# Improved Cloud resource allocation: how INDIGO-DataCloud is overcoming the
current limitations in Cloud schedulers

Alvaro Lopez Garcia1 Lisa Zangrando2 Massimo Sgaravatto2 Vincent Llorens3 Sara
Vallero4 Valentina Zaccolo4 Stefano Bagnasco4 Sonia Taneja5 Stefano Dal Pra5
Davide Salomoni5 Giacinto Donvito6 1 IFCA (CSIC–UC), Avda. los Castros s/n,
39005 Santander, Spain
2 INFN Padova, Via Marzolo 8, 35131 Padova, Italy
3 CC-IN2P3 (CNRS), 21 Avenue Pierre de Coubertin, 69627 Villeurbanne CEDEX,
France
4 INFN Torino, Via Pietro Giuria 1, 10125 Torino, Italy
5 INFN CNAF, Viale Berti Pichat 6/2, 40127 Bologna, Italy
6 INFN Bari, Via E. Orabona 4, 70125 Bari, Italy<EMAIL_ADDRESS>
###### Abstract

Performing efficient resource provisioning is a fundamental aspect for any
resource provider. Local Resource Management Systems (LRMS) have been used in
data centers for decades in order to obtain the best usage of the resources,
providing their fair usage and partitioning for the users. In contrast,
current cloud schedulers are normally based on the immediate allocation of
resources on a first-come, first-served basis, meaning that a request will
fail if there are no resources (e.g. OpenStack) or it will be trivially queued
ordered by entry time (e.g. OpenNebula). Moreover, these scheduling strategies
are based on a static partitioning of the resources, meaning that existing
quotas cannot be exceeded, even if there are idle resources allocated to other
projects. This is a consequence of the fact that cloud instances are not
associated with a maximum execution time and leads to a situation where the
resources are under-utilized. These facts have been identified by the INDIGO-
DataCloud project as being too simplistic for accommodating scientific
workloads in an efficient way, leading to an under-utilization of the
resources, a non desirable situation in scientific data centers. In this work,
we will present the work done in the scheduling area during the first year of
the INDIGO project and the foreseen evolutions.

##  1 Introduction

The Cloud computing model is being progressively adopted by the scientific
community as a promising paradigm. Cloud infrastructures are now common in the
computing resources portfolio that any scientific resource provider offers to
its user communities. However, in spite of its advantages, the cloud still has
several areas that need to be improved in order to expose to the researchers
its fully functional power. In this context, INDIGO-DataCloud111Referred to as
INDIGO from now. (INtegrating Distributed data Infrastructures for Global
ExplOitation) [1], a project funded under the Horizon 2020 framework program
of the European Union, addresses the challenge of developing an advanced data
and computing platform targeted at scientific communities, deployable on
multiple hardware, and provisioned over hybrid e-Infrastructures.

The project is built by putting together three different layers that integrate
together providing a scientific computing infrastructure, namely the
infrastructure layer, the platform layer and the user interface layer. In this
paper we will describe how INDIGO tackles the identified gaps in terms of
scheduling and resource provisioning at the Infrastructure as a Service layer,
overcoming the current limitations in the cloud management frameworks adopted
by the project: OpenStack and OpenNebula (ONE).

Although the Cloud model heavily improves the resource usage in the data
centers, there are still open issues in the resource management area. In a
Cloud environment, the concept of “task” or “job” (that, implicitly has an
associated duration) does not exist. Instead, the Cloud model focuses on
“resources” (that is, networks, servers, storage, etc.) that are provisioned
on demand by the users. This self-service of resources assumes that there is
no duration associated to them, thus making difficult to perform an efficient
resource provisioning from the provider standpoint.

Moreover, in the existing Cloud Management Frameworks the implemented
scheduling strategy is just based on the immediate First Come First Served
model [2, 3]; consequently a request will be rejected if there are no
resources immediately available: it is up to the client to later re-issue the
request. It must be also stressed that the adopted resource allocation model
is based on a static partitioning, so that every user group is assigned an
agreed and fixed quota of resources which cannot be exceeded even if there are
idle resources available but allocated to other groups.

Today the efficient management of Cloud resources is therefore an open issue.
The INDIGO project is addressing it through several, complementary approaches
that will be explained in the upcoming sections. In Section 2 we will explain
how we implemented fair-share scheduling strategies in both OpenStack and
OpenNebula. In Section 2.3 we present an implementation of preemptible
instances for OpenStack clouds. In Section 3 we describe the Partition
Director, a tool that esas the transition of computing nodes between different
infrastructures.

##  2 Advanced Scheduling

On large public cloud providers, in which resources are ideally infinite,
multi-tenant applications are able to scale in and out driven only by their
functional requirements and tenants are normally charged a posteriori for
their resource consumption.

Smaller private Cloud for scientific computing mostly operates in a saturated
regime. In this case, tenants are charged a priori for their computing needs
by paying for a fraction of the computing (or storage) resources. To optimize
the use of the data center an advanced resource allocation policy is needed.

We consider a scenario in which the available resources are divided into
statically and dynamically partitioned. Statically assigned resources are
partitioned among projects according to fixed shares, while dynamical assets
follow the effective requests per project, granting efficient and fair access
to such resources to all projects.

The general topic of advanced resource scheduling is addressed by several
components of the INDIGO project. On the one hand we have implemented fair
share policies by means of the FairShare Scheduler (FaSS) for OpenNebula and
Synergy for OpenStack. On the other hand we have developed an extension for
OpenStack allowing the execution of preemptible instances.

###  2.1 Synergy

Synergy [4] has been develeped in the INDIGO-DataCloud project as an
extensible general purpose management service designed for executing tasks in
OpenStack. Its functionalities are provided by a collection of managers which
are specific and independent pluggable tasks executed periodically, like cron
jobs, or interactively through a RESTful API. Different managers can coexist
and they can interact with each other or with different OpenStack services in
a loosely coupled way.

To enable a more effective and flexible resource allocation and utilization in
OpenStack, six specific managers (as presented in Section 2.1.1) have been
implemented. Together they provide an advanced scheduling and resource
allocation strategy based on a fair-share algorithm and a persistent priority
queue. Synergy can maximize the resource utilization by allowing the OpenStack
projects to consume extra shared resources in addition to those statically
assigned. Therefore the projects can access two different kind of quotas: the
private quota and the shared one. On the one hand, the private quota is
composed of resources statically allocated and managed using the “standard”
OpenStack policies. On the other hand, the shared quota refers to the
resources that can be shared among different projects: it is composed of
resources not statically allocated and they are fairly distributed among the
different users by Synergy. The size of such quota is calculated as the
difference between the total amount of cloud resources and the total resources
allocated to the private quotas (Figure 1).

Figure 1: Synergy quota structure.

Only the projects selected by the administrator can consume the shared
resources and, unlike the private quota, the user requests that cannot be
immediately satisfied are not rejected but instead inserted in a persistent
priority queue. The priority is periodically recalculated by Synergy by
enforcing specific fair-share policies defined by the administrator which
consider the historical resource usage in a well defined time window, the
project and user shares and some specific weights (e.g. age, decay, cpu usage,
memory).

The chosen fair-share algorithm for Synergy is the Priority Multifactor
implemented by the SLURM batch scheduler [5]. The enqueued requests are
processed according to their priority ordering. In case there are no resources
available for the selected request, it is skipped and Synergy will process the
next one in the queue (i.e. backfilling): this allows to maximize the resource
utilization and, at the same time, avoid to block the queue for long time.

####  2.1.1 Architecture

Synergy’s architecture is depicted in Figure 2. The Nova-Manager intercepts
all incoming user requests. The FairShare-Manager calculates and assigns to
each of them the proper priority value. Such requests are immediately inserted
in a persistent priority queue by the Queue-Manager. The role of the
Scheduler-Manager is to process the user requests by fetching them from the
queue according to their priority based ordering. The selected request is
finally sent to the OpenStack Compute Scheduler through Synergy’s Nova-
Manager. In the scenario of full resource utilization for a specific quota,
the Quota-Manager advises the Scheduler-Manager to wait until the compute
resources become available again. In case of failure, the Scheduler-Manager
provides a retry mechanism which handles the failed requests by inserting them
again into the queue (up to a given number of retries). The priority of the
queued requests is periodically recalculated in order to increase the priority
of older requests.

To prevent any possible interaction issue with the existing OpenStack clients,
no new states have been added so that from the client point of view the queued
requests remain in “Scheduling” state till the compute resources are
available.

Figure 2: Synergy’s architecture.

Whenever applicable Synergy has been developed by adopting the full OpenStack
style according to the specific development guidelines, including the coding
style and the technologies to be used: the source code is in launchpad [6, 7]
and the OpenStack Continuous Integration system is used. Synergy is a full
python service accessed by clients through a well defined RESTful API which
provides either management commands for handling the plugged managers (e.g.
activate manager, suspend, get status, etc) or executing specific commands on
the selected manager (e.g. get/set quota, list queues, etc). The API can be
extended for including new specific interfaces. For what concerns the internal
interfaces, Synergy interacts with the OpenStack services as Nova and
Keystone, by invoking the exposed RESTful interfaces or using the internal RPC
APIs. All Synergy default configuration options are defined in the
synergy.conf file.

###  2.2 FaSS: A fair-share scheduler for OpenNebula

The current OpenNebula scheduler is first-in-first-out (FIFO) and it is based
only on static resources partitioning among the projects, which is not
efficient for a scientific datacenter. FaSS grants fair access to dynamic
resources prioritizing tasks assigned according to an initial weight and to
the historical resource usage, irrespective of the number of tasks running on
the system.

The software was designed to be fully synchronized with the ONE
authentication, quota, monitoring and accounting systems, without being
intrusive in the ONE code and keeping to the minimum the dependencies on the
ONE implementation details. In such a way, the code is fairly independent on
future ONE releases.

The ONE original scheduler is kept to match the resources to requests and FaSS
is structured as a self-contained module interacting only with the ONE XML-RPC
interface.

####  2.2.1 Architecture

The architecture of the FaSS service is depicted in Figure 3. It has been
conceived following the ONE conceptual design [8]. In the figure, white blocks
are native ONE components, while the newly developed components are shown in
blue. New tools are pictured in green, and interfaces/API in gray. Purple
arrrows represent internal component interactions. Blue arrows represent the
API-Sunstone GUI interaction.

Figure 3: FaSS architecture.

The main component of FaSS is the Priority Manager. Its task is to calculate
periodically priorities for queued jobs. It receives input data from the ONE
core through XML-RPC calls, and from the Priority Database through a no-SQL
interface. This choice has been made in order to avoid storing large XML
strings, which could easily be corrupted in a single SQL entry.

The Priority Database holds the modules internal data: initial priority
values, historical information on resource usage and recalculated priority
values. The Priority Manager interacts with a set of pluggable algorithms,
such as MultiFactor [5] and FairTree [9], to calculate priorities. It exposes
an XML-RPC interface, independent from the one exposed by the ONE core, and
uses the Priority Database as data back-end.

The second fundamental component of FaSS is the XML-RPC interface of the
Priority Manager, which replaces the ONE XML-RPC interface in the
communication with the FIFO scheduler. The queue of pending jobs to be
processed must be the one ordered according to the priorities calculated by
the Priority Manager. The XML-RPC server, then, catches the scheduler calls,
provides a reordered queue and redirects unknown methods to the core of ONE
XML-RPC server.

The Sunstone web-based GUI service is extended to allow monitoring and
operating the FaSS module functionalities. The Priority Manager will be
accessed by clients, interfacing to the module core through a set of bindings
analogous to the ONE Cloud API (OCA) [10].

A prototype of the scheduler with limited functionalities is ready [11].
Namely, the prototype comprises the Priority Manager and its XML-RPC
interface. For the next INDIGO project release, a fully featured prototype,
including some algorithms and installation tools, will be made available.

###  2.3 Preemptible Instances support

Resources in a cloud environment do not have an associated lifespan prevents
resource providers to perform advanced scheduling strategies (like
backfilling) to increase their infrastructure utilization. Implementing fair
share policies (as depicted in Section 2) can alleviate the infrastructure
utilization problem. Another, complementary, approach is the implementation of
preemptible instances, that is instances that can be terminated whenever the
resource provider decides so.

In commercial cloud world, providers are offering this functionality to their
users, but there is not a similar implementation in the open source Cloud
Management Frameworks. For example, in the Amazon EC2 Spot Instances [12]
users are able to select how much they are willing to pay for their resources
in a market where the price fluctuates accordingly to the demand. Whenever the
reference price goes above the user bid, those requests are terminated. The
Google Cloud Engine (GCE) [13] has released a new product branded as
_Preemptible Virtual Machines_ [14], where machines are limited to 24h of
execution and can be terminated without prior notice.

Resource providers can offer these resources to the users at a fraction of the
original price, increasing their resource utilization without preventing
higher priority instances to be executed (since the preemptible instances
would be terminated whenever needed). Users would get access to a higher
number of resources that they can use for fault-tolerant tasks like batch
processing. We have proposed this concept to be included in the upstream
OpenStack Compute component [15], and an implementation (described below) is
available [16].

####  2.3.1 OPIE: OpenStack Preemptible Instances Extension

We have developed OPIE [16] as an extension for OpenStack, allowing to execute
preemptible instances in an existing infrastructure. In the current
implementation, preemptible instances can run within the user quota and as far
as there are not preventing normal requests from being executed. Whenever the
scheduler detects that a normal instance cannot be executed because of a
preemptible instance, it triggers its termination, according to several filter
and weight functions, configurable by the resource provider. OPIE is composed
of the following modules:

  * •

A modified scheduler, that augments the OpenStack FilterScheduler with the
ability to execute normal and preemptible instances, configurable and
extensible via additional filter and raking functions so that each resource
provider can implement its policies.

  * •

A special host manager, able to report different usage values so that the OPIE
scheduler is able to take the appropriate decision.

  * •

A plugin for the OpenStack clients that allows booting preemptible instances
in a cloud infrastructure with OPIE enabled.

##  3 Partition Director

A typical Grid scientific computing centre is organized and dimensioned
towards a quite specific usage pattern. Such computing resources are normally
managed by a batch system according to some sort of fair share policy so that
a convenient resource usage is enforced. User communities are now asking for
the provisioning of resources exposing a Cloud interface from such
infrastructures. Examples of some relevant use cases are:

  * •

A group would like to dedicate a subquota of its pledges at the centre to a
“cloud computation campaign”, then eventually go on as usual with batch mode.

  * •

A group wants to gradually migrate its computing resources to the cloud, few
at a time.

  * •

One user group needs some resources for interactive usage, which currently is
handled by providing powerful hosts “User Interface”, bypassing the official
pledge count.

The Partition Director (PD) is a tool developed to offer the ability to
convert a set of Worker Nodes in the batch cluster (currrently LSF) to Cloud
compute nodes (currently OpenStack), and vice versa. The tool can be used by a
site administrator, but can also be driven by a user group: it can trigger
node conversion to add more resources toward a partition with many more
requests. The remainder of this section describes its functionalities and
working principles.

###  3.1 Functionality

The Partition Director eases the management of a hybrid data center providing
both batch system and cloud based services. Partition Director provides site
administrators with an instrument to dynamically resize sub-quotas of a given
user group among different infrastructures (HPC, Grid, local batch systems,
cloud resources). Physical computing resources, in fact, can work as member of
batch system cluster or as compute node on cloud resources.

At its basic level, the PD is in charge of switching the role of selected
computing machines (Worker Nodes) from a batch cluster to a cloud one,
enabling them as Compute Node (CN) and vice versa, managing the intermediate
transition states, thus ensuring consistency, Moreover, it adjusts the shares
at the batch side, to enforce an overall uniform quota to groups using both
cloud and batch resources.

####  3.1.1 Node transitions

The role conversion of nodes between Batch and Cloud (WN $\leftrightarrow$
CN), are managed by a finite state machine as represented in Figure 4
exploiting a partitioning concept [17] implemented as a partition driver
script. Each physical machine in the cluster has one and only one status at
any point in time and only two of them are stable: $B$ for nodes active as WN
and $C$ for those active as CN. Transitory states are intended for validation
or draining phases: the validation step is required to evaluate the
consistency of each transition request; the draining phase is needed to ensure
that a node is actually free and clean before activating it with the new role.
It is worth noting that draining from cloud to batch requires us to define a
time limit, because there is no maximum lifetime for VMs running in a CN as
there is for jobs running in a WN. To address this problem, the Partition
Director sets a Time To live (TTL) value as defined by the Machine/Job
Features Task Force [18]. VMs can check whether a TTL is defined and stop
themselves gracefully. After the TTL has expired, remaining VMs are destroyed
and the role conversion can complete.

####  3.1.2 Share adjustment

Whenever nodes are moved from the partition they belong to, the size of both
the batch and cloud cluster changes. However, when resources are moved from
batch to cloud, they are assigned to a single tenant (user group). The
consequence is that the shares are to be rebalanced at the batch system side,
to guarantee that at the overall pledges of all the user communities working
with the centre are unaffected.

$B:1$$B2CR:1$$B2C:2$$C2B:2$$C2BR:2$$C:2$ Figure 4: The Status Transition Map.
Each node publishes a dynp “load index” with value 1 (pink, available for
batch jobs) or 2 (green, no new batch tasks). The batch system alters each new
job to require nodes having dynp==1. When a transition from batch ($B$) or
cloud ($C$) is triggered for a node, it goes through different states: $B2CR$
or $C2BR$ for validation, $B2C$ or $C2B$ for Draining tasks of the partition
being left, $C$ or $B$ to become active in the new partition.

##  4 Conclusions

INDIGO has tackled the existing cloud resource provisioning and scheduling
limitations in different, complementary ways. We have focused on the
enhancement of the scheduling policies through the development of services
that makes it possible to deliver fair-share policies and through the
introduction of preemptible instances. Furthermore we have developed a tool
that eases the transition of nodes from one infrastructure managed by a LRMS
to a Cloud one, preserving the user share and quota.

These components have been deployed at several production sites (like INFN-
Padova, INFN-CNAF in Italy and IFCA-CSIC in Spain) in order to validate its
functionalities, its robustness as well as its stability under different real
usage conditions. Some test results have confirmed the expected limitation of
the SLURM MultiFactor algorithm as documented at [5], thus Synergy developers
are working on the integration of the SLURM FairTree [9], a more sophisticated
algorithm which fully solves the observed issue.

##  5 Acknowledgments

The authors want to acknowledge the support of the INDIGO-DataCloud (grant
number 653549) project, funded by the European Commission’s Horizon 2020
Framework Programme.

## References

  * [1] Davide Salomoni et al.  “INDIGO-Datacloud: foundations and architectural description of a Platform as a Service oriented to scientific computing”  In _arXiv preprint arXiv:1603.09536_ , 2016
  * [2] Borja Sotomayor, Ruben S. Montero, Ignacio M. Llorente and Ian Foster  “Virtual infrastructure management in private and hybrid clouds”  In _IEEE Internet Computing_ 13, 2009, pp. 14–22  DOI: 10.1109/MIC.2009.119
  * [3] Oleg Litvinski and Abdelouahed Gherbi  “Experimental evaluation of OpenStack compute scheduler”  In _Procedia Computer Science_ 19.Ant Elsevier B.V., 2013, pp. 116–123  DOI: 10.1016/j.procs.2013.06.020
  * [4] Lisa Zangrando, Marco Verlato, F Fanzago and Massimo Sgaravatto  “Synergy: a service for optimising the resource allocation in the cloud based environments”  In _International Symposium on Grids and Clouds (ISGC) 2016_ PoS(ISGC 2016)032, 2016
  * [5] “Slurm Multifactor Priority Plugin”  URL: https://slurm.schedmd.com/priority_multifactor.html
  * [6] “Synergy Service”  URL: https://launchpad.net/synergy-service
  * [7] “Synergy Manager”  URL: https://launchpad.net/synergy-scheduler-manager
  * [8] “OpenNebula Introduction”, 2016  URL: http://docs.opennebula.org/5.2/intro_release_notes/concepts_terminology/intro.html
  * [9] Ryan Cox and Levy Morrison  “Fair Tree: Fairshare Algorithm for Slurm”  URL: slurm.schedmd.com/SUG14/fair_tree.pdf
  * [10] “OpenNebula APIs”, 2016  URL: http://docs.opennebula.org/5.2/integration/system_interfaces/introapis.html
  * [11] “FairShare Scheduler for OpenNebula”  URL: https://github.com/indigo-dc/one-fass
  * [12] Amazon  “Amazon EC2 Spot Instances”, 2016  URL: https://aws.amazon.com/ec2/spot/
  * [13] “Google Compute Engine”, 2016  URL: https://cloud.google.com/compute/
  * [14] “Google Compute Engine Preemptible VMs”, 2016  URL: https://cloud.google.com/preemptible-vms/
  * [15] “OpenStack proposal to support preemptible instance”  URL: https://review.openstack.org/#/c/104883/
  * [16] “OPIE”  URL: https://github.com/indigo-dc/opie
  * [17] Stefano Dal Pra  “Efficient provisioning for multi-core applications with LSF”  In _Journal of Physics: Conference Series_ 664.5, 2015, pp. 052008
  * [18] M Alef et al.  “Machine/Job Features”, 2016


# Fine Structure of Pentaquark Multiplets in the Dynamical Diquark Model

Jesse F. Giron<EMAIL_ADDRESS>Computational Physics Division (XCP), Los
Alamos National Laboratory, Los Alamos, NM 87545, USA  Richard F. Lebed
<EMAIL_ADDRESS>Department of Physics, Arizona State University, Tempe,
AZ 85287, USA

(October, 2021)

###### Abstract

We apply the dynamical diquark model to predict the spectrum of hidden-charm
pentaquark states in both unflavored and open-strange sectors. Using only
Hamiltonian parameters introduced in the tetraquark $S$\- and $P$-wave
multiplets, the model naturally produces the level spacing supported by the
most recent LHCb results for $P_{c}$ structures. Furthermore, using model
inputs obtained from data of hidden-charm, open-strange tetraquarks
($Z_{cs}$), we predict the spectrum of $P_{cs}$ states, including the recently
observed $P_{cs}(4459)$. We find all pentaquark candidates observed to date to
belong to the $1P$ multiplet and hence have positive parity.

Exotic hadrons, pentaquarks, diquarks

##  I Introduction

A recent burst of discoveries in the sector of heavy-quark exotic hadrons has
now pushed the number of observed candidates to over 50\. Multiple detailed
reviews of these states have been published in recent years Lebed  _et al._
(2017); Chen  _et al._ (2016); Hosaka  _et al._ (2016); Esposito  _et al._
(2017); Guo  _et al._ (2018); Ali  _et al._ (2017); Olsen  _et al._ (2018);
Karliner  _et al._ (2018); Yuan (2018); Liu  _et al._ (2019); Brambilla  _et
al._ (2020), but even such comprehensive reports have been unable to keep pace
with the spectacular rate of new experimental findings that have occurred with
remarkable regularity to the present day.

This paper focuses upon an analysis of the hidden-charm pentaquark states,
labeled $P_{c}$ and $P_{cs}$ in the nonstrange and strange sectors,
respectively. The first candidates [$P_{c}(4450)$, $P_{c}(4380)$] were
observed in 2015 by the LHCb Collaboration Aaij  _et al._ (2015) as structures
in the $J/\psi\,p$ spectrum in the decay $\Lambda_{b}\\!\to\\!J/\psi\,pK^{-}$.
Then in 2019, LHCb resolved $P_{c}(4450)$ into two peaks, $P_{c}(4440)$ and
$P_{c}(4457)$, and observed a further $J/\psi\,p$ structure, $P_{c}(4312)$
Aaij  _et al._ (2019). Very recently, one more $J/\psi\,p$ structure
[$P_{c}(4437)$] has been observed by LHCb in the decay
$B_{s}^{0}\\!\to\\!J/\psi\,p\bar{p}$ Aaij  _et al._ (2021a), but at a somewhat
lower statistical significance ($>\\!3\sigma$) than the other $P_{c}$ states.
The only known strange candidate to date, $P_{cs}(4459)$, was also very
recently observed at LHCb as a $J/\psi\,\Lambda$ structure in the decay
$\Xi_{b}^{-}\\!\to\\!J/\psi\,\Lambda K^{-}$ Aaij  _et al._ (2021b) (also at
just over $3\sigma$). The measured masses and widths of the states are
collected in Table 1.

Table 1: Pentaquark candidate masses and as widths measured by the LHCb Collaboration Aaij  _et al._ (2015, 2019, 2021a, 2021b). State | Mass (MeV) |  Width (MeV)
---|---|---
$P_{c}(4312)$ | $4311.9\pm 0.7^{+6.8}_{-0.6}$ | $9.8\pm 2.7^{+3.7}_{-4.5}$
$P_{c}(4337)$ | $4337^{+7\ +2}_{-4\ -2}$ | $29^{+26\ +14}_{-12\ -14}$
$P_{c}(4380)$ | $4380\pm 8\pm 29$ | $205\pm 18\pm 86$
$P_{c}(4440)$ | $4440.3\pm 1.3^{+4.1}_{-4.7}$ | $20.6\pm 4.9^{+8.7}_{-10.1}$
$P_{c}(4457)$ | $4457.3\pm 0.6^{+4.1}_{-1.7}$ | $6.4\pm 2.0^{+5.7}_{-1.9}$
$P_{cs}(4459)$ | $4458.8\pm 2.9^{+4.7}_{-1.1}$ | $17.3\pm 6.5^{+8.0}_{-5.7}$

Note that all of these measurements have been made by LHCb. The only
independent evidence for $P_{c}$ states to date comes from the D0
Collaboration, which observes events consistent with the unresolved
$P_{c}(4440)\\!\to\\!J/\psi\,p$ structure at 3.2$\sigma$ Abazov  _et al._
(2019).

Much about the pentaquark states remains unknown, starting with their $J^{P}$
quantum numbers. Indeed, their parity eigenvalues alone would already reveal a
great deal about their structure, since the parity of an $S$-wave $J/\psi\,p$
(or $J/\psi\,\Lambda$) system is $-1$. The possibility of molecules composed
of weakly bound $\Sigma^{(*)}_{c}$-${\bar{D}}^{(*)}$ pairs (also in an $S$
wave) was examined even before the first LHCb paper Wu  _et al._ (2010); Wang
_et al._ (2011); Yang  _et al._ (2012); Karliner and Rosner (2015). The
proximity of the $\Sigma_{c}\,\bar{D}^{*}$ threshold to $m_{P_{c}(4440)}$ and
$m_{P_{c}(4457)}$, the $\Sigma_{c}\bar{D}$ threshold to $m_{P_{c}(4312)}$, and
the $\Xi_{c}{\bar{D}}^{*}$ threshold to $m_{P_{cs}(4459)}$, has been noted
prominently in many publications, not least of which in LHCb’s own papers Aaij
_et al._ (2019, 2021b). The relevant thresholds and $J^{P}$ values for these
and related $S$-wave molecular states are conveniently tabulated in Ref. Dong
_et al._ (2021).111It should be noted that molecules in higher partial waves
($P,D,\ldots$) are not logically precluded from forming, although the presence
of a centrifugal potential barrier for $L\\!>\\!0$ may interfere with the
binding. In addition, it seems likely that if $P$-wave molecules exist, then
the lower-energy (and more prominent) $S$-wave molecules would be expected to
be observed first. In particular, no $P\\!=\\!+$ hidden-charm pentaquark
molecular thresholds occur below 4700 MeV; thus, if any of the known $P_{c}$
states is found to have $P\\!=\\!+$, it is not easily understood as a hadronic
molecule. Since the structure $P_{c}(4380)$ is found to carry opposite parity
to the original $P_{c}(4450)$ Aaij  _et al._ (2015), then if it persists as a
state, one must conclude that at least one of the $P_{c}$’s is not
molecular.222Other issues with molecular interpretations of $P_{c}$ involve
the $J^{P}$ of the particle responsible for binding. In the case of
$P_{c}(4312)$, a $0^{-}$ meson like $D$ cannot support a trilinear coupling to
another $0^{-}$ meson like $\pi$ or $\eta$; at minimum, a $\rho$-like exchange
would be necessary. And if nonstandard meson coupling fields like $\rho$ are
sufficient for binding hadronic molecules, then one might expect states, bound
by other light mesons, that include the isoscalar $\Lambda_{c}$ to have
appeared in the existing data.

A large body of literature has examined the $P_{c}$ states using a number of
approaches: not only as hadronic molecules, but also through quark-potential
models, diquark models, QCD sum rules, and others; for a discussion and
numerous references, see the previously cited reviews Lebed  _et al._ (2017);
Chen  _et al._ (2016); Hosaka  _et al._ (2016); Esposito  _et al._ (2017); Guo
_et al._ (2018); Ali  _et al._ (2017); Olsen  _et al._ (2018); Karliner  _et
al._ (2018); Yuan (2018); Liu  _et al._ (2019); Brambilla  _et al._ (2020).
Most of the relevant QCD sum-rule calculations Wang (2021, 2020); Azizi  _et
al._ (2021a, b, 2017), but not all (note Ref. Chen  _et al._ (2015)), postdate
these reviews.

In this work we apply a different approach, that of the dynamical diquark
model, to the quantitative study of the spectroscopic fine structure of
hidden-heavy-flavor pentaquarks. The dynamical diquark model is based upon the
idea that multiquark systems occasionally form configurations in which the
attraction of two quarks in a color-triplet channel is greater than that of
either quark to the nearest antiquark in a color-singlet channel. One may then
describe the full hadron in terms of compounds of diquark
[$\delta^{\vphantom{1}}\\!\equiv\\!(Qq)_{\bar{\bf 3}}$] Brodsky  _et al._
(2014) and triquark [${\bar{\theta}}\\!\equiv\\!(\bar{Q}_{\bar{\bf
3}}(q_{1}q_{2})_{\bar{\bf 3}})_{\bf 3}$] Lebed (2015) quasiparticles. In order
for this organization to be sensible, the quasiparticles must achieve
sufficient spatial separation to be described as interacting through a
potential $V(r)$. If each color-triplet quasiparticle component carries at
least one heavy quark (labeled above as $Q,\bar{Q}$), then $V(r)$ may be
modeled using the same static potentials as the ones that are calculated in
lattice simulations of quarkonium and its hybrid excitations, leading to a
description of exotic hadrons in terms of the Born-Oppenheimer (BO)
approximation. This dynamical diquark model and the spectroscopy of tetraquark
and pentaquark states obtained from it were first described in Ref. Lebed
(2017).

With a predictive model in hand, it becomes possible to study the multiplet
band structure for $c\bar{c}q\bar{q}^{\prime}$ and $c\bar{c}qqq$ states
numerically, as was first done in Ref. Giron  _et al._ (2019). The fine
structure of the ground-state ($S$-wave) and first excited-state ($P$-wave)
tetraquark multiplets were first studied in Refs. Giron  _et al._ (2020) and
Giron and Lebed (2020a), respectively. The application to
$b\bar{b}q\bar{q}^{\prime}$ and $c\bar{c}s\bar{s}$ states appears in Ref.
Giron and Lebed (2020b), $c\bar{c}c\bar{c}$ states in Ref. Giron and Lebed
(2020c), and $c\bar{c}q\bar{s}$ states in Ref. Giron  _et al._ (2021).

However, the fine structure of the hidden-charm pentaquark states has not yet
been analyzed in this model, largely due to the absence of any particular
$P_{c}$ state for which the $J^{P}$ quantum numbers are definitively known. In
all the tetraquark cases, a specific state [e.g., $X(3872)$, $Z_{b}(10610)$,
$X(4140)$] can be identified as the cornerstone upon which the rest of the
multiplet is built, but the pentaquark sector to date lacks such a candidate.
However, one expects that the same fine-structure Hamiltonian applies to the
tetraquark $\delta^{\vphantom{1}}$-${\bar{\delta}}$ and pentaquark
$\delta^{\vphantom{1}}$-${\bar{\theta}}$ sectors, and moreover, that comparing
the nonstrange $c\bar{c}q\bar{q}^{\prime}$ and strange $c\bar{c}q\bar{s}$
tetraquarks allows one to determine properties of the $P_{cs}$ states from the
corresponding $P_{c}$ states Giron  _et al._ (2021).

The observation of the new $P_{c}(4337)$ state produces a very interesting
spectrum (see Table 1): Two narrow, closely spaced pairs of states
[$P_{c}(4312),P_{c}(4337)$ and $P_{c}(4440),P_{c}(4457)$] with nearly the same
mass splitting. The dynamical diquark model, as shown in this paper, produces
a unique combination of such states in its pentaquark spectrum: The lower pair
are $J^{P}\\!=\\!{\frac{1}{2}}^{+}$, and the upper pair are
$J^{P}\\!=\\!{\frac{3}{2}}^{+}$. Since all of these are $Q\bar{Q}qqq$ states
with $P\\!=\\!+$, they are $P$-wave states; we predict masses for the other
states in this multiplet, as well as those in the lower $S$-wave multiplet.

We further use these results, as discussed above, to predict the masses of the
corresponding $P_{cs}$ states. In this case, the crucial ingredient of the
analysis is a comparison Giron  _et al._ (2021) of the well-known hidden-charm
nonstrange states and the newly observed open-strange, hidden-charm tetraquark
states $Z_{cs}$:

$\displaystyle m_{Z_{cs}(3985)}$ $\displaystyle=$ $\displaystyle
3982.5^{+1.8}_{-2.6}\pm 2.1\ {\rm MeV},$ $\displaystyle\Gamma_{Z_{cs}(3985)}$
$\displaystyle=$ $\displaystyle 12.8^{+5.3}_{-4.4}\pm 3.0\ {\rm MeV},$ (1)

from the BESIII Collaboration Ablikim  _et al._ (2021) in the process
$e^{+}e^{-}\\!\to\\!K^{+}(D_{s}^{-}D^{*0}\\!+\\!D_{s}^{*-}D^{0})$, and

$\displaystyle m_{Z_{cs}(4000)}$ $\displaystyle=$ $\displaystyle 4003\pm 6^{+\
4}_{-14}\ {\rm MeV},$ $\displaystyle\Gamma_{Z_{cs}(4000)}$ $\displaystyle=$
$\displaystyle\ 131\pm 15\pm 26\ {\rm MeV},$ $\displaystyle m_{Z_{cs}(4220)}$
$\displaystyle=$ $\displaystyle 4216\pm 24^{+43}_{-30}\ {\rm MeV},$
$\displaystyle\Gamma_{Z_{cs}(4220)}$ $\displaystyle=$ $\displaystyle\ 233\pm
52^{+97}_{-73}\ {\rm MeV},$ (2)

from the LHCb Collaboration in the process $B^{+}\\!\to\phi(J/\psi\,K^{+})$
Aaij  _et al._ (2021c).

Other diquark-based pentaquark models for have appeared in the literature. For
example, Ref. Zhu and Qiao (2016) was based upon the diquark-triquark proposal
of Ref. Lebed (2015), its model containing spin-dependent, but not flavor-
dependent, couplings. The model of Ref. Ali  _et al._ (2019) is also of
diquark-triquark type, but both of the heavy quarks $c\bar{c}$ reside in the
triquark. The very recent Ref. Shi  _et al._ (2021a) appeared subsequent to
the discovery of $P_{cs}(4459)$, but is a diquark-diquark-antiquark model.
This work is the first one to treat $P_{c}(4337)$ as a diquark-based state.

This paper is organized as follows: In Sec. II, we reprise the notation of the
model for identifying the pentaquark states within their BO multiplets.
Section III introduces the mass Hamiltonian for $S$-wave and $P$-wave
multiplets, and presents expressions for the masses of all their component
states. In Sec. IV, we analyze these expressions numerically, using inputs
from our previous work in the tetraquark sector, and predict values for all
unknown $S$\- and $P$-wave hidden-charm pentquarks in both light and open-
strange sectors. Section V presents our conclusions.

##  II States of the Model

A cataloguing of ${Q\bar{Q}}q\hskip 1.0ptq_{1}q_{2}$ (and
${Q\bar{Q}}q_{1}\bar{q}_{2}$) states in the dynamical diquark model, where
$q\\!\in\\!\\{u,d,s\\}$ and $q_{i}\\!\in\\!\\{u,d\\}$, first appears in Ref.
Lebed (2017). The same notation, with small modifications, is applied to
$c\bar{c}s\bar{s}$ in Ref. Giron and Lebed (2020b) and to $c\bar{c}c\bar{c}$
in Ref. Giron and Lebed (2020c). All confirmed exotic candidates to date have
successfully been accommodated within the lowest ($\Sigma^{+}$) Born-
Oppenheimer (BO) potential of the gluon field connecting the heavy diquark
[$\delta^{\vphantom{1}}\\!\equiv(Qq)$]-triquark
[${\bar{\theta}}\\!\equiv\\!(\bar{Q}(q_{1}q_{2}))$] quasiparticles. The
pentaquark BO potentials lack one discrete quantum number ($g,u$) compared to
those for $\delta^{\vphantom{1}}$-${\bar{\delta}}$ tetraquarks Lebed (2017).
In all cases, $\delta^{\vphantom{1}},{\bar{\theta}}$ are assumed to transform
as color triplets (or antitriplets), and each quasiparticle contains no
internal orbital angular momentum.

In the case of ${Q\bar{Q}}q\hskip 1.0ptq_{1}q_{2}$, the classification scheme
then begins with 3 possible core states in which the
$\delta^{\vphantom{1}}$-${\bar{\theta}}$ pair lie in a relative $S$ wave. The
most minimal form of the model also restricts to states in which the diquark
${\delta^{\prime\vphantom{1}}}\\!=\\!(q_{1}q_{2})$ internal to
${\bar{\theta}}$, consisting of only light $u,d$ quarks, is a “good” diquark:
isosinglet and $s_{{\delta^{\prime\vphantom{1}}}}\\!=\\!0$, which is expected
to be the most tightly bound combination from light-hadron phenomenology Jaffe
(2005) and from lattice simulations Francis  _et al._ (2021). In principle,
the “bad” (isotriplet, $s_{{\delta^{\prime\vphantom{1}}}}\\!=\\!1$)
combination could also appear, but if so would form pentaquark states
substantially higher in mass. Such states may be so broad as to have escaped
detection at LHCb. Indicating the total spin $s$ of
$\delta^{\vphantom{1}},{\bar{\theta}}$ by
$s_{\delta}^{\vphantom{1}},s_{\bar{\theta}}$, respectively, and including a
subscript on the full state to indicate its total spin, one obtains the
spectrum

$\displaystyle J^{P}=\frac{1}{2}^{-}:$ $\displaystyle
P_{\frac{1}{2}}\equiv\left|{\displaystyle{\frac{1}{2}}}_{\bar{\theta}},0_{\delta}^{\vphantom{1}}\right>_{\frac{1}{2}}\,,\
\
P_{\frac{1}{2}}^{\prime}\equiv\left|{\displaystyle{\frac{1}{2}}}_{\bar{\theta}},1_{\delta}^{\vphantom{1}}\right>_{\frac{1}{2}}\,,$
$\displaystyle J^{P}=\frac{3}{2}^{-}:$
$\displaystyle{P_{\frac{3}{2}}}\equiv\left|\frac{1}{2}_{\bar{\theta}},1_{\delta}^{\vphantom{1}}\right>_{\frac{3}{2}}\,,$
(3)

where again, the diquark ${\delta^{\prime\vphantom{1}}}$ spin
$s_{\delta^{\prime\vphantom{1}}}$ is fixed to zero, and so the triquark
${\bar{\theta}}$ spin $s_{{\bar{\theta}}}\\!=\\!\frac{1}{2}$. Since 4 quark
angular momenta ($s_{q},s_{\delta^{\prime\vphantom{1}}},s_{Q},s_{\bar{Q}}$)
are combined here, one may transform these states into other convenient bases
by means of $9j$ angular momentum recoupling coefficients. In particular, in
the basis of good total heavy-quark (${Q\bar{Q}}$) and all-light baryonic
($B^{\vphantom{1}}\\!\equiv\\!q{\delta^{\prime\vphantom{1}}}\\!=\\!q\hskip
1.0ptq_{1}q_{2}$) spin, the transformation reads

$\displaystyle\left<(s_{q}\,s_{{\delta^{\prime\vphantom{1}}}})s_{B}^{\vphantom{1}},(s_{Q}\,s_{\bar{Q}})s_{Q\bar{Q}},S\,\right|\left.(s_{q}\,s_{Q})s_{\delta}^{\vphantom{1}},(s_{\bar{q}}\,s_{\bar{Q}})s_{\bar{\theta}},S\right>$
$\displaystyle=\left([s_{B}^{\vphantom{1}}][s_{Q\bar{Q}}][s_{\delta}^{\vphantom{1}}][s_{\bar{\theta}}]\right)^{1/2}\begin{Bmatrix}s_{q}&s_{{\delta^{\prime\vphantom{1}}}}&s_{B}^{\vphantom{1}}\\\
s_{Q}&s_{\bar{Q}}&s_{Q\bar{Q}}\\\
s_{\delta}^{\vphantom{1}}&s_{\bar{\theta}}&S\end{Bmatrix}\,,\ \ $ (4)

using the abbreviation $[s]\\!\equiv\\!2s+1$ for the multiplicity of a
spin-$s$ state here and below. Combining Eqs. (3) and (4), one then obtains

$\displaystyle J^{P}=\frac{1}{2}^{-}:$ $\displaystyle
P_{\frac{1}{2}}=-\frac{1}{2}\left|\frac{1}{2}_{B}^{\vphantom{1}},0_{Q\bar{Q}}\right>_{\frac{1}{2}}+\frac{\sqrt{3}}{2}\left|\frac{1}{2}_{B}^{\vphantom{1}},1_{Q\bar{Q}}\right>_{\frac{1}{2}}\,,$
$\displaystyle
P_{\frac{1}{2}}^{\prime}=\frac{\sqrt{3}}{2}\left|\frac{1}{2}_{B}^{\vphantom{1}},0_{Q\bar{Q}}\right>_{\frac{1}{2}}+\frac{1}{2}\left|\frac{1}{2}_{B}^{\vphantom{1}},1_{Q\bar{Q}}\right>_{\frac{1}{2}}\,,$
$\displaystyle J^{P}=\frac{3}{2}^{-}:$
$\displaystyle{P_{\frac{3}{2}}}=\left|\frac{1}{2}_{B}^{\vphantom{1}},1_{Q\bar{Q}}\right>_{\frac{3}{2}}\,.$
(5)

In this work, it is especially convenient to employ a basis of states carrying
a unique value of $s_{Q\bar{Q}}$ and of $s_{B}^{\vphantom{1}}$ (the latter
always being $\frac{1}{2}$ since $s_{{\delta^{\prime\vphantom{1}}}}\\!=\\!0$
and $s_{q}\\!=\\!\frac{1}{2}$). These states are the combinations

$\displaystyle{\tilde{P}_{\frac{1}{2}}}$ $\displaystyle\equiv$
$\displaystyle\left\lvert\frac{1}{2}_{B}^{\vphantom{1}},0_{Q\bar{Q}}\right\rangle_{\frac{1}{2}}=-\frac{1}{2}P_{\frac{1}{2}}+\frac{\sqrt{3}}{2}P_{\frac{1}{2}}^{\prime}\,,$
$\displaystyle{\tilde{P}^{\prime\vphantom{1}}_{\frac{1}{2}}}$
$\displaystyle\equiv$
$\displaystyle\left\lvert\frac{1}{2}_{B}^{\vphantom{1}},1_{Q\bar{Q}}\right\rangle_{\frac{1}{2}}=+\frac{\sqrt{3}}{2}P_{\frac{1}{2}}+\frac{1}{2}P_{\frac{1}{2}}^{\prime}\,,$
$\displaystyle{P_{\frac{3}{2}}}$ $\displaystyle=$
$\displaystyle\left|\frac{1}{2}_{B}^{\vphantom{1}},1_{Q\bar{Q}}\right>_{\frac{3}{2}}\,.$
(6)

The ${Q\bar{Q}}q\hskip 1.0ptq_{1}q_{2}$ states in the multiplets
$\Sigma^{+}(1S)$ and $\Sigma^{+}(1P)$ are sufficient to accommodate all
particles considered in this work. However, we note that Ref. Lebed (2017)
also provides a classification of higher radial and orbital excitations [such
as $\Sigma^{+}(nD)$], as well as states in excited-glue BO potentials such as
$\Pi^{+}$ (which are exotic analogues to hybrid hadrons). The corresponding
${Q\bar{Q}}q_{1}\bar{q}_{2}$ states in the $\Sigma^{+}_{g}(1S)$ and
$\Sigma^{+}_{g}(1P)$ multiplets were first studied in Refs. Giron  _et al._
(2020) and Giron and Lebed (2020a), respectively, while radiative transitions
between them are computed in Ref. Gens  _et al._ (2021).

##  III Pentaquark Mass Hamiltonian

###  III.1 $S$-Wave Hamiltonian and Mass Expressions

The $\Sigma^{+}(1S)$ (ground-state) pentaquark multiplet contains only 3
states, those listed in Eqs. (3) or (6). For this multiplet, the Hamiltonian
closely follows the form used for the $\Sigma^{+}_{g}(1S)$ tetraquark
multiplet in Ref. Giron  _et al._ (2020), and reads

$\displaystyle H$ $\displaystyle=$ $\displaystyle M_{0}\\!+\Delta
H_{{\kappa_{qQ}}}+\Delta
H_{{\kappa_{{\delta^{\prime\vphantom{1}}}\bar{Q}}}}+\Delta H_{{V_{8}}}$ (7)
$\displaystyle=$ $\displaystyle
M_{0}+2\left[{\kappa_{qQ}}\left(\vec{s}_{q}\cdot\vec{s}_{Q}\right)+{\kappa_{{\delta^{\prime\vphantom{1}}}\bar{Q}}}\left(\vec{s}_{\delta^{\prime\vphantom{1}}}\cdot\vec{s}_{\bar{Q}}\right)\right]$
$\displaystyle+{V_{8}}\left(\lambda^{8}_{a}\vec{\sigma}_{\delta^{\prime\vphantom{1}}}\right)\cdot\left(\lambda^{8}_{a}\vec{\sigma}_{q}\right)\,.$

Since the diquark ${\delta^{\prime\vphantom{1}}}$ is isoscalar by assumption,
this Hamiltonian lacks the nontrivial isospin dependence of its tetraquark
analogue. Under the assumption that ${\delta^{\prime\vphantom{1}}}$ maintains
its flavor content (i.e., does not exchange quarks with the other diquark),
then $\pi$-like and $K$-like exchanges do not occur.333Models allowing for all
such SU(3)flavor exchanges do appear in the literature Shi  _et al._ (2021b,
a).

Its matrix elements are

$\displaystyle M$ $\displaystyle=$ $\displaystyle
M_{0}+\frac{{\kappa_{qQ}}}{2}\left[2s_{\delta}^{\vphantom{1}}\left(s_{\delta}^{\vphantom{1}}+1\right)-3\right]+\frac{{\kappa_{{\delta^{\prime\vphantom{1}}}\bar{Q}}}}{4}\left[4s_{\bar{\theta}}\left(s_{\bar{\theta}}+1\right)-3\right]$
(8)
$\displaystyle+\frac{1}{6}{V_{8}}C_{1}\left[4s_{B}^{\vphantom{1}}\left(s_{B}^{\vphantom{1}}+1\right)-3\right],$

where

$\displaystyle C_{1}$ $\displaystyle=$
$\displaystyle\begin{cases}+2&q_{1},q_{2},q\in\\{u,d\\},\\\
-4&q_{1},q_{2}\in\\{u,d\\}\cup q\in s.\\\ \end{cases}$ (9)

Because we consider only $s_{\bar{\theta}}=s_{B}^{\vphantom{1}}=\frac{1}{2}$
(since $s_{{\delta^{\prime\vphantom{1}}}}\\!=\\!0$ in the dynamical diquark
model), one can see that both the triquark internal spin-spin coupling
($\Delta H_{\kappa_{{\delta^{\prime\vphantom{1}}}\bar{Q}}}$) and the
$\eta$-like coupling ($\Delta H_{{V_{8}}}$, first introduced in Ref. Giron
_et al._ (2021)), are zero. Only the common mass coefficient $M_{0}(1S)$ and
the diquark $\delta^{\vphantom{1}}$ internal spin coupling $\kappa_{qQ}$
survive. In the order of the $s_{{Q\bar{Q}}}$ eigenstates
${\tilde{P}_{\frac{1}{2}}},{\tilde{P}^{\prime\vphantom{1}}_{\frac{1}{2}}},{P_{\frac{3}{2}}}$
of Eqs. (6), one finds

$\displaystyle\tilde{M}_{\frac{1}{2}^{-}}$ $\displaystyle=$ $\displaystyle
M_{0}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}+\frac{1}{2}{\kappa_{qQ}}\begin{pmatrix}0&\sqrt{3}\\\
\sqrt{3}&-2\end{pmatrix}\,,$ $\displaystyle M_{\frac{3}{2}^{-}}$
$\displaystyle=$ $\displaystyle M_{0}+\frac{1}{2}{\kappa_{qQ}}\,.$ (10)

The diagonalized form of $M_{\frac{1}{2}}$, which also corresponds to the
basis of good $s_{\delta}^{\vphantom{1}}$ eigenvalues in the order given by
Eqs. (3) ($P_{\frac{1}{2}},P_{\frac{1}{2}}^{\prime},P_{\frac{3}{2}}$), reads

$\displaystyle M_{\frac{1}{2}^{-}}$ $\displaystyle=$ $\displaystyle
M_{0}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}+\frac{1}{2}{\kappa_{qQ}}\begin{pmatrix}-3&0\\\
0&1\end{pmatrix}\,,$ $\displaystyle M_{\frac{3}{2}^{-}}$ $\displaystyle=$
$\displaystyle M_{0}+\frac{1}{2}{\kappa_{qQ}}\,.$ (11)

One finds (assuming ${\kappa_{qQ}}\\!>\\!0$) a unique
$J^{P}\\!=\\!{\frac{1}{2}}^{-}$ ground state and a degenerate pair with
$J^{P}\\!=\\!{\frac{1}{2}}^{-}$ and ${\frac{3}{2}}^{-}$.

###  III.2 $P$-Wave Hamiltonian and Mass Expressions

The $P$-wave states are obtained by combining those in the bases of Eqs. (3)
or (6) with a unit $L\\!=\\!1$ of orbital angular momentum. Using the latter
basis (unique $s_{{Q\bar{Q}}}$ eigenvalues), one obtains the 7 states in the
multiplet $\Sigma^{+}(1P)$:

$\displaystyle J^{P}=\frac{1}{2}^{+}:$
$\displaystyle{\tilde{P}_{\frac{1}{2}}}^{(L=1)},\
{\tilde{P}_{\frac{1}{2}}}^{\prime(L=1)},\ P_{\frac{3}{2}}^{(L=1)}\,,$
$\displaystyle J^{P}=\frac{3}{2}^{+}:$
$\displaystyle{\tilde{P}_{\frac{1}{2}}}^{(L=1)},\
{\tilde{P}_{\frac{1}{2}}}^{\prime(L=1)},\ P_{\frac{3}{2}}^{(L=1)}\,,$
$\displaystyle J^{P}=\frac{5}{2}^{+}:$ $\displaystyle
P_{\frac{3}{2}}^{(L=1)}\,.$ (12)

Table 2 collects the quantum numbers of these states, including the total spin
$s_{B}$ carried by the baryonic combination $B$ of light quarks $(q\hskip
1.0ptq_{1}q_{2})$ [which always equals $\frac{1}{2}$ since
${\delta^{\prime\vphantom{1}}}\\!\equiv\\!(q_{1}q_{2})$ has
$s_{{\delta^{\prime\vphantom{1}}}}\\!=\\!0$], and the total spin $S$ carried
by all quarks. In addition, we tabulate for completeness the relative
amplitude ${\cal M}_{J_{B}^{\vphantom{1}}}$ within each state for each allowed
eigenvalue $J_{B^{\vphantom{1}}}$ of total angular momentum
${\bm{J}}_{B^{\vphantom{1}}}\\!\equiv\\!{\bm{L}}+{\bm{s}_{B}}$ carried by the
light degrees of freedom. These recoupling amplitudes are given by

$\displaystyle\mathcal{M}_{J_{B}^{\vphantom{1}}}\equiv\langle\left(L,s_{B}^{\vphantom{1}}\right),J_{B}^{\vphantom{1}},s_{Q\bar{Q}},J|L,\left(s_{B}^{\vphantom{1}},s_{Q\bar{Q}}\right),S,J\rangle$
(13) $\displaystyle=$
$\displaystyle(-1)^{L+s_{B}^{\vphantom{1}}+s_{Q\bar{Q}}+J}\sqrt{\left[J_{B}^{\vphantom{1}}\right]\left[S\right]}\begin{Bmatrix}L&s_{B}^{\vphantom{1}}&J_{B}^{\vphantom{1}}\\\
s_{Q\bar{Q}}&J&S\end{Bmatrix}\,.$

While a decomposition in $J_{B^{\vphantom{1}}}$ is not needed for the
following mass analysis, we anticipate its potential usefulness for computing
transition matrix elements, as is done for the tetraquark states in Ref. Gens
_et al._ (2021).

Table 2: The 7 pentaquark ${Q\bar{Q}}q\hskip 1.0ptq_{1}q_{2}$ states in the $\Sigma^{+}(1P)$ multiplet of the dynamical diquark model, expressed in the basis of good heavy-quark spin $s_{{Q\bar{Q}}}$. The model restriction that the component diquark $\delta^{\prime}\\!\equiv\\!(q_{1}q_{2})$ carries zero spin leads to the baryonic combination $B\\!=\\!(q\hskip 1.0ptq_{1}q_{2})$ of the light quarks carrying $s_{B}\\!=\\!\frac{1}{2}$ for all states. $S$ is the total spin carried by quarks (the same as the state subscript). Also tabulated are the allowed values of light degree-of-freedom total angular momentum $J_{B}$ and their amplitude contribution $\mathcal{M}_{J_{B}}$ to the full state. State | $J^{P}$ | $s_{B}^{\vphantom{1}}$ | $s_{Q\bar{Q}}$ | $S$ | $J_{B}^{\vphantom{1}}$ | $\mathcal{M}_{J_{B}^{\vphantom{1}}}$
---|---|---|---|---|---|---
${\tilde{P}_{\frac{1}{2}}}^{(L=1)}$ | $\frac{1}{2}^{+}$ | $\frac{1}{2}$ | $0$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $+1$
${\tilde{P}_{\frac{1}{2}}}^{\prime(L=1)}$ | $\frac{1}{2}^{+}$ | $\frac{1}{2}$ | $1$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $-\frac{1}{3}$
|  |  |  |  | $\frac{3}{2}$ | $+\frac{2\sqrt{2}}{3}$
$P_{\frac{3}{2}}^{(L=1)}$ | $\frac{1}{2}^{+}$ | $\frac{1}{2}$ | $1$ | $\frac{3}{2}$ | $\frac{1}{2}$ | $+\frac{2\sqrt{2}}{3}$
|  |  |  |  | $\frac{3}{2}$ | $+\frac{1}{3}$
${\tilde{P}_{\frac{1}{2}}}^{(L=1)}$ | $\frac{3}{2}^{+}$ | $\frac{1}{2}$ | $0$ | $\frac{1}{2}$ | $\frac{3}{2}$ | $+1$
${\tilde{P}_{\frac{1}{2}}}^{\prime(L=1)}$ | $\frac{3}{2}^{+}$ | $\frac{1}{2}$ | $1$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $-\frac{2}{3}$
|  |  |  |  | $\frac{3}{2}$ | $+\frac{\sqrt{5}}{3}$
$P_{\frac{3}{2}}^{(L=1)}$ | $\frac{3}{2}^{+}$ | $\frac{1}{2}$ | $1$ | $\frac{3}{2}$ | $\frac{1}{2}$ | $+\frac{\sqrt{5}}{3}$
|  |  |  |  | $\frac{3}{2}$ | $+\frac{2}{3}$
$P_{\frac{3}{2}}^{(L=1)}$ | $\frac{5}{2}^{+}$ | $\frac{1}{2}$ | $1$ | $\frac{3}{2}$ | $\frac{3}{2}$ | $+1$

The minimal Hamiltonian for the first excited pentaquark multiplet,
$\Sigma^{+}(1P)$, closely follows the one used for the tetraquark multiplet
$\Sigma^{+}_{g}(1P)$ in Ref. Giron and Lebed (2020a):

$\displaystyle H$ $\displaystyle=$ $\displaystyle M_{0}+\Delta
H_{{\kappa_{qQ}}}+\Delta
H_{{\kappa_{{\delta^{\prime\vphantom{1}}}\bar{Q}}}}+\Delta H_{{V_{LS}}}+\Delta
H_{{V_{T}}}+\Delta H_{{V_{8}}}$ (14) $\displaystyle=$ $\displaystyle
M_{0}+2\left[{\kappa_{qQ}}\left(\vec{s}_{q}\cdot\vec{s}_{Q}\right)+{\kappa_{{\delta^{\prime\vphantom{1}}}\bar{Q}}}\left(\vec{s}_{\delta^{\prime\vphantom{1}}}\cdot\vec{s}_{\bar{Q}}\right)\right]$
$\displaystyle+{V_{LS}}\vec{L}\cdot\vec{S}+{V_{T}}\hat{S}^{(\delta^{\vphantom{1}}{\bar{\theta}})}_{12}+{V_{8}}\left(\lambda^{8}_{a}\vec{\sigma}_{\delta^{\prime\vphantom{1}}}\right)\cdot\left(\lambda^{8}_{a}\vec{\sigma}_{q}\right)\,.$

In addition to a common mass [$M_{0}(1P)$ here], we allow for internal spin-
spin couplings for the diquark $\delta^{\vphantom{1}}$ (${\kappa_{qQ}}$) and
the triquark ${\bar{\theta}}$
(${\kappa_{{\delta^{\prime\vphantom{1}}}\bar{Q}}}$), as well as a spin-orbit
term $V_{LS}$, a tensor term $V_{T}$, and an $\eta$-like exchange term labeled
by $V_{8}$ Giron  _et al._ (2021). In addition, the numerical value of
${\kappa_{qQ}}$ in the $1P$ multiplet is expected to differ from that in the
$1S$ multiplet Giron and Lebed (2020a). The matrix elements of the mass
Hamiltonian read:

$\displaystyle M$ $\displaystyle=$ $\displaystyle
M_{0}+\frac{{\kappa_{qQ}}}{2}\left[2s_{\delta}^{\vphantom{1}}\left(s_{\delta}^{\vphantom{1}}+1\right)-3\right]+\frac{{\kappa_{{\delta^{\prime\vphantom{1}}}\bar{Q}}}}{4}\left[4s_{\bar{\theta}}\left(s_{\bar{\theta}}+1\right)-3\right]$
(15)
$\displaystyle+\frac{{V_{LS}}}{2}\left[J\left(J+1\right)-L\left(L+1\right)-S\left(S+1\right)\right]$
$\displaystyle+{V_{T}}\langle
S^{(\delta^{\vphantom{1}}{\bar{\theta}})}_{12}\rangle+\frac{1}{6}{V_{8}}C_{1}\left[4s_{B}^{\vphantom{1}}\left(s_{B}^{\vphantom{1}}+1\right)-3\right]\,.$

The tensor term labeled by $V_{T}$ differs somewhat from the primary one
studied in Ref. Giron and Lebed (2020a), for which the spins entering the
operator are those of the individual light quarks $q,\bar{q}$ within the
diquarks. Instead, the tensor operator used here couples directly to the full
quasiparticle $\delta^{\vphantom{1}},{\bar{\theta}}$ spins, and thus is the
analogue of the secondary possible tensor operator studied in Ref. Giron and
Lebed (2020a), Appendix A. Its matrix elements are computed as

$\displaystyle\langle
L^{\prime},S^{\prime},J\rvert\hat{S}^{(\delta^{\vphantom{1}}{\bar{\theta}})}_{12}\lvert
L,S,J\rangle=(-1)^{S+J}\sqrt{30[L][L^{\prime}][S][S^{\prime}]}$
$\displaystyle\times\begin{Bmatrix}J&S^{\prime}&L^{\prime}\\\
2&L&S\end{Bmatrix}\begin{pmatrix}L^{\prime}&2&L\\\
0&0&0\end{pmatrix}\begin{Bmatrix}s_{{\bar{\theta}}}&s_{\delta}^{\vphantom{1}}&S\\\
s^{\prime}_{{\bar{\theta}}}&s^{\prime}_{\delta}&S^{\prime}\\\
1&1&2\end{Bmatrix}$ $\displaystyle\times\langle
s^{\prime}_{{\bar{\theta}}}\,\rvert|\,\bm{\sigma}_{\bar{\theta}}\,|\lvert\,s_{{\bar{\theta}}}\rangle\langle
s^{\prime}_{\delta}\,\rvert|\,\bm{\sigma}_{\delta}^{\vphantom{1}}\,|\lvert\,s_{\delta}^{\vphantom{1}}\rangle\,.$
(16)

Here, ${\bm{\sigma}}$ denotes not just spin-$\frac{1}{2}$ Pauli matrices, but
more generally twice the canonically normalized generators ${\bf s}$ for
arbitrary spin $s$, and the reduced matrix elements of the angular momentum
generators are given by

$\left<j^{\prime}||\,{\bf
j}\,||\,j\right>=\sqrt{j(2j+1)(j+1)}\,\delta_{j^{\prime}j}\,.$ (17)

In particular, if $s_{\delta}\\!=\\!0$, then the reduced matrix element (and
hence $\langle\hat{S}^{(\delta^{\vphantom{1}}{\bar{\theta}})}_{12}\rangle$)
vanishes.

Just as in the case of the $S$-wave states, one sees that both $\Delta
H_{\kappa_{{\delta^{\prime\vphantom{1}}}\bar{Q}}}\\!=\\!\Delta
H_{V_{8}}\\!=\\!0$, and the surviving matrix elements of Eq. (14) for the
$\Sigma^{+}(1P)$ multiplet, expressed in the same order as the states listed
in Eqs. (12) or Table 2, then read

$\displaystyle\tilde{M}_{\frac{1}{2}^{+}}$ $\displaystyle=$ $\displaystyle
M_{0}\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{pmatrix}+\frac{{\kappa_{qQ}}}{2}\begin{pmatrix}-2&\sqrt{3}&0\\\
\sqrt{3}&0&0\\\ 0&0&1\end{pmatrix}-\frac{{V_{LS}}}{2}\begin{pmatrix}2&0&0\\\
0&2&0\\\ 0&0&5\end{pmatrix}+{V_{T}}\begin{pmatrix}0&0&\sqrt{6}\\\
0&0&\sqrt{2}\\\ \sqrt{6}&\sqrt{2}&-4\end{pmatrix}\,,$
$\displaystyle\tilde{M}_{\frac{3}{2}^{+}}$ $\displaystyle=$ $\displaystyle
M_{0}\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{pmatrix}+\frac{{\kappa_{qQ}}}{2}\begin{pmatrix}-2&\sqrt{3}&0\\\
\sqrt{3}&0&0\\\ 0&0&1\end{pmatrix}+\frac{{V_{LS}}}{2}\begin{pmatrix}1&0&0\\\
0&1&0\\\ 0&0&-2\end{pmatrix}+\frac{{V_{T}}}{5}\begin{pmatrix}0&0&-\sqrt{15}\\\
0&0&-\sqrt{5}\\\ -\sqrt{15}&-\sqrt{5}&16\end{pmatrix}\,,$ $\displaystyle
M_{\frac{5}{2}^{+}}$ $\displaystyle=$ $\displaystyle
M_{0}+\frac{1}{2}{\kappa_{qQ}}+\frac{3}{2}{V_{LS}}-\frac{4}{5}{V_{T}}\,.$ (18)

These matrices, once diagonalized, provide the mass eigenvalues:

$\displaystyle M_{\frac{1}{2}^{+}}$ $\displaystyle=$ $\displaystyle
M_{0}\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{pmatrix}+\frac{{\kappa_{qQ}}}{2}\begin{pmatrix}1&0&0\\\ 0&-3&0\\\
0&0&1\end{pmatrix}-\frac{1}{4}{V_{LS}}\begin{pmatrix}7&0&0\\\ 0&4&0\\\
0&0&7\end{pmatrix}-{V_{T}}\begin{pmatrix}2&0&0\\\ 0&0&0\\\
0&0&2\end{pmatrix}+\frac{\sqrt{3}}{4}\tilde{V}_{1}\begin{pmatrix}-1&0&0\\\
0&0&0\\\ 0&0&1\end{pmatrix}\,,$ $\displaystyle M_{\frac{3}{2}^{+}}$
$\displaystyle=$ $\displaystyle M_{0}\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{pmatrix}+\frac{{\kappa_{qQ}}}{2}\begin{pmatrix}1&0&0\\\ 0&-3&0\\\
0&0&1\end{pmatrix}-\frac{1}{4}{V_{LS}}\begin{pmatrix}1&0&0\\\ 0&-2&0\\\
0&0&1\end{pmatrix}+\frac{8}{5}{V_{T}}\begin{pmatrix}1&0&0\\\ 0&0&0\\\
0&0&1\end{pmatrix}+\frac{\sqrt{3}}{20}\tilde{V}_{2}\begin{pmatrix}-1&0&0\\\
0&0&0\\\ 0&0&1\end{pmatrix}\,,$ $\displaystyle M_{\frac{5}{2}^{+}}$
$\displaystyle=$ $\displaystyle
M_{0}+\frac{1}{2}{\kappa_{qQ}}+\frac{3}{2}{V_{LS}}-\frac{4}{5}{V_{T}}\,,$ (19)

where

$\displaystyle\tilde{V}_{1}$ $\displaystyle=$
$\displaystyle\sqrt{3{V_{LS}}^{2}+16{V_{LS}}{V_{T}}+64{V_{T}}^{2}}\,,$
$\displaystyle\tilde{V}_{2}$ $\displaystyle=$
$\displaystyle\sqrt{75{V_{LS}}^{2}-320{V_{LS}}{V_{T}}+448{V_{T}}^{2}}\,.$ (20)

The elements of these diagonalized matrices are presented in the order of
increasing mass eigenvalues in each $J^{P}$ sector, under the assumption (as
found for tetraquarks in Ref. Giron and Lebed (2020a)) that the contribution
from $V_{LS}$ dominates the contribution from ${\kappa_{qQ}}$ (and from
$V_{T}$).

Anticipating results from the analysis in the next section, we also present
the corresponding expressions to Eqs. (19) when $V_{T}\\!\to\\!0$. Including
up to linear order in $V_{T}$, one finds

$\displaystyle M_{\frac{1}{2}^{+}}$ $\displaystyle=$ $\displaystyle
M_{0}\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{pmatrix}+\frac{{\kappa_{qQ}}}{2}\begin{pmatrix}1&0&0\\\ 0&-3&0\\\
0&0&1\end{pmatrix}-\frac{1}{2}{V_{LS}}\begin{pmatrix}5&0&0\\\ 0&2&0\\\
0&0&2\end{pmatrix}-4{V_{T}}\begin{pmatrix}1&0&0\\\ 0&0&0\\\
0&0&0\end{pmatrix}\,,$ $\displaystyle M_{\frac{3}{2}^{+}}$ $\displaystyle=$
$\displaystyle M_{0}\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{pmatrix}+\frac{{\kappa_{qQ}}}{2}\begin{pmatrix}1&0&0\\\ 0&-3&0\\\
0&0&1\end{pmatrix}+\frac{1}{2}{V_{LS}}\begin{pmatrix}-2&0&0\\\ 0&1&0\\\
0&0&1\end{pmatrix}+\frac{16}{5}{V_{T}}\begin{pmatrix}1&0&0\\\ 0&0&0\\\
0&0&0\end{pmatrix}\,,$ $\displaystyle M_{\frac{5}{2}^{+}}$ $\displaystyle=$
$\displaystyle
M_{0}+\frac{1}{2}{\kappa_{qQ}}+\frac{3}{2}{V_{LS}}-\frac{4}{5}{V_{T}}\,.$ (21)

It is worth emphasizing that the Hamiltonian of Eq. (14) with $V_{T}\\!=\\!0$
is diagonal in the basis of good $s_{\delta}^{\vphantom{1}}$ defined in Eqs.
(3) [as is apparent from Eq. (15), since the
${\kappa_{{\delta^{\prime\vphantom{1}}}\bar{Q}}}$ and $V_{8}$ terms are also
absent when $s_{\delta^{\prime\vphantom{1}}}\\!=\\!0$]. The mass expressions
of Eqs. (21) with $V_{T}\\!=\\!0$ therefore refer to the diquark-spin basis of
Eqs. (3), specifically, in the order
$P_{\frac{3}{2}},P_{\frac{1}{2}},P^{\prime}_{\frac{1}{2}}$.

##  IV Analysis

Using the results of the previous section, particularly Eqs. (11), (19), and
(20), we compute the mass eigenvalues for all pentaquarks with flavor content
$c\bar{c}qq_{1}q_{2}$ and $c\bar{c}uds$ in the ground-state multiplet
[$\Sigma^{+}(1S)$] given in Eqs. (3) or (6), and the first excited-state
multiplet [$\Sigma^{+}(1P)$] given in Eqs. (12). The measured masses of the 4
narrow states [$P_{c}(4312)$, $P_{c}(4337)$, $P_{c}(4440)$, and $P_{c}(4457)$]
in Table 1 are used to assign these states to the multiplet $\Sigma^{+}(1P)$,
while we defer a discussion of the problematic wide $P_{c}(4380)$ until later
in this section.

The assignment of the known $P_{c}$ states to $\Sigma^{+}(1P)$ rather than
$\Sigma^{+}(1S)$ in the dynamical diquark model was first noted in Ref. Giron
_et al._ (2019) to be much more natural, despite the lack of clear
experimental evidence for the lighter $S$-wave states. In that work, the
argument rested upon the opposite-parity nature of the (unresolved)
$P_{c}(4450)$ and $P_{c}(4380)$. Now, the $\Sigma^{+}(1P)$ assignment can be
made based upon the sheer multiplicity of $P_{c}$ states: $\Sigma^{+}(1S)$ has
only 3 states [Eqs. (3)], so that at least some of the $P_{c}$’s must belong
to a higher multiplet. However, the model predicts a $1P$-$1S$ mass splitting
of $\sim\\!400$ MeV, rendering a mixed $1P$-$1S$ assignment of the known
$P_{c}$ states untenable. In fact, the closely (and almost equally) spaced
pairs $P_{c}(4312)$-$P_{c}(4337)$ and $P_{c}(4440)$-$P_{c}(4457)$ have a
completely natural identification within $\Sigma^{+}(1P)$: Anticipating our
result that $V_{T}$ turns out to be numerically small, and using the
$V_{T}\\!\to\\!0$ expressions of Eqs. (21), one sees that the heaviest
${\frac{1}{2}}^{+}$ and lightest ${\frac{3}{2}}^{+}$ states are nearly
degenerate:

$m^{(1)}_{{\frac{3}{2}}^{+}}-m^{(3)}_{{\frac{1}{2}}^{+}}=\frac{36}{5}V_{T}\,,$
(22)

and that, for $V_{T}\\!=\\!0$, one finds two equally spaced pairs:

$m^{(2)}_{{\frac{1}{2}}^{+}}-m^{(1)}_{{\frac{1}{2}}^{+}}=m^{(2)}_{{\frac{3}{2}}^{+}}-m^{(1)}_{{\frac{3}{2}}^{+}}=\frac{3}{2}V_{LS}-2{\kappa_{qQ}}\,.$
(23)

In contrast, the other 3 mass splittings of consecutive states in
$\Sigma^{+}(1P)$ equal $V_{LS}$ or $\frac{3}{2}V_{LS}$. The analysis of
hidden-charm $P$-wave tetraquarks in Ref. Giron and Lebed (2020a) produces
values $V_{LS}\\!\simeq\\!45$–60 MeV and ${\kappa_{qc}}(1P)\\!=\\!40$–45 MeV,
so that typically,
$\left|\frac{3}{2}V_{LS}\\!-\\!2{\kappa_{qc}}\right|\\!\ll\\!V_{LS}$. Assuming
that the corresponding hidden-charm nonstrange $P$-wave pentaquarks produce
similar numerical values for the coefficients $V_{LS}$ and ${\kappa_{qc}}(1P)$
(as they must, since they arise from the same dynamics), one expects the 7
masses in the $\Sigma^{+}(1P)$ spectrum to appear as two nearly degenerate
values [Eq. (22)], two closely spaced pairs [Eq. (23)], and two heavier
masses, $m^{(3)}_{{\frac{3}{2}}^{+}}$ and $m_{{\frac{5}{2}}^{+}}$.

###  IV.1 The $c\bar{c}qq_{1}q_{2}$ Sector, $P$ Wave

Having thus identified specific candidates for the 4 known $P_{c}$ states:

$\displaystyle P_{c}(4312)$ $\displaystyle=$ $\displaystyle
P_{\frac{1}{2}}^{(1)}\,,$ $\displaystyle P_{c}(4337)$ $\displaystyle=$
$\displaystyle P_{\frac{1}{2}}^{(2)}\,,$ $\displaystyle P_{c}(4440)$
$\displaystyle=$ $\displaystyle P_{\frac{3}{2}}^{(1)}\ {\rm and}\
P_{\frac{1}{2}}^{(3)}\,,$ $\displaystyle P_{c}(4457)$ $\displaystyle=$
$\displaystyle P_{\frac{3}{2}}^{(2)}\,,$ (24)

we first perform a least-squares fit to the $V_{T}\\!=\\!0$ mass expressions
of Eqs. (21) and obtain values for $M_{0}(1P)$, $V_{LS}$, and
${\kappa_{qc}}(1P)$. We then predict the masses for the remaining states of
$\Sigma^{+}(1P)$. The results are presented in Table 3.

Since the $V_{T}\\!=\\!0$ fit produces a figure of merit $\chi^{2}_{\rm
min}\\!<\\!1$, we conclude that the parameter $V_{T}$ is not actually needed
for a complete description of the currently available $P_{c}$ data.
Nevertheless, we perform a fit to the full expressions of Eqs. (19)–(20),
which include $V_{T}$ nonlinearly. Since 4 parameters are fit by 4 masses in
that case, the solution is unique, and we therefore do not propagate the
uncertainties on the parameters or masses for that fit. These results are also
presented in Table 3. We find the essential result that $V_{T}$ is not merely
statistically unimportant (as shown by the $V_{T}\\!=\\!0$ fit), but also that
its value in a fit with no free parameters is numerically small compared to
that of the other Hamiltonian parameters [$M_{0},V_{LS},{\kappa_{qc}}(1P)$].
The most incisive test for the existence of a nonzero $V_{T}$, as indicated by
Eqs. (22) and (24), would be the resolution of $P_{c}(4440)$ into a very
closely spaced ${\frac{1}{2}}^{+}\\!,{\frac{3}{2}}^{+}$ pair. Indeed, the
difference between the ${\frac{3}{2}}^{+}$ and ${\frac{1}{2}}^{+}$ sides of
Eq. (23), using Eqs. (19)–(20), is $O(V_{T}^{2}/V_{LS})$; in comparison, its
experimental value using the masses in Table 1 is $-8.1\\!\pm\\!11.9$ MeV.

The hidden-charm $P$-wave pentaquarks studied here share several similarities
with the hidden-charm $P$-wave tetraquarks studied in Ref. Giron and Lebed
(2020a) within the dynamical diquark model, but also feature some significant
differences. Both are modeled as heavy (and therefore effectively semi-static)
color-triplet quasiparticles connected by the same orbitally excited
($L\\!=\\!1$) color flux tube, whose excitation energies are computed as BO
potentials obtained from specific lattice simulations, labeled here as JKM
(Refs. Juge  _et al._ (2003); Mor ) and CPRRW (Ref. Capitani  _et al._
(2019)). In both pentaquark and tetraquark cases, the known states occupy the
$1P$ levels of the ground-state BO potential $\Sigma^{+}$, but since each
tetraquark consists of a
[$\delta^{\vphantom{1}}\\!=\\!(cq)$]-[${\bar{\delta}}\\!=\\!(\bar{c}\bar{q})$]
pair, the tetraquark BO potential is labeled by an additional $CP$ eigenvalue:
$\Sigma^{+}_{g}$. In addition, the nonstrange pentaquark states in this
analysis all have $I\\!=\\!\frac{1}{2}$ because the diquark
${\delta^{\prime\vphantom{1}}}$ internal to the triquark ${\bar{\theta}}$ is
assumed to be isoscalar, so that the overall isospin of the state is carried
by the light quark in $\delta^{\vphantom{1}}$. The tetraquarks, in contrast,
have isospin dependence via interactions between the light quarks in the
$\delta^{\vphantom{1}}$-${\bar{\delta}}$ pair (via both spin-spin and tensor
terms).

The two systems also feature some of the same operators in their Hamiltonians,
specifically the internal diquark spin coupling ${\kappa_{qc}}(1P)$ and the
spin-orbit coupling $V_{LS}$. The analysis of the hidden-charm $P$-wave
tetraquarks is challenging because the 4 predicted $1^{--}$ states in
$\Sigma^{+}_{g}(1P)$ can be assigned to observed states [e.g., $Y(4220)$] in a
variety of ways Giron and Lebed (2020a), while as noted above, the analysis of
the hidden-charm $P$-wave pentaquarks is challenging due to a lack of measured
$J^{P}$ quantum numbers for any of the states. Nevertheless, the numerical
values of $V_{LS}$ and ${\kappa_{qc}}(1P)$ for the two cases are quite
comparable: $V_{LS}\\!\simeq\\!45$-60 MeV and ${\kappa_{qc}}(1P)\\!=\\!40$–45
MeV for the most plausible fits in Ref. Giron and Lebed (2020a), as compared
with $V_{LS}\\!\simeq\\!80$ MeV and ${\kappa_{qc}}(1P)\\!\simeq\\!50$ MeV from
Table 3.

Predictions for the masses of all 7 states in the $\Sigma^{+}(1P)$ multiplet
are presented in Table 3. $P_{c}(4312)$ is seen to be the lightest state of
the multiplet, $P_{c}(4337)$ is the next lightest, followed by the nearly
degenerate pair coinciding with $P_{c}(4440)$, and then $P_{c}(4457)$.
Notably, no lower or intermediate states appear. Finally, the heaviest
${\frac{3}{2}}^{+}$ and the ${\frac{5}{2}}^{+}$ lie much higher in mass; LHCb
does present $J/\psi\,p$ data from $\Lambda_{b}$ decays up to almost 5 GeV
Aaij  _et al._ (2019), but the statistics appear insufficient to resolve
states beyond about 4500 MeV.

Notably, no signal for $P_{c}(4337)$ is apparent in the LHCb $\Lambda_{b}$
decay data, but only in their $B_{s}^{0}$ decay data Aaij  _et al._ (2021a).
On the other hand, Ref. Aaij  _et al._ (2021a) sees no signal for
$P_{c}(4312)$, and limited phase space precludes observation of the higher
$P_{c}$ states. These curious results from the same facility, taken at face
value, suggest different internal wave-function structures for $P_{c}(4312)$
and $P_{c}(4337)$ being accessed through different processes. Recalling that
the $V_{T}\\!=\\!0$ mass eigenstates are those of good diquark spin
$s_{\delta}^{\vphantom{1}}$ [Eqs. (3)], the LHCb data can be explained if
$\Lambda_{b}$ decays preferentially couple to states with
$s_{\delta}^{\vphantom{1}}\\!=\\!1$ [and hence to $P_{\frac{3}{2}}$, the core
quark state of $P_{c}(4312)$], while $B_{s}^{0}$ decays preferentially couple
to states with $s_{\delta}^{\vphantom{1}}\\!=\\!0$ [and hence to
$P_{\frac{1}{2}}$, the core quark state of $P_{c}(4337)$]. While the dynamical
explanations for these couplings are not immediately clear, one may observe
that the decay $B_{s}^{0}\\!\to\\!J/\psi\,p\bar{p}$ requires the annihilation
of the initial valence quarks $\bar{b}s\\!\to\\!\bar{c}c$ through the
$t$-channel exchange of a single virtual $W$ boson, while the initial
$\Lambda_{b}$ light valence quarks $ud$ in the dynamical diquark model persist
through the decay as the “good” diquark ${\delta^{\prime\vphantom{1}}}$. Such
differences could certainly have a pronounced effect upon the internal spin
structure of the produced states.

Information can also be obtained from the quarkonium decays of the states,
assuming the conservation of heavy-quark spin $s_{{Q\bar{Q}}}$. The underlying
quark states $P_{\frac{1}{2}},P_{\frac{1}{2}}^{\prime},P_{\frac{3}{2}}$ of
Eqs. (3), which we have found to coincide with the mass eigenstates in the
limit $V_{T}\\!=\\!0$, are decomposed in terms of $s_{{Q\bar{Q}}}$ eigenvalues
in Eqs. (5). Thus, for example, $P_{c}(4457)$ coincides with
$P_{\frac{3}{2}}$, which has only an $s_{{Q\bar{Q}}}\\!=\\!1$ component.
Therefore, $P_{c}(4457)$ should (and does) decay prominently to $J/\psi\,p$,
but not to $\eta_{c}\,p$.

Lastly, we consider the troublesome $P_{c}(4380)$ opposite-parity signal. With
the narrow $P_{c}$ states filling the positive-parity $\Sigma^{+}(1P)$
multiplet, the very broad $P_{c}(4380)$ presumably belongs to a negative-
parity $S$-wave multiplet. However, calculations of multiplet-average masses
in the dynamical diquark model Giron  _et al._ (2019) predict the
$\Sigma^{+}(1S)$ masses to be nearly 400 MeV lower and the $\Sigma^{+}(2S)$
masses to be nearly 200 MeV higher than those in $\Sigma^{+}(1P)$. We confirm
these results in this latest analysis, incorporating the best determination of
the $(cq)$ diquark mass from the latest analysis of
$c\bar{c}q^{\prime}\\!\bar{q}$ tetraquarks in Ref. Giron  _et al._
(2021),444These values are quite comparable to those obtained from other
determinations, such as $m_{\delta^{\vphantom{1}}(cq)}\\!=\\!1975$ MeV from a
constituent-quark approach Maiani  _et al._ (2005) or $1860\\!\pm\\!50$ MeV
from QCD sum rules Kleiv  _et al._ (2013).

$\displaystyle m_{\delta^{\vphantom{1}}(cq)}$ $\displaystyle=$ $\displaystyle
1938.0\pm 0.9\ {\rm MeV}\ {\rm(JKM)}\,,$ (25) $\displaystyle=$ $\displaystyle
1916.2\pm 0.9\ {\rm MeV}\ {\rm(CPRRW)}\,,$

to obtain the triquark masses $m_{\bar{\theta}}$ presented in Table 3. If
$P_{c}(4380)$ survives further analysis as a distinct state, it cannot be of
the same diquark-triquark structure as the other $P_{c}$ states; the most
likely candidate would then be a $\bar{D}\Sigma^{*}_{c}$ threshold effect or
molecule Shen  _et al._ (2016), which in its relative $S$ wave has the
required negative parity.

Table 3: Calculations of Hamiltonian parameters and masses for the $c\bar{c}qq_{1}q_{2}$ $\Sigma^{+}(1P)$ states. All masses are in units of MeV. The fit in the first column sets the tensor coupling $V_{T}\\!=\\!0$. Boldface indicates best-fit masses to measured values from Table 1.  $\chi^{2}_{\mathrm{min}}$ | $0.468$ | $0.000$
---|---|---
$M_{0}(1P)$ | $4495.4\pm 6.4$ | $4492.0$
$m_{\delta^{\vphantom{1}}=(cq)}$ | $1927.1\pm 11.0$ | $1927.1\pm 11.0$
$m_{\bar{\theta}}$ | $2077.2\pm 11.6$ | $2073.5\pm\ 9.0$
${\kappa_{qc}}(1P)$ | $52.4\pm 6.3$ | $49.9$
${V_{LS}}$ | $82.8\pm 5.6$ | $80.2$
${V_{T}}$ | $0.0$ | $1.1$
$M_{\frac{1}{2}^{+}}$ | $\bm{4314.6}\pm 6.8$ | $\bm{4311.9}$
| $\bm{4334.1}\pm 9.4$ | $\bm{4337.0}$
| $4438.8\ \pm 4.9$ | $4436.8$
$M_{\frac{3}{2}^{+}}$ | $\bm{4438.8}\pm 4.9$ | $\bm{4440.3}$
| $\bm{4458.2}\pm 4.1$ | $\bm{4457.3}$
| $4563.0\pm 11.9$ | $4557.3$
$M_{\frac{5}{2}^{+}_{\vphantom{y}}}$ | $4645.8\pm 17.3$ | $4636.3$

###  IV.2 The $c\bar{c}qq_{1}q_{2}$ Sector, $S$ Wave

With specific diquark masses $m_{\delta}^{\vphantom{1}}$ [Eqs. (25)] and
triquark masses $m_{\bar{\theta}}$ (Table 3) in hand, one can solve the
coupled Schrödinger equations in the dynamical diquark model for all BO
potentials, as is done in Ref. Giron  _et al._ (2019). Here, we extract the
value of $M_{0}(1S)$, which is one of the two parameters appearing in the mass
expressions of Eqs. (11), and present it in Table 4. The only additional
parameter then needed to predict all state masses in the $\Sigma^{+}(1S)$
multiplet is ${\kappa_{qc}}(1S)$, which can be obtained from the latest
hidden-charm tetraquark analysis of Ref. Giron  _et al._ (2021):

${\kappa_{qc}}(1S)=25.6\pm 5.0\ {\rm MeV}\,.$ (26)

The masses of the 3 states of $\Sigma^{+}(1S)$ obtained from these values are
presented in Table 4.

Of course, no hidden-charm pentaquark states with such low masses have yet
been observed. However, one notes two important points in this regard: First,
the region below about 4100 MeV in the data of Ref. Aaij  _et al._ (2019) has
rapidly vanishing $J/\psi\,p$ phase space (threshold at 4035 MeV), meaning
that the ground state at $\simeq\\!4085$ MeV might be difficult to discern.
Even so, the LHCb data from about 4100–4200 MeV appears as a broad enhancement
that could easily contain signals of resonances. The $\Sigma^{+}(1S)$ states,
carrying negative parity, are not protected by a centrifugal barrier against
$S$-wave fall-apart decays into $J/\psi\,p$, and therefore can have
substantial widths. Second, the proposed suppression of $\Lambda_{b}$ decays
to $\Sigma^{+}(1P)$ states with $s_{\delta}^{\vphantom{1}}\\!=\\!0$, if it
holds for the multiplet $\Sigma^{+}(1S)$, eliminates the ground state
$P_{\frac{1}{2}}$ in Eqs. (3) from appearing in $\Lambda_{b}$ decays, leaving
only the (nearly degenerate) heavier
${\frac{1}{2}}^{-}\\!$-${\frac{3}{2}}^{-}$ pair, $P_{\frac{1}{2}}^{\prime}$
and $P_{\frac{3}{2}}$. The LHCb data from $B_{s}^{0}$ decays Aaij  _et al._
(2021a) has substantially larger statistical uncertainties than that from
$\Lambda_{b}$ decays, but even so, hints at some possible structures in the
range of 4100–4200 MeV.

Table 4: Calculations of Hamiltonian parameters and masses for the $c\bar{c}qq_{1}q_{2}$ $\Sigma^{+}(1S)$ states. All masses are in units of MeV. The fit in the first (second) column uses values in which the corresponding $M_{0}(1P)$ value (as well as $m_{\delta}^{\vphantom{1}},m_{\bar{\theta}}$) is obtained from the first (second) column of Table 3, respectively.  $M_{0}(1S)$ | $4123.6\pm\ 1.7$ | $4127.0\pm\ 1.7$
---|---|---
$m_{\delta^{\vphantom{1}}=(cq)}$ | $1927.1\pm 11.0$ | $1927.1\pm 11.0$
$m_{\bar{\theta}}$ | $2077.2\pm 11.6$ | $2073.5\pm\ 9.0$
${\kappa_{qc}}(1S)$ | $25.6\pm 5.0$ | $25.6\pm 5.0$
$M_{\frac{1}{2}^{-}}$ | $4085.2\pm 7.7$ | $4088.6\pm 7.7$
| $4136.4\pm 3.0$ | $4139.8\pm 3.0$
$M_{\frac{3}{2}^{-}_{\vphantom{y}}}$ | $4136.4\pm 3.0$ | $4139.8\pm 3.0$

###  IV.3 The $c\bar{c}uds$ Sector

Using the same triquark masses $m_{\bar{\theta}}$ as in Table 4 and the $(cs)$
diquark mass $m_{\delta^{\vphantom{1}}(cs)}$ obtained from the hidden-charm,
open-strange tetraquark states $Z_{cs}$ in Ref. Giron  _et al._ (2021),

$\displaystyle m_{\delta^{\vphantom{1}}(cs)}$ $\displaystyle=$ $\displaystyle
2080.2\pm 1.5\ {\rm MeV}\ {\rm(JKM)}\,,$ (27) $\displaystyle=$ $\displaystyle
2058.5\pm 1.5\ {\rm MeV}\ {\rm(CPRRW)}\,,$

one may immediately compute the $c\bar{c}uds$ multiplet-average masses
$M_{0}(1P)$ and $M_{0}(1S)$ for $\Sigma^{+}(1P)$ and $\Sigma^{+}(1S)$,
respectively, just as done in the previous subsections. We also have the value
of ${\kappa_{sc}}(1S)$ from Ref. Giron  _et al._ (2021),

${\kappa_{sc}}(1S)=109.8\pm 1.1\ {\rm MeV}\,,$ (28)

which permits the immediate computation of the 3 $c\bar{c}uds$ states in
$\Sigma^{+}(1S)$.

However, we possess no independent determination of ${\kappa_{sc}}(1P)$, since
no $P$-wave $c\bar{c}q\bar{s}$ candidates have yet been observed. The large
value of ${\kappa_{sc}}(1S)$ in Eq. (28), as compared to the value of
${\kappa_{qc}}(1S)$ in Eq. (26), is argued in Ref. Giron and Lebed (2020a) to
result from the heavier $s$ quark forming a $(cs)$ diquark that is more
compact than $(cq)$, which therefore subjects the $c,s$ quarks to much larger
spin-spin couplings. Meanwhile, the larger value of ${\kappa_{qc}}(1P)$
($\simeq\\!50$ MeV from Table 3), as compared to that of ${\kappa_{qc}}(1S)$
in Eq. (26), has been discussed in Ref. Giron and Lebed (2020a) as a possible
result of finite diquark sizes leading to a sensitivity of their internal spin
couplings on the angular momentum of the flux tube connecting them. One
expects this sensitivity to be greater for diquarks containing a light quark
$q$ than an $s$ quark. Additionally, the $P$-wave Hamiltonian of Eq. (14)
includes the spin-orbit coupling $V_{LS}$, whose value might very well depend
upon whether $(cs)$ or $(cq)$ diquarks are present. For the analysis of the
$\Sigma^{+}(1P)$ $c\bar{c}uds$ states, we simply adopt the numerical values of
the corresponding Hamiltonian couplings from the $\Sigma^{+}(1P)$
$c\bar{c}qq_{1}q_{2}$ system given in Table 3. The results are presented in
Table 5.

One notes immediately from Table 5 that the mass predictions for the lightest
two ${\frac{1}{2}}^{+}$ states are separated by only about 20 MeV, and they
bracket the measured mass of $P_{cs}(4459)$ given in Table 1. These states are
the strange analogues of $P_{c}(4312)$ and $P_{c}(4337)$. Indeed, since the
production of $P_{cs}(4459)$ via $\Xi_{b}^{-}\\!\to\\!(J/\psi\,\Lambda)K^{-}$
Aaij  _et al._ (2021b) is a strange analogue to the production of
$P_{c}(4312)$ via $\Lambda_{b}\\!\to\\!(J/\psi\,p)K^{-}$, one might expect
that the absence of $P_{c}(4337)$ in the latter predicts the absence of the
second-lightest ${\frac{1}{2}}^{+}$ $P_{cs}$ state in the former. The
identification of $P_{cs}(4459)$ as the strange analogue of $P_{c}(4312)$ in
the dynamical diquark model was proposed in Ref. Giron  _et al._ (2021).
Alternately, since the $J/\psi\,\Lambda$ data does not yet have the same level
of statistics as the $J/\psi\,p$ data, the possibility of two closely spaced
${\frac{1}{2}}^{+}$ states near 4459 MeV awaiting resolution is a very real
possibility.555A similar scenario is proposed in the molecular picture Du  _et
al._ (2021), except in that case referring to a
${\frac{1}{2}}^{-}\\!$-${\frac{3}{2}}^{-}$ pair. The other $\Sigma^{+}(1P)$
$P_{cs}$ states in this calculation are over 100 MeV heavier, and thus far
only appear in the LHCb data Aaij  _et al._ (2021b) as tantalizing hints.

The predictions for the $\Sigma^{+}(1S)$ states, using Eq. (28), are presented
in Table 6. Again, the statistics of the LHCb data in Ref. Aaij  _et al._
(2021b) are not yet sufficient to determine whether significant $P_{cs}$ peaks
reside in this mass range. Similar comments to those for the $\Sigma^{+}(1S)$
$P_{c}$ states apply in this case (noting that the $J/\psi\,\Lambda$ threshold
is 4213 MeV).

Table 5: Calculations of Hamiltonian parameters and masses for the $c\bar{c}sq_{1}q_{2}$ $\Sigma^{+}(1P)$ states. All masses are in units of MeV. The fit in the first column sets the tensor coupling $V_{T}\\!=\\!0$. The triquark mass $m_{\bar{\theta}}$ values are obtained from the corresponding column of Table 3.  $M_{0}(1P)$ | $4624.7\pm\ 0.0$ | $4621.3\pm\ 0.0$
---|---|---
$m_{\delta^{\vphantom{1}}=(cs)}$ | $2069.4\pm 10.9$ | $2069.4\pm 10.9$
$m_{\bar{\theta}}$ | $2077.2\pm 11.6$ | $2073.5\pm\ 9.0$
${\kappa_{qc}}(1P)$ | $52.4\pm 6.3$ | $49.9$
${V_{LS}}$ | $82.8\pm 5.6$ | $80.2$
${V_{T}}$ | $0.0$ | $1.1$
$M_{\frac{1}{2}^{+}}$ | $4444.0\pm 14.4$ | $4441.3$
| $4463.4\pm 11.0$ | $4466.3$
| $4568.1\pm 6.4$ | $4566.2$
$M_{\frac{3}{2}^{+}}$ | $4568.1\pm 6.4$ | $4569.6$
| $4587.6\pm 9.9$ | $4586.6$
| $4692.3\pm 4.2$ | $4686.4$
$M_{\frac{5}{2}^{+}_{\vphantom{y}}}$ | $4775.1\pm 9.0$ | $4765.7$
Table 6: Calculations of Hamiltonian parameters and masses for the $c\bar{c}sq_{1}q_{2}$ $\Sigma^{+}(1S)$ states. All masses are in units of MeV. The fit in the first (second) column uses values in which the corresponding $M_{0}(1P)$ value is obtained from the first (second) column of Table 5, respectively.  $M_{0}(1S)$ | $4258.6\pm 1.8$ | $4255.2\pm 1.7$
---|---|---
$\kappa_{sc}(1S)$ | $109.8\pm\ 1.1$ | $109.8\pm\ 1.1$
$m_{\delta^{\vphantom{1}}=(cs)}$ | $2069.4\pm 10.9$ | $2069.4\pm 10.9$
$m_{\bar{\theta}}$ | $2077.2\pm 11.6$ | $2073.5\pm\ 9.0$
$M_{\frac{1}{2}^{-}}$ | $4093.9\pm 2.4$ | $4090.5\pm 2.3$
| $4313.5\pm 1.9$ | $4310.1\pm 1.8$
$M_{\frac{3}{2}^{-}_{\vphantom{y}}}$ | $4313.5\pm 1.9$ | $4310.1\pm 1.8$

##  V Conclusions

In this paper, we have shown that the dynamical diquark model produces a
spectrum of $c\bar{c}qq_{1}q_{2}$ pentaquark states that agrees very well with
the 4 narrow $P_{c}$ resonances observed by the LHCb Collaboration. Their
spectrum, as two closely spaced pairs with nearly the same mass splitting,
fits neatly with the expected levels of the first excited-state multiplet
$\Sigma^{+}(1P)$ of the model, all members of which have positive parity.
Specifically, $P_{c}(4312)$ and $P_{c}(4337)$ are predicted to be
${\frac{1}{2}}^{+}$ states, $P_{c}(4440)$ is predicted to be a yet-unresolved
${\frac{1}{2}}^{+}\\!$-${\frac{3}{2}}^{+}$ pair, and $P_{c}(4457)$ is
predicted to be ${\frac{3}{2}}^{+}$. Moreover, the numerical values of the
Hamiltonian parameters (spin-spin, spin-orbit) in this calculation are found
to be closely comparable to those obtained in a study of the negative-parity
hidden-charm tetraquark states [$Y(4220)$, etc.].

The 2 yet-unobserved states of the $\Sigma^{+}(1P)$ multiplet are predicted to
have much higher masses. The broad, opposite-parity $P_{c}(4380)$ structure
does not fit into the model, and if it persists, it is much more likely a
threshold effect or molecule, e.g., caused by $\bar{D}\Sigma^{*}_{c}$.

The presence of $P_{c}(4312)$ and the absence of $P_{c}(4337)$ in one
production channel ($\Lambda_{b}$ decay), and vice versa in another production
channel ($B_{s}^{0}$ decay)—both seen at the same experiment—is a remarkable
feature. Such a pattern can be explained (if not yet fully understood) in this
model by the fact that the two states are composed of the same $[\bar{c}(ud)]$
color-triplet triquark, but have distinct internal spin structures for their
color-antitriplet diquark $(cq)$ component. The heavy-quark spin content of
the states, as indicated, for example, by the relative branching fractions of
the resonances into $J/\psi$ or $\eta_{c}$ final states, can also serve as a
useful diagnostic in uncovering the internal structure of the states.

Of exceptional importance in this analysis is the parity quantum number for
all of these states. All $S$-wave molecules formed from a ground-state charmed
baryon ($\Sigma^{+}_{c}$ or $\Lambda_{c}$: parity $+$) and a ground-state
charmed meson ($\bar{D}^{(*)}$: parity $-$) have negative parity. If any of
the narrow $P_{c}$ states is found to have negative parity, then virtually all
the analysis in this paper becomes invalidated, or at least must be radically
modified.

The 3 states of the ground-state multiplet $\Sigma^{+}(1S)$ in the model,
which do carry negative parity, are predicted to have masses above the
$J/\psi\,p$ threshold, but could have escaped detection by LHCb either due to
lower statistics in the near-threshold region, or due to the states having
large widths (as $S$-wave states, via their $J/\psi\,p$ fall-apart modes).

The same model applied to the open-strange sector predicts $P_{cs}(4459)$ to
be the strange analogue to the ${\frac{1}{2}}^{+}$ $P_{c}(4312)$. We also
present predictions for all other $c\bar{c}uds$ states in the $\Sigma^{+}(1P)$
and $\Sigma^{+}(1S)$ multiplets, for which the latest data gives only the
vaguest indications. Future refinements of the statistics for channels like
$\Xi_{b}^{-}\\!\to\\!(J/\psi\,\Lambda)K^{-}$ will almost certainly reveal the
existence of further $P_{cs}$ resonances, and the spectrum of these new states
will provide crucial information for unraveling their internal structure.

###### Acknowledgements.

This work was supported by the National Science Foundation (NSF) under Grant
Nos. PHY-1803912 and PHY-2110278.

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