Identification of newly observed singly charmed baryons using relativistic flux tube model

Singly charmed baryons provide the best environment to investigate the dynamics of light quarks in the presence of a heavy charm quark. In recent years, a significant experimental effort has been made to measure the singly charmed baryons. Many experimental groups such as LHCb, Belle, BaBar, and CLEO, have provided data and are expected to produce more precise results in near future [1].

The latest review of particle physics by Particle Data Group (PDG) [1] confirms the six states of $\Lambda_{c}$ baryon with their respective spin and parity (see Table 1). But, $\Lambda_{c}(2765)^{+}$/$\Sigma_{c}(2765)^{+}$ is still a controversial state. It was first observed in $\Lambda_{c}\pi^{+}\pi^{-}$ decay channel by CLEO Collaboration [2] and later confirmed by Belle in $\Sigma_{c}\pi$ decay mode [3]. We are uncertain of the identity of the observed state because both $\Lambda_{c}^{+}$ and $\Sigma_{c}^{+}$ can decay through these two channels. However, Belle Collaboration [4] predicts its isospin to be zero, and the particle is predicted as a state of $\Lambda_{c}$.

For $\Xi_{c}$ baryon family, states belonging to 1S and 1P wave with $J^{P}=\frac{1}{2}^{+},\frac{1}{2}^{-},\frac{3}{2}^{-}$ have been well established. Besides of these states, six other states are also included in the PDG [1] as shown in the Table 1. The spin and parity of these states, with the exception of $\Xi_{c}(2970)$ state, are still unknown. The $\Xi_{c}(3055)^{+}$ and $\Xi_{c}(3123)^{+}$ states were first observed by BaBar Collaboration in $\Sigma_{c}(2455)^{++}K^{-}$ and $\Sigma_{c}(2520)^{++}K^{-}$ channel respectively [5]. Belle confirmed $\Xi_{c}(3055)^{+}$ state, but no signal was found for $\Xi_{c}(3123)^{+}$ state [6]. The findings of the $\Xi_{c}(3080)$ state was first reported by Belle [7] and then verified by BaBar [5]. In 2020, LHCb observed three excited $\Xi_{c}^{0}$ resonances called $\Xi_{c}(2923)^{0}$, $\Xi_{c}(2939)^{0}$, and $\Xi_{c}(2965)^{0}$ in the $\Lambda_{c}^{+}K^{-}$ mass spectrum [8]. $\Xi_{c}(2923)^{0}$ and $\Xi_{c}(2939)^{0}$ states are observed for the first time. This study indicates that the broad peak observed by Belle [9, 10] and BaBar [11] for the $\Xi_{c}(2930)^{0}$ state resolves into two separate peaks for the $\Xi_{c}(2923)^{0}$ and $\Xi_{c}(2939)^{0}$ states. But, $\Xi_{c}(2965)^{0}$ state lies in the vicinity of previously observed state $\Xi_{c}(2970)^{0}$ [7, 12, 5]. Thus, further study is required to  establish whether or not the states $\Xi_{c}(2965)^{0}$ and $\Xi_{c}(2970)^{0}$ are equivalent. More recently in 2021, Belle reported first experimental determination of spin-parity of $\Xi_{c}(2970)^{+}$ using angular distribution of decay products in chain $\Xi_{c}(2970)^{+}\rightarrow\Xi_{c}(2645)^{0}\pi^{+}\rightarrow\Xi_{c}^{+}\pi^{-}\pi^{+}$ and ratio of branching fraction of two decays $\Xi_{c}(2970)^{+}\rightarrow\Xi_{c}(2645)^{0}\pi^{+}/\Xi_{c}^{\prime 0}\pi^{+}$ [13]. Their analysis favour $J^{P}=\frac{1}{2}^{+}$ over other possibilities with the zero spin of the light-quark degrees of freedom for $\Xi_{c}(2970)^{+}$.

For $\Sigma_{c}$, $\Xi_{c}^{\prime}$, and $\Omega_{c}$ baryons, despite multiple theoretical and experimental attempts, only states belonging to 1S-wave with $J^{P}=\frac{1}{2}^{+}$ and $\frac{3}{2}^{+}$ have been discovered, and higher excited states still need to be established. So far, just one excited state of $\Sigma_{c}$ named $\Sigma_{c}$(2800) has been discovered by Belle and BaBar collaborations in the channel of $\Lambda_{c}^{+}\pi$ [14, 15]. Its spin and parity are not identified yet. In 2017, LHCb declared the first observation of five narrow excited states of $\Omega_{c}^{0}$ in $\Xi_{c}^{+}K^{-}$channel: $\Omega_{c}(3000)^{0}$, $\Omega_{c}(3050)^{0}$, $\Omega_{c}(3065)^{0}$, $\Omega_{c}(3090)^{0}$, and $\Omega_{c}(3120)^{0}$ [16]. Later, except for $\Omega_{c}(3120)^{0}$, the other four states were confirmed by Belle collaboration [17]. Recently, observation of two new excited states, $\Omega_{c}(3185)^{0}$ and $\Omega_{c}(3327)^{0}$ in the $\Xi_{c}^{+}K^{-}$ invariant-mass spectrum, was revealed by LHCb collaboration [18]. These latest findings motivate us to identify the spin and parity of these seven states of $\Omega_{c}^{0}$ baryon so that they can be fitted into their mass spectrum. To achieve this, the sufficient experimental information about the $\Lambda_{c}$ and $\Xi_{c}$ baryonic states can be used to study the nature of other singly charmed baryons, such as the $\Sigma_{c}$, $\Xi_{c}^{\prime}$, and $\Omega_{c}$ baryons.

The spectra of singly charmed baryons have been examined by numerous theoretical models, particularly quark model [19, 20, 21, 22, 23, 24, 25, 26, 27, 28], heavy hadron chiral perturbation theory [29, 30, 31], lattice QCD [32], light cone QCD sum rules [33], QCD sum rules [34, 35, 36, 37, 38, 39, 40, 41], Regge phenomenology [42] and relativistic flux tube model [43, 44, 45]. In Ref. [46], the authors studied the masses of baryons containing one heavy quark in a combined expansion in 1/$\mathbf{m_{Q}}$ , 1/$\mathbf{N_{c}}$ , and SU(3) flavour symmetry breaking.

Five extremely narrow excited $\Omega_{c}$ baryons that were recently detected were analyzed by the authors of Ref. [47]. As well as possible spin assignments and the relation between the masses of different states are examined.

In our previous work [42], we employed Regge phenomenology to calculate ground state and excited state masses of $\Omega_{c}^{0}$, $\Omega_{cc}^{+}$ and $\Omega_{ccc}^{*++}$. However, this model is unable to predict all possible states of a system. We aim to uncover another model that is capable of predicting the entire spectrum. In ref. [44] authors have analytically derived linear Regge relation in a relativistic flux tube model for a heavy-light baryonic system. This relation is used to predict the complete spectrum of $\Lambda_{c}$ and $\Xi_{c}$ baryons. But, they exclude the study of other singly charmed baryonic systems($\Sigma_{c}$, $\Xi^{\prime}_{c}$, and $\Omega_{c}$), with vector diquark, due to the complexity of spin-dependent interactions.

In the present calculation, we apply linear Regge relation, introduced in ref. [44], to all singly charmed baryons in the quark-diquark picture. Diquark, which is thought to be in its ground state, is assumed to excite orbitally or radially with respect to charm quark. We incorporate spin-orbit interaction, spin-spin contact hyperfine interaction, and tensor interaction in the j-j coupling scheme to find spin-dependent splitting, and obtain the complete mass spectra of singly charmed baryons. We aim to identify the masses of fairly high orbital and radial excited states as well as assign possible quantum numbers to experimentally observed states of singly charmed baryons.

Following a brief overview of the experimental as well as theoretical progress, in Sec. II we discuss the details of the theoretical framework we have used to calculate the mass spectra of singly charmed baryons. It involves the derivation of the spin average mass formula in the relativistic flux tube model. Later, spin-dependent interactions are introduced. The parameters involved in this framework are calculated to obtain the mass spectra. In Sec. III, the obtained results are discussed to assign the spin-parity to experimentally available states and to compare it with other theoretical predictions. Finally, we outline our conclusion in Section IV.

The relativistic flux tube model [48, 49, 50, 51, 52, 53, 54, 55, 56] has achieved phenomenological success in explaining the Regge trajectory behavior of hadrons. It is based on Nambu’s idea of a dynamical gluonic string, which is responsible for the confinement of quarks within hadrons [57]. This model has been extended in a variety of ways to study mesons [58, 59, 60, 61, 62, 63], baryons [64, 44, 45, 65] as well as exotic hadrons [58, 66, 67, 68, 69]. The model is also derived from QCD based Wilson area law[70].

Singly charmed baryons are composed of charm quark($c$) with two light quarks($u,$ $d$ or $s$). According to heavy quark symmetry, the coupling between the $c$ quark and the light quark is predicted to be weak [71] and strong correlation between two light quarks ($u,$ $d$ or $s$) permits the formation of a diquark. Apart from this, quark-diquark picture of baryons is supported by a number of theoretical arguments. A string model represents baryons as pieces of open strings connected at one common point[72]. This model predicts that a baryonic state in which three quarks are coupled to one another by three open strings that are joined at a single point in a Y-shape is unstable. One of the three arms will eventually vanish, releasing its energy into the excitation modes of the other two arms. The classically stable configuration is made up of one open string connecting two quarks at the end points and one quark travelling along the string. However, an attraction between two quarks into a $\bar{3}$ bound state, one has a configuration of one quark at one end and a diquark at the other end of a single open string. Moreover ref.[73] concludes that the quark-diquark structure minimises baryon energy and is therefore preferred over the structure in which light quarks orbit around stationary heavy quark. The author in ref. [74] has shown that the singly heavy baryonic states with orbital angular momentum between a heavy quark and two light quarks are energetically favoured compared to the states with orbital angular momentum between two light quarks. This suggests that the states where two light quarks do not excite orbitally relative to each other and behave as a bound system, i.e., a diquark, are preferred. Inspired from these theoretical evidences, we take singly charmed baryons as a pair of diquark and charm quark.

In the context of the relativistic flux tube(RFT) model, a gluonic field connecting diquark with charm quark is proposed to lie in a straight tube-like structure called a flux tube. The color confinement is accomplished via this tube. The whole system of the charm quark, diquark, and flux tube, is constantly rotating with angular speed $\omega$ around its center of mass. Along with the energy of a flux tube, this model also includes the angular momentum of a flux tube having string tension $T$.

The relativistic Lagrangian $\mathfrak{L}$, of the $cqq$ baryon in RFT model is [49]

$$\mathfrak{L}=\displaystyle\sum_{i=1}^{2}\left[m_{i}(1-r_{i}^{2}\omega^{2})^{\frac{1}{2}}+T\int_{0}^{r_{i}}dr(1-r^{2}\omega^{2})^{\frac{1}{2}}\right],$$

(1)

where $m_{1}$ and $m_{2}$ can be accounted for current quark masses of diquark and charm quark, and $r_{i}$ denotes its position from the centre of mass. For simplicity, we have chosen speed of light c=1, in natural units.

We write the total orbital angular momentum $L$ as

$$L=\frac{\partial\mathfrak{L}}{\partial\omega}=\displaystyle\sum_{i=1}^{2}\left[\frac{m_{i}v_{i}^{2}}{\omega\sqrt{1-v_{i}^{2}}}+\frac{T}{\omega^{2}}\int_{0}^{v_{i}}\frac{v^{2}dv}{(1-v)^{2}}\right],$$

(2)

where $v_{i}=\omega r_{i}$ and $v=\omega r$.

The Hamiltonian $H$, which is equivalent to the mass of the $cqq$ baryon, $\bar{M}$, is given by

$$H=\bar{M}=\displaystyle\sum_{i=1}^{2}\left[\frac{m_{i}}{\sqrt{1-v_{i}^{2}}}+\frac{T}{\omega}\int_{0}^{v_{i}}\frac{dv}{(1-v)^{2}}\right].$$

(3)

We now define the effective mass of diquark by $m_{d}=m_{1}/\sqrt{1-v_{1}^{2}}$, and that of charm quark by $m_{c}=m_{2}/\sqrt{1-v_{2}^{2}}$, where $v_{1}$ and $v_{2}$ are speed of diquark and charm quark, respectively. Then performing integration, Eq.(2) and Eq.(3) simplifies to

$$L=\frac{1}{\omega}(m_{d}v_{1}^{2}+m_{c}v_{2}^{2})+\frac{T}{2\omega^{2}}\displaystyle\sum_{i=1}^{2}(sin^{-1}v_{i}-v_{i}\sqrt{1-v_{i}^{2}}),$$

(4)

and

$$\bar{M}=m_{d}+m_{c}+\frac{T}{\omega}\displaystyle\sum_{i=1}^{2}sin^{-1}v_{i}.$$

(5)

The boundary condition at the end of the flux tube with charm quark gives

$$\frac{T}{\omega}=\frac{m_{c}v_{2}}{\sqrt{1-v_{2}^{2}}}=m_{c}v_{2}[1+\frac{v_{2}^{2}}{2}+\frac{3v_{2}^{4}}{8}+...]\simeq m_{c}v_{2},$$

(6)

where higher order terms of $v_{2}$ are neglected.

For singly charmed baryons $m_{d}\ll m_{c}$. With this limit of a very small mass of diquark, we take the speed of light diquark $v_{1}\approx 1$ for approximation. Expanding Eq. (4) and (5) in terms of $v_{2}$ up to second order and using Eq. (6), mass relation can be obtained as [44]

$$(\bar{M}-m_{c})^{2}=\frac{\sigma}{2}L+(m_{d}+m_{c}v_{2}^{2}),$$

(7)

where $\sigma=2\pi T$. This gives Regge-like relation between mass and angular momentum. The method used in this study can be thought of as a semi-classical approximation of the quantized theory of strings as L is taken as the angular momentum quantum number. Despite the fact that the we did not account for the quantum correction, the obtained Regge relation shows the basic features predicted by a fully quantum mechanical approach[52, 53, 56] such as linear nature of Regge trajectory and non zero intercept for L=0 state in $(L,(\bar{M}-m_{c})^{2})$ plane.

Now, the distance between a heavy and light component of baryon can be given as

$$r=\frac{v_{1}+v_{2}}{\omega}=(v_{1}+v_{2})\sqrt{\frac{8L}{\sigma}},$$

(8)

where the relation between the angular speed of rotation of flux tube and orbital angular momentum, $\omega=\sqrt{\sigma/8L}$, is obtained by combining Eq.(4) and (7).

In Ref. [52, 53, 56], the RFT model is solved for heavy-light mesons with a quantum mechanical approach, which gives a nearly straight leading Regge trajectory in $(L,(\bar{M}-m_{c})^{2})$ plane, which is followed by nearly parallel and equally spaced daughter trajectories. In our model, the singly charmed baryons are pictured as two body system with one heavy quark, the same as heavy-light mesons, where one quark is heavy. So, for singly charmed baryons with two-body picture, we will also get a nearly straight, parallel and equally spaced Regge trajectories. In light of these quantum mechanical studies that show Regge trajectories in$(L,(\bar{M}-m_{c})^{2})$ plane with various values of $n$ (radial excitation quantum number) to be parallel to one another, we extend our semi-classical relation (7)and (8) for radially excited states by replacing $L$ with ($\lambda n+L$), as

$$(\bar{M}-m_{c})^{2}=\frac{\sigma}{2}[\lambda n_{r}+L]+(m_{d}+m_{c}v_{2}^{2}),$$

(9)

and

$$r=(v_{1}+v_{2})\sqrt{\frac{8[\lambda n_{r}+L]}{\sigma}},$$

(10)

where $\lambda$ is unknown parameter that we extracted using experimental data. Here, $n_{r}=n-1$ where $n$ represents the principal quantum number of the baryon state.

The RFT model considers the quarks to be spinless; hence, we now incorporate spin-dependent interactions to get the complete mass spectra. The resulting mass takes the form

$$M=\bar{M}+\Delta{M},$$

(11)

where spin average mass $\bar{M}$ can be obtained from Eq.(10), and contribution to mass from spin-dependent interaction is given by

$$\Delta{M}=H_{so}+H_{t}+H_{ss}.$$

(12)

Here, the first term is spin-orbit interaction, with the form

$$\begin{split}H_{so}&=[(\frac{2\alpha}{3r^{3}}-\frac{b^{\prime}}{2r})\frac{1}{m_{d}^{2}}+\frac{4\alpha}{3r^{3}}\frac{1}{m_{c}m_{d}}]\mathbf{L}.\mathbf{S_{d}}\\ &\hskip 8.5359pt+[(\frac{2\alpha}{3r^{3}}-\frac{b^{\prime}}{2r})\frac{1}{m_{c}^{2}}+\frac{4\alpha}{3r^{3}}\frac{1}{m_{c}m_{d}}]\mathbf{L}.\mathbf{S_{c}}\\ &=a_{1}\mathbf{L}.\mathbf{S_{d}}+a_{2}\mathbf{L}.\mathbf{S_{c}},\end{split}$$

(13)

which results from the short-range one-gluon exchange contribution and the long-range Thomas-precession term [75]. The second spin-dependent interaction

$$\begin{split}H_{t}&=\frac{4\alpha}{3r^{3}}\frac{1}{m_{c}m_{d}}[\frac{3(\mathbf{S_{d}}.\mathbf{r})(\mathbf{S_{c}}.\mathbf{r})}{r^{2}}-\mathbf{S_{d}}.\mathbf{S_{c}}]\\ &=b\hat{B},\end{split}$$

(14)

arises from magnetic-dipole-magnetic-dipole color hyperfine interaction, is a tensor term. Here, $\hat{B}$=$3(\mathbf{S_{d}}.\mathbf{r})(\mathbf{S_{c}}.\mathbf{r})/r^{2}-\mathbf{S_{d}}.\mathbf{S_{c}}$. The last term

$$\begin{split}H_{ss}&=\frac{32\alpha\sigma_{0}^{3}}{9\sqrt{\pi}m_{c}m_{d}}e^{-\sigma_{0}^{2}r^{2}}\mathbf{S_{d}}.\mathbf{S_{c}}\\ &=c\mathbf{S_{d}}.\mathbf{S_{c}},\end{split}$$

(15)

is spin-spin contact hyperfine interaction. The parameters $b^{\prime}$ and $\sigma_{0}$ can be fixed using experimental data. $\alpha$ is the coupling constant. $\mathbf{S_{c}}$ and $\mathbf{S_{d}}$ denote spin of charm quark and diquark, respectively.

As per Pauli’s exclusion principle, the total symmetry of a diquark under an exchange of two quarks is antisymmetric. Diquark has a symmetric spatial state and an antisymmetric color state [76]. Flavor and spin states of diquark can either be symmetric or antisymmetric, such that $|flavor\rangle$$\times$$|spin\rangle$ is symmetric. As shown in Fig. 1, flavor symmetries of light quarks organize singly charmed baryons into two groups, antisymmetric antitriplet (${\bar{3}}_{F}$) and symmetric sextet($6_{F}$), as: $3\otimes 3=6_{F}\oplus{\bar{3}}_{F}$. The symmetric flavor state of a diquark requires a symmetric spin state, whereas the antisymmetric flavor state of a diquark requires an antisymmetric spin state. As a result, baryons that belong to the antitriplet($\Lambda_{c}$ and $\Xi_{c}$ baryons) have spin zero diquark (also known as scalar diquark $[q_{1},q_{2}]$), whereas those that belong to the sextet($\Sigma_{c}$, $\Xi^{\prime}_{c}$ and $\Omega_{c}$ baryons) have spin one diquark (also known as vector diquark $\{q_{1},q_{2}\}$). Here, $q_{1}$ and $q_{2}$ are one of $u,$ $d$ or $s$ quark.

There are two ways by which $\mathbf{S_{c}}$, $\mathbf{S_{d}}$ and $\mathbf{L}$ can couple to give total angular momentum $\mathbf{J}$. First is the L-S coupling scheme in which spin of diquark $\mathbf{S_{d}}$ first couple with a spin of charm quark $\mathbf{S_{c}}$ to form $\mathbf{S}$ , later $\mathbf{S}$ couple with $\mathbf{L}$ to give $\mathbf{J}$. The second one is the j-j coupling scheme, which is the dominant one in heavy quark limit, where a spin of diquark $\mathbf{S_{d}}$ first couple with $\mathbf{L}$ and results in total angular momentum of diquark $\mathbf{j}$, and then $\mathbf{j}$ couple with a spin of charm quark $\mathbf{S_{c}}$ to give $\mathbf{J}$. Since $m_{c}\gg m_{d}$ for single charmed baryons, we assume that heavy quark symmetry is followed. This allows us to refer to baryonic states as j-j coupling state $|J,j\rangle$, where both $\mathbf{j}$ and $\mathbf{J}$ are conserved.

For $\Lambda_{c}$ and $\Xi_{c}$ baryons, with scalar diquark($S_{d}=0$), both L-S and j-j coupling scheme gives identical states. Spin interactions are much simpler as only the second term of the spin-orbit interaction survives and tensor, as well as spin-spin contact hyperfine interactions, become zero. The expectation value of $\mathbf{L}.\mathbf{S_{c}}$ in any coupling scheme for $\Lambda_{c}$and $\Sigma_{c}$ is given by

$$\langle\mathbf{L}.\mathbf{S_{c}}\rangle=\frac{1}{2}[J(J+1)-L(L+1)-S_{c}(S_{c}+1)].$$

(16)

The spin interactions for $\Sigma_{c}$, $\Xi^{\prime}_{c}$ and $\Omega_{c}$ baryons, with vector diquark($S_{d}=1$), are more complex than those for $\Lambda_{c}$ and $\Xi_{c}$ baryons. Detailed calculation of mass splitting operators, in j-j coupling scheme, that are involved in spin-dependent interactions is shown in the appendix and results are listed in a Table 2.