aainstitutetext: Deutsches Elektronen-Synchrotron DESY, D-22607 Hamburg,
Germanybbinstitutetext: National Centre for Physics, Quaid-i-Azam University
Campus, Islamabad 45320, Pakistanccinstitutetext: Physics Department, Quaid-i-
Azam University, Islamabad 45320, Pakistanddinstitutetext: P. G. Demidov
Yaroslavl State University, Sovietskaya 14, 150003 Yaroslavl, Russia

#  Mass spectrum of the hidden-charm pentaquarks in the compact diquark model

Ahmed Ali,111Corresponding author. b Ishtiaq Ahmed c M. Jamil Aslam d
Alexander Ya. Parkhomenko b and Abdur Rehman<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The LHCb collaboration have recently updated their analysis of the resonant
$J/\psi\,p$ mass spectrum in the decay $\Lambda_{b}^{0}\to J/\psi\,p\,K^{-}$,
making use of their combined Run 1 and Run 2 data. In the updated analysis,
three narrow states, $P_{c}(4312)^{+}$, $P_{c}(4440)^{+}$, and
$P_{c}(4457)^{+}$, are observed. The spin-parity assignments of these states
are not yet known. We interpret these narrow resonances as compact hidden-
charm diquark-diquark-antiquark pentaquarks. Using an effective Hamiltonian,
based on constituent quarks and diquarks, we calculate the pentaquark mass
spectrum for the complete $SU(3)_{F}$ lowest $S$\- and $P$-wave multiplets,
taking into account dominant spin-spin, spin-orbit, orbital and tensor
interactions. The resulting spectrum is very rich and we work out the quark
flavor compositions, masses, and $J^{P}$ quantum numbers of the pentaquarks.
However, heavy quark symmetry restricts the observable states in
$\Lambda_{b}$-baryon, as well as in the decays of the other weakly-decaying
$b$-baryons, $\Xi_{b}$ and $\Omega_{b}$. In addition, some of the pentaquark
states are estimated to lie below the $J/\psi\,p$ threshold in
$\Lambda_{b}$-decays (and corresponding thresholds in $\Xi_{b}$\- and
$\Omega_{b}$-decays). They decay via $c\bar{c}$ annihilation into light
hadrons or a dilepton pair, and are expected to be narrower than the
$P_{c}$-states observed. We anticipate their discovery, as well as of the
other pentaquark states present in the spectrum at the LHC, and in the long-
term future at a Tera-$Z$ factory.

††preprint: DESY 19-052

##  1 Introduction

Recently, the LHCb collaboration has presented an updated account of the
resonant $J/\psi\,p$ mass spectrum in the decay $\Lambda_{b}^{0}\to
J/\psi\,p\,K^{-}$, based on the combined Run 1 and Run 2 data, adding up to 9
fb-1 Aaij:2019vzc . In this analysis, which supersedes their earlier findings
from 2015 Aaij:2015tga , nominal fits of the data have been performed with an
incoherent sum of Breit-Wigner amplitudes, which have resulted in the
observation of three peaks, whose masses, decay widths (with 95% C.L. upper
limits), and the ratio ${\cal R}$, defined as

${\cal R}\equiv\frac{{\cal B}(\Lambda_{b}\to P_{c}^{+}\,K^{-})\,{\cal
B}(P_{c}^{+}\to J/\psi\,p)}{{\cal B}(\Lambda_{b}\to J/\psi\,p\,K^{-})},$ (1)

are given in table 1. The state $P_{c}(4450)^{+}$ in the 2015 data
Aaij:2015tga , is now replaced by two narrow states, $P_{c}(4440)^{+}$ and
$P_{c}(4457)^{+}$. In addition, a third narrow peak, $P_{c}(4312)^{+}$, having
the mass $M=(4311.9\pm 0.7^{+6.8}_{-0.6})$ MeV, is also observed. The spin-
parity, $J^{P}$, assignments of the three narrow states, which are crucial to
decipher the underlying dynamics of the pentaquark states, are not yet
determined. The broad peak $P_{c}(4380)^{+}$ from the earlier data
Aaij:2015tga  is neither confirmed nor refuted, as the current LHCb analysis
is not sensitive to broad resonances. Hence, it is entirely conceivable that
more $P_{c}^{+}$-like structures are present in the $J/\psi\,p$ invariant-mass
spectrum, anticipated in the compact pentaquark interpretation, similar to the
excited $\Lambda^{*}$-baryon spectrum in the $K\,p$ channel, expected in the
quark model and confirmed in data Tanabashi:2018oca . For the discussion of
the 2015 LHCb data on pentaquarks and other multiquark hadrons and references
to the earlier works see the reviews Tanabashi:2018oca ; Ali:2017jda ;
Esposito:2016noz ; Chen:2016qju ; Guo:2017jvc ; Olsen:2017bmm ; Lebed:2016hpi
; Hosaka:2016pey ; Karliner:2017qhf ; Yuan:2018inv ; Liu:2019zoy ;
Brambilla:2019esw  and Ali:2019roi .

The new pentaquarks reported by LHCb Aaij:2019vzc  have triggered a surge in
theoretical papers interpreting the three narrow resonances as loosely-bound
hadronic-molecule states in various incarnations Chen:2019bip ; Chen:2019asm ;
Guo:2019fdo ; Liu:2019tjn ; He:2019ify ; Xiao:2019aya ; Huang:2019jlf ;
Shimizu:2019ptd ; Xiao:2019mvs ; Zhang:2019xtu . A more up to date list of
references can be seen in the proceedings of a recent Workshop on Exotic
Hadrons Skwarnicki:2019 . Indeed, in the mass region of interest, a number of
kinematical thresholds are present, such as $\Sigma_{c}^{+}\bar{D}^{0}$
($E_{\rm thr}=4317.73\pm 0.41$ MeV), $\Sigma_{c}^{*+}\bar{D}^{0}$ ($E_{\rm
thr}=4382.3\pm 2.4$ MeV), $\Lambda_{c}^{*+}\bar{D}^{0}$ ($E_{\rm
thr}=4457.09\pm 0.35$ MeV), and $\Sigma_{c}^{+}\bar{D}^{*0}$ ($E_{\rm
thr}=4459.9\pm 0.9$ MeV) Tanabashi:2018oca . The masses of the observed
resonances in the hadron molecule interpretation are essentially the sums of
their respective hadronic constituents, which is striking. Thus, not only the
three observed narrow pentaquarks can be accommodated, having the spin-parity
and isospin ($J^{P}=1/2^{-}$, $I=1/2$) and ($J^{P}=3/2^{-}$, $I=1/2$), but
several other states, also with a negative parity due to their assumed
$S$-wave character, are predicted. These emphatic claims are at best tentative
and await a confirmation of the spin-parity assignments of the three observed
states. The nature of the $P_{c}(4312)^{+}$ as a candidate for the
$\Sigma_{c}^{+}\bar{D}$ molecular state was also analyzed based on the
$S$-matrix principles Fernandez-Ramirez:2019koa  and it was found that the
attraction in this system is not enough for a bound state.

We emphasize that the decays $\Lambda_{b}\to\Sigma_{c}^{(*)}+X$ are
anticipated to be suppressed by Heavy Quark Symmetry (HQS), due to the
mismatch of the spectator diquark spin (a conserved quantity in the HQS limit)
in $\Lambda_{b}$\- and $\Sigma_{c}^{(*)}$-baryons. This suppression is well-
known, but is often ignored in the literature on the hadron molecular approach
for the new $P_{c}$-states, though there are some rare exceptions, see, for
example, Weng:2019ynv . There is an ample evidence of this suppression in the
PDG tables, in which the non-leptonic transitions
$\Lambda_{b}\to\Lambda_{c}^{(*)}+X$ dominate and the
$\Lambda_{b}\to\Sigma_{c}^{(*)}+X$ transitions are rather sparse
Tanabashi:2018oca . In the exact HQS limit, such decays are forbidden, but as
the HQS is not an exact symmetry and is broken by power ($1/m_{b}$) and QCD
($\mathcal{O}(\alpha_{s}(m_{b}))$) corrections, they are allowed but have
reduced decay rates. It has important implications for the analysis of the
$P_{c}$-states in the hadron molecule approach as well. In particular, it
implies that the branching ratios ${\cal B}(\Lambda_{b}\to P_{c}^{+}\,K^{-})$,
which in the hadron-molecule interpretations of the $P_{c}$-states are
supposed to be induced by the intermediate
$(\Sigma_{c}\,\bar{D}^{(*)})\,K^{-}$ states, followed by rescattering, are
subject to the HQS-implied suppression. Since the ratio ${\cal R}$ in (1),
which is the product ${\cal B}(\Lambda_{b}\to P_{c}^{+}\,K^{-})\,\times{\cal
B}(P_{c}^{+}\to J/\psi\,p)$ in units of ${\cal B}(\Lambda_{b}\to
J/\psi\,p\,K^{-})$, is well-measured for all three states by LHCb (see table
1), the branching ratios ${\cal B}(P_{c}^{+}\to J/\psi\,p)$ are not expected
to be small in the hadron-molecule interpretation. It is relevant to point out
that model-dependent 90% C.L. upper limits on ${\cal B}(P_{c}^{+}\to
J/\psi\,p)$ of 4.6%, 2.3%, and 3.8% for $P_{c}^{+}(4312)$, $P_{c}^{+}(4440)$,
and $P_{c}^{+}(4457)$, respectively, have been posted by the GlueX
collaboration Ali:2019lzf , assuming they have $J^{P}=3/2^{-}$ (i. e.,
$S$-wave $J/\psi\,p$ states). While this is currently not robust enough an
argument against the $P_{c}$’s being hadron molecules, due to inherent
assumptions about the theoretical estimates of the photoproduction cross
sections, but improved measurements will soon test this interpretation in the
$P_{c}$-photoproduction experiments Ali:2019lzf ; Blin:2016dlf . The role of
the photoproduction process $\gamma\,p\to J/\psi\,p$ in constraining the
models of the new $P_{c}$-states is noted in Cao:2019kst ; Wang:2019krd .
Another process to measure the $P_{c}$ production is the antiproton-deuterium
collisions, in which case the charm quark and antiquark entering the
charmonium state $J/\psi$ or $\eta_{c}$ are the result of the proton-
antiproton annihilation Voloshin:2019wxx . Thus, the internal consistency of
the hadron molecule approach for the new $P_{c}$ states remains to be checked.

In the preceding Letter Ali:2019npk , two of us have argued that there also
exists a prima facie case for the three narrow $P_{c}^{+}$-resonances to be
considered as candidates for compact pentaquarks. The basic idea of this
approach is that highly correlated colored diquarks play a key role in the
physics of multiquark states Maiani:2004vq ; Lipkin:1987sk ; Jaffe:2003sg ,
and they are at work in the underlying dynamics of the $P_{c}^{+}$-states. A
brief account of the new pentaquarks was presented in Ali:2019npk  in the
effective Hamiltonian framework based on constituent quarks and diquarks,
using isospin and heavy quark symmetry. In this paper, we extend the compact
diquark template to cover the complete $SU(3)_{F}$-multiplets of compact
hidden-charm pentaquarks in the lowest $S$\- and $P$-wave states. Some of them
can be searched for in the decays of the $\Lambda_{b}$-baryon, but some others
can only be reached in the decays of the other weakly-decaying $b$-baryons,
$\Xi_{b}$ and $\Omega_{b}$.

State | Mass [MeV] | Width [MeV] | (95% CL) | $\mathcal{R}\,[\%]$
---|---|---|---|---
$P_{c}(4312)^{+}$ | $4311.9\pm 0.7^{+6.8}_{-0.6}$ | $9.8\pm 2.7^{+3.7}_{-4.5}$ | ($<27$) | $0.30\pm 0.07^{+0.34}_{-0.09}$
$P_{c}(4440)^{+}$ | $4440.3\pm 1.3^{+4.1}_{-4.7}$ | $20.6\pm 4.9^{+8.7}_{-10.1}$ | ($<49$) | $1.11\pm 0.33^{+0.22}_{-0.10}$
$P_{c}(4457)^{+}$ | $4457.3\pm 0.6^{+4.1}_{-1.7}$ | $6.4\pm 2.0^{+5.7}_{-1.9}$ | ($<20$) | $0.53\pm 0.16^{+0.15}_{-0.13}$
Table 1:  Masses, decay widths (with 95% C.L. upper limits), and the ratio
$\mathcal{R}$, of the three narrow $J/\psi\,p$ resonances observed by the LHCb
collaboration in the $\Lambda_{b}\to J/\psi\,p\,K^{-}$ decay Aaij:2019vzc .

The pentaquark dynamics depends upon how the five constituents, i. e., the 4
quarks and an antiquark, are structured. Since quarks transform as a triplet
$\tt 3$ of the color $SU(3)$-group, the diquarks resulting from the direct
product $\tt 3\otimes 3=\bar{3}\oplus 6$, are thus either a color anti-triplet
$\tt\bar{3}$ or a color sextet $\tt 6$. Of these only the color $\tt\bar{3}$
configuration is kept, as suggested by perturbative arguments. Another
justification is that the color sextet diquarks are heavier than the
corresponding color anti-triplet ones, and can be integrated out for
spectroscopic considerations. This remains to be quantified, which can be
eventually done in lattice QCD. Both spin-0 and spin-1 diquarks are, however,
allowed. In the case of a diquark $[q^{\prime}q^{\prime\prime}]$ consisting of
two light quarks ($q^{\prime},\,q^{\prime\prime}=u,\,d,\,s$), the spin-0
diquarks are believed to be more tightly bound than the spin-1, and this
hyperfine splitting has implications for the spectroscopy. For the heavy-light
diquarks, such as $[cq]$ or $[bq]$, this splitting is suppressed by $1/m_{c}$
for the $[cq]$-diquark or by $1/m_{b}$ for the $[bq]$-diquark, and hence both
spin configurations are treated at par. Thus, the constituents of the hidden-
charm pentaquarks in the compact diquark model are $[cq]_{\bar{3}}$,
$\bar{c}_{\bar{3}}$, and $[q^{\prime}q^{\prime\prime}]_{\bar{3}}$, which make
up a color singlet. However, it is still a three-body problem to solve and
there are several dynamical possibilities to model their interconnections. We
follow here the intuitive picture in which the heavier components form a
nucleus and the lighter one is in an orbit around this nucleus, as it is
energetically easier to excite light degrees of freedom. Among the three
constituents, the light diquark $[q^{\prime}q^{\prime\prime}]_{\bar{3}}$ is
the lightest. Moreover, its spin-parity quantum numbers are fixed due to
heavy-quark symmetry constraints. So, we keep the light diquark as emerging
intact in the $b$-baryon decays, and put it in the orbit for the $P$-wave
states, with the heavier components, carrying a charm quark or a charm
antiquark, acting as a nucleus of (an almost) static color source. This is
shown in fig. 1 and the color and flavor structure in this approach can be
written as
[$[\bar{c}_{\bar{3}}(cq)_{\bar{3}}]_{3}(q^{\prime}q^{\prime\prime})_{\bar{3}}$].
Such a description is closer to the doubly-charm tetraquarks, with the quark
content [$(\bar{c}\bar{c})_{3}(q^{\prime}q^{\prime\prime})_{\bar{3}}$], and
doubly-charm antibaryons, having [$(\bar{c}\bar{c})_{3}\bar{q}_{\bar{3}}$], as
all three systems have two charm (anti)quarks at the center Manohar:1992nd ;
Esposito:2013fma ; Luo:2017eub ; Karliner:2017qjm ; Eichten:2017ffp ;
Francis:2016hui ; Bicudo:2017szl ; Junnarkar:2017sey ; Mehen:2017nrh ;
Czarnecki:2017vco ; Maiani:2019cwl ; Maiani:2019lpu . We note that a colored
diquark-triquark template for pentaquarks
$\bar{\theta}\,\delta\equiv\left[\bar{Q}\left(q_{1}q_{2}\right)_{\bar{3}}\right]_{3}\left(Qq_{3}\right)_{\bar{3}}$,
through the color-triplet binding mechanism, was first used by Lebed
Lebed:2015tna . There is also the third possibility to introduce a model for
the internal structure of a pentaquark Ali:2016dkf ; Ali:2017ebb  in which the
light and heavy diquarks bind first into a tetraquark in a color-triplet $\tt
3$ state which further interacts with the charm antiquark,
[$[(cq)_{\bar{3}}(q^{\prime}q^{\prime\prime})_{\bar{3}}]_{3}\bar{c}_{\bar{3}}$].
Which of the three internal structures discussed provides a realistic template
for the bound charm-anticharm pentaquarks remains to be seen.

In this paper we use the $L$ \- $S$ scheme for describing the angular
structure of pentaquarks. The spin of the heavy $[cq]$-diquark is
$\mathbf{S}_{hd}=\mathbf{S}_{c}+\mathbf{S}_{q}$ with the allowed values
$S_{hd}=0,\,1$, where $\mathbf{S}_{c}$ and $\mathbf{S}_{q}$ are the spins of
the charm- and light $u$-, $d$\- or $s$-quark, respectively. Combining the
heavy diquark spin with the spin $\mathbf{S}_{\bar{c}}$ of the charm
antiquark, we get the spin of the doubly-heavy triquark,
$\mathbf{S}_{t}=\mathbf{S}_{hd}+\mathbf{S}_{\bar{c}}$, having the values
$S_{t}=1/2,\,3/2$. The spin of the light
$[q^{\prime}q^{\prime\prime}]$-diquark is
$\mathbf{S}_{ld}=\mathbf{S}_{q^{\prime}}+\mathbf{S}_{q^{\prime\prime}}$, where
$S_{ld}=0,\,1$. The total spin of the pentaquark is the sum of the triquark
and the light-diquark spins, $\mathbf{S}=\mathbf{S}_{t}+\mathbf{S}_{ld}$ with
the set of values $S_{t}=1/2,\,3/2,\,5/2$. In the case of the orbitally
excited pentaquarks, the orbital angular momentum $\mathbf{L}$ of the
pentaquark is the sum of two terms: $\mathbf{L}_{t}$, arising from the
triquark system, and $\mathbf{L_{ld}}$, which determines the relative motion
of the light diquark around the doubly heavy triquark. The total orbital
angular momentum $\mathbf{L}$ of the pentaquark is then obtained with the help
of the momentum sum rules from quantum mechanics, i. e., it takes a value in
the range $L=|L_{t}-L_{ld}|,\ldots,L_{t}+L_{ld}$. After $\mathbf{L}$ and
$\mathbf{S}$ are constructed, the total angular momentum $\mathbf{J}$ of the
pentaquark follows from $\mathbf{J}=\mathbf{L}+\mathbf{S}$. The parity of the
states depends on the orbital angular momentum as $P=(-1)^{L+1}$. It is
negative for the $S$-wave states and positive for the states with one unit of
the orbital angular momentum.

Mass estimates are worked out in an effective Hamiltonian approach, which
apart from the constituent quark and diquark masses, includes dominant spin-
spin, spin-orbit, orbital, and tensor interactions. The effective Hamiltonian
for the $S$-wave states in this formulation is the same as the one used
earlier in the analysis of the 2015 LHCb data on pentaquarks Ali:2016dkf ;
Ali:2017ebb , but the $P$-wave (and higher orbital angular) states are
packaged differently. In addition to this, we incorporate the tensor
interaction in the effective Hamiltonian, which affect the $P$-states, and
work out the detailed mass spectrum in this framework, using the parameters
fixed from earlier studies of baryons and hidden-charm tetraquarks. The
resulting pentaquark spectrum is very rich. However, imposing the spin
conservation in the heavy-quark symmetry limit, we argue that only that part
of the pentaquark spectrum is expected to be observed in $\Lambda_{b}$-decays,
in which the pentaquarks have a “good” light diquark, i. e., having spin
$S_{ld}=0$. This reduces the number of anticipated pentaquark states in
$\Lambda_{b}$-decays greatly. The same is also true for the decays of the
$\Xi_{b}^{0,-}$-baryons which are from the same $SU(3)_{F}$-triplet as the
$\Lambda_{b}$-baryon. The other weakly decaying $b$-baryon, $\Omega_{b}$, in
which the light diquark has the spin-1, belongs to the $SU(3)_{F}$-sextet. Its
decays will yield pentaquarks from the $SU(3)_{F}$-decuplets, i. e., have a
spin-1 light diquark ($S_{ld}=1$) as a constituent Li:2015gta ; Ali:2016dkf .
Some of the unflavored pentaquark states are estimated to lie below the
$J/\psi\,p$ (even below $\eta_{c}\,p$) threshold and they will decay via the
$c\bar{c}$-annihilation into light hadrons or a lepton pair ($e^{+}e^{-}$ and
$\mu^{+}\mu^{-}$), giving rise to states narrower than the observed peaks,
which can also be searched in the $P_{c}^{+}\to p\,\ell^{+}\ell^{-}$ modes in
$\Lambda_{b}\to P_{c}\,K^{-}$ decays. The same holds for the decays of some
other pentaquark states, produced in the decays of the $\Xi_{b}$\- and
$\Omega_{b}$-baryons, in which case the corresponding pentaquarks with one,
two, or three strange quarks will give rise to narrow
$\Lambda^{0}\,\ell^{+}\ell^{-}$, $\Xi^{\prime 0}\,\ell^{+}\ell^{-}$, and
$\Omega^{-}\,\ell^{+}\ell^{-}$ final states, respectively.

We also note that a detailed spectrum of the tetraquark and pentaquark states
has also been worked out in the dynamical diquark model Giron:2019bcs , using
Born-Oppenheimer potentials calculated numerically on the lattice. Likewise,
the $P_{c}^{+}$-states have been studied in different color-bindings, such as
$\left\\{\left(c\bar{c}\right)\left(\left(ud\right)u\right);\,L=0\right\\}$
Weng:2019ynv  and as $\left(c\bar{c}\right)\left(uud\right)$ hadrocharmonia
Eides:2019tgv .

This paper is organized as follows. In section 2, we introduce the doubly-
heavy triquark — light diquark model of pentaquark, and define the state
vectors having a total angular momentum quantum number $J$ by
$\left|S_{hd},S_{t},L_{t};S_{ld},L_{ld};S,L\right\rangle_{J}$, where $S_{hd}$,
$S_{t}$, and $S_{ld}$ are the spins of the heavy diquark, doubly-heavy
triquark and light diquark, respectively. The corresponding sets of the
$S$-wave state vectors (with $L_{t}=L_{ld}=L=0$) with the “good” ($S_{ld}=0$)
and “bad” ($S_{ld}=1$) light diquarks are presented in tables 3 and 3,
respectively. State vectors of the $P$-wave pentaquarks with the ground-state
triquark ($L_{t}=0$, $L_{ld}=L=1$) and “good” light diquark with the spin
$S_{ld}=0$ (“bad” light diquark with the spin $S_{ld}=1$), are given in table
5 (table 5). In section 3, we give the effective Hamiltonian used to work out
the pentaquark mass spectrum. In section 4, analytical expressions for
effective Hamiltonian matrix elements are presented taking into account the
dominant spin-spin interactions in the heavy and light diquarks and in the
hidden-charm triquark. For the $P$-wave states, additional contributions and
mixings due to the orbital, spin-orbit and tensor interactions are included.
In all the cases, we diagonalize the mass matrices and the analytical
equations for all the $S$\- and $P$-wave pentaquark masses are presented.
Section 5 contains the values of various input parameters, with the
constituent diquark masses given in table 6 and the spin-spin couplings,
$\mathcal{K}_{\bar{Q}Q^{\prime}}$ and $(\mathcal{K}_{QQ^{\prime}})_{\bar{3}}$,
extracted from spectra of mesons and baryons, are given in table 7. Our
predictions for the unflavored pentaquark masses having the quark flavors
$(\bar{c}[cq][q^{\prime}q^{\prime\prime}])$, with $q$, $q^{\prime}$, and
$q^{\prime\prime}$ being $u$\- and $d$-quarks, assuming the isospin symmetry,
are given in table 8. Masses of the hidden-charm strange pentaquarks are shown
in tables 10 — 14. In this section, a comparison of our approach with the
dynamical diquark model Giron:2019bcs  is also presented. We continue with a
discussion of dominant decay channels of ground-state pentaquarks containing a
scalar light diquark in section 6 and conclude in section 7. In appendix A,
corrections due to the spin-spin interactions between the constituents of the
doubly-heavy triquark and the light diquark are detailed; in appendix B,
cumbersome mass derivations for the $P$-wave pentaquark states are presented,
and in appendix C the details and results of the $\chi^{2}$-analysis of the
orbitally-excited $\Omega_{c}^{*}$-baryons are shown.

##  2 Doubly-heavy triquark — light diquark model of pentaquark

In the pentaquark picture shown in fig. 1, there are two color flux tubes,
with the first stretched between the charm diquark and the charm antiquark
from which the diquark is in the color-antitriplet state $\bar{3}$. With the
antiquark being also a color antitriplet state $\bar{3}$, their product is
decomposed into two irreducible representations,
$\bar{3}\times\bar{3}=3+\bar{6}$, from which the color triplet $3$ is kept.
This color-triplet triquark makes a color-singlet bound state with the light
diquark through the second flux tube in much the same way as the quark and
diquark bind in an ordinary baryon. This approach differs from the one
Ali:2016dkf ; Ali:2017ebb  used previously to get the mass spectrum of the
$S$\- and $P$-wave pentaquark states with the spin-parities $J^{P}=1/2^{\pm}$,
$3/2^{-}$, and $5/2^{\pm}$, where the two diquarks are assumed to be organized
first into a color-triplet tetraquark state which afterwards interacts with
the charm antiquark.

The double-charm triquark system $[[cq]\bar{c}]_{3}$ is expected to be
dynamically similar to the double-charm antidiquark,
$\\{\bar{c}\bar{c}\\}_{3}$, as far as their color and masses are concerned.
The role of the latter, forming an almost static source, has been discussed at
great length in a number of papers for the doubly-heavy anti-baryons, such as
$\bar{\Xi}_{cc}(=\bar{c}\bar{c}\bar{q})$, and tetraquarks
$T(\bar{c}\bar{c}qq^{\prime})$ Manohar:1992nd ; Karliner:2017qjm ;
Eichten:2017ffp ; Francis:2016hui ; Bicudo:2017szl ; Junnarkar:2017sey ;
Mehen:2017nrh . The doubly-heavy triquark and the doubly-heavy diquark also
differ in that the former has a light quark and the latter none. However,
under the assumption that the diquark $[cq]_{\bar{3}}$ is bound, and the
dynamics is essentially determined by the color configuration and masses, the
two are expected not to deviate from each other, at least as a first
approximation. With this in mind, we first detail the doubly-heavy triquark —
light diquark picture, modify the effective Hamiltonian for the $S$-wave
pentaquarks Ali:2016dkf ; Ali:2017ebb  by keeping the most relevant terms for
the mass determination, and then extend it for the $P$-states by including the
orbital, spin-orbit and tensor interactions between the hidden-charm triquark
and light diquark.

The effective Hamiltonian for the ground-state pentaquarks is described in
terms of two constituent diquark masses of the heavy diquark $m_{[cq]}\equiv
m_{hd}$ and of the light one $m_{[q^{\prime}q^{\prime\prime}]}\equiv m_{ld}$,
the spin-spin interactions between the quarks in each diquark shell, and spin-
spin interactions between the diquarks. To these are added the constituent
mass $m_{c}$ of the charm antiquark and its spin-spin interactions with each
of the diquarks.

Figure 1:  A picture of pentaquarks in the diquark model involving the heavy
diquark $[cq]_{\bar{3}}$ and charm antiquark $(\bar{c})_{\bar{3}}$, which form
the triquark system, which combines with the light diquark
$[q^{\prime}q^{\prime\prime}]_{\bar{3}}$ to make a color singlet pentaquark.
The subscripts indicate that all three constituents are color anti-triplets;
$q$, $q^{\prime}$, and $q^{\prime\prime}$ are light quarks each of which can
be $u$-, $d$-, or $s$-quark.

Using the quantum numbers introduced above, we specify the complete orthogonal
set of basis vectors for the hidden-charm pentaquark states. As already
stated, we define the state vectors having a total angular momentum quantum
number $J$ by

$\left|S_{hd},S_{t},L_{t};S_{ld},L_{ld};S,L\right\rangle_{J}.$

The corresponding sets of the $S$-wave state vectors (with $L_{t}=L_{ld}=L=0$)
with the “good” ($S_{ld}=0$) and “bad” ($S_{ld}=1$) light diquark are
presented in tables 3 and 3, respectively.

$J^{P}$ | $\left|S_{hd},S_{t},L_{t};S_{ld},L_{ld};S,L\right\rangle_{J}$
---|---
$1/2^{-}$ | $\left|0,1/2,0;0,0;1/2,0\right\rangle_{1/2}$
$1/2^{-}$ | $\left|1,1/2,0;0,0;1/2,0\right\rangle_{1/2}$
$3/2^{-}$ | $\left|1,3/2,0;0,0;3/2,0\right\rangle_{3/2}$
Table 2:  Spin-parity $J^{P}$ and state vectors of the $S$-wave pentaquarks
containing the “good” light diquark with the spin $S_{ld}=0$. The horizontal
line demarcates the spin $S_{hd}$ of the heavy diquark.

$J^{P}$ | $\left|S_{hd},S_{t},L_{t};S_{ld},L_{ld};S,L\right\rangle_{J}$
---|---
$1/2^{-}$ | $\left|0,1/2,0;1,0;1/2,0\right\rangle_{1/2}$
$3/2^{-}$ | $\left|0,1/2,0;1,0;3/2,0\right\rangle_{3/2}$
$1/2^{-}$ | $\left|1,1/2,0;1,0;1/2,0\right\rangle_{1/2}$
$3/2^{-}$ | $\left|1,1/2,0;1,0;3/2,0\right\rangle_{3/2}$
$1/2^{-}$ | $\left|1,3/2,0;1,0;1/2,0\right\rangle_{1/2}$
$3/2^{-}$ | $\left|1,3/2,0;1,0;3/2,0\right\rangle_{3/2}$
$5/2^{-}$ | $\left|1,3/2,0;1,0;5/2,0\right\rangle_{5/2}$
Table 3:  Spin-parity $J^{P}$ and state vectors of the $S$-wave pentaquarks
containing the “bad” light diquark with the spin $S_{ld}=1$. The horizontal
line demarcates the spin $S_{hd}$ of the heavy diquark.

Among the overall ten states with a fixed light-quark flavor content, there
are five states with $J^{P}=1/2^{-}$, four states with $J^{P}=3/2^{-}$ and one
with $J^{P}=5/2^{-}$. In view of the propensity of these states, we note that
in the heavy-quark symmetry limit, the production of the states with the
quantum numbers shown in table 3 is forbidden in $\Lambda_{b}$\- or
$\Xi_{b}$-decays. Nevertheless, they are present in the pentaquark mass
spectrum, and they are reachable in other processes.

For the orbitally excited states, one needs to specify which part of the
pentaquark wave function is excited. In the triquark-diquark template used
here, and shown in fig. 1, the heavy triquark state consists of the charm
diquark and charm antiquark. Since both are heavy, the triquark is an (almost)
static system and, hence, $L_{t}=0$ is its most probable orbital quantum
number. Thus, the orbital excitation is generated by the light diquark (i. e.,
$L=L_{ld}$). With this, we list the lowest-lying orbitally excited states
($L=1$) with the “good” ($S_{ld}=0$) and the “bad” ($S_{ld}=1$) light diquark
in tables 5 and 5, respectively. In total, there are 25 pentaquark states with
a fixed light-quark flavor content which are divided into nine states with
$J^{P}=1/2^{+}$, ten with $J^{P}=3/2^{+}$, five with $J^{P}=5/2^{+}$ and one
with $J^{P}=7/2^{+}$.

$J^{P}$ | $\left|S_{hd},S_{t},L_{t};S_{ld},L_{ld};S,L\right\rangle_{J}$
---|---
$1/2^{+}$ | $\left|0,1/2,0;0,1;1/2,1\right\rangle_{1/2}$
$3/2^{+}$ | $\left|0,1/2,0;0,1;1/2,1\right\rangle_{3/2}$
$1/2^{+}$ | $\left|1,1/2,0;0,1;1/2,1\right\rangle_{1/2}$
$3/2^{+}$ | $\left|1,1/2,0;0,1;1/2,1\right\rangle_{3/2}$
$1/2^{+}$ | $\left|1,3/2,0;0,1;3/2,1\right\rangle_{1/2}$
$3/2^{+}$ | $\left|1,3/2,0;0,1;3/2,1\right\rangle_{3/2}$
$5/2^{+}$ | $\left|1,3/2,0;0,1;3/2,1\right\rangle_{5/2}$
Table 4:  Spin-parity $J^{P}$ and state vectors of the $P$-wave pentaquarks
with the ground-state triquark ($L_{t}=0$) and “good” light diquark with the
spin $S_{ld}=0$. The horizontal line demarcates the spin $S_{hd}$ of the heavy
diquark.

$J^{P}$ | $\left|S_{hd},S_{t},L_{t};S_{ld},L_{ld};S,L\right\rangle_{J}$
---|---
$1/2^{+}$ | $\left|0,1/2,0;1,1;1/2,1\right\rangle_{1/2}$
$3/2^{+}$ | $\left|0,1/2,0;1,1;1/2,1\right\rangle_{3/2}$
$1/2^{+}$ | $\left|0,1/2,0;1,1;3/2,1\right\rangle_{1/2}$
$3/2^{+}$ | $\left|0,1/2,0;1,1;3/2,1\right\rangle_{3/2}$
$5/2^{+}$ | $\left|0,1/2,0;1,1;3/2,1\right\rangle_{5/2}$
$1/2^{+}$ | $\left|1,1/2,0;1,1;1/2,1\right\rangle_{1/2}$
$3/2^{+}$ | $\left|1,1/2,0;1,1;1/2,1\right\rangle_{3/2}$
$1/2^{+}$ | $\left|1,1/2,0;1,1;3/2,1\right\rangle_{1/2}$
$3/2^{+}$ | $\left|1,1/2,0;1,1;3/2,1\right\rangle_{3/2}$
$5/2^{+}$ | $\left|1,1/2,0;1,1;3/2,1\right\rangle_{5/2}$
$1/2^{+}$ | $\left|1,3/2,0;1,1;1/2,1\right\rangle_{1/2}$
$3/2^{+}$ | $\left|1,3/2,0;1,1;1/2,1\right\rangle_{3/2}$
$1/2^{+}$ | $\left|1,3/2,0;1,1;3/2,1\right\rangle_{1/2}$
$3/2^{+}$ | $\left|1,3/2,0;1,1;3/2,1\right\rangle_{3/2}$
$5/2^{+}$ | $\left|1,3/2,0;1,1;3/2,1\right\rangle_{5/2}$
$3/2^{+}$ | $\left|1,3/2,0;1,1;5/2,1\right\rangle_{3/2}$
$5/2^{+}$ | $\left|1,3/2,0;1,1;5/2,1\right\rangle_{5/2}$
$7/2^{+}$ | $\left|1,3/2,0;1,1;5/2,1\right\rangle_{7/2}$
Table 5:  Spin-parity $J^{P}$ and state vectors of the $P$-wave pentaquarks
with the ground-state triquark ($L_{t}=0$) and “bad” light diquark with the
spin $S_{ld}=1$. The horizontal line demarcates the spin $S_{hd}$ of the heavy
diquark.

The pentaquark spectrum emerging from the underlying diquark picture is very
rich. Contrasting it with the current experimental situation, with only three
known hidden-charm pentaquark states $P_{c}(4312)^{\pm}$, $P_{c}(4440)^{\pm}$,
and $P_{c}(4457)^{\pm}$ observed in $\Lambda_{b}$-baryon decays, some
selection rules have to be applied to restrict the number of observable
pentaquarks. Since in the heavy-quark symmetry limit, the spin of the light
quarks is conserved in heavy-baryon decays, and $S_{ld}=0$ in
$\Lambda_{b}$-baryon, the states with the light-diquark spin $S_{ld}=1$ are
suppressed. This means that we need to consider only the states in tables 3
and 5, for the $S$\- and $P$-wave pentaquarks, respectively. This reduces the
number of states to three ($S$-wave) and seven ($P$-wave). The states with
$S_{ld}=1$ can, however, be produced in $\Omega_{b}$-baryon decays or in
prompt production processes.

A related issue is the presence of two identical light quarks in the Fock
space of a hidden-charm pentaquark. Among the light quark $q$ present in the
triquark, and the light quarks $q^{\prime}$ and $q^{\prime\prime}$ in the
diquark, there are two identical quarks. Treated as free quarks, they are
subject to Pauli’s exclusion principle, affecting the resulting spectrum of
hadrons. This, for example, would be the case in the hadroquarkonium picture
of the hidden-charm pentaquarks. However, if the
$q^{\prime}q^{\prime\prime}$-pair is bound in a diquark, which is a boson,
then this exclusion does not apply so long as the diquark is not broken into
its fermionic constituents. We illustrate this situation by focusing on the
discovery mode $\Lambda_{b}\to P_{c}^{+}K^{-}$. This decay is induced at the
quark level by the $b\left[ud\right]\to
c\,\bar{c}\,s\left[ud\right]+\left(u\bar{u}\right)_{\rm vac}$ transition. In
this, the $\left[ud\right]$-diquark, which has well-defined quantum numbers
(spin $S_{[ud]}=0$ and isospin $I_{[ud]}=0$), is a boson. In the heavy-quark
symmetry limit, assumed in our paper, the light degrees of freedom remain
conserved, and the light $\left[ud\right]$-diquark emerges in tact from the
decay. Indeed, this imposes restrictions on the quantum numbers of final
states, as we have emphasized earlier. If this symmetry is broken, say, by
$1/m_{b}$ terms, then the $u$\- and $d$-quarks in the
$\left[ud\right]$-diquark are no longer bound, and in the example here, one
has two identical quarks, the $u$-quark produced from the vacuum excitation
and the $u$-quark from the break-up of the spectator diquark, and the Pauli’s
exclusion principle would apply. In the heavy-quark symmetry limit, we have
the spin-$1/2$ $u$-quark from the vacuum excitation, and the spin-0
$\left[ud\right]$-diquark, and they are distinctly different building blocks.
In this case, no exclusion restrictions from identical quarks apply, which is
what we have used in building the pentaquark spectrum.

##  3 Effective Hamiltonian for pentaquark spectrum

We calculate the mass spectrum of pentaquark under the assumption that their
underlying structure is given by $\bar{c}$, $[cq]$, and
$[q^{\prime}q^{\prime\prime}]$, with $q$, $q^{\prime}$, and $q^{\prime\prime}$
being any of the light $u$-, $d$-, and $s$-quarks. We also assume the isospin
symmetry among the states. For this, we extend the effective Hamiltonian
proposed for the tetraquark spectroscopy by Maiani et al. Maiani:2014aja . The
effective Hamiltonian for the $S$-wave pentaquark mass spectrum can be written
as follows:

$H^{(L=0)}=H_{t}+H_{ld}.$ (2)

The first term in the Hamiltonian (2) is related with the color triquark:

$H_{t}=m_{c}+m_{hd}+2\,(\mathcal{K}_{cq})_{\bar{3}}\,(\mathbf{S}_{c}\cdot\mathbf{S}_{q})+2\,\mathcal{K}_{\bar{c}q}\left(\mathbf{S}_{\bar{c}}\cdot\mathbf{S}_{q}\right)+2\,\mathcal{K}_{\bar{c}c}\left(\mathbf{S}_{\bar{c}}\cdot\mathbf{S}_{c}\right),$
(3)

where $m_{c}$ and $m_{hd}$ are the constituent masses of the charm antiquark
and charm diquark, respectively. The last three terms describe the spin-spin
interactions in the charm diquark and between the diquark constituents and the
charm antiquark. Among the three spin-spin couplings
$(\mathcal{K}_{cq})_{\bar{3}}$, $\mathcal{K}_{\bar{c}q}$, and
$\mathcal{K}_{\bar{c}c}$, the spin-spin interaction inside the diquark
$(\mathcal{K}_{cq})_{\bar{3}}$ is argued to be the dominant one Maiani:2014aja
.

The second term $H_{ld}$ in the Hamiltonian (2) contains the operators
responsible for the spin-spin interaction in the light diquark and its
interaction with the triquark:

$H_{ld}=m_{ld}+2\,(\mathcal{K}_{q^{\prime}q^{\prime\prime}})_{\bar{3}}\,(\mathbf{S}_{q^{\prime}}\cdot\mathbf{S}_{q^{\prime\prime}})+H_{SS}^{t-ld},$
(4)

where $m_{ld}$ is the constituent mass of the light diquark, consisting of the
light quarks $q^{\prime}$ and $q^{\prime\prime}$, and the contribution of the
spin-spin interaction in this diquark to the pentaquark mass is determined by
the coupling $(\mathcal{K}_{q^{\prime}q^{\prime\prime}})_{\bar{3}}$. The last
term in $H_{ld}$ is responsible for all the possible spin-spin interactions
between the constituents of the light diquark and doubly-heavy triquark:

$\displaystyle H_{SS}^{t-ld}$ $\displaystyle=$ $\displaystyle
2\,(\tilde{\mathcal{K}}_{cq^{\prime}})_{\bar{3}}\,(\mathbf{S}_{c}\cdot\mathbf{S}_{q^{\prime}})+2\,(\tilde{\mathcal{K}}_{qq^{\prime}})_{\bar{3}}\,(\mathbf{S}_{q}\cdot\mathbf{S}_{q^{\prime}})+2\,\tilde{\mathcal{K}}_{\bar{c}q^{\prime}}\,(\mathbf{S}_{\bar{c}}\cdot\mathbf{S}_{q^{\prime}})$
(5) $\displaystyle+$ $\displaystyle
2\,(\tilde{\mathcal{K}}_{cq^{\prime\prime}})_{\bar{3}}\,(\mathbf{S}_{c}\cdot\mathbf{S}_{q^{\prime\prime}})+2\,(\tilde{\mathcal{K}}_{qq^{\prime\prime}})_{\bar{3}}\,(\mathbf{S}_{q}\cdot\mathbf{S}_{q^{\prime\prime}})+2\,\tilde{\mathcal{K}}_{\bar{c}q^{\prime\prime}}\,(\mathbf{S}_{\bar{c}}\cdot\mathbf{S}_{q^{\prime\prime}}).$

Here, the coefficients with the tilde differ from the ones introduced above as
the former show the strengths of the spin-spin interactions inside the compact
objects like diquarks and triquark while the later ones determine the
strengths between the constituents of two objects, the heavy triquark and
light diquark, and are strongly suppressed. The rationale of this is specific
to the diquark model, in which the hadronic size of the diquarks is much
smaller or compared to the overall size of the multiquark hadrons. Thus, only
local spin-spin interactions (within a diquark or triquark) are allowed. This
then accounts for all possible spin-spin interactions and completes the
content of the effective Hamiltonian (2) for the masses of the ground-state
pentaquarks. A comment is in order: Despite the fact that this picture, which
is physically more motivated, differs form the one used in Ali:2016dkf ;
Ali:2017ebb , for the analysis and predictions of the ground-state
pentaquarks, actually all the terms in the effective Hamiltonian (2) are the
same.

The general form of the effective Hamiltonian for the orbitally-excited
pentaquark mass spectrum can be written as follows:

$H=H^{(L=0)}+H_{L}+H_{T}.$ (6)

In addition to the spin-spin interactions introduced for the ground-state
pentaquarks described above, the extended effective Hamiltonian (6) includes
terms explicitly dependent on the internal orbital angular momentum $L_{t}$ of
the hidden-charm triquark and orbital momentum $L_{ld}$ of the light diquark
relative to the triquark system. The corresponding Hamiltonian is called
$H_{L}$ in (6). The terms relevant for the tensor interactions in each
subsystem specified above are subsumed in $H_{T}$ in (6). As already stated,
we assume that the triquark state is an $S$-wave (i. e., $L_{t}=0$). Thus,
$L=L_{ld}$. Inclusion of the triquark orbital angular momentum $L_{t}$ will be
commented later on. The terms in $H_{L}$ which contain the orbital angular
momentum operator $\mathbf{L}$ are as follows:

$H_{L}=2A_{t}\left(\mathbf{S}_{t}\cdot\mathbf{L}\right)+2A_{ld}\left(\mathbf{S}_{ld}\cdot\mathbf{L}\right)+\frac{1}{2}\,B\,\mathbf{L}^{2},$
(7)

where quantities $A_{t}$, $A_{ld}$, and $B$ parametrize the strengths of the
triquark spin-orbit, light-diquark spin-orbit and orbital momentum couplings,
respectively. The last term in (6) represents the tensor interaction among the
heavy triquark and light diquark:

$H_{T}=b\left[3\,\frac{({\bf S}_{t}\cdot{\bf R})\,({\bf S}_{ld}\cdot{\bf
R})}{R^{2}}-({\bf S}_{t}\cdot{\bf S}_{ld})\right],$ (8)

where ${\bf R}$ determines the position of the light diquark relative to the
heavy triquark and $b$ is the strength of the tensor interaction.

The inclusion of a possible excitation in the triquark system ($L_{t}\neq 0$)
results into additional terms in the effective Hamiltonian:

$\Delta H=\Delta H_{L}+\Delta H_{T}.$ (9)

The structure of $\Delta H_{L}$ which includes the angular momentum operator
$\mathbf{L}_{t}$ in the triquark system is similar to (7):

$\Delta
H_{L}=2A_{\bar{c}}\left(\mathbf{S}_{\bar{c}}\cdot\mathbf{L}_{t}\right)+2A_{hd}\left(\mathbf{S}_{hd}\cdot\mathbf{L}_{t}\right)+\frac{1}{2}\,B_{t}\,\mathbf{L}_{t}^{2},$
(10)

where quantities $A_{\bar{c}}$, $A_{hd}$, and $B_{t}$ parametrize the
strengths of the charm antiquark and diquark spin-orbit and triquark orbital
momentum couplings, respectively. As the triquark contains charm quark and
antiquark, one expects a suppression of $A_{\bar{c}}$, $A_{hd}$, and $B_{t}$
by the $c$-quark mass in comparison with the parameters $A_{t}$, $A_{ld}$, and
$B$ entering (7). In the numerical calculations, we set them to zero.

The second term in (9) represents the tensor interaction among the charm
diquark and antiquark in the triquark:

$\Delta H_{T}=b_{t}\left[3\,\frac{({\bf S}_{\bar{c}}\cdot{\bf r})\,({\bf
S}_{hd}\cdot{\bf r})}{r^{2}}-({\bf S}_{\bar{c}}\cdot{\bf S}_{hd})\right],$
(11)

where ${\bf r}$ determines the position of the charm diquark relative to the
charm antiquark and $b_{t}$ is the strength of this interaction. In the
phenomenological analysis of the hidden-charm orbitally excited
$Y$-tetraquarks, it was shown Ali:2017wsf  that a similar strength entering
the tensor interaction between the charm diquark and antidiquark is consistent
with zero. Following Ali:2017wsf , it is reasonable to assume that $b_{t}$ in
(11) is also small, and hence we neglect this interaction in the analysis
reported below by setting $\Delta H=0$.

Returning to the discussion of the tensor contribution, the expression in eq.
(8) is of the following general form:

$\displaystyle Q({\bf S}_{1},{\bf S}_{2})$ $\displaystyle=$ $\displaystyle
3\left({\bf S}_{1}\cdot{\bf n}\right)\left({\bf S}_{2}\cdot{\bf
n}\right)-\left({\bf S}_{1}\cdot{\bf S}_{2}\right)=3S_{1i}S_{2j}\,N_{ij},$
(12)

where ${\bf S}_{1}$ and ${\bf S}_{2}$ are the spins of two particles and ${\bf
n}={\bf R}/R$ is the unit vector in the direction of ${\bf R}$. This notation
was used by us previously Ali:2017wsf  in the analysis of the orbitally
excited ($L=1$) $\Omega_{c}$-baryons and $Y$-tetraquarks. In both cases ${\bf
S}_{1}$ was identified with the spin of the diquark, ${\bf S}_{Q}$, and ${\bf
S}_{2}$ is either the spin of the charm quark, ${\bf S}_{c}$, in the
$\Omega_{c}$-baryon, or the spin of the antidiquark, ${\bf S}_{\bar{Q}}$, in
the $Y$-tetraquark.

The scalar operator $Q({\bf S}_{1},{\bf S}_{2})$ in eq. (12) is expressed as
the convolution of spins with the tensor operator:

$N_{ij}=n_{i}n_{j}-\frac{1}{3}\,\delta_{ij}.$ (13)

For further applications, we need the matrix elements of the operator $N_{ij}$
between the states with the same fixed value $L$ of the orbital angular
momentum operator ${\bf L}$ which can be obtained with the help of the
identity from Landau and Lifshitz Landau:1977 :

$\langle
N_{ij}\rangle=-\frac{1}{(2L-1)(2L+3)}\left[L_{i}L_{j}+L_{j}L_{i}-\frac{2}{3}\,\delta_{ij}\,L\left(L+1\right)\right].$
(14)

It is obvious that this matrix element is trivial when the two-particle
quantum state is in the $S$-wave, $L=0$.

In terms of $Q({\bf S}_{1},{\bf S}_{2})$, eq. (8) for $L=1$ can be expressed
as follows

$H_{T}=b\,\langle Q({\bf S}_{t},{\bf
S}_{ld})\rangle=-\frac{3}{5}\,b\left\langle\left({\bf L}\cdot{\bf
S}_{t}\right)\left({\bf L}\cdot{\bf S}_{ld}\right)+\left({\bf L}\cdot{\bf
S}_{ld}\right)\left({\bf L}\cdot{\bf S}_{t}\right)-\frac{4}{3}\left({\bf
S}_{t}\cdot{\bf S}_{ld}\right)\right\rangle.$ (15)

and similarly $\Delta H_{T}=b_{t}\,Q({\bf S}_{hd},{\bf S}_{\bar{c}})$, if
$b_{t}\neq 0$. In eq. (15), the mixed-spin $Q$-operator can be reexpressed in
terms of the single spin ones:

$Q({\bf S}_{t},{\bf S}_{ld})=Q({\bf S}_{ld},{\bf
S}_{t})=\frac{1}{2}\left[Q({\bf S},{\bf S})-Q({\bf S}_{t},{\bf S}_{t})-Q({\bf
S}_{ld},{\bf S}_{ld})\right],$ (16)

where ${\bf S}={\bf S}_{t}+{\bf S}_{ld}$. To calculate the single-spin
$Q$-operators inside the brackets, it is necessary to account for appropriate
commutation relations of the components of ${\bf L}$ and ${\bf S}_{X}$, where
${\bf S}_{X}={\bf S}_{t}$, ${\bf S}_{ld}$, and ${\bf S}$ are the spins of the
triquark, light-diquark, and the total spin of the pentaquark, respectively.
Setting $L=1$, one finds easily Karliner:2017kfm :

$\left\langle Q({\bf S}_{X},{\bf
S}_{X})\right\rangle=-\frac{3}{5}\,\left\langle 2\left({\bf L}\cdot{\bf
S}_{X}\right)^{2}+\left({\bf L}\cdot{\bf S}_{X}\right)-\frac{4}{3}\left({\bf
S}_{X}\cdot{\bf S}_{X}\right)\right\rangle.$ (17)

The matrix elements can be computed directly by applying the operators
$\left({\bf L}\cdot{\bf S}_{X}\right)$ to the products of states corresponding
to the individual spins and angular momenta. More effectively, one can use
Wigner $6j$-symbols, as is customary for analogous cases in atomic and nuclear
physics. An explicit example of their application to the analysis of the
orbitally-excited $\Omega_{c}$-baryons and $Y$-tetraquarks is presented in the
appendix of ref. Ali:2017wsf . So, to get the mass predictions one needs to
specify a concrete basis of the orbitally-excited pentaquark states in which
the mass predictions can be done.

##  4 Mass formulae for pentaquark spectrum

###  4.1 $S$-wave pentaquarks

With the basis vectors of the pentaquark states chosen, one can derive
analytical expressions for calculating the pentaquark spectrum. They are the
matrix elements of the effective Hamiltonian presented above in sec. 3. Note
that this is simpler for the Model II by Maiani et al Maiani:2014aja , but
becomes more involved in the Model I Maiani:2004vq , where additional mixings
between the spins of (anti)quarks in compact shells are included. As the later
couplings are suppressed in comparison with the spin-spin interactions inside
the shells, we neglect the later mixings and restrict ourselves with the Model
II.

Restricting ourselves to the light $u$\- and $d$-quarks, and assuming isospin
symmetry, the universal contribution entering all the pentaquark states is
defined as $M_{0}$, which is the sum of the constituent masses of the heavy
and light diquarks and charm antiquark:

$M_{0}\equiv m_{hd}+m_{ld}+m_{c}.$ (18)

Apart from this, there are two terms in the effective Hamiltonian explicitly
related with the spins of the diquarks [see eqs. (3) and (4)]:

$\displaystyle{}_{J}\langle
S_{hd},S_{t},L_{t};S_{ld},L_{ld};S,L|\,2\,(\mathcal{K}_{cq})_{\bar{3}}\,(\mathbf{S}_{c}\cdot\mathbf{S}_{q})\,|S_{hd},S_{t},L_{t};S_{ld},L_{ld};S,L\rangle_{J}$
(19) $\displaystyle\hskip
48.36967pt=(\mathcal{K}_{cq})_{\bar{3}}\left[S_{hd}\left(S_{hd}+1\right)-\frac{3}{2}\right]=\frac{1}{2}\,(\mathcal{K}_{cq})_{\bar{3}}\times\left\\{\begin{array}[]{cc}-3,&S_{hd}=0,\\\
1,&S_{hd}=1,\end{array}\right.$ (22) $\displaystyle{}_{J}\langle
S_{hd},S_{t},L_{t};S_{ld},L_{ld};S,L|\,2\,(\mathcal{K}_{q^{\prime}q^{\prime\prime}})_{\bar{3}}\,(\mathbf{S}_{q^{\prime}}\cdot\mathbf{S}_{q^{\prime\prime}})\,|S_{hd},S_{t},L_{t};S_{ld},L_{ld};S,L\rangle_{J}$
(23) $\displaystyle\hskip
48.36967pt=(\mathcal{K}_{q^{\prime}q^{\prime\prime}})_{\bar{3}}\left[S_{ld}\left(S_{ld}+1\right)-\frac{3}{2}\right]=\frac{1}{2}\,(\mathcal{K}_{q^{\prime}q^{\prime\prime}})_{\bar{3}}\times\left\\{\begin{array}[]{cc}-3,&S_{ld}=0,\\\
1,&S_{ld}=1.\end{array}\right.$ (26)

The two terms left in eq. (3) are responsible for the contributions of the
spin-spin interactions between the charm antiquark and the quarks inside the
heavy diquark, which together form the triquark state. To calculate their
impact, it is convenient to use the Wigner $6j$-symbols, which allows us to
describe a recoupling between three angular momenta, say $j_{1}$, $j_{2}$ and
$j_{3}$, which are combined into the state with the total momentum $J$.
Following the angular momentum sum rules in quantum mechanics, one combines
two of the three momenta first and then adds the third momentum. Thus, there
are three possibilities: in the the first, the momenta $j_{1}$ and $j_{2}$
couple together in the state with the angular momentum $j_{12}$, which after
combining with $j_{3}$ results in the state with total angular momentum $J$;
in the second, the momenta $j_{1}$ and $j_{3}$ produce the state with the
momentum $j_{13}$ and it is combined with $j_{2}$ to get the state with the
momentum $J$; and in the third, the momenta $j_{2}$ and $j_{3}$ produce the
state with the momentum $j_{23}$ which then combines with $j_{1}$ to obtain
the state with the momentum $J$. Each way of summing up the momenta results in
its own set of basis vectors. If, for example, the momentum $j_{2}$ is coupled
initially to $j_{1}$ and one wants to recouple it to $j_{3}$, there are two
extra momenta $j_{12}$ and $j_{23}$, each of which has its own set of basis
vectors, $|(j_{1},\,j_{2})_{j_{12}},\,j_{3};\,J\rangle$ and
$|j_{1},\,(j_{2},\,j_{3})_{j_{23}};\,J\rangle$. The Wigner $6j$-symbols
describe the transformation from one basis to the other as follows
Edmonds:1957 :

$|j_{1},\,(j_{2},\,j_{3})_{j_{23}};\,J\rangle=\sum_{j_{12}}(-1)^{j_{1}+j_{2}+j_{3}+J}\sqrt{(2j_{12}+1)\,(2j_{23}+1)}\left\\{\begin{array}[]{rrr}j_{1}&j_{2}&j_{12}\\\
j_{3}&J&j_{23}\end{array}\right\\}|(j_{1},\,j_{2})_{j_{12}},\,j_{3};\,J\rangle.$
(27)

Here, Wigner $6j$-symbols are represented by the curly brackets. To find the
matrix elements of the spin-spin operators
$2\,\mathcal{K}_{\bar{c}q}\left(\mathbf{S}_{\bar{c}}\cdot\mathbf{S}_{q}\right)$
and
$2\,\mathcal{K}_{\bar{c}c}\left(\mathbf{S}_{\bar{c}}\cdot\mathbf{S}_{c}\right)$
in eq. (3) with the help of the Wigner $6j$-symbols, it is convenient to
rewrite the pentaquark vector state
$|S_{hd},S_{t},L_{t};S_{ld},L_{ld};S,L\rangle_{J}$ as
$|S_{\bar{c}},\,(S_{q},\,S_{c})_{S_{hd}};\,S_{t}\rangle$ and
$|S_{\bar{c}},\,(S_{c},\,S_{q})_{S_{hd}};\,S_{t}\rangle$, respectively, (here,
all the unnecessary fixed quantum numbers are omitted from the vector and the
charm antiquark spin, $S_{\bar{c}}$, is shown explicitly) and make the angular
momenta recoupling as in eq. (27):

$\displaystyle|S_{\bar{c}},\,(S_{q},\,S_{c})_{S_{hd}};\,S_{t}\rangle=\sum_{S_{\bar{c}q}}(-1)^{S_{\bar{c}}+S_{q}+S_{c}+S_{t}}\sqrt{(2S_{\bar{c}q}+1)\,(2S_{hd}+1)}$
$\displaystyle\hskip
82.51299pt\times\left\\{\begin{array}[]{rrr}S_{\bar{c}}&S_{q}&S_{\bar{c}q}\\\
S_{c}&S_{t}&S_{hd}\end{array}\right\\}|(S_{\bar{c}},\,S_{q})_{S_{\bar{c}q}},\,S_{c};\,S_{t}\rangle,$
(30)
$\displaystyle|S_{\bar{c}},\,(S_{c},\,S_{q})_{S_{hd}};\,S_{t}\rangle=\sum_{S_{\bar{c}c}}(-1)^{S_{\bar{c}}+S_{q}+S_{c}+S_{t}}\sqrt{(2S_{\bar{c}c}+1)\,(2S_{hd}+1)}$
$\displaystyle\hskip
82.51299pt\times\left\\{\begin{array}[]{rrr}S_{\bar{c}}&S_{c}&S_{\bar{c}c}\\\
S_{q}&S_{t}&S_{hd}\end{array}\right\\}|(S_{\bar{c}},\,S_{c})_{S_{\bar{c}c}},\,S_{q};\,S_{t}\rangle.$
(33)

The required matrix elements can be written in the form:

$\displaystyle{}_{J}\langle
S_{hd}^{\prime},S_{t},L_{t};S_{ld},L_{ld};S,L|\,2\,\mathcal{K}_{\bar{c}q}\,(\mathbf{S}_{\bar{c}}\cdot\mathbf{S}_{q})\,|S_{hd},S_{t},L_{t};S_{ld},L_{ld};S,L\rangle_{J}$
$\displaystyle\hskip
19.91692pt=\mathcal{K}_{\bar{c}q}\,(-1)^{2S_{t}+1}\sqrt{(2S_{hd}+1)\,(2S_{hd}^{\prime}+1)}\,\sum_{S_{\bar{c}q}}\left(2S_{\bar{c}q}+1\right)\left[S_{\bar{c}q}\left(S_{\bar{c}q}+1\right)-\frac{3}{2}\right]$
$\displaystyle\hskip
19.91692pt\times\left\\{\begin{array}[]{rrr}1/2&1/2&S_{\bar{c}q}\\\
1/2&S_{t}&S_{hd}\end{array}\right\\}\left\\{\begin{array}[]{rrr}1/2&1/2&S_{\bar{c}q}\\\
1/2&S_{t}&S_{hd}^{\prime}\end{array}\right\\},$ (38)
$\displaystyle{}_{J}\langle
S_{hd}^{\prime},S_{t},L_{t};S_{ld},L_{ld};S,L|\,2\,\mathcal{K}_{\bar{c}c}\,(\mathbf{S}_{\bar{c}}\cdot\mathbf{S}_{c})\,|S_{hd},S_{t},L_{t};S_{ld},L_{ld};S,L\rangle_{J}$
$\displaystyle\hskip
19.91692pt=\mathcal{K}_{\bar{c}c}\,(-1)^{2S_{t}+1}\sqrt{(2S_{hd}+1)\,(2S_{hd}^{\prime}+1)}\,\sum_{S_{\bar{c}c}}\left(2S_{\bar{c}c}+1\right)\left[S_{\bar{c}c}\left(S_{\bar{c}c}+1\right)-\frac{3}{2}\right]$
$\displaystyle\hskip
19.91692pt\times\left\\{\begin{array}[]{rrr}1/2&1/2&S_{\bar{c}c}\\\
1/2&S_{t}&S_{hd}\end{array}\right\\}\left\\{\begin{array}[]{rrr}1/2&1/2&S_{\bar{c}c}\\\
1/2&S_{t}&S_{hd}^{\prime}\end{array}\right\\},$ (43)

where spins of all the quarks are replaced by their numerical values,
$S_{\bar{c}}=S_{c}=S_{q}=1/2$. Up to the factors $\mathcal{K}_{\bar{c}q}$ and
$\mathcal{K}_{\bar{c}c}$, both equations coincide, so the contribution of
these two terms entering the effective Hamiltonian to the mass formulae is
proportional to the sum of the coupling strengths,
$\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}$.

The matrix elements of the operators in eq. (5) have in general six different
coupling strengths. For the pentaquark having the light $u$\- and $d$-quarks
only, isospin symmetry results into relations among the couplings. This
simplifies the spin-spin interaction (5) substantially. For the pentaquarks
having one or more strange quarks, some of the relations no longer hold, as
$SU(3)_{F}$-symmetry is broken. Details of the matrix element calculation in
this case are presented in appendix A. This contribution will be neglected in
the numerical analysis as we are working in the Model II Maiani:2014aja .

We apply the above formalism for the calculation of the mass spectrum
beginning from the pentaquark states from table 3. From three states presented
there, two states with $J^{P}=1/2^{-}$ mix due to the spin-spin interaction of
the charm antiquark and the heavy diquark, and the third one with
$J^{P}=3/2^{-}$ remains unmixed. For the later state, the mass $m^{S0}_{3}$
(here, the superscript denotes the $S$-wave pentaquark with the “good” light
diquark, $S_{ld}=0$) is the average of the effective Hamiltonian over this
state:

$m^{S0}_{3}=M_{0}+\frac{1}{2}\,(\mathcal{K}_{cq})_{\bar{3}}-\frac{3}{2}\,(\mathcal{K}_{q^{\prime}q^{\prime\prime}})_{\bar{3}}+\frac{1}{2}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right),$
(44)

where $M_{0}$ is defined in (18). The former two states with $J^{P}=1/2^{-}$,
after sandwiching the effective Hamiltonian, yield the following $(2\times 2)$
mass matrix:

$M^{S0}_{J=1/2}=M_{0}-\frac{1}{2}\,(\mathcal{K}_{cq})_{\bar{3}}-\frac{3}{2}\,(\mathcal{K}_{q^{\prime}q^{\prime\prime}})_{\bar{3}}-(\mathcal{K}_{cq})_{\bar{3}}\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)+\frac{1}{2}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left(\begin{array}[]{cc}0&\sqrt{3}\\\
\sqrt{3}&-2\end{array}\right).$ (45)

Diagonalizing this matrix yields the masses of the other two states:

$\displaystyle
m^{S0}_{1}=M_{0}-\frac{1}{4}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left[2+r_{hd}+3r_{ld}+2\sqrt{3+(1-r_{hd})^{2}}\right],$
(46) $\displaystyle
m^{S0}_{2}=M_{0}-\frac{1}{4}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left[2+r_{hd}+3r_{ld}-2\sqrt{3+(1-r_{hd})^{2}}\right],$
(47)

where two ratios $r_{hd}$ and $r_{ld}$ of the couplings are defined as:

$r_{hd}\equiv\frac{2(\mathcal{K}_{cq})_{\bar{3}}}{\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}},\qquad
r_{ld}\equiv\frac{2(\mathcal{K}_{q^{\prime}q^{\prime\prime}})_{\bar{3}}}{\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}}.$
(48)

The averaged mass of the states with masses given in (46) and (47) has the
following value:

$\bar{m}^{S0}_{12}=\frac{1}{2}\left[m^{S0}_{1}+m^{S0}_{2}\right]=M_{0}-\frac{1}{4}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left(2+r_{hd}+3r_{ld}\right),$
(49)

and the mass of the third state lies above by an amount

$\Delta
m^{S0}_{3,\overline{12}}=m^{S0}_{3}-\bar{m}^{S0}_{12}=\frac{1}{2}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left(2+r_{hd}\right).$
(50)

We continue calculations of the mass spectrum taking the $S$-wave pentaquark
states with the “bad” light diquark from table 3. From the seven states
presented there, two states with $J^{P}=1/2^{-}$ and the other two with
$J^{P}=3/2^{-}$, both pairs having the triquark spin $S_{t}=1/2$, mix due to
the spin-spin interaction of the charm antiquark and the heavy diquark. The
other three states with the triquark spin $S_{t}=3/2$ remain unmixed due to
this interaction but can mix through the spin-spin interactions between
(anti)quarks in the two separated shells — the heavy triquark and light
diquark. As mentioned earlier, such types of spin-spin interactions are
suppressed and neglected in this analysis. For the later three states,
enumerated according to their entries in table 3, the masses $m^{S1}_{5,6,7}$
are the averages of the effective Hamiltonian over the states considered
(here, the superscript denotes the $S$-wave pentaquark with the “bad” light
diquark, $S_{ld}=1$):

$\displaystyle
m^{S1}_{5,6,7}=M_{0}+\frac{1}{2}\,(\mathcal{K}_{cq})_{\bar{3}}+\frac{1}{2}\,(\mathcal{K}_{q^{\prime}q^{\prime\prime}})_{\bar{3}}+\frac{1}{2}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right).$
(51)

So, one can see that taking into account only the spin-spin interactions
inside the light diquark and heavy triquark, these states are degenerated in
mass.

The pairs of states with $J^{P}=1/2^{-}$ and $J^{P}=3/2^{-}$, after
sandwiching the effective Hamiltonian, yield identical $(2\times 2)$ mass
matrices:

$M^{S1}_{J=1/2,3/2}=M_{0}-\frac{1}{2}\,(\mathcal{K}_{cq})_{\bar{3}}+\frac{1}{2}\,(\mathcal{K}_{q^{\prime}q^{\prime\prime}})_{\bar{3}}-(\mathcal{K}_{cq})_{\bar{3}}\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)+\frac{1}{2}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left(\begin{array}[]{cc}0&\sqrt{3}\\\
\sqrt{3}&-2\end{array}\right).$ (52)

Diagonalizing the matrix, the states, being twice degenerate, have the masses:

$\displaystyle
m^{S1}_{1,3}=M_{0}-\frac{1}{4}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left[2+r_{hd}-r_{ld}+2\sqrt{3+(1-r_{hd})^{2}}\right],$
(53) $\displaystyle
m^{S1}_{2,4}=M_{0}-\frac{1}{4}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left[2+r_{hd}-r_{ld}-2\sqrt{3+(1-r_{hd})^{2}}\right],$
(54)

where $r_{hd}$ and $r_{ld}$ are defined in eq. (48). The averaged masses of
the states with masses in (53) and (54) have the following value:

$\displaystyle\bar{m}^{S1}_{12}=\frac{1}{2}\left[m^{S1}_{1}+m^{S1}_{2}\right]=M_{0}-\frac{1}{4}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left(2+r_{hd}-r_{ld}\right),$
(55)
$\displaystyle\bar{m}^{S1}_{34}=\frac{1}{2}\left[m^{S1}_{3}+m^{S1}_{4}\right]=M_{0}-\frac{1}{4}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left(2+r_{hd}-r_{ld}\right),$
(56)

and the mass gap between these average values and the mass $\bar{m}^{S0}_{12}$
(49) is completely determined by the strength of the spin-spin interaction in
the light diquark:

$\Delta\bar{m}^{S1,S0}_{12}=\bar{m}^{S1}_{12}-\bar{m}^{S0}_{12}=\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)r_{ld}=2(\mathcal{K}_{q^{\prime}q^{\prime\prime}})_{\bar{3}}.$
(57)

The degeneracy is lifted if the spin-spin interactions between the triquark
and the light-diquark constituents are taken into account as suggested by
Maiani et al. in the Model I Maiani:2004vq .

###  4.2 $P$-wave pentaquarks

As it was discussed earlier in sec. 3, one needs to include the terms
dependent on the internal angular momenta of the pentaquark system which can
be written as additional terms (7) and (8) in the pentaquark effective
Hamiltonian.

Let the vector states $|S_{hd},S_{t},L_{t};S_{ld},L_{ld};S,L\rangle_{J}$,
appearing in tables 5 and 5, be denoted as
$|L,\,(S_{t},\,S_{ld})_{S};\,J\rangle$ and transformed to the state
$|(L,\,S_{t})_{J_{t}},\,S_{ld};\,J\rangle$. After the transformation (27) is
applied, we get:

$|L,\,(S_{t},\,S_{ld})_{S};\,J\rangle=\sum_{J_{t}}(-1)^{S_{t}+S_{ld}+L+J}\sqrt{(2S+1)\,(2J_{t}+1)}\left\\{\begin{array}[]{ccc}L&S_{t}&J_{t}\\\
S_{ld}&J&S\end{array}\right\\}|(L,\,S_{t})_{J_{t}},\,S_{ld};\,J\rangle.$ (58)

The required matrix elements can be written in the form:

$\displaystyle{}_{J}\langle
S_{hd},S_{t},L_{t};S_{ld},L_{ld};S^{\prime},L|\,2\,A_{t}\,(\mathbf{L}\cdot\mathbf{S}_{t})\,|S_{hd},S_{t},L_{t};S_{ld},L_{ld};S,L\rangle_{J}$
$\displaystyle\hskip
48.36967pt=A_{t}\,(-1)^{2S_{t}+2J}\sqrt{(2S+1)\,(2S^{\prime}+1)}\,\sum_{J_{t}}\left(2J_{t}+1\right)$
$\displaystyle\hskip
48.36967pt\times\left[J_{t}\left(J_{t}+1\right)-S_{t}\left(S_{t}+1\right)-2\right]\left\\{\begin{array}[]{ccc}1&S_{t}&J_{t}\\\
S_{ld}&J&S\end{array}\right\\}\left\\{\begin{array}[]{ccc}1&S_{t}&J_{t}\\\
S_{ld}&J&S^{\prime}\end{array}\right\\},$ (63)

where the value of the orbital angular momentum, $L=1$, is substituted.

Next, we denote the state $|S_{hd},S_{t},L_{t};S_{ld},L_{ld};S,L\rangle_{J}$
as $|L,\,(S_{ld},\,S_{t})_{S};\,J\rangle$ and transform it to the state
$|(L,\,S_{ld})_{J_{ld}},\,S_{t};\,J\rangle$. After the transformation (27) is
applied, we get:

$|L,\,(S_{ld},\,S_{t})_{S};\,J\rangle=\sum_{J_{ld}}(-1)^{S_{t}+S_{ld}+L+J}\sqrt{(2S+1)\,(2J_{ld}+1)}\left\\{\begin{array}[]{ccc}L&S_{ld}&J_{ld}\\\
S_{t}&J&S\end{array}\right\\}|(L,\,S_{ld})_{J_{ld}},\,S_{t};\,J\rangle.$ (64)

With this, the matrix elements can be written as:

$\displaystyle{}_{J}\langle
S_{hd},S_{t},L_{t};S_{ld},L_{ld};S^{\prime},L|\,2\,A_{ld}\,(\mathbf{L}\cdot\mathbf{S}_{ld})\,|S_{hd},S_{t},L_{t};S_{ld},L_{ld};S,L\rangle_{J}$
$\displaystyle\hskip
48.36967pt=A_{ld}\,(-1)^{2S_{t}+2J}\sqrt{(2S+1)\,(2S^{\prime}+1)}\,\sum_{J_{ld}}\left(2J_{ld}+1\right)$
$\displaystyle\hskip
48.36967pt\times\left[J_{ld}\left(J_{ld}+1\right)-S_{ld}\left(S_{ld}+1\right)-2\right]\left\\{\begin{array}[]{ccc}1&S_{ld}&J_{ld}\\\
S_{t}&J&S\end{array}\right\\}\left\\{\begin{array}[]{ccc}1&S_{ld}&J_{ld}\\\
S_{t}&J&S^{\prime}\end{array}\right\\},$ (69)

where again the value of the orbital angular momentum, $L=1$, is substituted.

We calculate the mass spectrum of the $P$-states, starting with the pentaquark
states with the “good” light diquark from table 5. From the seven states
presented there, two states with $J^{P}=1/2^{+}$ and the other two with
$J^{P}=3/2^{+}$, both pairs having the triquark spin $S_{t}=1/2$, mix due to
the spin-spin interaction of the charm antiquark and heavy diquark. The other
three states with the triquark spin $S_{t}=3/2$ remain unmixed due to this
interaction but can mix through the spin-spin interactions between the
(anti)quarks entering two separated shells — the heavy triquark and light
diquark, as also discussed earlier for the $S$-wave states. As mentioned
earlier, such types of spin-spin interactions are suppressed and neglected in
this analysis. From the terms entering the angular-momentum dependent parts
(7) and (8) of the effective Hamiltonian, one obtains the spin-orbit, orbital,
and tensor contributions to matrix elements. For the three states, enumerated
as the fifth, sixth and seventh according to their rows in table 5, the masses
$m^{P0}_{5,6,7}$ are the matrix elements of the effective Hamiltonian over
these states (here, the superscript denotes the $P$-wave pentaquark with the
“good” light diquark, $S_{ld}=0$):

$\displaystyle
m^{P0}_{5}=M_{0}+\frac{1}{2}\,(\mathcal{K}_{cq})_{\bar{3}}-\frac{3}{2}\,(\mathcal{K}_{q^{\prime}q^{\prime\prime}})_{\bar{3}}+\frac{1}{2}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)+B-5A_{t},$
(70) $\displaystyle m^{P0}_{6}=m^{P0}_{5}+3A_{t},\qquad
m^{P0}_{7}=m^{P0}_{5}+8A_{t}.$

These states are non-degenerate due to the triquark spin-orbit interaction, i.
e., $A_{t}\neq 0$.

The pair of states with $J^{P}=1/2^{-}$, after sandwiching the effective
Hamiltonian, yields the following $(2\times 2)$ mass matrix:

$M^{P0}_{J=1/2}=M_{0}-\frac{1}{2}\,(\mathcal{K}_{cq})_{\bar{3}}-\frac{3}{2}\,(\mathcal{K}_{q^{\prime}q^{\prime\prime}})_{\bar{3}}-(\mathcal{K}_{cq})_{\bar{3}}\left(\begin{array}[]{rr}1&0\\\
0&-1\end{array}\right)+\frac{1}{2}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left(\begin{array}[]{cc}0&\sqrt{3}\\\
\sqrt{3}&-2\end{array}\right)+B-2A_{t},$ (71)

while the masses of the pair with $J^{P}=3/2^{-}$ are determined by the
matrix:

$M^{P0}_{J=3/2}=M_{0}-\frac{1}{2}\,(\mathcal{K}_{cq})_{\bar{3}}-\frac{3}{2}\,(\mathcal{K}_{q^{\prime}q^{\prime\prime}})_{\bar{3}}-(\mathcal{K}_{cq})_{\bar{3}}\left(\begin{array}[]{rr}1&0\\\
0&-1\end{array}\right)+\frac{1}{2}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left(\begin{array}[]{cc}0&\sqrt{3}\\\
\sqrt{3}&-2\end{array}\right)+B+A_{t}.$ (72)

Diagonalizing these matrices, we get the masses:

$\displaystyle
m^{P0}_{1}=M_{0}-\frac{1}{4}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left[2+r_{hd}+3r_{ld}+2\sqrt{3+(1-r_{hd})^{2}}\right]+B-2A_{t},$
(73) $\displaystyle
m^{P0}_{2}=M_{0}-\frac{1}{4}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left[2+r_{hd}+3r_{ld}-2\sqrt{3+(1-r_{hd})^{2}}\right]+B-2A_{t},$
(74) $\displaystyle m^{P0}_{3,4}=m^{P0}_{1,2}+3A_{t},$ (75)

where $r_{hd}$ and $r_{ld}$ are defined in eq. (48).

We continue the calculations of the mass spectrum for the pentaquark states
with the “bad” light diquark from table 5. From the eighteen states presented,
there are six states with $J^{P}=1/2^{+}$, seven with $J^{P}=3/2^{+}$, four
with $J^{P}=5/2^{+}$, and the last one with $J^{P}=7/2^{+}$. Except for the
$J^{P}=7/2^{+}$ state, all the others mix due to the spin-spin interactions,
but also because of the spin-orbit and tensor terms (7) and (8) in the
effective Hamiltonian. For the last state, enumerated as the eighteenth
according to table 5, the mass $m^{P1}_{18}$ is the average of the effective
Hamiltonian over this state (here, the superscript denotes the $P$-wave
pentaquark with the “bad” light diquark, $S_{ld}=1$):

$m^{P1}_{18}=M_{0}+\frac{1}{2}\,(\mathcal{K}_{cq})_{\bar{3}}+\frac{1}{2}\,(\mathcal{K}_{q^{\prime}q^{\prime\prime}})_{\bar{3}}+\frac{1}{2}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)+B+3A_{t}+2A_{ld}-\frac{3}{5}\,b.$
(76)

As mentioned earlier, there are four states with spin-parity $J^{P}=5/2^{+}$
which can be divided into two pairs according to their mixing mechanism: the
fifth and the tenth states in table 5 mix due to the spin-spin interaction
while for the fifteenth and the seventeenth states, their mixing is determined
by the spin-orbit and tensor interactions. We start from the first of the
mentioned two states. Their mass matrix is as follows:

$\displaystyle
M^{P1}_{J=5/2}=M_{0}-\frac{1}{2}\,(\mathcal{K}_{cq})_{\bar{3}}+\frac{1}{2}\,(\mathcal{K}_{q^{\prime}q^{\prime\prime}})_{\bar{3}}-(\mathcal{K}_{cq})_{\bar{3}}\left(\begin{array}[]{rr}1&0\\\
0&-1\end{array}\right)+\frac{1}{2}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left(\begin{array}[]{cc}0&\sqrt{3}\\\
\sqrt{3}&-2\end{array}\right)$ (81) $\displaystyle\hskip
54.06023pt+B+A_{t}+2A_{ld}-\frac{1}{5}\,b.$ (82)

Diagonalizing the matrix, we obtain the masses:

$m^{P1}_{14}=M_{0}-\frac{1}{4}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left[2+r_{hd}-r_{ld}+2\sqrt{3+(1-r_{hd})^{2}}\right]+B+A_{t}+2A_{ld}-\frac{1}{5}\,b,$
(83)
$m^{P1}_{15}=M_{0}-\frac{1}{4}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left[2+r_{hd}-r_{ld}-2\sqrt{3+(1-r_{hd})^{2}}\right]+B+A_{t}+2A_{ld}-\frac{1}{5}\,b.$
(84)

For the remaining two states, the mass matrix has the form:

$\displaystyle\tilde{M}^{P1}_{J=5/2}=M_{0}+\frac{1}{2}\,(\mathcal{K}_{cq})_{\bar{3}}+\frac{1}{2}\,(\mathcal{K}_{q^{\prime}q^{\prime\prime}})_{\bar{3}}+\frac{1}{2}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)+B$
(85) $\displaystyle\hskip
31.29802pt+\frac{1}{5}\,A_{t}\left(\begin{array}[]{cc}11&2\sqrt{21}\\\
2\sqrt{21}&-6\end{array}\right)+\frac{2}{5}\,A_{ld}\left(\begin{array}[]{cc}2&\sqrt{21}\\\
\sqrt{21}&-2\end{array}\right)-\frac{1}{250}\,b\left(\begin{array}[]{cc}584&15\sqrt{21}\\\
15\sqrt{21}&24\end{array}\right).$ (92)

So, the masses of the pentaquarks are two eigenvalues of this matrix:

$\displaystyle
m^{P1}_{16}=M_{0}+\frac{1}{4}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left[2+r_{hd}+r_{ld}\right]+B+\frac{1}{2}\,A_{t}-\frac{152}{125}\,b$
(93) $\displaystyle\hskip
48.36967pt-\frac{1}{50}\,\sqrt{\left(85A_{t}+40A_{ld}-56b\right)^{2}+21\left(20A_{t}+20A_{ld}-3b\right)^{2}},$
$\displaystyle
m^{P1}_{17}=M_{0}+\frac{1}{4}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left[2+r_{hd}+r_{ld}\right]+B+\frac{1}{2}\,A_{t}-\frac{152}{125}\,b$
(94) $\displaystyle\hskip
48.36967pt+\frac{1}{50}\,\sqrt{\left(85A_{t}+40A_{ld}-56b\right)^{2}+21\left(20A_{t}+20A_{ld}-3b\right)^{2}}.$

Further, there are seven states with the spin-parity $J^{P}=3/2^{+}$ which can
be separated according to the triquark spin $S_{t}=1/2$ (four states) and
$S_{t}=3/2$ (three states) as shown in table 5. The former four states mix
under the spin-spin, spin-orbit and tensor interactions while three states
with $S_{t}=3/2$ mix due to the spin-orbit and tensor interactions only. The
details of the mass derivation can be found in appendix B; here we present
explicit equations for the masses only:

$m^{P1}_{11,12,13}=M_{0}+\frac{1}{4}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left(2+r_{hd}+r_{ld}\right)+B-\frac{4}{3}\left(A_{t}+A_{ld}\right)-\frac{3622}{1125}\,b+\lambda_{3,2,1},$
(95)

where $\lambda_{1,2,3}$ are determined in eq. (270).

To derive the expressions for the masses of the four states with the spin-
parity $J^{P}=3/2^{+}$ and the triquark spin $S_{t}=1/2$, one needs to find
the eigenvalues of a non-diagonal symmetric $(4\times 4)$ matrix. The explicit
form of the mass matrix and the details of calculations can be found in
appendix B. The set of masses is as follows:

$m^{P1}_{7}=M_{0}+B-\frac{A_{t}}{2}-\frac{14}{15}\,b-\frac{1}{4}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left(2+r_{hd}-r_{ld}+2\sqrt{3+(1-r_{hd})^{2}}\right)-\mu^{(3/2)},$
(96)
$m^{P1}_{8}=M_{0}+B-\frac{A_{t}}{2}-\frac{14}{15}\,b-\frac{1}{4}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left(2+r_{hd}-r_{ld}+2\sqrt{3+(1-r_{hd})^{2}}\right)+\mu^{(3/2)},$
(97)
$m^{P1}_{9}=M_{0}+B-\frac{A_{t}}{2}-\frac{14}{15}\,b-\frac{1}{4}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left(2+r_{hd}-r_{ld}-2\sqrt{3+(1-r_{hd})^{2}}\right)-\mu^{(3/2)},$
(98)
$m^{P1}_{10}=M_{0}+B-\frac{A_{t}}{2}-\frac{14}{15}\,b-\frac{1}{4}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left(2+r_{hd}-r_{ld}-2\sqrt{3+(1-r_{hd})^{2}}\right)+\mu^{(3/2)},$
(99)

where

$\mu^{(3/2)}=\frac{1}{30}\,\sqrt{\left(5A_{t}+40A_{ld}-12b\right)^{2}+5\left(20A_{t}+20A_{ld}+3b\right)^{2}}.$
(100)

Finally, there are six states with the spin-parity $J^{P}=1/2^{+}$ which can
be divided according to the triquark spin $S_{t}=1/2$ (four states) and
$S_{t}=3/2$ (two states) as shown in table 5. The four states mix under the
spin-spin, spin-orbit and tensor interactions while the two states with
$S_{t}=3/2$ mix through the spin-orbit and tensor interactions only. The mass
matrix of the later two states is as follows:

$\displaystyle\tilde{M}^{P1}_{J=1/2}=M_{0}+\frac{1}{2}\,(\mathcal{K}_{cq})_{\bar{3}}+\frac{1}{2}\,(\mathcal{K}_{q^{\prime}q^{\prime\prime}})_{\bar{3}}+\frac{1}{2}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)+B-\frac{4}{3}\,b$
(101) $\displaystyle\hskip
31.29802pt-\frac{1}{3}\,A_{t}\left(\begin{array}[]{cc}10&-2\sqrt{5}\\\
-2\sqrt{5}&11\end{array}\right)+\frac{2}{3}\,A_{ld}\left(\begin{array}[]{cc}2&\sqrt{5}\\\
\sqrt{5}&-2\end{array}\right)+\frac{7}{2\sqrt{5}}\,b\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right).$ (108)

The pentaquark masses are the two eigenvalues of this matrix:

$\displaystyle
m^{P1}_{5}=M_{0}+\frac{1}{4}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left[2+r_{hd}+r_{ld}\right]+B-\frac{7}{2}\,A_{t}-\frac{4}{3}\,b$
(109) $\displaystyle\hskip
48.36967pt-\frac{1}{6\sqrt{5}}\,\sqrt{5\left(A_{t}+8A_{ld}\right)^{2}+\left(20A_{t}+20A_{ld}+21b\right)^{2}},$
$\displaystyle
m^{P1}_{6}=M_{0}+\frac{1}{4}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left[2+r_{hd}+r_{ld}\right]+B-\frac{7}{2}\,A_{t}-\frac{4}{3}\,b$
(110) $\displaystyle\hskip
48.36967pt+\frac{1}{6\sqrt{5}}\,\sqrt{5\left(A_{t}+8A_{ld}\right)^{2}+\left(20A_{t}+20A_{ld}+21b\right)^{2}}.$

For the last four states with the spin-parity $J^{P}=1/2^{+}$ and triquark
spin $S_{t}=1/2$, the mass matrix is again a non-diagonal symmetric $(4\times
4)$ matrix. It is written explicitly in appendix B and details of calculations
of its eigenvalues can be also found there. The resulting set of masses is
presented below:

$\displaystyle m^{P1}_{1}=M_{0}+B-\frac{A_{t}}{2}-3A_{ld}-\frac{31}{30}\,b$
$\displaystyle\hskip
48.36967pt-\frac{1}{4}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left(2+r_{hd}-r_{ld}+2\sqrt{3+(1-r_{hd})^{2}}\right)-\mu^{(1/2)},$
(111) $\displaystyle
m^{P1}_{2}=M_{0}+B-\frac{A_{t}}{2}-3A_{ld}-\frac{31}{30}\,b$
$\displaystyle\hskip
48.36967pt-\frac{1}{4}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left(2+r_{hd}-r_{ld}+2\sqrt{3+(1-r_{hd})^{2}}\right)+\mu^{(1/2)},$
(112) $\displaystyle
m^{P1}_{3}=M_{0}+B-\frac{A_{t}}{2}-3A_{ld}-\frac{31}{30}\,b$
$\displaystyle\hskip
48.36967pt-\frac{1}{4}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left(2+r_{hd}-r_{ld}-2\sqrt{3+(1-r_{hd})^{2}}\right)-\mu^{(1/2)},$
(113) $\displaystyle
m^{P1}_{4}=M_{0}+B-\frac{A_{t}}{2}-3A_{ld}-\frac{31}{30}\,b$
$\displaystyle\hskip
48.36967pt-\frac{1}{4}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left(2+r_{hd}-r_{ld}-2\sqrt{3+(1-r_{hd})^{2}}\right)+\mu^{(1/2)},$
(114)

where

$\mu^{(1/2)}=\frac{1}{30}\,\sqrt{25\left(7A_{t}+2A_{ld}+3b\right)^{2}+2\left(20A_{t}+20A_{ld}+21b\right)^{2}}.$
(115)

With these expressions, the analytical calculations of the pentaquark mass
spectrum in Model II is done and we present the numerical estimates of the
pentaquark masses in the next section.

##  5 Hidden-charm pentaquark mass predictions

###  5.1 Input parameters

Working within the Constituent Quark-Diquark Model Maiani:2014aja , input
parameters are the masses of the constituents, charm quark and two diquarks,
spin-spin couplings, and other parameters related with the orbital or radial
excitations. In the present paper we analyse the ground-state pentaquarks and
their first orbital excitations only. To estimate the constituent quark
masses, there are two possibilities: either extract them from the masses of
known mesons or from baryons Ali:2019roi . Quark masses obtained from the
baryon spectrum are larger by typically $50$ MeV Ali:2019roi . This can be
exemplified by the charm quark mass, which is estimated as $m_{c}^{m}=1667$
MeV from the $D$-meson spectrum, as opposed to $m_{c}^{b}=1710$ MeV from the
charm baryon masses, yielding a mass difference of $43$ MeV Ali:2019roi . In
particular, with the $m_{c}^{b}$ value as an input, predictions for the charm
baryon masses were obtained Karliner:2014gca  which differ from the
experimentally observed masses Tanabashi:2018oca  by about $10$ MeV, which may
be viewed as an error on the input charm-quark mass. We use $m_{c}^{b}$ in the
numerical analysis. Based on the same arguments, we accept $m_{q}^{b}=(362\pm
10)$ MeV and $m_{s}^{b}=(540\pm 10)$ MeV as the light and strange quark
masses, respectively Ali:2019roi .

The diquark masses are presented in table 6. In contrast to quarks, which have
spin $S_{q}=1/2$, diquarks, being composite objects, have two possible spin
configurations from which the antisymmetric one corresponding to the diquark
spin $S=0$ is energetically more favorable. Both configurations are allowed if
the flavors of the quarks are different, indicated by right brackets,
$[\ldots]$, in table 6 (for $S=0$), and by curly brackets, $\\{\ldots\\}$,
(for $S=1$). For the diquarks, having two quarks of the same flavor, only the
spin-symmetric configuration is allowed by Bose statistics, as indicated for
the case of the diquark with two $s$-quarks in table 6.

In the limit of the exact isospin symmetry, the light $u$\- and $d$-quarks are
mass-degenerate and denoted by $q$ in table 6. For the lightest diquark,
$[ud]$, and its $SU(3)_{F}$-symmetry partners, $[su]$ and $[sd]$, the masses
are taken from Karliner:2018bms  as well as the mass of the double strange
diquark, $\\{ss\\}$. The masses of the charm diquarks, $[cq]$ and $[cs]$, are
borrowed from Ali:2019roi . To get the diquark-mass error estimates, one
should compare baryon masses predicted within this model and in experiment.
Based on the analysis presented in Karliner:2014gca , in which the predicted
and measured mass differences for unflavored (i. e., having $u$\- or
$d$-quarks), strange and charm baryons do not exceed $15$ MeV, we take it as a
measure of the uncertainty in diquark masses.

Diquark content | $[ud]$ | $[sq]$ | $\\{ss\\}$ | $[cq]$ | $[cs]$
---|---|---|---|---|---
Diquark mass (MeV) | 576 | 800 | 1099 | 1976 | 2105
Table 6:  Constituent diquark masses (in MeV) used for calculating the hidden-
charm pentaquark mass spectrum. Here, $q^{(\prime)}$ denotes $u$\- and
$d$-quarks and isospin symmetry is assumed. The brackets $[\ldots]$ and
$\\{\ldots\\}$ indicate the antisymmetric ($S=0$) and symmetric ($S=1$) spin
configurations of the quarks in the diquark, respectively. (Taken from
Karliner:2018bms  for light diquarks and from Ali:2019roi  for the heavy-light
ones).

The spin-spin couplings, $\mathcal{K}_{\bar{Q}Q^{\prime}}$ and
$(\mathcal{K}_{QQ^{\prime}})_{\bar{3}}$, extracted from the spectra of mesons
and baryons, respectively, are presented in table 7. These values are taken
from Ali:2019roi  except for $(\mathcal{K}_{cc})_{\bar{3}}$, which is from
Karliner:2014gca . The factor two difference in $\mathcal{K}_{\bar{c}c}$ and
$(\mathcal{K}_{cc})_{\bar{3}}$ reflects the fact that the one-gluon exchange
assumption is used to relate these couplings. Concerning the
$(\mathcal{K}_{cq})_{\bar{3}}$-coupling, it follows from table 7 that the
value $(\mathcal{K}_{cq})_{\bar{3}}=15$ MeV extracted from the charm baryon
spectroscopy is numerically substantially smaller than all others. In fact,
this is the spin-spin interaction between the charm quark and the two quarks
from a light diquark, but not between the quarks within the charm diquark
$[cq]$, which is estimated as $(\mathcal{K}_{cq})_{\bar{3}}=67$ MeV
Maiani:2014aja ; Ali:2019roi . The later value was obtained from the
Constituent Quark-Diquark Model analysis of the hidden-charm exotic mesons in
which the underlying structure is assumed to be a charm diquark and charm
antidiquark. In our numerical estimates, we use the later value. The spin-spin
coupling obtained from the charm baryons can be used to determine
$(\tilde{\mathcal{K}}_{cq})_{\bar{3}}$ in eq. (5), associated with the spin-
spin interactions between the charm quark in the doubly-heavy triquark and
quarks from the light diquark. The same is also true for the spin-spin
couplings between the charm and strange quarks, for which the value
$(\mathcal{K}_{cs})_{\bar{3}}=2$ MeV Karliner:2017kfm ; Ali:2017wsf 222This
value differs by a factor of two from the original estimate presented in
Karliner:2017kfm ; Ali:2017wsf  due to the coupling redefinition, i. e.,
$(\mathcal{K}_{cs})_{\bar{3}}=c/2$. follows from the analysis of the narrow
orbitally-excited $\Omega_{c}^{*}$-baryons recently observed by the LHCb
Aaij:2017nav  and Belle Yelton:2017qxg  collaborations. To understand the
uncertainty of $(\mathcal{K}_{cs})_{\bar{3}}$, we performed the
$\chi^{2}$-analysis based on the masses of these baryons Aaij:2017nav ;
Yelton:2017qxg , which yields $(\mathcal{K}_{cs})_{\bar{3}}=(2.01\pm 0.20)$
MeV. The details of this analysis are in appendix C. This yields 10% error on
$(\mathcal{K}_{cs})_{\bar{3}}$, which we assume as an uncertainty on all the
spin-spin couplings.

Couplings (MeV) | $qq^{\prime}$ | $sq$ | $ss$ | $cq$ | $cs$ | $cc$
---|---|---|---|---|---|---
$\mathcal{K}_{\bar{Q}Q^{\prime}}$ | 318 | 200 | 103 | 70 | 72 | 113
$(\mathcal{K}_{QQ^{\prime}})_{\bar{3}}$ | 98 | 59 | 23 | 15 | 50 | 57
Table 7:  Spin-spin couplings, $\mathcal{K}_{\bar{Q}Q^{\prime}}$ and
$(\mathcal{K}_{QQ^{\prime}})_{\bar{3}}$, extracted from the spectra of mesons
and baryons, respectively (borrowed from Ali:2019roi  and for $cc$ from
Karliner:2014gca ). Here, $q^{(\prime)}$ denotes $u$\- and $d$-quarks and
isospin symmetry is assumed.

The pentaquark masses involve the ratios of the couplings (48) which for the
input values are evaluated as: $r_{hd}=0.73$ and $r_{ld}=1.07$. According to
the Model II, these ratios should be of order one which explicitly
demonstrates that the strengths of all possible spin-spin interactions inside
the diquarks and triquark are approximately the same.

In estimating the $P$-wave pentaquark mass spectrum, values of the couplings
in the orbital angular momentum term and the spin-orbit ones are required. In
the previous analysis Ali:2016dkf ; Ali:2017ebb , the spin-orbit coupling was
taken to be $A_{\mathcal{P}}=52$ MeV, which follows from the analysis of the
vector hidden-charm $Y$-tetraquarks. Updated analysis of these states in
Ali:2017wsf , which takes into account in addition the tensor interaction
between the diquark and antidiquark, results in the smaller value,
$a_{Y}=(22\pm 3)$ MeV. A similar analysis Karliner:2017kfm ; Ali:2017wsf ,
performed for the orbitally-excited $\Omega_{c}$-baryons, yields even smaller
values,333 The original values ($a_{1}=26.95$ MeV and $a_{2}=25.75$ MeV) are
twice the ones in the text because of different definitions in the effective
Hamiltonian. $a_{1}=(13.45\pm 0.13)$ MeV and $a_{2}=(12.94\pm 0.36)$ MeV, for
the spin-orbit interactions of the $\\{ss\\}$-diquark and charm antiquark,
respectively. The uncertainties in these values result from the
$\chi^{2}$-analysis of the $\Omega_{c}^{*}$-baryon masses Aaij:2017nav ;
Yelton:2017qxg  and the details of this analysis are also presented in
appendix C. The errors in these couplings are typically a few percent and are
substantially smaller than the relative error from the $Y$-tetraquark spectrum
which is approximately 14%.

For $P$-wave pentaquarks, the spin-orbit couplings are responsible for the
interactions of the triquark and the light diquark, denoted by $A_{t}$ and
$A_{ld}$, respectively, which are, in general, different. However, based on
the observation that the couplings of the spin-orbit interaction in
$\Omega_{c}^{*}$-baryons are close to each other, $a_{1}\simeq a_{2}\simeq 13$
MeV, one can use the same approximate relation $A_{t}\simeq A_{ld}=13$ MeV in
the pentaquark system but, as we shall show later, this is not supported by
the data on the newly observed pentaquark states Aaij:2019vzc  for the
plausible spin-parity assignment, as discussed in Ali:2019npk .

For the mass spectrum of orbitally-excited pentaquarks with the “bad” light
diquark, the tensor coupling $b$ is required. It was extracted from the
spectrum of the orbitally-excited $\Omega_{c}$-baryons and $Y$-tetraquarks,
yielding $b_{\Omega_{c}}=13.5$ MeV Karliner:2017kfm ; Ali:2017wsf 444 The
$\chi^{2}$-analysis of the $\Omega_{c}^{*}$-baryons presented in appendix C
gives slightly smaller value $b_{\Omega_{c}}=(13.30\pm 0.48)$ MeV. and
$b_{Y}=(-136\pm 6)$ MeV Ali:2017wsf , respectively. Thus, this coupling is
quite different in the charm baryons and in the hidden-charm tetraquarks and,
moreover, the central values are of opposite signs in these hadrons. Here,
more experimental input is needed. To be definite, we use $b=(13.3\pm 0.5)$
MeV, which is obtained from the orbitally-excited $\Omega_{c}$-baryons rather
than from the $Y$-tetraquarks.

The last parameter to be specified is the orbital coupling $B$, several
estimates of which are available from different hadrons. Some representative
values are: $B(\bar{c}c)=457$ MeV Ali:2017wsf , obtained from charmonia,
$B(\Omega_{c})=325$ MeV, from $\Omega_{c}$-baryons, $B(Y)=362$ MeV and
$B(Y)=505$ MeV, from the hidden-charm $Y$-tetraquarks.555 The values presented
are determined by two possible assignments of the vector tetraquarks discussed
in Ali:2017wsf . In the analysis Ali:2016dkf ; Ali:2017ebb , the coupling $B$
was derived from the experimentally observed mass of the $P_{c}(4450)^{+}$
having the (preferred) spin-parity assignment $J^{P}=5/2^{+}$. The orbital
coupling was determined from the $P_{c}(4450)^{+}$ mass to be $B(P_{c})=220$
MeV for the Model II Maiani:2014aja . With the observation of three narrow
pentaquarks in the updated LHCb data, the experimental picture is changed
drastically. Following the interpretation of these peaks as suggested in
Ali:2019npk , which assumes that two resonances are orbitally-excited states
and the lowest mass peak is the $S$-wave state, it is possible to determine
the orbital coupling $B$ phenomenologically, discussed in the next subsection.

###  5.2 Predictions for the hidden-charm unflavored pentaquarks

To get an estimate of the mass spectrum of the ground-state hidden-charm
unflavored pentaquarks, it is enough to know the diquark and charm quark
masses and spin-spin couplings, discussed in detail in the previous
subsection. For the $P$-wave pentaquark masses, the values of the orbital
angular momentum coupling $B$, the spin-orbit couplings, $A_{t}$ and $A_{ld}$,
and the tensor coupling $b$ are required. Some of these couplings can be
determined from the measured masses of the observed resonances Aaij:2019vzc .
In this subsection we consider two possible assignments, both assume that the
heavier states, $P_{c}(4440)^{+}$ and $P_{c}(4457)^{+}$, are the $P$-wave
pentaquarks, while the lowest mass $P_{c}(4312)^{+}$ can be either $S$\- or
$P$-wave state, and show that the second scenario is physically unacceptable.
Assuming that the $P_{c}(4312)^{+}$ pentaquark is an $S$-wave state, we
predict the masses and $J^{P}$ quantum numbers of yet unmeasured resonances.

We discuss first the pentaquarks $P_{c}(4440)^{+}$ and $P_{c}(4457)^{+}$. As
they replace the former narrow $P_{c}(4450)^{+}$ state with the preferred
spin-parity $J^{P}=5/2^{+}$, we tentatively assign this spin-parity to one of
the observed two states, $P_{c}(4457)^{+}$. In this case, the lighter partner,
$P_{c}(4440)^{+}$, most probably has the spin-parity $J^{P}=3/2^{+}$. In the
Model II by Maiani et al. Maiani:2014aja , the mass splitting of the two
positive-parity states is related to the spin-orbit coupling $A_{t}$ of the
triquark:

$M[P_{c}(4457)^{+}]-M[P_{c}(4440)^{+}]=\left(17^{+6.4}_{-4.5}\right)~{}\mathrm{MeV}=5A_{t},$
(116)

where the error is obtained by adding the experimental errors on the masses in
quadrature. From (116), the value of the coefficient $A_{t}$ follows
immediately:

$A_{t}=\left(3.4^{+1.3}_{-0.9}\right)~{}\mathrm{MeV}.$ (117)

It is not surprising that $A_{t}$ is found numerically small, as the doubly-
heavy triquark is almost static.

The third narrow state $P_{c}(4312)^{+}$ can have several $J^{P}$ assignments.
Identifying it with the $J^{P}=3/2^{-}$ state, one can work out the mass
difference between this state and the heavier pentaquarks, $P_{c}(4440)^{+}$
and $P_{c}(4457)^{+}$. This is determined by the orbital $B$ and the triquark
spin-orbit $A_{t}$ couplings. The strength of the latter is already known from
the $P_{c}(4440)^{+}$ and $P_{c}(4457)^{+}$ mass splitting (117), and the mass
difference, say, between $P_{c}(4312)^{+}$ and $P_{c}(4457)^{+}$, allows us to
read off the strength of the orbital interaction:

$M[P_{c}(4457)^{+}]-M[P_{c}(4312)^{+}]=\left(145.4^{+4.2}_{-7.1}\right)~{}\mathrm{MeV}=B+3A_{t}.$
(118)

With $A_{t}$ from (117), we get $B=135$ MeV. This is too small in comparison
with the strengths of the orbital excitations in other hadrons Ali:2017wsf ,
and, in particular, $B(\Omega_{c})=325$ MeV, obtained from the
$\Omega_{c}^{*}$-baryons. Moreover, the theoretically predicted masses of the
$P_{c}(4440)^{+}$ and $P_{c}(4457)^{+}$ states with the value of $B\simeq 135$
MeV are found to be $\sim 70$ MeV below the experimental values. An error on
$B$ can be obtained by performing the $\chi^{2}$-analysis of the experimental
data. Assuming that the parameters $M_{0}$, $B$, and $A_{t}$ are free
variables and all the spin-spin couplings are fixed, we obtain:

$M_{0}=\left(4333.9\pm 3.9\right)~{}\mathrm{MeV},\quad B=\left(135.2\pm
4.6\right)~{}\mathrm{MeV},\quad A_{t}=\left(3.4\pm 1.1\right)~{}\mathrm{MeV}.$
(119)

The best-fit value of $M_{0}$ comes out about $72$ MeV higher than the sum of
the diquarks’ and charm quark masses, $M_{0}=4262$ MeV.

Alternatively, assuming that the spin-spin couplings and the constituent
(quark and diquarks) masses are known, the strength of $B$ can also be
determined from the masses of $P_{c}(4440)^{+}$ and $P_{c}(4457)^{+}$ only, as
follows:

$B=\frac{1}{5}\left\\{3M[P_{c}(4440)^{+}]+2M[P_{c}(4457)^{+}]\right\\}-M_{0}-\frac{1}{4}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left(2+r_{hd}-3r_{ld}\right).$
(120)

With the values of the other parameters already assigned, $B=207$ MeV
reproduces the masses of the observed $P_{c}(4440)^{+}$ and $P_{c}(4457)^{+}$
states, shown in table 8 as the last two entries situated in the part labeled
by $S_{ld}=0$, $L=1$ and inserted into the solid boxes. This value of $B$ is
closer to the estimates in the hidden and open charm hadrons Ali:2017wsf .
With these parameters, the mass of the third pentaquark $M=4240$ MeV with
$J^{P}=3/2^{-}$, also shown in the solid box, is somewhat lower than the mass
of the observed $P_{c}(4312)^{+}$ peak, but is still in the right ball-park.

The second possibility is to assign the lowest mass state $P_{c}(4312)^{+}$
with the one having the spin-parity $J^{P}=3/2^{+}$ and mass $M=4360$ MeV, or
with $J^{P}=1/2^{+}$ and $M=4351$ MeV. Both predictions are rather close to
the observed pentaquark mass. In the case of $J^{P}=1/2^{+}$ assignment for
$P_{c}(4312)^{+}$, Model II predicts for the mass difference:

$\displaystyle
M[P_{c}(4440)^{+}]-M[P_{c}(4312)^{+}]=\left(128.4^{+4.4}_{-8.4}\right)~{}\mathrm{MeV}$
$\displaystyle\hskip
54.06023pt=\frac{1}{2}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left(2+r_{hd}-\sqrt{3+\left(1-r_{hd}\right)^{2}}\right),$
(121)

which does not depend on $B$ and $A_{t}$ but is determined by the spin-spin
couplings from the doubly-heavy triquark. If we fix the spin-spin coupling
$(\mathcal{K}_{cq})_{\bar{3}}=67$ MeV, eq. (121) can be solved analytically
with the roots:

$r_{hd}^{(1)}=0,\qquad
r_{hd}^{(2)}=\frac{2\left(3-2\rho\right)}{\rho\left(2-\rho\right)},$ (122)

where
$\rho=\\{M[P_{c}(4440)^{+}]-M[P_{c}(4312)^{+}]\\}/(\mathcal{K}_{cq})_{\bar{3}}$.
The first solution is unacceptable, as it implies
$\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\to\infty$. The second solution
in (122) yields a negative and, hence, unphysical value for $r_{hd}$, if the
ratio $\rho$ lies in the interval: $3/2<\rho<2$. With the pentaquark mass
difference in (121) and $(\mathcal{K}_{cq})_{\bar{3}}=67$ MeV, one obtains the
value $\rho\simeq 1.92$, which is in the unphysical interval (the
corresponding value of the spin-spin coupling ratio
$r_{hd}^{(2)}\simeq-10.4$). The situation does not improve if we assign
$J^{P}=3/2^{+}$ to the third state. In this case, it has a higher mass than
the $J^{P}=1/2^{+}$ pentaquark by $3A_{t}\simeq 10$ MeV, thereby the r.h.s. of
(121) gets an additional factor $-3A_{t}$. There are again two solutions as in
(122) but in $r_{hd}^{(2)}$ one should make the following replacement:
$\rho\to\tilde{\rho}=\\{M[P_{c}(4440)^{+}]-M[P_{c}(4312)^{+}]+3A_{t}\\}/(\mathcal{K}_{cq})_{\bar{3}}$.
Numerically $\tilde{\rho}\simeq 2.07$, and with this value $r_{hd}^{(2)}$
becomes positive but too large, $r_{hd}^{(2)}\simeq 16$, which is again
unacceptable. So, the assignment of the third state, $P_{c}(4312)^{+}$, with
the orbitally-excited pentaquark is not tenable in the Model II.

It should be noted that the assignment of all the three observed states with
the $P$-wave pentaquarks was also discussed in Ref. Stancu:2019qga . In
contrast to the one-gluon-exchange model, from which the effective Hamiltonian
used in our paper follows, the effective Hamiltonian in Stancu:2019qga  is
based on a confining potential and an attractive flavor-spin interaction,
resulting from the underlying chiral dynamics. Under the assumption that the
flavor-spin interaction is dominant, the three LHCb states are then identified
with the lowest-mass orbitally-excited states. Moreover, in this model, the
lowest $S$-wave state lies above the $P$-wave states. This obviously
contradicts the predictions of our model discussed above.

We remark that the assignments we have discussed satisfy the heavy-quark
symmetry, which means that the spin $S_{ld}=0$ of the light diquark in
$\Lambda_{b}$-baryon is conserved in the decay to the charged pentaquark and
$K^{-}$-meson. That is why, only pentaquarks with the “good” light diquark are
considered. As the heavy-quark symmetry is not exact, pentaquarks with
$S_{ld}=1$ can also be produced in the $\Lambda_{b}$-baryon decay, but being
suppressed by $1/m_{b}$, they are rather unlikely.

Masses of the hidden-charm unflavored pentaquarks are presented in table 8,
and compared with the results obtained in Ali:2016dkf ; Ali:2017ebb . For the
$P$-wave pentaquarks, the values of the orbital, spin-orbit and tensor
couplings are taken from table 9.

$J^{P}$ |  This work |  refs. Ali:2016dkf ; Ali:2017ebb  | $J^{P}$ |  This work |  refs. Ali:2016dkf ; Ali:2017ebb
---|---|---|---|---|---
$S_{ld}=0$, $L=0$ |  $S_{ld}=1$, $L=1$
$1/2^{-}$ | $3830\pm 34$ | $4086\pm 42$ | $1/2^{+}$ | $4144\pm 37$ | $3970\pm 50$
| $4150\pm 29$ | $4162\pm 38$ |  | $4209\pm 37$ | $4174\pm 44$
$3/2^{-}$ | $4240\pm 29$ | $4133\pm 55$ |  | $4465\pm 32$ | $4198\pm 50$
$S_{ld}=1$, $L=0$ |  | $4530\pm 32$ | $4221\pm 40$
$1/2^{-}$ | $4026\pm 31$ | $4119\pm 42$ |  | $4564\pm 33$ | $4240\pm 50$
| $4346\pm 25$ | $4166\pm 38$ |  | $4663\pm 32$ | $4319\pm 43$
| $4436\pm 25$ | $4264\pm 41$ | $3/2^{+}$ | $4187\pm 37$ |
$3/2^{-}$ | $4026\pm 31$ | $4072\pm 40$ |  | $4250\pm 37$ |
| $4346\pm 25$ | $4300\pm 40$ |  | $4508\pm 32$ |
| $4436\pm 25$ | $4342\pm 40$ |  | $4570\pm 32$ |
$5/2^{-}$ | $4436\pm 25$ | $4409\pm 40$ |  | $4511\pm 33$ |
$S_{ld}=0$, $L=1$ |  | $4566\pm 32$ |
$1/2^{+}$ | $4030\pm 39$ | $4030\pm 62$ |  | $4656\pm 32$ |
| $4351\pm 35$ | $4141\pm 44$ | $5/2^{+}$ | $4260\pm 37$ | $4450\pm 44$
| $4430\pm 35$ | $4217\pm 40$ |  | $4581\pm 32$ | $4524\pm 41$
$3/2^{+}$ | $4040\pm 39$ |  |  | $4601\pm 32$ | $4678\pm 44$
| $4361\pm 35$ |  |  | $4656\pm 32$ | $4720\pm 44$
| $4440\pm 35$ |  | $7/2^{+}$ | $4672\pm 32$ |
$5/2^{+}$ | $4457\pm 35$ | $4510\pm 57$ |  |  |
Table 8:  Masses of the hidden-charm unflavored pentaquarks (in MeV) and their
comparison with the results presented in Ali:2016dkf ; Ali:2017ebb . For the
$P$-wave pentaquarks, the values of the orbital, spin-orbit and tensor
couplings are taken from table 9.

The threshold for the observed pentaquarks in the $P_{c}^{+}\to J/\psi+p$
decay mode is $M^{\rm thr}_{J/\psi\,p}=m_{J/\psi}+m_{p}=4035.17$ MeV
Tanabashi:2018oca . With the masses given in table 8, there are two states
with the “good” light diquark ($S_{ld}=0$), the $J^{P}=1/2^{-}$ state with the
mass $M=3830$ MeV and the $J^{P}=1/2^{+}$ state with the mass $M=4030$ MeV,
which lie below the $M^{\rm thr}_{J/\psi\,p}$ threshold. Also, the third state
having $J^{P}=3/2^{+}$, with the mass $M=4040$ MeV, may also lie below $M^{\rm
thr}_{J/\psi\,p}$. There are also two states, $J^{P}=1/2^{-}$ and
$J^{P}=3/2^{-}$, with the “bad” light diquark ($S_{ld}=1$), having the
degenerate masses $M=4026$ MeV, which also lie below the $M^{\rm
thr}_{J/\psi\,p}$ threshold. One of these states (with the mass $M=3830$ MeV)
is even below the threshold for the decay $P_{c}^{+}\to\eta_{c}+p$ with
$M^{\rm thr}_{\eta_{c}\,p}=m_{\eta_{c}}+m_{p}=3922$ MeV Tanabashi:2018oca .
They will decay through the annihilation of the $c\bar{c}$-pair into light
hadrons or a lepton pair, and hence will be narrower the
$P_{c}^{+}$-resonances observed. It is conceivable that a dedicated search of
narrow states in the LHCb data with improved statistics may reveal the
existence of these pentaquark states. It is interesting to note that the
virtual $J/\psi$-meson decay will also lead to an off-shell dilepton pair.
Hence, a search for a resonance in the $p\,\mu^{+}\mu^{-}$ (as well as
$p\,e^{+}e^{-}$) may result into observing narrow structures. In addition, the
other states shown in table 8 are also reachable in the $\Lambda_{b}\to
J/\psi\,p\,K^{-}$ decay, in particular, our model predicts a peak with the
mass $M=(4150\pm 29)$ MeV below the lowest-mass observed pentaquark,
$P_{c}(4312)^{+}$, and the other peak with a mass near $4350$ MeV, being a
combination of two $J^{P}=1/2^{+}$ and $J^{P}=3/2^{+}$ pentaquarks with close
masses. The measured spectrum by the LHCb collaboration Aaij:2019vzc  has
indications for such states, though more data are needed for a definite
conclusion.

$B$ | $A_{t}$ | $A_{ld}$ | $b$
---|---|---|---
$207\pm 20$ | $3.4\pm 1.1$ | $13.5\pm 0.4$ | $13.3\pm 0.5$
Table 9:  Input values (in MeV) for the couplings required for the mass
estimations of the $P$-wave pentaquarks. For the coupling $B$, the error is
taken from (371).

###  5.3 Comparisons with the dynamical diquark model

Before presenting the strange pentaquark spectrum, we compare some of our
estimates in the non-strange sector with those obtained in the dynamical
diquark model Giron:2019bcs .

The common feature of our model and the dynamical diquark model is the
assumption of point-like diquarks in the color-antitriplet representation,
with the pentaquarks built from a diquark and a color-triplet triquark.
However, the two approaches differ in their dynamical details and in assigning
the flavors of the quarks in the diquarks and triquarks. In our model, we
assume that the light diquark $[q^{\prime}q^{\prime\prime}]_{\bar{3}}$ is
bound to the doubly-heavy triquark
$\left[\bar{c}_{\bar{3}}\left[cq\right]_{\bar{3}}\right]_{3}$ and the orbital
excitations in the pentaquark are due to the light-diquark orbital excitations
(the doubly-heavy triquark is assumed to be in the $S$-wave). In the dynamical
diquark model Giron:2019bcs , both the diquark $\left[cq\right]_{\bar{3}}$ and
the triquark
$\left[\bar{c}_{\bar{3}}\left[q^{\prime}q^{\prime\prime}\right]_{\bar{3}}\right]_{3}$
are heavy objects, and orbital excitations are between them Giron:2019bcs .
More importantly, in our analysis we take into account all sizable spin-spin,
spin-orbit, orbital and tensor interactions between the charm antiquark and
the diquarks, except for the spin-spin interaction between the constituents of
the doubly-heavy triquark and the light diquark, which are assumed to be
strongly suppressed. All these interactions are omitted in the current
formulation of the dynamical diquark model and left for further study
Giron:2019bcs . So, a lot of pentaquark states which differ in their masses in
our model (see, for example, tables 3 and 5) are mass degenerate in the
dynamical diquark model, which makes a numerical comparison rather difficult.

We also differ in the $J^{P}$ assignments of the states. In the dynamical
diquark model Giron:2019bcs , two possible assignments are suggested: The
first one identifies the two pentaquark states with larger masses,
$P_{c}(4440)^{+}$ and $P_{c}(4457)^{+}$, with the states from the first
orbitally-excited multiplet called $\Sigma_{g}^{+}(1P)$ (see table 7 in
Giron:2019bcs ). With this assignment, there is no place in the mass spectrum
for the third narrow state, $P_{c}(4312)^{+}$, observed recently by the LHCb
collaboration. This is not surprising, as also remarked in Giron:2019bcs ,
since the strength of the orbital excitation in this assignment is about $\sim
400$ MeV for the states with the principle quantum number $n=1$, in accord
with the corresponding excitation energy in charmonia. The observed mass
splitting $\sim 140$ MeV is, therefore, impossible to accommodate with these
assignment. So, the second possibility discussed in Giron:2019bcs  is to put
the lowest-mass state, $P_{c}(4312)^{+}$, in the first orbitally-excited
multiplet $\Sigma_{g}^{+}(1P)$, and in this case the two other states
$P_{c}(4440)^{+}$ and $P_{c}(4457)^{+}$ are the members of the radially
excited multiplet $\Sigma_{g}^{+}(2S)$. The suggested spin-parity of the
$P_{c}(4380)^{+}$-state is $J^{P}=5/2^{+}$ (this means that the bottom of the
$\Sigma_{g}^{+}(1P)$-multiplet is fixed to the mass of the
$P_{c}(4312)^{+}$-pentaquark) while one of the $P_{c}(4440)^{+}$\- and
$P_{c}(4457)^{+}$-states has $J^{P}=3/2^{-}$. In our model, all three observed
pentaquark states have the principle quantum number $n=1$ and have the
indicated parities: $J^{P}=3/2^{-}$ for $P_{c}(4312)^{+}$ and $J^{P}=3/2^{+}$
and $J^{P}=5/2^{+}$ for $P_{c}(4440)^{+}$ and $P_{c}(4457)^{+}$, respectively.
We are able to explain quantitatively the mass splitting in the higher mass
pentaquarks by the spin-orbit interaction, and the difference between the
lower- and higher-mass states, to a large extent, by the orbital interaction.
The strength of the latter interaction was determined as $\sim 200$ MeV, which
is smaller by $\sim 100$ MeV compared to charmonia, but still acceptable as
the size of the pentaquark and, hence, the distance from the doubly-heavy
triquark to the light diquark, is anticipated to be significantly larger than
the hadronic size of the charmonium system.

Thus, despite the similarity in the general approach about the constituents
(colored diquark and triquark) of the pentaquarks, the two model are different
in detail and lead to different spectra. Both models require estimates of the
diquark-size effects on the spectroscopy, a point also discussed later in this
paper.

$J^{P}$ |  This work |  refs. Ali:2016dkf ; Ali:2017ebb  | $J^{P}$ |  This work |  refs. Ali:2016dkf ; Ali:2017ebb
---|---|---|---|---|---
$S_{ld}=0$, $L=0$ |  $S_{ld}=1$, $L=1$
$1/2^{-}$ | $3961\pm 34$ | $4318\pm 42$ | $1/2^{+}$ | $4275\pm 37$ | $4202\pm 50$
| $4292\pm 29$ | $4392\pm 38$ |  | $4341\pm 37$ | $4406\pm 44$
$3/2^{-}$ | $4362\pm 29$ | $4365\pm 55$ |  | $4607\pm 32$ | $4430\pm 50$
$S_{ld}=1$, $L=0$ |  | $4672\pm 32$ | $4453\pm 40$
$1/2^{-}$ | $4157\pm 31$ | $4351\pm 42$ |  | $4685\pm 33$ | $4472\pm 50$
| $4488\pm 25$ | $4398\pm 38$ |  | $4784\pm 32$ | $4551\pm 43$
| $4558\pm 25$ | $4496\pm 41$ | $3/2^{+}$ | $4319\pm 37$ |
$3/2^{-}$ | $4157\pm 31$ | $4304\pm 55$ |  | $4381\pm 37$ |
| $4488\pm 25$ | $4532\pm 40$ |  | $4650\pm 32$ |
| $4558\pm 25$ | $4574\pm 40$ |  | $4712\pm 32$ |
$5/2^{-}$ | $4558\pm 25$ | $4641\pm 40$ |  | $4633\pm 33$ |
$S_{ld}=0$, $L=1$ |  | $4687\pm 32$ |
$1/2^{+}$ | $4161\pm 39$ | $4262\pm 63$ |  | $4778\pm 32$ |
| $4492\pm 35$ | $4373\pm 44$ | $5/2^{+}$ | $4391\pm 37$ | $4682\pm 57$
| $4552\pm 35$ | $4449\pm 40$ |  | $4723\pm 32$ | $4756\pm 41$
$3/2^{+}$ | $4171\pm 39$ |  |  | $4723\pm 32$ | $4910\pm 44$
| $4503\pm 35$ |  |  | $4777\pm 32$ | $4952\pm 44$
| $4562\pm 35$ |  | $7/2^{+}$ | $4794\pm 32$ |
$5/2^{+}$ | $4579\pm 35$ | $4742\pm 57$ |  |  |
Table 10:  Masses of the hidden-charm strange pentaquarks (in MeV) with a strange-heavy diquark, i. e. having the structure $(\bar{c}_{\bar{3}}[cs]_{\bar{3}}[qq^{\prime}]_{\bar{3}})$, where $q^{(\prime)}$ is $u$\- or $d$-quark, and their comparison with the results presented in Ali:2016dkf ; Ali:2017ebb . For the $P$-wave pentaquarks, the values of the orbital, spin-orbit and tensor couplings are taken from table 9.  $J^{P}$ |  This work |  refs. Ali:2016dkf ; Ali:2017ebb  | $J^{P}$ |  This work |  refs. Ali:2016dkf ; Ali:2017ebb
---|---|---|---|---|---
$S_{ld}=0$, $L=0$ |  $S_{ld}=1$, $L=1$
$1/2^{-}$ | $4112\pm 32$ | $4094\pm 44$ | $1/2^{+}$ | $4348\pm 36$ | $3929\pm 53$
| $4433\pm 26$ | $4132\pm 43$ |  | $4414\pm 36$ | $4183\pm 45$
$3/2^{-}$ | $4523\pm 26$ | $4172\pm 47$ |  | $4669\pm 32$ | $4159\pm 53$
$S_{ld}=1$, $L=0$ |  | $4735\pm 32$ | $4189\pm 44$
$1/2^{-}$ | $4230\pm 30$ | $4128\pm 44$ |  | $4768\pm 32$ | $4201\pm 53$
| $4551\pm 25$ | $4134\pm 42$ |  | $4867\pm 32$ | $4275\pm 45$
| $4641\pm 25$ | $4220\pm 43$ | $3/2^{+}$ | $4392\pm 36$ |
$3/2^{-}$ | $4230\pm 30$ | $4031\pm 43$ |  | $4454\pm 36$ |
| $4551\pm 25$ | $4262\pm 43$ |  | $4713\pm 32$ |
| $4641\pm 25$ | $4303\pm 43$ |  | $4775\pm 32$ |
$5/2^{-}$ | $4641\pm 25$ | $4370\pm 43$ |  | $4716\pm 32$ |
$S_{ld}=0$, $L=1$ |  | $4770\pm 32$ |
$1/2^{+}$ | $4312\pm 37$ | $4069\pm 56$ |  | $4861\pm 32$ |
| $4633\pm 33$ | $4149\pm 45$ | $5/2^{+}$ | $4465\pm 36$ | $4409\pm 47$
| $4713\pm 33$ | $4187\pm 44$ |  | $4786\pm 32$ | $4486\pm 45$
$3/2^{+}$ | $4323\pm 37$ |  |  | $4806\pm 32$ | $4639\pm 47$
| $4643\pm 33$ |  |  | $4860\pm 32$ | $4681\pm 47$
| $4723\pm 33$ |  | $7/2^{+}$ | $4877\pm 32$ |
$5/2^{+}$ | $4740\pm 33$ | $4549\pm 51$ |  |  |
Table 11:  Masses of the hidden-charm strange pentaquarks (in MeV) with a
strange-light diquark, i. e. having the structure
$(\bar{c}_{\bar{3}}[cq]_{\bar{3}}[sq^{\prime}]_{\bar{3}})$, where
$q^{(\prime)}$ is $u$\- or $d$-quark, and their comparison with the results
presented in Ali:2016dkf ; Ali:2017ebb . For the $P$-wave pentaquarks, the
values of the orbital, spin-orbit and tensor couplings are taken from table 9.

###  5.4 Mass predictions for the strange hidden-charm pentaquarks

Including the $s$-quark, the hidden-charm pentaquark spectrum becomes much
richer. Depending on the strange content of the pentaquarks, they can be
classified into singly-, doubly-, and triple-strange pentaquarks and their
$SU(3)_{F}$-multiplets are in one to one correspondence with the ordinary and
strange baryons Ali:2016dkf . The mass estimates given below assume mass-
degenerate $u$\- and $d$-quarks. The constituent masses of the quarks are
accepted as specified in subsec. 5.1. The diquark masses and spin-spin
couplings are not $SU(3)_{F}$-invariant, and their breaking is taken into
account (see tables 6 and 7). For the $P$-wave pentaquarks, the orbital, spin-
orbit and tensor couplings are assumed equal for all three light quarks and
are taken from table 9.

The masses of the singly-strange pentaquarks are presented in tables 10 and 11
and compared with the results of Ali:2016dkf ; Ali:2017ebb . The difference
between the entries in the two tables lies in the $s$-quark attachment either
to the heavy diquark or to the light diquark, i. e., the pentaquark structures
are $(\bar{c}_{\bar{3}}[cs]_{\bar{3}}[qq^{\prime}]_{\bar{3}})$ and
$(\bar{c}_{\bar{3}}[cq]_{\bar{3}}[sq^{\prime}]_{\bar{3}})$, respectively, with
$q^{(\prime)}$ being $u$\- or $d$-quark. One should also remember that the
states presented in tables 10 and 11 with the same set of quantum numbers:
spin-parity $J^{P}$ and isospin $I$, having strangeness $S=-1$, are
experimentally indistinguishable and, hence, mix with each other. Thus, the
mass eigenstates, resulting from this mixing, and the weak interaction
eigenstates, in which the decay of the $\Xi_{b}$-baryon would lead dominantly
to the states with the internal quantum number
$(\bar{c}_{\bar{3}}[cq]_{\bar{3}}[sq^{\prime}]_{\bar{3}})$, are different. In
the present analysis, we have not taken into account this mixing.

As for the discovery modes of such pentaquarks in weak decays of bottom
baryons, they were studied in Ali:2016dkf  in the heavy-quark symmetry limit,
assuming $SU(3)_{F}$-symmetry. Both Cabibbo-allowed $b\to c\bar{c}s$ (with
$\Delta I=0$, $\Delta S=-1$) and Cabibbo-suppressed $b\to c\bar{c}d$ (with
$\Delta I=1/2$, $\Delta S=0$) transitions were considered. Among the possible
modes of the weakly-decaying $b$-baryons, the most promising modes in
searching for the hidden-charm singly-strange pentaquarks at the LHC are:
$\Xi_{b}^{-}\to P_{\Lambda}^{0}\,K^{-}\to J/\psi\,\Lambda^{0}\,K^{-}$ and
$\Xi_{b}^{0,-}\to P_{\Sigma}^{+,0}\,K^{-}\to J/\psi\,\Sigma^{+,0}\,K^{-}$,
where the symbol in the subscript on the pentaquarks denotes its light quark
content, expressed as the corresponding baryon in the $SU(3)_{F}$-multiplet.

Assuming a similarity in the unflavored and strange pentaquark production, we
anticipate the observation of the strange partners of the known pentaquarks.
We have marked the masses of these states by enclosing them in solid boxes in
table 11. The threshold for the strange pentaquarks in the $P_{\Lambda}^{0}\to
J/\psi\,\Lambda$ decay mode is $M^{\rm
thr}_{J/\psi\,\Lambda}=m_{J/\psi}+m_{\Lambda}=4212.58$ MeV Tanabashi:2018oca .
With the masses given in table 11, there is only one state with the “good”
light diquark ($S_{ld}=0$), the $J^{P}=1/2^{-}$ with the mass $M=(4112\pm 32)$
MeV, which lies $100$ MeV below the $M^{\rm thr}_{J/\psi\,\Lambda}$ threshold.
This will result into a state narrower than one should expect due to a strong
decay, with the $c\bar{c}$-pair annihilating into light hadrons or a lepton
pair. This makes a good case for a search of narrow $\Lambda\,\mu^{+}\mu^{-}$
(and $\Lambda\,e^{+}e^{-}$) resonant structure. The thresholds in other two
modes $P_{\Sigma}^{+,0}\to J/\psi\,\Sigma^{+,0}$ are higher: $M^{\rm
thr}_{J/\psi\,\Sigma^{+}}=m_{J/\psi}+m_{\Sigma^{+}}=4286.27$ MeV and $M^{\rm
thr}_{J/\psi\,\Sigma^{0}}=m_{J/\psi}+m_{\Sigma^{0}}=4289.54$ MeV
Tanabashi:2018oca , and there are two mass-degenerate states with the mass
$M=4230$ MeV and spin-parities $J^{P}=1/2^{-}$ and $J^{P}=3/2^{-}$ which are
below these thresholds. One should remember that these states contain the
“bad” light diquark and their production in the $\Xi_{b}$-baryon decays is
suppressed by heavy quark symmetry.

$J^{P}$ |  This work |  refs. Ali:2016dkf ; Ali:2017ebb  | $J^{P}$ |  This work |  refs. Ali:2016dkf ; Ali:2017ebb
---|---|---|---|---|---
$S_{ld}=0$, $L=0$ |  $S_{ld}=1$, $L=1$
$1/2^{-}$ | $4243\pm 32$ | $4326\pm 44$ | $1/2^{+}$ | $4480\pm 37$ | $4161\pm 53$
| $4575\pm 26$ | $4364\pm 43$ |  | $4545\pm 37$ | $4415\pm 45$
$3/2^{-}$ | $4644\pm 26$ | $4404\pm 47$ |  | $4811\pm 32$ | $4391\pm 53$
$S_{ld}=1$, $L=0$ |  | $4877\pm 32$ | $4421\pm 44$
$1/2^{-}$ | $4361\pm 31$ | $4360\pm 43$ |  | $4890\pm 32$ | $4433\pm 52$
| $4693\pm 25$ | $4366\pm 43$ |  | $4989\pm 32$ | $4507\pm 45$
| $4762\pm 25$ | $4452\pm 43$ | $3/2^{+}$ | $4523\pm 37$ |
$3/2^{-}$ | $4361\pm 31$ | $4263\pm 43$ |  | $4585\pm 37$ |
| $4693\pm 25$ | $4494\pm 43$ |  | $4855\pm 32$ |
| $4762\pm 25$ | $4535\pm 43$ |  | $4917\pm 32$ |
$5/2^{-}$ | $4762\pm 25$ | $4602\pm 43$ |  | $4837\pm 32$ |
$S_{ld}=0$, $L=1$ |  | $4892\pm 32$ |
$1/2^{+}$ | $4443\pm 38$ | $4301\pm 56$ |  | $4982\pm 32$ |
| $4775\pm 33$ | $4381\pm 45$ | $5/2^{+}$ | $4596\pm 37$ | $4641\pm 47$
| $4834\pm 33$ | $4419\pm 44$ |  | $4928\pm 32$ | $4718\pm 45$
$3/2^{+}$ | $4454\pm 38$ |  |  | $4927\pm 32$ | $4871\pm 47$
| $4785\pm 33$ |  |  | $4982\pm 32$ | $4913\pm 47$
| $4844\pm 33$ |  | $7/2^{+}$ | $4998\pm 32$ |
$5/2^{+}$ | $4861\pm 33$ | $4781\pm 51$ |  |  |
Table 12:  Masses of the hidden-charm doubly-strange pentaquarks (in MeV) with both the heavy and light strange diquarks, i. e. having the structure $(\bar{c}_{\bar{3}}[cs]_{\bar{3}}[sq]_{\bar{3}})$, where $q$ is $u$\- or $d$-quark, and their comparison with the results presented in Ali:2016dkf ; Ali:2017ebb . For the $P$-wave pentaquarks, the values of the orbital, spin-orbit and tensor couplings are taken from table 9.  $J^{P}$ |  This work |  refs. Ali:2016dkf ; Ali:2017ebb  | $J^{P}$ |  This work |  refs. Ali:2016dkf ; Ali:2017ebb
---|---|---|---|---|---
$S_{ld}=1$, $L=0$ |  $S_{ld}=1$, $L=1$
$1/2^{-}$ | $4511\pm 30$ | $4598\pm 44$ | $3/2^{+}$ | $4673\pm 36$ |
| $4832\pm 25$ | $4666\pm 43$ |  | $4735\pm 36$ |
| $4922\pm 25$ | $4775\pm 44$ |  | $4994\pm 32$ |
$3/2^{-}$ | $4511\pm 30$ | $4577\pm 43$ |  | $5056\pm 32$ |
| $4832\pm 25$ | $4810\pm 43$ |  | $4997\pm 32$ |
| $4922\pm 25$ | $4851\pm 43$ |  | $5051\pm 32$ |
$5/2^{-}$ | $4922\pm 25$ | $4918\pm 47$ |  | $5142\pm 32$ |
$S_{ld}=1$, $L=1$ | $5/2^{+}$ | $4746\pm 36$ | $4954\pm 47$
$1/2^{+}$ | $4629\pm 36$ | $4474\pm 53$ |  | $5067\pm 32$ | $5033\pm 47$
| $4695\pm 36$ | $4653\pm 45$ |  | $5087\pm 32$ | $5187\pm 47$
| $4950\pm 32$ | $4707\pm 53$ |  | $5141\pm 32$ | $5228\pm 47$
| $5016\pm 32$ | $4721\pm 44$ | $7/2^{+}$ | $5158\pm 32$ |
| $5049\pm 32$ | $4748\pm 52$ |  |  |
| $5148\pm 32$ | $4830\pm 45$ |  |  |
Table 13:  Masses of the hidden-charm doubly-strange pentaquarks (in MeV) with
the doubly-strange light diquark, i. e. having the structure
$(\bar{c}_{\bar{3}}[cq]_{\bar{3}}\\{ss\\}_{\bar{3}})$, where $q$ is $u$\- or
$d$-quark and curly brackets denote the spin-1 light diquark, and their
comparison with the results presented in Ali:2016dkf ; Ali:2017ebb . For the
$P$-wave pentaquarks, the values of the orbital, spin-orbit and tensor
couplings are taken from table 9.

The masses of doubly-strange pentaquarks are presented in tables 12 and 13 and
compared with the results of Ali:2016dkf ; Ali:2017ebb . The difference
between the entries in these tables is as follows: the states from table 12
contain both the strange heavy diquark and the strange light diquark, while
those from table 13 have hidden-charm unflavored triquark and doubly-strange
light diquark, i. e., the pentaquark structures are
$(\bar{c}_{\bar{3}}[cs]_{\bar{3}}[sq]_{\bar{3}})$ and
$(\bar{c}_{\bar{3}}[cq]_{\bar{3}}\\{ss\\}_{\bar{3}})$, respectively, with $q$
being $u$\- or $d$-quark. The number of states with the doubly-strange light
diquark is smaller666 In refs. Ali:2016dkf ; Ali:2017ebb  similar analysis was
done for the hidden-charm doubly-strange pentaquarks and mass predictions for
the states with the doubly-strange “good” light diquark were presented
erroneously. The corresponding entries should be excluded from tables V and VI
(the last values in the columns entitled $\mathcal{P}_{X_{2}}$ and
$\mathcal{P}_{Y_{2}}$, respectively) in Ali:2016dkf  and from tables V (the
last values in the columns entitled $\mathcal{P}_{X_{1}}$ and
$\mathcal{P}_{X_{3}}$), VI (similar values in the columns
$\mathcal{P}_{Y_{1}}$ and $\mathcal{P}_{Y_{3}}$), and VII (the last value in
the columns entitled $\mathcal{P}_{Y_{7}}$) in Ali:2017ebb .  because such a
diquark, being flavor-symmetric, can be only the “bad” diquark with the spin
$S_{ld}=1$. So, this means that the pentaquarks with the “good” light diquark
from table 12 remain unmixed while the states with the “bad” light diquark
mix, in general, with the corresponding states (with the same set of quantum
numbers: spin-parity $J^{P}$ and isospin $I$) from table 13. In this paper, we
neglect this mixing.

Based on the analysis in Ali:2016dkf , we specify the most promising decay
mode of the $\Omega_{b}$-baryon for searching of hidden-charm doubly-strange
pentaquarks at the LHC as $\Omega_{b}^{-}\to P_{\Xi_{10}}^{0}\,K^{-}\to
J/\psi\,\Xi^{\prime 0}\,K^{-}$. The threshold for the doubly-strange
pentaquarks in the $P_{\Xi_{10}}^{0}\to J/\psi\,\Xi^{\prime 0}$ decay mode is
$M^{\rm thr}_{J/\psi\,\Xi^{\prime 0}}=m_{J/\psi}+m_{\Xi^{\prime
0}}=(4628.70\pm 0.32)$ MeV Tanabashi:2018oca . With the masses given in table
13, there are two mass degenerate states, having $J^{P}=1/2^{-}$ and
$J^{P}=3/2^{-}$, with the mass $M=4511$ MeV, which lie below the $M^{\rm
thr}_{J/\psi\,\Xi^{\prime 0}}$ threshold. Also, the third orbitally-excited
state with the spin-parity $J^{P}=1/2^{+}$ and the mass $M=4629$ MeV could
also lie below this threshold. All the other states presented in table 13 can
be produced in the $\Omega_{b}$-baryon decays and their observation will
herald the era of the as yet unknown field of doubly-strange pentaquarks.

The masses of the triple-strange pentaquarks are presented in table 14. As
these pentaquarks do not contain $u$\- or $d$-quarks, they are isospin
singlets. All these states have the doubly-strange “bad” light diquark only,
i. e., the pentaquark structure is
$(\bar{c}_{\bar{3}}[cs]_{\bar{3}}\\{ss\\}_{\bar{3}})$. They can be produced at
the LHC in the weak decays of the $\Omega_{b}$-baryons, in particular, due to
the $b\to c\bar{c}s$ transition which gives rise to the quasi-two-body decay
mode $\Omega_{b}^{-}\to P_{\Omega_{10}}^{-}\,\phi\to
J/\psi\,\Omega^{-}\,\phi$. The threshold for the triple-strange pentaquarks in
the $P_{\Omega_{10}}^{-}\to J/\psi\,\Omega^{-}$ decay is $M^{\rm
thr}_{J/\psi\,\Omega^{-}}=m_{J/\psi}+m_{\Omega^{-}}=(4769.35\pm 0.29)$ MeV
Tanabashi:2018oca . With the masses given in table 14, there are two mass
degenerate states, having $J^{P}=1/2^{-}$ and $J^{P}=3/2^{-}$, with the mass
$M=(4642\pm 31)$ MeV, which lie $\sim 100$ MeV below the $M^{\rm
thr}_{J/\psi\,\Omega^{-}}$ threshold. Also, the third orbitally-excited state
with the spin-parity $J^{P}=1/2^{+}$ and the mass $M=(4761\pm 37)$ MeV could
also lie below this threshold. All the other states presented in table 14 can
be produced in the $\Omega_{b}$-baryon decays. The other possibility is to
produce such pentaquarks promptly, but their production is highly suppressed
due to their triple-strange content.

$J^{P}$ |  Mass | $J^{P}$ |  Mass
---|---|---|---
$S_{ld}=1$, $L=0$ |  $S_{ld}=1$, $L=1$
$1/2^{-}$ | $4642\pm 31$ | $3/2^{+}$ | $4804\pm 37$
| $4974\pm 25$ |  | $4866\pm 37$
| $5043\pm 25$ |  | $5136\pm 32$
$3/2^{-}$ | $4642\pm 31$ |  | $5198\pm 32$
| $4974\pm 25$ |  | $5118\pm 32$
| $5043\pm 25$ |  | $5173\pm 32$
$5/2^{-}$ | $5043\pm 25$ |  | $5263\pm 32$
$S_{ld}=1$, $L=1$ | $5/2^{+}$ | $4877\pm 37$
$1/2^{+}$ | $4761\pm 37$ |  | $5209\pm 32$
| $4826\pm 37$ |  | $5208\pm 32$
| $5092\pm 32$ |  | $5263\pm 32$
| $5158\pm 32$ | $7/2^{+}$ | $5279\pm 32$
| $5171\pm 32$ |  |
| $5270\pm 32$ |  |
Table 14:  Masses of the hidden-charm triple-strange pentaquarks (in MeV)
containing the strange heavy diquark and doubly-strange light diquark, i. e.
having the structure $(\bar{c}_{\bar{3}}[cs]_{\bar{3}}\\{ss\\}_{\bar{3}})$,
where curly brackets denote the spin-1 light diquark. For the $P$-wave
pentaquarks, the values of the orbital, spin-orbit and tensor couplings are
taken from table 9.

##  6 Pentaquark decay widths

In the compact diquark picture, the quarks in a diquark are bound and not
free. In the present context it means that there is a barrier (or bound-state
effect) which reduces the probability of the $\bar{c}$-quark and the charm
quark in the $[uc]$-diquark to form a charmonium state. This is seen also in
the decays of the $X,\,Y,\,Z$ states, which are tetraquark candidates in the
compact diquark picture Maiani:2017kyi ; Maiani:2019cwl ; Brodsky:2014xia ;
Lebed:2017min ; Giron:2019bcs ; Esposito:2018cwh . In the case of the hidden-
charm tetraquarks, it is the radius of the tetraquark $R_{4q}\sim 1-2$ fm (or,
perhaps somewhat larger) compared to the size of the compact heavy-light
diquark, typically $R_{Qq}\sim 0.5$ fm, which acts as a barrier to form a
$J/\psi$\- or $\eta_{c}$-meson. For example, it is argued in Maiani:2017kyi ,
that the relative ratio $\lambda$ is expected to be $\lambda\equiv
R_{4q}/R_{Qq}\geq 3$. At long distance (but within the confined tetraquark
radius), the potential is attractive, which changes as the internal structure
of the diquark (and anti-diquark) gets resolved. The tunneling probability
depends on the mass of the quark, with the probability exponentially
suppressed the heavier the quark is. This can be expressed through the semi-
classical approximation of the tunneling amplitude $A_{M}\sim
e^{-\sqrt{2ME}\,R_{4q}}$, where $M$ is the quark mass and the energy $E$ is
typically 100 MeV Maiani:2017kyi . This suppresses the formation of the
charmonium states, leading to a much reduced $P_{c}\to J/\psi\,p$ decay width.
In all these cases, the size of the diquark $R_{Qq}$ plays an important role
in reducing the decay widths, though the extent of this suppression may be
different in the tetraquark and pentaquark cases. To work it out requires a
dynamical theory, which is currently not at hand. As some of the states are
orbitally excited states, anticipated in the compact diquark picture, the
decay widths are further reduced due to the angular momentum barrier. Diquark-
size effects are also present in the tetraquark and pentaquark mass estimates
in the diquark model, and they have been calculated in a particular case,
namely double-bottom tetraquark An:2018cln .

For four quarks ($q$, $q^{\prime}$, $q^{\prime\prime}$ and $c$) and the charm
antiquark, present in a pentaquark $P_{c}$, there are different alternatives
for clustering in color neutral states. The first possibility is the triquark-
diquark alternative:

$\displaystyle\Psi^{D}_{1}=\frac{1}{\sqrt{3}}\left[\frac{1}{\sqrt{2}}\,\epsilon_{ijk}\bar{c}^{i}\left[\frac{1}{\sqrt{2}}\,\epsilon^{jlm}c_{l}q_{m}\right]\right]\left[\frac{1}{\sqrt{2}}\,\epsilon^{knp}q^{\prime}_{n}q^{\prime\prime}_{p}\right]\equiv\left[\bar{c}\left[cq\right]\right]\left[q^{\prime}q^{\prime\prime}\right],$
(123)
$\displaystyle\Psi^{D}_{2}=\frac{1}{\sqrt{3}}\left[\frac{1}{\sqrt{2}}\,\epsilon_{ikj}\bar{c}^{i}\left[\frac{1}{\sqrt{2}}\,\epsilon^{knp}q^{\prime}_{n}q^{\prime\prime}_{p}\right]\right]\left[\frac{1}{\sqrt{2}}\,\epsilon^{jlm}c_{l}q_{m}\right]\equiv\left[\bar{c}\left[q^{\prime}q^{\prime\prime}\right]\right]\left[cq\right].$
(124)

From the color algebra point of view, these states are related,
$\Psi^{D}_{2}=-\Psi^{D}_{1}$, but other internal dynamical properties can be
different. The square parenthesis on the r.h.s. of the second equality in each
equation denote color antisymmetrization of the constituents. The color
connection of the quarks in $\Psi^{D}_{1}$ is assumed in the present paper,
while the $\Psi^{D}_{2}$ color structure is employed in the dynamical diquark
model of multiquark exotic hadrons Lebed:2015tna ; Lebed:2017min ;
Giron:2019bcs . The other color-singlet combinations are the meson-baryon
alternatives:

$\displaystyle\Psi^{H}_{1}=\left(\frac{1}{\sqrt{3}}\,\bar{c}^{i}c_{i}\right)\left[\frac{1}{\sqrt{6}}\,\epsilon^{jkl}q_{j}q^{\prime}_{k}q^{\prime\prime}_{l}\right]\equiv\left(\bar{c}c\right)\left[qq^{\prime}q^{\prime\prime}\right],$
(125)
$\displaystyle\Psi^{H}_{2}=\left(\frac{1}{\sqrt{3}}\,\bar{c}^{i}q_{i}\right)\left[\frac{1}{\sqrt{6}}\,\epsilon^{jkl}c_{j}q^{\prime}_{k}q^{\prime\prime}_{l}\right]\equiv\left(\bar{c}q\right)\left[cq^{\prime}q^{\prime\prime}\right],$
(126)
$\displaystyle\Psi^{H}_{3}=\left(\frac{1}{\sqrt{3}}\,\bar{c}^{i}q^{\prime}_{i}\right)\left[\frac{1}{\sqrt{6}}\,\epsilon^{jkl}c_{j}q_{k}q^{\prime\prime}_{l}\right]\equiv\left(\bar{c}q^{\prime}\right)\left[cqq^{\prime\prime}\right],$
(127)
$\displaystyle\Psi^{H}_{4}=\left(\frac{1}{\sqrt{3}}\,\bar{c}^{i}q^{\prime\prime}_{i}\right)\left[\frac{1}{\sqrt{6}}\,\epsilon^{jkl}c_{j}q_{k}q^{\prime}_{l}\right]\equiv\left(\bar{c}q^{\prime\prime}\right)\left[cqq^{\prime}\right].$
(128)

Among these four combinations, only $\Psi^{H}_{1}$ and $\Psi^{H}_{2}$ satisfy
the heavy-quark-symmetry condition, which implies that the light
$\left[q^{\prime}q^{\prime\prime}\right]$-diquark is transmitted intact,
retaining its spin quantum number, from the $b$-baryon decays to the
pentaquark.

It is possible to reduce the states in the color configurations (123) or (124)
to a linear combination of the states in the set (125)–(128). Keeping the
color of the light diquark unchanged, the convolution of two Levi-Civita
tensors entering the triquark in (123) gives the following decomposition:

$\Psi^{D}_{1}=-\frac{\sqrt{3}}{2}\left[\Psi^{H}_{1}+\Psi^{H}_{2}\right],$
(129)

as mentioned earlier. However, a color reconnection is not enough to reexpress
a pentaquark operator as a direct product of the meson and baryon operators.
As hadrons are determined by their spin-parities, one should also project
spins of the quarks and diquarks onto definite hadronic spin states. This
requires to know the Dirac structure of the pentaquark operators to undertake
the Fierz transformations in the Dirac space. This is a rather involved
problem, already for the $S$-wave pentaquarks, and requires a dedicated study,
to which we hope to return in a future publication.

We exemplify it here by considering the three $S$-wave pentaquarks from table
3 with a light “good” diquark, following similar considerations for the
hidden-charm tetraquarks Maiani:2017kyi . The heavy diquark in them, being
also in $S$-wave, can have both spins, $S_{hd}=0$ and $S_{hd}=1$. We further
simplify the problem by assuming that the charm quark and antiquark are non-
relativistic particles, keeping them as point-like objects at different
positions in the Minkowski space-time. The diquark-diquark-antiquark operators
of the first state from table 3, containing the heavy-diquark with $S_{hd}=0$,
after color reconnection (the spin structure is determined by eq. (123)) and
making the color indices explicit, are as follows:

$\displaystyle\Psi_{1}^{H(1)}(x,y,z)=\frac{1}{3}\left(\tilde{c}^{i}(x)\,\sigma_{2}\right)\left(c_{i}(y)\,\sigma_{2}\,q_{k}(y)\right)d_{0}^{k}(z),$
(130)
$\displaystyle\Psi_{2}^{H(1)}(x,y,z)=\frac{1}{3}\left(\tilde{c}^{i}(x)\,\sigma_{2}\right)\left(c_{k}(y)\,\sigma_{2}\,q_{i}(y)\right)d_{0}^{k}(z),$
(131)

where $\tilde{c}(x)$, $c(y)$ and $q(y)$ are the charm-antiquark, charm-quark
and light-quark spinors in the non-relativistic limit, respectively, $i$ and
$k$ are the color indices, the light diquark operator is denoted as
$d_{0}(z)$, and the usual notation for the product of matrices is employed.
With the help of the Fierz transform for the Pauli matrices Ali:2019roi :

$\left(\sigma_{2}\right)^{\alpha}_{\beta}\left(\sigma_{2}\right)^{\gamma}_{\delta}=\frac{1}{2}\left[\left(\sigma_{2}\right)^{\alpha}_{\delta}\left(\sigma_{2}\right)^{\gamma}_{\beta}+\left(\sigma_{2}\sigma_{a}\right)^{\alpha}_{\delta}\left(\sigma_{2}\sigma_{a}\right)^{\gamma}_{\beta}\right],$
(132)

we reduce the states (130) and (131) to the forms:

$\Psi_{1}^{H(1)}(x,y,z)=-\frac{1}{2}\left[\left(\tilde{c}(x)\,\sigma_{2}\,c(y)\right)\left(q(y)\,\sigma_{2}\,d_{0}(z)\right)+\left(\tilde{c}(x)\,\sigma_{2}\,\boldsymbol{\sigma}\,c(y)\right)\left(q(y)\,\sigma_{2}\,\boldsymbol{\sigma}\,d_{0}(z)\right)\right],$
(133)
$\Psi_{2}^{H(1)}(x,y,z)=-\frac{1}{2}\left[\left(\tilde{c}(x)\,\sigma_{2}\,q(y)\right)\left(c(y)\,\sigma_{2}\,d_{0}(z)\right)+\left(\tilde{c}(x)\,\sigma_{2}\,\boldsymbol{\sigma}\,q(y)\right)\left(c(y)\,\sigma_{2}\,\boldsymbol{\sigma}\,d_{0}(z)\right)\right],$
(134)

where the quark-antiquark and quark-diquark combinations are properly
normalized in the color space.

The transition to hadrons needs specifying the flavors of all the quarks. If,
as an example, we consider pentaquarks with the same quark content as the
observed ones, this immediately specifies $q=u$ and
$d_{0}=\left[u\,C\,\gamma_{5}\,d\right]$, being the color-antitriplet scalar
state777 In the diquark, operator $C$ is the charge-conjugation matrix. .
Neglecting the position dependence of the operators, the quark operators (133)
and (134) can be rewritten in terms of the hadrons:

$\displaystyle\Psi_{1}^{H(1)}=-\frac{i}{\sqrt{2}}\,p\left[a\,\eta_{c}+b\left(\boldsymbol{\sigma}\,\mathbf{J}\boldsymbol{/\psi}\right)\right],$
(135)
$\displaystyle\Psi_{2}^{H(1)}=-\frac{i}{\sqrt{2}}\,\Lambda_{c}^{+}\left[A\,\bar{D}^{0}+B\left(\boldsymbol{\sigma}\,\mathbf{\bar{D}^{*0}}\right)\right],$
(136)

where $A$ and $B$ ($a$ and $b$) are non-perturbative coefficients associated
with different barrier penetration amplitudes for different light-quark
(heavy-quark) spin configurations. They are equal in the limit of naive Fierz
coupling. As mentioned in subsec. 5.2, the predicted mass of this state,
$M=(3830\pm 34)$ MeV, is below the $J/\psi\,p$ and $\eta_{c}\,p$ thresholds,
so its decay to these states is forbidden by phase space, and it will decay
through the charm-anticharm pair annihilation, having a width much smaller
than the widths of the newly observed pentaquarks in table 1. The decays of
this pentaquark to the $\Lambda_{c}^{+}\,\bar{D}^{0}$ and
$\Lambda_{c}^{+}\,\bar{D}^{*0}$ final states are also forbidden by phase
space, as these channels have higher thresholds than the proton-charmonium
modes.

Along similar lines, assigning $q=s$, but keeping the light diquark $d_{0}$ as
above, we get the first state with the internal spins $S_{hd}=S_{ld}=0$ from
table 11. In terms of the physical hadrons, the quark operators (133) and
(134) can be written as follows:

$\displaystyle\Psi_{1}^{H(1)}=-\frac{i}{\sqrt{2}}\,\Lambda\left[a_{s}\,\eta_{c}+b_{s}\left(\boldsymbol{\sigma}\,\mathbf{J}\boldsymbol{/\psi}\right)\right],$
(137)
$\displaystyle\Psi_{2}^{H(1)}=-\frac{i}{\sqrt{2}}\,\Lambda_{c}^{+}\left[A_{s}\,\bar{D}_{s}^{0}+B_{s}\left(\boldsymbol{\sigma}\,\mathbf{\bar{D}_{s}^{*0}}\right)\right],$
(138)

where $A_{s}$ and $B_{s}$ ($a_{s}$ and $b_{s}$) are non-perturbative
coefficients associated with barrier penetration amplitudes. The predicted
mass of this state, $M=(4112\pm 32)$ MeV, (see table 11 in subsec. 5.4) is
below the $J/\psi\,\Lambda$ and $\eta_{c}\,\Lambda$ thresholds, so its decay
to these states is forbidden by phase space and it will decay through the
charm-anticharm pair annihilation. As the thresholds of
$\Lambda_{c}^{+}\,\bar{D}_{s}^{0}$ and $\Lambda_{c}^{+}\,\bar{D}_{s}^{*0}$ are