# Revisiting the nature of the $P_{c}$ pentaquarks

Meng-Lin Du<EMAIL_ADDRESS>Helmholtz-Institut für Strahlen- und
Kernphysik and
Bethe Center for Theoretical Physics, Universität Bonn, D-53115 Bonn, Germany
Vadim Baru<EMAIL_ADDRESS>Institut für Theoretische Physik II, Ruhr-
Universität Bochum, D-44780 Bochum, Germany  Institute for Theoretical and
Experimental Physics NRC “Kurchatov Institute”, Moscow 117218, Russia  P.N.
Lebedev Physical Institute of the Russian Academy of Sciences, 119991,
Leninskiy Prospect 53, Moscow, Russia  Feng-Kun Guo<EMAIL_ADDRESS>CAS Key
Laboratory of Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, Zhong Guan Cun East Street 55, Beijing 100190,
China  School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, China  Christoph Hanhart<EMAIL_ADDRESS>Institute
for Advanced Simulation, Institut für Kernphysik and
Jülich Center for Hadron Physics, Forschungszentrum Jülich, D-52425 Jülich,
Germany  Ulf-G. Meißner<EMAIL_ADDRESS>Helmholtz-Institut für
Strahlen- und Kernphysik and
Bethe Center for Theoretical Physics, Universität Bonn, D-53115 Bonn, Germany
Institute for Advanced Simulation, Institut für Kernphysik and
Jülich Center for Hadron Physics, Forschungszentrum Jülich, D-52425 Jülich,
Germany  Tbilisi State University, 0186 Tbilisi, Georgia  José A. Oller
<EMAIL_ADDRESS>Departamento de Física, Universidad de Murcia, E-30071 Murcia,
Spain  Qian Wang<EMAIL_ADDRESS>Guangdong Provincial Key Laboratory
of Nuclear Science,
Institute of Quantum Matter, South China Normal University, Guangzhou 510006,
China  Guangdong-Hong Kong Joint Laboratory of Quantum Matter,
Southern Nuclear Science Computing Center, South China Normal University,
Guangzhou 510006, China  Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing 100049, China

###### Abstract

The nature of the three narrow hidden-charm pentaquark $P_{c}$ states, i.e.,
$P_{c}(4312)$, $P_{c}(4440)$ and $P_{c}(4457)$, is under intense discussion
since their discovery from the updated analysis of the process
$\Lambda_{b}^{0}\to J/\psi pK^{-}$ by LHCb. In this work we extend our
previous coupled-channel approach [Phys. Rev. Lett. 124, 072001 (2020)], in
which the $P_{c}$ states are treated as $\Sigma_{c}^{(*)}\bar{D}^{(*)}$
molecules, by including the $\Lambda_{c}\bar{D}^{(*)}$ and $\eta_{c}p$ in
addition to the $J/\psi p$ as explicit inelastic channels as required by
unitarity and heavy quark spin symmetry (HQSS), respectively. Since inelastic
parameters are very badly constrained by the current data, three calculation
schemes are considered: (a) scheme I with pure contact interactions between
the elastic, i.e., $\Sigma_{c}^{(*)}\bar{D}^{(*)}$, and inelastic channels and
without the $\Lambda_{c}\bar{D}^{(*)}$ interactions, (b) scheme II, where the
one-pion exchange (OPE) is added to scheme I, and (c) scheme III, where the
$\Lambda_{c}\bar{D}^{(*)}$ interactions are included in addition. It is shown
that to obtain cutoff independent results, OPE in the multichannel system is
to be supplemented with $S$-wave-to-$D$-wave mixing contact terms. As a
result, in line with our previous analysis, we demonstrate that the
experimental data for the $J/\psi p$ invariant mass distribution are
consistent with the interpretation of the $P_{c}(4312)$ and
$P_{c}(4440)/P_{c}(4457)$ as $\Sigma_{c}\bar{D}$ and $\Sigma_{c}\bar{D}^{*}$
hadronic molecules, respectively, and that the data show clear evidence for a
new narrow state, $P_{c}(4380)$, identified as a $\Sigma_{c}^{*}\bar{D}$
molecule, which should exist as a consequence of HQSS. However, now two
solutions are found in all schemes which describe the data equally well, and
thus no unambiguous conclusion about the quantum numbers of the $P_{c}(4440)$
and $P_{c}(4457)$ from data in the $J/\psi p$ channel alone is possible. It is
argued though that one of these solutions, in which the quantum numbers of the
$P_{c}(4440)$ and $P_{c}(4457)$ are $J^{P}=3/2^{-}$ and $1/2^{-}$,
respectively, is theoretically preferred. Moreover, we demonstrate that the
line shapes related to the $P_{c}(4440)$ and the $P_{c}(4457)$ in the
$\Sigma_{c}^{(*)}\bar{D}$ and $\eta_{c}p$ mass distributions from
$\Lambda_{b}^{0}\to\Sigma_{c}^{(*)}\bar{D}K^{-}$ and
$\Lambda_{b}^{0}\to\eta_{c}pK^{-}$ will allow one to pin down the quantum
numbers in the hadronic molecular picture, once the data are available. We
also investigate possible pentaquark signals in the $\Lambda_{c}\bar{D}^{(*)}$
final states.

##  I Introduction

A quantitative understanding of hadron spectra plays a key role in our
understanding of the strong interaction as well as its fundamental underlying
theory, i.e., quantum chromodynamics (QCD). For a long time, hadrons were
believed to be composed of either a pair of quark and antiquark ($q\bar{q}$)
or three quarks ($qqq$) according to the conventional quark model. However,
the confinement property of QCD in principle allows for the existence of any
color neutral object, such as multiquarks, hybrids, glueballs, hadronic
molecules and so on, which are usually called exotic states. Up to now, a
tremendous number of candidates for having an exotic structure has been
observed Chen  _et al._ (2016, 2017); Esposito  _et al._ (2017); Hosaka  _et
al._ (2016); Dong  _et al._ (2017); Olsen  _et al._ (2018); Guo  _et al._
(2018); Cerri  _et al._ (2019); Altmannshofer  _et al._ (2019); Brambilla  _et
al._ (2020); Guo  _et al._ (2020); Liu  _et al._ (2019a); Yuan and Olsen
(2019). Among them the observation of the $P_{c}(4380)$ and $P_{c}(4450)$ Aaij
_et al._ (2015), as the first observation of pentaquarks, in 2015 builds a
landmark. They decay strongly into $J/\psi p$ and thus contain at least five
quarks, $c\bar{c}uud$. The existence of these two pentaquark states is
consistent with a model-independent reanalysis of the same data sample Aaij
_et al._ (2016a) and a full amplitude analysis of the $\Lambda_{b}^{0}\to
J/\psi p\pi^{-}$ decay Aaij  _et al._ (2016b). A later analysis based on an
order-of-magnitude larger data sample, i.e., the combined data set collected
in Run 1 and Run 2 by the LHCb Collaboration, shows that the $P_{c}(4450)$
structure consists of two narrow overlapping peaks, i.e., $P_{c}(4440)$ and
$P_{c}(4457)$, and a third narrow peak $P_{c}(4312)$ emerges Aaij  _et al._
(2019). However, the broad $P_{c}(4380)$ reported in 2015 lost its
significance and its verification awaits a complete amplitude analysis.
Numerous theoretical interpretations of the nature of the pentaquarks followed
these discoveries, including hadronic molecules Chen  _et al._ (2019a, b); Guo
_et al._ (2019); Liu  _et al._ (2019b); He (2019); Guo and Oller (2019);
Shimizu  _et al._ (2019); Xiao  _et al._ (2019a, b); Wang  _et al._ (2020a);
Meng  _et al._ (2019); Wu  _et al._ (2019); Xiao  _et al._ (2019c); Voloshin
(2019); Sakai  _et al._ (2019); Wang and Wang (2020); Yamaguchi  _et al._
(2020); Liu  _et al._ (2019c); Lin and Zou (2019); Wang  _et al._ (2019);
Gutsche and Lyubovitskij (2019); Burns and Swanson (2019); Du  _et al._
(2020a); Wang  _et al._ (2020b); Xu  _et al._ (2020); Kuang  _et al._ (2020);
Peng  _et al._ (2020, 2021a); Xiao  _et al._ (2020); Dong  _et al._ (2021);
Peng  _et al._ (2021b), compact pentaquarks states Ali and Parkhomenko (2019);
Zhu  _et al._ (2019); Wang (2020); Giron  _et al._ (2019); Cheng and Liu
(2019); Stancu (2019); Kuang  _et al._ (2020), hadro-charmonia Eides  _et al._
(2020), and cusp effects Kuang  _et al._ (2020).

The proximity of the $\Sigma_{c}\bar{D}^{(*)}$ thresholds to these narrow
pentaquark structures suggests that the corresponding two-hadron states play
an important role in the dynamics of the pentaquarks, hinting at an
interpretation of their structure as hadronic molecules. In the most common
hadronic molecular picture, the $P_{c}(4312)$ is an $S$-wave
$\Sigma_{c}\bar{D}$ bound state, while the $P_{c}(4440)$ and $P_{c}(4457)$ are
bound states of $\Sigma_{c}\bar{D}^{*}$ with different spin structures, see
e.g., Refs. Liu  _et al._ (2019b); Xiao  _et al._ (2019b); Sakai  _et al._
(2019); Du  _et al._ (2020a); Xiao  _et al._ (2020). The origin of the peak
from the $P_{c}(4312)$ is attributed to a virtual state of $\Sigma_{c}\bar{D}$
in Ref. Fernández-Ramírez  _et al._ (2019) based on an amplitude
analysis,111As a consequence of being a near-threshold virtual state, the peak
for the $P_{c}(4312)$ should behave as a sharp threshold cusp. It is due to
the convolution with the energy resolution that the peak shown in Ref.
Fernández-Ramírez  _et al._ (2019) is smooth. which only fits to data around
the $\Sigma_{c}\bar{D}$ threshold. In Ref. Kuang  _et al._ (2020), final state
interactions are constructed based on a $K$-matrix including the channels
$J/\psi p$-$\Sigma_{c}\bar{D}$-$\Sigma_{c}\bar{D}^{*}$. The analysis suggests
that the $P_{c}(4312)$ is a $\Sigma_{c}\bar{D}$ molecule, while the
$P_{c}(4440)$ could be a compact pentaquark state, and the $P_{c}(4457)$ could
be caused by the cusp effect.222A strong threshold cusp effect normally
requires the existence of a near-threshold pole in an unphysical Riemann sheet
Guo  _et al._ (2015a, 2020); Dong  _et al._ (2020).

To make further progress it is necessary to systematically investigate the
implications of the symmetries of QCD, here, most importantly, heavy quark
spin symmetry (HQSS). In simple terms, under the assumption that the observed
pentaquarks are of molecular nature, the symmetry allows one to predict
additional states, the so-called spin partners. To be concrete, HQSS predicts
seven $P_{c}$ states which are classified into two heavy quark spin
multiplets. Three of these seven correspond to the ones reported by LHCb Xiao
_et al._ (2013); Liu  _et al._ (2018, 2019b); Du  _et al._ (2020a): While the
$P_{c}(4312)$ is unambiguously assigned to the $J^{P}=\frac{1}{2}^{-}$
$\Sigma_{c}\bar{D}$ bound state, there are two possible spin structures for
the $P_{c}(4440)$ and $P_{c}(4457)$ identified as the $\Sigma_{c}\bar{D}^{*}$
bound states, namely $J^{P}=\frac{1}{2}^{-}$ and $J^{P}=\frac{3}{2}^{-}$ Guo
and Oller (2019); Liu  _et al._ (2019b); Du  _et al._ (2020a), and their spin
assignment is not uniquely fixed by HQSS alone Chen  _et al._ (2019b); Pavon
Valderrama (2019); He (2019); Liu  _et al._ (2019c); Du  _et al._ (2020a). In
Ref. Du  _et al._ (2020a), the $J/\psi p$ invariant mass distribution of the
$\Lambda_{b}^{0}\to J/\psi pK^{-}$ process was described in the molecular
scenario. In particular, the $\Sigma_{c}^{(*)}\bar{D}^{(*)}$ channels (in what
follows called elastic channels, since their thresholds are close to the
$P_{c}$ states) and the $J/\psi p$ channel (inelastic channel) were included
dynamically. The interactions between elastic channels were constrained by
HQSS, which guarantees that there are only two momentum-independent contact
potentials, reflecting that the number of independent multiplets is two. The
effect from additional inelastic channels, such as the
$\Lambda_{c}\bar{D}^{(*)}$ and $\eta_{c}p$ channels and so on, was absorbed
into additional imaginary parts of the two contact potentials in the spirit of
an optical potential. For the case of only contact potentials, two different
solutions, corresponding to scenarios $A$ and $B$ in Ref. Liu  _et al._
(2019b), were found describing the data almost equally well. Each of them
gives seven poles in the $\Sigma_{c}^{(*)}\bar{D}^{(*)}$ scattering
amplitudes, however, with different pole locations. Meanwhile, as soon as the
one-pion-exchange (OPE) potentials was included, only one solution was found
which suggests that $P_{c}(4440)$ and $P_{c}(4457)$ couple dominantly to the
$\Sigma_{c}\bar{D}^{*}$ with quantum numbers $J^{P}=\frac{3}{2}^{-}$ and
$\frac{1}{2}^{-}$, respectively Du  _et al._ (2020a). In both fits, the
$P_{c}(4312)$ couples dominantly to the $\Sigma_{c}\bar{D}$ channel with
$J^{P}=\frac{1}{2}^{-}$. In addition, evidence for an additional narrow state,
also called for by HQSS, around $4.38$ GeV was found in the data with
$J^{P}=\frac{3}{2}^{-}$, which couples dominantly to the
$\Sigma_{c}^{*}\bar{D}$ (see also Ref. Xiao  _et al._ (2019a, 2020)).

The mentioned imaginary parts of the contact terms violate unitarity. In this
work we overcome this shortcoming by including explicitly the supposedly most
prominent additional inelastic channels, namely, $\Lambda_{c}\bar{D}^{(*)}$
and $\eta_{c}p$ making the mentioned imaginary parts of the contact terms
obsolete. The inclusion of the $\eta_{c}p$ channel explicitly is also
necessary from HQSS, since the $\eta_{c}$ and $J/\psi$ are in the same HQSS
multiplet. While the interactions of $\eta_{c}p$ and $J/\psi p$ with the
elastic channels are of a short-range character, $\Lambda_{c}\bar{D}^{(*)}$
can also interact with $\Sigma^{(*)}\bar{D}^{(*)}$ via one-pion exchange.
Thus, when the $\Lambda_{c}\bar{D}$ and $\Lambda_{c}\bar{D}^{*}$ channels are
included explicitly, three-body effects from the $\Lambda_{c}\bar{D}\pi$
intermediate states have to be taken into account in both the
$\Sigma^{*}\bar{D}^{(*)}\to\Lambda_{c}\bar{D}^{(*)}$ transition potentials and
the $\Sigma_{c}^{(*)}$ self-energy Aaron  _et al._ (1968) (for a recent
discussion on the subject see Ref. Mai  _et al._ (2017)). The
$\Sigma_{c}^{(*)}$ width may have a sizable effect on the widths of the
$\Sigma_{c}^{(*)}\bar{D}^{(*)}$ molecular states. This is especially important
for those states that have a prominent $\Sigma_{c}^{*}\bar{D}^{(*)}$
component, since the $\Sigma_{c}^{*}$ width, which is around 15 MeV Zyla  _et
al._ (2020), is relatively large. Among them, the narrow $P_{c}(4380)$
attracts special interest, since its confirmation would provide strong support
for the molecular picture. Equipped with the mentioned extensions of the
approach, in this work we are also in a position to investigate the invariant
mass distributions in all the elastic and inelastic channels to understand
what can be learned from the corresponding data, once they exist.

It was recently noticed in studies of the $X(3872)$ and $Z_{b}$ systems Baru
_et al._ (2017, 2016) that, in line with what is well-known for the nucleon-
nucleon interaction Ericson and Weise (1988), the most prominent contribution
from the OPE originates from the tensor force inducing significant $S$-$D$
transitions. However, the situation in the heavy quark sector is more
complicated than in nuclear effective field theory (EFT) because transitions
between the different channels connected by HQSS, can be separated by more
than 100 MeV due to the $D^{*}$-$D$ and $\Sigma^{*}_{c}$-$\Sigma_{c}$ mass
differences. The resulting sizable momentum scales enhance further the role of
the $S$-$D$ transitions. The analysis of the line shapes relevant for the
$Z_{b}(10610)/Z_{b}(10650)$ and their spin partners Wang  _et al._ (2018);
Baru  _et al._ (2019) showed that, to remove the strong regulator dependence
caused by the high-momentum contribution from the $S$-$D$ OPE transitions, the
formally next-leading-order (NLO) contact term for the $S$-$D$ transitions is
required to be promoted to leading order (LO). Meanwhile, the NLO $S$-$S$
contact terms are numerically marginal, consistent with the expectations. In
line with the observations of Refs. Wang  _et al._ (2018); Baru  _et al._
(2019), if we omit the $S$-$D$ counter terms in the pentaquark system, we also
observe a strong dependence on the regulator, as soon as the OPE is included
in the $\Sigma_{c}^{(*)}\bar{D}^{(*)}\to\Sigma_{c}^{(*)}\bar{D}^{(*)}$ and
$\Sigma_{c}^{(*)}\bar{D}^{(*)}\to\Lambda_{c}\bar{D}^{(*)}$ transition
potentials. Following Refs. Wang  _et al._ (2018); Baru  _et al._ (2019),
introducing the $S$-$D$ contact terms allows us to obtain regulator-
independent results and, at the same time, to arrive at a satisfying
description of the data. As a result, now even in the presence of the OPE
potential, two solutions describe the data equally well with different
assignments for the quantum numbers of the higher $P_{c}$ states, in contrast
to what was reported in Ref. Du  _et al._ (2020a). However, below we provide
arguments, why we regard one of the solutions preferable compared to the
other. Moreover, since two kinds of assignments correspond to different spin-
structures of the higher $P_{c}$ states, they have different features in their
decays into lower decay modes. Consequently, we propose a way to
experimentally distinguish these two scenarios even without a full amplitude
analysis, since the two solutions lead to different predictions for the line
shapes in the elastic and newly included inelastic channels.

The paper is organized as follows. In Sec. II we derive the effective
interaction potentials for the dynamical channels and parametrize the relevant
weak production vertices. In particular, in Subsection II.1, we discuss the
purely contact potentials in the basis of heavy-light spin degrees of freedom.
In Subsections II.2 and II.3, we derive the OPE and the $S$-$D$ contact terms
from the effective Lagrangian. The relativistic Lippmann-Schwinger equations
(LSEs) with the dynamical width of $\Sigma_{c}^{(*)}$ are given in Sec. III.
The fitting schemes to the $J/\psi p$ mass distributions are presented in Sec.
IV. Section IV.1 is devoted to a study, in which only contact potentials are
considered. The OPE potentials are included in Sec. IV.2 and the explicit
inclusion of the $\Lambda_{c}\bar{D}^{(*)}$ is investigated in Sec. IV.3. The
pole positions and their effective couplings are also explicitly presented for
schemes I and II. In addition, we also predict the line shapes for the
$\Sigma_{c}^{(*)}\bar{D}^{(*)}$ and $\eta_{c}p$ (and
$\Lambda_{c}\bar{D}^{(*)}$ for scheme III) mass distributions in the
$\Lambda_{b}^{0}\to\Sigma_{c}^{(*)}\bar{D}^{(*)}K^{-}$ and
$\Lambda_{b}^{0}\to\eta_{c}pK^{-}$ (and
$\Lambda_{b}^{0}\to\Lambda_{c}\bar{D}^{(*)}K^{-}$) processes. We summarize in
Sec. V.

##  II Effective potentials

In order to fully exploit the implications of HQSS and to study the role of
the $\Lambda_{c}\bar{D}^{(*)}$ channels and the effect of the
$\Sigma_{c}^{(*)}$ width stemming from the decay
$\Sigma_{c}^{(*)}\to\Lambda_{c}\pi$, we extend the framework of Ref. Du  _et
al._ (2020a) to dynamically include the $\Lambda_{c}\bar{D}^{(*)}$ and
$\eta_{c}p$ channels in addition to the $\Sigma_{c}^{(*)}\bar{D}^{(*)}$ and
$J/\psi p$ channels. The effective transition potentials for those channels
entering the LSEs contain short-range contact terms as well as OPE. In this
section, both the contact and OPE potentials are presented employing the
constraints from HQSS.

###  II.1 Contact potentials

The short-ranged contact potentials can be derived either from the effective
Lagrangians or from the decomposition of the spin structures of the charm-
anticharm and light degrees of freedom in the heavy-quark limit following,
e.g., Ref. Voloshin (2011). As the later one is more intuitive and easier to
be applied to the bare production amplitudes, we extract the LO contact
potentials and the bare production amplitudes based on the heavy-light spin
structures. Afterwards, we also show the construction based on the Lagrangian
method — while it provides identical results it is easier extendable to higher
orders.

Along this line, we expand the two-particle basis in terms of heavy-light spin
structure $|s_{Q}\otimes j_{\ell}\rangle$, with $s_{Q}$ and $j_{\ell}$
representing the total spin of the heavy quarks and the total angular momentum
of light degrees of freedom, respectively. In the heavy-light spin structure
basis, the $\Sigma_{c}^{(*)}$ and $\bar{D}^{(*)}$ are
$\left|\frac{1}{2}\otimes 1\right\rangle$ and
$\left|\frac{1}{2}\otimes\frac{1}{2}\right\rangle$ spin multiplets,
respectively, while the $\Lambda_{c}^{+}$ corresponds to a
$\left|\frac{1}{2}\otimes 0\right\rangle$ singlet. The higher partial-wave
amplitudes for the orbital angular momentum $\ell$ are suppressed by a factor
$(p/\Lambda)^{2\ell}$, where $p$ is the center-of-mass (c.m.) frame three-
momentum of the system and $\Lambda$ denotes some typical hard scale commonly
assumed to be of the order of 1 GeV. Accordingly, the short-ranged
$\Sigma_{c}^{(*)}\bar{D}^{(*)}$ and $\Lambda_{c}\bar{D}^{(*)}$ potentials
should be dominated by $S$ waves, although, given the large range of momenta
involved in the study, effects of the $S$-$D$ mixing may also become
important, as discussed below. The $S$-wave $\Sigma_{c}^{(*)}\bar{D}^{(*)}$
and $\Lambda_{c}\bar{D}^{(*)}$ systems can be cast in terms of $|s_{Q}\otimes
j_{\ell}\rangle$ as Sakai  _et al._ (2019); Xiao  _et al._ (2013); Du  _et
al._ (2020a)

$\displaystyle\left(\begin{array}[]{c}|\Sigma_{c}\bar{D}\rangle\\\
|\Sigma_{c}\bar{D}^{*}\rangle\\\ |\Sigma_{c}^{*}\bar{D}^{*}\rangle\\\
|\Lambda_{c}\bar{D}\rangle\\\
|\Lambda_{c}\bar{D}^{*}\rangle\end{array}\right)_{\frac{1}{2}}=\left(\begin{array}[]{ccccc}\frac{1}{2}&\frac{1}{2\sqrt{3}}&\sqrt{\frac{2}{3}}&0&0\\\
\frac{1}{2\sqrt{3}}&\frac{5}{6}&-\frac{\sqrt{2}}{3}&0&0\\\
\sqrt{\frac{2}{3}}&-\frac{\sqrt{2}}{3}&-\frac{1}{3}&0&0\\\
0&0&0&-\frac{1}{2}&\frac{\sqrt{3}}{2}\\\
0&0&0&\frac{\sqrt{3}}{2}&\frac{1}{2}\end{array}\right)\left(\begin{array}[]{c}|0\otimes\frac{1}{2}\rangle\\\
|1\otimes\frac{1}{2}\rangle\\\ |1\otimes\frac{3}{2}\rangle\\\
|0\otimes\frac{1}{2}\rangle^{\prime}\\\
|1\otimes\frac{1}{2}\rangle^{\prime}\end{array}\right),~{}~{}$ (16)
$\displaystyle\left(\begin{array}[]{c}|\Sigma_{c}\bar{D}^{*}\rangle\\\
|\Sigma_{c}^{*}\bar{D}\rangle\\\ |\Sigma_{c}^{*}\bar{D}^{*}\rangle\\\
|\Lambda_{c}\bar{D}^{*}\rangle\end{array}\right)_{\frac{3}{2}}=\left(\begin{array}[]{cccc}\frac{1}{\sqrt{3}}&-\frac{1}{3}&\frac{\sqrt{5}}{3}&0\\\
-\frac{1}{2}&\frac{1}{\sqrt{3}}&\frac{1}{2}\sqrt{\frac{5}{3}}&0\\\
\frac{1}{2}\sqrt{\frac{5}{3}}&\frac{\sqrt{5}}{3}&-\frac{1}{6}&0\\\
0&0&0&1\end{array}\right)\left(\begin{array}[]{c}|0\otimes\frac{3}{2}\rangle\\\
|1\otimes\frac{1}{2}\rangle\\\ |1\otimes\frac{3}{2}\rangle\\\
|1\otimes\frac{1}{2}\rangle^{\prime}\end{array}\right),~{}~{}$ (29)
$\displaystyle|\Sigma_{c}^{*}\bar{D}^{*}\rangle_{\frac{5}{2}}=\bigg{|}1\otimes\frac{3}{2}\bigg{\rangle},$
(30)

where the subscripts on the left-hand side represent the total angular
momentum $J=\frac{1}{2}$, $\frac{3}{2}$, $\frac{5}{2}$. The superscript ′ on
the right-hand side denotes the heavy-light structures of the
$\Lambda_{c}\bar{D}^{(*)}$ systems. The rotation matrices in Eqs. (16)-(30)
will be denoted as $R^{J}$ in the following. Because the $\Sigma_{c}^{(*)}$
and $\Lambda_{c}$ are in different light spin multiplets, one needs four
independent parameters to describe all transitions between various channels.
Therefore, we introduce

$\displaystyle
C_{\frac{1}{2}}\equiv\bigg{\langle}s_{Q}\otimes\frac{1}{2}\bigg{|}\hat{\mathcal{H}}_{I}\bigg{|}s_{Q}\otimes\frac{1}{2}\bigg{\rangle},\quad
C_{\frac{3}{2}}\equiv\bigg{\langle}s_{Q}\otimes\frac{3}{2}\bigg{|}\hat{\mathcal{H}}_{I}\bigg{|}s_{Q}\otimes\frac{3}{2}\bigg{\rangle},$
(31)

for the transitions between the $\Sigma_{c}^{(*)}\bar{D}^{(*)}$ channels, and

$\displaystyle
C_{\frac{1}{2}}^{\prime\prime}\equiv\prescript{\prime}{}{\bigg{\langle}}s_{Q}\otimes\frac{1}{2}\bigg{|}\hat{\mathcal{H}}_{I}\bigg{|}s_{Q}\otimes\frac{1}{2}\bigg{\rangle}^{\prime},$
(32)

for those between the $\Lambda_{c}\bar{D}^{(*)}$ channels with
$\hat{\mathcal{H}}_{I}$ the effective Hamiltonian respecting HQSS. For the
transition between $\Sigma_{c}^{(*)}\bar{D}^{(*)}$ and
$\Lambda_{c}\bar{D}^{(*)}$, we define

$\displaystyle
C_{\frac{1}{2}}^{\prime}\equiv\prescript{\prime}{}{\bigg{\langle}}s_{Q}\otimes\frac{1}{2}\bigg{|}\hat{\mathcal{H}}_{I}\bigg{|}s_{Q}\otimes\frac{1}{2}\bigg{\rangle}=\bigg{\langle}s_{Q}\otimes\frac{1}{2}\bigg{|}\hat{\mathcal{H}}_{I}\bigg{|}s_{Q}\otimes\frac{1}{2}\bigg{\rangle}^{\prime}.$
(33)

In the heavy quark limit, the contact interactions defined above are
independent of $s_{Q}=0$ or 1. By virtue of the decomposition of Eqs.
(16)-(30), the leading order contact potentials read

$\displaystyle
V^{\frac{1}{2}^{-}}_{C}=\left(\begin{array}[]{ccccc}\frac{1}{3}C_{\frac{1}{2}}+\frac{2}{3}C_{\frac{3}{2}}&\frac{2}{3\sqrt{3}}C_{\frac{1}{2}}-\frac{2}{3\sqrt{3}}C_{\frac{3}{2}}&\frac{1}{3}\sqrt{\frac{2}{3}}C_{\frac{1}{2}}-\frac{1}{3}\sqrt{\frac{2}{3}}C_{\frac{3}{2}}&0&\frac{1}{\sqrt{3}}C^{\prime}_{\frac{1}{2}}\\\
\frac{2}{3\sqrt{3}}C_{\frac{1}{2}}-\frac{2}{3\sqrt{3}}C_{\frac{3}{2}}&\frac{7}{9}C_{\frac{1}{2}}+\frac{2}{9}C_{\frac{3}{2}}&-\frac{\sqrt{2}}{9}C_{\frac{1}{2}}+\frac{\sqrt{2}}{9}C_{\frac{3}{2}}&\frac{1}{\sqrt{3}}C^{\prime}_{\frac{1}{2}}&\frac{2}{3}C^{\prime}_{\frac{1}{2}}\\\
\frac{1}{3}\sqrt{\frac{2}{3}}C_{\frac{1}{2}}-\frac{1}{3}\sqrt{\frac{2}{3}}C_{\frac{3}{2}}&-\frac{\sqrt{2}}{9}C_{\frac{1}{2}}+\frac{\sqrt{2}}{9}C_{\frac{3}{2}}&\frac{8}{9}C_{\frac{1}{2}}+\frac{1}{9}C_{\frac{3}{2}}&-\sqrt{\frac{2}{3}}C^{\prime}_{\frac{1}{2}}&\frac{\sqrt{2}}{3}C^{\prime}_{\frac{1}{2}}\\\
0&\frac{1}{\sqrt{3}}C^{\prime}_{\frac{1}{2}}&-\sqrt{\frac{2}{3}}C^{\prime}_{\frac{1}{2}}&C_{\frac{1}{2}}^{\prime\prime}&0\\\
\frac{1}{\sqrt{3}}C^{\prime}_{\frac{1}{2}}&\frac{2}{3}C^{\prime}_{\frac{1}{2}}&\frac{\sqrt{2}}{3}C^{\prime}_{\frac{1}{2}}&0&C_{\frac{1}{2}}^{\prime\prime}\end{array}\right),$
(39) $\displaystyle
V^{\frac{3}{2}^{-}}_{C}=\left(\begin{array}[]{cccc}\frac{1}{9}C_{\frac{1}{2}}+\frac{8}{9}C_{\frac{3}{2}}&-\frac{1}{3\sqrt{3}}C_{\frac{1}{2}}+\frac{1}{3\sqrt{3}}C_{\frac{3}{2}}&-\frac{\sqrt{5}}{9}C_{\frac{1}{2}}+\frac{\sqrt{5}}{9}C_{\frac{3}{2}}&-\frac{1}{3}C^{\prime}_{\frac{1}{2}}\\\
-\frac{1}{3\sqrt{3}}C_{\frac{1}{2}}+\frac{1}{3\sqrt{3}}C_{\frac{3}{2}}&\frac{1}{3}C_{\frac{1}{2}}+\frac{2}{3}C_{\frac{3}{2}}&+\frac{1}{3}\sqrt{\frac{5}{3}}C_{\frac{1}{2}}-\frac{1}{3}\sqrt{\frac{5}{3}}C_{\frac{3}{2}}&\frac{1}{\sqrt{3}}C^{\prime}_{\frac{1}{2}}\\\
-\frac{\sqrt{5}}{9}C_{\frac{1}{2}}+\frac{\sqrt{5}}{9}C_{\frac{3}{2}}&\frac{1}{3}\sqrt{\frac{5}{3}}C_{\frac{1}{2}}-\frac{1}{3}\sqrt{\frac{5}{3}}C_{\frac{3}{2}}&\frac{5}{9}C_{\frac{1}{2}}+\frac{4}{9}C_{\frac{3}{2}}&\frac{\sqrt{5}}{3}C^{\prime}_{\frac{1}{2}}\\\
-\frac{1}{3}C^{\prime}_{\frac{1}{2}}&\frac{1}{\sqrt{3}}C^{\prime}_{\frac{1}{2}}&\frac{\sqrt{5}}{3}C^{\prime}_{\frac{1}{2}}&C_{\frac{1}{2}}^{\prime\prime}\end{array}\right),$
(44) $\displaystyle V^{\frac{5}{2}^{-}}_{C}=C_{\frac{3}{2}}.$ (45)

The transition between the elastic channels and the inelastic channels $J/\psi
p$ and $\eta_{c}p$ can be obtained in a similar way. In the present work, we
include both $S$ and $D$ waves for the $J/\psi p$ and $\eta_{c}p$ systems,
since the three-momenta of the proton could be as large as 0.9 GeV and thus
the $D$ wave can be as important as the $S$ wave in the energy region of
interest. While the $\left|1\otimes\frac{1}{2}\right\rangle$ component only
couples to the $J/\psi p$ in $S$ wave in the heavy quark limit, the
$\left|1\otimes\frac{3}{2}\right\rangle$ only couples to the $J/\psi p$ in $D$
wave Du  _et al._ (2020a). The $\left|0\otimes\frac{1}{2}\right\rangle$ and
the $\left|0\otimes\frac{3}{2}\right\rangle$ components couple to the
$\eta_{c}p$ channel in $S$ wave and $D$ wave, respectively. The coupling
strengths of the $S$-wave $\Sigma_{c}^{(*)}\bar{D}^{(*)}$ to the $J/\psi p$
and $\eta_{c}p$ channels are related to each other via HQSS. We therefore need
to introduce only two coupling constants,

$\displaystyle g_{S}$ $\displaystyle\equiv$
$\displaystyle\left.\left.\left\langle
1\otimes\frac{1}{2}\right|\hat{\mathcal{H}}_{I}\right|J/\psi
p\right\rangle_{S}=\left.\left.\left\langle
0\otimes\frac{1}{2}\right|\hat{\mathcal{H}}_{I}\right|\eta_{c}p\right\rangle_{S},$
$\displaystyle g_{D}k^{2}$ $\displaystyle\equiv$
$\displaystyle\left.\left.\left\langle
1\otimes\frac{3}{2}\right|\hat{\mathcal{H}}_{I}\right|J/\psi
p\right\rangle_{D}=\left.\left.\left\langle
0\otimes\frac{3}{2}\right|\hat{\mathcal{H}}_{I}\right|\eta_{c}p\right\rangle_{D},$
(46)

where $k$ is the magnitude of the $J/\psi$ ($\eta_{c}$) momentum in the
$J/\psi p$ ($\eta_{c}p$) rest-frame. Employing Eqs. (16)-(30), the constants
$g_{S}$ and $g_{D}$ allow one to write the transition potentials
$\mathcal{V}_{\alpha i}$ and $\mathcal{V}_{\alpha i}^{\prime}$ between the
$\alpha$th elastic channel and the inelastic channels $J/\psi p$ and
$\eta_{c}p$, respectively, as

$\displaystyle\mathcal{V}^{J}_{\alpha 1}=g_{S}R_{\alpha 2}^{J},\quad$
$\displaystyle\mathcal{V}^{J}_{\alpha 2}(k)$
$\displaystyle=g_{D}k^{2}R_{\alpha 3}^{J},\quad$ $\displaystyle J$
$\displaystyle=\frac{1}{2},\,\frac{3}{2},$
$\displaystyle\mathcal{V}^{J}_{\alpha 1}=0,\quad$
$\displaystyle\mathcal{V}^{J}_{\alpha 2}(k)$ $\displaystyle=g_{D}k^{2},\quad$
$\displaystyle J$ $\displaystyle=\frac{5}{2},$
$\displaystyle\mathcal{V}^{\prime J}_{\alpha 1}=g_{S}R_{\alpha 1}^{J},\quad$
$\displaystyle\mathcal{V}^{\prime J}_{\alpha 2}(k)$ $\displaystyle=0,\quad$
$\displaystyle J$ $\displaystyle=\frac{1}{2},$
$\displaystyle\mathcal{V}^{\prime J}_{\alpha 1}=0$
$\displaystyle\mathcal{V}^{\prime J}_{\alpha 2}(k)$
$\displaystyle=g_{D}k^{2}R_{\alpha 1}^{J},\quad$ $\displaystyle J$
$\displaystyle=\frac{3}{2},$ $\displaystyle\mathcal{V}^{\prime J}_{\alpha
1}=0,\quad$ $\displaystyle\mathcal{V}^{\prime J}_{\alpha 2}(k)$
$\displaystyle=0,\quad$ $\displaystyle J$ $\displaystyle=\frac{5}{2},$ (47)

where $\alpha=1,2,3$ for $J=\frac{1}{2},\frac{3}{2}$ and $\alpha=1$ for
$J=\frac{5}{2}$, and $i=1$, 2 denote the $S$ and $D$ wave in the inelastic
channels, respectively. The direct $J/\psi p$ and $\eta_{c}p$ interactions can
be neglected since they are Okubo-Zweig-Iizuka suppressed and were found very
weak in a recent lattice QCD calculation Skerbis and Prelovsek (2019) (see
also the estimate based on a coupled-channel model in Ref. Du  _et al._
(2020b) leading to a $J/\psi p$ scattering length of the order of mfm). A
similar assumption about neglecting the direct interactions between the
inelastic $\Upsilon\pi$ and $h_{b}\pi$ channels was shown to be consistent
with data for the $Z_{b}(10610)$ and $Z_{b}(10650)$ in a combined analysis of
the $\Upsilon(10860)$ decays Wang  _et al._ (2018), and with the Dalitz plot
analysis of the $\Upsilon\pi\pi$ final states from the $\Upsilon(10860)$ Baru
_et al._ (2020). In addition, we assume that the effect from the interaction
of the $\Lambda_{c}\bar{D}^{(*)}$ with the $J/\psi p$ and $\eta_{c}p$ channels
is suppressed and can also be neglected. As long as the direct $J/\psi p$ and
$\eta_{c}p$ interactions are neglected, the effect of these channels on the
elastic channels can be included through an additional contribution Hanhart
_et al._ (2015); Albaladejo  _et al._ (2016); Guo  _et al._ (2016); Wang  _et
al._ (2018); Baru  _et al._ (2019); Du  _et al._ (2020a) to the potentials
among the elastic channels. While the real parts of this contribution can be
absorbed by redefining the contact terms $C_{\frac{1}{2}}$ and
$C_{\frac{3}{2}}$ Baru  _et al._ (2019), the imaginary parts are included into
the potentials as

$\displaystyle V_{\text{in},\alpha\beta}^{J}(E)=-\frac{i}{2\pi
E}\sum_{j=1}^{2}m_{J/\psi}m_{p}\mathcal{V}_{\alpha j}^{J}\mathcal{V}_{\beta
j}^{J}k-\frac{i}{2\pi E}\sum_{j=1}^{2}m_{\eta_{c}}m_{p}\mathcal{V}_{\alpha
j}^{\prime J}\mathcal{V}_{\beta j}^{\prime J}k.$ (48)

As a result, the full effective potential for the elastic channels is the sum
of the contact potential $V_{C}^{J}$, the effective potential from the
inelastic channel $V_{\text{in}}^{J}$ and the OPE potential
$V_{\text{OPE}}^{J}(E,p,p^{\prime})$

$\displaystyle
V^{J}(E,p,p^{\prime})=V_{C}^{J}+V_{\text{in}}^{J}(E)+V_{\text{OPE}}^{J}(E,p,p^{\prime}),$
(49)

with $E$ the total energy of the system, and $p$ and $p^{\prime}$ the incoming
and outgoing three-momenta. The OPE potential
$V_{\text{OPE}}^{J}(E,p,p^{\prime})$ will be discussed in the next subsection.

To describe the $\Lambda_{b}^{0}\to J/\psi pK^{-}$ decay, one needs the bare
weak production vertices as well. In this work, we only consider the $S$-wave
$\Sigma_{c}^{(*)}\bar{D}^{(*)}$ bare production vertices for the
$\Lambda_{b}^{0}\to\Sigma_{c}^{(*)}\bar{D}^{(*)}K^{-}$ process, since the
$\Sigma_{c}^{(*)}\bar{D}^{(*)}$ thresholds are close to the energy region of
interest. It is stated in Ref. Burns and Swanson (2019) that the decay of the
$\Lambda_{b}$ into a spectator and $\Sigma_{c}^{(*)}\bar{D}^{(*)}$ is
suppressed compared to that of $\Lambda_{c}\bar{D}^{(*)}$ due to isospin
breaking or color suppression in the isospin conserving case. However, the
color suppression factor is hard to quantify. In fact, with the $b$ quark
replaced by a $c$ quark, were the color suppression effective, one would
expect that the branching ratio of $\Lambda_{c}^{+}\to\Sigma^{0}K^{+}$ should
be suppressed compared to that of $\Lambda_{c}^{+}\to\Lambda K^{+}$, which,
however, is in conflict to data, since these branching ratios are almost equal
Zyla  _et al._ (2020). Reversely, this might suggest that the production
strength of $\Lambda_{b}^{0}\to\Sigma_{c}^{(*)}\bar{D}^{(*)}K^{-}$ could be
comparable to that of $\Lambda_{b}^{0}\to\Lambda_{c}\bar{D}^{(*)}K^{-}$. We
take this as the motivation to neglect the $\Lambda_{c}\bar{D}^{(*)}K^{-}$
bare production vertex, since in the molecular picture advocated here the
pentaquarks couple most strongly to the $\Sigma_{c}^{(*)}\bar{D}^{(*)}$
channels. Also, it was shown in Ref. Dong  _et al._ (2020) based on an EFT, if
the system contains only two channels, that a dip must appear near the elastic
threshold in the inelastic line shape if the elastic-channel interaction is
strongly attractive and the production process goes dominantly through
inelastic channels. On the other hand, data in the production channels driven
by elastic channels should have a near-threshold peak. The fact that there are
no obvious hints of dips in the vicinity of the
$\Sigma_{c}^{(*)}\bar{D}^{(*)}$ thresholds may then be regarded as yet another
indication for the dominance of the elastic production mechanisms in the
pentaquark system.

The weak bare production matrix elements can also be parametrized in terms of
the $|s_{Q}\otimes j_{\ell}\rangle$ basis, i.e.,
$\mathcal{F}_{n}^{J}=\langle\Lambda_{b}|\hat{\mathcal{H}}_{W}|(s_{Q}\otimes
j_{\ell})_{n}^{J}K^{-}\rangle$, where $(s_{Q}\otimes j_{\ell})_{n}^{J}$ refers
to the $n$th state in the $|s_{Q}\otimes j_{\ell}\rangle$ basis in Eqs.
(16)-(30). With the seven parameters $\mathcal{F}_{n}^{J}$ in total Du  _et
al._ (2020a), the bare weak production amplitude for the $\alpha$th elastic
channel for a given $J$ reads

$\displaystyle P_{\alpha}^{J}=\sum_{n}R_{\alpha n}^{J}\mathcal{F}_{n}^{J}.$
(50)

Therefore, the bare weak production amplitudes for
$\Lambda_{b}^{0}\to\Sigma_{c}^{(*)}\bar{D}^{(*)}K^{-}$ with the $S$ wave
$\Sigma_{c}^{(*)}\bar{D}^{(*)}$ in $J=\frac{1}{2}$, $\frac{3}{2}$, and
$\frac{5}{2}$ read

$\displaystyle P^{\frac{1}{2}}$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{1}{2}\mathcal{F}_{1}^{\frac{1}{2}}+\frac{1}{2\sqrt{3}}\mathcal{F}_{2}^{\frac{1}{2}}+\sqrt{\frac{2}{3}}\mathcal{F}_{3}^{\frac{1}{2}},\frac{1}{2\sqrt{3}}\mathcal{F}_{1}^{\frac{1}{2}}+\frac{5}{6}\mathcal{F}_{2}^{\frac{1}{2}}-\frac{\sqrt{2}}{3}\mathcal{F}_{3}^{\frac{1}{2}},\sqrt{\frac{2}{3}}\mathcal{F}_{1}^{\frac{1}{2}}-\frac{\sqrt{2}}{3}\mathcal{F}_{2}^{\frac{1}{2}}-\frac{1}{3}\mathcal{F}_{3}^{\frac{1}{2}}\bigg{)}^{T},$
$\displaystyle P^{\frac{3}{2}}$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{1}{\sqrt{3}}\mathcal{F}_{1}^{\frac{3}{2}}-\frac{1}{3}\mathcal{F}_{2}^{\frac{3}{2}}+\frac{\sqrt{5}}{3}\mathcal{F}_{3}^{\frac{3}{2}},-\frac{1}{2}\mathcal{F}_{1}^{\frac{3}{2}}+\frac{1}{\sqrt{3}}\mathcal{F}_{2}^{\frac{3}{2}}+\frac{1}{2}\sqrt{\frac{5}{3}}\mathcal{F}_{3}^{\frac{3}{2}},\frac{1}{2}\sqrt{\frac{5}{3}}\mathcal{F}_{1}^{\frac{3}{2}}+\sqrt{\frac{5}{3}}\mathcal{F}_{2}^{\frac{3}{2}}-\frac{1}{6}\mathcal{F}_{3}^{\frac{3}{2}}\bigg{)}^{T},$
$\displaystyle P^{\frac{5}{2}}$ $\displaystyle=$
$\displaystyle\mathcal{F}_{1}^{\frac{5}{2}}.$ (51)

###  II.2 One-pion-exchange potentials

The LO OPE potential can be obtained using the effective Lagrangian for the
axial coupling of the pions to the charmed mesons and baryons Wise (1992); Yan
_et al._ (1992),

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{g_{1}}{4}\langle\sigma\cdot
u_{ab}\bar{H}_{b}\bar{H}_{a}^{\dagger}\rangle+ig_{2}\epsilon_{ijk}S^{i{\dagger}}_{ab}u^{j}_{bc}S^{k}_{ca}-\frac{1}{\sqrt{2}}g_{3}\big{(}S_{ab}^{i{\dagger}}u_{bc}^{i}T_{ca}+T^{\dagger}_{ab}u_{bc}^{i}S^{i}_{ca}\big{)},$
(52)

where $\langle\cdot\rangle$ denotes the trace in the spinor space and $\sigma$
represents the Pauli matrices. The subindices $a$, $b$, $c$ denote the light-
flavor content, and the superindices $i$, $j$, $k$ denote the polarization.
$S^{i}$ and $\bar{H}$ are the heavy quark spin doublets for the ground states
$(\Sigma_{c},\Sigma_{c}^{*})$ and $(\bar{D},\bar{D}^{*})$ Manohar and Wise
(2000),

$\displaystyle
S^{i}=\frac{1}{\sqrt{3}}\sigma^{i}\Sigma_{c}+\Sigma^{*i}_{c},\qquad\bar{H}=-\bar{D}+\sigma\cdot\bar{D}^{*},$
(53)

with

$\displaystyle
S_{c}^{(*)}=\begin{pmatrix}\Sigma_{c}^{(*)++}&\frac{1}{\sqrt{2}}\Sigma_{c}^{(*)+}\\\
\frac{1}{\sqrt{2}}\Sigma_{c}^{(*)+}&\Sigma_{c}^{(*)0}\\\
\end{pmatrix},\quad\quad\bar{D}^{(*)}=\begin{pmatrix}\bar{D}^{(*)0}\\\
D^{(*)-}\\\ \end{pmatrix}.$ (54)

The $\Lambda_{c}^{+}$ is described by the spinor field $T$,

$\displaystyle T=\begin{pmatrix}0&\frac{1}{\sqrt{2}}\Lambda_{c}^{+}\\\
-\frac{1}{\sqrt{2}}\Lambda_{c}^{+}&0\\\ \end{pmatrix},$ (55)

and the pions are collected in
$\bm{u}=-\nabla\Phi/F_{\pi}+\mathcal{O}(\Phi^{3})$, where
$\Phi=\bm{\tau}\cdot\bm{\pi}$ with $\bm{\tau}$ and $\bm{\pi}$ the Pauli
matrices in the isospin space and the pion fields, and $F_{\pi}=92.1$ MeV is
the pion decay constant. From the width of $D^{*+}\to D^{0}\pi^{+}$ one gets
$g_{1}=0.57$, and the couplings $g_{2}=0.42$ and $g_{3}=0.71$ are taken from a
lattice QCD calculation Detmold  _et al._ (2012). The signs of these couplings
are fixed from the lattice results Detmold  _et al._ (2012).

Once the OPE is considered, the corresponding tensor force, whose importance
is well-known for few-nucleon systems, generates a mixture between $S$ waves
and $D$ waves, which can have a sizable impact on the line shapes between
thresholds Baru  _et al._ (2017); Wang  _et al._ (2018); Baru  _et al._
(2019). The quantum numbers of the $\Sigma_{c}^{(*)}\bar{D}^{(*)}$ and
$\Lambda_{c}\bar{D}^{(*)}$ systems taken into account in this work are listed
in Table 1.

Table 1: Coupled channels for both $S$\- and $D$-wave $\Sigma_{c}^{(\ast)}\bar{D}^{(\ast)}$, $\Lambda_{c}\bar{D}^{(*)}$ systems considered in the present work. Whenever the two particles can couple to different spins, we use a subindex to denote their total spin. The states given in this table appear as basis vectors for the potentials discussed in the text.  $J^{P}$ |  $S$ wave |  $D$ wave
---|---|---
$\bigg{(}\dfrac{1}{2}\bigg{)}^{-}$ |  $\Sigma_{c}\bar{D}$, $\Sigma_{c}\bar{D}^{\ast}$,$\Sigma_{c}^{\ast}\bar{D}^{\ast}$, $\Lambda_{c}\bar{D}$, $\Lambda_{c}\bar{D}^{*}$ |  $\Sigma_{c}\bar{D}^{\ast}_{\frac{3}{2}}$, $\Sigma_{c}^{\ast}\bar{D}$, $\Sigma_{c}^{\ast}\bar{D}^{\ast}_{\frac{3}{2}}$, $\Sigma_{c}^{\ast}\bar{D}^{\ast}_{\frac{5}{2}}$, $\Lambda_{c}\bar{D}^{*}_{\frac{3}{2}}$
$\bigg{(}\dfrac{3}{2}\bigg{)}^{-}$ |  $\Sigma_{c}\bar{D}^{\ast}$, $\Sigma_{c}^{\ast}\bar{D}$, $\Sigma_{c}^{\ast}\bar{D}^{\ast}$, $\Lambda_{c}\bar{D}^{*}$ |  $\Sigma_{c}\bar{D}$, $\Sigma_{c}\bar{D}^{\ast}_{\frac{1}{2}}$, $\Sigma_{c}\bar{D}^{\ast}_{\frac{3}{2}}$, $\Sigma_{c}^{\ast}\bar{D}$, $\Sigma_{c}^{\ast}\bar{D}^{\ast}_{\frac{1}{2}}$, $\Sigma_{c}^{\ast}\bar{D}^{\ast}_{\frac{3}{2}}$, $\Sigma_{c}^{\ast}\bar{D}^{\ast}_{\frac{5}{2}}$, $\Lambda_{c}\bar{D}$, $\Lambda_{c}\bar{D}^{*}_{\frac{1}{2}}$, $\Lambda_{c}\bar{D}^{*}_{\frac{3}{2}}$
$\bigg{(}\dfrac{5}{2}\bigg{)}^{-}$ | $\Sigma_{c}^{\ast}\bar{D}^{\ast}$ |  $\Sigma_{c}\bar{D}$, $\Sigma_{c}\bar{D}^{\ast}_{\frac{1}{2}}$, $\Sigma_{c}\bar{D}^{\ast}_{\frac{3}{2}}$, $\Sigma_{c}^{\ast}\bar{D}$, $\Sigma_{c}^{\ast}\bar{D}^{\ast}_{\frac{1}{2}}$, $\Sigma_{c}^{\ast}\bar{D}^{\ast}_{\frac{3}{2}}$, $\Sigma_{c}^{\ast}\bar{D}^{\ast}_{\frac{5}{2}}$, $\Lambda_{c}\bar{D}$, $\Lambda_{c}\bar{D}^{*}_{\frac{1}{2}}$, $\Lambda_{c}\bar{D}^{*}_{\frac{3}{2}}$

All pentaquarks so far were observed in the $J/\psi p$ final state and
therefore have isospin 1/2. The corresponding isospin wave functions can
therefore be related to the particle channels via

$\displaystyle
P_{c}^{+}=\sqrt{\frac{2}{3}}\Sigma_{c}^{(*)++}D^{(*)-}-\sqrt{\frac{1}{3}}\Sigma_{c}^{(*)+}\bar{D}^{(*)0}.$
(56)

Starting from the potentials

$\displaystyle V$
${}^{\text{\tiny{OPE}}}_{\Sigma_{c}^{(*)+}\bar{D}^{(*)0}\to\Sigma_{c}^{(*)+}\bar{D}^{(*)0}}=0,$
(57) $\displaystyle V$
${}^{\text{\tiny{OPE}}}_{\Sigma_{c}^{(*)++}D^{(*)-}\to\Sigma_{c}^{(*)++}D^{(*)-}}=-\frac{1}{\sqrt{2}}V^{\text{\tiny{OPE}}}_{\Sigma_{c}^{(*)+}\bar{D}^{(*)0}\to\Sigma_{c}^{(*)++}D^{(*)-}}=-\frac{1}{\sqrt{2}}V^{\text{\tiny{OPE}}}_{\Sigma_{c}^{(*)++}D^{(*)-}\to\Sigma_{c}^{(*)+}\bar{D}^{(*)0}},$
$\displaystyle V$
${}^{\text{\tiny{OPE}}}_{\Sigma_{c}^{(*)++}D^{(*)-}\to\Lambda_{c}^{+}\bar{D}^{(*)0}}=-\sqrt{2}V^{\text{\tiny{OPE}}}_{\Sigma_{c}^{(*)+}\bar{D}^{(*)0}\to\Lambda_{c}^{+}\bar{D}^{(*)0}},$

one thus gets

$\displaystyle
V_{\Sigma_{c}^{(*)}\bar{D}^{(*)}\to\Sigma_{c}^{(*)}\bar{D}^{(*)}}^{\text{\tiny{OPE}}}$
$\displaystyle=$ $\displaystyle
2V^{\text{\tiny{OPE}}}_{\Sigma_{c}^{(*)++}D^{(*)-}\to\Sigma_{c}^{(*)++}D^{(*)-}},$
$\displaystyle
V^{\text{\tiny{OPE}}}_{\Sigma_{c}^{(*)}\bar{D}^{(*)}\to\Lambda_{c}\bar{D}^{(*)}}$
$\displaystyle=$
$\displaystyle\sqrt{\frac{3}{2}}V^{\text{\tiny{{OPE}}}}_{\Sigma_{c}^{(*)++}D^{(*)-}\to\Lambda_{c}^{+}\bar{D}^{(*)0}}.$
(58)

In the framework of time-ordered-perturbation theory (TOPT), the OPE
potentials acquire two parts corresponding to the two contributions in Fig. 1,
where the vertical lines indicate the intermediate three-body states.

Figure 1: Diagrams in TOPT responsible for the two contributions to the OPE
potential. The solid line represents the $\Sigma_{c}^{(*)}$, $\Lambda_{c}$ and
$\bar{D}^{(*)}$, the dashed line denotes the pion field.

In some cases, for example, for the
$\Sigma_{c}\bar{D}\to\Lambda_{c}\bar{D}^{*}$ transition, the intermediate
state $\Lambda_{c}\bar{D}\pi$ can be on shell in the energy region where all
$P_{c}$ states reside. Thus, the effect of the three-body cut has to be taken
into account in the potential and in the $\Sigma_{c}^{(*)}$ self-energy loops,
see Eqs. (92)-(98) in Appendix A for the relevant potentials and Sec. III for
the self-energy loop.

###  II.3 Next-to-leading-order contact terms

In Subsection II.1, we construct the LO contact potentials using the heavy-
light spin structure decomposition $|s_{Q}\otimes j_{\ell}\rangle$.
Alternatively, one can also obtain those potentials using an effective
Lagrangian. For the $\Sigma_{c}^{(*)}\bar{D}^{(*)}$ and
$\Lambda_{c}\bar{D}^{(*)}$ channels, the effective Lagrangian reads Liu  _et
al._ (2018, 2019b); Sakai  _et al._ (2019)

$\displaystyle\mathcal{L}_{\text{LO}}$ $\displaystyle=$ $\displaystyle-
C_{a}\bm{S}^{\dagger}_{ab}\cdot\bm{S}_{ba}\langle\bar{H}_{c}^{\dagger}\bar{H}_{c}\rangle-
C_{b}i\epsilon_{jik}S_{ab}^{j{\dagger}}S_{ba}^{k}\langle\bar{H}_{c}^{\dagger}\sigma^{i}\bar{H}_{c}\rangle$
(59)
$\displaystyle+C_{c}\Big{(}S_{ab}^{i{\dagger}}T_{ca}\langle\bar{H}_{c}^{\dagger}\sigma^{i}\bar{H}_{b}\rangle-T^{\dagger}_{ca}S_{ab}^{i}\langle\bar{H}^{\dagger}_{b}\sigma^{i}\bar{H}_{c}\rangle\Big{)}-C_{d}T^{\dagger}_{ab}T_{ba}\langle\bar{H}_{c}^{\dagger}\bar{H}_{c}\rangle,$

where the contact potentials $C_{a}$, $C_{b}$, $C_{c}$ and $C_{d}$ are related
to those in Eqs. (39)-(45) as

$\displaystyle C_{a}$
$\displaystyle=\frac{1}{3}C_{\frac{1}{2}}+\frac{2}{3}C_{\frac{3}{2}},\qquad
C_{b}$ $\displaystyle=\frac{1}{3}(C_{\frac{3}{2}}-C_{\frac{1}{2}}),$
$\displaystyle C_{c}$
$\displaystyle=\frac{2}{3}C^{\prime}_{\frac{1}{2}},\qquad\qquad C_{d}$
$\displaystyle=C_{\frac{1}{2}}^{\prime\prime}.$ (60)

The transitions between the elastic and inelastic channels in Eqs. (II.1) can
also be obtained using the effective Lagrangian respecting the HQSS Sakai  _et
al._ (2019):

$\displaystyle\mathcal{L}_{\text{ine}}=\frac{g_{S}}{\sqrt{3}}N^{\dagger}\sigma^{i}\bar{H}J^{\dagger}S^{i}-\sqrt{3}g_{D}N^{\dagger}\sigma^{i}\bar{H}(\partial^{i}\partial^{j}-\frac{1}{3}\delta^{ij}\partial^{2})J^{\dagger}S^{j},$
(61)

where $N$ denotes the nucleon doublet, and
$J=-\eta_{c}+\bm{\sigma}\cdot\bm{\psi}$ contains the ground state charmonium
fields.

It was noticed in Refs. Baru  _et al._ (2017, 2016) for $B^{(*)}\bar{B}^{*}$
scattering that the NLO contact terms which contribute to the $S$-$D$
transitions need to be promoted to LO to tame the ultraviolet (UV) divergence
associated with the $S$-$D$ OPE potentials in the course of iterations. The
$S$-$S$ NLO contact terms, however, play a subleading role resulting only in a
marginal improvement in the fits to the $Z_{b}$ line shapes. In this work, we
proceed analogously, and we also find that once the full OPE potential is
included dynamically the $S$-$D$ NLO contact terms are required to obtain
regulator-independent results. The $S$-$S$ NLO contact terms are expected to
play a marginal role also in the pentaquark system. More details on this will
be given in Sec. IV.2. To this end, we construct the NLO effective Lagrangian

$\displaystyle\mathcal{L}_{\text{NLO}}$ $\displaystyle=$ $\displaystyle-
D_{a}^{SS}\Big{(}\partial^{i}\bm{S}^{\dagger}_{ab}\cdot\bm{S}_{ba}\langle\partial^{i}\bar{H}_{c}^{\dagger}\bar{H}_{c}\rangle+\bm{S}^{\dagger}_{ab}\cdot\partial^{i}\bm{S}_{ba}\langle\bar{H}_{c}^{\dagger}\partial^{i}\bar{H}_{c}\rangle\Big{)}$
(62) $\displaystyle-
D_{b}^{SS}i\epsilon_{jik}\Big{(}\partial^{\ell}S_{ab}^{j{\dagger}}S_{ba}^{k}\langle\partial^{\ell}\bar{H}_{c}^{\dagger}\sigma^{i}\bar{H}_{c}\rangle+S_{ab}^{j{\dagger}}\partial^{\ell}S_{ba}^{k}\langle\bar{H}_{c}^{\dagger}\sigma^{i}\partial^{\ell}\bar{H}_{c}\rangle\Big{)}$
$\displaystyle-
D_{b}^{SD}i\epsilon_{jik}\bigg{[}\partial_{i}S^{\dagger}_{j}S_{k}\langle\partial_{\ell}\bar{H}^{\dagger}\sigma_{\ell}\bar{H}\rangle+\partial_{\ell}S^{\dagger}_{j}S_{k}\langle\partial_{i}\bar{H}^{\dagger}\sigma_{\ell}\bar{H}\rangle-\frac{2}{3}\partial_{\ell}S^{\dagger}_{j}S_{k}\langle\partial_{\ell}\bar{H}^{\dagger}\sigma_{i}\bar{H}\rangle$
$\displaystyle+S^{\dagger}_{j}\partial_{i}S_{k}\langle\bar{H}^{\dagger}\sigma_{\ell}\partial_{\ell}\bar{H}\rangle+S^{\dagger}_{j}\partial_{\ell}S_{k}\langle\bar{H}^{\dagger}\sigma_{\ell}\partial_{i}\bar{H}\rangle-\frac{2}{3}S^{\dagger}_{j}\partial_{\ell}S_{k}\langle\bar{H}^{\dagger}\partial_{\ell}\sigma_{i}\bar{H}\rangle\bigg{]}$
$\displaystyle+\frac{2}{3}\sqrt{2}D_{c}^{SD}\bigg{[}\partial^{i}S^{i{\dagger}}_{ab}T_{ca}\langle\partial^{j}\bar{H}_{c}^{\dagger}\sigma^{j}\bar{H}_{b}\rangle+\partial^{j}S^{i{\dagger}}_{ab}T_{ca}\langle\partial^{i}\bar{H}_{c}^{\dagger}\sigma^{j}\bar{H}_{b}\rangle-\frac{2}{3}\partial^{j}S^{i{\dagger}}_{ab}T_{ca}\langle\partial^{j}\bar{H}_{c}^{\dagger}\sigma^{i}\bar{H}_{b}\rangle$
$\displaystyle-\partial^{i}T^{\dagger}_{ca}S_{ab}^{i}\langle\partial^{j}\bar{H}_{b}^{\dagger}\sigma^{j}\bar{H}_{c}\rangle-\partial^{j}T^{\dagger}_{ca}S_{ab}^{i}\langle\partial^{i}\bar{H}_{b}^{\dagger}\sigma^{j}\bar{H}_{c}\rangle+\frac{2}{3}\partial^{j}T^{\dagger}_{ca}S_{ab}^{i}\langle\partial^{j}\bar{H}_{b}^{\dagger}\sigma^{i}\bar{H}_{c}\rangle$
$\displaystyle+S^{i{\dagger}}_{ab}\partial^{i}T_{ca}\langle\bar{H}_{c}^{\dagger}\sigma^{j}\partial^{j}\bar{H}_{b}\rangle+S^{i{\dagger}}_{ab}\partial^{j}T_{ca}\langle\bar{H}_{c}^{\dagger}\sigma^{j}\partial^{i}\bar{H}_{b}\rangle-\frac{2}{3}S^{i{\dagger}}_{ab}\partial^{j}T_{ca}\langle\bar{H}_{c}^{\dagger}\sigma^{i}\partial^{j}\bar{H}_{b}\rangle$
$\displaystyle-T^{\dagger}_{ca}\partial^{i}S_{ab}^{i}\langle\bar{H}_{b}^{\dagger}\sigma^{j}\partial^{j}\bar{H}_{c}\rangle-T^{\dagger}_{ca}\partial^{j}S_{ab}^{i}\langle\bar{H}_{b}^{\dagger}\sigma^{j}\partial^{i}\bar{H}_{c}\rangle+\frac{2}{3}T^{\dagger}_{ca}\partial^{j}S_{ab}^{i}\langle\bar{H}_{b}^{\dagger}\sigma^{i}\partial^{j}\bar{H}_{c}\rangle\bigg{]}$
$\displaystyle+D_{d}^{SS}\Big{(}\partial^{i}T_{ab}^{\dagger}T_{ba}\langle\partial^{i}\bar{H}_{c}^{\dagger}\bar{H}_{c}\rangle+T_{ab}^{\dagger}\partial^{i}T_{ba}\langle\bar{H}_{c}^{\dagger}\partial^{i}\bar{H}_{c}\rangle\Big{)},$

where the $D_{a(b,d)}^{SS}$ and the $D_{b(c)}^{SD}$ are responsible for the
contact $S$-$S$ and $S$-$D$ transitions, respectively. Specifically,
$D_{b}^{SD}$ contributes to the $O(Q^{2})$ $S$-$D$ transitions
$\Sigma_{c}^{(*)}\bar{D}^{(*)}\to\Sigma_{c}^{(*)}\bar{D}^{(*)}$, while
$D_{c}^{SD}$ to $\Sigma_{c}^{(*)}\bar{D}^{(*)}\to\Lambda_{c}\bar{D}^{(*)}$.
The NLO contact potential matrix, on the basis of the $S$ and $D$ waves, for
the scattering process $12\to 1^{\prime}2^{\prime}$ with $p$ and $p^{\prime}$
for the incoming and outgoing three-momenta, respectively, reads

$\displaystyle
V_{\text{NLO}}^{J}(p,p^{\prime})=\begin{pmatrix}(p^{2}+p^{\prime
2})V_{SS}^{J}&p^{\prime 2}V_{SD}^{J}\\\
p^{2}\big{(}V_{SD}^{J}\big{)}^{T}&0\end{pmatrix}.$ (63)

The potential $V_{SD}^{J}$ for the channels listed in Table 1 can be written
in the matrix form (where the columns and rows are given for the channels in
order they appear in the table) as

$\displaystyle V_{SD}^{\frac{1}{2}}$ $\displaystyle=$
$\displaystyle-\frac{128}{3}\left(\begin{array}[]{ccccc}\frac{D_{b}^{SD}}{4\sqrt{6}}&0&-\frac{D_{b}^{SD}}{8\sqrt{30}}&-\frac{1}{8}\sqrt{\frac{3}{10}}D_{b}^{SD}&\frac{D_{c}^{SD}}{8\sqrt{3}}\\\
-\frac{D_{b}^{SD}}{12\sqrt{2}}&-\frac{D_{b}^{SD}}{8\sqrt{6}}&\frac{D_{b}^{SD}}{6\sqrt{10}}&-\frac{D_{b}^{SD}}{8\sqrt{10}}&-\frac{D_{c}^{SD}}{24}\\\
\frac{D_{b}^{SD}}{48}&-\frac{D_{b}^{SD}}{16\sqrt{3}}&\frac{7D_{b}^{SD}}{48\sqrt{5}}&\frac{D_{b}^{SD}}{8\sqrt{5}}&-\frac{D_{c}^{SD}}{24\sqrt{2}}\\\
\frac{D_{c}^{SD}}{8\sqrt{3}}&0&\frac{D_{c}^{SD}}{8\sqrt{15}}&\frac{1}{8}\sqrt{\frac{3}{5}}D_{c}^{SD}&0\\\
-\frac{D_{c}^{SD}}{24}&\frac{D_{c}^{SD}}{8\sqrt{3}}&-\frac{D_{c}^{SD}}{6\sqrt{5}}&\frac{D_{c}^{SD}}{8\sqrt{5}}&0\\\
\end{array}\right),$ (69) $\displaystyle V_{SD}^{\frac{3}{2}}$
$\displaystyle=$
$\displaystyle-\frac{128}{3}\left(\begin{array}[]{cccccccccc}-\frac{D_{b}^{SD}}{8\sqrt{3}}&\frac{D_{b}^{SD}}{24}&-\frac{D_{b}^{SD}}{12}&\frac{D_{b}^{SD}}{16\sqrt{3}}&-\frac{D_{b}^{SD}}{48\sqrt{2}}&-\frac{D_{b}^{SD}}{48\sqrt{5}}&\frac{1}{16}\sqrt{\frac{7}{10}}D_{b}^{SD}&-\frac{D_{c}^{SD}}{8\sqrt{6}}&\frac{D_{c}^{SD}}{24\sqrt{2}}&-\frac{D_{c}^{SD}}{12\sqrt{2}}\\\
0&\frac{D_{b}^{SD}}{16\sqrt{3}}&\frac{D_{b}^{SD}}{16\sqrt{3}}&0&\frac{D_{b}^{SD}}{16\sqrt{6}}&\frac{D_{b}^{SD}}{4\sqrt{15}}&\frac{1}{16}\sqrt{\frac{21}{10}}D_{b}^{SD}&0&-\frac{D_{c}^{SD}}{8\sqrt{6}}&-\frac{D_{c}^{SD}}{8\sqrt{6}}\\\
\frac{D_{b}^{SD}}{16\sqrt{15}}&-\frac{D_{b}^{SD}}{12\sqrt{5}}&-\frac{D_{b}^{SD}}{48\sqrt{5}}&\frac{D_{b}^{SD}}{4\sqrt{15}}&-\frac{7D_{b}^{SD}}{48\sqrt{10}}&-\frac{D_{b}^{SD}}{15}&\frac{1}{80}\sqrt{\frac{7}{2}}D_{b}^{SD}&-\frac{D_{c}^{SD}}{8\sqrt{30}}&\frac{D_{c}^{SD}}{6\sqrt{10}}&\frac{D_{c}^{SD}}{24\sqrt{10}}\\\
-\frac{D_{c}^{SD}}{8\sqrt{6}}&\frac{D_{c}^{SD}}{24\sqrt{2}}&-\frac{D_{c}^{SD}}{12\sqrt{2}}&-\frac{D_{c}^{SD}}{8\sqrt{6}}&\frac{D_{c}^{SD}}{48}&\frac{D_{c}^{SD}}{24\sqrt{10}}&-\frac{1}{16}\sqrt{\frac{7}{5}}D_{c}^{SD}&0&0&0\\\
\end{array}\right),$ (74) $\displaystyle V_{SD}^{\frac{5}{2}}$
$\displaystyle=$
$\displaystyle-\frac{128}{3}\left(\begin{array}[]{cccccccccc}-\frac{D_{b}^{SD}}{8\sqrt{10}}&-\frac{D_{b}^{SD}}{8\sqrt{30}}&-\frac{D_{b}^{SD}}{16}\sqrt{\frac{7}{15}}&-\frac{D_{b}^{SD}}{16}\sqrt{\frac{7}{5}}&\frac{D_{b}^{SD}}{8\sqrt{15}}&-\frac{D_{b}^{SD}}{80}\sqrt{\frac{7}{3}}&-\frac{D_{b}^{SD}}{20}\sqrt{\frac{7}{2}}&\frac{D_{c}^{SD}}{8\sqrt{5}}&\frac{D_{c}^{SD}}{8\sqrt{15}}&\frac{D_{c}^{SD}}{8}\sqrt{\frac{7}{30}}\\\
\end{array}\right).$ (76)

By comparing to the $S$-$D$ OPE potentials in Eqs. (92)-(98), it is easy to
see that the NLO $S$-$D$ contact potential has the same structure as the
corresponding OPE potentials. Indeed, the coefficients for the
$\Sigma_{c}^{(*)}\bar{D}^{(*)}\to\Sigma_{c}^{(*)}\bar{D}^{(*)}$ transitions
are exactly the same, while those for the transitions
$\Sigma_{c}^{(*)}\bar{D}^{(*)}\to\Lambda_{c}\bar{D}^{(*)}$ differ by an
overall coefficient, which can be absorbed into a redefined $D_{c}^{SD}$. The
same holds for the structure of $V_{SS}^{J}$ compared to the LO contact
potentials in Eqs. (39)-(45). Thus, $V_{SS}^{J}$ is not shown explicitly here.

##  III Lippmann-Schwinger equation with dynamic width of $\Sigma_{c}^{(*)}$

Figure 2: Diagram for the $\Lambda_{b}^{0}\to J/\psi pK^{-}$ decay through
$\Lambda_{b}^{0}\to\Sigma_{c}^{(*)}\bar{D}^{(*)}K^{-}$ following the final
state interactions.

With the above ingredients, one can obtain the production amplitude satisfying
unitarity by solving the LSEs. A diagrammatic representation thereof is shown
in Fig. 2. Here we treat the pion relativistically and employ the relativistic
mass-energy relations for $\Sigma_{c}^{(*)}$, $\bar{D}^{(*)}$ and
$\Lambda_{c}$. To be consistent, we employ the LSEs with relativistic Green
functions. The $T$-matrix for
$\Sigma_{c}^{(*)}\bar{D}^{(*)}\to\Sigma_{c}^{(*)}\bar{D}^{(*)}$ scattering is
given by the LSE,

$\displaystyle
T_{\alpha\gamma}^{J}(E,p,p^{\prime})=V_{\alpha\gamma}^{J}(E,p,p^{\prime})-\sum_{\beta}\int\frac{d^{3}\bm{q}}{(2\pi)^{3}}V_{\alpha\beta}^{J}(E,p,q)G_{\beta}(E,q)T_{\beta\gamma}^{J}(E,q,p^{\prime}),$
(77)

where the two-body propagator reads Baru  _et al._ (2015)

$\displaystyle
G_{\beta}(E,\bm{q})=\frac{m_{\Sigma_{c}^{(*)}}m_{D^{(*)}}}{E_{\Sigma_{c}^{(*)}}(\bm{q})E_{D^{(*)}}(\bm{q})}\frac{1}{E_{\Sigma_{c}^{(*)}}(\bm{q})+E_{D^{(*)}}(\bm{q})-E-{\tilde{\Sigma}_{R}(s,m_{\Sigma_{c}^{(*)}})}/{(2E_{\Sigma_{c}^{(*)}}(\bm{q})})},$
(78)

with

$\displaystyle s=\big{(}E-E_{D^{(*)}}(\bm{q})\big{)}^{2}-\bm{q}^{2}$ (79)

for the off-shell $\Sigma_{c}^{(*)}$ resonance and
$\tilde{\Sigma}_{R}(s,m_{\Sigma_{c}^{(*)}})$ the renormalized self-energy loop
diagram with $E_{i}=\sqrt{\bm{q}^{2}+m_{i}^{2}}$. The self-energy function
$\tilde{\Sigma}_{R}(s,m_{\Sigma_{c}^{(*)}})$ can be evaluated from the diagram
in Fig. 3, since the width of the $\Sigma_{c}^{(*)}$ is almost saturated by
the two-body decay into the $\Lambda_{c}\pi$ Zyla  _et al._ (2020). For
simplicity, here we use a more pragmatic way and consider only the unitarity-
driven part of the loop diagram, which generates a relevant three-body cut
from the $\Lambda_{c}\bar{D}\pi$ threshold. We expect that the principal value
part of the loop can be largely absorbed into renormalization of the
$\Sigma_{c}^{(*)}$ mass. Therefore, the self-energy function reduces to

$\displaystyle\tilde{\Sigma}_{R}(s,m_{\Sigma_{c}^{(*)}})=ig^{2}\frac{p^{3}}{\sqrt{s}},$
(80)

where $p=\lambda^{1/2}(s,m_{\Lambda_{c}}^{2},m_{\pi}^{2})/{2\sqrt{s}}$ with
Källén function $\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2ab-2bc-2ca$ and $g$ is set
to reproduce the physical width of the $\Sigma_{c}^{(*)}$, i.e.,
$\tilde{\Sigma}_{R}(m_{\Sigma_{c}^{(*)}}^{2},m_{\Sigma_{c}^{(*)}})=im_{\Sigma_{c}^{(*)}}\Gamma_{\Sigma_{c}^{(*)}}.$
The effect of the $\bar{D}^{*}$ width can be safely neglected as it is only
several tens of keV, i.e., it is three orders of magnitude smaller than that
of the $\Sigma_{c}^{(*)}$. The widths of the $\bar{D}$ and $\Lambda_{c}$ are
even smaller than that of $\bar{D}^{*}$ since both states decay only via the
weak interaction, and thus are also negligible. Neglecting the energy-
dependence of the self-energy, i.e., employing
$\tilde{\Sigma}_{R}(s,m_{\Sigma_{c}^{(*)}})=im_{\Sigma_{c}^{(*)}}\Gamma_{\Sigma_{c}^{(*)}}$
in the LSEs, the two-body propagator Eq. (78) reduces to the form used in Ref.
Du  _et al._ (2020a) in the nonrelativistic limit.

Figure 3: Loop diagrams of the self-energy for $\Sigma_{c}^{(*)}$.

To describe the $J/\psi p$ mass distribution, we work in terms of the
production amplitudes rather than the scattering amplitudes. The LSE for the
physical production amplitude of the $\alpha$th elastic channel
$U_{\alpha}(E,p)^{J}$ reads

$U^{J}_{\alpha}(E,p){=}P^{J}_{\alpha}{-}\\!\sum_{\beta}\\!\\!\int\\!\\!\frac{d^{3}\bm{q}}{(2\pi)^{3}}V^{J}_{\alpha\beta}(E,p,q)G_{\beta}(E,q)U^{J}_{\beta}(E,q),$
(81)

where $P_{\alpha}$ denotes the bare production vertices in Eq. (II.1). Note
that we only include the $S$-wave $\Sigma_{c}^{(*)}\bar{D}^{(*)}$ production
source in LSE, while the sources from the other channels are neglected, as
discussed above. The source corresponding to production of the $D$-wave
$\Sigma_{c}^{(*)}\bar{D}^{(*)}$ is also neglected due to the centrifugal
barrier as discussed above. As a result of these simplifying assumptions, the
production amplitude, $U_{i}^{(\prime)J}$, for the $i$th $J/\psi p$
($\eta_{c}p$) inelastic channel is simply given by

$\displaystyle
U^{(\prime)J}_{i}(E,k){=}{-}\\!\sum_{\beta}\\!\\!\int\\!\\!\frac{d^{3}\bm{q}}{(2\pi)^{3}}\mathcal{V}^{(\prime)J}_{i\beta}(k)G_{\beta}(E,q)U^{J}_{\beta}(E,q).$
(82)

To render the integrals in the LSEs well defined one needs to introduce a
regulator, with a cutoff larger than all typical three-momenta scales related
to coupled-channel dynamics. In what follows, we will discuss the results
obtained for a hard cutoff $\Lambda$ set to 1 GeV for the pure contact
potentials, namely scheme I, and to 1.3 GeV for the case including OPE
potentials, i.e., schemes II and III, as discussed below. However, we have
checked that the final results barely depend on the cutoff value, when
$\Lambda$ is varied from 0.7 GeV to 1.5 GeV for the contact scheme, and from
1.0 to 1.5 GeV for the schemes including OPE potentials, once the mentioned
$S$-$D$ counter terms are included.

##  IV Fit results

With the decay amplitude from Eq. (82), we are in a position to fit the
$J/\psi p$ invariant mass distribution of the decay $\Lambda_{b}^{0}\to J/\psi
pK^{-}$ by LHCb Aaij  _et al._ (2019). The possible contributions from
misidentified non-$\Lambda_{b}^{0}$ events, the $\Lambda^{*}$ resonances
coupled to the $pK^{-}$, and possibly additional broad $P_{c}^{+}$ structures
are modelled by a smooth incoherent background Du  _et al._ (2020a)

$\displaystyle
f_{\text{bgd}}(E)=b_{0}+b_{1}E^{2}+b_{2}E^{4}+\frac{g_{r}^{2}}{(m-E)^{2}+\Gamma^{2}/4},$
(83)

where $b_{0}$, $b_{1}$, $b_{2}$, $g_{r}$, $m$ and $\Gamma$ are the parameters
to be determined by fitting to the data. In this work, we consider three
different fit schemes:

  * •

scheme I: the LO contact potentials for the $\Sigma_{c}^{(*)}\bar{D}^{(*)}$
channels are implemented including the couplings of the elastic channels to
the $S$\- and $D$-wave $J/\psi p$ and $\eta_{c}p$ channels;

  * •

scheme II: the OPE potentials supplemented by the NLO $S$-$D$ contact terms
for the $\Sigma_{c}^{(*)}\bar{D}^{(*)}$ channels are added to the potential of
I;

  * •

scheme III: the explicit $\Lambda_{c}\bar{D}^{(*)}$ channels are added to
scheme II, including the LO contact potentials, the OPE potentials and the NLO
$S$-$D$ contact terms.

In this analysis, we do not consider isospin symmetry breaking effects, which
can be important to the isospin-breaking decay modes Burns (2015); Guo  _et
al._ (2019), but have little effect on the description of the line shapes in
the isospin-conserving $J/\psi p$ channel. The masses of particles used in the
calculation are collected in Eq. (B) in Appendix B. For all results shown a
Gaussian convolution of the theoretical distributions with the mass resolution
of the experiment Aaij  _et al._ (2019) is employed.

Following the fit schemes introduced above, we improve our previous framework
presented in Ref. Du  _et al._ (2020a), where only the
$\Sigma_{c}^{(*)}\bar{D}^{(*)}$ and $J/\psi p$ channels were considered
explicitly and all other inelastic channels were parametrized as imaginary
parts of the short ranged operators, in several steps.

###  IV.1 Scheme I: Contact potentials

####  IV.1.1 Including the $\eta_{c}p$ channel

The new $\eta_{c}p$ channel comes without additional unknown parameters, since
the transitions between the $\Sigma_{c}^{(*)}\bar{D}^{(*)}$ and $\eta_{c}p$
channels are related to those between $\Sigma_{c}^{(*)}\bar{D}^{(*)}$ and
$J/\psi p$ channels by HQSS, as discussed in Sec. II.1. In addition, a
relativistic form of the two-body propagator Eq. (78) and the energy-
dependence of the width for the $\Sigma_{c}^{(*)}$, Eq. (80), are employed in
this work. As the first step, we do not consider coupled-channel effects from
the $\Lambda_{c}\bar{D}^{(*)}$ channels, i.e., for the fits discussed here we
set both $C_{\frac{1}{2}}^{\prime}$ and $C_{\frac{1}{2}}^{\prime\prime}$ to
zero. From now on we further assume that the widths of the $P_{c}$ states are
saturated by the channels included — in this subsection those are $J/\psi p$,
$\eta_{c}p$, and the elastic channels (including those via the decays of the
$\Sigma_{c}^{(*)}$). Therefore, no additional imaginary contact potentials as
in Ref. Du  _et al._ (2020a) need to be introduced and unitarity is realized.

Three data sets of the $m_{J/\psi p}$ distribution given in Ref. Aaij  _et
al._ (2019) are used in fits. The $m_{J/\psi p}$ distribution with
$m_{Kp}>1.9$ GeV can effectively remove over $80\%$ of the $\Lambda^{*}$
contributions and the $P_{c}$ signals are enhanced in the $m_{J/\psi p}$
distribution by applying $\cos\theta_{P_{c}}$-dependent weights to each event
candidate, with $\theta_{P_{c}}$ the angle between the $K^{-}$ and the
$J/\psi$ in the $P_{c}$ rest frame Aaij  _et al._ (2019). In analogy to what
was reported in Ref. Du  _et al._ (2020a), also with our modified model, two
solutions, called $A$ and $B$, describing the data almost equally well are
found as shown in Fig. 4.

Figure 4: The best fits to the different experimental data Aaij  _et al._ (2019) for both solution $A$ (blue dashed curves) and solution $B$ (red solid curves) of scheme I. The corresponding backgrounds are shown as red-dotted and blue-dotted curves, respectively. The vertical dashed lines from left to right are the $\Sigma_{c}\bar{D}$, $\Sigma_{c}^{*}\bar{D}$, $\Sigma_{c}\bar{D}^{*}$, and $\Sigma_{c}^{*}\bar{D}^{*}$ thresholds, respectively. Table 2: The names of the states, their dominant channels (DCs) and thresholds, the Riemann sheet (RS) where they lie, their quantum numbers found, the pole positions, and the dimensionless couplings to their DCs, $g_{\text{DC}}$, from the $T$-matrix residues for the solutions $A$ and $B$ of scheme I are given. The uncertainties stated result from taking the fit to different data and the different backgrounds used. |  |  |  | solution | $A$ |  | solution | $B$
---|---|---|---|---|---|---|---|---
| DC ([MeV]) | RS | $J^{P}$ | Pole [MeV] | $g_{\text{DC}}$ | $J^{P}$ | Pole [MeV] | $g_{\text{DC}}$
$P_{c}(4312)$ | $\Sigma_{c}\bar{D}\;(4321.6)$ | I | $\frac{1}{2}^{-}$ | $4314(1)-4(1)i$ | $2.6(1)+0.4(2)i$ | $\frac{1}{2}^{-}$ | $4312(2)-4(2)i$ | $2.9(1)+0.4(2)i$
$P_{c}(4380)$ | $\Sigma_{c}^{*}\bar{D}\;(4386.2)$ | II | $\frac{3}{2}^{-}$ | $4377(1)-7(1)i$ | $2.8(1)+0.1(1)i$ | $\frac{3}{2}^{-}$ | $4375(2)-6(1)i$ | $3.0(1)+0.1(1)i$
$P_{c}(4440)$ | $\Sigma_{c}\bar{D}^{*}\;(4462.1)$ | III | $\frac{1}{2}^{-}$ | $4440(1)-9(2)i$ | $3.7(2)+0.6(1)i$ | $\frac{3}{2}^{-}$ | $4441(3)-5(2)i$ | $3.6(1)+0.3(1)i$
$P_{c}(4457)$ | $\Sigma_{c}\bar{D}^{*}\;(4462.1)$ | III | $\frac{3}{2}^{-}$ | $4458(2)-3(1)i$ | $2.1(2)+0.3(1)i$ | $\frac{1}{2}^{-}$ | $4462(4)-5(3)i$ | $2.0(2)+1.2(3)i$
$P_{c}$ | $\Sigma_{c}^{*}\bar{D}^{*}\;(4526.7)$ | IV | $\frac{1}{2}^{-}$ | $4498(2)-9(3)i$ | $4.0(1)+0.4(2)i$ | $\frac{1}{2}^{-}$ | $4526(3)-9(2)i$ | $1.5(2)+1.1(4)i$
$P_{c}$ | $\Sigma_{c}^{*}\bar{D}^{*}\;(4526.7)$ | IV | $\frac{3}{2}^{-}$ | $4510(2)-14(3)i$ | $3.3(2)+0.6(2)i$ | $\frac{3}{2}^{-}$ | $4521(2)-12(3)i$ | $2.5(2)+0.9(2)i$
$P_{c}$ | $\Sigma_{c}^{*}\bar{D}^{*}\;(4526.7)$ | IV | $\frac{5}{2}^{-}$ | $4525(2)-9(3)i$ | $1.9(2)+0.6(7)i$ | $\frac{5}{2}^{-}$ | $4501(3)-6(4)i$ | $3.9(2)+0.1(2)i$

To study the poles in an $n$-channel problem, a multi-sheet Riemann surface in
the complex energy plane is invoked. We only consider the Riemann sheets
associated to the four elastic channels $\Sigma_{c}^{(*)}\bar{D}^{(*)}$, since
all inelastic thresholds are remote and their impact on the poles near the
elastic thresholds is marginal. For this four-elastic-channel case, there are
$2^{4}=16$ Riemann sheets in total. However, only 5 of them are of interest
because they can be reached from the physical region by crossing the unitary
branch cut from $E+i\epsilon$ to $E-i\epsilon$ between the $(n-1)$th and the
$n$th threshold. As a result, the poles located on these Riemann sheets have
more significant impact physical observables than those on the more remote
sheets.333This statement is not necessarily applicable to resonances which
manifest as strong cusp effects, such as those that correspond to poles in the
Riemann sheet reached by crossing the real-$s$ axis right above the previous
threshold, but having a mass above the next one. This is not the case found
for the $P_{c}$ states here discussed. We use the sign of the imaginary part
of ${(T_{\alpha\alpha})}^{-1}$ above the $\alpha$th threshold to label the
Riemann sheets. The $N$th Riemann sheet (RS-N) can be reached from the
physical amplitude $T$ by the analytical continuation in a matrix form,

${(T_{N})}^{-1}={(T)}^{-1}-2i\rho_{N},$ (84)

where $\rho_{N}$ is a diagonal matrix with the first $N-1$ elements equal to
the two-body phase space factors for the corresponding channels and the others
zero, i.e., $\rho_{N}=\text{diag}(\rho_{1},\dots,\rho_{N-1},0,\dots)$ with
$\rho_{i}=m_{\Sigma_{c}^{(*)}}m_{D^{(*)}}q_{\text{cm},i}/(2\pi E)$ and
$q_{\text{cm},i}$ is the c.m. three-momentum in the $i$th channel. Moreover,
one can define the product of the effective couplings $g_{\alpha}g_{\beta}$ of
a given state to channels $\alpha$ and $\beta$ from the residue of the
scattering amplitude $T_{\alpha\beta}(E)$ at the pole of that state, i.e.,

$\displaystyle g_{\alpha}g_{\beta}=\lim_{E\to
E_{\text{pole}}}(E^{2}-E^{2}_{\text{pole}})T_{\alpha\beta}(E).$ (85)

The two solutions produce different values of the parameters, which are given
in Table 4 in Appendix B, in particular $C_{\frac{1}{2}}$ and
$C_{\frac{3}{2}}$, and thus give different pole locations. Both solutions give
seven poles in the $\Sigma_{c}^{(*)}\bar{D}^{(*)}$ amplitudes, i.e., seven
$P_{c}$ states: three with $J=1/2$, three with $J=3/2$ and one with $J=5/2$.
The pole positions, the dominant channels (DCs) having the largest effective
couplings and lying closer to the corresponding pole, and the effective
couplings of those for solutions $A$ and $B$ are listed in Table 2. In both
solutions, among the seven poles, the lowest one corresponds to the
$P_{c}(4312)$ with $J^{P}=\frac{1}{2}^{-}$. It appears as a
$\Sigma_{c}\bar{D}$ bound state, located on the physical Riemann sheet for the
elastic channel and on the unphysical Riemann sheet for the $J/\psi p$ and
$\eta_{c}p$ inelastic channels. It would become a true bound state, if the
$J/\psi p$ and $\eta_{c}p$ channels were switched off. There are two
$\Sigma_{c}\bar{D}^{*}$ bound states with quantum numbers $\frac{1}{2}^{-}$
and $\frac{3}{2}^{-}$, corresponding to the $P_{c}(4440)$ and the
$P_{c}(4457)$, respectively, in solution $A$ and with the quantum numbers
interchanged in solution $B$, as that in Ref. Du  _et al._ (2020a). The
interchange of the spin assignments for the $P_{c}(4440)$ and $P_{c}(4457)$ in
the two solutions is a result of the proximity of the values of
$C_{\frac{1}{2}}$ ($C_{\frac{3}{2}}$) in solution $A$ to those of
$C_{\frac{3}{2}}$ ($C_{\frac{1}{2}}$) in solution $B$, see e.g. Table 4 in
Appendix B. There are three $P_{c}$ states dominated by the
$\Sigma_{c}^{*}\bar{D}^{*}$ channel with quantum numbers $\frac{1}{2}^{-}$,
$\frac{3}{2}^{-}$, and $\frac{5}{2}^{-}$. Analogous to that of the
$\Sigma_{c}\bar{D}^{*}$ channel, the mass pattern of the three $P_{c}$ states
is $m_{\frac{1}{2}^{-}}<m_{\frac{3}{2}^{-}}<m_{\frac{5}{2}^{-}}$ for solution
$A$ and the opposite, i.e.,
$m_{\frac{1}{2}^{-}}>m_{\frac{3}{2}^{-}}>m_{\frac{5}{2}^{-}}$ for solution
$B$. The narrow state $P_{c}(4380)$, required by HQSS, with
$J^{P}=\frac{3}{2}^{-}$ predicted in Refs. Xiao  _et al._ (2019b); Sakai  _et
al._ (2019); Du  _et al._ (2020a); Xiao  _et al._ (2020) is found around 4.38
GeV in both solutions. As a consequence of HQSS, it is as narrow as the other
three $P_{c}$ states (see e.g. Table 2) and different from the broad
$P_{c}(4380)$ reported by LHCb in 2015 Aaij  _et al._ (2015).

####  IV.1.2 Predictions for the line shapes

With the parameters extracted from fits to the $J/\psi p$ channel and under
the assumption that production of the inelastic channels goes through the
elastic ones, i.e., $\Lambda_{b}^{0}\to\Sigma_{c}^{(*)}\bar{D}^{(*)}K^{-}$,
one can predict line shapes for other channels. This is particularly
interesting because the two solutions found here correspond to different
heavy-light spin structure decompositions of the higher $P_{c}$ states. They
are expected to have distinct consequences in their decays into channels other
than the $J/\psi p$ whose $s_{Q}$ is 1. Consequences of HQSS on the decays of
$P_{c}$ states into the $\Lambda_{c}\bar{D}^{(*)}$ and $\eta_{c}p$ considering
only $S$-waves have been reported in Refs. Voloshin (2019); Sakai  _et al._
(2019). The results below on these channels go beyond the previous results in
giving line shapes, instead of ratios of partial widths, and including
$D$-waves.

Among all possible decay channels other than the $J/\psi p$ of the $P_{c}$
states, the $\Sigma_{c}^{(*)}\bar{D}^{(*)}$ are of particular interest since
their thresholds are close to the $P_{c}$ states and are sensitive to their
nature Guo  _et al._ (2015a). One can predict the line shapes of the
$\Sigma_{c}^{(*)}\bar{D}^{(*)}$ mass distributions from the decay
$\Lambda_{b}\to\Sigma_{c}^{(*)}\bar{D}^{(*)}K$. They are shown in Fig. 5. The
existence of $S$-wave bound states below the thresholds leads to several
manifestations in the line shapes. First, one sees the threshold enhancements
for the line shapes of the $\Sigma_{c}^{(*)}\bar{D}^{(*)}$ mass distributions.
The threshold enhancement above the threshold of the $\Sigma_{c}\bar{D}$ is
due to the $P_{c}(4312)$, and that of the $\Sigma_{c}^{*}\bar{D}$ is due to
the narrow $P_{c}(4380)$. Second, the $P_{c}(4440)$ and $P_{c}(4457)$ show
prominent signals in the $\Sigma_{c}\bar{D}$ and $\Sigma_{c}^{*}\bar{D}$
invariant mass distributions only if they couple to these channels in $S$
waves. Indeed, the $J=1/2$ $P_{c}(4440)$ ($P_{c}(4457)$) and the $J=3/2$
$P_{c}(4457)$ ($P_{c}(4440)$ ) are clearly seen in the $\Sigma_{c}\bar{D}$ and
$\Sigma_{c}^{*}\bar{D}$ invariant mass distributions for solution $A$ ($B$),
respectively. These observations can be used to single out the unique solution
from the two once data in these channels become available. Furthermore, since
there are no $J=3/2$ elastic channels below the $\Sigma_{c}^{*}\bar{D}$
threshold, the $J=3/2$ $\Sigma_{c}^{*}\bar{D}$ molecular state $P_{c}(4380)$
can only manifest itself as a threshold enhancement in the
$\Sigma_{c}^{*}\bar{D}$ channel but not as peaks in the other elastic
channels. The higher $P_{c}$ states show up as structures in all channels for
both solutions though the strength of the peaks is quite uncertain.

Figure 5: Predictions for the line shapes of $\Sigma_{c}\bar{D}$ (blue dashed
curves), $\Sigma_{c}^{*}\bar{D}$ (orange dotted curves),
$\Sigma_{c}\bar{D}^{*}$ (green dot-dashed curves), and
$\Sigma_{c}^{*}\bar{D}^{*}$ (red solid curves) mass distributions based on the
fit results in Fig. 4 for solutions $A$ and $B$ of scheme I without the
$\Lambda_{c}\bar{D}^{(*)}$ channels.

With the parameters extracted from fits to the $J/\psi p$ channel and under
the assumption that production of the inelastic channels goes through the
elastic ones, i.e.,
$\Lambda_{b}^{0}\to\Sigma_{c}^{(*)}\bar{D}^{(*)}K^{-}\to\eta_{c}pK^{-}$, one
can also predict line shapes for the $\eta_{c}p$ channels. In Fig. 6, we show
predictions for the $\eta_{c}p$ invariant mass distributions, which also allow
one to make several important observations. First, this inelastic channel
shows quite pronounced signals from all $P_{c}$ states observed by LHCb and
their spin partners predicted in our analysis. Second, the centrifugal-barrier
suppression of the $D$-wave coupling of $J/\psi p$ to the $P_{c}$’s is no
longer active because the $J/\psi p$ threshold is much lower than the relevant
$\Sigma_{c}^{(*)}\bar{D}^{(*)}$ thresholds. Therefore, the coupling of the
elastic channels to $\eta_{c}p$ in the $D$ wave, which is related to that for
$J/\psi p$ via HQSS, is of a comparable strength or even larger than the
$S$-wave coupling, as shown in Table 4 in Appendix B. This observation can be
used to account for the fact that, contrary to the elastic channels, the
$J=3/2$ $P_{c}(4457)$ for solution $A$ with its $D$-wave coupling to
$\eta_{c}p$ is more pronounced in this line shape than the $J=1/2$
$P_{c}(4440)$, though the latter can couple to $\eta_{c}p$ in $S$-wave.
Similarly, the $J=3/2$ $P_{c}(4440)$ coupled to $\eta_{c}p$ in $D$ wave
dominates in the line shape for solution $B$. Thus, the shape of the
$\Sigma_{c}^{*}\bar{D}$ and $\eta_{c}p$ spectra provides a straightforward
criterion for distinguishing between the two solutions of the simplified model
including contact interactions only, in the experiment.

Figure 6:  Predictions for the line shapes of the $\eta_{c}p$ invariant mass
distributions based on the fit results in Fig. 4 for solutions $A$ and $B$ of
scheme I without the $\Lambda_{c}\bar{D}^{(*)}$ channels. The blue dotted,
orange dashed and green dot-dashed curves correspond to the predictions from
the three fit results ($\cos\theta_{P_{c}}$-weighted, $m_{Kp}>1.9$ GeV and
$m_{Kp}$-all) of Fig. 4, in order.

###  IV.2 Scheme II: Including the OPE potential

The importance of the OPE potential is well known for the nucleon interaction
as its tensor force leads to the mixing between the $S$ and $D$ waves and can
leave a significant impact on the line shapes. In this section, we investigate
the role of the OPE for the elastic channels without considering the
$\Lambda_{c}\bar{D}^{(*)}$ channels under the assumption that the widths of
the $P_{c}$ states are saturated by the $J/\psi p$, $\eta_{c}p$ and elastic
channels. We also note that the inclusion of the OPE potential does not
involve any additional parameters, see Sec. II.2 for the discussion of the
coupling constants used, however, it calls for the inclusion of the $S$-$D$
counter terms that come with unknown strengths.

####  IV.2.1 How to renormalize the OPE and the role of $S$-$D$ transitions

Figure 7: Left panel: Best fit results for different cutoffs $\Lambda$ using a
potential of scheme I plus OPE in the elastic channels. Only solution $B$
exists in this case. The background is shown as the orange dot-dashed curve.
Right panel: Predictions for the line shapes of the $\Sigma_{c}\bar{D}$
(blue), $\Sigma_{c}^{*}\bar{D}$ (orange), $\Sigma_{c}\bar{D}^{*}$ (green), and
$\Sigma_{c}^{*}\bar{D}^{*}$ (red) mass distributions.

As noticed in Refs. Baru  _et al._ (2017, 2016), because of the large mass of
the heavy system and the large splittings between the thresholds treated
dynamically, the typical involved momenta are much larger than those for low-
energy two–nucleon scattering, e.g., in the deuteron. In our case, the typical
momenta $p_{\text{typ}}\sim\sqrt{2\mu\delta}$, with $\delta$ and $\mu$ the
largest threshold splitting and the reduced mass of the system, are about
$670~{}\mathrm{MeV}$ when the energy range between the
$\Sigma_{c}^{(*)}\bar{D}^{(*)}$ thresholds is considered and can reach
$900~{}\mathrm{MeV}$ when the energy is extended down to the
$\Lambda_{c}\bar{D}$ threshold. These large typical momenta make the
contribution from the tensor force of the OPE that leads to the $S$-$D$
transitions even more important than that in the $NN$ case, as it was already
pointed out in similar studies of the $X(3872)$ and $Z_{b}$ systems Baru  _et
al._ (2017, 2016). In previous studies, the iteration of such a potential
within the integral equations was shown to yield a strong regulator dependence
for the observable quantities in the $b$-quark sector Wang  _et al._ (2018);
Baru  _et al._ (2019). This is also what we observe in the current study, as
can be seen from the left panel in Fig. 7. In line with Ref. Du  _et al._
(2020a), once the OPE is included, only solution $B$ survives when the cutoff
$\Lambda$ in the LS equations is varied from 0.8 to 1.4 GeV. Although the
$\chi^{2}/\text{dof}$ changes barely with the cutoff variation, a closer look
into the line shapes in Fig. 7 shows visible deviations in the results:
indeed, the peak corresponding to the $P_{c}(4440)$ disappears in the results
with increasing the cutoff. Moreover, the difference in the predicted line
shapes – see the right panel in Fig. 7 – becomes even more pronounced, in line
with the observations of Ref. Baru  _et al._ (2019). In particular, the
disappearance of the $P_{c}(4440)$ in the $J/\psi p$ spectrum for
$\Lambda=1.4$ GeV leads to the same effect in the $\Sigma_{c}\bar{D}$ and
$\Sigma_{c}^{*}\bar{D}$ line shapes in Fig. 7.

Figure 8:  First row: Best fit results for solutions $A$ (left) and $B$
(right) for different cutoffs $\Lambda$ using a potential of scheme II. The
background is shown as red-dotted curve. Second row: Predictions for the line
shapes of $\Sigma_{c}\bar{D}$ (blue), $\Sigma_{c}^{*}\bar{D}$ (orange),
$\Sigma_{c}\bar{D}^{*}$ (green), and $\Sigma_{c}^{*}\bar{D}^{*}$ (red) mass
distributions. Third row: Predictions for the line shapes for the $\eta_{c}p$
mass distributions.

In order to cure the cutoff dependence, caused by the short-range part of the
$S$-$D$ OPE transitions, it was suggested to promote the formally NLO
$O(Q^{2})$ $S$-$D$ contact terms444$Q$ denotes the soft scale for the given
system. to LO Wang  _et al._ (2018); Baru  _et al._ (2019). Here, we follow
the same procedure, namely we include the $S$-$D$ contact terms in the elastic
channels ($D_{b}^{SD}$ terms in Eq. (69)) together with the OPE. In what
follows, this formulation will be referred to as scheme II. As expected, the
line shapes obtained within scheme II demonstrate only mild regulator
dependence, especially for the cutoffs $\Lambda>1$ GeV, which provide a larger
separation between the soft and hard scales. Specifically, our line shapes
obtained from fitting to data in the $J/\psi p$ channel (see the line shapes
in the first row in Fig. 8) are almost cutoff independent. More importantly,
the line shapes predicted in the elastic and inelastic channels also exhibit a
very mild regulator dependence (see the results in the second and third rows
in Fig. 8). We expect that the residual cutoff dependence can be further
reduced if the $S$-wave momentum dependent $O(Q^{2})$ contact terms are added,
in line with a related study of Ref. Baru  _et al._ (2019). Since these
contact terms obviously cannot be fixed on the basis of current data, further
studies of the regulator dependence within scheme II will be postponed to
future work when new data become available.

Since the focus of the discussion above in Sec. IV.2 was put on the
renormalization of the OPE, the results presented in Figs. 7 and 8 were
obtained using some fixed background. It is also worth mentioning that while
the separation of scales calls for larger cutoffs, the cutoff larger than the
$c$-quark mass, $m_{c}\simeq 1.5$ GeV, may introduce additional HQSS breaking
effects.

####  IV.2.2 Description of the data in $\Lambda_{b}^{0}\to J/\psi pK^{-}$

Figure 9: The best fits to the different experimental data Aaij  _et al._
(2019) for both solution $A$ (blue dashed curves) and solution $B$ (red solid
curves) of scheme II. The corresponding backgrounds are shown as red-dotted
and blue-dotted curves, respectively. The vertical dashed lines from left to
right are the $\Sigma_{c}\bar{D}$, $\Sigma_{c}^{*}\bar{D}$,
$\Sigma_{c}\bar{D}^{*}$, and $\Sigma_{c}^{*}\bar{D}^{*}$ thresholds,
respectively.

In Ref. Du  _et al._ (2020a), it was found that the inclusion of the OPE
singles out a unique solution from the two almost equivalent ones present for
the pure contact potentials. However, no effects from the $S$-$D$ contact
terms were taken into account. As long as the OPE is renormalized along the
lines discussed in the previous paragraph, one again arrives at two solutions,
which describe the data almost equally well. This is illustrated in Fig. 9.

A general feature of both schemes I and II is that the source couplings
$P_{\alpha}^{J}$ are badly determined, especially for those of the three
$\Sigma_{c}^{*}\bar{D}^{*}$ bound states, which are almost invisible in the
line shapes, as can be seen from Tables 4 and 5 in Appendix B. Also, the
couplings to the inelastic channels have sizeable uncertainties. However, the
elastic channel interactions, i.e., $C_{\frac{1}{2}}$ and $C_{\frac{3}{2}}$,
are well determined and thus the pole positions and their effective couplings
to the dominant channels are quite stable for different fits, as shown in
Table 3. The two solutions lead to poles which are similar to those in scheme
I (see also Refs. Pavon Valderrama (2019); Liu  _et al._ (2019c)). The results
are insensitive to the form of the background. Similar to scheme I, the
$P_{c}(4312)$ couples dominantly to the $\Sigma_{c}\bar{D}$ with
$J^{P}=\frac{1}{2}^{-}$, and the $P_{c}(4440)$ and $P_{c}(4457)$ couple
dominantly to the $\Sigma_{c}\bar{D}^{*}$ with quantum numbers
$\frac{1}{2}^{-}$ and $\frac{3}{2}^{-}$, respectively, in solution $A$, and
vice versa in solution $B$. They are all (quasi-)bound state poles and should
be understood as hadronic molecules of the corresponding dominant channels Guo
_et al._ (2018). In both schemes, there is a narrow $P_{c}(4380)$ with
$J^{P}=\frac{3}{2}^{-}$ located at around $4.38$ GeV as a
$\Sigma_{c}^{*}\bar{D}$ bound state. Its width is smaller than that obtained
using a constant width of $\Sigma_{c}^{(*)}$ in Ref. Du  _et al._ (2020a) by
about a factor of 2 ( see e.g. Table I in Ref. Du  _et al._ (2020a) for
solution $B$). The reduction of the width obtained within our dynamical
framework including a three body cut is consistent with the findings of Ref.
Baru  _et al._ (2011), where the same effect was found for the partial width
$X(3872)\to D\bar{D}\pi$.

Table 3:  The names of the states, their dominant channels (DCs) and thresholds, the Riemann sheet (RS) where they lie, their quantum numbers found, the pole positions, and the dimensionless couplings to their DCs, $g_{\text{DC}}$, from the $T$-matrix residues for the solutions $A$ and $B$ of scheme II are given. The uncertainties stated result from taking the fit to different data and from the different backgrounds used. The statistical uncertainties from the fit for a given background are negligible.  |  |  |  | solution | $A$ |  | solution | $B$
---|---|---|---|---|---|---|---|---
| DC ([MeV]) | RS | $J^{P}$ | Pole [MeV] | $g_{\text{DC}}$ | $J^{P}$ | Pole [MeV] | $g_{\text{DC}}$
$P_{c}(4312)$ | $\Sigma_{c}\bar{D}\;(4321.6)$ | I | $\frac{1}{2}^{-}$ | $4313(2)-4(2)i$ | $2.5(2)+0.3(2)i$ | $\frac{1}{2}^{-}$ | $4313(2)-3(2)i$ | $2.8(2)+0.3(2)i$
$P_{c}(4380)$ | $\Sigma_{c}^{*}\bar{D}\;(4386.2)$ | II | $\frac{3}{2}^{-}$ | $4376(2)-7(2)i$ | $2.6(3)+0.1(2)i$ | $\frac{3}{2}^{-}$ | $4376(2)-6(2)i$ | $3.6(5)+0.2(2)i$
$P_{c}(4440)$ | $\Sigma_{c}\bar{D}^{*}\;(4462.1)$ | III | $\frac{1}{2}^{-}$ | $4441(2)-10(3)i$ | $0.6(4)+2.2(9)i$ | $\frac{3}{2}^{-}$ | $4441(2)-4(2)i$ | $3.8(2)+0.2(2)i$
$P_{c}(4457)$ | $\Sigma_{c}\bar{D}^{*}\;(4462.1)$ | III | $\frac{3}{2}^{-}$ | $4460(2)-2(1)i$ | $1.8(1)+0.1(2)i$ | $\frac{1}{2}^{-}$ | $4461(2)-5(2)i$ | $1.6(4)+2.0(3)i$
$P_{c}$ | $\Sigma_{c}^{*}\bar{D}^{*}\;(4526.7)$ | IV | $\frac{1}{2}^{-}$ | $4500(2)-14(3)i$ | $-0.2(1)+1.9(6)i$ | $\frac{1}{2}^{-}$ | $4525(5)-9(2)i$ | $1.5(3)+0.4(4)i$
$P_{c}$ | $\Sigma_{c}^{*}\bar{D}^{*}\;(4526.7)$ | IV | $\frac{3}{2}^{-}$ | $4510(4)-10(3)i$ | $1.2(4)+0.9(4)i$ | $\frac{3}{2}^{-}$ | $4520(3)-12(3)i$ | $2.6(4)+1.1(3)i$
$P_{c}$ | $\Sigma_{c}^{*}\bar{D}^{*}\;(4526.7)$ | IV | $\frac{5}{2}^{-}$ | $4524(5)-9(7)i$ | $1.4(5)+2.6(5)i$ | $\frac{5}{2}^{-}$ | $4500(2)-8(5)i$ | $4.3(3)+0.2(2)i$

####  IV.2.3 Predictions for the line shapes in the elastic and inelastic
channels

Our predictions for the elastic line shapes within scheme II are given in Fig.
10. In order to see the effect from the renormalized OPE, these results should
be compared to the analogous predictions of scheme I in the contact potential
framework given in Fig. 5. A general feature of the line shapes in the
$\Sigma_{c}\bar{D}$ channel is that they look different for schemes I and II
when solution $A$ is considered, while they are much more consistent with each
other for solution $B$. As already discussed in the contact framework, the
state $P_{c}(4440)$ couples to the $\Sigma_{c}\bar{D}$ channel in $S$ wave for
solution $A$ and, consequently, is clearly seen in the $\Sigma_{c}\bar{D}$
line shape of scheme I. However, this state is not at all seen in scheme II.
Moreover, the state $P_{c}(4457)$, which can couple to $\Sigma_{c}\bar{D}$
only in $D$ wave for solution $A$, has a prominent signal in the corresponding
line shape. Similarly, the $D$-wave structure generated by the $P_{c}(4440)$
state is seen in the $\Sigma_{c}^{*}\bar{D}$ line shapes in scheme II, though
it is not so pronounced as in the previous case. The presence of two poles
from the $P_{c}(4440)$ and $P_{c}(4457)$ enhanced by $D$ waves leads to an
unnaturally strong enhancement in the $\Sigma_{c}\bar{D}^{*}$ line shape right
above the threshold, as compared to the results of scheme I.

Unlike solution $A$, the line shapes predicted in scheme II for solution $B$
behave naturally and generate the peaking structures only in $S$ waves, in
agreement with the results of scheme I. The effect from the OPE on the line
shapes for solution $B$ is also qualitatively consistent with that found on
the $Z_{b}$ states and their spin partners in Refs. Wang  _et al._ (2018);
Baru  _et al._ (2019), where the structures in the line shapes were also
dominated by the amplitudes in $S$ waves. Because of the unnatural behavior of
solution $A$ discussed above, we tend to consider solution $B$ as our
preferred solution based on the current data. Data in the
$\Sigma_{c}^{*}\bar{D}$ are called for to resolve this issue experimentally.
Indeed, the presence of the $P_{c}(4440)$ as a prominent peak in this line
shape would vote for solution $B$ while the $P_{c}(4457)$ as a dominant signal
would be in favour of solution $A$. This feature can be used as a guideline
for determining the quantum numbers of the $P_{c}(4440)$ and $P_{c}(4457)$, in
analogy to the discussion in the context of scheme I.

Figure 10: Predictions for the line shapes of $\Sigma_{c}\bar{D}$ (blue dashed
curves), $\Sigma_{c}^{*}\bar{D}$ (orange dotted curves),
$\Sigma_{c}\bar{D}^{*}$ (green dot-dashed curves), and
$\Sigma_{c}^{*}\bar{D}^{*}$ (red solid curves) mass distributions based on the
fit results in Fig. 9 for solutions $A$ and $B$ of scheme II.

Predictions for the ${\eta_{c}p}$ invariant mass distribution within scheme II
are shown in Fig. 11. Also here one finds that, for solution $A$, the
inclusion of the OPE together with the $S$-$D$ contact terms leads to a
drastic modification of the line shapes, which also depend on the data set
used in the fits. As a general pattern, the comparison to scheme I in Fig. 6
reveals a very strong enhancement of the $P_{c}(4312)$, a weaker signal from
the $P_{c}(4457)$, and a strong reduction of the rate from the $P_{c}(4380)$.
Like in the elastic channels, for solution $B$, the line shapes with and
without the OPE are qualitatively consistent with each other, though
quantitative changes in the magnitudes of the peaks are visible. Thus, also
based on the ${\eta_{c}p}$ invariant mass distribution, we would conclude that
solution $B$ shows a much more natural pattern than $A$. As already proposed
for scheme I, data in the ${\eta_{c}p}$ channel would be very useful to
disentangle between these two solutions experimentally, and thus to pin down
the quantum numbers of the $P_{c}(4440)$ and $P_{c}(4457)$ unambiguously.
Indeed, the experimental observation of a pronounced peak from the
$P_{c}(4440)$ ($P_{c}(4457)$) would be a clear signature of solution $B$
($A$). For this channel it appears promising, that the first data for the
reaction $\Lambda_{b}\to\eta_{c}pK^{-}$ are already available Aaij  _et al._
(2020), although the quality of the spectra is not yet sufficient for the kind
of analysis indicated above.

Figure 11:  Predictions for the line shapes of the $\eta_{c}p$ invariant mass
distributions based on various fit results for solutions $A$ and $B$ of scheme
II. The blue dotted, orange dashed and green dot-dashed curves correspond to
the predictions from the three fit results ($\cos\theta_{P_{c}}$-weighted,
$m_{Kp}>1.9$ GeV and $m_{Kp}$-all) of Fig. 9, in order.

###  IV.3 Scheme III: Including the $\Lambda_{c}\bar{D}^{(*)}$ channels

As discussed in schemes I and II, the $J/\psi p$ invariant mass distribution
can be well described without considering the coupled-channel effects from the
$\Lambda_{c}\bar{D}^{(*)}$ channels and thus the widths of the $P_{c}$ states
can be accounted for by the channels $J/\psi p$, $\eta_{c}p$ and the elastic
channels. However, the lack of experimental information at present does not
mean that the $\Lambda_{c}\bar{D}^{(*)}$ channels can be neglected. Indeed,
the proximity of the $\Lambda_{c}\bar{D}^{*}$ channel to the elastic
thresholds, where the $P_{c}$ states reside, implies that the
$\Lambda_{c}\bar{D}^{(*)}$ channels potentially can be even more important
than the distant channels composed of a charmonium and a pion. Unlike other
inelastic channels, the $\Lambda_{c}\bar{D}^{(*)}$ channels can couple to the
elastic channels via the OPE in addition to the short-range operators, which
provides further support for their potentially important role. In addition,
some model calculations without the unitarity constraint concluded large
branching fractions of the $P_{c}$ states decaying into the
$\Lambda_{c}\bar{D}^{*}$ channel Lin  _et al._ (2017); Lin and Zou (2019). As
discussed above, as the $\Sigma_{c}^{(*)}$ states dominantly decay to
$\Lambda_{c}\pi$, the simultaneous inclusion of the $\Lambda_{c}\bar{D}^{(*)}$
channels and the OPE potential will lead to $\Lambda_{c}\bar{D}^{(*)}\pi$
three-body cuts555 Strictly speaking, $\Lambda_{c}\bar{D}^{*}\pi$ is a four-
body state, since $D^{*}$ can decay to $D\pi$. However, neglecting this width
which is tiny, i.e., under the assumption that the $D^{*}$ is a stable
particle, that we use here, $\Lambda_{c}\bar{D}^{*}\pi$ corresponds
effectively to a three-body channel. , when the energy goes above the
corresponding three-body threshold. Therefore, once the
$\Lambda_{c}\bar{D}^{(*)}$ channels are included one is also in a position to
investigate the effects from the associated $\Lambda_{c}\bar{D}^{(*)}\pi$
three-body cuts on the line shapes and the $P_{c}$ states.

Considering these points as a motivation, we developed a coupled-channel
framework that incorporates the $\Lambda_{c}\bar{D}^{(*)}$ channels explicitly
including the OPE. While the currently available data do not allow us to
adjust the parameters in the $\Lambda_{c}\bar{D}^{(*)}$ potentials reliably,
this framework, being the most advanced and well defined formulation of the
relevant coupled-channel problem on the market, should be very useful in the
future in revealing the role of $\Lambda_{c}\bar{D}^{(*)}$ in the formation of
the $P_{c}$ states and their decays. It should be mentioned that preliminary
LHCb data on the $\Lambda_{b}^{0}\to\Lambda_{c}^{+}\bar{D}^{0}K^{-}$ are
already available, but the statistics is not high enough to see clear $P_{c}$
signals Piucci (2019).

####  IV.3.1 Parameters in the fits

As discussed in scheme II, because of the large mass of the heavy system and
the large splittings between the dynamic thresholds, the typical involved
momenta $p_{\text{typ}}$ can reach up to $900~{}\mathrm{MeV}$ when the energy
is extended down to the $\Lambda_{c}\bar{D}$ threshold. These large typical
momenta make the contribution from the tensor force of the OPE that leads to
the $S$-$D$ transitions even more important than that in scheme II. As a
result, to obtain regulator-independent line shapes, promoting the formally
NLO $O(Q^{2})$ $S$-$D$ contact terms (see $D_{b}^{SD}$ and $D_{c}^{SD}$ in Eq.
(69)) to LO is mandatory in line with the procedure discussed in Sec. IV.2.1.

From the currently available data it is impossible to determine the strength
of the $\Lambda_{c}\bar{D}^{(*)}\to\Lambda_{c}\bar{D}^{(*)}$ interactions,
i.e., $C_{\frac{1}{2}}^{\prime\prime}$. Indeed, since the effect of
$\Lambda_{c}\bar{D}^{(*)}$ on the elastic channels appears as a combination of
$C_{\frac{1}{2}}^{\prime}$ and $C_{\frac{1}{2}}^{\prime\prime}$, a stronger
(weaker) $\Lambda_{c}\bar{D}^{(*)}$ interaction can be always compensated by a
weaker (stronger) transition
$\Lambda_{c}\bar{D}^{(*)}\to\Sigma_{c}^{(*)}\bar{D}^{(*)}$. In addition,
because there is no isospin-conserving $\Lambda_{c}\Lambda_{c}\pi$ vertex, the
long-range OPE potentials for the
$\Lambda_{c}\bar{D}^{(*)}\to\Lambda_{c}\bar{D}^{(*)}$ transitions vanish (see
Appendix A), and thus only short-ranged direct interactions could be possible.
Therefore, in what follows, we use these arguments as a motivation to neglect
direct interactions in the $\Lambda_{c}\bar{D}^{(*)}$ channels, as it was also
done for other inelastic channels. This approximation can be relaxed in the
future easily, once new data, e.g. in the $\Lambda_{c}\bar{D}^{(*)}$ channels
become available. In the current study, the effect of
$\Lambda_{c}\bar{D}^{(*)}$ channels is therefore included through their
coupling to the elastic channels. The inclusion of the OPE potential will not
introduce additional parameters as the coupling constants are either fixed
from experimental data or from lattice QCD Detmold  _et al._ (2012). In this
case, under the assumption that the $J/\psi p$, $\eta_{c}p$,
$\Lambda_{c}\bar{D}^{(*)}$ and $\Sigma_{c}^{(*)}\bar{D}^{(*)}$ saturate the
widths of the $P_{c}$ states, we have 7 parameters to describe the final state
interactions:

  * •

$C_{\frac{1}{2}}$ and $C_{\frac{3}{2}}$ in Eq.(39) for the contact potentials
among the elastic $\Sigma_{c}^{(*)}\bar{D}^{(*)}$ channels;

  * •

$C_{\frac{1}{2}}^{\prime}$ in Eq. (45) for the transition between the elastic
$\Sigma_{c}^{(*)}\bar{D}^{(*)}$ channels and the inelastic
$\Lambda_{c}\bar{D}^{(*)}$ ones;

  * •

$g_{S}$ and $g_{D}$ in Eq. (69) for the $S$\- and $D$-wave inelastic $J/\psi
p$ and $\eta_{c}p$ channels;

  * •

$D_{b}^{SD}$ and $D_{c}^{SD}$ in Eq. (69), the NLO $S$-$D$ contact terms, for
the transitions among the $\Sigma_{c}^{(*)}\bar{D}^{(*)}$ and
$\Lambda_{c}\bar{D}^{(*)}$ channels.

In addition, there are the parameters that parametrize the background (Eq.
(83)) and those of the bare production amplitudes (Eq. (II.1)).

####  IV.3.2 Description of the data in $\Lambda_{b}^{0}\to J/\psi pK^{-}$

Figure 12: The best fits to the experimental data of $m_{Kp}>1.9$ GeV Aaij
_et al._ (2019) with three different fixed backgrounds for solution $B$ (blue
dashed curves) of scheme III. The corresponding backgrounds are shown as
dotted curves. The vertical dashed lines from left to right are the
$\Sigma_{c}\bar{D}$, $\Sigma_{c}^{*}\bar{D}$, $\Sigma_{c}\bar{D}^{*}$, and
$\Sigma_{c}^{*}\bar{D}^{*}$ thresholds, respectively.

There are three prominent narrow $P_{c}$ structures in the $J/\psi p$
spectrum, which are used to extract the overall strength of the inelastic
parameters and the production constants. As the fits without the
$\Lambda_{c}\bar{D}^{(*)}$ channels have already yielded values for
$\chi^{2}/$dof less than one, it is not surprising that the explicit inclusion
of $\Lambda_{c}\bar{D}^{(*)}$ does not improve the fit quality. As a result,
the fitted inelastic parameters, as well as the bare production parameters
$P_{\alpha}^{J}$, are very sensitive to the data sets used in the fits and to
the background employed. Indeed, as the system is overdetermined, it is hard
to discriminate the contributions from the $J/\psi p$ ($\eta_{c}p$) and
$\Lambda_{c}\bar{D}^{(*)}$ to the widths of the $P_{c}$ states from just the
$J/\psi p$ mass distributions.666We checked also that the inclusion of just
the $S$-wave contact interaction $C_{\frac{1}{2}}^{\prime}$ between
$\Lambda_{c}\bar{D}^{(*)}$ and $\Sigma_{c}^{(*)}\bar{D}^{(*)}$ to scheme I
does not allow one to reliably extract $C_{\frac{1}{2}}^{\prime}$ from data.
To illustrate this more clearly, instead of fitting to the three data sets as
used in schemes I and II, we make fits to the data with $m_{Kp}>1.9$ GeV for
solution $B$ only with three different, fixed backgrounds777Since at this
stage the results in scheme III can be given for illustration only, we
postpone a detailed comparison of solutions $A$ and $B$ to the future studies,
once data sensitive to the poorly determined parameters become available., as
shown in Fig. 12. All the fits have values for $\chi^{2}/$dof very close to
the best fit results. However, their inelastic and bare production parameters
are very different, as shown in Table 6 in Appendix B. Accordingly they lead
to different predictions for the line shapes in the elastic and inelastic
channels, as illustrated in Fig. 13. It follows in particular from Table 6
that the product of the bare production vertices and the inelastic constants
$J/\psi p$ is much better constrained by the $J/\psi p$ mass distributions
than these quantities individually. For example, the fit with the background-2
(orange lines in Fig. 12) yields values for the $J/\psi p$ constants which are
an order of magnitude smaller than for the other two fits (cf. the couplings
$g_{S}$ and $g_{D}$ for the fit in the middle of Table 6 with the similar
couplings in other fits), which is balanced by a corresponding increase in
bare production vertices. Consequently, the production rates to
$\Sigma_{c}^{(*)}\bar{D}^{(*)}$ and $\Lambda_{c}\bar{D}^{(*)}$ channels for
this fit are an order of magnitude larger than for the other fits, as can be
seen from Fig. 13 (cf. the rates in the middle column with the others).

On the other hand, the poles around the $\Sigma_{c}^{(*)}\bar{D}^{(*)}$
thresholds are sensitive neither to the form of the background nor to the
choice of the experimental data set in the fits. The poles in scheme III are
consistent with those in scheme II and thus not explicitly presented here.
Also, it is worth mentioning that the line shapes in the $\eta_{c}p$ channels,
which are related to the $J/\psi p$ distributions via HQSS, have comparable
rates for all fits considered here (see Fig. 14). Furthermore, the shape of
the $\eta_{c}p$ distributions is generally consistent with the predictions of
schemes I and II, reported above. Specifically, the peak from the
$P_{c}(4440)$, which is clearly seen in all line shapes for solution $B$, is
an experimentally testable prediction.

Figure 13:  Predictions for the line shapes in the
$\Sigma_{c}^{(*)}\bar{D}^{(*)}$ and $\Lambda_{c}\bar{D}^{(*)}$ channels based
on the fit results of scheme III (solution $B$) for three different
backgrounds. The left, middle and right columns show the results that are
related to the blue, orange and green lines in Fig. 12, respectively. First
row: Mass distributions in the $\Sigma_{c}\bar{D}$ (blue dashed curves),
$\Sigma_{c}^{*}\bar{D}$ (orange dotted curves), $\Sigma_{c}\bar{D}^{*}$ (green
dot-dashed curves), and $\Sigma_{c}^{*}\bar{D}^{*}$ (red solid curves)
channels. Second row: Mass distributions in the $\Lambda_{c}\bar{D}$ (blue
dashed curves) and $\Lambda_{c}\bar{D}^{*}$ (orange dotted curves) channels.
Note the different scales on the $y$-axes of the different plots. Figure 14:
Predictions for the line shapes of the $\eta_{c}p$ invariant mass
distributions based on the fit results of Scheme III (solution $B$) in Fig. 12
for three different backgrounds. The blue dotted, orange dashed and green dot-
dashed curves correspond to the predictions from the three fit results in Fig.
12 of blue, orange and green colors, in order.

##  V Summary

In this work we present an updated coupled-channel analysis of the process
$\Lambda_{b}^{0}\to J/\psi pK^{-}$ for the hidden-charm pentaquark states
discovered by LHCb. In Ref. Du  _et al._ (2020a), an analysis of the $J/\psi
p$ invariant mass distributions in this process was performed within a
coupled-channel approach including interactions between the elastic
$\Sigma_{c}^{(*)}\bar{D}^{(*)}$ channels and transitions from the elastic to
the $S$\- and $D$-wave $J/\psi p$ inelastic channels in a way consistent with
HQSS. To account for other inelastic channels not included explicitly, the LO
contact interactions in the elastic channels were augmented with imaginary
parts in the spirit of optical potentials. The investigation presented here
improves and extends the study reported in Ref. Du  _et al._ (2020a) in
several aspects. In particular, now we include the $\eta_{c}p$ and
$\Lambda_{c}\bar{D}^{(*)}$ channels dynamically in order to restore unitarity
and investigate the effect from these inelastic channels. The inclusion of
$\eta_{c}p$ is also required by HQSS. To account for the effect of the width
of $\Sigma_{c}^{(*)}$ in the Lippmann-Schwinger equations, the self-energy of
the $\Sigma_{c}^{(*)}$ is considered dynamically, which generates three-body
cuts from the $\Lambda_{c}\bar{D}^{(*)}\pi$ intermediate states. Additional
three-body cut contributions appear from the transition potentials
$\Sigma_{c}^{(*)}\bar{D}^{(*)}\to\Lambda_{c}\bar{D}^{(*)}$ driven by OPE.
Apart from OPE, these transition potentials also involve one leading-order
momentum-independent contact term as a consequence of HQSS.

We emphasize that in the fits performed in this study we take into account all
the data sets of the $J/\psi p$ invariant mass distributions provided in Ref.
Aaij  _et al._ (2019). To perform the fits, we consider three strategies:
employing the momentum-independent LO contact potentials without including the
$\Lambda_{c}\bar{D}^{(*)}$ channels (scheme I), including the OPE potential
supplemented by the $S$-$D$ contact terms to scheme I (scheme II), and
explicitly including in addition the $\Lambda_{c}\bar{D}^{(*)}$ channels
(scheme III).

For scheme I, we assume that the widths of the $P_{c}$ states are saturated by
the $J/\psi p$, $\eta_{c}p$ and $\Sigma_{c}^{(*)}\bar{D}^{(*)}$ channels. Two
solutions (denoted by $A$ and $B$) are found describing the data almost
equally well, with $\chi^{2}$-values almost coincident to those of scheme I in
Ref. Du  _et al._ (2020a). Both solutions give seven poles corresponding to
seven $P_{c}$ states. Among these states, the lowest one corresponds to the
$P_{c}(4312)$ with $J^{P}=\frac{1}{2}^{-}$. It appears as a
$\Sigma_{c}\bar{D}$ bound state in both solutions. There are two
$\Sigma_{c}\bar{D}^{*}$ bound states with quantum numbers
$J^{P}=\frac{1}{2}^{-}$ and $\frac{3}{2}^{-}$, corresponding to the
$P_{c}(4440)$ and the $P_{c}(4457)$, respectively, in solution $A$ and with
interchanged quantum numbers in solution $B$, as in Ref. Du  _et al._ (2020a).
Also the three predicted $\Sigma_{c}^{*}\bar{D}^{*}$ states show different
mass patterns: $m_{\frac{1}{2}^{-}}<m_{\frac{3}{2}^{-}}<m_{\frac{5}{2}^{-}}$
for solution $A$ and reversed for solution $B$. However, the three
$\Sigma_{c}^{*}\bar{D}^{*}$ states are almost invisible in the spectra. In
addition, HQSS calls for a seventh state, located close to the
$\Sigma_{c}^{*}\bar{D}$ threshold with $J^{P}=\frac{3}{2}^{-}$ and with a
width comparable to those of the other states. As in Ref. Du  _et al._
(2020a), in the data we found clear evidence for this state, dubbed
$P_{c}(4380)$, in both solutions. Testable predictions for the line shapes in
the $\Sigma_{c}^{(*)}\bar{D}^{(*)}$ and $\eta_{c}p$ channels are proposed. For
the $\Sigma_{c}^{(*)}\bar{D}^{(*)}$ line shapes, the threshold enhancement at
the $\Sigma_{c}^{(*)}\bar{D}^{(*)}$ thresholds, which is seen in these line
shapes, signals the existence of the corresponding shallow bound states.

Once the OPE potentials for the $\Sigma_{c}^{(*)}\bar{D}^{(*)}$ channels are
included in scheme II, an enhanced $S$-wave-to-$D$-wave mixing is induced that
calls for a promotion of formally NLO contact terms to LO to yield regulator-
independent results. Once these $S$-$D$ contact terms are included, also in
the approach with OPE two solutions ($A$ and $B$) are found with almost equal
$\chi^{2}$ — in contrast to this, the study of Ref. Du  _et al._ (2020a),
where no NLO contact terms were promoted to LO, only allowed for one solution,
once OPE was included. These two solutions lead to poles similar to those in
scheme I. Meanwhile, useful insights are possible from the predicted line
shapes. For one of the two solutions, namely for solution $A$, some
predictions are found to be very unnatural such as the predicted spectrum in
the $\Sigma_{c}\bar{D}$ channel, where the $P_{c}(4457)$, with a $D$-wave
coupling to $\Sigma_{c}\bar{D}$, has a prominent peak while the $P_{c}(4440)$
in $S$ wave is not seen at all. Also, the line shapes in scheme II for
solution $A$ are substantially different from the results in scheme I. This is
very different from solution $B$, where the predictions of the two schemes
considered behave naturally and are consistent with each other. This may
indicate that solution $B$ is more preferred theoretically than solution $A$,
which is also consistent with the finding of Ref. Du  _et al._ (2020a), where
only solution $B$, with the $P_{c}(4440)$ ($P_{c}(4457)$) identified as a
$\frac{3}{2}^{-}$ ($\frac{1}{2}^{-}$) $\Sigma_{c}\bar{D}^{*}$ bound state,
remained once the OPE was included. However, no strict conclusion about the
quantum numbers of the $P_{c}(4440)$ and the $P_{c}(4457)$ is possible from
the $J/\psi p$ data alone. The nontrivial shape of the $\Sigma_{c}^{*}\bar{D}$
and $\eta_{c}p$ distributions can be used to discriminate between the two
solutions experimentally. In particular, the $P_{c}(4457)$ has a clear peak
structure in these spectra for solution $A$, while the $P_{c}(4440)$ is
prominent for solution $B$. Apart from this observation, the $\eta_{c}p$
invariant mass distributions are very interesting since in most fits they show
significant signals from the $P_{c}(4312)$ and also from higher states around
4500 MeV. Also, the $P_{c}(4380)$ is seen in the $\eta_{c}p$ spectrum in both
fit schemes though in scheme II only for solution $B$. Thus, data in the
elastic and inelastic channels discussed here are not only needed to further
refine our approach, but also to determine the quantum numbers of the
pentaquark states and to unravel their nature.

In scheme III, we developed the most advanced version of the coupled-channel
approach for the pentaquark states reported by LHCb, which also incorporates
the $\Lambda_{c}\bar{D}^{(*)}$ channels as well as the OPE. It is
demonstrated, however, that, because of the lack of structures around the
$\Lambda_{c}\bar{D}^{(*)}$ thresholds in the $J/\psi p$ channel and the
absence of data in the other channels, the $\Lambda_{c}\bar{D}^{(*)}$
interaction strength cannot be reliably extracted from the fits to the $J/\psi
p$ data only. Once new data become available, this approach should not only
allow one to determine the pentaquarks quantum numbers but also to reveal
their leading decay properties. Meanwhile, even based on the current data we
find that the effect of including a three-body cut from the
$\Lambda_{c}\bar{D}\pi$ channel in the OPE potentials and especially in the
self-energy $\Sigma_{c}^{(*)}$ in the two-body Green function reduces the
width of the narrow $P_{c}(4380)$ by about a factor 2 compared to the one
extracted using a constant complex mass of $\Sigma_{c}^{(*)}$ in Ref. Du  _et
al._ (2020a).

It still remains to be understood why the $P_{c}$ states near 4.5 GeV are
almost invisible in the data for the $J/\psi p$ distributions, or why the
$P_{c}(4440)$ and $P_{c}(4457)$ are more pronounced than the others. This must
be related to the production mechanism, which in this work is assumed to
proceed entirely through the $\Sigma_{c}\bar{D}^{(*)}$ channels at the primary
production vertex. There can also be effects from nearby triangle
singularities discussed in Refs. Guo  _et al._ (2015b); Liu  _et al._ (2016);
Bayar  _et al._ (2016); Aaij  _et al._ (2019) enhancing or interfering with
the production of some of the $P_{c}$ states.

###### Acknowledgements.

This work is supported in part by the National Natural Science Foundation of
China (NSFC) and the Deutsche Forschungsgemeinschaft (DFG) through the funds
provided to the Sino-German Collaborative Research Center TRR110 “Symmetries
and the Emergence of Structure in QCD” (NSFC Grant No. 12070131001, DFG
Project-ID 196253076), by the NSFC under Grants No. 11835015, No. 12047503,
No. 11961141012, and No.12035007, by the Chinese Academy of Sciences (CAS)
under Grants No. QYZDB-SSW-SYS013 and No. XDB34030000, by the CAS Center for
Excellence in Particle Physics (CCEPP), and by the Munich Institute for Astro-
and Particle Physics (MIAPP) of the DFG cluster of excellence “Origin and
Structure of the Universe.” Q.W. is also supported by the Guangdong Major
Project of Basic and Applied Basic Research under Grant No. 2020B0301030008,
and by the Science and Technology Program of Guangzhou under Grant No.
2019050001. The work of U.G.M. is further supported by the CAS President’s
International Fellowship Initiative (PIFI) under Grant No. 2018DM0034 and by
the VolkswagenStiftung under Grant No. 93562. The work of V.B. is supported by
the Russian Science Foundation under Grant No. 18-12-00226. J.A.O. would like
to acknowledge partial financial support by the MINECO (Spain) and FEDER (EU)
under Grant No. FPA2016-77313-P and by the MICINN (Spain) under Grant No.
PID2019-106080GB-C22.

##  Appendix A One-pion-exchange potentials

Since the pion couples to the charmed mesons and baryons in the $P$ wave, see
the Lagrangian in Eq. (52), the OPE potential mixes the $S$ wave and $D$ wave.
To calculate the partial wave projection of the OPE potentials Oller (2019),
we start from a two-body state $\left|JM,L\Sigma\right\rangle$ with $J$ ($M$)
the total angular momentum (its third component), $L$ the orbital angular
momentum and $\Sigma$ the total spin

$\displaystyle\left|JM,L\Sigma\right\rangle=\frac{1}{\sqrt{4\pi}}\sum_{\sigma_{1},\sigma_{2},\Sigma_{3},L_{3}}\int
d\Omega_{\bm{p}}Y_{L}^{L_{3}}\left(\Omega_{\bm{p}}\right)\langle
s_{1}s_{2}\sigma_{1}\sigma_{2}|\Sigma\Sigma_{3}\rangle\langle L\Sigma
L_{3}\Sigma_{3}|JM\rangle\left|\bm{p}\sigma_{1}\sigma_{2}\right\rangle,$ (86)

where $\left|\bm{p}\sigma_{1}\sigma_{2}\right\rangle$ is the direct product of
the one-particle states $\left|\bm{p},\sigma_{1}\right\rangle$ and
$\left|-\bm{p},\sigma_{2}\right\rangle$ with $\sigma_{i}$ the third component
of spin $s_{i}$ for the $i^{\rm th}$ particle in that channel; $\bm{p}$ is the
momentum in the c.m. frame. The partial wave transition between the states
with the same $J$ and $M$ reads

$\displaystyle T_{L\Sigma,L^{\prime}\Sigma^{\prime}}^{J}$ $\displaystyle=$
$\displaystyle\langle JM,L\Sigma|\hat{T}|JM,L^{\prime}\Sigma^{\prime}\rangle$
(87) $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\sum_{\sigma_{1}\sigma_{2}\Sigma_{3}L_{3}}\sum_{\sigma_{1}^{\prime}\sigma_{2}^{\prime}\Sigma_{3}^{\prime}L_{3}^{\prime}}\int
d\Omega_{\bm{p}}Y_{L}^{L_{3}}\left(\Omega_{\bm{p}}\right)\langle
s_{1}s_{2}\sigma_{1}\sigma_{2}|\Sigma\Sigma_{3}\rangle\langle L\Sigma
L_{3}\Sigma_{3}|JM\rangle$ $\displaystyle\times\int
d\Omega_{\bm{p}^{\prime}}Y_{L^{\prime}}^{L_{3}^{\prime}*}\left(\Omega_{\bm{p}^{\prime}}\right)\langle
s_{1}^{\prime}s_{2}^{\prime}\sigma_{1}^{\prime}\sigma_{2}^{\prime}|\Sigma^{\prime}\Sigma_{3}^{\prime}\rangle\langle
L^{\prime}\Sigma^{\prime}L_{3}^{\prime}\Sigma_{3}^{\prime}|JM\rangle\left\langle\bm{p}^{\prime}\sigma_{1}^{\prime}\sigma_{2}^{\prime}\right|\hat{T}\left|\bm{p}\sigma_{1}\sigma_{2}\right\rangle,$

where $\hat{T}$ is the transition operator. In the framework of the TOPT, the
OPE potential acquires two contributions as shown in Fig. 1. For the
scattering process $12\to 1^{\prime}2^{\prime}$ with $E$ the total energy of
the system, and $p$, $p^{\prime}$ the incoming and outgoing three-momenta, we
define

$\displaystyle V_{SS}(E,p,p^{\prime})$ $\displaystyle\equiv$
$\displaystyle-\int_{-1}^{1}d\cos\theta\frac{p^{\prime
2}+p^{2}-2p^{\prime}p\cos\theta}{2E_{\pi}(p_{\pi})}\big{[}D_{a}^{\pi}(E,p,p^{\prime},\theta)+D_{b}^{\pi}(E,p,p^{\prime},\theta)\big{]},$
$\displaystyle V_{SD}(E,p,p^{\prime})$ $\displaystyle\equiv$
$\displaystyle-\int_{-1}^{1}d\cos\theta\frac{4p^{\prime
2}+p^{2}-8p^{\prime}p\cos\theta+3p^{2}\cos(2\theta)}{2E_{\pi}(p_{\pi})}\big{[}D_{a}^{\pi}(E,p,p^{\prime},\theta)+D_{b}^{\pi}(E,p,p^{\prime},\theta)\big{]},$
$\displaystyle V_{DS}(E,p,p^{\prime})$ $\displaystyle\equiv$
$\displaystyle-\int_{-1}^{1}d\cos\theta\frac{p^{\prime
2}+4p^{2}-8p^{\prime}p\cos\theta+3p^{\prime
2}\cos(2\theta)}{2E_{\pi}(p_{\pi})}\big{[}D_{a}^{\pi}(E,p,p^{\prime},\theta)+D_{b}^{\pi}(E,p,p^{\prime},\theta)\big{]},$
$\displaystyle V_{DD}^{(c_{1},c_{2},c_{3},c_{4})}(E,p,p^{\prime})$
$\displaystyle\equiv$
$\displaystyle-\int_{-1}^{1}d\cos\theta\frac{c_{1}(p^{\prime
2}+p^{2})-c_{2}\,p^{\prime}p\cos\theta+c_{3}(p^{\prime
2}+p^{2})\cos(2\theta)-c_{4}\,p^{\prime}p\cos(3\theta)}{2E_{\pi}(p_{\pi})}$
(88)
$\displaystyle\times\big{[}D_{a}^{\pi}(E,p,p^{\prime},\theta)+D_{b}^{\pi}(E,p,p^{\prime},\theta)\big{]},$

where $\theta$ denotes the angle between the three-momenta $\bm{p}^{\prime}$
and $\bm{p}$. The values for the $c_{i}(i=1,\ldots,4)$ coefficients are
specified below for each value of the total angular momentum $J$. Moreover,

$\displaystyle D_{a}^{\pi}(E,p,p^{\prime},\theta)$ $\displaystyle=$
$\displaystyle\frac{1}{E_{\pi}(p_{\pi})+E_{1^{\prime}}(p^{\prime})+E_{2}(p)-E},$
$\displaystyle D_{b}^{\pi}(E,p,p^{\prime},\theta)$ $\displaystyle=$
$\displaystyle\frac{1}{E_{\pi}(p_{\pi})+E_{1}(p)+E_{2^{\prime}}(p^{\prime})-E}$

are the contributions of the two TOPT orderings with
$p_{\pi}=\sqrt{p^{2}+p^{\prime 2}-2pp^{\prime}\cos\theta}$ and
$E_{i}=\sqrt{p_{i}^{2}+m_{i}^{2}}$.

The OPE potentials in the elastic channels listed in Table 1 for
$J^{P}=\frac{1}{2}^{-}$ can be written in a matrix form (where the columns and
rows are given by the channels listed in order in Table 1) as

$\displaystyle
V_{\frac{1}{2}^{-}}^{\text{OPE}}=\frac{g_{1}}{f_{\pi}^{2}}\left(\begin{array}[]{cc}V_{\frac{1}{2}^{-}}^{SS}&V_{\frac{1}{2}^{-}}^{SD}\\\
V_{\frac{1}{2}^{-}}^{DS}&V_{\frac{1}{2}^{-}}^{DD}\end{array}\right),$ (92)

with

$V_{\frac{1}{2}^{-}}^{SS}=\left(\begin{array}[]{ccccc}0&\frac{g_{2}}{2\sqrt{3}}V_{SS}&\frac{g_{2}}{2\sqrt{6}}V_{SS}&0&\frac{g_{3}}{8}V_{SS}\\\
\frac{g_{2}}{2\sqrt{3}}V_{SS}&\frac{g_{2}}{3}V_{SS}&-\frac{g_{2}}{6\sqrt{2}}V_{SS}&\frac{g_{3}}{8}V_{SS}&\frac{g_{3}}{4\sqrt{3}}V_{SS}\\\
\frac{g_{2}}{2\sqrt{6}}V_{SS}&-\frac{g_{2}}{6\sqrt{2}}V_{SS}&\frac{5g_{2}}{12}V_{SS}&-\frac{g_{3}}{4\sqrt{2}}V_{SS}&\frac{g_{3}}{4\sqrt{6}}V_{SS}\\\
0&\frac{g_{3}}{8}V_{SS}&-\frac{g_{3}}{4\sqrt{2}}V_{SS}&0&0\\\
\frac{g_{3}}{8}V_{SS}&\frac{g_{3}}{4\sqrt{3}}V_{SS}&\frac{g_{3}}{4\sqrt{6}}V_{SS}&0&0\\\
\end{array}\right),$

$V_{\frac{1}{2}^{-}}^{SD}=\left(\begin{array}[]{ccccc}\frac{g_{2}}{4\sqrt{6}}V_{SD}&0&-\frac{g_{2}}{8\sqrt{30}}V_{SD}&-\frac{g_{2}}{8}\sqrt{\frac{3}{10}}V_{SD}&\frac{g_{3}}{16\sqrt{2}}V_{SD}\\\
-\frac{g_{2}}{12\sqrt{2}}V_{SD}&-\frac{g_{2}}{8\sqrt{6}}V_{SD}&\frac{g_{2}}{6\sqrt{10}}V_{SD}&-\frac{g_{2}}{8\sqrt{10}}V_{SD}&-\frac{g_{3}}{16\sqrt{6}}V_{SD}\\\
\frac{g_{2}}{48}V_{SD}&-\frac{g_{2}}{16\sqrt{3}}V_{SD}&\frac{7g_{2}}{48\sqrt{5}}V_{SD}&\frac{g_{2}}{8\sqrt{5}}V_{SD}&-\frac{g_{3}}{32\sqrt{3}}V_{SD}\\\
\frac{g_{3}}{16\sqrt{2}}V_{SD}&0&\frac{g_{3}}{16\sqrt{10}}V_{SD}&\frac{3g_{3}}{16\sqrt{10}}V_{SD}&0\\\
-\frac{g_{3}}{16\sqrt{6}}V_{SD}&\frac{g_{3}}{16\sqrt{2}}V_{SD}&-\frac{g_{3}}{4\sqrt{30}}V_{SD}&\frac{g_{3}}{16}\sqrt{\frac{3}{10}}V_{SD}&0\\\
\end{array}\right),$
$V_{\frac{1}{2}^{-}}^{DS}=\left(\begin{array}[]{ccccc}\frac{g_{2}}{4\sqrt{6}}V_{DS}&-\frac{g_{2}}{12\sqrt{2}}V_{DS}&\frac{g_{2}}{48}V_{DS}&\frac{g_{3}}{16\sqrt{2}}V_{DS}&-\frac{g_{3}}{16\sqrt{6}}V_{DS}\\\
0&-\frac{g_{2}}{8\sqrt{6}}V_{DS}&-\frac{g_{2}}{16\sqrt{3}}V_{DS}&0&\frac{g_{3}}{16\sqrt{2}}V_{DS}\\\
-\frac{g_{2}}{8\sqrt{30}}V_{DS}&\frac{g_{2}}{6\sqrt{10}}V_{DS}&\frac{7g_{2}}{48\sqrt{5}}V_{DS}&\frac{g_{3}}{16\sqrt{10}}V_{DS}&-\frac{g_{3}}{4\sqrt{30}}V_{DS}\\\
-\frac{g_{2}}{8}\sqrt{\frac{3}{10}}V_{DS}&-\frac{g_{2}}{8\sqrt{10}}V_{DS}&\frac{g_{2}}{8\sqrt{5}}V_{DS}&\frac{3g_{3}}{16\sqrt{10}}V_{DS}&\frac{g_{3}}{16}\sqrt{\frac{3}{10}}V_{DS}\\\
\frac{g_{3}}{16\sqrt{2}}V_{DS}&-\frac{g_{3}}{16\sqrt{6}}V_{DS}&-\frac{g_{3}}{32\sqrt{3}}V_{DS}&0&0\\\
\end{array}\right),$
$V_{\frac{1}{2}^{-}}^{DD}=\left(\begin{array}[]{ccccc}\frac{g_{2}}{24}V_{\text{DD}}^{\text{(1,8,3,0)}}&-\frac{g_{2}V_{\text{DD}}^{\text{(4,23,12,9)}}}{32\sqrt{3}}&-\frac{g_{2}V_{\text{DD}}^{\text{(8,37,24,27)}}}{96\sqrt{5}}&\frac{g_{2}V_{\text{DD}}^{\text{(2,13,6,3)}}}{32\sqrt{5}}&\frac{g_{3}V_{\text{DD}}^{\text{(1,8,3,0)}}}{32\sqrt{3}}\\\
-\frac{g_{2}V_{\text{DD}}^{\text{(4,23,12,9)}}}{32\sqrt{3}}&0&\frac{g_{2}V_{\text{DD}}^{\text{(1,-1,3,9)}}}{16\sqrt{15}}&\frac{g_{2}}{32}\sqrt{\frac{3}{5}}V_{\text{DD}}^{\text{(2,13,6,3)}}&\frac{g_{3}}{64}V_{\text{DD}}^{\text{(4,23,12,9)}}\\\
-\frac{g_{2}V_{\text{DD}}^{\text{(8,37,24,27)}}}{96\sqrt{5}}&\frac{g_{2}V_{\text{DD}}^{\text{(1,-1,3,9)}}}{16\sqrt{15}}&\frac{g_{2}}{120}V_{\text{DD}}^{\text{(13,77,39,27)}}&\frac{g_{2}}{160}V_{\text{DD}}^{\text{(2,13,6,3)}}&\frac{g_{3}V_{\text{DD}}^{\text{(8,37,24,27)}}}{64\sqrt{15}}\\\
\frac{g_{2}V_{\text{DD}}^{\text{(2,13,6,3)}}}{32\sqrt{5}}&\frac{g_{2}}{32}\sqrt{\frac{3}{5}}V_{\text{DD}}^{\text{(2,13,6,3)}}&\frac{g_{2}}{160}V_{\text{DD}}^{\text{(2,13,6,3)}}&\frac{3g_{2}}{80}V_{\text{DD}}^{\text{(1,9,3,-1)}}&-\frac{g_{3}}{64}\sqrt{\frac{3}{5}}V_{\text{DD}}^{\text{(2,13,6,3)}}\\\
\frac{g_{3}V_{\text{DD}}^{\text{(1,8,3,0)}}}{32\sqrt{3}}&\frac{g_{3}}{64}V_{\text{DD}}^{\text{(4,23,12,9)}}&\frac{g_{3}V_{\text{DD}}^{\text{(8,37,24,27)}}}{64\sqrt{15}}&-\frac{g_{3}}{64}\sqrt{\frac{3}{5}}V_{\text{DD}}^{\text{(2,13,6,3)}}&0\\\
\end{array}\right).$

The OPE potentials for $J^{P}=\frac{3}{2}^{-}$ read

$\displaystyle
V_{\frac{3}{2}^{-}}^{\text{OPE}}=\frac{g_{1}}{f_{\pi}^{2}}\left(\begin{array}[]{cc}V_{\frac{3}{2}^{-}}^{SS}&V_{\frac{3}{2}^{-}}^{SD}\\\
V_{\frac{3}{2}^{-}}^{DS}&V_{\frac{3}{2}^{-}}^{DD}\end{array}\right),$ (95)

where888As the $V_{\frac{3}{2}^{-}}^{DD}$ potential is symmetric, here only
the elements on the upper triangle are presented. The case is similar for the
$V_{\frac{5}{2}^{-}}^{DD}$ potential.

$V_{\frac{3}{2}^{-}}^{SS}=\left(\begin{array}[]{cccc}-\frac{g_{2}}{12}V_{SS}&-\frac{g_{2}}{8\sqrt{3}}V_{SS}&-\frac{\sqrt{5}g_{2}}{24}V_{SS}&-\frac{g_{3}}{12\sqrt{2}}V_{SS}\\\
-\frac{g_{2}}{8\sqrt{3}}V_{SS}&0&\frac{g_{2}}{8}\sqrt{\frac{5}{3}}V_{SS}&\frac{g_{3}}{4\sqrt{6}}V_{SS}\\\
-\frac{\sqrt{5}g_{2}}{24}V_{SS}&\frac{g_{2}}{8}\sqrt{\frac{5}{3}}V_{SS}&\frac{g_{2}}{12}V_{SS}&\frac{g_{3}}{12}\sqrt{\frac{5}{2}}V_{SS}\\\
-\frac{g_{3}}{12\sqrt{2}}V_{SS}&\frac{g_{3}}{4\sqrt{6}}V_{SS}&\frac{g_{3}}{12}\sqrt{\frac{5}{2}}V_{SS}&0\\\
\end{array}\right),$
$V_{\frac{3}{2}^{-}}^{SD}=V_{SD}\left(\begin{array}[]{cccccccccc}-\frac{g_{2}}{8\sqrt{3}}&\frac{g_{2}}{24}&-\frac{g_{2}}{12}&\frac{g_{2}}{16\sqrt{3}}&-\frac{g_{2}}{48\sqrt{2}}&-\frac{g_{2}}{48\sqrt{5}}&\frac{g_{2}}{16}\sqrt{\frac{7}{10}}&-\frac{g_{3}}{32}&\frac{g_{3}}{32\sqrt{3}}&-\frac{g_{3}}{16\sqrt{3}}\\\
0&\frac{g_{2}}{16\sqrt{3}}&\frac{g_{2}}{16\sqrt{3}}&0&\frac{g_{2}}{16\sqrt{6}}&\frac{g_{2}}{4\sqrt{15}}&\frac{g_{2}}{16}\sqrt{\frac{21}{10}}&0&-\frac{g_{3}}{32}&-\frac{g_{3}}{32}\\\
\frac{g_{2}}{16\sqrt{15}}&-\frac{g_{2}}{12\sqrt{5}}&-\frac{g_{2}}{48\sqrt{5}}&\frac{g_{2}}{4\sqrt{15}}&-\frac{7g_{2}}{48\sqrt{10}}&-\frac{g_{2}}{15}&\frac{g_{2}}{80}\sqrt{\frac{7}{2}}&-\frac{g_{3}}{32\sqrt{5}}&\frac{g_{3}}{8\sqrt{15}}&\frac{g_{3}}{32\sqrt{15}}\\\
-\frac{g_{3}}{32}&\frac{g_{3}}{32\sqrt{3}}&-\frac{g_{3}}{16\sqrt{3}}&-\frac{g_{3}}{32}&\frac{g_{3}}{32\sqrt{6}}&\frac{g_{3}}{32\sqrt{15}}&-\frac{g_{3}}{32}\sqrt{\frac{21}{10}}&0&0&0\\\
\end{array}\right),$
$V_{\frac{3}{2}^{-}}^{DS}=V_{DS}\left(\begin{array}[]{cccccccccc}-\frac{g_{2}}{8\sqrt{3}}&\frac{g_{2}}{24}&-\frac{g_{2}}{12}&\frac{g_{2}}{16\sqrt{3}}&-\frac{g_{2}}{48\sqrt{2}}&-\frac{g_{2}}{48\sqrt{5}}&\frac{g_{2}}{16}\sqrt{\frac{7}{10}}&-\frac{g_{3}}{32}&\frac{g_{3}}{32\sqrt{3}}&-\frac{g_{3}}{16\sqrt{3}}\\\
0&\frac{g_{2}}{16\sqrt{3}}&\frac{g_{2}}{16\sqrt{3}}&0&\frac{g_{2}}{16\sqrt{6}}&\frac{g_{2}}{4\sqrt{15}}&\frac{g_{2}}{16}\sqrt{\frac{21}{10}}&0&-\frac{g_{3}}{32}&-\frac{g_{3}}{32}\\\
\frac{g_{2}}{16\sqrt{15}}&-\frac{g_{2}}{12\sqrt{5}}&-\frac{g_{2}}{48\sqrt{5}}&\frac{g_{2}}{4\sqrt{15}}&-\frac{7g_{2}}{48\sqrt{10}}&-\frac{g_{2}}{15}&\frac{g_{2}}{80}\sqrt{\frac{7}{2}}&-\frac{g_{3}}{32\sqrt{5}}&\frac{g_{3}}{8\sqrt{15}}&\frac{g_{3}}{32\sqrt{15}}\\\
-\frac{g_{3}}{32}&\frac{g_{3}}{32\sqrt{3}}&-\frac{g_{3}}{16\sqrt{3}}&-\frac{g_{3}}{32}&\frac{g_{3}}{32\sqrt{6}}&\frac{g_{3}}{32\sqrt{15}}&-\frac{g_{3}}{32}\sqrt{\frac{21}{10}}&0&0&0\\\
\end{array}\right)^{T},$

${V_{\frac{3}{2}^{-}}^{DD}=g_{2}\left(\begin{array}[]{cccccccccc}0&\frac{V_{\text{DD}}^{\text{(1,5,3,3)}}}{8\sqrt{3}}&\frac{V_{\text{DD}}^{\text{(2,
13,6,3)}}}{16\sqrt{3}}&0&\frac{V_{\text{DD}}^{\text{(1,5,3,3)}}}{8\sqrt{6}}&\frac{-V_{\text{DD}}^{\text{(2,13,6,3)}}}{32\sqrt{15}}&\frac{\sqrt{\frac{3}{70}}V_{\text{DD}}^{\text{(2,13,6,3)}}}{16}&0&\frac{g_{3}V_{\text{DD}}^{\text{(1,5,3,3)}}}{32g_{2}}&\frac{g_{3}V_{\text{DD}}^{\text{(2,13,6,3)}}}{64g_{2}}\\\
&\frac{V_{\text{DD}}^{\text{(1,5,3,3)}}}{12}&\frac{-V_{\text{DD}}^{\text{(2,13,6,3)}}}{48}&\frac{-V_{\text{DD}}^{\text{(2,13,6,3)}}}{32\sqrt{3}}&\frac{-V_{\text{DD}}^{\text{(1,5,3,3)}}}{24\sqrt{2}}&\frac{V_{\text{DD}}^{\text{(2,
13,6,3)}}}{24\sqrt{5}}&\frac{V_{\text{DD}}^{\text{(2,13,6,3)}}}{16\sqrt{70}}&\frac{g_{3}V_{\text{DD}}^{\text{(1,5,3,3)}}}{32g_{2}}&\frac{g_{3}V_{\text{DD}}^{\text{(1,5,3,
3)}}}{16\sqrt{3}g_{2}}&\frac{-g_{3}V_{\text{DD}}^{\text{(2,13,6,3)}}}{64\sqrt{3}g_{2}}\\\
&&\frac{-V_{\text{DD}}^{\text{(1,5,3,3)}}}{24}&\frac{-V_{\text{DD}}^{\text{(1,5,3,3)}}}{16\sqrt{3}}&\frac{V_{\text{DD}}^{\text{(2,13,6,3)}}}{96\sqrt{2}}&\frac{-\sqrt{5}V_{\text{DD}}^{\text{(1,5,3,3)}}}{48}&\frac{\sqrt{\frac{5}{14}}V_{\text{DD}}^{\text{(2,13,6,3)}}}{32}&\frac{g_{3}V_{\text{DD}}^{\text{(2,13,6,3)}}}{64g_{2}}&\frac{-g_{3}V_{\text{DD}}^{\text{(2,13,6,3)}}}{64\sqrt{3}g_{2}}&\frac{-g_{3}V_{\text{DD}}^{\text{(1,5,3,3)}}}{32\sqrt{3}g_{2}}\\\
&&&0&\frac{-V_{\text{DD}}^{\text{(2,13,6,3)}}}{32\sqrt{6}}&\frac{\sqrt{\frac{5}{3}}V_{\text{DD}}^{\text{(1,5,3,3)}}}{16}&\frac{\sqrt{\frac{15}{14}}V_{\text{DD}}^{\text{(2,13,6,3)}}}{32}&0&\frac{g_{3}V_{\text{DD}}^{\text{(2,13,6,3)}}}{64g_{2}}&\frac{g_{3}V_{\text{DD}}^{\text{(1,5,3,3)}}}{32g_{2}}\\\
&&&&\frac{5V_{\text{DD}}^{\text{(1,5,3,3)}}}{48}&\frac{7V_{\text{DD}}^{\text{(2,13,6,3)}}}{96\sqrt{10}}&\frac{-V_{\text{DD}}^{\text{(2,13,6,3)}}}{16\sqrt{35}}&\frac{-g_{3}V_{\text{DD}}^{\text{(1,5,3,3)}}}{16\sqrt{2}g_{2}}&\frac{g_{3}V_{\text{DD}}^{\text{(1,
5,3,3)}}}{16\sqrt{6}g_{2}}&\frac{-g_{3}V_{\text{DD}}^{\text{(2,13,6,3)}}}{64\sqrt{6}g_{2}}\\\
&&&&&\frac{V_{\text{DD}}^{\text{(1,5,3,3)}}}{24}&\frac{V_{\text{DD}}^{\text{(2,13,6,3)}}}{32\sqrt{14}}&\frac{g_{3}V_{\text{DD}}^{\text{(2,13,6,3)}}}{64\sqrt{5}g_{2}}&\frac{-g_{3}V_{\text{DD}}^{\text{(2,13,6,3)}}}{16\sqrt{15}g_{2}}&\frac{\sqrt{\frac{5}{3}}g_{3}V_{\text{DD}}^{\text{(1,5,3,3)}}}{32g_{2}}\\\
&&&&&&\frac{-3V_{\text{DD}}^{\text{(1,3,3,5)}}}{112}&\frac{-3g_{3}V_{\text{DD}}^{\text{(2,13,6,3)}}}{32\sqrt{70}g_{2}}&\frac{-\sqrt{\frac{3}{70}}g_{3}V_{\text{DD}}^{\text{(2,13,6,3)}}}{32g_{2}}&\frac{\sqrt{\frac{15}{14}}g_{3}V_{\text{DD}}^{\text{(2,13,6,3)}}}{64g_{2}}\\\
&&&&&&&0&0&0\\\ &&&&&&&&0&0\\\ &&&&&&&&&0\\\ \end{array}\right).}$

The OPE potentials for the channels in order listed in Table 1 for
$J^{P}=\frac{5}{2}^{-}$ read

$\displaystyle
V_{\frac{5}{2}^{-}}^{\text{OPE}}=\frac{g_{1}}{f_{\pi}^{2}}\left(\begin{array}[]{cc}V_{\frac{5}{2}^{-}}^{SS}&V_{\frac{5}{2}^{-}}^{SD}\\\
V_{\frac{5}{2}^{-}}^{DS}&V_{\frac{5}{{}^{2}}^{-}}^{DD}\end{array}\right),$
(98)

where

$V_{\frac{5}{2}^{-}}^{SS}=-\frac{1}{4}g_{2}V_{SS},$
$V_{\frac{5}{2}^{-}}^{SD}=\left(\begin{array}[]{cccccccccc}-\frac{g_{2}}{8\sqrt{10}}&-\frac{g_{2}}{8\sqrt{30}}&-\frac{g_{2}}{16}\sqrt{\frac{7}{15}}&-\frac{g_{2}}{16}\sqrt{\frac{7}{5}}&\frac{g_{2}}{8\sqrt{15}}&-\frac{g_{2}}{80}\sqrt{\frac{7}{3}}&-\frac{g_{2}}{20}\sqrt{\frac{7}{2}}&\frac{g_{3}}{16}\sqrt{\frac{3}{10}}&\frac{g_{3}}{16\sqrt{10}}&\frac{g_{3}}{32}\sqrt{\frac{7}{5}}\\\
\end{array}\right)V_{SD},$
$V_{\frac{5}{2}^{-}}^{DS}=\left(\begin{array}[]{cccccccccc}-\frac{g_{2}}{8\sqrt{10}}&-\frac{g_{2}}{8\sqrt{30}}&-\frac{g_{2}}{16}\sqrt{\frac{7}{15}}&-\frac{g_{2}}{16}\sqrt{\frac{7}{5}}&\frac{g_{2}}{8\sqrt{15}}&-\frac{g_{2}}{80}\sqrt{\frac{7}{3}}&-\frac{g_{2}}{20}\sqrt{\frac{7}{2}}&\frac{g_{3}}{16}\sqrt{\frac{3}{10}}&\frac{g_{3}}{16\sqrt{10}}&\frac{g_{3}}{32}\sqrt{\frac{7}{5}}\\\
\end{array}\right)^{T}V_{DS},$

${V_{\frac{5}{2}^{-}}^{DD}=g_{2}\left(\begin{array}[]{cccccccccc}0&\frac{V_{\text{DD}}^{\text{(1,5,3,3)}}}{8\sqrt{3}}&\frac{-V_{\text{DD}}^{\text{(2,13,6,3)}}}{8\sqrt{42}}&0&\frac{V_{\text{DD}}^{\text{(1,5,3,3)}}}{8\sqrt{6}}&\frac{V_{\text{DD}}^{\text{(2,13,6,3)}}}{16\sqrt{210}}&\frac{V_{\text{DD}}^{\text{(2,13,6,3)}}}{8\sqrt{35}}&0&\frac{g_{3}V_{\text{DD}}^{\text{(1,5,3,3)}}}{32g_{2}}&\frac{-g_{3}V_{\text{DD}}^{\text{(2,13,6,3)}}}{32\sqrt{14}g_{2}}\\\
&\frac{V_{\text{DD}}^{\text{(1,5,3,3)}}}{12}&\frac{V_{\text{DD}}^{\text{(2,13,6,3)}}}{24\sqrt{14}}&\frac{V_{\text{DD}}^{\text{(2,13,6,3)}}}{16\sqrt{42}}&\frac{-V_{\text{DD}}^{\text{(1,5,3,3)}}}{24\sqrt{2}}&\frac{-V_{\text{DD}}^{\text{(2,13,6,3)}}}{12\sqrt{70}}&\frac{V_{\text{DD}}^{\text{(2,13,6,3)}}}{8\sqrt{105}}&\frac{g_{3}V_{\text{DD}}^{\text{(1,5,3,3)}}}{32g_{2}}&\frac{g_{3}V_{\text{DD}}^{\text{(1,5,3,3)}}}{16\sqrt{3}g_{2}}&\frac{g_{3}V_{\text{DD}}^{\text{(2,13,6,3)}}}{32\sqrt{42}g_{2}}\\\
&&\frac{-V_{\text{DD}}^{\text{(17,100,51,36)}}}{168}&\frac{-V_{\text{DD}}^{\text{(4,5,12,27)}}}{224\sqrt{3}}&\frac{-V_{\text{DD}}^{\text{(2,13,6,3)}}}{96\sqrt{7}}&\frac{-\sqrt{5}V_{\text{DD}}^{\text{(16,83,48,45)}}}{672}&\frac{\sqrt{\frac{5}{6}}V_{\text{DD}}^{\text{(2,13,6,3)}}}{112}&\frac{-g_{3}V_{\text{DD}}^{\text{(2,13,6,3)}}}{32\sqrt{14}g_{2}}&\frac{g_{3}V_{\text{DD}}^{\text{(2,13,6,3)}}}{32\sqrt{42}g_{2}}&\frac{-g_{3}V_{\text{DD}}^{\text{(17,100,51,36)}}}{224\sqrt{3}g_{2}}\\\
&&&0&\frac{V_{\text{DD}}^{\text{(2,13,6,3)}}}{32\sqrt{21}}&\frac{\sqrt{\frac{5}{3}}V_{\text{DD}}^{\text{(11,61,33,27)}}}{112}&\frac{\sqrt{\frac{5}{2}}V_{\text{DD}}^{\text{(2,13,6,3)}}}{112}&0&\frac{-g_{3}V_{\text{DD}}^{\text{(2,13,6,3)}}}{32\sqrt{14}g_{2}}&\frac{g_{3}V_{\text{DD}}^{\text{(4,5,12,27)}}}{448g_{2}}\\\
&&&&\frac{5V_{\text{DD}}^{\text{(1,5,3,3)}}}{48}&\frac{-\sqrt{\frac{7}{5}}V_{\text{DD}}^{\text{(2,13,6,3)}}}{96}&\frac{-V_{\text{DD}}^{\text{(2,13,6,3)}}}{4\sqrt{210}}&\frac{-g_{3}V_{\text{DD}}^{\text{(1,5,3,3)}}}{16\sqrt{2}g_{2}}&\frac{g_{3}V_{\text{DD}}^{\text{(1,5,3,3)}}}{16\sqrt{6}g_{2}}&\frac{g_{3}V_{\text{DD}}^{\text{(2,13,6,3)}}}{64\sqrt{21}g_{2}}\\\
&&&&&\frac{-V_{\text{DD}}^{\text{(1,17,3,-9)}}}{168}&\frac{V_{\text{DD}}^{\text{(2,13,6,3)}}}{112\sqrt{6}}&\frac{-g_{3}V_{\text{DD}}^{\text{(2,13,6,3)}}}{32\sqrt{70}g_{2}}&\frac{g_{3}V_{\text{DD}}^{\text{(2,13,6,3)}}}{8\sqrt{210}g_{2}}&\frac{g_{3}\sqrt{\frac{5}{3}}V_{\text{DD}}^{\text{(16,83,48,45)}}}{448g_{2}}\\\