Investigating excited $\Omega_{c}$ states from pentaquark perspective

In the last few decades, significant experimental progress has been made in the sector of heavy baryons. Many heavy baryons have been reported, such as $\Lambda_{c}$ and $\Sigma_{c}$ family Knapp:1976qw ; ARGUS:1993vtm ; CLEO:1994oxm ; ARGUS:1997snv ; E687:1993bax ; CLEO:2000mbh ; LHCb:2017jym ; BaBar:2006itc ; Belle:2021qip ; Belle:2014fde ; Ammosov:1993pi ; CLEO:1996czm ; Belle:2004zjl ; Belle:2022hnm , $\Xi_{c}$ family LHCb:2020gge ; ALICE:2021bli ; Belle:2016lhy ; CLEO:1998wvk ; Belle:2013htj ; Belle:2020ozq ; LHCb:2020iby ; Belle:2020tom ; Belle:2016tai and $\Omega_{c}$ family Belle:2021gtf ; LHCb:2017uwr ; BaBar:2006pve ; LHCb:2021ptx ; Belle:2017ext . These observations have stimulated extensive interest in understanding the structures of these baryons. For one thing, verifying these heavy baryons could deepen our understanding of the non-perturbative behavior of quantum chromodynamics (QCD) Chen:2016qju ; Swanson:2006st ; Voloshin:2007dx ; Chen:2016heh ; Huang:2023jec ; Esposito:2016noz ; Lebed:2016hpi ; Guo:2017jvc . For another thing, with the appearance of heavy baryons that are difficult to be interpreted simply as traditional three-quark baryons, the study of multi-quark explanation has become a non-negligible subject.

Among them, the excited $\Omega_{c}$ baryons were considerably enriched by the LHCb Collaboration in 2017 LHCb:2017uwr . Five narrow $\Omega_{c}^{0}$ states were observed in the $\Xi_{c}^{+}K^{-}$ invariant mass spectrum, which are $\Omega_{c}^{0}(3000)$, $\Omega_{c}^{0}(3050)$, $\Omega_{c}^{0}(3065)$, $\Omega_{c}^{0}(3090)$ and $\Omega_{c}^{0}(3120)$. The narrow width of these states, along with their unknown quantum numbers and structures, has attracted broad interest in theoretical work. A classical way to describe these $\Omega_{c}$ baryons is considering that they are conventional three-quark excitations, and another way is treating them as multi-quark states.

On the basis of three-quark configuration, $\Omega_{c}$ states have been studied in the framework of the Lattice QCD Padmanath:2017lng ; Bahtiyar:2020uuj , the QCD sum rules Agaev:2017jyt ; Wang:2017zjw ; Aliev:2017led ; Agaev:2017lip ; Wang:2017xam , the light cone QCD sum rules Chen:2017sci ; Aliev:2018uby , the heavy hadron chiral perturbation theory Cheng:2017ove , the Regge phenomenology Oudichhya:2021kop ; Oudichhya:2023awb , the chiral quark model Yang:2017rpg , the constituent quark model Wang:2017hej ; Wang:2017kfr ; Yao:2018jmc , the quark-diquark model Wang:2017vnc ; Ali:2017wsf , the quark pair creation model Chen:2017gnu , the ${}^{3}P_{0}$ model Zhao:2017fov ; Garcia-Tecocoatzi:2022zrf , the chiral quark-soliton model Kim:2017jpx ; Kim:2017khv , the holographic model Liu:2017frj , the string model Sonnenschein:2017ylo , the harmonic oscillator based model Santopinto:2018ljf , the light-front quark model Chua:2019yqh , the relativized potential quark model Jia:2020vek , the non-relativistic potential model Luo:2023sra , the relativistic flux tube model Jakhad:2023mni and other quark models Karliner:2017kfm ; Ortiz-Pacheco:2020hmj .

On the other hand, the pentaquark interpretation of $\Omega_{c}$ states has been investigated in the framework of the QCD sum rules Wang:2018alb ; Wang:2021cku , the chiral quark model Yang:2017rpg ; Huang:2017dwn , the constituent quark model An:2017lwg , the diquark-diquark-antiquark model Anisovich:2017aqa , the heavy-quark spin symmetry model Nieves:2017jjx , the one boson exchange model Liu:2018bkx , the vector meson exchange model Montana:2018teh ; Montana:2017kjw , the meson-baryon interaction model Ramos:2020bgs , the chiral quark-soliton model Praszalowicz:2022hcp , the extended local hidden gauge approach Debastiani:2017ewu ; Debastiani:2018adr , the Bethe–Salpeter formalism Wang:2017smo , the effective Lagrangian approach Huang:2018wgr and the quasipotential Bethe-Salpeter equation approach Zhu:2022fyb .

Very recently, two new excited states, $\Omega_{c}^{0}(3185)$ and $\Omega_{c}^{0}(3327)$ are observed in the $\Xi_{c}^{+}K^{-}$ invariant-mass spectrum by the LHCb Collaboration LHCb:2023zpu . Bisides, the five narrow $\Omega_{c}$ states obtained before LHCb:2017uwr are also confirmed. The measured masses and widths of the two newly found states are

	$\displaystyle M_{\Omega_{c}(3185)}$	$\displaystyle=3185.1\pm 1.7_{-0.9}^{+7.4}\pm 0.2\mathrm{MeV},$
	$\displaystyle\Gamma_{\Omega_{c}(3185)}$	$\displaystyle=50\pm 7_{-20}^{+10}\mathrm{MeV},$
	$\displaystyle M_{\Omega_{c}(3327)}$	$\displaystyle=3327.1\pm 1.2_{-1.3}^{+0.1}\pm 0.2\mathrm{MeV},$
	$\displaystyle\Gamma_{\Omega_{c}(3327)}$	$\displaystyle=20\pm 5_{-1}^{+13}\mathrm{MeV}.$		(1)

So far, there have been a few studies on the two newly discovered states. In Ref. Luo:2023sra , in the framework of the non-relativistic potential model with Gaussian Expansion Method, the authors’ results imply that the $\Omega_{c}^{0}(3327)$ is a good candidate of $\Omega_{c}^{0}(1D)$ state with $J^{P}=5/2^{+}$. In Ref. Yu:2023bxn , the ${}^{3}P_{0}$ model calculation results support assigning the observed $\Omega_{c}^{0}(3185)$ and $\Omega_{c}^{0}(3327)$ as the $2S(3/2^{+}$) and $1D(3/2^{+}$) states, respectively. In Ref. Wang:2023wii , using the QCD sum rules, the $\Omega_{c}^{0}(3327)$ is assigned to be $D$-wave $\Omega_{c}$ state with $J^{P}=1/2^{+},3/2^{+}$ or $5/2^{+}$. In Ref. Feng:2023ixl , $\Omega_{c}^{0}(3185)$ and $\Omega_{c}^{0}(3327)$ are studied in the effective Lagrangian approach by assuming they are molecular states. The results support $\Omega_{c}^{0}(3327)$ as a $J^{P}=3/2^{-}$ $D^{*}\Xi$ molecular state, and the $\Omega_{c}^{0}(3185)$ may be a meson-baryon molecule with a big $D\Xi$ component. In Ref. Karliner:2023okv , the assignment of $\Omega_{c}^{0}(3185)$ and $\Omega_{c}^{0}(3327)$ to $2S_{1/2}$ and $2S_{3/2}$ is discussed. In Ref. Yan:2023ttx , within a simple contact-range theory in which the couplings are saturated by light-meson exchanges, $\Omega_{c}^{0}(3185)$ and $\Omega_{c}^{0}(3327)$ match the masses of $J=1/2$ $\Xi D$ and $J=3/2$ $\Xi D^{*}$, respectively. In Ref. Jakhad:2023mni , based on the quark-diquark configuration with relativistic flux tube model, $\Omega_{c}^{0}(3185)$ and $\Omega_{c}^{0}(3327)$ is assigned to be $\left|2S,3/2^{+}\right\rangle$ and $\left|1D,3/2^{+}\right\rangle$. In Ref. Xin:2023gkf , via the QCD sum rules, the numerical results favor assigning $\Omega_{c}^{0}(3185)$ as the $D\Xi$ molecular state with the $J^{P}=1/2^{-}$, assigning $\Omega_{c}^{0}(3327)$ as the $D^{*}\Xi$ molecular state with the $J^{P}=3/2^{-}$.

In addition to the above theoretical methods, quark delocalization color screening model (QDCSM) is a reliable approach, which was developed in the 1990s with the aim of explaining the similarities between nuclear and molecular forces Wu:1996fm . The model gives a good description of $NN$ and $YN$ interactions and the properties of deuteron Ping:2000dx ; Ping:1998si ; Wu:1998wu ; Pang:2001xx . It is also employed to calculate the baryon-baryon and baryon-meson scattering phase shifts, and the exotic hadronic states are also studied in this model. Studies show that color screening is an effective description of the hidden-color channel coupling ChenLZ ; Huang:2011kf . So it is feasible and meaningful to extend this model to investigate the pentaquark interpretation of excited $\Omega_{c}$ states.

In this work, we systematically investigate the $ssc\bar{q}q$ systems in order to find out if there are $\Omega_{c}$ states that are possible to be interpreted as pentaquark states. The five-body system is calculated by means of the resonating group method to search for bound states. The strong decay channels of the $ssc\bar{q}q$ systems are investigated to determine the resonance states, based on the conservation of the quantum numbers and the limit of phase space. In order to ensure the reliability and stability of the calculation results, the parameters used in this work are the same as those used in the previous work Yan:2022nxp .

This paper is organized as follows. After introduction, the details of QDCSM are presented in section II. The calculation of the bound state and scattering phase shift is presented in Section III, along with the discussion and analysis of the results. Finally, the paper ends with summary in Section IV.

Herein, the QDCSM is employed to investigate the properties of $ssc\bar{q}q$ systems. The QDCSM is an extension of the native quark cluster model DeRujula:1975qlm ; Isgur:1978xj ; Isgur:1978wd ; Isgur:1979be . It has been developed to address multi-quark systems. The detail of the QDCSM can be found in Refs. Wu:1996fm ; Huang:2011kf ; ChenLZ ; Ping:1998si ; Wu:1998wu ; Pang:2001xx ; Ping:2000cb ; Ping:2000dx ; Ping:2008tp . In this sector, we mainly introduce the salient features of this model. The general form of the pentaquark Hamiltonian is given by

$\displaystyle H=$

$\displaystyle\sum_{i=1}^{5}\left(m_{i}+\frac{\boldsymbol{p}_{i}^{2}}{2m_{i}}\right)-T_{CM}+\sum_{j>i=1}^{5}V(\boldsymbol{r}_{ij}),$

(2)

where $m_{i}$ is the quark mass, $\boldsymbol{p}_{i}$ is the momentum of the quark, and $T_{CM}$ is the center-of-mass kinetic energy. The dynamics of the pentaquark system is driven by a two-body potential

$\displaystyle V(\boldsymbol{r}_{ij})=$

$\displaystyle V_{CON}(\boldsymbol{r}_{ij})+V_{OGE}(\boldsymbol{r}_{ij})+V_{\chi}(\boldsymbol{r}_{ij}).$

(3)

The most relevant features of QCD at its low energy regime: color confinement ($V_{CON}$), perturbative one-gluon exchange interaction ($V_{OGE}$), and dynamical chiral symmetry breaking ($V_{\chi}$) have been taken into consideration.

Here, a phenomenological color screening confinement potential ($V_{CON}$) is used as

$\displaystyle V_{CON}(\boldsymbol{r}_{ij})=$

$\displaystyle-a_{c}\boldsymbol{\lambda}_{i}^{c}\cdot\boldsymbol{\lambda}_{j}^{c}\left[f(\boldsymbol{r}_{ij})+V_{0}\right],$

(4)

$\displaystyle f(\boldsymbol{r}_{ij})=$

$\displaystyle\left\{\begin{array}[]{l}\boldsymbol{r}_{ij}^{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}i,j~{}\text{occur in the same cluster }\\ \frac{1-e^{-\mu_{q_{i}q_{j}}\boldsymbol{r}_{ij}^{2}}}{\mu_{q_{i}q_{j}}},~{}~{}~{}i,j~{}\text{occur in different cluster }\end{array}\right.$

(7)

where $a_{c}$, $V_{0}$ and $\mu_{q_{i}q_{j}}$ are model parameters, and $\boldsymbol{\lambda}^{c}$ stands for the SU(3) color Gell-Mann matrices. Among them, the color screening parameter $\mu_{q_{i}q_{j}}$ is determined by fitting the deuteron properties, nucleon-nucleon scattering phase shifts, and hyperon-nucleon scattering phase shifts, respectively, with $\mu_{qq}=0.45~{}$fm^-2, $\mu_{qs}=0.19~{}$fm^-2 and $\mu_{ss}=0.08~{}$fm^-2, satisfying the relation, $\mu_{qs}^{2}=\mu_{qq}\mu_{ss}$ ChenM . Besides, we found that the heavier the quark, the smaller this parameter $\mu_{q_{i}q_{j}}$. When extending to the heavy quark system, the hidden-charm pentaquark system, we took $\mu_{cc}$ as a adjustable parameter from $0.01~{}$fm^-2 to $0.001~{}$fm^-2, and found that the results were insensitive to the value of $\mu_{cc}$ HuangPc1 . Moreover, the $P_{c}$ states were well predicted in the work of Refs. HuangPc1 ; HuangPc2 . So here we take $\mu_{cc}=0.01~{}$fm^-2 and $\mu_{qc}=0.067~{}$fm^-2, also satisfy the relation $\mu_{qc}^{2}=\mu_{qq}\mu_{qc}$.

In the present work, we mainly focus on the low-lying negative parity $ssc\bar{q}q$ pentaquark states of $S$-wave, so the spin-orbit and tensor interactions are not included. The one-gluon exchange potential ($V_{OGE}$), which includes coulomb and chromomagnetic interactions, is written as

	$\displaystyle V_{OGE}(\boldsymbol{r}_{ij})=$	$\displaystyle\frac{1}{4}\alpha_{s_{q_{i}q_{j}}}\boldsymbol{\lambda}_{i}^{c}\cdot\boldsymbol{\lambda}_{j}^{c}$		(8)
		$\displaystyle\cdot\left[\frac{1}{r_{ij}}-\frac{\pi}{2}\delta\left(\mathbf{r}_{ij}\right)\left(\frac{1}{m_{i}^{2}}+\frac{1}{m_{j}^{2}}+\frac{4\boldsymbol{\sigma}_{i}\cdot\boldsymbol{\sigma}_{j}}{3m_{i}m_{j}}\right)\right],$

where $\boldsymbol{\sigma}$ is the Pauli matrices and $\alpha_{s_{q_{i}q_{j}}}$ is the quark-gluon coupling constant.

However, the quark-gluon coupling constant between quark and anti-quark, which offers a consistent description of mesons from light to heavy-quark sector, is determined by the mass differences between pseudoscalar mesons (spin-parity $J^{P}=0^{-}$) and vector (spin-parity $J^{P}=1^{-}$), respectively. For example, from the model Hamiltonian, the mass difference between $D$ and $D^{*}$ is determined by the chromomagnetic interaction in Eq. (II), so the parameter $\alpha_{s_{qc}}$ is determined by fitting the mass difference between $D$ and $D^{*}$.

The dynamical breaking of chiral symmetry results in the SU(3) Goldstone boson exchange interactions appear between constituent light quarks $u,d$ and $s$. Hence, the chiral interaction is expressed as

$\displaystyle V_{\chi}(\boldsymbol{r}_{ij})=$

$\displaystyle V_{\pi}(\boldsymbol{r}_{ij})+V_{K}(\boldsymbol{r}_{ij})+V_{\eta}(\boldsymbol{r}_{ij}).$

(9)

Among them

	$\displaystyle V_{\pi}\left(\boldsymbol{r}_{ij}\right)=$	$\displaystyle\frac{g_{ch}^{2}}{4\pi}\frac{m_{\pi}^{2}}{12m_{i}m_{j}}\frac{\Lambda_{\pi}^{2}}{\Lambda_{\pi}^{2}-m_{\pi}^{2}}m_{\pi}\left[Y\left(m_{\pi}\boldsymbol{r}_{ij}\right)\right.$
		$\displaystyle\left.-\frac{\Lambda_{\pi}^{3}}{m_{\pi}^{3}}Y\left(\Lambda_{\pi}\boldsymbol{r}_{ij}\right)\right]\left(\boldsymbol{\sigma}_{i}\cdot\boldsymbol{\sigma}_{j}\right)\sum_{a=1}^{3}\left(\boldsymbol{\lambda}_{i}^{a}\cdot\boldsymbol{\lambda}_{j}^{a}\right),$		(10)

	$\displaystyle V_{K}\left(\boldsymbol{r}_{ij}\right)=$	$\displaystyle\frac{g_{ch}^{2}}{4\pi}\frac{m_{K}^{2}}{12m_{i}m_{j}}\frac{\Lambda_{K}^{2}}{\Lambda_{K}^{2}-m_{K}^{2}}m_{K}\left[Y\left(m_{K}\boldsymbol{r}_{ij}\right)\right.$
		$\displaystyle\left.-\frac{\Lambda_{K}^{3}}{m_{K}^{3}}Y\left(\Lambda_{K}\boldsymbol{r}_{ij}\right)\right]\left(\boldsymbol{\sigma}_{i}\cdot\boldsymbol{\sigma}_{j}\right)\sum_{a=4}^{7}\left(\boldsymbol{\lambda}_{i}^{a}\cdot\boldsymbol{\lambda}_{j}^{a}\right),$		(11)

	$\displaystyle V_{\eta}\left(\boldsymbol{r}_{ij}\right)=$	$\displaystyle\frac{g_{ch}^{2}}{4\pi}\frac{m_{\eta}^{2}}{12m_{i}m_{j}}\frac{\Lambda_{\eta}^{2}}{\Lambda_{\eta}^{2}-m_{\eta}^{2}}m_{\eta}\left[Y\left(m_{\eta}\boldsymbol{r}_{ij}\right)\right.$
		$\displaystyle\left.-\frac{\Lambda_{\eta}^{3}}{m_{\eta}^{3}}Y\left(\Lambda_{\eta}\boldsymbol{r}_{ij}\right)\right]\left(\boldsymbol{\sigma}_{i}\cdot\boldsymbol{\sigma}_{j}\right)\left[\cos\theta_{p}\left(\boldsymbol{\lambda}_{i}^{8}\cdot\boldsymbol{\lambda}_{j}^{8}\right)\right.$
		$\displaystyle\left.-\sin\theta_{p}\left(\boldsymbol{\lambda}_{i}^{0}\cdot\boldsymbol{\lambda}_{j}^{0}\right)\right],$		(12)

where $Y(x)=e^{-x}/x$ is the standard Yukawa function. The physical $\eta$ meson is considered by introducing the angle $\theta_{p}$ instead of the octet one. The $\boldsymbol{\lambda}^{a}$ are the SU(3) flavor Gell-Mann matrices. The values of $m_{\pi}$, $m_{k}$ and $m_{\eta}$ are the masses of the SU(3) Goldstone bosons, which adopt the experimental values ParticleDataGroup:2020ssz . The chiral coupling constant $g_{ch}$, is determined from the $\pi NN$ coupling constant through

$\displaystyle\frac{g_{ch}^{2}}{4\pi}$

$\displaystyle=\left(\frac{3}{5}\right)^{2}\frac{g_{\pi NN}^{2}}{4\pi}\frac{m_{u,d}^{2}}{m_{N}^{2}}.$

(13)

Assuming that flavor SU(3) is an exact symmetry, it will only be broken by the different mass of the strange quark. The other symbols in the above expressions have their usual meanings. All the parameters shown in Table 1 are fixed by masses of the ground baryons and mesons. Table 2 shows the masses of the baryons and mesons used in this work.

In the QDCSM, quark delocalization was introduced to enlarge the model variational space to take into account the mutual distortion or the internal excitations of nucleons in the course of interaction. It is realized by specifying the single particle orbital wave function of the QDCSM as a linear combination of left and right Gaussians, the single particle orbital wave functions used in the ordinary quark cluster model

	$\displaystyle\psi_{\alpha}(\boldsymbol{S_{i}},\epsilon)$	$\displaystyle=$	$\displaystyle\left(\phi_{\alpha}(\boldsymbol{S_{i}})+\epsilon\phi_{\alpha}(-\boldsymbol{S_{i}})\right)/N(\epsilon),$
	$\displaystyle\psi_{\beta}(-\boldsymbol{S_{i}},\epsilon)$	$\displaystyle=$	$\displaystyle\left(\phi_{\beta}(-\boldsymbol{S_{i}})+\epsilon\phi_{\beta}(\boldsymbol{S_{i}})\right)/N(\epsilon),$
	$\displaystyle N(S_{i},\epsilon)$	$\displaystyle=$	$\displaystyle\sqrt{1+\epsilon^{2}+2\epsilon e^{-S_{i}^{2}/4b^{2}}}.$		(14)

It is worth noting that the mixing parameter $\epsilon$ is not an adjusted one but determined variationally by the dynamics of the multi-quark system itself. In this way, the multi-quark system chooses its favorable configuration in the interacting process. This mechanism has been used to explain the cross-over transition between hadron phase and quark-gluon plasma phase Xu .

In addition, the dynamical calculation is carried out using the resonating group method and the generating coordinates method. The details of the two methods can be seen in Appendix A, and the way of constructing wave functions are presented in Appendix B.

In this work, we investigate the $S-$wave $ssc\bar{q}q$ pentaquark systems in the framework of QDCSM with resonating group method. The quantum numbers of the pentaquark system are $I=0$, $J^{P}=1/2^{-},3/2^{-}$ and $5/2^{-}$. Three structures $qss-\bar{q}c$, $qsc-\bar{q}s$ and $ssc-\bar{q}q$, as well as the coupling of these structures are taken into consideration. To find out if there exists any bound state, we carry out a dynamic bound-state calculation. The scattering process is also studied to obtain the genuine resonance state. The introduction of the bound state calculation and scattering process can be seen in Appendix A. Moreover, the calculation of root mean square (RMS) of cluster spacing is helpful to explore the structure of the bound state or resonance state on the one hand, and to further estimate whether the observed states are resonance state or scattering state on the other hand.

The single-channel results of different systems are listed in Tables 3, 5 and 7, respectively. The first column headed with Structure incluedes $qss-\bar{q}c$, $qsc-\bar{q}s$ and $ssc-\bar{q}q$ three kinds. The second and third columns, headed with $\chi^{f_{i}}$ and $\chi^{\sigma_{j}}$, denote the way how wave functions constructed, which can be seen in Appendix B. The forth column headed with Channel gives the physical channels involved in the present work. The fifth column headed with $E_{th}^{Theo}$ refers to the theoretical value of non-interacting baryon-meson threshold. The sixth column headed with $E_{sc}$ shows the energy of each single channel. The values of binding energies $E_{B}$= $E_{sc}-E_{th}^{Theo}$ are listed in the eighth column only if $E_{B}<0$ MeV. Finally, the experimental thresholds $E_{th}^{Exp}$ (the sum of the experimental masses of the corresponding baryon and meson) along with corrected energies $E^{\prime}=E_{th}^{Exp}+E_{B}$ are given in last two columns.

As for coupled-channel, the results are listed in the Tables 4, 6 and 8, respectively. The first column represents the structures involved in the channel coupling and the second column is the theoretical value of the lowest threshold. The third column headed with $E_{cc}$ shows the energy of coupled-channel. The definitions of $E_{B}$, $E_{th}^{Exp}$ and $E^{\prime}$ in coupled-channel calculation are similar to their definitions in single-channel calculation.