the experimental bin by
convoluting the signal with a function to consider the effects of the spread
in energy, typically with a Gaussian whose $\sigma$ is 2.6 MeV (around half of
the bin size).

For every $T$ matrix obtained in our studies we calculate the resonance pole
positions and their residues. The resonance poles lie in the complex energy
plane of an unphysical RS, which can be accessed via the analytical
extrapolation of the $G_{j}(s)$ functions. The expression given in Eq. (3)
represents $G_{j}(s)$ in the first or physical RS, and the corresponding
formula on its unphysical RS is given in Eq. (12) with the square root
calculated in its first RS (so that the phase of its argument is taken between
0 and $2\pi$ in the complex $s$ plane). The imaginary part of
$G_{j}(s+i\epsilon)^{\rm II}$ is opposite to the one of $G_{j}(s+i\epsilon)$
for $s$ taking physical values (real ones above threshold), with $\epsilon\to
0^{+}$. Different unphysical RSs of the coupled-channel scattering amplitudes
in Eq. (20) can be accessed by properly taking $G_{j}(s)$ or $G_{j}(s)^{\rm
II}$ for different channels. The second RS can be labeled as $(-,+)$, where
the plus(minus) sign in the $j$th entry indicates taking
$G_{j}(s)$($G_{j}(s)^{\rm II})$ in the $j$th channel. In this convention, the
first, third, and fourth Riemann sheets are labeled as $(+,+)$, $(-,-)$, and
$(+,-)$, respectively. The most relevant resonance poles are found to lie in
the second RS, which connects continuously with the physical RS below the
$\Xi_{c}\bar{D}^{\ast}$ threshold.

The matrix elements of the scattering matrix in the unphysical RS around the
resonance pole position can be written as

$\displaystyle\left.\mathcal{T}_{J}\right|_{kj}(s)=-\frac{g_{k}g_{j}}{s-s_{\rm
pole}}+\cdots\,,$ (38)

where $g_{k,j=1,2}$ are the effective couplings of the resonance to the
corresponding channels. Here, we follow the usual procedure, perfectly
justifiable for narrow resonances, of identifying $s_{\rm
pole}=(M_{R}-i\Gamma_{R}/2)^{2}$. The omitted terms in Eq. (38) are the
regular parts in the Laurent expansion around $s_{\rm pole}$.

###  4.1 Fits with the $J/\psi\Lambda$ and $\Xi_{c}\bar{D}^{*}$ channels

First we present fits for $dN(s)/d\sqrt{s}$ including only the $J/\psi\Lambda$
and $\Xi_{c}\bar{D}^{*}$ channels in which we add progressively $J=1/2$ and
then $3/2$ (both required by HQSS, cf. Eq. (19)). We also separate the
discussion between the cases in which only a constant potential is used and
when we also consider a linear $s$ dependence.

Firstly, the potential is taken as the constant matrix

$\displaystyle V_{\frac{1}{2}}$ $\displaystyle=\left(\begin{array}[]{ll}0&g\\\
g&C_{\frac{1}{2}}\end{array}\right)~{},$ (41)

and we use Eq. (35) with $d_{\frac{1}{2}}$ for $J=1/2$. Here we follow two
strategies. One of them consists of fixing the only two free parameters in
$V_{\frac{1}{2}}$ by reproducing the experimental $P_{cs}(4459)$ resonance
pole in $1/\Delta_{1/2}$ in the second RS. The other strategy is to fit
directly the data and obtain $g$, $C_{1/2}$ and $d_{1/2}$.

The equation that results by imposing $\Delta_{1/2}(s_{R})=0$ is directly
obtained from Eq. (36) by taking $G_{1}(s)$ evaluated in its second RS,
$G_{1}^{\rm II}(s)$, with $s_{R}=(M_{R}-i\Gamma_{R}/2)^{2}$ determined from
the experimental central values, $M_{R}=4458.8$ MeV and $\Gamma_{R}=17.3$ MeV
[16]. The resulting event distribution is shown by the dot-dashed line in the
top left panel of Fig. 1, and we see that it provides a qualitative agreement
with the experimental data.

Now, we proceed with the direct fits to data by minimizing a standard
$\chi^{2}$. We show the fit results in Fig. 1 and provide the fitted values of
the parameters in Table 4. Every type of fit is described in the left-most
column of the table by $(abc)$ where: $a$ refers to the number of channels, 2
($J/\psi\Lambda$-$\Xi_{c}\bar{D}^{*}$ or
$J\psi\Lambda$-$\Xi^{\prime}_{c}\bar{D}$) or 3
($J/\psi\Lambda$-$\Xi^{\prime}_{c}\bar{D}$-$\Xi_{c}\bar{D}^{*}$ described in
Sec. 4.3); $b$ to the number of partial waves, 1 ($J=1/2$ or $3/2$) or 2
($J=1/2$ and $3/2$); and $c$ indicates whether we use a constant potential (0)
or include a linear dependence in $s$ (1) for the near-threshold coupled-
channel potential. We use a cross when an entry is not meaningful to be given
(typically in those instances with only 2 channels involved). For each fit we
also give the position of the resonance pole and the residues of the $T$
matrix, collected in Table 5, following the same conventions as just
explained. The energy region actually fitted is shown by the shaded area in
Fig. 1, comprising eleven experimental points around the $P_{cs}(4459)$ peak.

In the first panel of Fig. 1 the fit $(210)$ corresponds to the solid line,
which provides a good overall fit to data given the experimental
uncertainties. Nonetheless, we also notice that it lies somewhat low compared
to data in the region to the left of the peak. The values of the fit
parameters are given in Table 4.

We have also tested the case of allowing a linear $s$ dependence in the
$\Xi_{c}\bar{D}^{*}$ interacting channels, so that $C_{\frac{1}{2}}\to
C_{\frac{1}{2}}(\frac{s}{M_{\rm CDD}^{2}}-1)$ in Eq. (41). The position of the
zero is called $M_{\rm CDD}^{2}$, because in the case of the
$\Xi_{c}\bar{D}^{*}$ one-channel scattering it would be a CDD pole [33, 58].
However, the solution found with the smallest $\chi^{2}$ always gives rise to
a pole in the first or physical RS, which is not admissible. This is related
to the fact that these fits of smallest $\chi^{2}$ produce
$C_{\frac{1}{2}}<0$, which allows the vanishing of the denominator of the one-
channel $T$ matrix,

$\displaystyle\frac{1}{C_{1/2}(s/M_{\rm CDD}^{2}-1)}+G_{2}(s)~{}.$ (42)

This can be easily seen because $\Im G_{2}(s)<0$ for $\Im s>0$ in the first
RS, and then it has opposite sign to $s^{*}/C_{\frac{1}{2}}$ for
$C_{\frac{1}{2}}<0$. The absence of an acceptable solution with a nearby CDD
pole is in favor of interpreting the $P_{cs}(4459)$ as a dynamically generated
resonance, i.e., a $\Xi_{c}\bar{D}^{*}$ molecular state.

Next, we include together the two angular momenta $J=1/2$ and $3/2$, and then
we have two more free parameters $C_{\frac{3}{2}}$ and $d_{\frac{3}{2}}$,

$\displaystyle V_{\frac{3}{2}}=\left(\begin{array}[]{ll}0&g\\\
g&C_{\frac{3}{2}}\end{array}\right)~{},~{}P_{\frac{3}{2}}$
$\displaystyle=\left(\begin{array}[]{l}0\\\
d_{\frac{3}{2}}\end{array}\right)~{}.$ (47)

According to Eq. (37) we have now
$|F_{\frac{1}{2}}(s)|^{2}+|F_{\frac{3}{2}}(s)|^{2}$. Notice, that both
$F_{\frac{1}{2}}$ and $F_{\frac{3}{2}}$ obey the same type of expression, so
that they are interchangeable. This is important when listing the poles found
for the $(220)$ fit because it is then not really possible to discern which is
actually the angular momentum of each of them. The results of the fit are
given by the entry $(220)$ in Tables 4 and 5. The resulting $\chi^{2}=2.95$ is
much lower than for the fit of type $(210)$, and this is clearly observed by
the curve obtained for $(220)$ in the second panel of Fig. 1, which runs much
closer to the central values of data and there are now two peaks corresponding
to two poles, listed in Table 5. However, there is the caveat that
$C_{\frac{1}{2}}$ and $C_{\frac{3}{2}}$ are very different and are not
compatible. The large deviation between the $C_{\frac{1}{2}}$ and
$C_{\frac{3}{2}}$ can not be simply accounted for by the HQSS breaking which
implies their difference should be of order of
$\mathcal{O}(\frac{\Lambda_{\text{QCD}}}{m_{c}})$. This issue is a strong
reason for exploring three-channel scattering including the $J/\psi\Lambda$,
$\Xi_{c}\bar{D}^{*}$ and $\Xi_{c}^{\prime}\bar{D}$ states, as we do in the
Sec. 4.3.

###  4.2 Fits including only the $J/\psi\Lambda$ and $\Xi^{\prime}_{c}\bar{D}$
channels

The decomposition of the $|\Xi_{c}^{\prime}\bar{D}\rangle$ in the HQSS basis
is

$\displaystyle|\Xi_{c}^{\prime}\bar{D}\rangle$
$\displaystyle=-\frac{1}{2}|0\otimes\frac{1}{2}\rangle_{\frac{1}{2}}^{\prime}+\frac{\sqrt{3}}{2}|1\otimes\frac{1}{2}\rangle_{\frac{1}{2}}^{\prime}~{},$
(48)

and we can only have the $J=1/2$ contribution in this scenario. When referring
to this case we include primes to indicate that $\Xi_{c}^{\prime}\bar{D}$ is
considered instead of $\Xi_{c}\bar{D}^{*}$.

We have the same number of free parameters as when including the
$J/\psi\Lambda$ and $\Xi_{c}\bar{D}^{*}$ states. However, when the
$J/\psi\Lambda$ and $\Xi_{c}^{\prime}\bar{D}$ are the ones considered instead
the resulting fits with either constant potentials or allowing a linear $s$
dependence in the $\Xi^{\prime}_{c}\bar{D}$ direct transition potential are
not able to reproduce the data even qualitatively. This is clear from the
resulting curve shown in the third panel of Fig. 1, where the curve labeled by
“Cons” and “CDD” correspond to the constant potentials and allowing a linear
$s$ dependence, respectively. Therefore, in light of the results here and in
Sec. 4.1 one can clearly conclude that the $P_{cs}(4459)$ is dominated by the
$\Xi_{c}\bar{D}^{*}$ interactions, and much less by the
$\Xi_{c}^{\prime}\bar{D}$ channel. This is also in agreement with the results
in Secs. 2 and 3.

###  4.3 Fits including the three channels $J/\psi\Lambda$,
$\Xi_{c}^{\prime}\bar{D}$ and $\Xi_{c}\bar{D}^{*}$

All the three channels $J/\psi\Lambda$ (1), $\Xi_{c}^{\prime}\bar{D}$(2), and
$\Xi_{c}\bar{D}^{*}$(3), with the channel numbers given between brackets,
interact among them only in $J=1/2$. The interaction potential
$V_{\frac{1}{2}}$ is taken as

$\displaystyle V_{\frac{1}{2}}$
$\displaystyle=\left(\begin{array}[]{lll}0&g^{\prime}&g\\\
g^{\prime}&C^{\prime}_{\frac{1}{2}}&C_{\rm mx}\\\ g&C_{\rm
mx}&C_{\frac{1}{2}}\end{array}\right)~{},$ (52)

where all the matrix entries are constants. We also have the elementary
production vertices
$P_{\frac{1}{2}}=(0\,,d_{\frac{1}{2}}^{\prime}\,,d_{\frac{1}{2}})$. The
$J=3/2$ potential is left formally unchanged, cf. Eq. (47), but now having
included the three channels we impose the HQSS constraint
$C_{\frac{1}{2}}=C_{\frac{3}{2}}$. Once the $\Xi^{\prime}\bar{D}$ is
explicitly included, it could provide a mechanism for the mass split of the
two $P_{cs}$ resonant states. That is, we assume that the main source of HQSS
breaking in the relation $C_{\frac{1}{2}}\neq C_{\frac{3}{2}}$ in Sec. 4.1,
fit $(220)$, is due to the neglected nearby $\Xi^{\prime}_{c}\bar{D}$ channel.
All in all, we have 8 free parameters, $g$, $g^{\prime}$, $C_{\frac{1}{2}}$,
$C_{\frac{1}{2}}^{\prime}$, $C_{\rm mx}$, $d_{\frac{1}{2}}$,
$d_{\frac{1}{2}}^{\prime}$ and $d_{\frac{3}{2}}$.

We explore first a perturbative inclusion of the $\Xi^{\prime}_{c}\bar{D}$
channel so that we take
$g^{\prime}=C^{\prime}_{\frac{1}{2}}=d^{\prime}_{\frac{1}{2}}=0$, and then its
inclusion is similar to that for the $J/\psi\Lambda$ channel, though
$G_{2}(s)$ is peaked in the $\Xi^{\prime}_{c}\bar{D}$ threshold. We then
obtain the fit $(320)$. Even though we now impose that
$C_{\frac{1}{2}}=C_{\frac{3}{2}}$, the $\chi^{2}=3.06$ is almost equal to that
for the fit $(220)$, and the fit (320) reproduces very well the data as shown
in the last panel of Fig. 1. The values for the $\chi^{2}$ and the fit
parameters are given in Tables 4 and 6. In addition we also notice that
$C_{\frac{1}{2}}=C_{\frac{3}{2}}$ is almost identical to the values for
$C_{\frac{1}{2}}$ registered for the fits $(210)$ and $(220)$, which then
indicates that this parameter is well determined within our approach. Related
to this, we conclude then that our hypothesis that the lack of the channel
$\Xi^{\prime}_{c}\bar{D}$ is the reason for the HQSS breakdown in the relation
$C_{\frac{1}{2}}=C_{\frac{3}{2}}$, as obtained in the fit (220), is well
confronted to phenomenology.

The nomenclature for the RSs with three channels is arranged in increasing
values for the thresholds. Namely, the RSII $(-++)$ is the one connecting with
the RSI $(+++)$ above the $J/\psi\Lambda$ and below the
$\Xi_{c}^{\prime}\bar{D}$ thresholds, the RSIII $(--+)$ connects with the
physical sheet between the $\Xi_{c}^{\prime}\bar{D}$ and the
$\Xi_{c}\bar{D}^{*}$ thresholds, and the RSIV connects with RSI above the
$\Xi_{c}\bar{D}^{*}$ threshold. In this case the scattering in the $J=1/2$ and
$J=3/2$ angular momenta is not interchangeable and we can give the
corresponding spin of the resonances found in Table 7.

The two poles observed for the fit $(320)$, collected in Table 7, one with
$J=1/2$ (first peak in the last panel of Fig. 1) and another with $J=3/2$ (the
higher one in the same plot), have each of them an imaginary part that is
smaller in absolute value than the one in the first two poles of Table 5. This
is indeed related with the size of the coupling $g$ which is also smaller for
the $(320)$ fits as compared with the $(220)$. Since this coupling controls
how strongly the relatively-light channel $J/\psi\Lambda$ couples to
$\Xi_{c}\bar{D}^{*}$, and the resonances are strongly affected by the latter
channel, the smaller the value of $g$ the smaller the resulting widths because
the residue $g_{1}$ is proportional to it.

We have also explored new fits in which $g^{\prime}$,
$C^{\prime}_{\frac{1}{2}}$ and/or $d^{\prime}_{\frac{1}{2}}$ are allowed to
float. However, we do not observe any significant change, either qualitative
or quantitative, in the description of the data. For instance, the resulting
$\chi^{2}$ in every case barely improves compared to the fit $(320)$. Of
course, the resulting fits are typically over-determined so that some of the
parameters are completely undetermined (having errorbars that are much bigger
than their central values).

Let us discuss two of such fits. If we still fix $C^{\prime}_{\frac{1}{2}}=0$
but release $g^{\prime}$ we have the fit $(320)_{1}$ in Tables 4 and 6. We
observe only a slight improvement in the $\chi^{2}$,
$g^{\prime}=59.6^{+160.7}_{-249.2}$ is compatible with zero and basically
undetermined, while the rest of fit parameters are very similar to the
reference fit with three channels $(320)$. However, if we fix $g^{\prime}=0$
and release $C^{\prime}_{\frac{1}{2}}$ the resulting fit $(320)_{2}$ in Tables
4 and 6, gives rise to a very uncertain coupling $g$. Though, it is still
compatible within errors with the determinations from the other three-channel
fits, its central value is very different, while for the other fits $(320)$
and $(320)_{1}$ their central values are quite similar. Notice also that again
the $\chi^{2}=2.81$ has improved only by a basically negligible amount
compared to $(320)$. The pole content for these fits is not shown because it
is very similar to the one already given in Table 7.

In summary, our reference fit for the three-channel case is the $(320)$ one,
in which the $\Xi^{\prime}_{c}\bar{D}$ channel is introduced perturbatively
($g^{\prime}=C^{\prime}_{\frac{1}{2}}=d^{\prime}_{\frac{1}{2}}=0$) and we also
take into account both $J=1/2$ and $J=3/2$. Releasing either $g^{\prime}$ or
$C^{\prime}_{\frac{1}{2}}$ does not really provide better fits and the values
for the fit parameters are compatible with those from $(320)$. This indicates
a perturbative role of the $\Xi_{c}^{\prime}\bar{D}$ channel, which is also in
agreement with our previous results in Sec. 4.2 when considering only the
$J/\psi\Lambda$ and $\Xi^{\prime}_{c}\bar{D}$ channels in $J=1/2$. Let us
recall that  in that case it was not possible to account for the
$P_{cs}(4459)$ resonance signal.

Fit | $\chi^{2}$ | $g$ | $C_{\frac{1}{2}}$ | $d_{\frac{1}{2}}$ | $C_{\frac{3}{2}}$ | $d_{\frac{3}{2}}$
---|---|---|---|---|---|---
$(210)$ | 6.08 | $316.0^{+86.2}_{-69.3}$ | $1126.6^{+327.8}_{-214.9}$ | $290.5^{+71.1}_{-77.2}$ | $\times$ | $\times$
$(220)$ | 2.95 | $217.8^{+80.6}_{-79.3}$ | $1125.5^{+190.6}_{-185.9}$ | $174.7^{+86.5}_{-77.3}$ | $3862.7^{+1466.1}_{-1003.3}$ | $97.6^{+37.9}_{-35.5}$
$(320)$ | 3.06 | $124.5^{+130.4}_{-164.7}$ | $1105.8^{+191.9}_{-132.5}$ | $250.8^{+62.0}_{-40.2}$ | $C_{\frac{3}{2}}=C_{\frac{1}{2}}$ | $82.1^{+292.1}_{-128.0}$
$(320)_{1}$ | $3.00$ | $145.8^{+122.9}_{-414.5}$ | $1108.7_{-142.9}^{+192.3}$ | $237.8^{+84.6}_{-72.5}$ | $C_{\frac{3}{2}}=C_{\frac{1}{2}}$ | $100.1_{-144.2}^{+125.1}$
$(320)_{2}$ | $2.81$ | $0.06^{+289.5}_{-288.6}$ | $1098.0^{+175.7}_{-137.5}$ | $197.9_{-78.3}^{+177.8}$ | $C_{\frac{3}{2}}=C_{\frac{1}{2}}$ | $19.1^{+90.2}_{-90.2}$
Table 4: Coupling parameters in $V_{\frac{1}{2}}$ and $V_{\frac{3}{2}}$ related to $\Xi_{c}\bar{D}^{*}$ for several fits. The type of fit and the $\chi^{2}$ are also given in the first and second columns, respectively. For the meaning of different fits, see the text for details. Note that the values of $C_{\frac{1}{2}}$($d_{\frac{1}{2}}$) and $C_{\frac{3}{2}}$($d_{\frac{3}{2}}$) are interchangeable in Fit (220).  Fit | RS |  $\sqrt{s_{R}}$ (MeV) |  $|g_{1}|$ (MeV) |  $|g_{2}|$ (MeV)
---|---|---|---|---
$(210)$ | $(-+)$ | $4463.2^{+2.8}_{-4.4}-i\,7.1^{+2.5}_{-2.8}$ | $3.29^{+0.64}_{-0.68}$ | $13.81^{+0.87}_{-0.68}$
$(220)$ | $(-+)$ | $4465.5^{+2.3}_{-2.3}-i\,3.8^{+2.3}_{-3.4}$ | $1.20^{+0.46}_{-0.44}$ | $13.01^{+0.65}_{-0.62}$
$(220)$ | $(-+)$ | $4452.1^{+2.4}_{-2.0}-i\,0.5^{+0.3}_{-0.7}$ | $0.88^{+0.46}_{-0.33}$ | $15.73^{+0.33}_{-0.41}$
Table 5: Poles in the second RS for the studies with the $J/\psi\Lambda$ and $\Xi_{c}\bar{D}^{*}$ channels included.  Fit | $\chi^{2}$ | $C_{\rm mx}$ | $g^{\prime}$ | $C^{\prime}_{\frac{1}{2}}$ | $d^{\prime}_{\frac{1}{2}}$
---|---|---|---|---|---
$(320)$ | 3.06 | $851.1^{+341.0}_{-148.1}$ | 0 | 0 | 0
$(320)_{1}$ | 3.00 | $885.6^{+353.4}_{-204.9}$ | $59.6^{+160.7}_{-249.2}$ | 0 | 0
$(320)_{2}$ | 2.81 | $579.8^{+1014.2}_{-2201.5}$ | $0$ | $187.6^{+174.8}_{-831.8}$ | 0
Table 6: Coupling parameters in $V_{\frac{1}{2}}$ related to the $\Xi_{c}^{\prime}\bar{D}$ channel in the three-coupled channel fits.  |
---|---
|
Figure 1: The panels are ordered from top to bottom and left to right. The solid lines (and dashed line in the third figure) give the result of the fits, and the type of fit is indicated at the right top corner of every panel. The greenish area singled out in the figures corresponds to the energy region actually fitted. We also indicate by the vertical lines the thresholds from left to right of the $\Xi_{c}^{\prime}\bar{D}$ and $\Xi_{c}\bar{D}^{*}$, respectively. The experimental data from Ref. [16] are the points with errorbars.  Type | $J$ | RS | $\sqrt{s_{R}}$ | $|g_{1}|$ | $|g_{2}|$ | $|g_{3}|$
---|---|---|---|---|---|---
|  |  | (MeV) | (MeV) | (MeV) | (MeV)
$(320)$ | 3/2 | RSII | $4466.6^{+1.9}_{-2.7}-i\,1.3^{+1.3}_{-3.7}$ | $1.4^{+1.4}_{-1.4}$ | $\times$ | $12.6^{+0.8}_{-0.6}$
| $(-+)$ |  |  |  |  |
$(320)$ | 1/2 | RSIII | $4453.8^{+2.4}_{-3.3}-i\,2.8^{+0.9}_{-0.8}$ | $0.6^{+0.6}_{-0.6}$ | $4.2^{+0.2}_{-0.4}$ | $15.0^{+0.5}_{-0.3}$
| $(--+)$ |  |  |  |  |
Table 7: The poles for the fit $(320)$ with three channels. Only the poles
lying in the energy region fitted and in the RSs connected to the physical
sheet are presented. These are the poles that reflect in the peaks visible in
the fits. We also gather the couplings to all channels included. For the
meaning of different fits and RSs, see the text for details.

###  4.4 The three-coupled channel fit (320) and the revisit of saturation of
the width and compositeness

With the determined pole parameters in Table 7 for the fit (320) involving the
three-coupled channels $J/\psi\Lambda$, ${\Xi^{\prime}}_{c}\bar{D}$ and
$\Xi_{c}\bar{D}^{*}$, we can revise the discussions of the compositeness and
partial decay widths, and make comparisons with the studies based on the
saturation of the compositeness coefficient and decay width in the Sec. 3. For
the partial decay width of the $P_{cs}(4459)$ into the $J/\psi\Lambda$
channel, one can safely use the standard decay width formula in Eq. (16). For
the partial widths into the near-threshold channels
${\Xi^{\prime}}_{c}\bar{D}$ and $\Xi_{c}\bar{D}^{*}$, we will rely on the
Lorentzian spectral integration formula in Eq. (17). By taking $n=8$, the
partial decay widths for the lighter resonance pole with $J=1/2$ in Table 7
are found to be

$\displaystyle\Gamma_{1}=0.5_{-0.5}^{+1.9}~{}{\rm
MeV}\,,\quad\Gamma_{2}=4.3_{-1.4}^{+1.2}~{}{\rm
MeV}\,,\quad\Gamma_{3}=0.9_{-0.6}^{+1.2}~{}{\rm MeV}\,,$ (53)

and the sum of the three widths agrees nicely with the width of the
corresponding resonance pole, i.e. minus twice the imaginary part of the pole.
This is also the main reason to take $n=8$ here as well as in Sec. 3. The
partial compositeness coefficients contributed by the $J/\psi\Lambda$ and
${\Xi^{\prime}}_{c}\bar{D}$ based on the pole parameters in Table 7 are

$\displaystyle X_{1}=0.0\pm 0.0\,,\quad X_{2}=0.15\pm 0.05\,.$ (54)

It should be noted that for the $\Xi_{c}\bar{D}^{*}$ channel the pole position
of the lighter resonance state with $J=1/2$ does not meet the working
condition of Ref. [37], and as a result it is not very meaningful to interpret
the partial $X_{2}$ (corresponding to $X_{3}$ in the three-channel case)
calculated in Eq. (10) as the probability of the $\Xi_{c}\bar{D}^{*}$
component in that resonance pole. In fact, had we blindly used Eq. (10) to
calculate $X_{2}$ its value found would be 0.94, so that the sum of
compositeness for the three channels would turn out to be slightly larger than
1. This is a manifestation that one should follow the working conditions
argued in Ref. [37] for giving a probabilistic interpretation of the
compositeness $X$ of a resonance.

For the heavier resonance pole with $J=3/2$ in Table 7 , the partial widths
calculated for the $J/\psi\Lambda$ and $\Xi_{c}\bar{D}^{*}$ are

$\displaystyle\Gamma_{1}=2.6_{-2.6}^{+8.2}~{}{\rm
MeV}\,,\quad\Gamma_{3}=0.4_{-0.4}^{+2.5}~{}{\rm MeV}\,,$ (55)

and the resulting partial compositeness coefficients turn out to be

$\displaystyle X_{1}=0.0\pm 0.0\,,\quad X_{3}=1.0_{-0.2}^{+0.2}\,.$ (56)

Despite that the mass of the $J=3/2$ pole lies clearly below the threshold of
$\Xi_{c}\bar{D}^{*}$ we have given $X_{3}$ calculated from Eq. (10) as an
indication that this resonance is clearly dominated by the
$\Xi_{c}\bar{D}^{*}$ channel component. A similar comment could be also made
in relation to $X_{3}$ for the $J=1/2$ pole. Indeed, we also find for both
poles that the coupling $|g_{3}|$ is huge in comparison with $|g_{1}|$ and
much larger, by around a factor 7/2, than $|g_{2}|$ for $J=1/2$. We point out
that a direct comparison of the results for the partial widths and
compositeness coefficients in this section and the ones in Sec. 3 is not
straightforward. The point is that in Sec. 3 the discussions are based on the
masses and widths of the $P_{cs}(4459)$ from the single-state determination of
the LHCb collaboration [16]. However, in the present section our reference fit
prefers the two-resonance solutions and the widths of the two resonance poles
taking separately are found to be much smaller than the single-state
determinations from the LHCb, though the combination of the two nearby peaks,
convoluted with the experimental resolution in energy, gives rise to a total
signal as a double bump with a width in agreement with the one pole
determination for the $P_{cs}(4459)$ [16]. Nonetheless, both the results here
and in Sec. 3 are essentially compatible, as one concludes by comparing the
couplings $|g_{i}|$ in the last and first columns of Tables 2 and 3,
respectively, with those determined by fitting directly the experimental data
in Table 7.

##  5 Conclusions

In this work we focus on the narrow peak around 4459 MeV observed in the
$J/\psi\Lambda$ event distributions from the LHCb measurements [16], which is
the first discovered charmonium pentaquark with strangeness. We have
considered three different methods, based on the elastic effective-range
expansion, the saturation of the width and the compositeness of the resonance,
and a fit to data employing unitarized phenomenological amplitudes with
constraints from heavy-quark-spin symmetry.

It is important to notice that the different methods provide compatible
results for those magnitudes to which they can be applied for their
calculation. In this sense, our analyses clearly favor a dominant role of the
$\Xi_{c}\bar{D}^{*}$ state in the composition of the $P_{cs}(4459)$. For
instance, the elastic effective-range analysis gives an effective range of
$-0.94\pm 0.13$ fm when considering only the $\Xi_{c}\bar{D}^{*}$, while it is
much larger in absolute value, around $-4~{}$fm, when taking only the
$\Xi_{c}^{\prime}\bar{D}$ channel. When considering the fits to data we then
obtain that the compositeness factor $X$ of the $\Xi_{c}\bar{D}^{*}$ channel
is around one, implying that it is saturated almost entirely by the
$\Xi_{c}\bar{D}^{*}$ channel alone. Finally, even if allowing the presence of
a CDD pole in the two-channel case, with $J/\psi\Lambda$ and either
$\Xi_{c}\bar{D}^{*}$ or $\Xi_{c}^{\prime}\bar{D}$, no acceptable fits result,
which is also a clear indication of the molecular scenario.

Another important outcome of our fits is that when requiring heavy-quark-spin
symmetry, such that the $J=1/2$ and 3/2 couplings of the $\Xi_{c}\bar{D}^{*}$
are equal, our analysis requires to include also the $\Xi^{\prime}_{c}\bar{D}$
channel to fit data properly. Furthermore, in such circumstances, the
resulting fit contains two poles in the energy region of interest around the
resonant peak of the $P_{cs}(4459)$, a $J=1/2$ pole with mass
$4453.8^{+2.4}_{-3.3}$ MeV, and a $J=3/2$ one with mass $4466.6^{+1.9}_{-2.7}$
MeV. These two poles are narrow, the half-width obtained for the lighter one
is $2.8^{+0.9}_{-0.8}$ MeV, and that for the heavier one is determined much
less precisely as $1.3^{+1.3}_{-3.7}$ MeV. With the present energy resolution
and statistics of the LHCb collaboration [16] there is only a small hint that
there are actually two resonances involved, though this could be disentangled
in the future by experiments with improved statistics.

## Acknowledgements

This work is funded in part by the Natural Science Foundation of China under
Grant Nos. 11975090 and 11575052, the MICINN AEI (Spain) Grant No.
PID2019-106080GB-C22/AEI/10.13039/501100011033, and the Fundamental Research
Funds for the Central Universities with contract No. 2242021R10099. The work
of M.L.D. is supported by the Spanish Ministerio de Economía y Competitividad
(MINECO) and the European Regional Development Fund (ERDF) under contract
FIS2017-84038-C2-1-P, by the EU Horizon 2020 research and innovation
programme, STRONG-2020 project, under grant agreement No. 824093, by
Generalitat Valenciana under contract PROMETEO/2020/023.

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