# Experimental constraints on the properties of $P_{c}$ states

T. J. Burns  Department of Physics, Swansea University, Singleton Park,
Swansea, SA2 8PP, UK.  E. S. Swanson  Department of Physics and Astronomy,
University of Pittsburgh, Pittsburgh, PA 15260, USA.

###### Abstract

Using new experimental data on photoproduction and $\Lambda_{b}^{0}$ decays,
we derive constraints on the properties of the LHCb $P_{c}$ states. We
conclude that $P_{c}(4312)$, $P_{c}(4380)$ and $P_{c}(4440)$ can be described
as $\Sigma_{c}\bar{D}$, $\Sigma_{c}^{*}\bar{D}$ and $\Sigma_{c}\bar{D}^{*}$
molecules, but that $P_{c}(4457)$ does not fit into the same picture. Based on
the apparent absence of additional partner states, and the striking disparity
between $\Lambda_{c}^{+}\bar{D}^{0}$ and $\Lambda_{c}^{+}\bar{D}^{*0}$ decays,
we conclude that $P_{c}(4440)$ has $3/2^{-}$ quantum numbers. Using heavy-
quark symmetry we predict large experimental signals for $P_{c}$ states in
$\eta_{c}pK^{-}$, $\Lambda_{c}^{+}\bar{D}^{*0}K^{-}$, and
$\Sigma_{c}^{(*)}\bar{D}K^{-}$. We also argue that current experimental data
on photoproduction is almost at the level of sensitivity required to observe
$P_{c}$ states.

##  I Introduction

The LHCb $P_{c}$ states have so far been observed only in the $J/\psi\,p$
spectrum of $\Lambda_{b}\to J/\psi\,p\,K^{-}$ decays Aaij:2015tga ;
Aaij:2016phn ; Aaij:2019vzc , but experimental data from other processes can
also usefully constrain their properties. In this paper we explore the
consequences of some recent experimental observations whose implications for
the phenomenology of the $P_{c}$ states have not been recognised in the
literature. In particular, we consider: tighter constraints on photoproduction
and $J/\psi\,p$ decays from the $J/\psi$-007 experiment, measurements of
$\Lambda_{b}^{0}$ decays to $\Lambda_{c}^{+}\bar{D}^{0}K^{-}$,
$\Lambda_{c}^{+}\bar{D}^{*0}K^{-}$ and $\eta_{c}pK^{-}$ (including limits on
$P_{c}$ fit fractions), and the apparent absence of $P_{c}$ partners with
higher mass.

In the molecular scenario, we find that these new experimental results imply
constraints on the quantum numbers of $P_{c}$ states, and give striking
predictions for prominent production channels that can be tested in
experiment. Our observations follow directly from experiment, with minimal
theoretical assumptions. At most, we rely only on symmetry principles which
are well-justified theoretically and empirically: heavy-quark symmetry, and
the dominance of colour-favoured processes in weak transitions.

We begin with an overview of molecular scenarios for the $P_{c}$ states (Sec.
II), and then review the recent experimental data which forms the basis of our
analysis (Sec. III). We then consider the resulting phenomenology associated
with various decay modes, namely $J/\psi\,p$ (Sec. IV), $\eta_{c}p$ (Sec. V),
$\Lambda_{c}^{+}\bar{D}^{0}$ (Sec. VI), $\Lambda_{c}\bar{D}^{(*)}\pi$ (Sec.
VII), $\Lambda_{c}^{+}\bar{D}^{*0}$ (Sec. VIII) and $\Sigma_{c}^{(*)}\bar{D}$
(Sec. IX). We finally consider the implications of the apparent absence of
heavier partners to the $P_{c}$ states (Sec. X), and summarise our main
results (Sec. XI).

##  II Scenarios

In the discovery paper Aaij:2015tga , an amplitude analysis of $\Lambda_{b}\to
J/\psi\,p\,K^{-}$ identified two states decaying to $J/\psi\,p$: the broad
$P_{c}(4380)$ and narrower $P_{c}(4450)$. A model-independent analysis
Aaij:2016phn  confirmed the importance of including these contributions in
adequately describing the data. A subsequent analysis, with an order of
magnitude more data Aaij:2019vzc , discovered an additional new state
$P_{c}(4312)$, and resolved the initial $P_{c}(4450)$ into two distinct,
narrow peaks, $P_{c}(4440)$ and $P_{c}(4457)$.

The original $P_{c}(4380)$ did not feature in the newer analysis, which was
sensitive only the narrow features in the spectrum. Because its discovery
relied on an amplitude model which has been superseded, the existence or
otherwise of $P_{c}(4380)$ remains to be determined experimentally, as
explained in the note in the appendix of ref. Aaij:2019vzc . In our discussion
we will refer to the “$P_{c}(4380)$”, although we will not insist that its
measured properties are consistent with those extracted from the original
analysis, which is regarded as obsolete.

Because of their proximity to thresholds, the $P_{c}$ structures are widely
interpreted as molecular states whose wavefunctions are dominated by S-wave
combinations of $\Sigma_{c}^{(*)}\bar{D}^{(*)}$ constituents Wu:2010vk ;
Roca:2015dva ; He:2015cea ; Chen:2015loa ; Karliner:2015ina ; Ortega:2016syt ;
Shimizu:2016rrd ; Yamaguchi:2016ote ; Yamaguchi:2017zmn ; Shimizu:2017xrg ;
Shimizu:2018ran ; Du:2019pij ; Gutsche:2019mkg ; Liu:2019zvb ;
Valderrama:2019chc ; Sakai:2019qph ; Guo:2019kdc ; He:2019ify ; Liu:2019tjn ;
Chen:2019asm ; Burns:2019iih ; He:2019rva ; Xiao:2019aya ; Peng:2020xrf ;
Xu:2020gjl ; Yalikun:2021bfm ; Du:2021fmf . So $P_{c}(4312)$ is a $1/2^{-}$
$\Sigma_{c}\bar{D}$ state, while $P_{c}(4380)$ is $3/2^{-}$
$\Sigma_{c}^{*}\bar{D}$ state, albeit with significantly smaller width that
the state observed in the original LHCb analysis. The $P_{c}(4440)$ and
$P_{c}(4457)$ are both $\Sigma_{c}\bar{D}^{*}$ states, and can be assigned to
either $1/2^{-}$ and $3/2^{-}$, or $3/2^{-}$ and $1/2^{-}$, respectively. We
summarise these assignments in the first two rows of Table 1, where the
Scenarios “A” and “B” correspond to those of ref. Liu:2019tjn .

|  $P_{c}(4312)$ |  $P_{c}(4380)$ |  $P_{c}(4440)$ |  $P_{c}(4457)$
---|---|---|---|---
Scenario A |  $1/2^{-}~{}\Sigma_{c}\bar{D}$ |  $3/2^{-}~{}\Sigma_{c}^{*}\bar{D}$ |  $1/2^{-}~{}\Sigma_{c}\bar{D}^{*}$ |  $3/2^{-}~{}\Sigma_{c}\bar{D}^{*}$
Scenario B |  $1/2^{-}~{}\Sigma_{c}\bar{D}$ |  $3/2^{-}~{}\Sigma_{c}^{*}\bar{D}$ |  $3/2^{-}~{}\Sigma_{c}\bar{D}^{*}$ |  $1/2^{-}~{}\Sigma_{c}\bar{D}^{*}$
Scenario C |  $1/2^{-}~{}\Sigma_{c}\bar{D}$ |  $3/2^{-}~{}\Sigma_{c}^{*}\bar{D}$ |  $3/2^{-}~{}\Sigma_{c}\bar{D}^{*}$ |  see text
Table 1: Quantum numbers and dominant degrees of freedom in the various
scenarios.

The implicit assumption in most models is that the states are bound with
respect to the nearest $\Sigma_{c}^{(*)}\bar{D}^{(*)}$ threshold, but
experimentally, this has only been established for $P_{c}(4440)$. In Fig. 1 we
show the binding energies of $P_{c}(4312)$, $P_{c}(4440)$ and $P_{c}(4457)$
with respect to their nearest thresholds. While $P_{c}(4440)$ is unambiguously
bound with respect to $\Sigma_{c}^{+}\bar{D}^{*0}$, the masses of the other
states are consistent, within uncertainties, with the thresholds.

The case of $P_{c}(4457)$ is particularly noteworthy. Its mass is consistent
not only with the the $\Sigma_{c}^{+}\bar{D}^{*0}$ threshold but also, more
strikingly, with $\Lambda_{c}(2595)\bar{D}$ threshold: the difference in the
central values is just 0.2 MeV. This naturally suggests that for
$P_{c}(4457)$, the $\Sigma_{c}\bar{D}^{*}$ bound state scenario is not the
only possibility. In a forthcoming paper we explore a number of viable
alternatives inprep . Given its proximity to threshold, the most prosaic
option is that it is a threshold cusp arising from $\Sigma_{c}\bar{D}^{*}\to
J/\psi\,p$ or $\Lambda_{c}(2595)\bar{D}\to J/\psi\,p$, or some combination of
the two. Indeed the cusp scenario is favoured in the analysis of ref.
Kuang:2020bnk . Alternatively, the peak could arise as a triangle singularity
in $\Lambda_{c}(2595)\bar{D}$ or, as in our previous paper Burns:2019iih , as
a genuine resonance, but with $1/2^{+}$ quantum numbers and
$\Lambda_{c}(2595)\bar{D}$ degrees of freedom. We find that all of these
scenarios give a very good fit to the $\Lambda_{b}\to J/\psi\,p\,K^{-}$ data
inprep .

Figure 1: The binding energies of $P_{c}(4312)$, $P_{c}(4440)$ and
$P_{c}(4457)$ with respect to their nearest thresholds. The $P_{c}(4380)$ is
not shown, due to the large uncertainty in its mass (which overlaps
considerably with the $\Sigma_{c}^{*}\bar{D}$ threshold).

It turns out to be very liberating to abandon the hypothesis that
$P_{c}(4457)$ is a bound $\Sigma_{c}\bar{D}^{*}$ state, and not only because
the alternative scenarios give a good fit to data. As we argue below, there
are some phenomenological problems with the usual modelling, all of which can
be traced to the assumption that $P_{c}(4457)$ is a bound
$\Sigma_{c}\bar{D}^{*}$ state. By abandoning this assumption we avoid these
problems.

Our alternative Scenario “C”, also shown in the Table 1, differs from the
other two in no longer assuming that $P_{c}(4457)$ is a
$\Sigma_{c}\bar{D}^{*}$ bound state. The assignments of the other states are
consistent with Scenario B; the rationale for preferring $3/2^{-}$ quantum
numbers for $P_{c}(4440)$ is explained below. Note that our Scenario C
actually incorporates several possibilities, corresponding to different
explanations for $P_{c}(4457)$.

Of course other scenarios are also discussed in the literature. In particular,
we highlight refs. Nakamura:2021qvy ; Nakamura:2021dix  in which non-resonant
interpretations of $P_{c}(4312)$ have been discussed.

##  III Experimental data

In Table 2 we show the experimental data which forms the basis of much of our
discussion. For each of four possible $\Lambda_{b}$ decays (indicated in the
top row), we show the three-body branching fraction ($\mathcal{B}$) and, where
data are available, a measure ($\mathcal{R}$) of the contribution of $P_{c}$
states in the fit, defined below. The data in the last three columns are quite
new, and their significance has not yet been appreciated in literature.

The three-body branching fractions $\mathcal{B}$ for $\Lambda_{b}^{0}\to
J/\psi\,pK^{-}$ and $\Lambda_{b}^{0}\to\eta_{c}pK^{-}$ are from LHCb
Aaij:2015fea ; Aaij:2020mlx ; ParticleDataGroup:2020ssz , while those of
$\Lambda_{b}^{0}\to\Lambda_{c}^{+}\bar{D}^{0}K^{-}$ and
$\Lambda_{b}^{0}\to\Lambda_{c}^{+}\bar{D}^{*0}K^{-}$ are from Stahl
Stahl:2018eme , combining the measured ratios

$\displaystyle\frac{\mathcal{B}(\Lambda_{b}^{0}\to\Lambda_{c}^{+}\bar{D}^{0}K^{-})}{\mathcal{B}(\Lambda_{b}^{0}\to\Lambda_{c}^{+}D_{s}^{-})}$
$\displaystyle=(14.04\pm 0.58\pm 0.33\pm 0.45)\%,$ (1)
$\displaystyle\frac{\mathcal{B}(\Lambda_{b}^{0}\to\Lambda_{c}^{+}\bar{D}^{*0}K^{-})}{\mathcal{B}(\Lambda_{b}^{0}\to\Lambda_{c}^{+}D_{s}^{-})}$
$\displaystyle=(43.5\pm 1.4^{+1.2}_{-0.8}\pm 1.4)\%,$ (2)

from LHCb data, with the Particle Data Group (PDG) value for the denominators
ParticleDataGroup:2020ssz

$\mathcal{B}(\Lambda_{b}^{0}\to\Lambda_{c}^{+}D_{s}^{-})=(1.1\pm 0.1)\%.$ (3)

|  $\Lambda_{b}^{0}\to J/\psi\,pK^{-}$ |  $\Lambda_{b}^{0}\to\eta_{c}pK^{-}$ |  $\Lambda_{b}^{0}\to\Lambda_{c}^{+}\bar{D}^{0}K^{-}$ |  $\Lambda_{b}^{0}\to\Lambda_{c}^{+}\bar{D}^{*0}K^{-}$
---|---|---|---|---
$\mathcal{B}~{}(/10^{-4})$ |  $3.2^{+0.6}_{-0.5}$ |  $1.06\pm 0.26$ |  $15\pm 2$ |  $48\pm 5$
$\mathcal{R}~{}(/10^{-2})$ |  |  |  $1/2^{-}\qquad 3/2^{-}$ |
$P_{c}(4312)$ |  $0.30\pm 0.07{}^{+0.34}_{-0.09}$ |  $<24$ |  $<0.4$ |
$P_{c}(4440)$ |  $1.11\pm 0.33{}^{+0.22}_{-0.10}$ |  |  $<0.6\phantom{0}\quad<0.45$ |
$P_{c}(4457)$ |  $0.53\pm 0.16{}^{+0.15}_{-0.13}$ |  |  $<0.45\quad<0.8$ |
Table 2: Experimental data on the three-body branching fractions
$\mathcal{B}$, and $P_{c}$ fit fractions $\mathcal{R}$, in $\Lambda_{b}^{0}$
decays. The sources of experimental data are described in the text. All upper
limits are at 95% confidence level. As shown, in the case of
$\Lambda_{b}^{0}\to\Lambda_{c}^{+}\bar{D}^{0}K^{-}$, the limits on fit
fractions depend on the assumed quantum numbers ($1/2^{-}$ or $3/2^{-}$) of
the $P_{c}$ states.

The quantity $\mathcal{R}$ is a measure of the contribution of the $P_{c}$
states in the fit, although the experimental procedures in how this number is
computed differ. For $\Lambda_{b}^{0}\to J/\psi\,pK^{-}$ and
$\Lambda_{b}^{0}\to\eta_{c}pK^{-}$, the numbers are from refs. Aaij:2019vzc ;
Aaij:2020mlx , where they are described as “relative contributions”, and are
identified in terms of branching fractions as

$\mathcal{R}(J/\psi\,p)=\frac{{\mathcal{B}}(\Lambda_{b}^{0}\to
P_{c}^{+}K^{-})\,{\mathcal{B}}(P_{c}^{+}\to
J/\psi\,p)}{{\mathcal{B}}(\Lambda_{b}^{0}\to J/\psi\,p\,K^{-})},$ (4)

and

${\mathcal{R}(\eta_{c}p)}=\frac{{\mathcal{B}}(\Lambda_{b}^{0}\to
P_{c}^{+}K^{-})\,{\mathcal{B}}(P_{c}^{+}\to\eta_{c}p)}{{\mathcal{B}}(\Lambda_{b}^{0}\to\eta_{c}pK^{-})}.$
(5)

The numbers for $\Lambda_{b}^{0}\to\Lambda_{c}^{+}\bar{D}^{0}K^{-}$, from the
amplitude analysis of Piucci Piucci:2019vsk , are “fit fractions”, defined in
the usual way: the ratio, for a single resonance versus the full amplitude, of
the phase space integrals of the square of the matrix element. (In the
original $P_{c}$ discovery paper Aaij:2015tga , the contributions of
$P_{c}(4380)$ and $P_{c}(4450)$ were also quantified in terms of fit
fractions.) We adopt the usual interpretation of the fit fraction, which is
widespread in the experimental literature and the PDG
ParticleDataGroup:2020ssz , in terms of branching fractions, namely

${\mathcal{R}(\Lambda_{c}^{+}\bar{D}^{0})}=\frac{{\mathcal{B}}(\Lambda_{b}^{0}\to
P_{c}^{+}K^{-})\,{\mathcal{B}}(P_{c}^{+}\to\Lambda_{c}^{+}\bar{D}^{0})}{{\mathcal{B}}(\Lambda_{b}^{0}\to\Lambda_{c}^{+}\bar{D}^{0}K^{-})}.$
(6)

Note that the limits on the fit fractions depend on the assumed quantum
numbers ($1/2^{-}$ or $3/2^{-}$) of the $P_{c}$ states, as shown in the table.

The difference between the “relative contributions” of refs. Aaij:2019vzc ;
Aaij:2020mlx  and the “fit fractions” of refs. Aaij:2015tga ; Piucci:2019vsk
relates to the treatment of interference terms, and is discussed in ref.
Aaij:2019vzc . There is however no difference in the way the numbers are
interpreted in terms of branching fractions, as shown in equations (4), (5)
and (6). Hence we will use the generic term “fit fraction”, and the same
symbol ${\mathcal{R}}$, for both quantities.

We remark, however, that for both the “relative contributions” and “fit
fractions”, the standard interpretation of these experimental numbers in terms
of branching fractions has an implicit assumption, which is not necessarily
justified. The factorisation of the numerator into a product of branching
fractions is legitimate if the $P_{c}$ signal is due to a resonance, but not,
for example, if it arises from a kinematical singularity or threshold cusp, in
which case it is not possible (conceptually or mathematically) to separate the
production ($\Lambda_{b}^{0}\to P_{c}^{+}K^{-}$) and decay (such as
$P_{c}^{+}\to J/\psi\,p$). We discussed this point previously in relation to
the $X(2900)$ states Burns:2020xne , for which several competing
interpretations are possible Burns:2020epm .

Much of our analysis below involves manipulations of measured $\mathcal{R}$
values, and in particular it relies on the standard factorisation of the
branching fractions in the numerators. In our preferred Scenario C, we can
only assume this factorisation for $P_{c}(4312/4380/4440)$, which we are
treating as $\Sigma_{c}^{(*)}\bar{D}^{(*)}$ resonances. We cannot assume
factorisation for $P_{c}(4457)$ because it is not justified for some of the
scenarios applicable to this state (such as threshold cusp and triangle
singularity); hence in much of the discussion below we will not consider
$P_{c}(4457)$.

In computing the fit fractions ${\mathcal{R}(\Lambda_{c}^{+}\bar{D}^{0})}$,
the masses and widths of $P_{c}$ states, as determined in previous
experiments, are taken as inputs Piucci:2019vsk . For this reason we only
quote the results for $P_{c}(4312/4440/4457)$, rather than those of the
precursor states $P_{c}(4380/4450)$, whose mass and width measurements are
considered to be obsolete Aaij:2019vzc .

There is a caveat on the limits on
$\Lambda_{b}^{0}\to\Lambda_{c}^{+}\bar{D}^{0}K^{-}$ fit fractions that appear
in the table. These were obtained by testing separately for the presence of a
single state in the data – either $P_{c}(4312)$, $P_{c}(4440)$ or
$P_{c}(4457)$ – rather than all three simultaneously. (The latter approach was
found to be numerically unfeasible.)

While this suggests a degree of caution is warranted in interpreting the
figures, it is reassuring to consider the results obtained for the precursor
states $P_{c}(4380)$ and $P_{c}(4450)$, where both types of analyses were
performed. The model in which both states were included simultaneously gives
comparable, but somewhat tighter, upper limits on the branching fractions,
compared to the model in which the presence of each state was tested
separately. If the same applies in the case of $P_{c}(4312/4440/4457)$, the
upper limits quoted in the table would be conservative.

The discussion so far has concentrated on $P_{c}$ states in 3-body
$\Lambda_{b}$ decays. Another possibility, much discussed in the literature,
is to search for the states in photoproduction Wang:2015jsa ; Karliner2015a ;
Kubarovsky:2016whd ; Ali:2019lzf ; Blin:2016dlf ; Joosten2021 . Currently
there is no evidence for $P_{c}$ states in $\gamma p\to P_{c}\to J/\psi\,p$,
but the measured upper limits, which have recently become tighter, have
interesting implications.

In particular, recall that the absence of the states in $\gamma p\to P_{c}\to
J/\psi\,p$ implies upper limits, via vector meson dominance, on the $P_{c}\to
J/\psi\,p$ branching fractions. After the initial discovery of the original
LHCb states $P_{c}(4450)$ and $P_{c}(4380)$, analysis of photoproduction data
available at the time implied an upper limit on the $J/\psi\,p$ branching
fractions at the percent level Wang:2015jsa . More recently, the GlueX
experiment Ali:2019lzf  obtained similar upper limits from their new data,
using a variant of the JPAC model for the amplitude Blin:2016dlf . The GlueX
results assume $3/2^{-}$ quantum numbers for all the states, however choosing
a different assignment is not expected to change the results drastically. The
results are

$\mathcal{B}(P_{c}^{+}\to J/\psi\,p)<(2.3\%\div 4.6\%),$ (7)

depending on which of $P_{c}(4312/4440/4457)$ is being considered.

These stringent upper limits have recently become even tighter: preliminary
results from another Jefferson Lab experiment, $J/\psi$-007, indicate that the
$J/\psi\,p$ upper limits are smaller than the GlueX limits by a factor of
approximately 10 Joosten2021 .

We may summarise the situation, rather roughly, as

${\mathcal{B}}(P_{c}^{+}\to J/\psi\,p)<(\textrm{a few})\times 10^{-3}.$ (8)

This is adequate for present purposes, as our arguments below only require
estimates of the scales of the branching fractions, rather than their precise
values. (We are also limited in not having an estimate of the systematic
uncertainty associated with the assumed $3/2^{-}$ quantum numbers in the
JPAC/GlueX analysis.)

##  IV $J/\psi\,p$ decays

We now come to the main thrust of the paper, exploring the implications of the
experimental data discussed in the previous section. We begin with $J/\psi\,p$
decays, and note that the stringent upper limit (8) is in conflict with the
vast majority of models, in some cases by orders of magnitude Wu:2010jy ;
Wu:2010vk ; Roca:2015dva ; Shen:2016tzq ; Lu:2016nnt ; Ortega:2016syt ;
Lin:2017mtz ; Huang:2018wed ; Xiao:2019mvs ; Wang:2019spc ; Xu:2019zme ;
Eides:2019tgv ; Xiao:2019mvs ; Xiao:2020frg ; Ruangyoo:2021aoi . Our own
estimate Burns:2019iih , for $P_{c}(4440)$, is one of very few that is
consistent with (8). Similarly, Dong et al.  Dong:2020nwk  obtain a
$P_{c}(4312)$ branching fraction which is near to the upper limit (8). Eides
et al.  Eides:2018lqg  computed $J/\psi\,p$ branching fractions in both the
molecular and hadrocharmonium pictures, noting a very different pattern in
decays; their results for the molecular scenario, but not the hadrocharmonium
scenario, are consistent with (8).

Superficially, the tight upper limit on $J/\psi\,p$ decays may appear
surprising, considering that the $P_{c}$ states were of course discovered as
prominent peaks in the $J/\psi\,p$ spectrum. This illustrates the point that
in quantifying the prominence of a structure in the 2-body spectrum of a
3-body decay, the most relevant quantity is not the 2-body decay branching
fraction itself, but the fit fraction $\mathcal{R}$. Recalling equation (4),
the $P_{c}$ fit fractions are a measure of the fraction of all
$J/\psi\,pK^{-}$ events that are produced via $\Lambda_{b}^{0}\to
P_{c}^{+}K^{-},P_{c}^{+}\to J/\psi\,p$. Obviously, the fit fractions therefore
depend not only on the properties of the $P_{c}$ states themselves (the
production and decay branching fractions in the numerator), but also on the
sum total of all other production mechanisms resulting in the $J/\psi\,pK^{-}$
final state (the denominator). It turns out that tree-level contributions to
$J/\psi\,p\,K^{-}$ are suppressed, resulting in enhanced $P_{c}$ fit
fractions.

The suppression is explained in Fig. 2. There are three possible flavour
topologies at the weak vertex (top panel), and for each, there is a
corresponding three-body tree-level diagram that produces a kaon (bottom
panel). The weak vertex in Fig. 2(a) is colour-enhanced, whereas those of Fig.
2(b) and 2(c) are colour-suppressed Burns:2019iih . On this basis we expect,
very roughly,

$\displaystyle{\mathcal{B}}(\Lambda_{b}^{0}\to\Lambda_{c}^{+}\bar{D}^{(*)0}K^{-})$
$\displaystyle>>{\mathcal{B}(\Lambda_{b}^{0}\to
c\bar{c}\,p\,K^{-})}\approx{\mathcal{B}(\Lambda_{b}^{0}\to\Sigma_{c}^{(*)}\,\bar{D}^{(*)}\,K^{-})},$
(9)

which is consistent with the numbers in Table 2.

Figure 2: Tree level contributions to three-body $\Lambda_{b}^{0}$ decays.
Diagram (a) is colour-enhanced, whereas diagrams (b) and (c) are colour-
suppressed.

We conclude that the $P_{c}$ structures in $\Lambda_{b}^{0}\to
J/\psi\,p\,K^{-}$ are prominent (namely, have large fit fractions) in part
because of the suppression of tree-level $\Lambda_{b}^{0}\to J/\psi\,p\,K^{-}$
decays. Intriguingly, the situation is similar for several other exotic states
observed in 3-body weak decays. For example, $\chi_{c1}(3872)$ [formerly
$X(3872)$] has comparatively small branching fraction to
$J/\psi\pi^{+}\pi^{-}$ and $J/\psi\omega$ ParticleDataGroup:2020ssz , but was
discovered and extensively studied in $B\to J/\psi\pi^{+}\pi^{-}K$ and $B\to
J/\psi\omega K$, transitions for which the tree-level diagrams are also
colour-suppressed. (Note that $\chi_{c1}(3872)$ has much larger branching
fraction to $D^{0}\bar{D}^{*0}$, for example, but the tree-level diagram
$D^{0}\bar{D}^{*0}K$ is colour-enhanced, and much larger.) In ref.
Burns:2020xne  we noted a similar mechanism enhancing the $X(2900)$ signal in
$B^{+}\to D^{+}X,X\to D^{-}K^{+}$ Aaij:2020hon ; Aaij:2020ypa .

We return now to the tighter upper limit (8) on $J/\psi\,p$ decays, and note
that it has striking implications for $P_{c}$ production in $\Lambda_{b}^{0}$
decays. Upper limits on ${\mathcal{B}}(P_{c}^{+}\to J/\psi\,p)$ imply lower
limits on ${\mathcal{B}}(\Lambda_{b}^{0}\to P_{c}^{+}\,K^{-})$ Cao:2019kst .
Following our previous paper Burns:2019iih , in Fig. 3 we plot
${\mathcal{B}}(\Lambda_{b}^{0}\to P_{c}^{+}\,K^{-})$ as a function of
${\mathcal{B}}(P_{c}^{+}\to J/\psi\,p)$, using equation (4), and the
experimental values (Table 2) for ${\mathcal{B}}(\Lambda_{b}^{0}\to
J/\psi\,p\,K^{-})$ and ${\mathcal{R}}(J/\psi\,p)$. We have also included
$P_{c}(4380)$ in the plot, using the measured value Aaij:2015tga

$\displaystyle\mathcal{R}(J/\psi\,p)$ $\displaystyle=(8.4\pm 0.7\pm 4.2)\%,$
(10)

although as noted below, this part of the plot should be interpreted with some
caution. Note that in the plot we are showing the central values only; we do
not include error bars, because the discussion below concerns the overall
scale of the branching fractions, not their precise values.

Figure 3: The branching fractions $\mathcal{B}(\Lambda_{b}^{0}\to
P_{c}^{+}K^{-})$ as a function of $\mathcal{B}(P_{c}^{+}\to J/\psi\,p)$,
obtained as described in the text.

Combining Fig. 3 with equation (8) we conclude that
$\mathcal{B}(\Lambda_{b}^{0}\to P_{c}^{+}K^{-})$ is at least
${\mathcal{O}}(10^{-3})$ for $P_{c}(4312/4440/4457)$, and
${\mathcal{O}}(10^{-2})$ for $P_{c}(4380)$. (These are larger by a factor of
10 than our previous estimates Burns:2019iih , because the limit on the
$J/\psi\,p$ branching fraction is now smaller by a factor of 10.) Strikingly,
these numbers are comparable to the largest measured two-body branching
fraction for $\Lambda_{b}^{0}$ decays ParticleDataGroup:2020ssz

${\mathcal{B}}(\Lambda_{b}^{0}\to\Lambda_{c}^{+}D_{s}^{-})=(1.1\pm 0.1)\%.$
(11)

The comparison is quite awkward because naively one would expect the
production of multiquark states (whether molecular or compact) to be
suppressed compared to that of conventional hadrons, and indeed that is what
is observed in other sectors. The closest analogy is with $\chi_{c1}(3872)$,
for which the measured branching fraction ParticleDataGroup:2020ssz ,

${\mathcal{B}}(B^{+}\to\chi_{c1}(3872)K^{+})=(2.1\pm 0.7)\times 10^{-4},$ (12)

is around a factor of fifty smaller than the corresponding branching fractions
for conventional hadrons $B^{+}\to D_{s}^{(*)}\bar{D}^{(*)}$. Indeed, if
anything, we would expect an even stronger relative suppression for the
$P_{c}$ states because, unlike in the case of $\chi_{c1}(3872)$, their
dominant wavefunction components cannot be produced in colour-favoured
processes Burns:2019iih .

With this comparison in mind, the production branching fractions for
$P_{c}(4380)$ seem implausibly large, but as noted previously, the numbers in
this case should be treated with caution: we have used a measured fit fraction
which, along with other properties of this state, are now considered to be
obsolete Aaij:2019vzc .

But even the less dramatic numbers for $P_{c}(4312/4440/4457)$ present a
challenge for models, and of course, the challenge becomes more acute as the
limits on branching fractions ${\mathcal{B}}(P_{c}^{+}\to J/\psi\,p)$ become
tighter. This suggests that ${\mathcal{B}}(P_{c}^{+}\to J/\psi\,p)$ cannot be
much less than the current upper bound (8), otherwise
$\mathcal{B}(\Lambda_{b}^{0}\to P_{c}^{+}K^{-})$ will become implausibly
large. Indirectly, it suggests that the sensitivity required to observe
$P_{c}$ states in photoproduction is not much more than that of the
$J/\psi$-007 experiment.

Note that the production branching fractions implied by the above analysis are
orders of magnitude larger than those predicted by the effective Lagrangian
approach of ref. Wu:2019rog .

##  V $\eta_{c}p$ decays

For $1/2^{-}$ states the S-wave decays to $J/\psi\,p$ and $\eta_{c}p$ are
related by heavy-quark symmetry Voloshin:2019aut ; Sakai:2019qph . In our
Scenario C there is a single $1/2^{-}$ state, the $P_{c}(4312)$ with
$\Sigma_{c}\bar{D}$ constituents. Ignoring differences due to phase space, the
relation among branching fractions is

$\frac{{\mathcal{B}}[P_{c}(4312)\to\eta_{c}p]}{{\mathcal{B}}[P_{c}(4312)\to
J/\psi\,p]}=3.$ (13)

We conclude, by comparison to equation (8), that

${\mathcal{B}}[P_{c}(4312)\to\eta_{c}p]<{\mathcal{O}}(1\%)$ (14)

Nevertheless, the experimental prospects in $\Lambda_{b}\to\eta_{c}pK^{-}$
decays are encouraging: as noted previously, the relevant quantity is not the
branching fraction ${\mathcal{B}}(P_{c}\to J/\psi\,p)$, but the fit fraction
${\mathcal{R}}(\eta_{c}p)$. From equations (4) and (5) we find

$\frac{\mathcal{R}(\eta_{c}p)}{\mathcal{R}(J/\psi\,p)}=3~{}\frac{{\mathcal{B}}(\Lambda_{b}^{0}\to
J/\psi\,p\,K^{-})}{{\mathcal{B}}(\Lambda_{b}^{0}\to\eta_{c}pK^{-})}$ (15)

and thus, using the numbers from Table 2, we predict a substantial fit
fraction for $P_{c}(4312)$:

${\mathcal{R}}(\eta_{c}p)=(2.7^{+4.3}_{-2.0})\%$ (16)

This is large in comparison to the fit fractions for the discovery mode
$\Lambda_{b}^{0}\to J/\psi\,p\,K^{-}$, suggesting the experimental prospects
are encouraging. Note that the prediction is consistent with the experimental
upper limit in Table 2.

As for the other states, for $P_{c}(4457)$ we cannot make any prediction for
$\eta_{c}p$ decays in our Scenario C, without further modelling. (As noted
previously, there are several viable scenarios for this state.) However for
$P_{c}(4380)$ or $P_{c}(4440)$ we anticipate, on general grounds, negligible
$\eta_{c}p$ signals, as they both have $3/2^{-}$ quantum numbers and thus do
not couple to $\eta_{c}p$ in S-wave.

This is quite different from Scenarios A and B, in which one of $P_{c}(4440)$
or $P_{c}(4457)$ is a $\Sigma_{c}\bar{D}^{*}$ bound state with $1/2^{-}$
quantum numbers, and thus couples to $\eta_{c}p$ in S-wave. Drawing on the
results of ref. Voloshin:2019aut , we find

$\frac{\mathcal{R}(\eta_{c}p)}{\mathcal{R}(J/\psi\,p)}=\frac{3}{25}~{}\frac{{\mathcal{B}}(\Lambda_{b}^{0}\to
J/\psi\,p\,K^{-})}{{\mathcal{B}}(\Lambda_{b}^{0}\to\eta_{c}pK^{-})}.$ (17)

This implies, in comparison to equation (8), that if $P_{c}(4440)$ or
$P_{c}(4457)$ is a $1/2^{-}$ state,

${\mathcal{B}}[P_{c}(4440/4457)\to\eta_{c}p]<{\mathcal{O}}(10^{-3}).$ (18)

In terms of fit fractions, in Scenario A we predict

${\mathcal{R}}(\eta_{c}p)=\left(0.40^{+0.31}_{-0.29}\right)\%$ (19)

for $P_{c}(4440)$ and negligible $\eta_{c}p$ for $P_{c}(4457)$, whereas for
Scenario B we predict

${\mathcal{R}}(\eta_{c}p)=\left(0.19\pm 0.16\right)\%$ (20)

for $P_{c}(4457)$, and negligible $\eta_{c}p$ for $P_{c}(4440)$. The
${\mathcal{R}}(\eta_{c}p)$ fit fractions in these scenarios are comparable to
the measured ${\mathcal{R}}(J/\psi\,p)$ fit fractions in Table 2, suggesting
such measurements may be within reach in future analyses. Confronting these
predictions with data can discriminate among Scenarios A, B and C.

##  VI $\Lambda_{c}^{+}\bar{D}^{0}$ decays

We now turn to $\Lambda_{c}^{+}\bar{D}^{0}$ decays, which in many models are
expected to be prominent channels. As we will see, this is not borne out by
experimental data.

For a given $P_{c}$ state, combining equations (4) and (6) yields a relation
between the $\Lambda_{c}^{+}\bar{D}^{0}$ and $J/\psi\,p$ branching fractions

$\frac{{\mathcal{B}}(P_{c}^{+}\to\Lambda_{c}^{+}\bar{D}^{0})}{{\mathcal{B}}(P_{c}^{+}\to
J/\psi\,p)}=\frac{{\mathcal{R}}(\Lambda_{c}^{+}\bar{D}^{0})}{{\mathcal{R}}(J/\psi\,p)}\frac{{\mathcal{B}(\Lambda_{b}^{0}\to\Lambda_{c}^{+}\bar{D}^{0}K^{-})}}{{\mathcal{B}(\Lambda_{b}^{0}\to
J/\psi\,p\,K^{-})}}.$ (21)

Taking the experimental data from Table 2 we obtain

$\frac{{\mathcal{B}}(P_{c}^{+}\to\Lambda_{c}^{+}\bar{D}^{0})}{{\mathcal{B}}(P_{c}^{+}\to
J/\psi\,p)}<(1.9\div 7.1),$ (22)

depending on which of the states $P_{c}(4312/4440/4457)$ is considered, and
its assumed quantum numbers. In combination with the photoproduction upper
limit (8), we arrive at the surprising result

${\mathcal{B}}(P_{c}^{+}\to\Lambda_{c}^{+}\bar{D}^{0})\lesssim\mathcal{O}(1\%).$
(23)

We will show later that this limit argues in favour of Scenario C.

##  VII $\Lambda_{c}\bar{D}^{(*)}\pi$ decays

In previous sections we observed that the $P_{c}$ branching fractions to
$J/\psi\,p$, $\eta_{c}p$ and $\Lambda_{c}^{+}\bar{D}^{0}$ are tiny. This
implies that their measured decay widths,

$\displaystyle\Gamma[P_{c}(4312)]$ $\displaystyle=9.8\pm
2.7^{+3.7}_{-4.5}~{}\textrm{MeV},$ (24) $\displaystyle\Gamma[P_{c}(4440)]$
$\displaystyle=20.6\pm 4.9^{+8.7}_{-10.1}~{}\textrm{MeV},$ (25)
$\displaystyle\Gamma[P_{c}(4457)]$ $\displaystyle=6.4\pm
2.0^{+5.7}_{-1.9}~{}\textrm{MeV},$ (26)

must be dominated by other modes. One possibility is three-body decays which,
in the molecular scenario, would arise via the decay of a constituent hadron.
Given that $\bar{D}$ is stable and $\bar{D}^{*}$ has negligible decay width,
molecular three-body decays are presumably dominated by
$\Sigma_{c}\to\Lambda_{c}\pi$ or $\Sigma_{c}^{*}\to\Lambda_{c}\pi$
Burns:2015dwa . Following refs. Swanson:2006st ; Voloshin:2019aut ;
Burns:2019iih , we assume that the three-body width is determined by the width
of the $\Sigma_{c}^{(*)}$ constituent, resulting in the following predicted
partial widths

$\displaystyle\Gamma[P_{c}(4312)\to\Lambda_{c}^{+}\bar{D}\pi]$
$\displaystyle=1.9~{}\textrm{MeV}$ (27)
$\displaystyle\Gamma[P_{c}(4380)\to\Lambda_{c}^{+}\bar{D}\pi]$
$\displaystyle=15~{}\textrm{MeV}$ (28)
$\displaystyle\Gamma[P_{c}(4440/4457)\to\Lambda_{c}^{+}\bar{D}^{*}\pi]$
$\displaystyle=1.9~{}\textrm{MeV}$ (29)

Evidently these three-body decays cannot account entirely for the measured
$P_{c}$ decay widths. Given the tiny $J/\psi\,p$ branching fractions, we also
disregard more esoteric possibilities like $\chi_{c0}p$ and $J/\psi N\pi$.

##  VIII $\Lambda_{c}^{+}\bar{D}^{*0}$ decays

We have established that the partial widths of $P_{c}$ states to $J/\psi\,p$,
$\eta_{c}p$ and $\Lambda_{c}^{+}\bar{D}^{0}$ are tiny, and that three-body
modes, while not negligible, cannot account for the measured decay widths. The
rest of the decay width must be accounted for by other modes. We assume that
decays to exclusively light-flavoured hadrons are small; this is because of
the OZI rule, and the apparent absence of prominent light-flavoured decays in
other hidden-charm states.

In the case of $P_{c}(4312)$ we conclude, by a process of elimination, that
the dominant decay must be $\Lambda_{c}^{+}\bar{D}^{*0}$, as this is the only
two-body mode with hidden charm which is kinematically accessible. Ignoring
the small contributions from $J/\psi\,p$, $\eta_{c}p$ and
$\Lambda_{c}^{+}\bar{D}^{0}$, we estimate the $\Lambda_{c}^{+}\bar{D}^{*0}$
partial width by taking the difference of equations (24) and (27),

$\Gamma[P_{c}^{+}(4312)\to\Lambda_{c}^{+}\bar{D}^{*0}]=2.7\div
12.5~{}\textrm{MeV},$ (30)

where we have combined the statistical and systematic uncertainties in (24) in
quadrature. We have not accounted for the uncertainties in equation (27),
because of the difficulty in quantifying the systematic uncertainty on the
underlying assumption, namely that the three-body width is equal to the width
of the free constituents. Most of our arguments below relate to the scale of
the experimental numbers, not their precise values, so we do not expect the
conclusions to be compromised by the unquantified uncertainties.

One may question the validity of predicting prominent
$\Lambda_{c}^{+}\bar{D}^{*0}$ modes on the basis of a process of elimination.
However we also note that our conclusion is supported by model calculations
which confirm the prominence of $\Lambda_{c}^{+}\bar{D}^{*0}$ decays in
molecular models Ortega:2016syt ; Shimizu:2017xrg ; Lin:2017mtz ;
Huang:2018wed ; He:2019rva ; Dong:2020nwk .

From our estimate (30), the corresponding branching fraction

${\mathcal{B}}[P_{c}^{+}(4312)\to\Lambda_{c}^{+}\bar{D}^{*0}]=59\%\div 87\%,$
(31)

is enormous in comparison to the closely related $\Lambda_{c}^{+}\bar{D}^{0}$
mode, equation (23). The challenge for models is to explain the striking
disparity between these two modes which would naively be expected to be
comparable. We will see that this issue argues in favour of our Scenario C.

Heavy quark symmetry implies relations among all couplings of the type
$\Sigma_{c}^{(*)}\bar{D}^{(*)}\to\Lambda_{c}^{+}\bar{D}^{0(*)}$. In Table 3 we
show the relative matrix elements for S-wave transitions obtained assuming
heavy-quark symmetry, taken from the potentials in ref. Du:2021fmf . Note that
the relative matrix elements apply not only to effective field theory models,
where the S-wave potentials correspond to contact terms that are fit to data,
but also to models where the transition is due to the central potential from
one-pion exchange (since such models satisfy heavy-quark symmetry).

|  $\Sigma_{c}\bar{D}$ |  $\Sigma_{c}^{*}\bar{D}$ |  $\Sigma_{c}\bar{D}^{*}$ |  $\Sigma_{c}\bar{D}^{*}$
---|---|---|---|---
|  $1/2^{-}$ |  $3/2^{-}$ |  $1/2^{-}$ |  $3/2^{-}$
$\Lambda_{c}\bar{D}$ |  0 |  |  $\sqrt{3}$ |
$\Lambda_{c}\bar{D}^{*}$ |  $\sqrt{3}$ |  $-\sqrt{3}$ |  $-2$ |  1
Scenario A |  $P_{c}(4312)$ |  $P_{c}(4380)$ |  $P_{c}(4440)$ |  $P_{c}(4457)$
Scenario B |  $P_{c}(4312)$ |  $P_{c}(4380)$ |  $P_{c}(4457)$ |  $P_{c}(4440)$
Scenario C |  $P_{c}(4312)$ |  $P_{c}(4380)$ |  |  $P_{c}(4440)$
Table 3: Relative matrix elements for S-wave transitions
$\Sigma_{c}^{(*)}\bar{D}^{(*)}\to\Lambda_{c}\bar{D}^{(*)}$, according to
heavy-quark symmetry. The values are extracted from ref. Du:2021fmf , but
apply not only to contact terms in effective field theory, but also to the
central potentials due to one-pion exchange. The lower part of the table
identifies the various scenarios.

The first thing to notice is that for $P_{c}(4312)$, modelled as a $1/2^{-}$
$\Sigma_{c}\bar{D}$ state, the striking disparity in the magnitudes of
$\Lambda_{c}\bar{D}$ and $\Lambda_{c}\bar{D}^{*}$ has a natural explanation,
and confirms a selection rule predicted by Voloshin Voloshin:2019aut .
Although conservation of angular momentum allows for decays to both
$\Lambda_{c}\bar{D}$ (in S-wave) and $\Lambda_{c}\bar{D}^{*}$ (S-wave and
D-wave), the $\Lambda_{c}\bar{D}$ decay is forbidden by heavy-quark symmetry.
In this sense the experimental data are nicely consistent with the molecular
model for $P_{c}(4312)$. (The suppression of the $\Lambda_{c}\bar{D}$ mode is
also a feature of the chiral constituent quark model Dong:2020nwk , and the
chromomagnetic pentaquark model Weng:2019ynv , but not the model of ref.
He:2019rva .)

The tight upper limit (23) on $\Lambda_{c}\bar{D}$ decays also has a natural
explanation for $3/2^{-}$ states, both $P_{c}(4380)$ ($\Sigma_{c}^{*}\bar{D}$)
and $P_{c}(4440/4457)$ ($\Sigma_{c}\bar{D}^{*}$). In these cases the
$\Lambda_{c}\bar{D}$ decay would be D-wave, hence suppressed compared to the
S-wave decay $P_{c}(4312)\to\Lambda_{c}\bar{D}^{*}$.

On the other hand, for a $1/2^{-}$ $\Sigma_{c}\bar{D}^{*}$ state, the tight
upper limit on $\Lambda_{c}^{+}\bar{D}^{0}$ is a problem. The previous
argument relies on the assumption (which is very natural) that D-wave decays
are suppressed compared to S-wave decays. In that case, the
$\Lambda_{c}\bar{D}$ and $\Lambda_{c}\bar{D}^{*}$ partial widths for the
various $P_{c}$ states are dominated by the S-wave matrix elements, so we can
estimate the relative strengths using the numbers in Table 3. We notice in
particular that for $1/2^{-}$, the
$\Sigma_{c}\bar{D}^{*}\to\Lambda_{c}^{+}\bar{D}^{0}$ matrix element is
identical to $\Sigma_{c}\bar{D}\to\Lambda_{c}^{+}\bar{D}^{*0}$, whose scale is
set by equation (30). This is a problem for Scenarios A and B, because it
implies that if either of $P_{c}(4440)$ or $P_{c}(4457)$ were a $1/2^{-}$
$\Sigma_{c}\bar{D}^{*}$ state, then its $\Lambda_{c}^{+}\bar{D}^{0}$ partial
width would be $2.7\div 12.5$ MeV, indeed even larger because of the enhanced
phase space compared to the $P_{c}(4312)$ decay. This is wildly inconsistent
with the upper limit of eq. (23).

To avoid this problem, we have to abandon the assumption that there is a
$1/2^{-}$ $\Sigma_{c}\bar{D}^{*}$ bound state, which leads us to Scenario C.
Since for $P_{c}(4457)$ there are several viable alternatives, we assume that
only $P_{c}(4440)$ is $\Sigma_{c}\bar{D}^{*}$ bound state, and that its
quantum numbers are $3/2^{-}$. We will argue later that there are other
reasons to favour this over the alternative Scenarios A and B.

In summary, there is an apparent tension between the dominance of the
$P_{c}(4312)\to\Lambda_{c}^{+}\bar{D}^{*0}$ decays, and the tight experimental
upper limits on $P_{c}\to\Lambda_{c}^{+}\bar{D}^{0}$. This is a problem for
Scenarios A and B, but not for Scenario C, where the
$\Lambda_{c}^{+}\bar{D}^{0}$ decays are small either due to their D-wave
nature (for the $3/2^{-}$ states) or because of heavy quark symmetry (for
$P_{c}(4312)$).

From now on we concentrate on Scenario C. We may estimate the
$\Lambda_{c}\bar{D}^{*}$ partial widths of $P_{c}(4380)$ and $P_{c}(4440)$
using equation (30) and the numbers in Table 3:

$\displaystyle\Gamma[P_{c}^{+}(4380)\to\Lambda_{c}^{+}\bar{D}^{*0}]$
$\displaystyle=2.7\div 12.5~{}\textrm{MeV},$ (32)
$\displaystyle\Gamma[P_{c}^{+}(4440)\to\Lambda_{c}^{+}\bar{D}^{*0}]$
$\displaystyle=0.9\div 4.2~{}\textrm{MeV}.$ (33)

Here we have ignored the effect of the mass differences between $P_{c}(4312)$,
$P_{c}(4380)$ and $P_{c}(4440)$. (Phase space would enhance the decays of the
heavier states, but this is to some extent mitigated by a corresponding form
factor suppression in the matrix element.)

For $P_{c}(4380)$ we assume the decay width is saturated by
$\Lambda_{c}^{+}\bar{D}^{*0}$ and, from equation (28),
$\Lambda_{c}\bar{D}\pi$. This is because of the tight upper limit on
$J/\psi\,p$, and the absence of other two-body S-wave decays. (We are assuming
that the D-wave decay to $\Sigma_{c}\bar{D}$ is small.) This suggests a total
width

$\Gamma[P_{c}(4380)]=17.7\div 27.5~{}\textrm{MeV},$ (34)

considerably smaller than the measured width Aaij:2015tga  which, however, is
considered to be obsolete Aaij:2019vzc . Other approaches also find a narrower
$P_{c}(4380)$ Du:2019pij ; Du:2021fmf . Assuming our estimated total width,
the $\Lambda_{c}\bar{D}^{*}$ branching fraction is

${\mathcal{B}}[P_{c}(4380)\to\Lambda_{c}\bar{D}^{*}]=15\%\div 45\%.$ (35)

If neglected decays (notably the $\Sigma_{c}\bar{D}$ D-wave) are significant,
this will be an overestimate.

For $P_{c}(4440)$, the $\Lambda_{c}\bar{D}^{*}$ branching fraction is

${\mathcal{B}}[P_{c}(4440)\to\Lambda_{c}\bar{D}^{*}]=3\%\div 45\%,$ (36)

where we have used the measured total width, eq. (31), with statistical and
systematic uncertainties combined in quadrature.

Given the predicted large $\Lambda_{c}^{+}\bar{D}^{*0}$ branching fractions
for $P_{c}(4312)$, $P_{c}(4380)$ and $P_{c}(4440)$, searching for these states
in $\Lambda_{b}^{0}\to\Lambda_{c}^{+}\bar{D}^{*0}K^{-}$ is warranted. We note
that this decay has already been observed at LHCb Stahl:2018eme , although an
amplitude analysis has not been carried out.

To understand the experimental prospects in these channels, we consider now
the fit fraction

${\mathcal{R}(\Lambda_{c}^{+}\bar{D}^{*0})}=\frac{{\mathcal{B}}(\Lambda_{b}^{0}\to
P_{c}^{+}K^{-})\,{\mathcal{B}}(P_{c}^{+}\to\Lambda_{c}^{+}\bar{D}^{*0})}{{\mathcal{B}}(\Lambda_{b}^{0}\to\Lambda_{c}^{+}\bar{D}^{*0}K^{-})}.$
(37)

We may eliminate the unknown production branching fraction
${\mathcal{B}}(\Lambda_{b}^{0}\to P_{c}^{+}K^{-})$ by taking a ratio with
equation (4), to give

$\frac{\mathcal{R}(\Lambda_{c}^{+}\bar{D}^{*0})}{\mathcal{R}(J/\psi\,p)}=\frac{{\mathcal{B}}(\Lambda_{b}^{0}\to
J/\psi\,p\,K^{-})}{{\mathcal{B}}(\Lambda_{b}^{0}\to\Lambda_{c}^{+}\bar{D}^{*0}K^{-})}\frac{{\mathcal{B}}(P_{c}^{+}\to\Lambda_{c}^{+}\bar{D}^{*0})}{{\mathcal{B}}(P_{c}^{+}\to
J/\psi\,p)}.$ (38)

Using the experimental numbers in Table 2, and our estimates of
${\mathcal{B}}(P_{c}^{+}\to\Lambda_{c}^{+}\bar{D}^{*0})$, we have a relation
between the fit fractions $\mathcal{R}(\Lambda_{c}^{+}\bar{D}^{*0})$ and the
branching fraction ${\mathcal{B}}(P_{c}^{+}\to J/\psi\,p)$, which we plot in
Fig. 4. Because we are interested in the scale of the numbers rather than
their precise values, for illustration we plot the central values only. For
$P_{c}(4380)$, we have used the obsolete value of the fit fraction, equation
(10).

Figure 4: The fit fractions ${\mathcal{R}}(\Lambda_{c}^{+}\bar{D}^{*0})$ as a
function of ${\mathcal{B}}[P_{c}(4312)\to J/\psi\,p]$, obtained as described
in the text.

The message of this plot, when combined with the upper limit (8), is that the
fit fractions ${\mathcal{R}(\Lambda_{c}^{+}\bar{D}^{*0})}$ are enormous.
Indeed for $P_{c}(4380)$ the predicted fit fractions are implausibly large. We
attribute this to the use of the obsolete value (10) when constructing the
plot. (We encountered a similar problem when interpreting Figure 3.)

For $P_{c}(4312)$ and $P_{c}(4440)$, the fits fractions of order
${\mathcal{O}}(10\%)$ or larger are still plausible, but are strikingly large
when compared to the ${\mathcal{O}}(1\%)$ fit fractions in the observed
$J/\psi\,p$ mode (Table 2). It suggests of course that there are strong
prospects for the experimental observation of these states in
$\Lambda_{b}^{0}\to\Lambda_{c}^{+}\bar{D}^{*0}K^{-}$.

A corollary of this plot is that the branching fraction
${\mathcal{B}}[P_{c}(4312)\to J/\psi\,p]$ cannot be much less than the current
bound in equation (8), otherwise the fit fractions
${\mathcal{R}}(\Lambda_{c}^{+}\bar{D}^{*0})$ become implausibly (or
impossibly) large. We arrived at the same conclusion when interpreting Fig. 3.
As discussed there, this also implies that observation of $P_{c}$ states in
photoproduction requires not much more sensitivity than that of the
$J/\psi$-007 experiment.

##  IX $\Sigma_{c}^{(*)}\bar{D}$ decays

In molecular models, couplings of the type
$\Sigma_{c}^{(*)}\bar{D}^{(*)}\to\Sigma_{c}^{(*)}\bar{D}^{(*)}$ are
responsible for binding. The same couplings will also lead to decays into
$\Sigma_{c}^{(*)}\bar{D}$ final states, where phase space is available. The
$P_{c}(4312)$ is too light, but $P_{c}(4380)$ could decay into
$\Sigma_{c}\bar{D}$ (in D-wave), and $P_{c}(4440)$ could decay into both
$\Sigma_{c}\bar{D}$ (in D-wave) and $\Sigma_{c}^{*}\bar{D}$ (in S-wave and
D-wave), where we assume the $3/2^{-}$ quantum numbers of Scenario C. We
further assume that S-wave decays dominate over D-waves, in which case the
most prominent decay will be $P_{c}(4440)\to\Sigma_{c}^{*}\bar{D}$. Note that
having accounted for $\Lambda_{c}^{+}\bar{D}^{*0}$ and
$\Lambda_{c}\bar{D}^{*}\pi$ decays, in equations (29) and (33), respectively,
the measured width (25) still allows for a considerable
$\Sigma_{c}^{*}\bar{D}$ mode. We also note that various models find
substantial $\Sigma_{c}^{*}\bar{D}$ decays, for example refs Lin:2017mtz ;
He:2019rva .

The experimental prospects for observing $P_{c}(4380)$ or $P_{c}(4440)$ in
$\Lambda_{b}^{0}\to\Sigma_{c}^{(*)}\bar{D}K^{-}$ depend on the fit fraction

${\mathcal{R}(\Sigma_{c}^{(*)}\bar{D})}=\frac{{\mathcal{B}}(\Lambda_{b}^{0}\to
P_{c}^{+}K^{-})\,{\mathcal{B}}(P_{c}^{+}\to\Sigma_{c}^{(*)}\bar{D})}{{\mathcal{B}}(\Lambda_{b}^{0}\to\Sigma_{c}^{(*)}\bar{D}K^{-})}.$
(39)

Taking a ratio with equation (4), we get a relation between
$\mathcal{R}(\Sigma_{c}^{(*)}\bar{D})$ and $\mathcal{R}(J/\psi\,p)$,

$\frac{\mathcal{R}(\Sigma_{c}^{(*)}\bar{D})}{\mathcal{R}(J/\psi\,p)}=\frac{{\mathcal{B}}(\Lambda_{b}^{0}\to
J/\psi\,p\,K^{-})}{{\mathcal{B}}(\Lambda_{b}^{0}\to\Sigma_{c}^{(*)}\bar{D}K^{-})}\frac{{\mathcal{B}}(P_{c}^{+}\to\Sigma_{c}^{(*)}\bar{D})}{{\mathcal{B}}(P_{c}^{+}\to
J/\psi\,p)}.$ (40)

As far as we know,
${\mathcal{B}}(\Lambda_{b}^{0}\to\Sigma_{c}^{(*)}\bar{D}K^{-})$ has not been
measured, but since the tree-level contributions to this mode are colour
suppressed (Fig. 2), we expect it to be comparable to
${\mathcal{B}}(\Lambda_{b}^{0}\to J/\psi\,p\,K^{-})$, as noted in equation
(9). In this case, we expect the first ratio on the right-hand side of
equation (40) to be a number of order 1. Conversely, because of the stringent
upper limit (8) on $P_{c}^{+}\to J/\psi\,p$, we expect the second ratio on the
right-hand side to be large, at least for
$P_{c}(4440)\to\Sigma_{c}^{*}\bar{D}$, where the S-wave is expected to be
prominent. We conclude that for $P_{c}(4440)^{+}$, the fit fraction
$\mathcal{R}(\Sigma_{c}^{*}\bar{D})$ ought to be large in comparison to the
measured fit fraction $\mathcal{R}(J/\psi\,p)$, implying strong experimental
prospects. Depending on the magnitude of the D-wave decays, by a similar
argument we may also expect considerable (albeit smaller) fit fractions
$\mathcal{R}(\Sigma_{c}\bar{D})$ for $P_{c}(4380)$ and $P_{c}(4440)$.

This is another example of the general phenomenon discussed in Section IV.
Where tree-level three-body decays are suppressed, two-body fit fractions can
be large.

Finally, we remark that, assuming the $P_{c}$ states are isospin 1/2, the
different charge modes $\Lambda_{b}^{0}\to\Sigma_{c}^{(*)+}\bar{D}^{0}K^{-}$
and $\Lambda_{b}^{0}\to\Sigma_{c}^{(*)++}D^{-}K^{-}$ have relative rates
$1:2$. Deviations from this would be an indication of isospin mixing
Burns:2015dwa .

##  X Partner states

Given the existence of states near the thresholds for $\Sigma_{c}\bar{D}$,
$\Sigma_{c}^{*}\bar{D}$ and $\Sigma_{c}\bar{D}^{*}$, the apparent absence of
states near $\Sigma_{c}^{*}\bar{D}^{*}$ threshold is conspicuous. In this
section we show that this is a problem for Scenarios A and B, but not for
Scenario C.

Interactions among $\Sigma_{c}^{(*)}\bar{D}^{(*)}$ channels are constrained by
heavy-quark symmetry. This applies both to models based on meson exchange, and
also effective field theory approaches where the long-range contribution to
the potential is due to pion-exchange, and the short-range part is modelled
via contact terms that are fit to data. The pattern of binding in such models
can be understood qualitatively with reference to the simplest approach, where
the potentials are due to S-wave contact terms only (no meson exchange). The
elastic potentials, from ref. Liu:2018zzu , are given in Table 4.

|  $\Sigma_{c}\bar{D}$ |  $\Sigma_{c}^{*}\bar{D}$ |  $\Sigma_{c}\bar{D}^{*}$ |  $\Sigma_{c}^{*}\bar{D}^{*}$
---|---|---|---|---
$1/2^{-}$ |  $C_{a}$ |  |  $C_{a}-\frac{4}{3}C_{b}$ |  $C_{a}-\frac{5}{3}C_{b}$
$3/2^{-}$ |  |  $C_{a}$ |  $C_{a}+\frac{2}{3}C_{b}$ |  $C_{a}-\frac{2}{3}C_{b}$
$5/2^{-}$ |  |  |  |  $C_{a}+C_{b}$
Table 4: S-wave elastic potentials for $\Sigma_{c}^{(*)}\bar{D}^{(*)}$
molecules constrained by heavy quark symmetry.

Clearly, binding in $\Sigma_{c}\bar{D}$ and $\Sigma_{c}^{*}\bar{D}$ requires
$C_{a}<0$. The potentials in these channels are identical, so apart from small
differences due to their masses, and coupled-channel effects, binding in one
channel implies binding in the other. Thus a model that accounts for
$P_{c}(4312)$ inevitably implies a partner state $P_{c}(4380)$.

To get binding in both of the $\Sigma_{c}\bar{D}^{*}$ channels, and thus
account for both $P_{c}(4440)$ and $P_{c}(4457)$, requires $C_{a}$ to be large
enough (in magnitude) compared to $C_{b}$. Scenarios A and B are then
distinguished by having $C_{b}>0$ and $C_{b}<0$, respectively.

With $C_{a}$ large (in magnitude) compared to $C_{b}$, on the basis of the
potentials in Table 4 it seems likely that all three of the
$\Sigma_{c}^{*}\bar{D}^{*}$ states ($1/2^{-}$, $3/2^{-}$ and $5/2^{-}$) will
also bind, and that expectation is confirmed by calculation Liu:2019tjn . This
immediately raises the question of why there is apparently no evidence for
such states in the experimental data. The absence of the $5/2^{-}$ state may
be understood from its lack of S-wave coupling to $J/\psi\,p$; it decays in
D-wave, which may be expected to be suppressed. However the missing $1/2^{-}$
and $3/2^{-}$ states are more problematic.

It is possible that Scenarios A and B may be rescued with the idea that the
extra $\Sigma_{c}^{*}\bar{D}^{*}$ states do exist, but are just not seen in
$\Lambda_{b}\to J/\psi\,p\,K^{-}$ because of suppressed production or decays.
But in the model we propose inprep  the situation is precisely the opposite.
From heavy quark symmetry, the production and decay of the $3/2^{-}$
$\Sigma_{c}^{*}\bar{D}^{*}$ state, in particular, would be enhanced compared
to other observed states, so if there is binding in this channel, it ought to
be visible as a prominent peak in $\Lambda_{b}\to J/\psi\,p\,K^{-}$.
Similarly, the $1/2^{-}$ $\Sigma_{c}^{*}\bar{D}^{*}$ state would be expected
to decay prominently $\Lambda_{c}\bar{D}$; there is no obvious indication of
structure in the $\Lambda_{b}^{0}\to\Lambda_{c}^{+}\bar{D}^{0}K^{-}$ spectrum
Piucci:2019vsk , but more detailed analysis with more statistics could be
revealing.

The inevitability of binding in all three $\Sigma_{c}^{*}\bar{D}^{*}$ channels
($1/2^{-}$, $3/2^{-}$ and $5/2^{-}$) is a problem that is intrinsic to
Scenarios A and B. As noted, to get binding in both $\Sigma_{c}\bar{D}^{*}$
channels ($1/2^{-}$ and $3/2^{-}$) implies that $C_{a}$ must be large in
magnitude compared to $C_{b}$, which in turn renders all of the
$\Sigma_{c}^{*}\bar{D}^{*}$ channels sufficiently attractive to cause binding.

It all plays out very differently if only one of the two
$\Sigma_{c}\bar{D}^{*}$ were required to bind. In this case it is no longer
necessary to have $C_{a}$ large (in magnitude) in comparison to $C_{b}$, with
the consequence that binding is not automatic in all channels. Indeed, this
scenario was already considered after the initial LHCb paper, when the
experimental data suggested one $\Sigma_{c}\bar{D}^{*}$ state, not two
Liu:2018zzu .

If only one of the $\Sigma_{c}\bar{D}^{*}$ channels binds, it is natural to
associate this with $P_{c}(4440)$, since this is unambiguously below
$\Sigma_{c}\bar{D}^{*}$ threshold. As discussed previously, there are several
viable alternative scenarios for $P_{c}(4457)$, because its mass is consistent
with $\Sigma_{c}\bar{D}^{*}$ and $\Lambda_{c}(2595)\bar{D}$ thresholds. The
next question is whether $P_{c}(4440)$ has $1/2^{-}$ or $3/2^{-}$ quantum
numbers. In Section VI we argued in favour of $3/2^{-}$ quantum numbers (what
we called Scenario C), noting that the tight upper limits on
$\Lambda_{c}^{+}\bar{D}^{0}$ decays are not consistent with a $1/2^{-}$
$\Sigma_{c}\bar{D}^{*}$ state. We will now show that the pattern of binding
also argues in favour of the $3/2^{-}$ assignment.

With reference to Table 4, to achieve $\Sigma_{c}\bar{D}^{*}$ binding in
$3/2^{-}$, but not $1/2^{-}$, implies $C_{b}<0$. This in turn suggests that
out of the $\Sigma_{c}^{*}\bar{D}^{*}$ states, there is binding in $5/2^{-}$,
but not $1/2^{-}$ or $3/2^{-}$, and indeed this expectation is borne out in
explicit calculation Liu:2018zzu . This spectrum of states is nicely
consistent with the absence of $\Sigma_{c}^{*}\bar{D}^{*}$ states in the data.
The $1/2^{-}$ and $3/2^{-}$ states simply don’t exist, and the $5/2^{-}$ state
is not visible because its decay is suppressed. (In our model inprep  we find
that not only the decay, but also the production, of the $5/2^{-}$ state is
suppressed.)

The alternative scenario does not work so well. If instead we assume
$\Sigma_{c}\bar{D}^{*}$ binding in $1/2^{-}$ (but not $3/2^{-}$), we need
$C_{b}>0$, which implies $\Sigma_{c}^{*}\bar{D}^{*}$ binding in $1/2^{-}$ and
$3/2^{-}$ (but not $5/2^{-}$). In this scenario one needs an explanation for
why the $1/2^{-}$ and $3/2^{-}$ $\Sigma_{c}^{*}\bar{D}^{*}$ states are not
visible in $\Lambda_{b}\to J/\psi\,p\,K^{-}$. (Note that some authors do argue
in favour of the $C_{b}>0$ pattern, for other reasons Chen:2021cfl ;
Meng:2019ilv .)

Another reason to prefer binding $\Sigma_{c}\bar{D}^{*}$ in $3/2^{-}$ is that
it is also consistent with expectations from one-pion exchange. In Table 5 we
show the central potentials due to pion-exchange obtained from the quark model
Liu:2018zzu ; Burns:2019iih , where $C(r)$ behaves as a Yukawa function (with
positive sign) for large $r$ (i.e. it is repulsive at large $r$). It follows
that the attractive channels are those in which $C(r)$ comes with a negative
coefficient, namely $\Sigma_{c}\bar{D}^{*}$ in $3/2^{-}$ and
$\Sigma_{c}^{*}\bar{D}^{*}$ in $5/2^{-}$; this is consistent with the pattern
of binding in our Scenario C.

Of course the arguments about attraction and repulsion in one-pion exchange
are admittedly simplistic, particularly because they ignore the tensor
potential, which is known to be important for binding. Nevertheless, explicit
calculation including the tensor terms confirms that the channels that bind
most easily (requiring the smallest form factor cutoff) are those identified
above Liu:2018zzu ; Burns:2019iih ; Valderrama:2019chc . The pattern also
applies to models where the potential is due to exchange of not only pions,
but also other light mesons Liu:2019zvb ; Peng:2020xrf .

|  $\Sigma_{c}\bar{D}$ |  $\Sigma_{c}^{*}\bar{D}$ |  $\Sigma_{c}\bar{D}^{*}$ |  $\Sigma_{c}^{*}\bar{D}^{*}$
---|---|---|---|---
$1/2^{-}$ |  0 |  |  $4C(r)$ |  $5C(r)$
$3/2^{-}$ |  |  0 |  $-2C(r)$ |  $2C(r)$
$5/2^{-}$ |  |  |  |  $-3C(r)$
Table 5: S-wave elastic potentials for $\Sigma_{c}^{(*)}\bar{D}^{(*)}$
molecules from one-pion exchange in the quark model, where $C(r)$ is the
potential defined in ref. Burns:2019iih .

(Notice that the pion exchange potentials in Table 5 follow the same pattern
as the “$C_{b}$” contact terms in Table 4, under the replacement
$C_{b}\to-3C(r)$. This is ultimately because both contributions are
proportional to an operator of the form $\mathbf{S}_{1}\cdot\mathbf{S}_{2}$,
where $\mathbf{S}_{1}$ and $\mathbf{S}_{2}$ are vectors acting on the spin
degrees of freedom of the $\Sigma_{c}^{(*)}$ and $\bar{D}^{(*)}$,
respectively.)

##  XI Conclusions

We have shown that, with minimal theoretical assumptions, current experimental
data on $\Lambda_{b}$ decays and photoproduction imply constraints on the
quantum numbers of $P_{c}$ states, and lead to predictions for their decay
branching fractions, and fit fractions in $\Lambda_{b}^{0}$ decays to
$\eta_{c}pK^{-}$, $\Lambda_{c}^{+}\bar{D}^{*0}K^{-}$, and
$\Sigma_{c}^{(*)}\bar{D}K^{-}$.

Unlike the usual Scenarios A and B, where $P_{c}(4440)$ and $P_{c}(4457)$ are
both $\Sigma_{c}\bar{D}^{*}$ bound states, in our Scenario C only
$P_{c}(4440)$ is a bound state, and its quantum numbers are $3/2^{-}$. Because
the $P_{c}(4457)$ mass coincides with both $\Sigma_{c}\bar{D}^{*}$ and
$\Lambda_{c}(2595)\bar{D}$ thresholds, there are several viable alternatives
for this state, which we explore in future work inprep . The cusp
interpretation of $P_{c}(4457)$ is favoured by ref. Kuang:2020bnk .

One of the advantages of Scenario C is that, unlike Scenarios A and B, it is
consistent with the striking observation that the $P_{c}$ states hardly decay
to $\Lambda_{c}^{+}\bar{D}^{0}$, but apparently decay prominently to
$\Lambda_{c}^{+}\bar{D}^{*0}$. The suppression of $\Lambda_{c}^{+}\bar{D}^{0}$
is natural for $P_{c}(4312)$, where it is forbidden by heavy-quark symmetry,
and for $P_{c}(4380)$ and $P_{c}(4440)$ which, as $3/2^{-}$ states, decay to
$\Lambda_{c}^{+}\bar{D}^{0}$ in D-wave. The problem with Scenarios A and B is
that each has a $1/2^{-}$ $\Sigma_{c}\bar{D}^{*}$ state which, from heavy
quark symmetry, should decay prominently to $\Lambda_{c}^{+}\bar{D}^{0}$, in
conflict with experimental data.

Another issue with Scenarios A and B is that they imply the existence of
$\Sigma_{c}^{*}\bar{D}^{*}$ partner states, with quantum numbers $1/2^{-}$,
$3/2^{-}$ and $5/2^{-}$, which are not apparent in $\Lambda_{b}\to
J/\psi\,p\,K^{-}$ data. This is an inevitable consequence of fixing the
potential parameters in order to generate two $\Sigma_{c}\bar{D}^{*}$ bound
states, with both $1/2^{-}$ and $3/2^{-}$ quantum numbers. Our Scenario C
naturally avoids this issue, as we only require a single
$\Sigma_{c}\bar{D}^{*}$ state with $3/2^{-}$ quantum numbers. In this case the
potentials are such that the $1/2^{-}$ and $3/2^{-}$
$\Sigma_{c}^{*}\bar{D}^{*}$ partners do not bind, and while the $5/2^{-}$
$\Sigma_{c}^{*}\bar{D}^{*}$ partner does bind, its absence in experiment can
be understood as a consequence of D-wave suppression both in decay and, as
discussed in our next paper inprep , production.

A further advantage of Scenario C is that it reproduces the pattern of binding
expected on the basis of one-pion exchange.

We have given many predictions for the branching fractions and
$\Lambda_{b}^{0}$ fit fractions of $P_{c}(4312)$, $P_{c}(4380)$ and
$P_{c}(4440)$ in various modes. The most striking prediction is for strong
experimental signals for all the states in
$\Lambda_{b}^{0}\to\Lambda_{c}^{+}\,\bar{D}^{*0}\,K^{-}$. This is a
particularly sharp prediction when juxtaposed against the absence of signals
in the closely related decay
$\Lambda_{b}^{0}\to\Lambda_{c}^{+}\,\bar{D}^{0}\,K^{-}$. The decay
$\Lambda_{b}^{0}\to\Lambda_{c}^{+}\,\bar{D}^{*0}\,K^{-}$ has been observed at
LHCb Stahl:2018eme , and we suggest an amplitude analysis to test for the
existence of $P_{c}$ states in the $\Lambda_{c}^{+}\,\bar{D}^{*0}$ spectrum.

In $\Lambda_{b}^{0}\to\eta_{c}\,p\,K^{-}$, we predict a substantial
$P_{c}(4312)$ fit fraction, which can be tested in future experimental
analyses. (Our prediction is consistent with the current upper bound.) In our
Scenario C, we do not expect prominent signals for the other $P_{c}$ states in
$\Lambda_{b}\to\eta_{c}\,p\,K^{-}$. We showed that, by contrast, Scenarios A
and B could be distinguished by the relative fit fractions of $P_{c}(4440)$
and $P_{c}(4457)$.

We also predict strong experiment signals for $P_{c}(4440)$ in
$\Lambda_{b}^{0}\to\Sigma_{c}^{*}\bar{D}K^{-}$, and smaller signals for both
$P_{c}(4380)$ and $P_{c}(4440)$ in $\Lambda_{b}^{0}\to\Sigma_{c}\bar{D}K^{-}$.

We also pointed out that the branching fractions ${\mathcal{B}}(P_{c}^{+}\to
J/\psi\,p)$ cannot be much less than the current upper bound (8), otherwise
the production branching fractions $\mathcal{B}(\Lambda_{b}^{0}\to
P_{c}^{+}K^{-})$ and fit fractions
${\mathcal{R}}(\Lambda_{c}^{+}\bar{D}^{*0})$ would be implausibly large. (See
Figs. 3 and 4.) An interesting consequence is that existing experimental
measurements are almost at the level of sensitivity required to observe
$P_{c}$ states in photoproduction.

We want to emphasise that many of the constraints and predictions we have
derived are very general in nature, and so if it turns out they are not
satisfied by future experimental data, it could be an indication not of the
failure of our particular realisation of the molecular model, but of the
molecular approach all together. As an example, consider again the observation
(Fig. 3) that if the upper limit on ${\mathcal{B}}(P_{c}^{+}\to J/\psi\,p)$ is
much less than equation (8), it would imply implausibly large production
branching fractions $\mathcal{B}(\Lambda_{b}^{0}\to P_{c}^{+}K^{-})$. In this
case some new mechanism would need to be invoked to explain how, contrary to
intuition, molecular states are produced with comparable branching fraction to
conventional hadrons. Alternatively, since the observation is a direct
consequence of the measured fit fractions, one could question whether or not
it is appropriate, in the fit fraction, to factorise the production and decay
branching fractions $\mathcal{B}(\Lambda_{b}^{0}\to P_{c}^{+}K^{-})$ and
${\mathcal{B}}(P_{c}^{+}\to J/\psi\,p)$ in the numerator. The factorisation is
legitimate in the case that $P_{c}$ is a resonance, but not if the $P_{c}$
signal is an effect which is uniquely associated with a particular production
mechanism, such as a cusp or triangle singularity. Indeed we gave examples, in
the case of the $X(2900)$ states, where the factorisation is clearly not
appropriate Burns:2020xne . Many of our manipulations in this paper rely on
this factorisation, and so our predictions should be interpreted with this in
mind.

###### Acknowledgements.

Swanson’s research was supported by the U.S. Department of Energy under
contract DE-SC0019232.

## References

  * (1) LHCb, R. Aaij et al.,  Phys. Rev. Lett. 115, 072001 (2015), arXiv:1507.03414.
  * (2) LHCb, R. Aaij et al.,  Phys. Rev. Lett. 117, 082002 (2016), arXiv:1604.05708.
  * (3) LHCb, R. Aaij et al.,  Phys. Rev. Lett. 122, 222001 (2019), arXiv:1904.03947.
  * (4) J.-J. Wu, R. Molina, E. Oset, and B. S. Zou,  Phys. Rev. C84, 015202 (2011), arXiv:1011.2399.
  * (5) L. Roca, J. Nieves, and E. Oset,  Phys. Rev. D92, 094003 (2015), 1507.04249.
  * (6) J. He,  Phys. Lett. B753, 547 (2015), 1507.05200.
  * (7) R. Chen, X. Liu, X.-Q. Li, and S.-L. Zhu,  Phys. Rev. Lett. 115, 132002 (2015), 1507.03704.
  * (8) M. Karliner and J. L. Rosner,  Phys. Rev. Lett. 115, 122001 (2015), 1506.06386.
  * (9) P. G. Ortega, D. R. Entem, and F. Fernández,  Phys. Lett. B764, 207 (2017), arXiv:1606.06148.
  * (10) Y. Shimizu, D. Suenaga, and M. Harada,  Phys. Rev. D93, 114003 (2016), arXiv:1603.02376.
  * (11) Y. Yamaguchi and E. Santopinto,  Phys. Rev. D96, 014018 (2017), arXiv:1606.08330.
  * (12) Y. Yamaguchi et al.,  Phys. Rev. D96, 114031 (2017), arXiv:1709.00819.
  * (13) Y. Shimizu and M. Harada,  Phys. Rev. D96, 094012 (2017), arXiv:1708.04743.
  * (14) Y. Shimizu, Y. Yamaguchi, and M. Harada,  Phys. Rev. D98, 014021 (2018), arXiv:1805.05740.
  * (15) M.-L. Du et al.,  (2019), arXiv:1910.11846.
  * (16) T. Gutsche and V. E. Lyubovitskij,  Phys. Rev. D 100, 094031 (2019), arXiv:1910.03984.
  * (17) M.-Z. Liu et al.,  (2019), arXiv:1907.06093.
  * (18) M. P. Valderrama,  (2019), arXiv:1907.05294.
  * (19) S. Sakai, H.-J. Jing, and F.-K. Guo,  (2019), arXiv:1907.03414.
  * (20) Z.-H. Guo and J. A. Oller,  Phys. Lett. B793, 144 (2019), arXiv:1904.00851.
  * (21) J. He,  Eur. Phys. J. C79, 393 (2019), arXiv:1903.11872.
  * (22) M.-Z. Liu et al.,  Phys. Rev. Lett. 122, 242001 (2019), arXiv:1903.11560.
  * (23) R. Chen, Z.-F. Sun, X. Liu, and S.-L. Zhu,  Phys. Rev. D100, 011502 (2019), arXiv:1903.11013.
  * (24) T. J. Burns and E. S. Swanson,  Phys. Rev. D100, 114033 (2019), arXiv:1908.03528.
  * (25) J. He and D.-Y. Chen,  (2019), arXiv:1909.05681.
  * (26) C. W. Xiao, J. Nieves, and E. Oset,  Phys. Rev. D100, 014021 (2019), arXiv:1904.01296.
  * (27) F.-Z. Peng, M.-Z. Liu, M. Sánchez Sánchez, and M. Pavon Valderrama,  Phys. Rev. D 102, 114020 (2020), arXiv:2004.05658.
  * (28) H. Xu, Q. Li, C.-H. Chang, and G.-L. Wang,  Phys. Rev. D 101, 054037 (2020), arXiv:2001.02980.
  * (29) N. Yalikun, Y.-H. Lin, F.-K. Guo, Y. Kamiya, and B.-S. Zou,  (2021), arXiv:2109.03504.
  * (30) M.-L. Du et al.,  (2021), arXiv:2102.07159.
  * (31) T. J. Burns and E. S. Swanson,  in preparation.
  * (32) S.-Q. Kuang, L.-Y. Dai, X.-W. Kang, and D.-L. Yao,  Eur. Phys. J. C 80, 433 (2020), arXiv:2002.11959.
  * (33) S. X. Nakamura,  (2021), arXiv:2103.06817.
  * (34) S. X. Nakamura, A. Hosaka, and Y. Yamaguchi,  Phys. Rev. D 104, L091503 (2021), arXiv:2109.15235.
  * (35) LHCb, R. Aaij et al.,  Chin. Phys. C 40, 011001 (2015), arXiv:1509.00292.
  * (36) LHCb, R. Aaij et al.,  Phys. Rev. D 102, 112012 (2020), arXiv:2007.11292.
  * (37) Particle Data Group, P. A. Zyla et al.,  PTEP 2020, 083C01 (2020).
  * (38) M. Stahl,  First observation of the decay $\Lambda^{0}_{b}\to\Lambda^{+}_{c}\bar{D}^{(*)0}K^{-}$ in preparation of a pentaquark search in the $\Lambda^{+}_{c}\bar{D}^{(*)0}$ system at the LHCb experiment,  PhD thesis, U. Heidelberg (main), 2018.
  * (39) A. Piucci,  Amplitude analysis of $\Lambda_{b}^{0}\to\Lambda_{c}^{+}\bar{D}^{0}K^{-}$ decays and pentaquark searches in the $\Lambda_{c}^{+}\bar{D}^{0}$ system at the LHCb experiment.,  PhD thesis, Heidelberg U., 2019.
  * (40) T. J. Burns and E. S. Swanson,  Phys. Rev. D 103, 014004 (2021), arXiv:2009.05352.
  * (41) T. J. Burns and E. S. Swanson,  Phys. Lett. B 813, 136057 (2021), arXiv:2008.12838.
  * (42) Q. Wang, X.-H. Liu, and Q. Zhao,  Phys. Rev. D92, 034022 (2015), arXiv:1508.00339.
  * (43) M. Karliner and J. L. Rosner,  (2015), 1508.01496.
  * (44) V. Kubarovsky and M. B. Voloshin,  (2016), arXiv:1609.00050.
  * (45) GlueX, A. Ali et al.,  (2019), arXiv:1905.10811.
  * (46) A. N. Hiller Blin et al.,  Phys. Rev. D94, 034002 (2016), arXiv:1606.08912.
  * (47) $J/\psi-007$ collaboration, S. Joosten,  Talk at 9th Workshop of the APS Topical Group on Hadronic Physics (2021).
  * (48) J.-J. Wu, R. Molina, E. Oset, and B. S. Zou,  Phys. Rev. Lett. 105, 232001 (2010), arXiv:1007.0573.
  * (49) C.-W. Shen, F.-K. Guo, J.-J. Xie, and B.-S. Zou,  Nucl. Phys. A954, 393 (2016), arXiv:1603.04672.
  * (50) Q.-F. Lü and Y.-B. Dong,  Phys. Rev. D93, 074020 (2016), arXiv:1603.00559.
  * (51) Y.-H. Lin, C.-W. Shen, F.-K. Guo, and B.-S. Zou,  Phys. Rev. D95, 114017 (2017), arXiv:1703.01045.
  * (52) H. Huang and J. Ping,  Phys. Rev. D99, 014010 (2019), arXiv:1811.04260.
  * (53) C.-J. Xiao, Y. Huang, Y.-B. Dong, L.-S. Geng, and D.-Y. Chen,  Phys. Rev. D100, 014022 (2019), arXiv:1904.00872.
  * (54) G.-J. Wang et al.,  (2019), arXiv:1911.09613.
  * (55) Y.-J. Xu, C.-Y. Cui, Y.-L. Liu, and M.-Q. Huang,  (2019), arXiv:1907.05097.
  * (56) M. I. Eides, V. Y. Petrov, and M. V. Polyakov,  (2019), arXiv:1904.11616.
  * (57) C. Xiao, J. Lu, J. Wu, and L. Geng,  (2020), arXiv:2007.12106.
  * (58) W. Ruangyoo, K. Phumphan, C.-C. Chen, A. Limphirat, and Y. Yan,  (2021), arXiv:2105.14249.
  * (59) Y. Dong, P. Shen, F. Huang, and Z. Zhang,  Eur. Phys. J. C 80, 341 (2020), arXiv:2002.08051.
  * (60) M. I. Eides and V. Yu. Petrov,  Phys. Rev. D98, 114037 (2018), arXiv:1811.01691.
  * (61) LHCb, R. Aaij et al.,  Phys. Rev. Lett. 125, 242001 (2020), arXiv:2009.00025.
  * (62) LHCb, R. Aaij et al.,  Phys. Rev. D 102, 112003 (2020), arXiv:2009.00026.
  * (63) X. Cao and J.-p. Dai,  Phys. Rev. D 100, 054033 (2019), arXiv:1904.06015.
  * (64) Q. Wu and D.-Y. Chen,  Phys. Rev. D 100, 114002 (2019), arXiv:1906.02480.
  * (65) M. B. Voloshin,  Phys. Rev. D 100, 034020 (2019), arXiv:1907.01476.
  * (66) T. J. Burns,  Eur. Phys. J. A51, 152 (2015), arXiv:1509.02460.
  * (67) E. S. Swanson,  Phys. Rept. 429, 243 (2006), arXiv:hep-ph/0601110.
  * (68) X.-Z. Weng, X.-L. Chen, W.-Z. Deng, and S.-L. Zhu,  Phys. Rev. D100, 016014 (2019), arXiv:1904.09891.
  * (69) M.-Z. Liu, F.-Z. Peng, M. Sanchez Sanchez, and M. P. Valderrama,  Phys. Rev. D98, 114030 (2018), arXiv:1811.03992.
  * (70) K. Chen, R. Chen, L. Meng, B. Wang, and S.-L. Zhu,  (2021), arXiv:2109.13057.
  * (71) L. Meng, B. Wang, G.-J. Wang, and S.-L. Zhu,  Phys. Rev. D100, 014031 (2019), arXiv:1905.04113.


For the hurricane movement, we assume that the hurricane originates at the bottom row of the grid (i.e., $m^y = 0$ for the initial MC state) and that $m^y$ increases by one in each period (i.e., the hurricane advances upwards by one step in each period) and will reach land in exactly $T$ periods. The hurricane can also move on the x-axis by either staying in the same x-coordinate, moving one cell to the left or one cell to the right. To determine the transition probabilities of each movement, we assign random weights to each movement and normalize them. In particular, we assign a uniform weight between 30 and 40 for staying in the same x-coordinate and a weight between 20 and 40 for moving to the right and left cells.  If the current x-coordinate is at one of the grid borders (e.g., $m^x=0$), then the probability of stepping out of the grid is zero. As an example, consider a $4\times 5$ grid, current hurricane location $(m^x,m^y)=(1,1)$, and weights 25, 36, and 22, for moving to locations $(0,2)$, $(1,2)$, $(2,2)$, respectively. Then, the transition probability of moving from location $(1,1)$ to $(0,2)$ is $\frac{25}{83}\approx 0.3$.

The hurricane movement transition probability matrix is generated at random for each instance. The initial $x$-coordinate is chosen uniformly at random considering all possible options. The initial intensity level is also generated with a uniform random distribution considering hurricane intensity levels of two or higher.

§.§ Demand Generation

The demand for each shelter depends on the current MC state of the hurricane (i.e., location and intensity). We consider a maximum demand parameter $d^{\max}$ and a maximum distance parameter $\delta^{\max}$ to model this dependency. The maximum demand $d^{\max}$ is chosen at random between values 1000 and 1500 for each instance and  represents the maximum demand for all shelters in a cell. The maximum demand at a cell is randomly distributed across all shelters in that cell. Therefore, each shelter $j\in \shelters$ has a maximum demand $d^{\max}_j$ and the total maximum demand of all shelters in the same cell is equal to $d^{\max}$.

The maximum distance $\delta^{\max}=150$ represents the distance of influence of the hurricane, i.e., all shelters which are farther than  $\delta^{\max}$ from the hurricane have zero demand. If the distance between a shelter $j \in \shelters$ and the hurricane is less than $\delta^{\max}$, then the shelters demand for MC state $m=(m^x, m^y, m^i) \in \markov$ is given by:

\[ d_{j,m} = d^{\max}_j \left(1 - \frac{\delta_{m,j}}{\delta^{\max}}\right)\cdot \left(\frac{m^i}{5}\right)^2,   \]

where $\delta_{m,j}$ is the Euclidean distance between the current location of the hurricane and the shelter. Thus, the demand generation is linear with respect to the hurricane location and quadratic with respect to the hurricane intensity. Note that the demand is zero if the intensity level is 0 and that the maximum demand is achieved when the hurricane is on top of the shelter and has  the highest intensity level (i.e., $\delta_{m,j} = 0$ and $m^i=5$).

§.§ Initial Capacity, Contingency Modality Options, and Capacity Expansion

The initial capacity of a DC depends on both $d^{\max}$ and the initial capacity percentage defined in Section <ref>. The total initial capacity over all DCs in a cell is given by $C^\text{ini}= d^{\max}\cdot p$, where $p \in \{0.2, 0.25, 0.3\}$ is the initial capacity percentage. The total initial capacity of a cell $C^\text{ini}$ is then distributed uniformly at random across all DCs in that cell. Thus, the initial capacity $C_j$ of a particular DC $j \in \centers$ is given by a portion of $C^\text{ini}$.

We generate different contingency modality options depending on the grid size (i.e., number of land cells) and the modality Type (i.e., either Type-1 or Type-2, see Section <ref>). Our modality set $\modality$ considers options that increase DCs capacity in adjacent cells and options that increase the capacity for all DCs. For example, in a $4\times5$ grid we consider modality options that increase capacity for DCs in cells 0 and 1, 1 and 2, 2 and 3, and options that increase the capacity in all DCs, that is, a total of four different contingency modality options based on location.

In addition, we consider capacity increments of Type-1 or Type-2, each one with four different options: Type-1 has options $\{10\%, 20\%, 30\%, 40\%\}$ and Type-2 has options $\{15\%, 30\%, 45\%, 60\%\}$. Then each modality option in $\modality$ corresponds to a combination of the affected DCs based on location and the capacity increase. Considering a $4\times5$ grid example with Type-1 modalities, a possible modality option is to increase the capacity of DCs in cells 2 and 3 by 20% increments. In this example, we have a total of 16 modality options (i.e., four location options and four increments options).

The capacity increments for each modality are based on the initial capacity of the DCs. For example, consider the  $4\times5$ grid with Type-1 modalities and active modality $\ell \in \modality$ affecting  cells 2 and 3 by 20% increments. Then, a DC $j \in \centers$ in cell 0 has no capacity increments given the current active modality (i.e., $K_{j\ell} =0$), but a DC $j' \in \centers$ in cell 2 with initial capacity $C_{j'}=100$ has capacity increments  $K_{j'\ell}= 20$.

§ ADDITIONAL RESULTS FOR APPROXIMATION METHODS

Solution time and quality of 2SLDR and SDDP bounds.  Results for $\hn$ over mid-size instances.

4c|Average Time (sec)     5cRelative Difference (%)

Modality     1l|Cap.     1c     1l     1l|     1c|     1c     1l     1l     1c|     1c

3[2]*Type-1     1l|20%     193.7     103.6     71.7     76.5      0.04      0.05      0.00     0.06      0.44

1l|25%     132.2     31.3     33.1      15.2      0.05      0.07      0.00     0.00     0.63

1l|30%     82.5      20.0      19.0     9.9       0.10      0.21      0.00     0.00     1.36

3[2]*Type-2     1l|20%     262.8     64.3     76.0      61.6      0.03      0.05      0.00     0.03      0.46

1l|25%     131.7     29.3     35.8      12.4      0.05      0.07      0.00     0.01      0.62

1l|30%     80.4      19.7      19.0     10.0      0.10      0.21      0.00     0.00      1.36

2c|Average     147.2     44.7      42.4     30.9      0.06      0.11      0.00     0.02      0.81

Solution time and quality of 2SLDR and SDDP bounds. Results for $\hn$ over large-size instances.

4c|Average Time (sec)     3cOpt. Gap (%)

Modality     Cap.      1c     1c     1c     (opt)     1c     1c     1c

3[2]*Type-1     1l|20%         2,100.9          1,961.8             449.3     (10)      0.51      0.33      0.13

1l|25%         1,679.4          1,628.6             288.0     (10)      0.78      0.39      0.15

1l|30%            965.1             864.2               56.6     (10)      0.61      0.48      0.18

3[2]*Type-2     1l|20%         2,501.3          1,794.3             730.9     (10)      0.56      0.31      0.12

1l|25%         1,660.5          1,764.6             203.7     (10)      0.78      0.40      0.13

1l|30%            950.1          1,035.3               57.9     (10)      0.62      0.49      0.18

2c|Av. (Total)         1,642.9          1,508.1             297.7     (60)      0.64      0.40      0.15

Solution time and  quality of 2SLDR and SDDP bounds. Results for $\ma$ over mid-size instances.

4c|Average Time (sec)     5cRelative Difference (%)

Modality     1l|Cap.     1c     1l     1l|     1c|     1c     1l     1l     1c|     1c

3[2]*Type-1     1l|20%     431.1     104.2     95.2     367.3     0.26      0.29      0.23     0.08      0.19

1l|25%     368.3     58.4     49.2      167.4     0.14      0.18      0.11     0.22     1.11

1l|30%     119.3     25.7      25.4     17.1      0.11      0.22      0.02     0.00     1.56

3[2]*Type-2     1l|20%     630.6     119.4     117.1     439.4     0.12      0.14      0.10     0.09      0.26

1l|25%     464.3     49.3     46.9      111.2     0.15      0.18      0.11     0.18      1.15

1l|30%     113.8     22.7      23.1     16.4      0.10      0.21      0.01     0.00      1.58

2c|Average     354.5     63.3      59.5     186.5     0.15      0.21      0.09     0.10      0.97

Solution time and quality of 2SLDR and SDDP bounds. Results for $\ma$ over large-size instances.

4c|Average Time (sec)     3cOpt. Gap (%)

Modality     Cap.      1c     1c     1c     (opt)     1c     1c     1c

3[2]*Type-1     1l|20%         2,266.5          2,855.8          1,913.0     (10)      0.50      0.32      0.13

1l|25%         3,067.1          2,458.6          1,295.8     (10)      0.84      0.46      0.19

1l|30%         1,051.1          1,028.4             133.1     (10)      0.62      0.49      0.19

3[2]*Type-2     1l|20%         4,065.5          3,488.9         3,500.0      (10)      0.59      0.34      0.14

1l|25%         4,295.9          4,107.4          1,960.6     (10)      0.89      0.40      0.18

1l|30%         1,272.7          1,252.9             150.3     (10)      0.65      0.52      0.20

2c|Av. (Total)         2,669.8          2,532.0          1,492.1     (60)      0.68      0.42      0.17

Solution time and quality of 2SLDR and SDDP bounds. Results for $\mm$ over mid-size instances.

4c|Average Time (sec)     5cRelative Difference (%)

Modality     1l|Cap.     1c     1l     1l|     1c|     1c     1l     1l     1c|     1c

3[2]*Type-1     1l|20%     2766.2     262.2     277.0     3521.5     0.12      0.26      0.11      0.00     0.39

1l|25%     1272.7     176.6     154.9     12447.8     0.09      0.13      0.04     0.25      0.98

1l|30%     4049.5     235.3     310.3     14746.8     0.13      0.25      0.00     1.37      1.77

3[2]*Type-2     1l|20%     3916.3     504.6     396.0     11703.3     0.13      0.26      0.13      0.02     0.22

1l|25%     1486.7     226.3     164.9     21226.9     0.09      0.12      0.01     0.38      2.30

1l|30%     8042.7     431.0     720.4     15323.7     0.13      0.24      0.01     0.95      1.80

2c|Average     3589.0     306.0     337.2     13161.7     0.11      0.21      0.05     0.50      1.24

Solution time and quality of 2SLDR and SDDP bounds. Results for $\mm$ over large-size instances.

4c|Average Time (sec)     3cOpt. Gap (%)

Modality     Cap.      1c     1c     1c     (opt)     1c     1c     1c

3[2]*Type-1     1l|20%         8,288.1          4,776.9       20,125.0      (1)       39.71     33.00     39.39

1l|25%       14,398.1        12,203.6       21,601.1      (0)       15.73     15.08     17.53

1l|30%         1,911.4         2,076.6        21,601.6      (0)       2.41      0.67     2.09

3[2]*Type-2     1l|20%       14,668.7        13,135.3       21,600.7      (0)       39.55     37.42     43.68

1l|25%       16,280.7        16,051.0       21,601.6      (0)       23.98     23.22     26.70

1l|30%         2,391.2          1,925.4       21,601.4      (0)       3.63      1.51     3.50

2c|Av. (Total)         9,656.4          8,361.5       21,355.2      (1)       20.84     18.48     22.15