DESY 19-051

March 2019

Interpretation of the Narrow $J/\psi\,p$ Peaks in $\Lambda_{b}\to
J/\psi\,p\,K^{-}$ Decay
in the Compact Diquark Model
Ahmed Ali1 and Alexander Ya. Parkhomenko2

1 Deutsches Elektronen-Synchrotron DESY, D-22607 Hamburg, Germany

2 Department of Theoretical Physics, P. G. Demidov Yaroslavl State University,
Sovietskaya 14, 150003 Yaroslavl, Russia

Abstract

Recently, the LHCb Collaboration have updated their analysis of the resonant
$J/\psi\,p$ mass spectrum in the decay $\Lambda_{b}^{0}\to J/\psi\,p\,K^{-}$.
In the combined Run 1 and Run 2 LHCb data, three peaks are observed, with the
former $P_{c}(4450)^{+}$ state split into two narrow states, $P_{c}(4440)^{+}$
and $P_{c}(4457)^{+}$, having the masses $M=(4440.3\pm 1.3^{+4.1}_{-4.7})$ MeV
and $M=(4457.3\pm 0.6^{+4.1}_{-1.7})$ MeV, and decay widths $\Gamma=(20.6\pm
4.9^{+8.7}_{-10.1})$ MeV and $\Gamma=(6.4\pm 2.0^{+5.7}_{-1.9})$ MeV,
respectively. In addition, a third narrow peak, $P_{c}(4312)^{+}$, having the
mass $M=(4311.9\pm 0.7^{+6.8}_{-0.6})$ MeV and decay width $\Gamma=(9.8\pm
2.7^{+3.7}_{-4.5})$ MeV is also observed. The LHCb analysis is not sensitive
to broad $J/\psi\,p$ contributions like the former $P_{c}(4380)^{+}$, implying
that there could be more states present in the data. Also the spin-parity,
$J^{P}$, assignments of the states are not yet determined. We interpret these
resonances in the compact diquark model as hidden-charm diquark-diquark-
antiquark baryons, having the following spin and angular momentum quantum
numbers:
$P_{c}(4312)^{+}=\\{\bar{c}[cu]_{s=1}[ud]_{s=0};L_{\mathcal{P}}=0,J^{P}=3/2^{-}\\}$,
the $S$-wave state, and the other two as $P$-wave states, with
$P_{c}(4440)^{+}=\\{\bar{c}[cu]_{s=1}[ud]_{s=0};L_{\mathcal{P}}=1,J^{P}=3/2^{+}\\}$
and
$P_{c}(4457)^{+}=\\{\bar{c}[cu]_{s=1}[ud]_{s=0};L_{\mathcal{P}}=1,J^{P}=5/2^{+}\\}$.
The subscripts denote the spins of the diquarks and $L_{\mathcal{P}}=0,\,1$ is
the orbital angular momentum quantum number of the pentaquark. These
assignments are in accord with the heavy-quark-symmetry selection rules for
$\Lambda_{b}$-baryon decays, in which the spin $S=0$ of the light diquark
$[ud]_{s=0}$ is conserved. The masses of observed states can be accommodated
in this framework and the two heaviest states have the positive parities as
opposed to the molecular-like interpretations. In addition, we predict several
more states in the $J/\psi\,p$ mass spectrum, and urge the LHCb Collaboration
to search for them in their data.

##  I Introduction

In 2015, the LHCb Collaboration reported the first observation of two hidden-
charm pentaquark states $P_{c}(4380)^{+}$ and $P_{c}(4450)^{+}$ in the decay
$\Lambda_{b}^{0}\to J/\psi\,p\,K^{-}$ Aaij:2015tga , having the masses
$M=(4380\pm 8\pm 29)$ MeV and $M=(4449.8\pm 1.7\pm 2.5)$ MeV, and decay widths
$\Gamma=(205\pm 18\pm 86)$ MeV and $\Gamma=(39\pm 5\pm 19)$ MeV, with the
preferred spin-parity assignments $J^{P}=3/2^{-}$ and $J^{P}=5/2^{+}$,
respectively, but the reversed spin-parity assignments were also tenable. This
was followed in a subsequent paper Aaij:2016ymb , in which evidence was
presented for the Cabibbo-suppressed decay $\Lambda_{b}^{0}\to
J/\psi\,p\,\pi^{-}$, found consistent with the production of the
$P_{c}(4380)^{+}$ and $P_{c}(4450)^{+}$ pentaquark states. These states have
the quark content $(c\bar{c}uud)$ and, like their tetraquark counterparts $X$,
$Y$, and $Z$, they lie close in mass to several (charmed meson-baryon)
thresholds Tanabashi:2018oca . This has led to a number of theoretical
proposals for their interpretation, which include rescattering-induced
kinematical effects Guo:2015umn ; Liu:2015fea ; Mikhasenko:2015vca ;
Meissner:2015mza , open charm-baryon and charm-meson bound states Chen:2015moa
; He:2015cea ; Roca:2015dva ; Chen:2015loa ; Xiao:2015fia , and baryocharmonia
Kubarovsky:2015aaa . They have also been interpreted as compact pentaquark
hadrons with the internal structure organised as diquark-diquark-antiquark
Maiani:2015vwa ; Li:2015gta ; Mironov:2015ica ; Anisovich:2015cia ;
Ghosh:2015ksa ; Wang:2015epa ; Wang:2015ava ; Wang:2015wsa ; Ali:2016dkf ;
Ali:2017ebb  or as diquark-triquark Lebed:2015tna ; Zhu:2015bba .

Very recently, the LHCb Collaboration has updated their analysis of the
$\Lambda_{b}\to J/\psi\,p\,K^{-}$ decay, making use of 9 times more data based
on Run 1 and Run 2 than in the Run 1 data alone Aaij:2019vzc . Thanks to the
impressive number (246 K) of $\Lambda_{b}$-baryon signal events, improved
$J/\psi\,p$ mass and momentum resolution, and excellent vertexing, narrow
$J/\psi\,p$ structures have been observed, which were insignificant in the
older Run 1 data 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 two narrow resonances, hence-after called
$P_{c}(4440)^{+}$ and $P_{c}(4457)^{+}$, replace the older resonance,
$P_{c}(4450)^{+}$, which is now found split, and the third narrow resonance,
called $P_{c}(4312)^{+}$, is a new addition to the pentaquark spectrum. The
situation with the broader $P_{c}(4380)^{+}$ state, observed in the 2015 data,
having the large width, $\Gamma=(205\pm 18\pm 86)$ MeV, is not clear, as the
current analysis is not sensitive to broad structures. Hence, a dedicated
analysis of the LHCb data may turn out to have more narrow and broad
resonances.

In this work we follow the compact pentaquark interpretation. The basic idea
of this approach is that highly correlated diquarks play a key role in the
physics of multiquark states Maiani:2004vq ; Lipkin:1987sk ; Jaffe:2003sg .
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. Both spin-1 and spin-0 diquarks are, however, allowed.
In the case of a diquark $[qq^{\prime}]$ consisting of two light quarks, the
spin-0 diquark is 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 $[cq]$ or by $1/m_{b}$ for the $[bq]$ diquark, and hence both spin
configurations are treated at par. For pentaquarks, the mass spectrum depends
upon how the five quarks, i. e., the 4 quarks and an antiquark, are
dynamically structured. A diquark-triquark picture, in which the two observed
pentaquarks consist of a rapidly separating pair of a color-$\tt\bar{3}$
$[cu]$-diquark and a color-$\tt 3$ triquark $\bar{\theta}=\bar{c}[ud]$, has
been presented in Lebed:2015tna ; Zhu:2015bba . A “Cornell”-type non-
relativistic linear-plus-Coulomb potential Eichten:1978tg  is used to
determine the diquark-triquark separation $R$ and the ensuring phenomenology
is worked out.

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 decay $\Lambda_{b}\to J/\psi\,p\,K^{-}$ Aaij:2019vzc . State | Mass [MeV] | Width [MeV] | (95% CL) | $\mathcal{R}\,[\%]$
---|---|---|---|---
$P_{c}(4312)^{+}$ | $4311.9\pm 0.7^{+6.8}_{-0.6}$ | $9.8\pm 2.7^{+3.7}_{-4.5}$ | ($<27$) | $0.30\pm 0.07^{+0.34}_{-0.09}$
$P_{c}(4440)^{+}$ | $4440.3\pm 1.3^{+4.1}_{-4.7}$ | $20.6\pm 4.9^{+8.7}_{-10.1}$ | ($<49$) | $1.11\pm 0.33^{+0.22}_{-0.10}$
$P_{c}(4457)^{+}$ | $4457.3\pm 0.6^{+4.1}_{-1.7}$ | $6.4\pm 2.0^{+5.7}_{-1.9}$ | ($<20$) | $0.53\pm 0.16^{+0.15}_{-0.13}$

In this paper, we employ the diquark-triquark picture shown in Fig. 1. Here,
the nucleus consists of the doubly-heavy triquark, with the $[cq]$-diquark and
$\bar{c}$-antiquark, and a light diquark $[q^{\prime}q^{\prime\prime}]$, where
$q$, $q^{\prime}$, and $q^{\prime\prime}$ are $u$-, $d$-, and $s$-quarks,
which is in an orbit for the $P$-wave (and higher orbitally-excited) states,
as it is easier to excite a light diquark to form an orbitally-excited
pentaquark. All three constituents are color anti-triplets, ${\tt\bar{3}}$,
making an overall color-singlet — the pentaquark. Such a description is closer
to the doubly-heavy tetraquarks and doubly-heavy baryons, in which the double-
heavy diquark may be considered as the static color source, which has received
a lot of theoretical attention lately Manohar:1992nd ; Esposito:2013fma ;
Luo:2017eub ; Karliner:2017qjm ; Eichten:2017ffp ; Francis:2016hui ;
Bicudo:2017szl ; Junnarkar:2017sey ; Mehen:2017nrh ; Czarnecki:2017vco ;
Maiani:2019cwl .

The resulting pentaquark spectrum in the diquark model is very rich. However,
imposing the spin conservation in the heavy-quark symmetry limit, which was
already advocated in the analysis of the older LHCb data Ali:2016dkf , we
argue that only that part of the pentaquark spectrum is reachable in
$\Lambda_{b}$-decays, in which the pentaquarks have a “good” light diquark, i.
e., having the spin $S_{ld}=0$, in their Fock space. This reduces the number
of observable pentaquark states greatly. There is an overwhelming support of
the heavy-quark symmetry constraints from the available data on
$\Lambda_{b}$-decays, which are dominated by the $\Lambda_{b}\to\Lambda_{c}+X$
transitions. Here, $X$ stands for a large number of leptonic and hadronic
final states, enlisted by Particle Data Group Tanabashi:2018oca . The decays
$\Lambda_{b}\to\Sigma_{c}+X$ are sparse, with PDG listing only two
Tanabashi:2018oca : $\Lambda_{b}\to\Sigma_{c}(2455)^{0}\,\pi^{+}\,\pi^{-}$ and
$\Lambda_{b}\to\Sigma_{c}(2455)^{++}\,\pi^{-}\,\pi^{-}$, having the branching
fractions ${\cal B}=(5.7\pm 2.2)\times 10^{-4}$ and ${\cal B}=(3.2\pm
1.6)\times 10^{-4}$, respectively. Clearly, there is the
$\Sigma_{c}/\Lambda_{c}$-suppression in the branching ratios of
$\Lambda_{b}$-decays. This also restricts the resonant $J/\psi\,p$ spectrum in
the $\Lambda_{b}\to J/\psi\,p\,K^{-}$ decay.

We show that the newly observed spectrum shown in Table 1 can be accommodated
in the diquark picture, though the spin-parity, $J^{P}$, assignment of the
observed states is tentative, as these quantum numbers are not yet measured.
There are yet more states present in the $J/\psi\,p$ mass spectrum in the
compact diquark model, whose masses and $J^{P}$ quantum numbers are given
here, and we urge the LHCb Collaboration to search for them in their data.

This paper is organised as follows. In Section II, we introduce the doubly-
heavy triquark — light diquark model of pentaquarks, and define the state
vectors having the 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, and $L_{t}$ and $L_{ld}$ are the
orbital quantum numbers of the triquark and light-diquark, which are combined
into the total orbital angular momentum $L$ of the pentaquark. The
corresponding set of the $S$-wave state vectors (with $L_{t}=L_{ld}=L=0$) with
the “good” ($S_{ld}=0$) light diquark is presented in Table 3. 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 $S_{ld}=0$ are given in
Table 3. In Section III, we give the effective Hamiltonian used to work out
the pentaquark mass spectrum. In Section IV, the analytical expressions for
the matrix elements of the effective Hamiltonian 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 and spin-orbit interactions are
included. In all the cases, we diagonalise the mass matrices, and analytical
equations for all the masses of the $S$\- and $P$-wave pentaquarks, which can
be reached in $\Lambda_{b}$-decays, are presented. Section V contains the
values of various input parameters and predictions for the pentaquark masses.
Possible assignments of the newly observed pentaquark states are also
discussed here. Decay widths of the pentaquarks are briefly discussed in
Section VI. We conclude in Section VII.

##  II Doubly-heavy triquark — light diquark model of pentaquarks

In the pentaquark picture considered here, there are two flux tubes, with the
first stretched between the charmed diquark and charmed antiquark from which
the diquark is in the color-antitriplet state $\bar{3}$. With the antiquark
being also a color anti-triplet 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. We
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 and spin-orbit
interactions between the hidden-charm triquark and light diquark. Tensor
interactions affect the $P$-wave pentaquark spectrum in which light diquarks
have the spin $S_{ld}=1$.

The effective Hamiltonian for the ground-state pentaquarks is described in
terms of two constituent masses of the heavy diquark $m_{[cq]}\equiv m_{hd}$
and the light diquark $m_{[q^{\prime}q^{\prime\prime}]}\equiv m_{ld}$, where
$q$, $q^{\prime}$, and $q^{\prime\prime}$ are light $u$-, $d$\- and
$s$-quarks, 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 charmed antiquark and its spin-spin
interactions with each of the diquarks.

In the case of the orbitally excited pentaquarks, the orbital angular momentum
$L$ of the pentaquark is the sum of two terms, $L_{t}$, arising from the
triquark system consisting of the heavy diquark and the charmed antiquark, and
$L_{ld}$, which determines the relative motion of the light diquark around the
doubly-heavy triquark. The total orbital angular momentum $L$ of the
pentaquark is then obtained with the help of the momentum sum rules from
quantum mechanics, i. e., it takes a value from the range
$L=|L_{t}-L_{ld}|,\ldots,L_{t}+L_{ld}$. The orbital angular momentum $L$ is
combined with the spin $S$ to get the total angular momentum $J$ of the
pentaquark using the $L-S$ coupling scheme. With 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
$\left|S_{hd},S_{t},L_{t};S_{ld},L_{ld};S,L\right\rangle_{J}$.

Figure 1:  A picture of pentaquarks in the diquark model involving the heavy
diquark $[cq]_{\bar{3}}$ and charmed 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.

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 charmed
diquark and charmed antiquark. Since both are heavy, the triquark is an
(almost) static system. Hence, $L_{t}=0$ is the most probable quantum state of
the triquark. 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” ($S_{ld}=0$) light diquark in Table 3.

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.  $J^{P}$ | $\left|S_{hd},S_{t},L_{t};S_{ld},L_{ld};S,L\right\rangle_{J}$
---|---
$1/2^{-}$ | $\left|0,1/2,0;0,0;1/2,0\right\rangle_{1/2}$
$1/2^{-}$ | $\left|1,1/2,0;0,0;1/2,0\right\rangle_{1/2}$
$3/2^{-}$ | $\left|1,3/2,0;0,0;3/2,0\right\rangle_{3/2}$

Table 3:  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.  $J^{P}$ | $\left|S_{hd},S_{t},L_{t};S_{ld},L_{ld};S,L\right\rangle_{J}$
---|---
$1/2^{+}$ | $\left|0,1/2,0;0,1;1/2,1\right\rangle_{1/2}$
$3/2^{+}$ | $\left|0,1/2,0;0,1;1/2,1\right\rangle_{3/2}$
$1/2^{+}$ | $\left|1,1/2,0;0,1;1/2,1\right\rangle_{1/2}$
$3/2^{+}$ | $\left|1,1/2,0;0,1;1/2,1\right\rangle_{3/2}$
$1/2^{+}$ | $\left|1,3/2,0;0,1;3/2,1\right\rangle_{1/2}$
$3/2^{+}$ | $\left|1,3/2,0;0,1;3/2,1\right\rangle_{3/2}$
$5/2^{+}$ | $\left|1,3/2,0;0,1;3/2,1\right\rangle_{5/2}$

##  III Effective Hamiltonian for Pentaquark Spectrum

We calculate the mass spectrum of pentaquarks 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. For this, we extend the
effective Hamiltonian proposed for the tetraquark spectroscopy 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 colored 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 charmed antiquark
and charmed diquark, respectively. The last three terms describe the spin-spin
interactions in the charmed diquark and between the diquark constituents and
the charmed antiquark, quantified by the couplings
$(\mathcal{K}_{cq})_{\bar{3}}$, $\mathcal{K}_{\bar{c}q}$, and
$\mathcal{K}_{\bar{c}c}$.

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 possible spin-spin interactions
between the constituents of the light diquark and 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 earlier
in Eq. (3), as they are the couplings of the spin-spin interactions inside the
compact objects, like diquarks and triquark, while the ones in Eq. (5) denote
the couplings between the constituents of the two objects, the heavy triquark
and light diquark, and are anticipated to be strongly suppressed. 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.

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 tensor interactions are subsumed in $H_{T}$ in (6). They
are relevant only for the pentaquarks in which the light-diquarks have the
spin $S_{ld}=1$. Since they are not anticipated to be produced in the
$\Lambda_{b}$-baryons decays, due to the heavy-quark symmetry constraints, we
set $H_{T}=0$ in this paper. They have to be included in the extended
pentaquark systems, detailed in a forthcoming paper Ali:2019xyz . As already
stated, we assume that the triquark state is an $S$-wave (i. e., $L_{t}=0$).
Thus, the total angular momentum of the pentaquark is determined by the
orbital excitation of the light diquark, $L=L_{ld}$. 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 $A_{t}$, $A_{ld}$, and $B$ parametrise the strengths of the triquark
spin-orbit, light-diquark spin-orbit and orbital momentum couplings,
respectively.

##  IV Mass Formulas for Pentaquark Spectrum

###  IV.1 $S$-wave pentaquarks

With the basis vectors of the pentaquark states chosen, one can derive
analytical expressions for calculating the pentaquark spectrum Ali:2019xyz .
They are the matrix elements of the effective Hamiltonian presented in Sec.
III. 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 interactions (5) between the spins of (anti)quarks in compact
shells are included. As the later couplings are suppressed in comparison with
the spin-spin ones inside the shells, we neglect the contributions from the
term (5) and restrict ourselves with the Model II Maiani:2014aja .

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 charmed antiquark:

$M_{0}\equiv m_{hd}+m_{ld}+m_{c}.$ (8)

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

$\displaystyle{}_{J}\langle
S_{hd},S_{t},L_{t};S_{ld},L_{ld};S,L|\,2\,(\mathcal{K}_{cq})_{\bar{3}}\,(\mathbf{S}_{c}\cdot\mathbf{S}_{q})\,|S_{hd},S_{t},L_{t};S_{ld},L_{ld};S,L\rangle_{J}$
(9) $\displaystyle\hskip
48.36967pt=(\mathcal{K}_{cq})_{\bar{3}}\left[S_{hd}\left(S_{hd}+1\right)-\frac{3}{2}\right]=\frac{1}{2}\,(\mathcal{K}_{cq})_{\bar{3}}\times\left\\{\begin{array}[]{cc}-3,&S_{hd}=0,\\\
1,&S_{hd}=1,\end{array}\right.$ (12) $\displaystyle{}_{J}\langle
S_{hd},S_{t},L_{t};S_{ld},L_{ld};S,L|\,2\,(\mathcal{K}_{q^{\prime}q^{\prime\prime}})_{\bar{3}}\,(\mathbf{S}_{q^{\prime}}\cdot\mathbf{S}_{q^{\prime\prime}})\,|S_{hd},S_{t},L_{t};S_{ld},L_{ld};S,L\rangle_{J}$
(13) $\displaystyle\hskip
48.36967pt=(\mathcal{K}_{q^{\prime}q^{\prime\prime}})_{\bar{3}}\left[S_{ld}\left(S_{ld}+1\right)-\frac{3}{2}\right]=\frac{1}{2}\,(\mathcal{K}_{q^{\prime}q^{\prime\prime}})_{\bar{3}}\times\left\\{\begin{array}[]{cc}-3,&S_{ld}=0,\\\
1,&S_{ld}=1.\end{array}\right.$ (16)

The remaining terms in Eq. (3) are responsible for the contributions of the
spin-spin interactions between the charmed 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 spins inside the triquark. The details of
calculations are given in Ali:2019xyz  and we present the results here.

From the three $S$-wave states presented in Table 3, the two states with
$J^{P}=1/2^{-}$ mix due to the spin-spin interaction of the charmed antiquark
and the heavy diquark, and the third one with $J^{P}=3/2^{-}$ remains unmixed.

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

$M^{S0}_{J=1/2}=M_{0}-\frac{1}{2}\,(\mathcal{K}_{cq})_{\bar{3}}-\frac{3}{2}\,(\mathcal{K}_{q^{\prime}q^{\prime\prime}})_{\bar{3}}-(\mathcal{K}_{cq})_{\bar{3}}\left(\begin{array}[]{rr}1&0\\\
0&-1\end{array}\right)+\frac{1}{2}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left(\begin{array}[]{rr}0&\sqrt{3}\\\
\sqrt{3}&-2\end{array}\right).$ (17)

Diagonalising this matrix yields the masses of these two states:

$\displaystyle
m^{S0}_{1}=M_{0}-\frac{1}{4}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left[2+r_{hd}+3r_{ld}+2\sqrt{3+(1-r_{hd})^{2}}\right],$
(18) $\displaystyle
m^{S0}_{2}=M_{0}-\frac{1}{4}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left[2+r_{hd}+3r_{ld}-2\sqrt{3+(1-r_{hd})^{2}}\right],$
(19)

where the superscript $S0$ denotes the $S$-wave pentaquark with the “good”
light diquark, $S_{ld}=0$, and the two ratios $r_{hd}$ and $r_{ld}$ of the
couplings are defined as:

$r_{hd}\equiv\frac{2(\mathcal{K}_{cq})_{\bar{3}}}{\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}},\qquad
r_{ld}\equiv\frac{2(\mathcal{K}_{q^{\prime}q^{\prime\prime}})_{\bar{3}}}{\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}}.$
(20)

For the state with $J^{P}=3/2^{-}$, the mass $m^{S0}_{3}$ is the average of
the effective Hamiltonian over this state:

$m^{S0}_{3}=M_{0}+\frac{1}{4}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left(2+r_{hd}-3r_{ld}\right).$
(21)

The case of $S$-wave pentaquark states with the “bad” light diquark is worked
out in detail elsewhere Ali:2019xyz .

###  IV.2 $P$-wave pentaquarks

As discussed earlier in Sec. III, one needs to include the terms dependent on
the internal angular momenta of the pentaquark and, in particular, the term
(7) from the pentaquark effective Hamiltonian.

From the seven $P$-wave pentaquark states with the “good” light diquark shown
in Table 3, 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 charmed 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 angular-momentum dependent term $H_{L}$ of the
effective Hamiltonian, one obtains the spin-orbit and orbital contributions to
matrix elements.

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

$M^{P0}_{J=1/2}=M_{0}-\frac{1}{2}\,(\mathcal{K}_{cq})_{\bar{3}}-\frac{3}{2}\,(\mathcal{K}_{q^{\prime}q^{\prime\prime}})_{\bar{3}}-(\mathcal{K}_{cq})_{\bar{3}}\left(\begin{array}[]{rr}1&0\\\
0&-1\end{array}\right)+\frac{1}{2}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left(\begin{array}[]{rr}0&\sqrt{3}\\\
\sqrt{3}&-2\end{array}\right)+B-2A_{t},$ (22)

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

$M^{P0}_{J=3/2}=M_{0}-\frac{1}{2}\,(\mathcal{K}_{cq})_{\bar{3}}-\frac{3}{2}\,(\mathcal{K}_{q^{\prime}q^{\prime\prime}})_{\bar{3}}-(\mathcal{K}_{cq})_{\bar{3}}\left(\begin{array}[]{rr}1&0\\\
0&-1\end{array}\right)+\frac{1}{2}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left(\begin{array}[]{rr}0&\sqrt{3}\\\
\sqrt{3}&-2\end{array}\right)+B+A_{t}.$ (23)

Diagonalising these matrices, we get the masses:

$\displaystyle
m^{P0}_{1}=M_{0}-\frac{1}{4}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left[2+r_{hd}+3r_{ld}+2\sqrt{3+(1-r_{hd})^{2}}\right]+B-2A_{t},$
(24) $\displaystyle
m^{P0}_{2}=m^{P0}_{1}+\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\sqrt{3+(1-r_{hd})^{2}},$
(25) $\displaystyle m^{P0}_{3,4}=m^{P0}_{1,2}+3A_{t},$ (26)

where $r_{hd}$ and $r_{ld}$ are defined in Eq. (20) and the superscript $P0$
means the $P$-wave pentaquark with the “good” light diquark, $S_{ld}=0$.

For the three states, enumerated as the fifth, sixth and seventh according to
the entries in Table 3, the masses $m^{P0}_{5,6,7}$ are simply the average
values of the effective Hamiltonian over these states:

$\displaystyle
m^{P0}_{5}=M_{0}+\frac{1}{4}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left(2+r_{hd}-3r_{ld}\right)+B-5A_{t},$
(27) $\displaystyle m^{P0}_{6}=m^{P0}_{5}+3A_{t},\qquad
m^{P0}_{7}=m^{P0}_{5}+8A_{t}.$

These states are non-degenerate due to the triquark spin-orbit interaction, i.
e., $A_{t}\neq 0$. Hence, the mass gap between them is a measure of the
coupling $A_{t}$, and the following relations hold:

$m^{P0}_{6}-m^{P0}_{5}=\frac{3}{8}\left(m^{P0}_{7}-m^{P0}_{5}\right)=\frac{3}{5}\left(m^{P0}_{7}-m^{P0}_{6}\right).$
(28)

If more states are experimentally observed and their quantum numbers
determined, the validity of these relations will be a strong argument in
favour of the compact diquark model.

##  V Hidden-charm Pentaquark Mass Predictions

Working within the constituent quark-diquark model Maiani:2014aja , input
parameters are the masses of the constituents, charmed quark and diquarks,
spin-spin couplings, and other parameters related with the orbital or radial
excitations. For the constituent quark masses, there are two possibilities:
Either extract them from the masses of the known mesons or from baryons
Ali:2019 . The charmed quark mass is estimated as $m_{c}^{m}=1667$ MeV from
the $D$-mesons spectrum, and $m_{c}^{b}=1710$ MeV from the charmed baryon
masses, yielding a difference of $43$ MeV Ali:2019 . In particular, with the
$m_{c}^{b}$ value as an input, predictions for the charmed baryon masses were
obtained in Karliner:2014gca , which are remarkably accurate and differ from
the experimentally measured masses Tanabashi:2018oca  by about $10$ MeV. We
use $m_{c}^{b}$ in the numerical analysis.

In contrast to quarks, which have the fixed spin $S_{q}=1/2$, for diquarks
being composite objects, there are two possible spin configurations from which
the antisymmetric one corresponding to the diquark spin $S=0$ is energetically
more favourable. Both configurations are allowed if the flavors of the quarks
are different. As we consider the pentaquarks from the $\Lambda_{b}$-decay in
the heavy-quark symmetry limit, we need to specify the mass of the light
$[ud]$-diquark, for which we use the value $m_{ld}=m_{[ud]}=576$ MeV
Karliner:2018bms . The charmed diquarks mass $m_{hd}=m_{[cq]}=1976$ MeV is
borrowed from Ali:2019 . To estimate the errors on the diquark-masses, one
should compare the baryon masses predicted within this (constituent quark-
diquark) model and the measured ones. Based on the analysis presented in
Karliner:2014gca , we infer that the computed masses for the charmed baryons
are accurate to within $15$ MeV. This value may be taken as indicative of the
uncertainty in diquark masses.

In addition, we need the spin-spin couplings,
$\mathcal{K}_{\bar{Q}Q^{\prime}}$ and $(\mathcal{K}_{QQ^{\prime}})_{\bar{3}}$,
which are extracted from the mass spectra of mesons and baryons, respectively.
The values of the spin-spin couplings used are:
$(\mathcal{K}_{cq})_{\bar{3}}=67$ MeV,
$(\mathcal{K}_{qq^{\prime}})_{\bar{3}}=98$ MeV, $\mathcal{K}_{\bar{c}q}=70$
MeV, and $\mathcal{K}_{\bar{c}c}=113$ MeV. They are taken from Ali:2019 ,
except for $\mathcal{K}_{\bar{c}c}$ which is from Karliner:2014gca . For
numerical analysis, one needs to show uncertainces in these couplings and we
assign it to be 10% of each value, the same as in the $\Omega_{c}^{*}$-baryons
Ali:2017wsf . The pentaquark masses involve the ratios of the couplings (20)
which for the input values are evaluated as: $r_{hd}=0.73$ and $r_{ld}=1.07$.
In Model II Maiani:2014aja , these ratios are of $O(1)$, which reflects the
dominance of the spin-spin interaction inside the diquarks and triquark over
other possible spin-spin interactions. These arguments give quantitative
support in favour of the Model II Maiani:2014aja .

In estimating the $P$-wave pentaquark mass spectrum, the values of the
couplings in the orbital angular momentum term $B$ and the spin-orbit coupling
$A_{t}$ are required. We discuss first the two heaviest 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, used here for
the pentaquark spectrum, their mass splitting is related to the spin-orbit
coupling $A_{t}$ of the triquark:

$M[P_{c}(4457)^{+}]-M[P_{c}(4440)^{+}]=\left(17^{+6.4}_{-4.5}\right)~{}\mathrm{MeV}=5A_{t},$
(29)

where the error is obtained by adding the experimental errors on the masses in
quadrature. From (29), the value of the coefficient $A_{t}$ follows
immediately:

$A_{t}=\left(3.4^{+1.3}_{-0.9}\right)~{}\mathrm{MeV}.$ (30)

It is not surprising that it is numerically small, as the doubly-heavy
triquark is almost static. Of course, as there are more states present in the
theoretical spectrum than the number of observed pentaquarks, there are also
other assignments possible, but we find them unrealistic and hence not
discussed here.

The third narrow state, $P_{c}(4312)^{+}$, can have several 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 later coupling is already
known from the $P_{c}(4440)^{+}$ and $P_{c}(4457)^{+}$ mass splitting (30),
and the mass difference, say, between $P_{c}(4312)^{+}$ and $P_{c}(4457)^{+}$,
allows us to read off the strength of the orbital interaction:

$M[P_{c}(4457)^{+}]-M[P_{c}(4312)^{+}]=\left(145.4^{+4.2}_{-7.1}\right)~{}\mathrm{MeV}=B+3A_{t}.$
(31)

With $A_{t}$ from (30), we get:

$B=\left(135.2^{+5.0}_{-8.1}\right)~{}\mathrm{MeV}.$ (32)

This is too small in comparison with the strengths of the orbital excitations
in other hadrons Ali:2017wsf , and, in particular, with $B(\Omega_{c})=325$
MeV obtained from $\Omega_{c}^{*}$-baryons. Moreover, the theoretically
predicted masses of the $P_{c}(4440)^{+}$ and $P_{c}(4457)^{+}$ states with
the value of $B$ from (32) are found to be $\sim 70$ MeV below the
experimental values. Alternatively, assuming that the spin-spin couplings and
constituent 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:

$B=\frac{1}{5}\left\\{3M[P_{c}(4440)^{+}]+2M[P_{c}(4457)^{+}]\right\\}-M_{0}-\frac{1}{4}\left(\mathcal{K}_{\bar{c}q}+\mathcal{K}_{\bar{c}c}\right)\left(2+r_{hd}-3r_{ld}\right).$
(33)

With the values of the other parameters already assigned, the value $B=207$
MeV reproduces the masses of the observed $P_{c}(4440)$ and $P_{c}(4457)$
states, as shown in Table 4 as the last two entries inserted into the solid
boxes. This value of $B$ is closer to the estimates in the hidden and open
charm and bottom hadrons Ali:2017wsf . This can be exemplified Maiani:2015vwa
by the mass difference in hyperons $M_{\Lambda(1405)}-M_{\Lambda(1116)}\simeq
290$ MeV and in the exotic $X,\,Y,\,Z$ states where $\Delta M(L=0\to 1)\simeq
280$ MeV, and similar differences in the charm baryons. As a conservative
estimate of the uncertainty in $B$, we took it as $\pm 20$ MeV, which is
approximately the same as the uncertainty on the orbital coupling for the
$\Omega_{c}^{*}$-baryons Ali:2019xyz . With these parameters, the mass of the
third pentaquark $M=(4240\pm 29)$ 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. A correlated error analysis on the
pentaquark masses will be undertaken once the spin-parity of the three
observed states is determined.

The second possibility is to assign the lowest mass state $P_{c}(4312)^{+}$
with the one from the orbitally-excited set of states, in particular, 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. However, in this case, some of the input parameters
have unphysical values, and we do not entertain this assignment here.

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 4, there are two states,
the $J^{P}=1/2^{-}$ with a mass $3830$ MeV, and $J^{P}=1/2^{+}$ with the mass
$4031$ MeV, which lie below the $M^{\rm thr}_{J/\psi\,p}$ threshold. Also, a
third state having $J^{P}=3/2^{+}$, with a mass 4040 MeV, may also lie below
$M^{\rm thr}_{J/\psi\,p}$. One of these states is even below the threshold for
the decay $P_{c}^{+}\to\eta_{c}+p$. Hence, it will decay weakly. The others
shown in Table 4 are reachable in the $\Lambda_{b}\to J/\psi\,p\,K^{-}$ decay,
and we urge the LHCb Collaboration to search for them.

Table 4:  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, following values of the parameters are used: orbital coupling $B=207\pm 20$ MeV and spin-orbit coupling of the heavy triquark $A_{t}=3.4\pm 1.1$ MeV.  $J^{P}$ |  This work |  Refs. Ali:2016dkf ; Ali:2017ebb
---|---|---
$S_{ld}=0$, $L=0$
$1/2^{-}$ | $3830\pm 34$ | $4086\pm 42$
| $4150\pm 29$ | $4162\pm 38$
$3/2^{-}$ | $4240\pm 29$ | $4133\pm 55$
$S_{ld}=0$, $L=1$
$1/2^{+}$ | $4030\pm 39$ | $4030\pm 62$
| $4351\pm 35$ | $4141\pm 44$
| $4430\pm 35$ | $4217\pm 40$
$3/2^{+}$ | $4040\pm 39$ |
| $4361\pm 35$ |
| $4440\pm 35$ |
$5/2^{+}$ | $4457\pm 35$ | $4510\pm 57$

##  VI 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 $\bar{c}$ 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. The tunneling probability depends on the mass of the quark, with the
probabilty exponentially suppressed the heavier the quark is, expressed
through the semi-classical approximation of the tunneling amplitude $A_{M}\sim
e^{-\sqrt{2MEl}}$, where $M$ is the quark mass, $l$ is the radius of a
tetraquark (or pentaquark), of order ($1-2$) fm, and $E$ is typically 100 MeV
Maiani:2017kyi . This suppresses the formation of the charmonium states,
leading to a much reduced decay width for $P_{c}\to J/\psi p$, and less so for
the decay $P_{c}\to\Lambda_{c}\bar{D}^{(*)}$, which remains to be tested. In
all these cases, the size of the diquark plays an important role in reducing
the decay widths. For example, it is argued in Maiani:2017kyi , that the
relative ratio $\lambda$ between the radius of a tetraquark $R_{4q}$ and that
of a compact diquark $R_{Qq}$ is expected to be $\lambda\equiv
R_{4q}/R_{Qq}\geq 3$. This presumably can also be taken as an estimate of this
quantity for the pentaquarks. This will be checked also in the photoproduction
experiments under way at the Jafferson Laboratory. 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 . We do expect reduced cross-
sections for the narrow $P_{c}$-states in the compact diquark picture, as
opposed to the one based on hadron molecule interpretation Kubarovsky:2015aaa
. Moreover, as some of the states are $P$-states, anticipated in the compact
diquark picture, the decay widths are further reduced due to the angular
momentum barrier.

##  VII Conclusions

We have presented the mass spectrum of the hidden-charm pentaquark states
$(c\bar{c}qq^{\prime}q^{\prime\prime})$, where $q$, $q^{\prime}$, and
$q^{\prime\prime}$ are light $u$\- and $d$-quarks, using the isospin symmetry.
In doing this, we have used an effective Hamiltonian, based on a doubly-heavy
triquark — light diquark picture of a pentaquark, shown in Fig. 1. Apart from
the constituent quark and diquark masses, the Hamiltonian incorporates
dominant spin-spin, spin-orbit and orbital interactions. Following this, we
interpret the three narrow resonances with the states having the following
spin and angular momentum quantum numbers:
$P_{c}(4312)^{+}=\\{\bar{c}[cu]_{s=1}[ud]_{s=0};L_{\mathcal{P}}=0,J^{P}=3/2^{-}\\}$,
the $S$-wave, and the other two as $P$-wave states, with
$P_{c}(4440)^{+}=\\{\bar{c}[cu]_{s=1}[ud]_{s=0};L_{\mathcal{P}}=1,J^{P}=3/2^{+}\\}$,
and
$P_{c}(4457)^{+}=\\{\bar{c}[cu]_{s=1}[ud]_{s=0};L_{\mathcal{P}}=1,J^{P}=5/2^{+}\\}$.
We have presented analytical expressions for the masses of the states having
well-defined quantum numbers. However, there are more states present in the
spectrum having the quark content $(c\bar{c}uud)$. They are listed in Tables 3
and 3. The details of the formalism are given elsewhere Ali:2019xyz , where
also the spectrum of the hidden-charm pentaquarks, having a non-vanishing
strangeness is worked out, as well as the states having spin-1 light diquarks,
i. e., with $S_{ld}=1$. Some of them can be reached in weak decays of bottom-
strange baryons, $\Xi_{b}$ and $\Omega_{b}$ Ali:2016dkf .

For the numerical estimates of the pentaquark masses, we infer the input
parametric values from the earlier studies of the hidden-charm tetraquarks and
charmed baryons. Since not all parameters in the effective Hamiltonian can be
uniquely determined from the existing resonances, as there are currently only
three such observed states, we use additionally the results of the known heavy
baryons, such as the orbitally-excited $\Omega_{c}^{*}$ states, since
$P_{c}(4312)^{+}$, $P_{c}(4440)^{+}$ and $P_{c}(4457)^{+}$ are also heavy
baryons. It is difficult to quantify the theoretical errors on our mass
estimates, as we have dropped sub-dominant spin-spin interactions, but we
trust that the masses of the pentaquarks given in Table 4 are in the right
ball-park. We note that, in addition to the observed ones, there are three
states (one with $J^{P}=1/2^{-}$ and two with positive parity, $J^{P}=1/2^{+}$
and $J^{P}=3/2^{+}$) which have $[cu]_{s=0}$-diquark in their Fock space.
Their masses are lower than their counterparts with $[cu]_{s=1}$-diquark. Two
of these states lie below the $J/\psi\,p$ threshold, probably the third one as
well, requiring a different search strategy. We anticipate other pentaquark
states waiting to be discovered with yet more data and a detailed study of the
$J/\psi\,p$ mass spectrum, including both narrow and broad resonances.

Finally, we stress that it is not enough to claim the observed narrow peaks in
the $J/\psi\,p$ mass spectrum as a confirmation of hadronic molecules, just
due to their kinematic vicinity to the charmed meson-charmed baryon
thresholds, $\Sigma_{c}\,\bar{D}$ and $\Sigma_{c}\,\bar{D}^{*}$, which in this
hypothesis are interpreted as loosely-bound $S$-wave states, and hence have
necessarily a negative parity. Crucially, the spin-parities of the
$P_{c}(4312)^{+}$, $P_{c}(4440)^{+}$ and $P_{c}(4457)^{+}$ remain to be
determined. In the compact pentaquark picture presented here, heavy-quark
symmetry requirements are built in and the mass estimates are in the right
ball-park. The broad resonance $P_{c}(4380)^{+}$, present in the older LHCb
analysis Aaij:2015tga , which was difficult to accommodate in the compact
pentaquark picture due to the conflict with the heavy quark symmetry
Ali:2016dkf , is presumably no longer there. So, the compact pentaquark
interpretation of the three $P_{c}$-resonances is very much in the game. In
particular, and as a crucial test, we anticipate the three narrow resonances
to have different parities, i. e., both negative and positive parities are
foreseen. Their determination would help greatly in discriminating the compact
pentaquark interpretation and the competing ones based on hadronic molecules
in various incarnations Chen:2019bip ; Chen:2019asm ; Guo:2019fdo ;
Liu:2019tjn ; He:2019ify ; Xiao:2019aya ; Huang:2019jlf ; Shimizu:2019ptd ;
Xiao:2019mvs .

Acknowledgments

We thank Luciano Maiani, Antonello Polosa and Sheldon Stone for reading the
manuscript and helpful discussions. A. P. thanks the Theory Group at DESY for
their kind hospitality, where the major part of this work was done, and
acknowledges financial support by the German Academic Exchange Service (DAAD),
by the Russian Foundation for Basic Research and National Natural Science
Foundation of China for the research project No. 19-52-53041, and by the “YSU
Initiative Scientific Research Activity” (Project No.
AAAA-A16-116070610023-3).

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