The LHCb Collaboration has recently reported strong evidences of the existence of pentaquark states in the hidden-charm baryon sector, the so-called and signals. Five-quark bound states in the hidden-charm sector were explored by us using, for the quark-quark interaction, a chiral quark model which successfully explains meson and baryon phenomenology, from the light to the heavy quark sector. We extend herein such study but to the hidden-bottom pentaquark sector, analyzing possible bound-states with spin-parity quantum numbers , and , and in the and isospin sectors. We do not find positive parity hidden-bottom pentaquark states; however, several candidates with negative parity are found with dominant baryon-meson structures . The calculated distances among any pair of quarks within the bound-state reflect that molecular-type bound-states are favored when only color-singlet configurations are considered in the coupled-channels calculation whereas compact pentaquarks, which are also deeply bound, can be found when hidden-color configurations are added. Finally, our findings resemble the ones found in the hidden-charm sector but, as expected, we find in the hidden-bottom sector larger binding energies and bigger contributions of the hidden-color configurations.
pacs:12.38.-t and 12.39.-x and 14.20.-c and 14.20.Pt
After decades of experimental and theoretical studies of hadrons, the conventional picture of mesons and baryons as, respectively, quark-antiquark and -quark bound states is being left behind. On one hand, Quantum Chromodynamics (QCD), the non-Abelian quantum field theory of the strong interactions, does not prevent to have exotic hadrons such as glueballs, quark-gluon hybrids and multiquark systems. On the other hand, more than two dozens of nontraditional charmonium- and bottomonium-like states, the so-called XYZ mesons, have been observed in the last 15 years at B-factories (BaBar, Belle and CLEO), -charm facilities (CLEO-c and BESIII) and also proton-(anti)proton colliders (CDF, D0, LHCb, ATLAS and CMS).
In , the LHCb Collaboration observed two hidden-charm pentaquark states in the invariant mass spectrum of the decay Aaij et al. (2015). One is with a mass of and a width of , and another is with a mass of and a width of . The preferred assignments are of opposite parity, with one state having spin and the other .
The discovery of the and has triggered a strong theoretical interest on multiquark systems. The interested reader is directed to the recent review Chen et al. (2016a) in order to have a global picture of the current progress; however, one can highlight those theoretical studies of the and in which different kind of quark arrangements are used such as diquark-triquark Wang et al. (2016); Zhu and Qiao (2016); Lebed (2015), diquark-diquark-antiquark Wang et al. (2016); Anisovich et al. (2015); Maiani et al. (2015); Ghosh et al. (2015); Wang and Huang (2016); Wang (2016a, b), and meson-baryon molecule Wang et al. (2016); Roca et al. (2015); Chen et al. (2015a); Huang et al. (2016); MeiÃner and Oller (2015); Xiao and MeiÃner (2015); He (2016); Chen et al. (2015b, 2016b); Yamaguchi and Santopinto (2017); He (2017); Ortega et al. (2017a); Azizi et al. (2017); Anwar et al. (2018). It is also noteworthy that some recent investigations have considered other possible physical mechanisms as the origin of the experimental signals like kinematic effects and triangle singularities Guo et al. (2015); Mikhasenko (2015); Liu et al. (2016); Liu and Oka (2016); Bayar et al. (2016).
The observation of hadrons containing valence -quarks is historically followed by the identification of similar structures with -quark content. Therefore, it is natural to expect a subsequent observation of the bottom analogues of the and resonances. In fact, the LHCb Collaboration has recently made an attempt to search for pentaquark states containing a single -quark, that decays weakly via the transition, in the final states , , , and Aaij et al. (2018); and thus reports about similar explorations in the hidden-bottom pentaquark sector should be expected in the near future.
Theoretical investigations of the spectrum of hidden-bottom pentaquarks as well as their electromagnetic, strong and weak decays help in the experimental hunt mentioned above. In addition to this, further theoretical studies supply complementary information on the internal structure and inter-quark interactions of pentaquarks with heavy quark content. In Ref. Shimizu et al. (2016), besides the state, the possible existence of hidden-bottom pentaquarks with a mass around and quantum numbers was emphasized; it was also indicated that there may exist some loosely-bound molecular-type pentaquarks in other heavy quark sectors. See also Refs. Wu et al. (2010); Wu and Zou (2012) for more information on the properties of the charmed and bottom pentaquark states using the coupled-channel unitary approach as well as Refs. Guo et al. (2015); Liu et al. (2016); Guo et al. (2016); Bayar et al. (2016) for illuminating discussions on the structure of pentaquarks and their possible relation with triangle singularities.
We study herein, within a chiral quark model formalism, the possibility of having pentaquark bound-states in the hidden-bottom sector with quantum numbers , and , and in the and isospin sectors. This work is a natural extension of the analysis performed in Ref. Yang and Ping (2017) in which similar structures were studied but in the hidden-charm sector. In Ref. Yang and Ping (2017), the was suggested to be a bound state of with quantum numbers whereas the nature of the structure was not clearly established because, despite of having a couple of possible candidates attending to the agreement between theoretical and experimental masses, there was an inconsistency between the parity of the state determined experimentally and those predicted theoretically. Further pentaquark bound-states which contain dominant and Fock-state components were also found in the region about .
All the details about our computational framework will be described later but let us sketch here some of its main features. Our chiral quark model (ChQM) is based on the fact that chiral symmetry is spontaneously broken in QCD and, among other consequences, it provides a constituent quark mass to the light quarks. To restore the chiral symmetry in the QCD Lagrangian, Goldstone-boson exchange interactions appear between the light quarks. This fact is encoded in a phenomenological potential which already contains the perturbative one-gluon exchange (OGE) interaction and a nonperturbative linear-screened confining term.111The interested reader is referred to Refs. Valcarce et al. (2005); Segovia et al. (2013) for detailed reviews on the naive quark model in which this work is based. It is worth to note that chiral symmetry is explicitly broken in the heavy quark sector and this translates in our formalism to the fact that the interaction terms between light-light, light-heavy and heavy-heavy quarks are not the same, i.e. while Goldstone-boson exchanges are considered when the two quarks are light, they do not appear in the other two configurations: light-heavy and heavy-heavy; however, the one-gluon exchange and confining potentials are flavor blindness.
The five-body bound state problem is solved by means of the Gaußian expansion method (GEM) Hiyama et al. (2003) which has been demonstrated to be as accurate as a Faddeev calculation (see, for instance, Figs. 15 and 16 of Ref. Hiyama et al. (2003)). As it is well know, the quark model parameters are crucial in order to describe particular physical observables. We have used values that have been fitted before through hadron Valcarce et al. (1996); Vijande et al. (2005); Segovia et al. (2008a, b); Ortega et al. (2016a); Yang et al. (2018), hadron-hadron Fernandez et al. (1993); Valcarce et al. (1994); Ortega et al. (2010, 2016b, 2017b) and multiquark Vijande et al. (2006); Yang and Ping (2017, 2018) phenomenology.
Ii Theoretical framework
Although Lattice QCD (LQCD) has made an impressive progress on understanding multiquark systems Alexandrou et al. (2002); Okiharu et al. (2005) and the hadron-hadron interaction Prelovsek et al. (2015); Lang et al. (2014); Briceno et al. (2018), the QCD-inspired quark models are still the main tool to shed some light on the nature of the multiquark candidates observed by experimentalists.
The general form of our five-body Hamiltonian is given by Yang and Ping (2017)
where is the center-of-mass kinetic energy and the two-body potential
includes the color-confining, one-gluon exchange and Goldstone-boson exchange interactions. Note herein that the potential could contain central, spin-spin, spin-orbit and tensor contributions; only the first two will be considered attending the goal of the present manuscript and for clarity in our discussion.
Color confinement should be encoded in the non-Abelian character of QCD. Studies of lattice-regularized QCD have demonstrated that multi-gluon exchanges produce an attractive linearly rising potential proportional to the distance between infinite-heavy quarks Bali et al. (2005). However, the spontaneous creation of light-quark pairs from the QCD vacuum may give rise at the same scale to a breakup of the created color flux-tube Bali et al. (2005). We have tried to mimic these two phenomenological observations by the expression:
where and are model parameters, and the SU(3) color Gell-Mann matrices are denoted as . One can see in Eq. (3) that the potential is linear at short inter-quark distances with an effective confinement strength , while it becomes constant at large distances.
The one-gluon exchange potential is given by
where is the quark mass and the Pauli matrices are denoted by . The contact term of the central potential has been regularized as
with a regulator that depends on , the reduced mass of the quark–(anti-)quark pair.
The wide energy range needed to provide a consistent description of mesons and baryons from light to heavy quark sectors requires an effective scale-dependent strong coupling constant. We use the frozen coupling constant of, for instance, Ref. Segovia et al. (2013)
in which , and are parameters of the model.
The central terms of the chiral quark–(anti-)quark interaction can be written as
where is the standard Yukawa function defined by . We consider the physical meson instead of the octet one and so we introduce the angle . The are the SU(3) flavor Gell-Mann matrices. Taken from their experimental values, , and are the masses of the SU(3) Goldstone bosons. The value of is determined through the PCAC relation Scadron (1982). Finally, the chiral coupling constant, , is determined from the coupling constant through
which assumes that flavor SU(3) is an exact symmetry only broken by the different mass of the strange quark.
The model parameters have been fixed in advance reproducing hadron Valcarce et al. (1996); Vijande et al. (2005); Segovia et al. (2008a, b); Ortega et al. (2016a); Yang et al. (2018), hadron-hadron Fernandez et al. (1993); Valcarce et al. (1994); Ortega et al. (2010, 2016b, 2017b) and multiquark Vijande et al. (2006); Yang and Ping (2017, 2018) phenomenology. For clarity, the ones involved in this calculation are listed in Table 1. They were used in Ref. Yang and Ping (2017) to study possible hidden-charm pentaquark bound-states with quantum numbers , and ; moreover, their properties were compared with those associated with the hidden-charm pentaquark signals observed by the LHCb Collaboration in Ref. Aaij et al. (2015).
The pentaquark wave function is a product of four terms: color, flavor, spin and space wave functions. Concerning the color degree-of-freedom, multiquark systems have richer structure than the conventional mesons and baryons. For instance, the -quark wave function must be colorless but the way of reaching this condition can be done through either a color-singlet or a hidden-color channel or both. The authors of Refs. Harvey (1981); Vijande et al. (2009) assert that it is enough to consider the color singlet channel when all possible excited states of a system are included. However, a more economical way of computing is considering both, the color singlet wave function:
and the hidden-color one:
where is an index which stands for the symmetric (anti-symmetric) configuration of two quarks in the -quark sub-cluster. All color configurations have been used herein, as in the case of the hidden-charm pentaquarks studied in Ref. Yang and Ping (2017).
In analogy to the study of the -type bound states in Ref. Yang and Ping (2017), we assume that the flavor wave function of the system is composed by and configurations. According to the SU(2) symmetry in isospin space, the flavor wave functions for the sub-clusters mentioned above are given by:
Consequently, the flavor wave-functions for the 5-quark system with isospin or are
where the third component of the isospin is set to be equal to the total one without loss of generality because there is no interaction in the Hamiltonian that can distinguish such component.
We consider herein 5-quark bound states with total spin ranging from to . Since our Hamiltonian does not have any spin-orbital coupling dependent potential, we can assume that third component of the spin is equal to the total one without loss of generality. Our spin wave function is given by:
for , and
for , and
for . These expressions can be obtained easily considering the 3-quark and quark-antiquark sub-clusters and using SU(2) algebra. They were derived in Ref. Yang and Ping (2017) for the hidden-charm pentaquarks.
Among the different methods to solve the Schrödinger-like 5-body bound state equation, we use the Rayleigh-Ritz variational principle which is one of the most extended tools to solve eigenvalue problems due to its simplicity and flexibility. However, it is of great importance how to choose the basis on which to expand the wave function. The spatial wave function of a -quark system is written as follows:
where the internal Jacobi coordinates are defined as
This choice is convenient because the center-of-mass kinetic term can be completely eliminated for a nonrelativistic system.
In order to make the calculation tractable, even for complicated interactions, we replace the orbital wave functions, ’s in Eq. (39), by a superposition of infinitesimally-shifted Gaussians (ISG) Hiyama et al. (2003):
where the limit must be carried out after the matrix elements have been calculated analytically. This new set of basis functions makes the calculation of 5-body matrix elements easier without the laborious Racah algebra Hiyama et al. (2003). Moreover, all the advantages of using Gaußians remain with the new basis functions.
Finally, in order to fulfill the Pauli principle, the complete antisymmetric wave-function is written as
where is the antisymmetry operator of the 5-quark system. This is needed because we have constructed an antisymmetric wave function for only two quarks of the 3-quark sub-cluster, the remaining (anti)-quarks of the system have been added to the wave function by simply considering the appropriate Clebsch-Gordan coefficients. Moreover, the antisymmetry operator has six terms but since we are considering that the system is made by the quark arrangements and , we have
for the configuration, and
for the structure.
|Index||; ;||Channel||; ;||Channel|
In the present calculation, we investigate the possible lowest-lying states of the pentaquark system taking into account the and configurations in which the considered baryons have always positive parity and the open- and hidden-bottom mesons are either pseudoscalars or vectors .222There may exist other baryon-meson structures which contain excited hadrons such as , and so on; all of them are beyond the scope of the present calculation. This means that, in our approach, a pentaquark state with positive parity should have at least one unity of angular momentum: , whereas the negative parity states have . Reference Yang and Ping (2017) showed that positive parity hidden-charm pentaquark states are always above its corresponding non-interacting baryon-meson threshold and the same situation is found within the hidden-bottom sector.
For negative parity hidden-bottom pentaquarks we assume that the angular momenta , , , , which appear in Eq. (39), are . In this way, the total angular momentum, , coincides with the total spin, , and can take values , and . The possible baryon-meson channels which are under consideration in the computation are listed in Table 2, they have been grouped according to total spin and isospin. The third and fifth columns of Table 2 show the necessary basis combination in spin , flavor , and color degrees-of-freedom. The physical channels with color-singlet (labeled with the superindex ) and hidden-color (labeled with the superindex ) configurations are listed in the fourth and sixth columns.
|Percentage (S;H): 98.5%; 1.5%|
|Percentage (S;H): 57.9%; 42.1%|
|Percentage (S;H): 15.8%; 84.2%|
|Percentage (S;H): 99.6%; 0.4%|
|Percentage (S;H): 55.5%; 44.5%|
|Percentage (S;H): 22.2%; 77.8%|
|Percentage (S;H): 99.6%; 0.4%|
|Percentage (S;H): 64.7%; 35.3%|
|Percentage (S;H): 18.4%; 81.6%|
|Percentage (S;H): 15.7%; 84.3%|
|Percentage (S;H): 19.9%; 80.1%|