# Hidden-flavor pentaquarks

H. Garcilazo<EMAIL_ADDRESS>Escuela Superior de Física y Matemáticas,
Instituto Politécnico Nacional, Edificio 9, 07738 Mexico D.F., Mexico  A.
Valcarce<EMAIL_ADDRESS>Departamento de Física Fundamental,
Universidad de Salamanca, E-37008 Salamanca, Spain

###### Abstract

We have recently studied hidden-charm pentaquarks, $c\bar{c}qqq$, using
dynamical correlations between the heavy quarks arising from the Coulomb-like
nature of the short-range interaction. A pattern was obtained that compares
well with the experimental data. We extend our description to other flavor
sectors which can be framed within the same type of structures discussed in
the original paper. A detailed comparison is made with other results in the
literature and with experimental data. Predictions will be a useful tool to
discriminate between different models of multiquark system dynamics.

##  I Introduction

The recent findings in the heavy-hadron spectra have become both a theoretical
challenge and a suitable test bench for trying to understand QCD realizations
in the non-perturbative regime Che16 ; Bri16 ; Ric16 ; Hos16 ; Che17 ; Leb17 ;
Ali17 ; Esp17 ; Guo18 ; Ols18 ; Kar18 ; Bra20 . It appears to be undeniable
that more complex quark structures allowed by QCD than the simplest quark-
antiquark (meson) or three-quark (baryon) clusters proposed by Gell-Mann Gel64
, the so-called multiquarks, are being found in the heavy-hadron experimental
data Aai21 ; Aaj21 .

For multiquark states with manifestly exotic quantum numbers, as it is the
case of doubly-heavy tetraquarks, both lattice QCD approaches Hug18 ; Hud20
and constituent models Sil93 ; Ric18  predict a very small number of states
restricted to very specific configurations. To give the big picture of
multiquark states with ordinary quantum numbers it has been suggested the
possibility of the existence of correlations between the constituents Mai15 ;
Gir19 ; Ali19 ; Shi21 ; Wul17 .

In a recent paper Gar22 , we have explored a theoretical scenario where the
dynamics of a multiquark system remains marked by correlations between heavy
flavors dictated by QCD Jaf05 , that turn the five-body problem into a more
tractable three-body problem. The most suitable system for developing and
testing our model was the hidden-charm pentaquarks, $c\bar{c}qqq$. We obtained
a pattern that compares well with the experimental data available in this
sector. This work is a natural extension of the analysis performed in Ref.
Gar22  to study hidden-flavor pentaquarks which present the same type of
structure in other flavor sectors and, therefore, can benefit from the
correlations induced by the color-Coulomb potential. In particular, we address
$b\bar{b}qqq$ states and also hidden-charm and hidden-bottom pentaquarks with
a strange quark, $c\bar{c}qqs$ and $b\bar{b}qqs$.

The structure of the paper is the following. In the next section we briefly
review the model: the interacting potential between the quarks, the Hilbert
space arising from the correlations between the heavy flavors, and the
solution of the Faddeev equations for the bound state three-body problem. In
Sec. III we present and discuss our results. Finally, our conclusions are
summarized in Sec. IV.

##  II The model

###  II.1 Pentaquarks wave function

We study hidden-flavor pentaquarks $Q\bar{Q}qqq^{\prime}$, with $Q=b$ or $c$,
$q=u$ or $d$, and $q^{\prime}=u$ $d$ or $s$. Models based on the attractive
character of a $qq$ pair in a color-${\bf\bar{3}}$ state have been widely
explored in the literature Mai15 ; Gir19 ; Ali19 ; Shi21 . If a $Qq$
color-${\bf\bar{3}}$ diquark has a binding proportional to $m_{q}$, in the
same units a $Q\bar{Q}$ color-${\bf 1}$ has a binding proportional to
$2M_{Q}$. Thus, the color Coulomb-like interaction between the components of a
hidden-flavor pentaquark favors a $Q\bar{Q}$ color singlet instead of a color
octet Jaf05 , uniquely determining its color wave function. Antisymetrization
constraints allow to identify the different vectors that contribute to any
$(I,J)$ pentaquark for the lowest lying states, i.e., in the case of a fully
symmetric radial wave function,

$\Psi^{(I,J)}_{\rm Pentaquark}\,\,=\,\,\\{{\bf
3}_{c},i_{1},s_{1}=1/2\\}_{q}\,\otimes\,\\{{\bf
1}_{c},i_{2}=0,s_{2}\\}_{(Q\bar{Q})}\,\otimes\,\\{{\bf\bar{3}}_{c},i_{3}=s_{3},s_{3}\\}_{(qq)}\,,$
(1)

where $i_{1}=1/2$ for $Q\bar{Q}qqq$ pentaquarks and $i_{1}=0$ for
$Q\bar{Q}qqs$ states.

Table 1 of Ref. Gar22  summarizes the possible value of the quantum numbers
leading to an allowed $(I,J)$ hidden-flavor pentaquark. Quark correlations
dominating the QCD phenomena Jaf05  hint to the most favorable states that can
be observed in nature. First, the very strong quark-antiquark correlation in
the color-, flavor-, and spin-singlet channel $\\{{\bf 1}_{c},{\bf
1}_{f},0_{s}\\}$ which can be viewed as the responsible for chiral symmetry
breaking. The attractive forces in this channel are so strong that condenses
in the vacuum, breaking $SU(N_{f})_{L}\times SU(N_{f})_{R}$ chiral symmetry.
The next most attractive channel in QCD seems to be the color antitriplet,
flavor antisymmetric, spin singlet
$\\{{\bf\bar{3}}_{c},{\bf\bar{3}}_{f},0_{s}\\}$, that would select the $qq$
configurations most important spectroscopically. Thus, we summarize in Table 1
those states that contain at least one the most attractive QCD channels, i.e.,
a diquark with spin zero. $s_{1}$ stands for the spin of the single light
quark (with isospin $1/2$ for $u,d$ and $0$ for $s$), $s_{2}$ denotes the spin
of the heavy quark-antiquark pair (with isospin zero) and finally $s_{3}$
represents the spin of the light quark pair (with the restrictions imposed by
the Pauli principle such that $s_{3}=i_{3}$). The notation in the last column
will later be used to identify the wave function of the different pentaquarks.

$I$ |  | $J$ |  | $s_{1}$ |  | $s_{2}$ |  | $s_{3}$ |  | Vector
---|---|---|---|---|---|---|---|---|---|---
$1/2(0)$ |  | $1/2$ |  | $1/2$ |  | $0$ |  | $0$ |  | $v_{1}$
| $1/2$ |  |  | $1$ |  | $0$ |  | $v_{2}$
| $1/2$ |  |  | $0$ |  | $1$ |  | $v_{3}$
| $3/2$ |  |  | $1$ |  | $0$ |  | $w_{1}$
$3/2(1)$ |  | $3/2$ |  | $1/2$ |  | $0$ |  | $1$ |  | $w_{3}$
Table 1: Quantum numbers of the different components resulting in a $(I,J)$
$Q\bar{Q}qqq$ hidden-flavor pentaquark containing one of the most attractive
QCD channels, according to Eq. (1). The numbers in parenthesis stand for the
isospin of $Q\bar{Q}qqs$ pentaquarks. See text for details.

###  II.2 Quark-quark interaction

To perform exploratory studies of systems with more than three-quarks it is of
basic importance to work with models that correctly describe the two- and
three-quark problems of which thresholds are made of. In Ref. Gar22  we have
adopted a generic constituent model, containing chromoelectric and
chromomagnetic contributions, tuned to reproduce the masses of the mesons and
baryons entering the various vectors, the so-called AL1 model by Semay and
Silvestre-Brac Sem94 . It has been widely used in a number of exploratory
studies of multiquark systems Sil93 ; Ric18 ; Jan04 ; Her20 ; Ric17 ; Hiy18 ;
Meg19 . It includes a standard Coulomb-plus-linear central potential,
supplemented by a smeared version of the chromomagnetic interaction,

$\displaystyle V(r)$
$\displaystyle=-\frac{3}{16}\,\tilde{\lambda}_{i}.\tilde{\lambda}_{j}\left[\lambda\,r-\frac{\kappa}{r}-\Lambda+\frac{V_{SS}(r)}{m_{i}\,m_{j}}\,\vec{\sigma}_{i}\cdot\vec{\sigma}_{j}\right]\,,$
(2) $\displaystyle V_{SS}$
$\displaystyle=\frac{2\,\pi\,\kappa^{\prime}}{3\,\pi^{3/2}\,r_{0}^{3}}\,\exp\left(-\frac{r^{2}}{r_{0}^{2}}\right)~{},\quad
r_{0}=A\left(\frac{2m_{i}m_{j}}{m_{i}+m_{j}}\right)^{-B}\,,$

where $\lambda=$ 0.1653 GeV2, $\Lambda=$ 0.8321 GeV, $\kappa=$ 0.5069,
$\kappa^{\prime}=$ 1.8609, $A=$ 1.6553 GeVB-1, $B=$ 0.2204, $m_{u}=m_{d}=$
0.315 GeV, $m_{s}=$ 0.577 GeV, $m_{c}=$ 1.836 GeV and $m_{b}=$ 5.227 GeV.
Here, $\tilde{\lambda}_{i}.\tilde{\lambda}_{j}$ is a color factor, suitably
modified for the quark-antiquark pairs. Note that the smearing parameter of
the spin-spin term is adapted to the masses involved in the quark-quark or
quark-antiquark pairs. The parameters of the AL1 potential are constrained in
a simultaneous fit of 36 well-established mesons and 53 baryons, with a
remarkable agreement with data, as could be seen in Table 2 of Ref. Sem94 . It
is worth to note that although the $\chi^{2}$ obtained in Ref. Sem94  with the
AL1 potential is slightly larger than the one obtained with other models, this
is essentially because a number of resonances with high angular momenta were
considered. The AL1 model is very well suited to study the low-energy hadron
spectra Sil96 . The spin-color algebra of the five-quark system has been
worked elsewhere Ale11 ; Ric17 .

###  II.3 Faddeev equations

The flavor-independence of the interacting potential makes the five-body
problem to factorize into a three-body problem that can be exactly solved by
means of the Faddeev equations. We follow the method developed in Ref. Gar03 ,
that it is described in detail in Ref. Gar22  for $S$\- and $P$-wave states.
Three-body states in which a particle has a given spin can only couple to
other three-body states in which that particle has the same spin, since the
spinors corresponding to different eigenvalues are orthogonal. The same
applies for isospin. This leads to a decoupling of the integral equations in
various sets in which the spin and isospin of each particle remains the same.
The different sets contributing to each $(I,J)$ state are listed in Ref. Gar22
.

For $S$-wave states one finally gets,

$T_{i;IJ}^{I_{i}S_{i}}(x_{i}q_{i})=\sum_{n}P_{n}(x_{i})T_{i;IJ}^{nI_{i}S_{i}}(q_{i})\,,$
(3)

where $T_{i;IJ}^{nI_{i}S_{i}}(q_{i})$ satisfies the one-dimensional integral
equation,

$T_{i;IJ}^{nI_{i}S_{i}}(q_{i})=\sum_{j\neq
i}\sum_{mI_{j}S_{j}}\int_{0}^{\infty}dq_{j}K_{ij;IJ}^{nI_{i}S_{i};mI_{j}S_{j}}(q_{i},q_{j};E)\;T_{j;IJ}^{mI_{j}S_{j}}(q_{j})\,,$
(4)

with

$\displaystyle K_{ij;IJ}^{nI_{i}S_{i};mI_{j}S_{j}}(q_{i},q_{j};E)=$
$\displaystyle\sum_{r}\tau_{i;I_{i}S_{i}}^{nr}(E-q_{i}^{2}/2\nu_{i})\frac{q_{j}^{2}}{2}$
(5) $\displaystyle\times\int_{-1}^{1}d{\rm
cos}\theta\;h_{ij;IJ}^{I_{i}S_{i};I_{j}S_{j}}\frac{P_{r}(x_{i}^{\prime})P_{m}(x_{j})}{E-p_{j}^{2}/2\eta_{j}-q_{j}^{2}/2\nu_{j}}\,.$

The three amplitudes $T_{1;IJ}^{rI_{1}S_{1}}(q_{1})$,
$T_{2;IJ}^{mI_{2}S_{2}}(q_{2})$, and $T_{3;IJ}^{nI_{3}S_{3}}(q_{3})$ in Eq.
(4) are coupled together.

In these equations $\tau_{i;I_{i}S_{i}}^{nr}(e)$ are expansion coefficients
given in terms of Legendre polynomials and the two-body amplitudes
$t_{i;I_{i}S_{i}}$,

$\tau_{i;I_{i}S_{i}}^{nr}(e)=\frac{2n+1}{2}\frac{2r+1}{2}\int_{-1}^{1}dx_{i}\int_{-1}^{1}dx_{i}^{\prime}\;P_{n}(x_{i})t_{i;I_{i}S_{i}}(x_{i},x_{i}^{\prime};e)P_{r}(x_{i}^{\prime})\,,$
(6)

$h_{ij;IJ}^{I_{i}S_{i};I_{j}S_{j}}$ are the spin–isospin coefficients

$\displaystyle h_{ij;IJ}^{I_{i}S_{i};I_{j}S_{j}}=$
$\displaystyle(-)^{I_{j}+i_{j}-I}\sqrt{(2I_{i}+1)(2I_{j}+1)}W(i_{j}i_{k}Ii_{i};I_{i}I_{j})$
(7)
$\displaystyle\times(-)^{S_{j}+s_{j}-J}\sqrt{(2S_{i}+1)(2S_{j}+1)}W(s_{j}s_{k}Js_{i};S_{i}S_{j})\,,$

where $W$ is a Racah coefficient and $i_{i}$, $I_{i}$, and $I$ ($s_{i}$,
$S_{i}$, and $J$) are the isospins (spins) of particle $i$, of the pair $jk$,
and of the three–body system. $\eta_{i}$ and $\nu_{i}$ are the corresponding
reduced masses,

$\displaystyle\eta_{i}$ $\displaystyle=$
$\displaystyle\frac{m_{j}m_{k}}{m_{j}+m_{k}}\,,$ $\displaystyle\nu_{i}$
$\displaystyle=$
$\displaystyle\frac{m_{i}(m_{j}+m_{k})}{m_{i}+m_{j}+m_{k}}\,,$ (8)

$\vec{p}_{i}^{\;\prime}$ is the momentum of the pair $jk$ (with $ijk$ an even
permutation of $123$) and $\vec{p}_{j}$ is the momentum of the pair $ki$ which
are given by,

$\displaystyle\vec{p}_{i}^{\;\prime}$ $\displaystyle=$
$\displaystyle-\vec{q}_{j}-\alpha_{ij}\vec{q}_{i}\,,$
$\displaystyle\vec{p}_{j}$ $\displaystyle=$
$\displaystyle\vec{q}_{i}+\alpha_{ji}\vec{q}_{j}\,,$ (9)

where,

$\displaystyle\alpha_{ij}$ $\displaystyle=$
$\displaystyle\frac{\eta_{i}}{m_{k}}\,,$ $\displaystyle\alpha_{ji}$
$\displaystyle=$ $\displaystyle\frac{\eta_{j}}{m_{k}}\,,$ (10)

so that,

$\displaystyle p_{i}^{\prime}$ $\displaystyle=$
$\displaystyle\sqrt{q_{j}^{2}+\alpha_{ij}^{2}q_{i}^{2}+2\alpha_{ij}q_{i}q_{j}{\rm
cos}\theta}\,,$ $\displaystyle p_{j}$ $\displaystyle=$
$\displaystyle\sqrt{q_{i}^{2}+\alpha_{ji}^{2}q_{j}^{2}+2\alpha_{ji}q_{i}q_{j}{\rm
cos}\theta}\,.$ (11)

Finally,

$x_{i}=\frac{p_{i}-b}{p_{i}+b}\,,$ (12)

where and $b$ is a scale parameter that has no effect on the solution.

To solve the Faddeev equations for $P$-wave states, we write them symbolically
as,

$T_{i}=t_{i}h_{ij}G_{0}T_{j},$ (13)

that has to be generalized to a matrix equation,

$\begin{pmatrix}T_{i}^{01}\\\
T_{i}^{10}\end{pmatrix}=\begin{pmatrix}t_{i}^{0}\\\
t_{i}^{1}\end{pmatrix}h_{ij}G_{0}\begin{pmatrix}\hat{q}_{i}\cdot\hat{q}_{j}&\hat{q}_{i}\cdot\hat{p}_{j}\\\
\hat{p}_{i}^{\;\prime}\cdot\hat{q}_{j}&\hat{p}_{i}^{\;\prime}\cdot\hat{p}_{j}\end{pmatrix}\begin{pmatrix}T_{j}^{01}\\\
T_{j}^{10}\end{pmatrix},$ (14)

where, from Eq. (9),

$\displaystyle\hat{q}_{i}\cdot\hat{q}_{j}$ $\displaystyle=$ $\displaystyle{\rm
cos}\theta\,,$ $\displaystyle\hat{q}_{i}\cdot\hat{p}_{j}$ $\displaystyle=$
$\displaystyle\frac{q_{i}^{2}+\alpha_{ji}q_{i}q_{j}{\rm
cos}\theta}{q_{i}p_{j}}\,,$
$\displaystyle\hat{p}_{i}^{\;\prime}\cdot\hat{q}_{j}$ $\displaystyle=$
$\displaystyle\frac{-q_{j}^{2}-\alpha_{ij}q_{i}q_{j}{\rm
cos}\theta}{p_{i}^{\prime}q_{j}}\,,$
$\displaystyle\hat{p}_{i}^{\;\prime}\cdot\hat{p}_{j}$ $\displaystyle=$
$\displaystyle\frac{-(1+\alpha_{ij}\alpha_{ji})q_{i}q_{j}{\rm
cos}\theta-\alpha_{ji}q_{j}^{2}-\alpha_{ij}q_{i}^{2}}{p_{i}^{\prime}p_{j}}\,,$
(15)

and $p_{i}^{\prime}$ and $p_{j}$ are given by Eq. (11).

In general, the two-body amplitudes $t_{i;I_{i}S_{i}}$ are obtained by solving
the Lippmann-Schwinger equation,

$t=V+VG_{0}t\,,$ (16)

where $V$ is the interaction given by Eq. (2). Due to the reduction from five
to three particles, some pairs of two-body amplitudes are coupled together.
Therefore, in this case one has to solve the coupled equations,

$\displaystyle t_{11}$ $\displaystyle=$ $\displaystyle
V_{11}+V_{11}G_{0}t_{11}+V_{12}G_{0}t_{21}\,,$ $\displaystyle t_{21}$
$\displaystyle=$ $\displaystyle V_{21}+V_{21}G_{0}t_{11}+V_{22}G_{0}t_{21}\,,$
(17)

where the diagonal interactions $V_{11}$ and $V_{22}$ show contributions from
the chromoelectric and chromomagnetic terms of the interaction, while the off-
diagonal interactions $V_{12}$ and $V_{21}$ contain only the contribution of
the chromomagnetic part of the interacting potential. As expected, the
confinement and Coulomb terms are the dominant ones such that the spin-spin
term is just a small perturbation. The effect of the non-diagonal terms is
very small and it can be safely neglected.

##  III Results

We have solved the three-body problem for the different $(I,J)$
$Q\bar{Q}qqq^{\prime}$ states as discussed in Sec. II. We show in Table 2 the
binding energy of the most favorable five-quark states that could be observed
in nature.

$Q$ | $q$ | $q^{\prime}$ | $v_{1}$ | $v_{2}$ | $v_{3}$ | $w_{1}$ | $w_{3}$
---|---|---|---|---|---|---|---
$c$ | $u,d$ | $u,d$ | 7 | 17 | 24 | 12 | 12
$c$ | $u,d$ | $s$ | 133 | 138 | 143 | 134 | 134
$b$ | $u,d$ | $u,d$ | 39 | 41 | 52 | 40 | 40
$b$ | $u,d$ | $s$ | 165 | 167 | 175 | 166 | 166
Table 2: Binding energy, in MeV, of the different $Q\bar{Q}qqq^{\prime}$
pentaquarks.

Let us first note the degeneracy existing between $I=1/2$ and $I=3/2$
$Q\bar{Q}qqq$ $J=3/2$ pentaquarks ($I=0$ and $I=1$ $Q\bar{Q}qqs$ $J=3/2$
pentaquarks), as could have been expected a priori due to the isospin
independence of the potential model in Eq. (2), although the result is not
trivial due to the requirements of the Pauli principle. Secondly, it has been
checked that the conclusions dealing with stability or instability of
multiquarks survive variations of the parameters, we have specifically checked
that the pattern remains for different strengths of the spin-spin interaction
by modifying the regularization parameter, $r_{0}$ in Eq. (2).

One can determine the general properties of the multiquarks favored by the
quark correlations dominating the QCD phenomena shown in Table 1. In the
charmonium sector, the mass difference between the $Q\bar{Q}$ $\\{{\bf
1}_{c},{\bf 1}_{f},1_{s}\\}$ and $\\{{\bf 1}_{c},{\bf 1}_{f},0_{s}\\}$
correlated states could be assimilated to the $J/\Psi-\eta_{c}$ mass
difference. Likewise, in the bottomonium sector it corresponds to the
$\Upsilon-\eta_{b}$ mass difference. The mass difference between the $qq$
$\\{{\bf\bar{3}}_{c},{\bf 6}_{f},1_{s}\\}$ and
$\\{{\bf\bar{3}}_{c},{\bf\bar{3}}_{f},0_{s}\\}$ has been estimated from full
lattice QCD simulations to be in the range of 100$-$200 MeV Fra21 ; Ale06 ;
Gre10 . Thus, we have fixed the effective mass difference of the correlated
structures considering the following realistic values,

$\displaystyle\Delta M^{c\bar{c}}=M^{c\bar{c}}_{\\{{\bf 1}_{c},{\bf
1}_{f},1_{s}\\}}-M^{c\bar{c}}_{\\{{\bf 1}_{c},{\bf 1}_{f},0_{s}\\}}$
$\displaystyle=86\,\,{\rm MeV}\,,$ $\displaystyle\Delta
M^{b\bar{b}}=M^{b\bar{b}}_{\\{{\bf 1}_{c},{\bf
1}_{f},1_{s}\\}}-M^{b\bar{b}}_{\\{{\bf 1}_{c},{\bf 1}_{f},0_{s}\\}}$
$\displaystyle=61\,\,{\rm MeV}\,,$ $\displaystyle\Delta
M^{qq}=M^{qq}_{\\{{\bf\bar{3}}_{c},{\bf
6}_{f},1_{s}\\}}-M^{qq}_{\\{{\bf\bar{3}}_{c},{\bf\bar{3}}_{f},0_{s}\\}}$
$\displaystyle=146\,\,{\rm MeV}\,.$ (18)

Then, denoting by $M_{0}^{Q\bar{Q},q}$ the sum of the masses of a spin zero
$Q\bar{Q}$ diquark, a spin zero $qq$ diquark and a light quark, the mass of
the $Q\bar{Q}qqq$ states in Table 1 would be given by,

$M_{i}=M_{0}^{Q\bar{Q},q}-B_{i}+\Delta M^{Q\bar{Q}}\,\delta_{s_{2},1}+\Delta
M^{qq}\,\delta_{s_{3},1}\,,$ (19)

where $B_{i}$ is the binding energy calculated above, see Table 2. For
$Q\bar{Q}qqs$ states we have a similar expression,

$M_{i}=M_{0}^{Q\bar{Q},s}-B_{i}+\Delta M^{Q\bar{Q}}\,\delta_{s_{2},1}+\Delta
M^{qq}\,\delta_{s_{3},1}\,.$ (20)

By taking $M_{0}^{c\bar{c},q}=4319$ MeV, one gets the results shown in Table
3.

Vector |  | $(I)J^{P}$ |  |  $M_{\rm Th}$ (MeV) |  | State |  |  $M_{\rm Exp}$ (MeV)
---|---|---|---|---|---|---|---|---
$v_{1}$ |  | $(1/2)1/2^{-}$ |  | 4312 |  | $P_{c}(4312)^{+}$ |  | $4311.9\pm 0.7^{+6.8}_{-0.6}$
$v_{2}$ |  | $(1/2)1/2^{-}$ |  | 4388 |  | $P_{c}(4380)^{+}$ |  | $4380\pm 8\pm 29$
$w_{1}$ |  | $(1/2)3/2^{-}$ |  | 4393 |  |
$v_{3}$ |  | $(1/2)1/2^{-}$ |  | 4441 |  | $P_{c}(4440)^{+}$ |  | $4440.3\pm 1.3^{+4.1}_{-4.7}$
$w_{3}$ |  | $(3/2)3/2^{-}$ |  | 4453 |  | $P_{c}(4457)^{+}$ |  | $4457.3\pm 0.6^{+4.1}_{-1.7}$
Table 3: Properties of the $c\bar{c}qqq$ pentaquarks in Table 1.

As can be seen, there is a good agreement between theoretical states showing
the most important correlations dictated by the QCD phenomena and the
experimental data Aai15 ; Aai19 . Thus, Table 3 presents a theoretical spin-
parity assignment for the existing hidden-charm pentaquarks. A careful
analysis of the results and a detailed comparison with other approaches in the
literature was performed in Ref. Gar22 .

Now, we can make parameter-free predictions for the lowest-lying hidden-bottom
pentaquarks, for which there is still no experimental. The results are shown
in Table 4, compared to other results available in the literature.

Vector |  | $(I)J^{P}$ |  | Our model |  | Ref. Wul17  |  | Ref. Yan19  |  | Ref. Fer19
---|---|---|---|---|---|---|---|---|---|---
$v_{1}$ |  | $(1/2)1/2^{-}$ |  | 11062 |  | 11137.1 |  | 11080 (11078) |  | 10605
$v_{2}$ |  | $(1/2)1/2^{-}$ |  | 11121 |  | 11148.9 |  | 11115 (11043) |  | 10629
$w_{1}$ |  | $(1/2)3/2^{-}$ |  | 11122 |  | 11237.5 |  | 11124 (11122) |  | 10629
$v_{3}$ |  | $(1/2)1/2^{-}$ |  | 11195 |  | 11205.0 |  | $-$ |  | $-$
$w_{3}$ |  | $(3/2)3/2^{-}$ |  | 11207 |  | 11370.6 |  | 11112 (10999) |  | $-$
Table 4: Predictions of different models for the mass, in MeV, of the
$b\bar{b}qqq$ pentaquarks in Table 1.

Ref. Wul17  considers a color-magnetic interaction to estimate the mass
splitting of the different states with respect to a reference mass adjusted to
experimental data. Ref. Yan19  makes use of a chiral quark model and solves
the five-body bound state problem by the Gaussian expansion method. We quote
the results corresponding to the color-singlet calculation, which, from a
theoretical point of view, would be the closest to our model, and between
parenthesis the coupled channel calculation including hidden-color channels.
Ref. Fer19  presents results of a hadro-quarkonium model with baryons of
$I=1/2$ and two different chromoelectric polarizability strengths. We show the
results of the model with lower attraction, in which the hidden-bottom
pentaquarks are located in the 10.6$-$10.9 GeV energy region 111The model with
higher attraction, derived by considering charmonia as a pure Coulombic
system, predicts the lowest-lying hidden-bottom pentaquarks in the 10.4$-$10.7
GeV energy region.. Positive parity states have smaller binding energies and
appear about 150 MeV above the negative parity states. In the first two
references, which use quark degrees of freedom, there is a rich spectra of
pentaquarks with different isospins, $I=1/2$ and $3/2$, and spins,
$J^{P}=1/2^{-}$, $3/2^{-}$ and $5/2^{-}$. We only show the lowest lying states
to compare with those obtained with our model. It is interesting to note that
most of the hidden-bottom pentaquarks predicted by quark substructure models
are in the same energy region, 11.0$-$11.2 GeV. The hadro-quarkonium model
finds more deeply bound states. These differences support a future
experimental effort to look for hidden-bottom pentaquarks in this energy
region, what would be a clear signal to discriminate between the different
dynamics that may drive to hidden-bottom pentaquarks.

Very recently the LHCb Collaboration announced the observation of a new
strange pentaquark $P_{\Psi s}^{\Lambda}(4338)$ in the decay $B^{-}\to
J/\Psi\Lambda\bar{p}$ as a resonance in the $J/\Psi\Lambda$ invariant mass
distribution Che22 . It has a mass of $4338.2\pm 0.7$ and a minimal quark
content $c\bar{c}uds$. One can take advantage of the recent discovery of this
hidden-charm pentaquark with strangeness to tune the free parameter in the
strange sector, by taking $M_{0}^{c\bar{c},s}=4471$ MeV. Thus, one gets the
results shown in Table 5.

Vector |  | $(I)J^{P}$ |  |  $M_{\rm Th}$ (MeV) |  | State |  |  $M_{\rm Exp}$ (MeV) |  |  $M^{\dagger}_{\rm Exp}$ (MeV)
---|---|---|---|---|---|---|---|---|---|---
$v_{1}$ |  | $(0)1/2^{-}$ |  | 4338 |  | $P_{\Psi s}^{\Lambda}(4338)$ |  | $4338.2\pm 0.7$ |  | $4338\pm 0.7$
$v_{2}$ |  | $(0)1/2^{-}$ |  | 4419 |  |  |  |  |  |
$w_{1}$ |  | $(0)3/2^{-}$ |  | 4423 |  |  |  |  |  |
$v_{3}$ |  | $(0)1/2^{-}$ |  | 4474 |  | $P_{cs}(4459)$ |  | $4458.8\pm 2.9^{+4.7}_{-1.1}$ |  | $4454.9\pm 2.7$
$w_{3}$ |  | $(1)3/2^{-}$ |  | 4483 |  |  |  |  |  | $4467.8\pm 3.7$
Table 5: Properties of the $c\bar{c}qqs$ pentaquarks in Table 1.

Moreover, the LHCb Collaboration has reported evidence of a structure in the
$J/\Psi\Lambda$ invariant mass distribution obtained from an amplitude
analysis of $\Xi^{-}_{b}\to J/\Psi\Lambda K^{-}$ decays Aak21 . The observed
structure, with mass $4458.8\pm 2.9^{+4.7}_{-1.1}$ MeV, is consistent with
being due to a charmonium pentaquark with strangeness, i.e., minimal quark
content $c\bar{c}uds$. These two states are collected in the penultimate
column, $M_{\rm Exp}$, of Table 5. However, the structure observed in Ref.
Aak21  is also consistent with being due to two resonances, with masses
$4454.9\pm 2.7$ MeV and $4467.8\pm 3.7$ MeV. This experimental situation is
reflected in the last column, $M^{\dagger}_{\rm Exp}$, of Table 5. The
existence of two states resembles the situation found in the nonstrange sector
with the $P_{c}(4440)^{+}$ and the $P_{c}(4457)^{+}$ Kar21 , two resonances
predicted as $J=1/2$ and $3/2$ states, but different isospin in our model, see
Table 3.

As can be seen, there is a good agreement between theoretical states showing
the most important correlations dictated by the QCD phenomena and the
experimental data Aak21 ; Che22 . Therefore, Table 5 presents a theoretical
spin-parity assignment for the existing strange hidden-charm pentaquarks
deduced from our model together with the prediction of a few new states in the
same energy region.

Vector |  | $(I)J^{P}$ |  | Our model |  | Ref. Wul17  |  | Ref. Wan20  |  | Ref. Hup22  |  | Ref. Fer20
---|---|---|---|---|---|---|---|---|---|---|---|---
$v_{1}$ |  | $(0)1/2^{-}$ |  | 4338 |  | 4362.3 |  | $4319.4^{+2.8}_{-3.0}$ |  | 4330 |  | 4474
$v_{2}$ |  | $(0)1/2^{-}$ |  | 4419 |  | 4548.2 |  | $4456.9^{+3.2}_{-3.3}$ |  | 4475 |  | 4522
$w_{1}$ |  | $(0)3/2^{-}$ |  | 4423 |  | 4556.1 |  | $4423.7^{+6.4}_{-6.8}$ |  | 4440 |  | 4522
$v_{3}$ |  | $(0)1/2^{-}$ |  | 4474 |  | 4571.4 |  | $4463.0^{+2.8}_{-3.0}$ |  | 4476 |  | $-$
$w_{3}$ |  | $(1)3/2^{-}$ |  | 4483 |  | 4846.4 |  | $-$ |  | $-$ |  | $-$
Table 6: Predictions of different models for the mass, in MeV, of the
$c\bar{c}qqs$ pentaquarks in Table 1.

In Table 6 we compare our results with others available in the literature. The
perturbative color-magnetic calculation of Ref. Wul17  predicts large
splittings among the lowest lying states. The chiral effective field theory
potentials of Ref. Wan20  are closer to the results of our model, with the
lowest lying states in the 4.3$-$4.4 GeV energy region. They do not find bound
state solutions for $I=1$ channels. Ref. Hup22  presents results of a chiral
quark model using a variational method with radial wave functions expanded in
terms of Gaussians. The model, successful in describing the nonstrange
pentaquarks, predicts very small binding energies in the different baryon-
meson channels. They focus on $I=0$ states and do not show results for $I=1$
systems. The hadro-quarkonium model of Ref. Fer20  obtains the larger masses
for the hidden-charm pentaquarks with strangeness, unlike non-strange and
strange hidden-bottom pentaquarks where they found much smaller masses that
quark based models.

Similarly to the hidden-charm sector, we can now make parameter-free
predictions for the lowest-lying strange hidden-bottom pentaquarks, for which
experimental data are not yet available. The results are shown in Table 7,
compared to other results in the literature.

Vector |  | $(I)J^{P}$ |  | Our model |  | Ref. Wul17  |  | Ref. Fer20
---|---|---|---|---|---|---|---|---
$v_{1}$ |  | $(0)1/2^{-}$ |  | 11088 |  | 11117.7 |  | 10671
$v_{2}$ |  | $(0)1/2^{-}$ |  | 11147 |  | 11183.8 |  | 10695
$w_{1}$ |  | $(0)3/2^{-}$ |  | 11148 |  | 11180.2 |  | 10695
$v_{3}$ |  | $(0)1/2^{-}$ |  | 11224 |  | 11301.2 |  | $-$
$w_{3}$ |  | $(1)3/2^{-}$ |  | 11233 |  | 11509.0 |  | $-$
Table 7: Predictions of different models for the mass, in MeV, of the
$b\bar{b}qqs$ pentaquarks in Table 1.

The strange hidden-bottom pentaquarks are obtained close to the non-strange
case. They are found to be in the 11.1$-$11.3 GeV energy region. This is the
same conclusion of the lowest lying states of the perturbative chromomagnetic
model of Ref. Wul17 . Analogously to the hidden-bottom non-strange case, the
hadro-quarkonium model of Ref. Fer20  reports smaller masses 222The model with
higher attraction, derived by considering charmonia as a pure Coulombic
system, predicts the lowest-lying hidden-bottom pentaquarks in the 10.4 GeV
energy region.. It also predicts smaller splittings than quark-based model
between the lowest-lying states. Thus, hidden-bottom pentaquarks, either with
or without strangeness, seems to be an adequate tool to discriminate among
different models for the dynamics of multiquark systems. Models based on the
quark substructure indicate that future searches of nonstrange and strange
hidden-bottom pentaquarks should be carried out in the 11 GeV energy region.

It is worth mentioning that other alternatives have been used for the study of
strange and non-strange hidden-flavor pentaquarks. Among them a diquark-
triquark model Zhu16  has been applied to the same systems studied in this
work. The model exploits the color attractive configurations of a not
pointlike triquark with a light cone distribution amplitude for the
pentaquark. An effective diquark-triquark Hamiltonian based on a spin-orbital
interaction is established. The results of this model suggest that the
$P_{c}(4380)^{+}$ could correspond to a $J^{P}=3/2^{-}$ state with a mass of
4349 MeV and the state originally reported as $P_{c}(4450)^{+}$ (see Ref.
Gar22  for details) would correspond to a $J^{P}=5/2^{+}$ with a mass of 4453
MeV. The lightest state is a $J^{P}=3/2^{-}$ hidden-charm pentaquark with a
mass of 4085 MeV. $J^{P}=1/2^{-}$ states are theoretically discussed but their
masses are not shown in the paper. Following the same strategy strange hidden-
charm and hidden-bottom pentaquarks were studied in Ref. Zhu16 . For the sake
of completeness we quote the lowest lying state for each quantum numbers in
the different flavor sectors: hidden-charm
$(3/2^{-},5/2^{-},5/2^{+})=(4085,4433,4453)$ MeV; strange hidden-charm
$(3/2^{-},5/2^{-},5/2^{+})=(4314,4624,4682)$ MeV; hidden-bottom
$(3/2^{-},5/2^{-},5/2^{+})=(10723,11045,1146)$ MeV; strange hidden-bottom
$(3/2^{-},5/2^{-},5/2^{+})=(10981,11264,11413)$ MeV.

Preliminary analysis of the experimental data in the hidden-charm sector
suggested the coexistence of negative and positive parity pentaquarks in the
same energy region Aai15 . We have calculated the mass of the lowest positive
parity state, the first orbital angular momentum excitation of the $v_{1}$
state. The technical details have been described in Sec. II.3. We chose this
state because it is made up of the most strongly correlated structures,
$Q\bar{Q}$ $\\{{\bf 1}_{c},{\bf 1}_{f},0_{s}\\}$ and $qq$
$\\{{\bf\bar{3}}_{c},{\bf\bar{3}}_{f},0_{s}\\}$. Then, it might have a similar
mass to negative parity states made up of spin 1 structures.

$Q$ |  | $q$ |  | $q^{\prime}$ |  | Mass (MeV)
---|---|---|---|---|---|---
$c$ |  | $u,d$ |  | $u,d$ |  | 4527
$c$ |  | $u,d$ |  | $s$ |  | 4552
$b$ |  | $u,d$ |  | $u,d$ |  | 11273
$b$ |  | $u,d$ |  | $s$ |  | 11291
Table 8: Mass of the first positive parity states, two degenerate states with
quantum numbers $J^{P}=1/2^{+}$ and $3/2^{+}$, for the different
$Q\bar{Q}qqq^{\prime}$ pentaquarks.

As can be seen in Table 8, the lowest lying positive parity hidden-charm
pentaquarks appear above 4.5 GeV, a mass slightly larger than that of the
states measured so far. The orbital excitation is smaller in the hidden-bottom
case such that positive parity states appear closer to the negative parity
ground states. Similarly, most of the theoretical works prefer to assign the
lowest lying pentaquarks to negative parity states. Almost degenerate negative
and positive parity states may occur for hidden-flavor pentaquarks that have
been detected in the same channel but that were formed by different pairs of
quarkonium-nucleon states Eid16 , one of them radially excited. The assignment
of negative and positive parity states to different parity Born-Oppenheimer
multiplets has already been suggested as a plausible solution in the triquark-
diquark picture of Ref. Gir19 . Nevertheless, this issue remains one of the
most challenging problems in the pentaquark phenomenology that should be first
confirmed experimentally.

Multiquark states would show very different decay patterns regarding its
internal structure Wul17 . The decays of the pentaquarks in Table 1 into an
heavy meson + heavy baryon are strongly suppressed since decays into open
flavor channels can go only via $t$-channel exchange by a heavy meson. Due to
the content of the pentaquarks states they would follow the decays of
quarkonium excited states, $\Psi(nS)[\Upsilon(nS)]$ and
$\eta_{c}(nS)[\eta_{b}(nS)]$. Thus, multiquarks containing a spin zero heavy
quark-antiquark pair: $v_{1}$, $v_{3}$ and $w_{3}$ in Table 1, would be
narrower than those with a spin one heavy quark-antiquark pair: $v_{2}$ and
$w_{1}$ in Table 1. This corresponds nicely with the experimental observations
up to now. However, besides the contribution to the width of the substructures
that form each pentaquark, one should also consider the width due to the bound
nature of the system. At this point it is worth to mention that the final
width of a resonance does not come only determined by its internal content,
but there are significant corrections due to an interplay between the phase
space for its decay to the detection channel and its mass with respect to the
hadrons generating the state Gar18 .

Finally, it is worth noting that the correlations used do not lead to stable
multiquarks for any quark substructure, in the same way the $NN$ short-range
repulsion induced by the one-gluon exchange dynamics is not universal and
disappears for other two-hadron channels Oka80 . Thus, for example, the QCD
correlations used in this work would not constraint the color wave function of
pentaquarks with anticharm or beauty, $\bar{Q}qqqq$. Therefore, such systems
would not present bound states, as recently discussed in Ref. Ric19 , due to a
non favorable interplay between chromoelectric and chromomagnetic effects.

##  IV Summary

In brief, we have studied hidden-flavor pentaquarks which can be framed within
the same type of structures discussed in our recent work of Ref. Gar22 . In
particular, we addressed hidden-flavor pentaquarks $Q\bar{Q}qqq^{\prime}$,
with $Q=b$ or $c$, $q=u$ or $d$, and $q^{\prime}=u$ $d$ or $s$. The color
Coulomb-like nature of the short-range one-gluon exchange interaction leads to
a frozen color wave function of the five-body system, which allows to reduce
the problem to a more tractable three-body problem. The three-body problem has
been exactly solved by means of the Faddeev equations. The interactions
between the constituents are deduced from a generic constituent model, the AL1
model, that gives a nice description of the low-energy baryon and meson
spectra.

For the non-strange and strange hidden-charm pentaquarks a pattern was
obtained that compares well with the experimental data. The tentative spin-
parity assignment of the different pentaquarks agrees well with other
approaches dedicated to study a particular set of states. Under the assumption
that nature favors multiquarks which are made up of correlated substructures
dictated by QCD, we have estimated the mass of the lowest lying pentaquarks.
We have considered realistic values for the mass difference of the correlated
quark pairs. A good description of the experimental data has been obtained.

For the non-strange and strange hidden-bottom pentaquarks we present
parameter-free predictions. The splitting between the lowest-lying states is
smaller than in the charm sector, due to the larger mass of the bottom quark
appearing in the chromomagnetic potential. The hidden-bottom pentaquarks are
found to be in the 11.0$-$11.2 GeV energy region. This is a general conclusion
for model based on the quark substructure. Hadro-quarkonium models predict
smaller masses for hidden-bottom pentaquarks, which makes them an excellent
test bench for testing the dynamics of multiquark systems.

The coexistence of negative and positive parity states in the same energy
region appears to be more likely in the bottom sector. It is worth noting that
the correlations used do not lead to stable multiquarks for any quark
substructure. Thus, for example, the QCD correlations used in this work would
not constraint the color wave function of pentaquarks with anticharm or
beauty.

Bound states and resonances are usually very sensitive to model details and
therefore theoretical investigations with different phenomenological models
are highly desirable. We have tried to minimize the influence of the
interacting potential by using a standard constituent model and we have
explored the consequences of dynamical correlations arising from the Coulomb-
like nature of the short-range potential. The pattern obtained could be
scrutinized against the future experimental results providing a great
opportunity for extending our knowledge to some unreached part of the hadron
spectra. More such exotic baryons are expected and needed to make reliable
hypotheses on the way the interactions in the system are shaping the spectra.

##  V Acknowledgments

This work has been partially funded by COFAA-IPN (México) and by Ministerio de
Ciencia e Innovación and EU FEDER under Contracts No. PID2019-105439GB-C22 and
RED2018-102572-T.

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