Study of the hidden-heavy pentaquarks and $P_{cs}$ states

In addition to the conventional hadronic states, such as mesons $q\bar{q}$ and baryons $qqq$ in quark configurations, the possible exotic tetraquarks $q^{2}\bar{q}^{2}$ and pentaquarks $q^{4}\bar{q}$ are also suggested at the birth of quark model Gell-Mann (1964); Zweig (1964). Later in the 1970s, the MIT bag model has been developed by Jaffe Jaffe (1977a, b); DeGrand et al. (1975) for the study of exotic multiquark states and notation of color confinement. In the past few decades, it has been applied to describe the doubly heavy baryons Fleck and Richard (1989); He et al. (2004); Bernotas and Simonis (2009, 2012) and exotic states, including light pentaquarks Strottman (1979) and hybrid mesons Barnes et al. (1983); Chanowitz and Sharpe (1983). Despite that these states are considered to be exotic beyond the conventional scheme of quark model, the existences of them are allowed by quantum chromodynamics (QCD).

Since the observation of exotic $X(3872)$ in 2003 by the Belle Choi et al. (2003), there are many candidates that have been discovered for tetraquarks, such as the $Z_{c}(3900)$ Ablikim et al. (2013); Liu et al. (2013) and the $T_{cc}(3875)$ Aaij et al. (2022a), as well as the fully charm systems $X(6600)$ Hayrapetyan et al. (2023) and $X(6900)$ Aaij et al. (2020). In 2015, the first evidence for pentaquark-like structures $P_{c}(4450)^{+}$ and $P_{c}(4380)^{+}$ with a minimal quark constituent of $uudc\bar{c}$ was reported by LHCb in $J/\psi p$ channel Aaij et al. (2015), for which the former exhibits a two-peak structure resolved into $P_{c}(4440)^{+}$ and $P_{c}(4457)^{+}$ in 2019. Recently, the LHCb Collaboration has reported two new hidden charm pentaquarks with a single strange flavor, $P_{\psi s}^{\Lambda}(4338)^{0}$ Aaij et al. (2023), and $P_{cs}(4459)^{0}$Aaij et al. (2021) both in $J/\psi\Lambda$ channel. These pentaquark candidates encourage the theoretical study on their mass spectrum, hadron properties and decay behaviors. There are various pictures and methods applied to analyze the hidden-charm pentaquarks, including molecular Chen et al. (2021); Chen (2021); Yang et al. (2021); Lu et al. (2021); Xiao et al. (2021); Zhu et al. (2021); Feijoo et al. (2023); Zhu et al. (2023); Chen et al. (2022); Wang and Liu (2022, 2023); Azizi et al. (2023) and compact scenario Weng et al. (2019); Shi et al. (2021); Ruangyoo et al. (2022); Li et al. (2021, 2023); Guo and Li (2023), as well as the hidden-bottom pentaquarks Yang et al. (2019); Zhu et al. (2020).

In our previous work Zhang et al. (2021), we performed a systematical study on the spectrum of mesons and baryons in their ground states, including light-flavor baryons and doubly heavy baryon $\Xi_{cc}$ Aaij et al. (2017), and extented the computation to the doubly heavy tetraquark $T_{cc}$. Considering the binding energies between heavy quarks, along side a running coupling constant, our predictions align with the experimental reports for baryons and tetraquarks with two heavy flavors, even for the fully heavy system $X(6600)$ Yan et al. (2023). Therefore, the MIT bag model serves as a capable method to study heavy hadrons on masses and magnetic moments.

Motivated by this, as demonstrated in Fig.(1), a light-flavor baryon $qqq$ ($q=u,d,s$) described as a sphere bag is extended into a hidden-heavy pentaquark $qqqQ\bar{Q}$, with the present of heavy $Q\bar{Q}$ ($Q=c,b$) in configuration. Due to the dynamics of heavy constituents and suppression of relativistic effects, it is possible to consider pentaquarks $qqqQ\bar{Q}$ in compact scenario.

The purpose of this work is to study masses, magnetic moments and ratios of partial widths for hidden-heavy pentaquarks, classified into strangeness $S=0,-1,-2,-3$, namely $nnnQ\bar{Q}$, $nnsQ\bar{Q}$, $ssnQ\bar{Q}$ and $sssQ\bar{Q}$, respectively. Numerical computations are based on the unified framework of MIT bag model, characterized by model parameters and fundamental relations. In order for describing inner-structures of pentaquarks, the color-spin wavefunctions are employed to the chromomagnetic interactions, which are expressed in terms of Young tableau and Young-Yamanouchi bases. For partial width study, we transform the color-spin wavefunctions into the form of color-singlet (baryon-meson coupling) and color-octet components. Finally, with the help of spin bases, it’s straightforward to derive the magnetic moments for possible pentaquark configurations.

This work is structured as follows. In the next section, we provide a brief introduction to the fundamental relations of the Hamiltonian and magnetic moment. Additionally, we outline the parameters utilized in our numerical calculations. The detailed formulations about the chromomagnetic structure and magnetic moment can be found in the Appendix. In Section III, we present a comprehensive numerical analysis of hidden-heavy pentaquarks, classified into subsections III.1, III.2, III.3, and III.4, corresponding to strangeness $S=0,-1,-2,-3$, respectively. Finally, this work ends with summaries and conclusions in Section IV.

In the framework of the MIT bag model, all valence quarks are confined within a spherical bag characterized by a radius $R$ and respective momentum $x_{i}$. These quarks are coupled with perturbative chromomagnetic interactions facilitated by the exchange of the lowest-order gluon, as described in prior works DeGrand et al. (1975); Johnson (1975). The mass formula associated with the bag, representing a hadronic state, is given by

Here, in Eq.(1), the first term represents the sum of the kinetic energy for the relativistic quark $i$ with mass $m_{i}$, the second is volume energy with bag constant $B$, which characterizes the difference between perturbative and non-perturbative QCD vacuum, the third is zero point energy with coefficient $Z_{0}$, the forth accounts for the binding energy between heavy quarks or heavy and strange quarks Karliner and Rosner (2014, 2017a, 2017b, 2020), and the fifth represents the chromomagnetic interaction De Rujula et al. (1975). The bag radius $R$ is the only variable to be determined by minimizing Eq.(1). The momentum $x_{i}$, given in units of $R^{-1}$ satisfies a boundary condition on the bag surface

which can be derived by quark spinor wave function.

The interactions among quarks involved in this work have two primary components: the lowest-order gluon exchange and short-range effects De Rujula et al. (1975). For the first part, the binding energies $M_{BD}$ primarily arise from short-range chromoelectric interactions between heavy quarks or heavy-strange systems, both of which are massive and moving nonrelativistically. Specifically, we incorporate four binding energies $B_{cs}$, $B_{cc}$, $B_{bs}$ and $B_{bb}$ in color $\bm{\bar{3}}_{c}$ representation for hidden-heavy pentaquarks, and scale them to other color configurations by color factors Karliner and Rosner (2014, 2017b). The second component involves chromomagnetic interaction $M_{CMI}$, which contributes to the overall dynamics of the system. The interaction arising from the magnetic moments of quark spin can be described by the following typical form:

where $i$ and $j$ represent the indices of the quarks (or anti-quarks), $\lambda$ denotes the Gell-Mann matrices, $\sigma$ signifies the Pauli matrices, and $C_{ij}$ corresponds to the chromomagnetic interaction (CMI) parameters as defined in Ref. Zhang et al. (2021). The color and spin factor (average value of Casimir operator) in Eq.(4) can be calculated utilizing the specific method in Refs.Zhang et al. (2021, 2023); Yan et al. (2023), with the color-spin wavefunctions given in Appendix A.

Next, we will provide a brief overview of the foundational equations of magnetic moment. In the context of MIT bag model, the magnetic moment for an individual quark is not determined by constant parameters but rather by variables that satisfy specific boundary conditions. Following variational calculations on mass equation (1), the model fulfills the quark spinor wavefunction DeGrand et al. (1975)

with parameters $(R,x_{i})$ and mass $m_{i}$ for quark $i$, which would help determine the physical quantities and interactions within the bag model. Subsequently, the operator $\bm{r\times\gamma}$ is applied to the wavefunction (5) to derive the magnetic moment, as the following expression:

where $Q_{i}$ is electric charge of quark and $\bm{\gamma}$ are Dirac matrices for spinor fields. Accordingly, it’s straightforward to apply Eq.(6) to calculate the magnetic moment for any flavor of quark with mass $m_{i}$ and parameters $(R,x_{i})$ evaluated through spectrum studies.

We calculate the magnetic moment of hadron by summing up Eq.(6) via color-spin wavefunction $\left|\psi\right\rangle$, resulting in the expression Wang et al. (2016)

with $g_{i}=2$ and $\hat{S}_{iz}$ denotes the third component of spin for an individual quark. The numerical results will be related to the magnetic moment of proton, and transformed into that in unit of $\mu_{N}$, with the help of measured data $\mu_{p}=2.79285\mu_{N}$ Workman et al. (2022); Tiesinga et al. (2021). Eq.(7) holds true for chromomagnetic mixing, as the spin wavefunctions expanding to basis vectors create matrix elements of magnetic moment. For further details, refer to Appendix B.

In numerical calculations, we proceed with model parameters from a previous work Zhang et al. (2021), that successfully reconcile ground-state mesons and baryons and provide descriptions of fully heavy tetraquarks and pentaquarks Zhang et al. (2023); Yan et al. (2023). The chosen constants and masses for quarks with flavor $n$ (light nonstrange flavor $n=u,d$), strange $s$, charm $c$ and bottom $b$ are as follows:

Meanwhile, the binding energies for color $\bm{\bar{3}}_{c}$ representation are considered:

These bindings will be scaled by color factors. With the parameters in Eqs.(8) and (9), the variational method can be performed to determine the bag radius $R$ and respective momentum $x_{i}$ using Eq.(3) for pentaquarks with quantum numbers $IJ^{P}$. Subsequently, the parameters $(R,x_{i})$ enable the calculations of masses and magnetic moments for the hidden-heavy pentaquarks.