Universal behavior of mass gaps existing in the single heavy baryon family

Abstract

The mass gaps existing in the discovered single heavy flavor baryons are analyzed, which show some universal behaviors. Under the framework of a constituent quark model, we quantitatively explain why such interesting phenomenon happens, when these established excited heavy baryons are regarded as the $\lambda$-mode excitations. Based on the universal behaviors of the discussed mass gaps, we may have three implications including the prediction of the masses of excited $\Xi _b^0$ baryons which are still missing in the experiment. For completeness, we also discuss the mass gaps of these $\rho$-mode excited single heavy flavor baryons.

1 Introduction

As an effective approach to solve the nonperturbative problem of strong interaction, studying hadron spectroscopy has become an active research issue with the abundant observations of new hadronic states in experiment (see Refs. [1,2,3,4,5,6,7,8] for the recent progress). Among different research aspects of the study of hadron spectroscopy, mass spectrum analysis is a crucial way to decode the property of hadronic states.

Until now, there have been different methods to perform mass spectrum analysis. If seriously depicting the mass spectrum, we have various versions of potential model [9,10,11,12,13,14,15,16], the flux tube model [17, 18], the quantum chromodynamics (QCD) sum rule [19, 20], the lattice QCD [21,22,23], and so on. Besides, some semi-quantitative methods were extensively applied to the mass spectrum analysis. A typical example is the Regge trajectory analysis [24,25,26,27] which has been adopted to investigate the mass spectrum of different kinds of hadrons [28,29,30,31,32,33,34,35]. Recently, Chen gave a mass formula for light meson and baryon, and discussed its implication [36].

On other hand, some mass gap relations existing in the hadron mass spectrum have also been realized by the theorists, which can become the simple but effective approach when scaling mass spectrum. For illustrating this point, we review several representative recent work. The similar dynamics of the $\omega$ and $\phi$ meson families requires the similarity between $\omega$ and $\phi$ meson families, where the mass gap of $\omega (782)$ and $\omega (1420)$ is similar to that of $\phi (1020)$ and $\phi (1680)$. Adopted this mass gap relation for higher states, the Y(1915) state as the partner of Y(2175) was predicted [37]. In Ref. [38], Lanzhou group predicted the existence of a narrow charmonium $\psi (4S)$ with the mass around 4.263 GeV, which was estimated by the mass gap between $\Upsilon (3S)$ and $\Upsilon (4S)$, if considering the mass gap relation $m_{\psi (4S)}-m_{\psi (3S)}=m_{\Upsilon (4S)}-m_{\Upsilon (3S)}$. In Ref. [39], the authors pointed out that the excitation energies of the first excited $J^P=1/2^+$ baryons with various flavor contents are about 500 MeV. As the last example, there exists a mass relation $M_{0^+}+3M_{1^+}+5M_{2^+}\simeq 9M_{1^{+\prime }}$ [40] for charmonium and bottomonium families, where the spin-parity quantum numbers are given to distinguish the different charmonium masses with the corresponding quantum number. Accordingly, Chang et al. [41] suggested a new mass relation $M_{0^+}+5M_{2^+}=3(M_{1^{+\prime }}+M_{1^+})$ for the P-wave $B_c$ mesons.

Fig. 1

All observed charm and bottom baryons [48,49,50, 60]

In the following, we need to pay attention to the single heavy flavor baryon. Among this fantastic hadron zoo, the single heavy flavor baryon family is being constructed step by step, which is due to the big progress on the observations of charm and bottom baryons, where the LHCb Collaboration has played an important role [42,43,44,45,46,47,48,49,50,51,52] in the past years. In Fig. 1, we list these reported single heavy flavor baryon states. Obviously, the present situation of single heavy flavor baryons shows that it is a good opportunity to check whether there exist some mass gap relations for single heavy flavor baryon family, which will be one of tasks of this work. We will analyze the mass gaps which are extracted by these measured mass spectrum of single heavy flavor baryons, by which we may find some universal behavior of mass gaps for these discovered single heavy baryons.

Facing such interesting mass gap phenomenon, we want to further study why there exists this universal behavior of mass gaps, which is involved in the dynamics of single heavy flavor baryon. In this work, we start with a constituent quark model [53], which has been successfully applied to depict the mass spectrum of single heavy baryon in our previous works [54,55,56,57]. Here, two light quarks in single heavy flavor baryon are treated as a cluster, and then the single heavy flavor baryon is simplified as a quasi-two-body system.

For understanding the universal behavior of mass gaps, we perform a study of mass gaps of single heavy baryon under the framework of constituent quark model, which shows that this universal behavior of mass gaps can be well explained. For phenomenological study, mass gap relation is always a welcome approach. Thus, in this work, we continue to apply this universal behavior of mass gaps to predict some higher single heavy baryon states, which can be tested by the future experiments.

The paper is organized as follows. After Introduction, we illustrate these mass gap relations by the measured mass spectrum of single heavy flavor baryon in Sect. 2. And we deduce these mass gap relations by a constituent quark model. In Sect. 3, its application to predict some higher states in the single heavy flavor baryon family will be given. Further discussions for the mass gaps of $\rho$-mode excited single heavy baryons will be presented in Sect. 4. Finally, the paper ends with a summary.

2 Universal behavior of mass gaps

2.1 Mass gaps of single heavy baryon

The single heavy baryon system, which contains one heavy quark (c or b quark) and two light quarks (u, d, or s quark), occupies a particular place in the whole hadron family [58]. The heavy quark in a single heavy flavor baryon system provides a quasi-static colour field for two surrounding light quarks [59]. Based on the flavor SU(3) symmetry of light quark cluster, i.e. $3_f\otimes 3_f = {\bar{3}}_f\oplus 6_f$, the $\Lambda _Q$ and $\Xi _Q$ baryon states belong to the ${\bar{3}}_f$ multiplet, while the $\Sigma _Q$, $\Xi _Q^\prime$, and $\Omega _Q$ baryon states can be grouped into the $6_f$ multiplet. Thus, in the following, we will give the mass gasps of single heavy flavor baryons in ${\bar{3}}_f$ and $6_f$ multiplets, and show their universal behaviors.

2.1.1 The ${{\bar{3}}}_f$ multiplet

With the joint effort of experimentalists and theorists, the family of single heavy baryons has been established step by step. In 2017, a new charm baryon state $\Lambda _c(2860)^+$ was discovered by the LHCb Collaboration in the process $\Lambda _b^0\rightarrow \Lambda _c(2860)^+\pi ^-\rightarrow {D^0p}\pi ^-$ [44]. Combining with the previously observed $\Lambda _c(2286)^+$, $\Lambda _c(2595)^+$, $\Lambda _c(2625)^+$, $\Lambda _c(2880)^+$, $\Xi _c(2470)$, $\Xi _c(2790)$, $\Xi _c(2815)$, $\Xi _c(3055)$, and $\Xi _c(3080)$ states [60], all 1S, 2S, 1P, 1D states of $\Lambda _c^+$ and $\Xi _c^{0,+}$ baryons have been discovered. In the past years, experiment has also made a big progress on searching for the excited $\Lambda _b^0$ states. In 2012, two narrow P-wave $\Lambda _b^0$ states, i.e., the $\Lambda _b(5912)^0$ and $\Lambda _b(5920)^0$, were first reported by the LHCb Collaboration in the $\Lambda _b^0\pi ^+\pi ^-$ invariant mass spectrum [42], which were confirmed by the further measurements from the CDF, CMS, and LHCb collaborations [50, 61, 62]. Besides, the D-wave states $\Lambda _b(6146)^0$ and $\Lambda _b(6152)^0$ have also been constructed in the recent years [50, 52, 62]. In last year, the $\Lambda _b(6072)^0$, which could be regarded as a good 2S $\Lambda _b^0$ candidate, was found by CMS [62] and LHCb [50]. And then, the low-lying $\Lambda _c^+$, $\Xi _c^{0,+}$, and $\Lambda _b^0$ baryons including the 1S, 2S, 1P, and 1D states have been reported by experiment. When checking the masses of these known heavy baryons (see Table 1), the interesting universal behavior of mass gaps can be found. Specifically, the obtained mass gaps for the discussed $\Lambda _c^+$ and $\Xi _c^+$ states are nearly about 180$\sim$200 MeV (see the fourth column in Table 1).

Table 1 The measured masses of 1S, 2S, 1P, 1D $\Lambda _c^+$ and $\Xi _c^+$ baryons [60] and the mass gaps of corresponding states (in MeV)

In the following, we define the following ratios

$$\begin{aligned} {\mathcal {R}}_1 = \frac{M_{2S}-M_{1S}}{{\bar{M}}_{1P}-M_{1S}},\quad {\mathcal {R}}_2 = \frac{{\bar{M}}_{1D}-M_{1S}}{{\bar{M}}_{1P}-M_{1S}}, \end{aligned}$$

(1)

where $M_{1S}$ and $M_{2S}$ refer to the masses of the S-wave ground state and its first radial excitation, respectively, while ${\bar{M}}_{1P}$ and ${\bar{M}}_{1D}$ denote the spin average masses of 1P and 1D states. As shown in Table 2, the values of ${\mathcal {R}}_1$ and ${\mathcal {R}}_2$ are nearly universal for the $\Lambda _c^+$ and $\Xi _c^{+}$ baryon systems. If further checking the $\Lambda _b$ baryons, the obtained ${\mathcal {R}}_1$ and ${\mathcal {R}}_2$ are also close to the corresponding values of the $\Lambda _c$ and $\Xi _c$ baryons,¹ which means the universal behavior of ${\mathcal {R}}_1$ and ${\mathcal {R}}_2$. To some extent, this novel phenomenon reflects the similar dynamics of charm and bottom baryons.

Table 2 The extracted ratios of ${\mathcal {R}}_1$ and ${\mathcal {R}}_2$ (see Eq. (1)) for the $\Lambda _Q$ and $\Xi _Q$ baryons. The 2S candidate of $\Lambda _b$ baryon with the mass around 6.07 GeV, which was found recently by the CMS [62] and LHCb [50] collaborations in the $\Lambda _b^0\pi ^+\pi ^-$ mass spectrum, is taken as an input

2.1.2 The $6_f$ multiplet

In Table 3, we extract the mass gaps according to the data of the measured $\Sigma _Q$, $\Xi _Q^\prime$, and $\Omega _Q$ baryons. Before discussing the mass gaps, we should briefly introduce the experimental progress on the $\Sigma _Q$, $\Xi _Q^\prime$, and $\Omega _Q$ baryons. In the past years, some P-wave $\Sigma _Q$, $\Xi _Q^\prime$, and $\Omega _Q$ candidates, including the $\Xi _c(2923)^0$, $\Xi _c(2939)^0$, $\Xi _c(2965)^0$, $\Omega _c(3000)^0$, $\Omega _c(3050)^0$, $\Omega _c(3065)^0$, $\Omega _c(3090)^0$, $\Sigma _b(6097)^0$, $\Xi ^\prime _b(6227)^0$, $\Omega _b(6316)^-$, $\Omega _b(6330)^-$, $\Omega _b(6340)^-$, and $\Omega _b(6350)^-$, have also been announced by the different experiments [45,46,47,48,49, 51, 65, 66]. These observed charm baryons can be categorized into the $6_f$ representation. Among these observed low-lying $6_f$ heavy baryon states, the $\Xi _c(2939)^0$, $\Omega _c(3065)^0$, $\Sigma _b(6097)^0$, $\Xi ^\prime _b(6227)^0$, and $\Omega _b(6350)^-$ are suggested to be the $J^P=3/2^-$ or $5/2^-$ states [54,55,56, 63, 67,68,69,70,71,72,73,74,75,76,77]. The $\Sigma _c(2800)^{++}$ state, which was discovered previously by the Belle Collaboration in the $e^+e^-$ collision [78], could also be regarded as a P-wave $3/2^-$ or $5/2^-$ state [53, 79]. Under these assignments, we obtain the mass gaps relevant to $\Sigma _c(2800)^{++}$, $\Xi _c(2939)^0$, and $\Omega _c(3065)^0$, which are about 130 MeV (see Table 3 for more details). Similar value for the mass gaps of the $\Sigma _b(6097)^-$, $\Xi ^\prime _b(6227)^-$, and $\Omega _b(6350)^-$ states can be found. Furthermore, the mass gaps of ground $\Sigma _Q$, $\Xi _Q^\prime$, and $\Omega _Q$ baryons are about 120 MeV. So the last column in Table 3 also reflects the universal behavior of mass gaps of low-lying single heavy flavor baryons in $6_f$ multiplet.

Table 3 The observed $\Sigma _Q$, $\Xi _Q^\prime$, and $\Omega _Q$ baryons [49, 60] and the corresponding mass gaps involved in these states (in MeV). In the last column, there are two values for each line, where the first value is the mass gap of the first and the second states listed in the second column, and the second value denotes the mass difference of the second and the third states. Here, $\Omega _b^-(\cdots )$ denotes the absent 1S $\Omega _b^-(3/2^+)$ state in the experiment

In the next section, we will explain the universal behaviors of mass gaps of single heavy flavor baryons.

2.2 Understanding the universal behavior of mass gaps by a constituent quark model

Although the universal behavior of mass gaps introduced in Sect. 2.1 has been mentioned for many years [67, 80, 81], it has never been investigated seriously. In the work, we will give a quantitative study of the mass gap of single heavy flavor baryons with the same $nL(J^P)$ quantum numbers but different strangeness under the framework of a non-relativistic constituent quark model. Here, n and L denote the radial and orbital angular quantum numbers of a baryon state, respectively while $J^P$ denotes its spin-parity.

The obtained $180\sim 200$ MeV mass gaps (see Table 1) of the ground and excited states of $\Lambda _Q$ and $\Xi _Q$ baryons in ${{\bar{3}}}_f$ representation reflect a universal behavior of these mass gaps. It implies that the similarity of mass gaps may be resulted from the same dynamics mechanism. Furthermore, the differences of the excited energy $E_{nL}$ for $\Lambda _Q$ and $\Xi _Q$ baryons with same nL quantum number can be roughly ignored. Such conclusion should also hold for the $6_f$ heavy baryons. Indeed, the mass gaps shown in Sect. 2.1 can be naturally interpreted as the mass differences of light quark cluster involved in the discussed heavy baryons, which will be proved in following by a non-relativistic constituent quark model.

If treating two light quarks as a cluster, the single heavy baryon system can be simplified as a quasi-two-body system [53]. And then, the Hamiltonian of a heavy baryon system reads as

$$\begin{aligned} {\hat{H}} = m_Q + m_{cluster } + \frac{{\hat{p}}^2}{2\mu } + V_{SI }(r) + V_{SD }(r), \end{aligned}$$

(2)

where the reduced mass $\mu$ is defined as $\mu = \frac{m_{cluster }m_Q}{m_{cluster }+m_Q}$. Different types of potentials can be taken for $V_{SI }(r)$, which describes the spin-independent interaction between the light quark cluster and the heavy quark. In Sect. 2.3, we will list four types of expression of $V_{SI }(r)$. By solving the Schrödinger equation with the concrete $V_{SI }(r)$, the excited energy $E_{nL}$ can be determined. The spin-dependent interactions $V_{SD }(r)$ in Eq. (2) include the hyperfine interaction, the spin-orbit forces, and the tensor force [53, 82]. Usually, the spin-dependent interactions are much weaker than the spin-independent interaction. Here, we may take P-wave charm baryon states $\Lambda _c(2595)^+$ and $\Lambda _c(2625)^+$ as an example to show it. The mass splitting of $\Lambda _c(2595)^+$ and $\Lambda _c(2625)^+$ is about 36 MeV, which is one order smaller than their excited energies. So, the spin-dependent interactions are usually treated as the leading-order perturbation contribution in the practical calculation.

Thus, we first ignore the spin-dependent interactions and ascribe the mass gaps between the single baryons with the different strangeness but with same $nL(J^P)$ quantum numbers to the mass difference of light quark clusters. In the quasi-two-body picture, the spin average mass of a nL heavy baryon multiplet could be denoted as follows

$$\begin{aligned} {\bar{M}}_{nL} = m_Q + m_{cluster } + E_{nL}. \end{aligned}$$

(3)

Since two light quarks in the cluster are in the ground state, we take the chromomagnetic model [83,84,85,86] to parameterize the mass of light quark cluster, where the mass of the light quark cluster could be written as

$$\begin{aligned} m_{cluster } = m_{1} + m_{2} + A\frac{{\mathbf{s }_1}\cdot {\mathbf{s }_2}}{m_1m_2} \end{aligned}$$

(4)

with

$$\begin{aligned} A = \frac{16\pi }{9}\langle \alpha _s(r)\delta ^3(\mathbf{r })\rangle . \end{aligned}$$

(5)

In the following analysis, assuming the coefficient A to be a positive value for simplicity and combing with Eqs. (3) and (4), we have

$$\begin{aligned} \begin{aligned} {\bar{M}}^{\Xi _Q}_{nL}-{\bar{M}}^{\Lambda _Q}_{nL}=~&m_s-m_q+\frac{3A}{4m_q}\left( \frac{1}{m_q}-\frac{1}{m_s}\right) =\delta {m}+3\Delta ,\\ {\bar{M}}^{\Xi _Q^\prime }_{nL}-{\bar{M}}^{\Sigma _Q}_{nL}=~&m_s-m_q-\frac{A}{4m_q}\left( \frac{1}{m_q}-\frac{1}{m_s}\right) =\delta {m}-\Delta ,\\ {\bar{M}}^{\Omega _Q}_{nL}-{\bar{M}}^{\Xi _Q^\prime }_{nL}=~&m_s-m_q-\frac{A}{4m_s}\left( \frac{1}{m_q}-\frac{1}{m_s}\right) =\delta {m}-\Delta ^\prime . \end{aligned} \end{aligned}$$

(6)

Here, $m_q$ and $m_s$ denote the constituent masses of up/down and strange quarks, respectively, while $\delta {m}$ denotes their mass difference. Under the situations $m_s>m_q$ and $A>0$, $\delta {m}$, $\Delta$, and $\Delta ^\prime$ defined in Eqs. (6) must be positive. Thus, we obtain the following relation

$$\begin{aligned} {\bar{M}}^{\Xi _Q}_{nL}-{\bar{M}}^{\Lambda _Q}_{nL} > {\bar{M}}^{\Xi _Q^\prime }_{nL}-{\bar{M}}^{\Sigma _Q}_{nL}, \end{aligned}$$

(7)

which explain why the mass gaps of the baryons in the ${\bar{3}}_f$ multiplet is larger than the $6_f$ multiplet well (see the comparison of the mass gap values shown in Tables 1 and 3). There, the measured mass gaps of the involved $\Lambda _Q$ and $\Xi _Q$ states are about 190 MeV (see Table 1), while the mass gaps of $\Sigma _Q$ and $\Xi _Q^\prime$ states are around 120 MeV (see Table 3).

Table 4 The measured mass splittings of the 1P and 1D states of $\Lambda _c$ and $\Xi _c$ baryons due to the spin-orbit interaction. The corresponding spin-parity quantum numbers of these states can be found in Table 1

Besides, we can do a further numerical analysis to illustrate why mass gaps of the discussed $\Omega _Q$ and $\Xi _Q^\prime$ states are around 120 MeV. One takes the average values of mass gaps of $\Lambda _Q$ and $\Xi _Q$ baryon states with the same nL to be 195 MeV, and the average values of the mass gaps of $\Sigma _Q$ and $\Xi _Q^\prime$ baryons with the same nL to be 122 MeV, by which we may fix the values of $\delta {m}$ and $\Delta$ in Eqs. (6), i.e., $\delta {m}=140.3$ MeV and $\Delta =18.3$ MeV. If further setting the masses of light quarks as $m_q=280$ MeV and $m_s=420$ MeV, the parameters A and $\Delta ^\prime$ can be fixed as 1.717$\times 10^7$ MeV$^3$ and 12.2 MeV, respectively. Finally, the mass gap between $\Xi _Q^\prime$ and $\Omega _Q$ states is estimated to be

$$\begin{aligned} {\bar{M}}^{\Omega _Q}_{nL}-{\bar{M}}^{\Xi _Q^\prime }_{nL} \simeq 128 ~MeV . \end{aligned}$$

(8)

This value is consistent with the measured mass gaps of $\Omega _Q$ and $\Xi _Q^\prime$ baryons with the same $nL(J^P)$ quantum numbers, as shown in Table 3.

In a word, the universal behavior of mass gaps of single heavy baryons can be well understood by the constituent quark model.

2.3 Further study of the equal mass splitting phenomenon involved in the excited ${\bar{3}}_f$ baryons

Our study already indicates that the nearly equal mass gap of $\Lambda _c$ and $\Xi _c$ baryons with the same nL can be reproduced. In this subsection, we discuss the case when the spin-dependent interaction is included for the ${\bar{3}}_f$ baryons.

Firstly, we present the experimental results in Table 4, where the $\Lambda _c(2595)/\Xi _c(2790)$ with $J^P=1/2^-$ and the $\Lambda _c(2625)/\Xi _c(2815)$ with $J^P=3/2^-$ are the 1P states. We may find that the mass splitting of $\Lambda _c(2595)$ and $\Lambda _c(2625)$ is comparable with the splitting of $\Xi _c(2790)$ and $\Xi _c(2815)$. Similar phenomenon happens for D-wave $\Lambda _c$ and $\Xi _c$ baryons which are listed in the second line of Table 4. This nearly equal mass splitting phenomenon can also be understood in our scheme.

For these $\Lambda _Q$ and $\Xi _Q$ baryons in ${{\bar{3}}}_f$ representation, the expression of the spin-dependent interaction is very simple since the involved spin of light quark cluster is zero. Thus, the only spin-dependent interaction is relevant to the following spin-orbit coupling

$$\begin{aligned} V_{\text {so}} = \frac{4}{3}\frac{\alpha _s}{r^3}\frac{1}{m_cluster m_Q}\mathbf{S }_Q\cdot {\mathbf{L }}. \end{aligned}$$

(9)

Here, the second and higher order contributions of $1/m_Q$ are ignored since the heavy quark mass in the baryon system is much heavier than the mass of the light quark cluster. The spin-orbital interaction which is given in Eq. (9) has been adopted in our previous work [80] and successfully predicted the mass splitting of D-wave $\Lambda _c^+$ baryons.

In fact, the equal mass splitting phenomenon shown in Table 4 requires the same value of $\xi _Q=\langle \frac{4}{3}\frac{\alpha _s}{r^3}\frac{1}{m_cluster m_Q}\rangle$ in Eq. (9) for the excited $\Lambda _Q$ and $\Xi _Q$ baryons. In the following, we will prove it.

Here, we adopt the scaling technique [87] to deal with the Schrödinger equation, which can help us to find how the eigenvalues $E_{nL}$ and the expectation value $\langle {r^n}\rangle$ vary with changing the parameters in the quark potential model.

The Schrödinger equation for the single heavy flavor baryon could be written as

$$\begin{aligned} \left[ -\frac{\nabla ^2}{2\mu } +V_{SI }(r) \right] \psi ^m_{nL} = E_{nL}\psi ^m_{nL}. \end{aligned}$$

(10)

As mentioned before, different kinds of potentials can be taken to depict the effective interaction between the light quark cluster and the heavy quark. We will show that the relationship $\xi _{\Lambda _Q}\simeq \xi _{\Xi _Q}$ may always stand for the usual effective potentials. To illustrate this point, we employ four typical kinds of potentials for the effective interaction $V_{SI }(r)$, i.e., the power-law potential [88], the Cornell potential [9], the logarithmic potential [89, 90], and the Indiana potential [91]. Their expressions are

$$\begin{aligned} \begin{aligned} V_1(r) =&b_1r^\nu -V_{10},\\ V_2(r) =&-\frac{4}{3}\frac{\alpha }{r}+b_2r-V_{20},\\ V_3(r) =&b_3\ln \left( \frac{r+t}{r_0}\right) ,\\ V_4(r) =&b_4\frac{(1-\Lambda r)^2}{r\ln r}. \end{aligned} \end{aligned}$$

(11)

More details of these phenomenological potentials could be found in Refs. [87, 92].

We should repeat the nearly equal excited energies of $\Lambda _Q$ and $\Xi _Q$ states, i.e.,

$$\begin{aligned} E_{nL}^{\Lambda _Q} \simeq E_{nL}^{\Xi _Q}, \end{aligned}$$

(12)

which has been implied in the discussion in Sect. 2.2 (refer to Eq. (3)). In the following, we take the power-law potential as an example to show how to obtain the following relationship

$$\begin{aligned} \frac{\xi _{\Lambda _Q}}{\xi _{\Xi _Q}} = \left( \frac{\mu _{\Lambda _Q}}{\mu _{\Xi _Q}}\right) ^{3/2}\frac{m_{[qs]}}{m_{[ud]}}, \end{aligned}$$

(13)

associated with the relation in Eq. (12). Here, the $m_{[ud]}$ and $m_{[qs]}$ denote the masses of light scalar quark clusters in the $\Lambda _Q$ and $\Xi _Q$ baryons, respectively.

With the power-law potential, the radical part of Eq. (10) could be written as

$$\begin{aligned} -\frac{{\text {d}}^2\chi _{nl}}{{\text {d}}r^2} + \left[ 2\mu b_1r^\nu +\frac{l(l+1)}{r^2}\right] \chi _{nl} = 2\mu (E_{nL}+V_{10})\chi _{nl}. \end{aligned}$$

(14)

Next, we set $z=\eta r$ and define $u_{nl}(z)\equiv \chi _{nl}(r)$. Then Eq. (14) can be translated into the following form

$$\begin{aligned} -\frac{{\text {d}}^2u_{nl}}{{\text {d}}z^2} + \left[ \frac{2\mu b_1}{\eta ^{\nu +2}}z^\nu +\frac{l(l+1)}{z^2}\right] u_{nl} = \frac{2\mu (E_{nL}+V_{10})}{\eta ^2}u_{nl}. \end{aligned}$$

(15)

When setting

$$\begin{aligned} \frac{2\mu b_1}{\eta ^{\nu +2}} =1,\quad \varepsilon _{nL} = \frac{2\mu (E_{nL}+V_{10})}{\eta ^2}, \end{aligned}$$

(16)

the following scaled Schrödinger equation is obtained

$$\begin{aligned} -\frac{{\text {d}}^2u_{nl}}{{\text {d}}z^2} + \left[ z^\nu +\frac{l(l+1)}{z^2}\right] u_{nl} = \varepsilon _{nL}u_{nl}, \end{aligned}$$

(17)

which can be solved by the numerical calculation or the approximation method when the parameter $\nu$ is given. For the single heavy flavor baryon, the parameter $\nu$ is constrained to be about 0.38 by the values of ${\mathcal {R}}_1$ and ${\mathcal {R}}_2$ listed in Table 2. The other parameters $m_\mathrm{cluster}$, $m_Q$, $b_1$, and $V_{10}$ can be further constrained by fitting the masses of these known $\Lambda _Q$ and $\Xi _Q$ baryons.

In the following discussion, however, we do not need to perform the detailed numerical calculation. As pointed above, the parameter $\nu$ could be regarded as a constant for both $\Lambda _Q$ and $\Xi _Q$ states. Then the eigenvalue $\varepsilon _{nL}$ and the expectation value $\langle {z^{-3}}\rangle$ given by Eq. (17) are same for the $\Lambda _Q$ and $\Xi _Q$ states. With Eq. (12), we get

$$\begin{aligned} A_1 = \frac{E_{nL}-E_{n^\prime L^\prime }}{\varepsilon _{nL}-\varepsilon _{n^\prime L^\prime }}=b_1(2b_1\mu )^{-\frac{\nu }{\nu +2}}, \end{aligned}$$

(18)

which can also be regarded as a constant for the $\Lambda _Q$ and $\Xi _Q$ states. Here, Eq. (18) can be derived with the help of Eq. (16). Then one can obtain the following expression

$$\begin{aligned} \frac{\xi _{\Lambda _Q}}{\xi _{\Xi _Q}} = \frac{\langle r^{-3}\rangle _{\Lambda _Q}}{\langle r^{-3}\rangle _{\Xi _Q}}\frac{m_{[qs]}}{m_{[ud]}} = \left( \frac{b_{1\Lambda _Q}\mu _{\Lambda _Q}}{b_{1\Xi _Q}\mu _{\Xi _Q}}\right) ^{\frac{3}{\nu +2}}\frac{m_{[qs]}}{m_{[ud]}} \end{aligned}$$

(19)

with help of $\langle r^{-3}\rangle =\eta ^3\langle z^{-3}\rangle =(2b\mu )^{3/(\nu +2)}\langle z^{-3}\rangle$. In addition, one deduces $b_1$ by Eq. (18) as

$$\begin{aligned} b_1 = A^{\frac{\nu +2}{2}}(2\mu )^{\nu /2}. \end{aligned}$$

(20)

Combining Eqs. (19) and (20), the result of Eq. (13) can be obtained. The similar derivations involved in other three kinds of potentials are presented in Appendix. With the definition of $\mu$, we further have the following relations

$$\begin{aligned} \frac{\xi _{\Lambda _Q}}{\xi _{\Xi _Q}} = \left( \frac{m_Q+m_{[qs]}}{m_Q+m_{[ud]}}\right) ^{3/2}\left( \frac{m_{[ud]}}{m_{[qs]}}\right) ^{1/2}. \end{aligned}$$

(21)

When taking the following values

$$\begin{aligned} \begin{aligned} m_c=1.55~\text {GeV},&\quad m_{[ud]}=0.71~\text {GeV},\\ m_b=4.65~\text {GeV},&\quad m_{[us]}=0.90~\text {GeV}, \end{aligned} \end{aligned}$$

(22)

for the masses of light quark clusters and heavy quarks in the single heavy flavor baryon, we estimate the values of two ratios

$$\begin{aligned} \frac{\xi _{\Lambda _c}}{\xi _{\Xi _c}} \simeq 1.003,\quad \frac{\xi _{\Lambda _b}}{\xi _{\Xi _b}} \simeq 0.936, \end{aligned}$$

(23)

which prove that $\xi _Q$ almost keeps same for the $\Lambda _Q$ and $\Xi _Q$ baryons.

3 Application

The universal behavior of mass gaps which was discussed above can give some valuable implications, especially, for predicting the masses of the undiscovered single heavy baryon states.

• (1) A neutral resonance was discovered by the BaBar Collaboration in the process $B^-\rightarrow \Sigma _c(2846)^0{\bar{p}}\rightarrow \Lambda _c^+\pi ^-{\bar{p}}$ [93]. Its mass and decay width are

  $$\begin{aligned} \begin{aligned} m(\Sigma _c(2846)^0) ~=~&2846\pm 8\pm 10 ~\text {MeV},\\ \Gamma (\Sigma _c(2846)^0) ~=~&86^{+33}_{-22}\pm 12 ~\text {MeV}. \end{aligned} \end{aligned}$$

  (24)

  With the higher mass and the weak evidence of $J = 1/2$, BaBar suggested the signal of $\Sigma _c(2846)^0$ to be different from the $\Sigma _c(2800)^{0,+,++}$ states which were discovered by the Belle Collaboration via the $e^+e^-$ collision [78]. For the limited information of the $J^P$ quantum numbers and branching ratios, however, PDG temporarily treated the $\Sigma _c(2846)^0$ and $\Sigma _c(2800)^{0,+,++}$ as a same state [60].

  The universal behavior of mass gap discussed in this work may provide some valuable clues for clarifying the puzzle of $\Sigma _c(2846)^0$ and $\Sigma _c(2800)^{0,+,++}$ states. We find that the recently reported $\Xi _c^\prime (2965)^0$ [48] and $\Omega _c(3090)^0$ [45, 66] could form the following chain

  $$\begin{aligned} \Sigma _c(2846)^0~\leftrightarrow ~~\Xi _c^\prime (2965)^0~\leftrightarrow ~~\Omega _c(3090)^0, \end{aligned}$$

  (25)

  if adding $\Sigma _c(2846)^0$, which satisfies the requirement from universal behavior of mass gaps. Additionally, the $\Sigma _c(2800)^{0,+,++}$ has been grouped into another chain with $\Xi _c^\prime (2923)^0$ and $\Omega _c(3050)^0$, as shown in Table 3. This analysis based on universal behavior of mass gap suggests that the $\Sigma _c(2846)^0$ and $\Sigma _c(2800)^{0,+,++}$ baryons should be two different states.

• (2) As the second example, we will apply the universal behavior of mass gap rule to predict the masses of 1P, 1D, and 2S $\Xi _b$ states. At present, only the ground states, i.e., the $\Xi _b(5792)^0$ and the $\Xi _b(5797)^-$, have been established. Since LHCb has shown its capability in accumulating the data sample of the excited bottom baryon resonances in the past years, we expect the following predicted masses of excited $\Xi _b^0$ baryons to be tested by the LHCb Collaboration in the near future.

  The mass gap between $\Xi _b(5792)^0$ and $\Lambda _b(5620)^0$ state is 172.3 MeV which is about 10 MeV smaller than the mass difference of $\Lambda _c(2286)^+$ and $\Xi _c(2470)^+$ states. Comparing the results in Table 1, we may take the mass gaps as 185 MeV for the excited $\Lambda _b^0$ and $\Xi _b^0$. With the measured masses of $\Lambda _b(6072)^0$ states, the mass of 2S $\Xi _b$ state could be predicted as 6257 MeV directly. We have shown that the mass splitting of nL $\Xi _Q$ states is nearly equal to the corresponding $\Lambda _Q$ baryons (see Eq. (23)). Therefore, the masses of two 1P $\Xi _b$ states are predicted to be 6097 MeV and 6105 MeV, while the masses of two 1D $\Xi _b$ states are about 6331 MeV and 6337 MeV.

• (3) As the last example, we point out that two resonance structures may exist in the previously observed signals of $\Sigma _b(6097)^\pm$ [47] and $\Xi _b^\prime (6227)^{0,-}$ [46, 51]. Two narrow bottom baryons $\Omega _b(6340)^-$ and $\Omega _b(6350)^-$ which were reported by LHCb in the $\Xi _b^0K^-$ decay channel [49] could be regarded as the good candidates of P-wave states with $J^P=3/2^-$ and $J^P=5/2^-$, respectively. With this assignment, the mass splitting of these two states can also be explained by the quark potential model [15, 56] and the QCD sum rule [94]. According to the universal behavior of mass gaps of single heavy flavor baryons, we could conjecture that the $\Sigma _b$ and $\Xi _b^\prime$ partners of $\Omega _b(6340)^-$ and $\Omega _b(6350)^-$ have been contained in the signals of $\Sigma _b(6097)^\pm$ and $\Xi _b^\prime (6227)^{0,-}$. The situation of charmed baryons is alike. The observed $\Xi _c(2923)^0$ and $\Omega _c(3050)^0$ can be explained as the P-wave states with $J^P=3/2^-$, while the $\Xi _c(2939)^0$ and $\Omega _c(3065)^0$ could be regarded as the $J^P=5/2^-$ partners. Thus, we may point out that the signal of $\Sigma _c(2800)^{0,+,++}$ states [78] may also contain the $J^P=3/2^-$ and $J^P=5/2^-$ states. With the higher statistical precision, we expect the LHCb and Belle II experiments to distinguish these two resonance structures in future.

4 Further discussion for the $\rho$-mode excited single heavy baryons

It is interesting to point out that none of $\rho$-mode excited single heavy baryons have been established by experiments. We have treated these discovered heavy baryons as the $\lambda$-mode excitations and explained their mass gaps well. One may naturally want to know how about the mass gaps of these $\rho$-mode excited single heavy baryons. As shown in Sect. 3, the mass gaps of these $\lambda$-mode excited heavy baryons arise from the mass difference of s and u/d quarks and the chromomagnetic interaction of two light quarks in the cluster. For the $\rho$-mode excited single heavy baryons, the chromomagnetic interaction of two light quarks becomes smaller and can be negligible. Then the mass gap of corresponding $\rho$-mode excited single heavy baryons mainly comes from the mass difference of s and u/d quarks. This means that the mass gap behavior of $\rho$-mode excited single heavy baryons should be different from these $\lambda$-mode excited heavy baryons.

For confirming our conjecture above, we will calculate the masses of charm baryons by a non-relativistic quark model in the three-body picture. The Hamiltonian of this model is given as [95]

$$\begin{aligned} {\hat{H}}_0=\sum \limits _{i=1}^{3}\left( m_i+\frac{p_i^2}{2m_i}\right) +\sum \limits _{i<j}V_{ij}, \end{aligned}$$

(26)

where $m_i$ and $p_i$ are the mass and momentum of i-th constituent quark. The $V_{ij}$ in Eq. (26) denotes the interactions between quark-quark in the baryon system, which contains both spin-independent and spin-dependent interactions. Specifically, the $V_{ij}$ is written as

$$\begin{aligned} V_{ij}= & {} -\frac{2}{3}\frac{\alpha _s}{r_{ij}}+\frac{1}{2}br_{ij}-C+\frac{16\pi \alpha _s}{9m_im_j}\left( \frac{\sigma }{\sqrt{\pi }}\right) ^3\mathrm{e}^{-\sigma ^2r^2}{} \mathbf{s}_i\cdot \mathbf{s}_j\nonumber \\&+V^{\mathrm{tens}}_{ij}+V^\mathrm{so}_{ij}, \end{aligned}$$

(27)

where the $\alpha _s$, b, $\sigma$, and C denote the coupling constant of one-gluon exchange (OGE), the strength of linear confinement, the Gaussian smearing parameter, and a mass-renormalized constant, respectively. The $V^{\mathrm{tens}}_{ij}$ and $V^{\mathrm{so}}_{ij}$ in Eq. (27) are the tensor and spin-orbit terms, respectively (see Ref. [11] for more details). The model parameters are collected in Table 5.

Table 5 The parameters of non-relativistic quark potential model. The consistent quark masses are presented in the last row

As shown in Table 6, the measured masses of well-established charm baryons can be reproduced by the non-relativistic quark potential model.

Table 6 A comparison of the predicted masses of established charm baryons with the measured results (in MeV). The measured masses are listed in the first row below the states while the predictions are given in the second row. The measured masses of $\Xi ^\prime _c(2923)$, $\Xi ^\prime _c(2939)$, and $\Xi ^\prime _c(2965)$ are taken from Ref. [48], while the others are borrowed from Ref. [60]

The non-relativistic quark model also allows us to obtain the masses of $\rho$-mode excited charm baryons simultaneously with the same parameters in Table 5. For illustrating the mass gaps of $\rho$-mode excited charmed and charmed-strange baryons, we list the predicted masses and mass gaps in Table 7.

Table 7 The predicted masses of P-wave $\rho$-mode excited charm baryons (in MeV). States with the same $J^P$ are distinguished by their different masses. Specifically, the states with the lower and higher masses are denoted by the subscripts “L” and “H”, respectively

For the P-wave $\rho$-mode excited charmed baryons, there are five $\Lambda _c$ states and two $\Sigma _c$ states (see Table 7). The case of charmed-strange baryons is alike. By comparing of the predicted masses, one may find that the mass gaps of P-wave $\rho$-mode excited charmed and charmed-strange baryons are all around 150 MeV. As shown in Table 5, the mass of s quark in our model is 140 MeV larger than the u/d quark. Then we may preliminarily conclude that the mass gaps of $\rho$-mode excited charm baryons mainly come from the mass difference of s and u/d quarks. Therefore, we confirmed our conjecture above.

5 Summary

Until now, about fifty single heavy baryons have been observed by experiment [60], where their isospin partners have not been counted. Facing such abundant experimental data of single heavy flavor baryons, we have a good chance to check the mass gaps existing in the established mass spectrum of single heavy flavor baryons.

In this work, we analysed the measured masses of these discovered single heavy baryons and pointed out that some universal behaviors of the mass gaps may exist in this kind of baryons. By a constituent quark model, we have revealed the underlying mechanism behind this universal phenomenon and explained the universal behaviors of mass gaps for the ${{\bar{3}}}_f$ and $6_f$ single heavy flavor baryons. Additionally, we also illustrate why there exists the nearly equal mass splitting for the orbital excited baryons in the ${{\bar{3}}}_f$ representation. It is important to note that these discovered heavy baryons are assigned as the $\lambda$-mode excitations.

Universal behaviors of mass gaps can be applied to the mass spectrum analysis. In this work, we give three implications: (1) We indicated that the neutral resonance $\Sigma _c(2846)^0$ reported from the BaBar Collaboration [93] is different from the $\Sigma _c(2800)^{0,+,++}$ states [78]. (2) The masses of 1P, 1D, and 2S $\Xi _b^0$ states were predicted. (3) We pointed out that two resonance structures may exist in the previously observed signals of $\Sigma _c(2800)^{0,+,++}$ [60], $\Sigma _b(6097)^\pm$ [47], and $\Xi _b^\prime (6227)^{0,-}$ [46, 51] states. These predictions based on the universal behaviors of mass gaps could be tested by the LHCb and Belle II experiments in future.

We also discuss the mass gaps of these $\rho$-mode excited single heavy flavor baryons. We find that the mass gaps of P-wave $\rho$-mode excited charmed baryons are around 150 MeV. Then we may preliminarily conclude that the mass gap behavior of $\rho$-mode excited single heavy baryons is quite different from these $\lambda$-mode excitations. So the mass gaps may provide some valuable clues for distinguishing the $\rho$-mode excited single heavy baryons from the $\lambda$-mode excitations in future.

Appendix

In this section, we follow the similar derivations presented in Sect. 2.3 and show how to obtain the relation in Eq. (13) by the Cornell potential, the logarithmic potential, and the Indiana potential.

For the Cornell potential, the Schrödinger equation is

$$\begin{aligned}&-\frac{{\text {d}}^2\chi _{nl}}{{\text {d}}r^2} + \left[ 2\mu \left( -\frac{4}{3}\frac{\alpha }{r}+b_2r\right) +\frac{l(l+1)}{r^2}\right] \chi _{nl}\nonumber \\&\quad = 2\mu (E_{nL}+V_{20})\chi _{nl}. \end{aligned}$$

(28)

When we set $z=\eta r$ and define $u_{nl}(z)\equiv \chi _{nl}(r)$, Eq. (28) can be translated into the following form

$$\begin{aligned}&-\frac{{\text {d}}^2u_{nl}}{{\text {d}}z^2} + \left[ \left( \frac{2\mu b_2}{\eta ^3}z-\frac{8\alpha \mu }{3\eta z}\right) +\frac{l(l+1)}{z^2}\right] u_{nl}\nonumber \\&\quad =\frac{2\mu (E_{nL}+V_{20})}{\eta ^2}u_{nl}. \end{aligned}$$

(29)

We further set $\frac{8\alpha \mu }{3\eta } = 1$, $\frac{2\mu b_2}{\eta ^3} = \lambda$, and $\varepsilon _{nL} = \frac{2\mu (E_{nL}+V_{20})}{\eta ^2}$, the scaled Schrödinger equation for the Cornell potential is given as

$$\begin{aligned} -\frac{{\text {d}}^2u_{nl}}{{\text {d}}z^2} + \left[ \left( -\frac{1}{z}+\lambda z\right) +\frac{l(l+1)}{z^2}\right] u_{nl} = \varepsilon _{nL}u_{nl}. \end{aligned}$$

(30)

The only parameter $\lambda$ in Eq. (30) could be fixed as $\lambda \simeq 1.70$ by the ${\mathcal {R}}_1$ and ${\mathcal {R}}_2$ (see Eq. (1) and Table 2) for $\Lambda _Q$ and $\Xi _Q$ baryons. With the variable substitutions above, we have

$$\begin{aligned}\left\{ \begin{array}{l} r = \dfrac{3}{8\alpha \mu }z,\\ b = \dfrac{2^8\mu ^2\alpha ^3}{27}\lambda ,\\ E_{nL}= \dfrac{32\mu \alpha ^2}{9}\varepsilon _{nL}-V_{20}. \end{array}\right. \end{aligned}$$

Due to the condition of Eq. (12), the following ratio

$$\begin{aligned} A_2 = \frac{E_{nL}-E_{n^\prime L^\prime }}{\varepsilon _{nL}-\varepsilon _{n^\prime L^\prime }} = \dfrac{32\mu \alpha ^2}{9}, \end{aligned}$$

(31)

could be regarded as a constant for the $\Lambda _Q$ and $\Xi _Q$ baryons. For the Cornell potential, we have

$$\begin{aligned} \frac{\xi _{\Lambda _Q}}{\xi _{\Xi _Q}} = \frac{\langle r^{-3}\rangle _{\Lambda _Q}}{\langle r^{-3}\rangle _{\Xi _Q}}\frac{m_{[qs]}}{m_{[ud]}} = \left( \frac{\mu _{\Xi _Q}\alpha _{\Xi _Q}}{\mu _{\Lambda _Q}\alpha _{\Lambda _Q}}\right) ^3\frac{m_{[qs]}}{m_{[ud]}}. \end{aligned}$$

(32)

Then Eq. (13) can be obtained directly by considering the relationship $\alpha =\frac{3}{4}\sqrt{\frac{A_2}{2\mu }}$.

For the logarithmic potential, the Schrödinger equation is given as

$$\begin{aligned} -\frac{{\text {d}}^2\chi _{nl}}{{\text {d}}r^2} + \left[ 2\mu b_3\ln \left( \frac{r+t}{r_0}\right) +\frac{l(l+1)}{r^2}\right] \chi _{nl} = 2\mu E_{nL}\chi _{nl}. \end{aligned}$$

(33)

Next, we set $z=\eta r$ and define $u_{nl}(z)\equiv \chi _{nl}(r)$. Then Eq. (33) can be translated into the following form

$$\begin{aligned} -\frac{{\text {d}}^2u_{nl}}{{\text {d}}z^2} + \left[ \ln \left( z+z_0\right) +\frac{l(l+1)}{z^2}\right] u_{nl} = \varepsilon _{nL}u_{nl}, \end{aligned}$$

(34)

where we have done the following variable substitutions

$$\begin{aligned}{\left\{ \begin{array}{ll} ~~\eta &{}= ~\sqrt{2b_3\mu },\\ ~~z_0 &{}= ~\sqrt{2b_3\mu }t,\\ ~E_{nL} &{}= ~b_3\left[ \varepsilon _{nL}-\ln \left( \sqrt{2b_3\mu }r_0\right) \right] . \end{array}\right. }\end{aligned}$$

For the $\Lambda _Q$ and $\Xi _Q$ baryons, the parameter $z_0$ in the scaled Schrödinger equation of Eq. (34) can be fixed as $z_0\simeq 1.80$ by the ${\mathcal {R}}_1$ and ${\mathcal {R}}_2$. Similar to the Eq. (18), one could treat the parameter $b_3$ as a constant for the $\Lambda _Q$ and $\Xi _Q$ baryons. Thus, Eq. (13) can be obtained directly.

Finally, the Schrödinger equation of the Indiana potential is given as

$$\begin{aligned} -\frac{{\text {d}}^2\chi _{nl}}{{\text {d}}r^2} + \left[ 2\mu b_4\frac{(1-\Lambda r)^2}{r\ln (\Lambda r)}+\frac{l(l+1)}{r^2}\right] \chi _{nl} = 2\mu E_{nL}\chi _{nl}. \end{aligned}$$

(35)

In the same way, we set $z=\eta r$ and define $u_{nl}(z)\equiv \chi _{nl}(r)$. Eq. (35) becomes as

$$\begin{aligned} -\frac{{\text {d}}^2u_{nl}}{{\text {d}}z^2} + \left[ \frac{2\mu b_4}{\eta }\frac{(1-\frac{\Lambda }{\eta }z)^2}{z\ln (\frac{\Lambda }{\eta }z)}+\frac{l(l+1)}{z^2}\right] u_{nl} = \frac{2\mu E_{nL}}{\eta ^2}u_{nl}. \end{aligned}$$

(36)

When one sets $\eta =\Lambda$ and does the following variable substitutions

$$\begin{aligned} \kappa = \frac{2\mu b_4}{\Lambda },\quad E_{nL} = \frac{\Lambda ^2}{2\mu }\varepsilon _{nL}, \end{aligned}$$

(37)

the scaled Schrödinger equation for the Indiana potential is given as

$$\begin{aligned} -\frac{{\text {d}}^2u_{nl}}{{\text {d}}z^2} + \left[ \kappa \frac{(1-z)^2}{z\ln {z}}+\frac{l(l+1)}{z^2}\right] u_{nl} = \varepsilon _{nL}u_{nl}. \end{aligned}$$

(38)

For the $\Lambda _Q$ and $\Xi _Q$ baryons, the parameter $\kappa$ in the scaled Schrödinger equation above should be fixed as $\kappa \simeq 0.80$ by the ${\mathcal {R}}_1$ and ${\mathcal {R}}_2$. Similar to the Eq. (18), the following ratio

$$\begin{aligned} A_4 = \frac{E_{nL}-E_{n^\prime L^\prime }}{\varepsilon _{nL}-\varepsilon _{n^\prime L^\prime }} = \frac{\Lambda ^2}{2\mu } \end{aligned}$$

(39)

could be treated as a constant for the $\Lambda _Q$ and $\Xi _Q$ baryons. With the $\eta =\Lambda =\sqrt{2A_4\mu }$, Eq. (13) could be obtained for the Indiana potential.