The newly observed $\Lambda _b(6072)^0$ structure and its $\rho$-mode nonstrange partners

Abstract

Inspired by the newly observed $\Lambda _b(6072)$ structure, we investigate its strong decay behaviors under various assignments within the $^3P_0$ model. Compared with the mass and total decay width, our results suggest that the $\Lambda _b(6072)$ can be regarded as the lowest $\rho$-mode excitation in $\Lambda _b$ family. Then, the strong decays of $\rho$-mode nonstrange partners for the $\Lambda _b(6072)$ are calculated. It is found that the $J^P=5/2^-$ $\Lambda _b$ and $\Lambda _c$ states are relatively narrow, and mainly decay into the $\Sigma _b^{(*)} \pi$ and $\Sigma _c^{(*)} \pi$ final states, respectively. These two states have good potentials to be observed in future experiments, which may help us to distinguish the three-quark model and diquark model.

1 Introduction

Recently, the CMS Collaboration observed an evidence for a broad excess of events in the $\Lambda _b^0 \pi ^+ \pi ^-$ mass spectrum in the region of $6040 \sim 6100$ MeV [1]. Subsequently, the LHCb Collaboration also found a new baryon resonance $\Lambda _b(6072)^0$ in the same final states. The measured mass and decay width are [2]:

$$\begin{aligned} m[\Lambda _b(6072)^0]= & {} 6072.3\pm 2.9~\pm 0.6\pm 0.2~\mathrm {MeV}, \end{aligned}$$

(1)

$$\begin{aligned} \Gamma [\Lambda _b(6072)^0]= & {} 72\pm 11~\pm 2~\mathrm {MeV}, \end{aligned}$$

(2)

which is consistent with the structure reported by the CMS Collaboration.

In recent years, a series of important progresses on the $\Lambda _{c(b)}$ and $\Sigma _{c(b)}$ families have been made experimentally. In 2017, a new charmed baryon $\Lambda _c(2860)$ was reported by the LHCb Collaboration in the $D^0 p$ final state, and the masses and decay widths of the $\Lambda _c(2880)$ and $\Lambda _c(2940)$ were measured [3]. In 2018, the LHCb Collaboration found an excited bottom baryon $\Sigma _b(6097)$ in the $\Lambda ^0_b \pi ^\pm$ invariant mass spectrum [4]. In 2019, the LHCb Collaboration reported the observation of $\Lambda _b(6146)$ and $\Lambda _b(6152)$ in the $\Lambda _b^0 \pi ^+ \pi ^-$ decay mode [5], which were confirmed by the CMS Collaboration subsequently [1]. These discoveries have attracted wide attentions of theorists, and plenty of works have been done to investigate the inner structures of observed $\Lambda _{c(b)}$ and $\Sigma _{c(b)}$ resonances [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40]. In the traditional quark model, for the low-lying $\lambda$-mode $\Lambda _c$ states, the $\Lambda _c(2595)$ and $\Lambda _c(2625)$ resonances can be assigned as the $\Lambda _c(1P)$ doublet, the $\Lambda _c(2860)$ and $\Lambda _c(2880)$ resonances can be categorized into the $\Lambda _c(1D)$ doublet, and the $\Lambda _c(2765)$ can be regarded as the $\Lambda _c(2S)$ singlet [21,22,23,24,25,26,27,28,29,30]. Similarly, for the low-lying $\lambda$-mode $\Lambda _b$ states, the $\Lambda _b(5912)$ and $\Lambda _b(5920)$ have been assigned as the $\Lambda _b(1P)$ doublet, and the two structures $\Lambda _b(6146)$ and $\Lambda _b(6152)$ can be clarified into the $\Lambda _b(1D)$ doublet [9,10,11, 27,28,29,30,31,32]. It can be seen that the spectra of the low-lying $\lambda$-mode $\Lambda _c$ and $\Lambda _b$ states have been established well except for the $\Lambda _b(2S)$ state. For the $\Sigma _c$ and $\Sigma _b$ states, the similar patterns should also exist, and two P-wave candidates $\Sigma _c(2800)$ and $\Sigma _b(6097)$ have been observed experimentally [4, 41]. Contrary to the abundant $\lambda$-mode states, no $\rho$-mode heavy baryon has been confirmed both theoretically and experimentally.

Considering the mass and width of $\Lambda _b(6072)$, the LHCb Collaboration suggested that it can be assigned as the first radial excitation of the $\Lambda _b$ baryon, the $\lambda$-mode $\Lambda _b(2S)$ state. Although the predicted mass of $\lambda$-mode $\Lambda _b(2S)$ state is consistent with the experimental observation [28,29,30,31,32], the predicted total decay width is actually smaller than $72\pm 11~\pm 2~\mathrm {MeV}$ in the literature [9, 10, 15]. Meanwhile, recent works suggested that it may be a Roper-like resonance [42] or a overlap of two $\Sigma _b(1P)$ states [43]. Before making a final conclusion, it is necessary to examine all possible interpretations carefully.

Based on the small predicted $\Lambda ^0_b \pi \pi$ decay widths of $\Sigma _b(1P)$ states [44] and no signal with 6072 MeV in $\Lambda ^0_b \pi ^\pm$ decay modes [4], the LHCb Collaboration suggested that the interpretation of this newly observed structure as an excited $\Sigma _b(1P)$ resonance is disfavoured. However, for the $\Lambda ^0_b \pi \pi$ decay mode, only the $\Sigma _b(1P) \rightarrow \Lambda ^0_b (\pi \pi )_{I=1} \rightarrow \Lambda ^0_b \pi \pi$ decay chain was calculated [44], and other decay chains, such as $\Sigma _b(1P) \rightarrow \Sigma _b^{(*)} \pi \rightarrow \Lambda ^0_b \pi \pi$, should also exist. Indeed, the decay chains via virtual heavy baryons play essential roles in the three-body decays of excited heavy baryons [42, 45,46,47]. From Refs. [9, 15,16,17, 27], there are five $\Sigma _b(1P)$ states and several of them have rather small $\Lambda ^0_b \pi ^\pm$ branching ratios. It is possible for a $\Sigma _b(1P)$ state to be not observed in the $\Lambda ^0_b \pi ^\pm$ final states experimentally. Hence, the observed structure with 6072 MeV as $\Sigma _b(1P)$ assignments need further investigations.

In the three-quark model, the excitations of the heavy baryons can be divided into two parts, the $\rho$-mode and $\lambda$-mode. In the diquark model, the two light quarks are usually treated as a cluster without excitation, and only the $\lambda$-mode excitation exists. It can be seen that the diquark model freezes the degree of freedom between two light quarks and have less states than that of the three-quark model. In the relativized quark model, the predicted lowest $\rho$-mode $\Lambda _b$ state lies about 6100 MeV [31], which is consistent with the experimental data. It is crucial to discuss the possibility of newly observed $\Lambda _b(6072)$ as a $\rho$-mode ${\tilde{\Lambda }}_b(1P)$ state. Moreover, it is a good opportunity to investigate these low-lying $\rho$-mode excitations in the nonstrange singly heavy baryon systems.

In this work, we tentatively assign the newly observed resonance $\Lambda _b(6072)$ as the $\lambda$-mode $\Lambda _b(2S)$, $\lambda$-mode $\Sigma _b(1P)$, and $\rho$-mode ${\tilde{\Lambda }}_b(1P)$ states, respectively, and calculate their strong decay behaviors. Our results indicate that the $\lambda$-mode $\Lambda _b(2S)$ and $\Sigma _b(1P)$ assignments are disfavored, while the $\Lambda _b(6072)$ as the lowest $\rho$-mode $\Lambda _b$ state is supported. With this interpretation, the $\Lambda _b(6072)$ may be the first observed $\rho$-mode excitation in singly heavy baryons. Then, the strong decays of the $\rho$-mode ${\tilde{\Lambda }}_c(1P)$, ${\tilde{\Sigma }}_b(1P)$ and ${\tilde{\Sigma }}_c(1P)$ states are calculated, and some of them have relatively narrow total decay widths. We hope these theoretical predictions on ${\tilde{\Lambda }}_{b(c)}(1P)$ and ${\tilde{\Sigma }}_{b(c)}(1P)$ states can provide valuable information for searching more $\rho$-mode excitations in future experiments.

The paper is organized as follows. In Sect. 2, we introduce the $^3P_0$ model and notations briefly. In Sect. 3, the strong decays of the newly observed $\Lambda _b(6072)$ state under various assignments are investigated. The strong decay behaviors of the $\rho$-mode ${\tilde{\Lambda }}_c(1P)$, ${\tilde{\Sigma }}_b(1P)$ and ${\tilde{\Sigma }}_c(1P)$ states are presented in Sect. 4. A summary is given in the last section.

2 Model and notations

In this issue, we adopt the $^3P_0$ model to study the strong decay behaviors of the singly heavy baryons. For a certain decay process, this model supposes that a quark–antiquark pair with the quantum number $J^{PC} =0^{++}$ is created from the vacuum and falls apart into the final states [48]. The transition operator T for a decay process $A \rightarrow BC$ can be expressed as

$$\begin{aligned} T= & {} -3\gamma \sum _m\langle 1m1-m|00\rangle \int d^3{\varvec{p}}_4d^3{\varvec{p}}_5\delta ^3({\varvec{p}}_4+{\varvec{p}}_5)\nonumber \\&\times \mathcal{{Y}}^m_1\left( \frac{{\varvec{p}}_4-{\varvec{p}}_5}{2}\right) \chi ^{45}_{1,-m}\phi ^{45}_0\omega ^{45}_0b^\dagger _{4i}({\varvec{p}}_4)d^\dagger _{4j}({\varvec{p}}_5). \end{aligned}$$

(3)

Here, the overall constant $\gamma$ reflects the $q_4{\bar{q}}_5$ pair-production strength, and ${\varvec{p}}_4$ and ${\varvec{p}}_5$ are the momenta of the created quark–antiquark pair. The solid harmonic polynomial $\mathcal{{Y}}^m_1({\varvec{p}})\equiv |p|Y^m_1(\theta _p, \phi _p)$ is the P-wave distribution of the created quark pair in the momentum representation. The $\phi ^{45}_{0}$, $\omega ^{45}_{0}$ and $\chi _{{1,-m}}^{45}$ are the flavor, color, and spin parts of the quark pair $q_4{\bar{q}}_5$.

The transition matrix element for this model can be given by

$$\begin{aligned} \langle BC|T|A\rangle =\delta ^3({\varvec{P}}_A - {\varvec{P}}_B - {\varvec{P}}_C)\mathcal{{M}}^{M_{J_A}M_{J_B}M_{J_C}}, \end{aligned}$$

(4)

where the $\mathcal{{M}}^{M_{J_A}M_{J_B}M_{J_C}}$ corresponds to the helicity amplitude of the decay process $A\rightarrow B+C$. With the $\mathcal{{M}}^{M_{J_A}M_{J_B}M_{J_C}}$, one can calculate the partial width of this process directly

$$\begin{aligned} \Gamma = \pi ^2\frac{p}{M^2_A}\frac{1}{2J_A+1}\sum _{M_{J_A},M_{J_B},M_{J_C}}|\mathcal{{M}}^{M_{J_A}M_{J_B}M_{J_C}}|^2, \end{aligned}$$

(5)

where $p=|{\varvec{p}}|$ is the momentum of the final hadrons. More details of the $^3P_0$ model and the explicit formula can be found in Refs. [49,50,51,52].

Table 1 Notations, quantum numbers and masses of the initial baryons. The masses are taken from the relativized quark model [31]. The Q stands for a charm or bottom quark, and the $\sim$ is used for the $\rho$-mode states. The units are in MeV

As mentioned above, there are two independent excitations in these baryon systems. The excitation between two light quarks is named as the $\rho$-mode, while the $\lambda$-mode stands for the excitation between the heavy quark and light quark subsystem. Then, the P-wave baryons can be divided into two groups, the $\rho$-mode states with $l_\rho =1$ and $l_\lambda =0$, and the $\lambda$-mode states with $l_\rho =0$ and $l_\lambda =1$. Here, we adopt a series of quantum numbers $n_\rho$, $l_\rho$, $n_\lambda$, $l_\lambda$, L, $S_\rho$, j, and $J^P$ to denote a theoretical state. The $n_\rho$ and $l_\rho$ are the radial and orbital quantum numbers between the two light quarks, respectively. Similarly, the $n_\lambda$ and $l_\lambda$ correspond to the radial and orbital quantum numbers between the heavy quark and light quark subsystem, respectively. The $S_\rho$ stands for the total spin of the two light quarks. The j represents total angular momentum of the L and $S_\rho$, where L is the total orbital angular momentum. $J^P$ is the spin-parity of a heavy baryon.

The notations of initial states and predicted masses within the relativized quark model are listed in Table 1. For the $\Lambda _b(6072)$ or $\Sigma _b(6072)$ resonance, we adopt its experimental mass under various assignments. For other $\rho$-mode baryons, we employ the theoretical masses to calculate their strong decays. All the parameters adopted here are the same as our previous works, which have been widely used for the strong decays of singly heavy baryon systems [8, 9, 52,53,54].

3 Strong decays of the $\Lambda _b(6072)$ or $\Sigma _b(6072)$

3.1 $\Lambda _b(2S)$

As mentioned in the introduction, LHCb Collaboration suggested that the $\Lambda _b(6072)$ may be regarded as $\Lambda _b(2S)$ state. The predicted masses of the $\Lambda _b(2S)$ state within quark models are also consistent with the experimental data. The strong decays of this assignment are calculated and listed in Table 2. The predicted width is about 9 MeV, which is significantly smaller than the experimental observation. Thus, our calculations do not support the $\Lambda _b(6072)$ as $\Lambda _b(2S)$ state.

Table 2 Strong decays of the $\Lambda _b(6072)$ as $\Lambda _b(2S)$ state in MeV

Other theoretical works also predicted smaller total decay widths of the $\Lambda _b(2S)$ state, which varies in the range of $2 \sim 36$ MeV [9, 10, 15]. These calculations indicate that the $\Lambda _b(2S)$ should be a narrow state. Also, the decay behaviors for $\Lambda _b(2S)$ state versus its initial mass is presented in Fig. 1. It can be seen that both $\Sigma _b \pi$ and $\Sigma _b^* \pi$ decay modes are important, which can be tested by future experiments.

Fig. 1

The decay behaviors for $\Lambda _b(2S)$ state versus its initial mass. The black line corresponds to the total decay width. The red dotted and blue dashed lines stand for the partial decay widths of $\Sigma _{b} \pi$ and $\Sigma _{b}^{*} \pi$, respectively

Table 3 Decay widths of the $\Sigma _b(6072)$ as $\Sigma _b(1P)$ states in MeV

3.2 $\Sigma _b(1P)$

In the traditional quark model, there are five $\lambda$-mode $\Sigma _b(1P)$ states, and their masses are predicted to be around 6070 $\thicksim$ 6090 MeV in relativized quark model. It is possible to regard the newly observed structure as the $\Sigma _b(1P)$ states. The strong decay behaviors of $\Sigma _b(1P)$ assignments are presented in Table 3. It should be mentioned that the $^3P_0$ model is actually a spectator model, where the quarks in the initial state carry their color, flavor, spin, and momenta into the final states. The only change of degrees of freedom arises from the created quark pair. Based on this spectator assumption, the $\Sigma _b^{(*)} \pi$ channels for the $j=0$ $\Sigma _b(1P)$ state is not allowed. It is shown that all the pure $\Sigma _b(1P)$ assignments are disfavored.

However, the physical resonance may correspond to the mixing of the quark model states. The mixing scheme of P-wave states can be expressed as

$$\begin{aligned} \left( {\begin{array}{c}\left| 1P~1/2^{-}\right\rangle _{1}\\ \left| 1P~1/2^{-}\right\rangle _{2}\end{array}}\right)= & {} \left( \begin{array}{cc} \cos \theta &{} \sin \theta \\ -\sin \theta &{} \cos \theta \end{array} \right) \left( \begin{array}{c} \left| 1/2^{-},j=0\right\rangle \\ \left| 1/2^{-},j=1\right\rangle \end{array} \right) ,\\ \left( {\begin{array}{c}\left| 1P~3/2^{-}\right\rangle _{1}\\ \left| 1P~3/2^{-}\right\rangle _{2}\end{array}}\right)= & {} \left( \begin{array}{cc} \cos \theta &{} \sin \theta \\ -\sin \theta &{} \cos \theta \end{array} \right) \left( \begin{array}{c} \left| 3/2^{-},j=1\right\rangle \\ \left| 3/2^{-},j=2\right\rangle \end{array} \right) . \end{aligned}$$

Due to the finite mass of bottom quark, a small mixing angle is allowed which breaks the heavy quark symmetry explicitly. The total decay widths of various assignments versus the mixing angle $\theta$ in the range of $-30^\circ \sim 30^\circ$ are plotted in Fig. 2. It can be seen that when the mixing angle $\theta$ goes close to $\pm 30^\circ$, the predicted width of $\Sigma _b(6072)$ is consistent with the experimental data. This mixing angle seems larger than that of previous works on the singly bottom baryons [9, 53]. With this mixing angle, the calculated total width of $\Sigma _b(6072)$ is about 52 MeV and the predicted branching ratios of dominating channels are

$$\begin{aligned} Br(\Lambda _b \pi , \Sigma ^*_b \pi )=12.4\%, 86.4\%, \end{aligned}$$

(6)

which indicates that the branching ratio of $\Lambda _b \pi$ final state is also significant. However, the LHCb Collaboration did not observe the $\Sigma _b(6072)$ signal in $\Lambda _b \pi$ mass spectrum [4]. Moreover, there already exists a good candidate $\Sigma _b(6097)$ for the $|1P~{3/2^-}\rangle _2$ state in the literatures [4, 9, 14,15,16,17,18,19]. Given the large mixing angle and significant branching ratio of $\Lambda _b \pi$ channel, the newly observed structure as $\Sigma _b(1P)$ assignments are disfavored.

Fig. 2

The total decay widths versus the mixing angle $\theta$. The red solid and blue dashed lines correspond to the $|1P~{J^P}\rangle _1$ and $|1P~{J^P}\rangle _2$ states, respectively, where the $J^P$ equals to $1/2^-$ or $3/2^-$. The green bands are the experimental total decay widths

3.3 ${\tilde{\Lambda }}_b(1P)$

According to the quark model, there are five $\rho$-mode ${\tilde{\Lambda }}_b(1P)$ states, and the predicted masses within relativized quark model vary from 6100 to 6205 MeV. Considering the uncertainties of quark model, it is possible to treat the $\Lambda _b(6072)$ as the lowest one of ${\tilde{\Lambda }}_b(1P)$ states. Meanwhile, the masses for the $\rho$-mode ${\tilde{\Lambda }}_b(1P)$ states can be estimated with the help of the $\lambda$-mode resonances. In the three-quark model with harmonic oscillator approximation [7], the masses between $\rho$-mode and $\lambda$-mode P-wave excitations can be expressed as

$$\begin{aligned} \frac{{{\bar{m}}}[{\tilde{\Lambda }}_b(1P)] - m[\Lambda _b(1S)]}{{{\bar{m}}}[\Lambda _b(1P)] - m[\Lambda _b(1S)]} = \sqrt{\frac{3m_b}{2m_q+m_b}}, \end{aligned}$$

(7)

where the ${{\bar{m}}}[{\tilde{\Lambda }}_b(1P)]$ and ${{\bar{m}}}[\Lambda _b(1P)]$ are the average masses of ${\tilde{\Lambda }}_b(1P)$ and $\Lambda _b(1P)$ states, respectively. With the masses of $\Lambda _b(5912)$ and $\Lambda _b(5920)$, one can obtain

$$\begin{aligned}&{{\bar{m}}}[\Lambda _b(1P)] \nonumber \\&\quad = \frac{2m[\Lambda _b(5912)]+4m[\Lambda _b(5920)]}{6}=5917.35~\mathrm {MeV}.\nonumber \\ \end{aligned}$$

(8)

The $m_b=4977~\mathrm {MeV}$ and $m_q=220~\mathrm {MeV}$ are the masses of bottom and light quarks, respectively [9, 53]. Then, the estimated ${{\bar{m}}}[{\tilde{\Lambda }}_b(1P)]$ equals to 6114 MeV, which is consistent with the predicted average mass from relativized quark model [31]. When the mass splitting is considered, the lowest ${\tilde{\Lambda }}_b(1P)$ state should lie around 6072 MeV. Moreover, this estimation is not sensitive to the parameters of quark masses because the bottom quark is much heavier than the light quark and the ratio approximately equals to $\sqrt{3}$.

Table 4 Strong decays of the $\Lambda _b(6072)$ as ${\tilde{\Lambda }}_b(1P)$ states in MeV

Here, the strong decays of the ${\tilde{\Lambda }}_b(1P)$ assignments are calculated and shown in Table 4. For the pure $j=0$ state, the strong decays are not allowed based on the spectator assumption in the $^3P_0$ model. The predicted widths of two $j=1$ states are extremely large, while the two $j=2$ assignments show smaller total decay widths. Our results indicate that the $\Lambda _b(6072)$ as pure ${\tilde{\Lambda }}_b(1P)$ states can be excluded.

The mixing of the states with same $J^P$ is taken into account. The mixing scheme is similar to the $\Sigma _b(1P)$ states, and the results are shown in Fig. 3. Obviously, the $\Lambda _b(6072)$ state can be regarded as the $J^P=1/2^-$ ${{\tilde{\Lambda }}}_b(1P)$ state with large $j=0$ component. Meanwhile, the strong decays also support it as the $J^P=3/2^-$ ${{\tilde{\Lambda }}}_b(1P)$ state with large $j=2$ component. From Table 1, the predicted masses of ${\tilde{\Lambda }}_{b0}(\frac{1}{2}^{-})$ and ${\tilde{\Lambda }}_{b2}(\frac{3}{2}^{-})$ states are 6100 and 6190 MeV, respectively. Given these predicted masses, the $|1P~1/2^- \rangle _1$ assignment is favored, while the $|1P~3/2^- \rangle _2$ assignment can be excluded.

Fig. 3

The total decay widths versus the mixing angle $\theta$. The red solid and blue dashed lines correspond to the $|1P~{J^P}\rangle _1$ and $|1P~{J^P}\rangle _2$ states, respectively, where the $J^P$ equals to $1/2^-$ or $3/2^-$. The green bands are the experimental total decay widths

When the mixing angle is $22^\circ$, the strong decay width of $|1P~1/2^- \rangle _1$ state is about 73 MeV, which is consistent with the experimental data. The branching ratios are predicted to be

$$\begin{aligned} Br(\Sigma _b \pi , \Sigma ^*_b \pi )=99.7\%, 0.3\%, \end{aligned}$$

(9)

which indicates that the $\Sigma _b \pi$ is the predominant decay channel and the three-body decay can occur via the $\Lambda (6072) \rightarrow \Sigma _b \pi \rightarrow \Lambda _b \pi \pi$ process. Also, we plot the decay widths as functions of the initial masses for the ${{\tilde{\Lambda }}}_b(1P)$ states in Fig. 4. For the $|1P~1/2^- \rangle _2$ and ${\tilde{\Lambda }}_{b1}(\frac{3}{2}^{-})$ states, it is difficult to find them experimentally due to the large total decay widths. However, for the ${\tilde{\Lambda }}_{b2}(\frac{3}{2}^{-})$ and ${\tilde{\Lambda }}_{b2}(\frac{5}{2}^-)$ states, their total widths are relatively narrow, which have good potentials to be observed by future experiments. Also, when the initial masses for the $j=2$ states increase, the total decay widths become larger and the dominating channels remain.

Fig. 4

The decay behaviors for the ${\tilde{\Lambda }}_b(1P)$ states versus their initial masses. The black lines correspond to the total decay widths. The red dotted, blue dashed, purple dot-dashed lines stand for the partial decay widths of $\Sigma _{b} \pi$, $\Sigma _{b}^{*} \pi$, and $\Lambda _{b} \eta$ channels, respectively. The $\Lambda _b \eta$ mode for $|1P~1/2^- \rangle _2$ state arises due to the allowed phase space and mixing scheme

Table 5 Strong decays of the ${\tilde{\Lambda }}_c(1P)$ states with the predicted masses in MeV

It should be mentioned that the ${\tilde{\Lambda }}_{b2}(\frac{5}{2}^-)$ state is a crucial point for distinguishing the three-quark model and diquark model. In the three-quark model, the lowest $J^P=5/2^-$ $\Lambda _b$ state belongs to the $\rho$-mode P-wave excitation. However, the $\rho$-mode excitations are frozen in the diquark model, and the lowest $J^P=5/2^-$ state is the F-wave $\Lambda _{b2}(\frac{5}{2}^-)$. The predicated masses of $\Lambda _{b2}(\frac{5}{2}^-)$ in diquark model lie in the region of $6346 \sim 6408$ MeV [29, 30, 32], which is significantly larger than that of ${{\tilde{\Lambda }}}_{b2}(\frac{5}{2}^-)$ state in the three-quark model [28, 31]. The future experimental searches can help us to distinguish these two types of models.

4 Strong decays of nonstrange partners

4.1 ${\tilde{\Lambda }}_c(1P)$

In the constituent quark model, there are five $\rho$-mode ${\tilde{\Lambda }}_c(1P)$ states, and their predicted masses in relativized quark model are about $2780 \sim 2900$ MeV. Similarly, one can estimate the average mass of $\rho$-mode ${\tilde{\Lambda }}_c(1P)$ states with the help of the $\lambda$-mode excitations. The average mass ${{\bar{m}}}[{\tilde{\Lambda }}_c(1P)]$ equals to 2793 MeV, which is consistent with the result of relativized quark model [31]. Within these masses, their strong decays are calculated and listed in Table 5. For the pure $j=0$ state, the strong decays are not allowed based on the spectator assumption in the $^3P_0$ model. For the two $j=1$ states, their total decay width are predicted to be several hundred MeV, and the main decay modes for ${\tilde{\Lambda }}_{c1}(\frac{1}{2}^-)$ and ${\tilde{\Lambda }}_{c1}(\frac{3}{2}^-)$ states are $\Sigma _c \pi$ and $\Sigma _c^* \pi$, respectively. Due to the large decay widths, these two states can hardly be observed experimentally. For the two $j=2$ states, the total decay widths for the $J^P=3/2^-$ and $5/2^-$ states are about 70 and 68 MeV, respectively. The main decay channels are $\Sigma _c \pi$ and $\Sigma _c^* \pi$, and the branching ratios are

$$\begin{aligned} Br(\Sigma _c \pi , \Sigma ^*_c \pi )= & {} 70.1\%, 28.4\%, \qquad J^P=3/2^-, \end{aligned}$$

(10)

$$\begin{aligned} Br(\Sigma _c \pi , \Sigma ^*_c \pi )= & {} 38.9\%, 58.0\%, \qquad J^P=5/2^-. \end{aligned}$$

(11)

In Ref. [30], the authors claimed that the lowest $J^P=5/2^-$ $\Lambda _c$ state is a nice criterion to test the three-quark model and diquark model for charmed baryons. Our results indicate that the ${\tilde{\Lambda }}_{c2}(\frac{5}{2}^{-})$ state is not a broad state and can be searched in the $\Sigma _c \pi$ and $\Sigma _c^* \pi$ final states.

Also, we plot the decay widths as functions of the initial masses for the ${{\tilde{\Lambda }}}_c(1P)$ states in Fig. 5. In constituent quark model, the features of these $\rho$-mode states may be mainly generated by the interactions from two light quarks. From Table 1, the mass gaps among five $\rho$-mode $\Lambda _b$ and $\Lambda _c$ states are 105 and 120 MeV, respectively, which suggests that $\rho$-mode $\Lambda _b$ and $\Lambda _c$ states may have similar characteristics. Thus, we take the same mixing angle as $\Lambda _b$ states to show the mass dependence of five $\rho$-mode $\Lambda _c$ states. It can be seen that the total decay widths increases with the rise of initial masses, but the branching ratios are almost unchanged. By assuming various observed $\Lambda _c$ baryons as the ${\tilde{\Lambda }}_{c}$ states, the authors discussed their strong decay behaviors [25], where the calculated branching ratios are consistent with ours. With mixing mechanism, the $|1P~1/2^- \rangle _1$ state in the charm sector is predicted to be relatively narrow, which can be searched in the $\Sigma _c \pi$ final state in future.

Fig. 5

The decay behaviors for the ${\tilde{\Lambda }}_c(1P)$ states versus their initial masses. The black lines correspond to the total decay widths. The red dotted, blue dashed, purple dot-dashed lines stand for the partial decay widths of $\Sigma _{c} \pi$, $\Sigma _{c}^{*} \pi$, and $\Lambda _{c} \eta$ channels, respectively. The $\Lambda _c \eta$ mode for $|1P~1/2^- \rangle _2$ state arises due to the allowed phase space and mixing scheme

4.2 ${\tilde{\Sigma }}_b(1P)$

From Table 1, two $\rho$-mode ${\tilde{\Sigma }}_b(1P)$ states exist in the conventional quark model, and their masses are predicted to be 6170 MeV and 6180 MeV in the relativized quark model. With the calculated masses, their strong decays are estimated and listed in Table 6. The total decay widths for these two states are about 260 MeV, which are rather large. The main decay modes are $\Sigma _b \pi$ and $\Sigma _b^* \pi$ for the $J^P=1/2^-$ and $3/2^-$ states, respectively. The branching ratios for these two states are

$$\begin{aligned} Br(\Sigma _b \pi , \Sigma ^*_b \pi )= & {} 91.3\%, 8.7\%, \qquad J^P=1/2^-, \end{aligned}$$

(12)

$$\begin{aligned} Br(\Sigma _b \pi , \Sigma ^*_b \pi )= & {} 7.2\%, 92.8\%, \qquad J^P=3/2^-. \end{aligned}$$

(13)

The total decay widths for the ${\tilde{\Sigma }}_b(1P)$ states versus their initial masses are plotted in Fig. 6. With the initial masses increase, the total decay widths also become larger, and the branching ratios are almost unchanged. In Ref. [17], by assuming the $\Sigma _b(6097)$ as the ${\tilde{\Sigma }}_b(1P)$ states, the authors estimated the strong decays, and the calculated total decay widths are about 168-235 MeV. Since large masses are adopted for the initial ${{\tilde{\Sigma }}}_b(1P)$ states in present work, the predicted decay widths are also larger than the previous work [17]. More theoretical efforts on masses and decays are needed to understand these two ${\tilde{\Sigma }}_b(1P)$ states.

Table 6 Strong decays of the ${\tilde{\Sigma }}_b(1P)$ states with the predicted masses in MeV

Fig. 6

The total decay widths for the ${\tilde{\Sigma }}_b(1P)$ states versus their initial masses. The black lines correspond to the total decay widths. The red dotted and blue dashed lines stand for the partial decay widths of $\Sigma _{b} \pi$ and $\Sigma _{b}^{*} \pi$ channels, respectively

4.3 ${\tilde{\Sigma }}_c(1P)$

The predicted masses of the ${\tilde{\Sigma }}_{c1}(\frac{1}{2}^-)$ and ${\tilde{\Sigma }}_{c1}(\frac{3}{2}^-)$ states in the relativized quark model are 2840 and 2865 MeV, respectively. Their calculated decay widths are presented in Table 7. It is shown that the total decay widths are quite large and the main decay modes are $\Sigma _c \pi$ and $\Sigma ^*_c \pi$ for the $J^P=1/2^-$ and $3/2^-$ states, respectively. Moreover, the total decay widths for the ${\tilde{\Sigma }}_c(1P)$ states versus their initial masses are plotted in Fig. 7 for reference. These results are similar with the ${\tilde{\Sigma }}_b(1P)$ states, which preserves the heavy quark symmetry well. In Ref. [49], the authors calculated the strong decays of the ${\tilde{\Sigma }}_c(1P)$ states with a mass of 2882 MeV, and the calculated total decay width are also large. However, the predicted branching ratios in present work are quite different with theirs, and more theoretical works are demanded to clarify this problem.

Table 7 Strong decays of the ${\tilde{\Sigma }}_c(1P)$ states with the predicted masses in MeV

Fig. 7

The total decay widths for the ${\tilde{\Sigma }}_c(1P)$ states versus their initial masses. The black lines correspond to the total decay widths. The red dotted and blue dashed lines stand for the partial decay widths of $\Sigma _{c} \pi$ and $\Sigma _{c}^{*} \pi$ channels, respectively

5 Summary

In this work, we study the strong decay behaviors of the newly observed resonance $\Lambda _b(6072)$ by the LHCb Collaboration. Given its mass and decay modes, this resonance can be tentatively assigned as the $\Lambda _b(2S)$, $\Sigma _b(1P)$ and ${\tilde{\Lambda }}_b(1P)$ states. The strong decay behaviors are investigated within the $^3P_0$ model, and our results suggest that the $\Lambda _b(6072)$ can be regarded as the lowest $\rho$-mode excitation in $\Lambda _b$ family.

Then, the strong decays for the nonstrange partners of $\Lambda _b(6072)$ resonance are calculated. The predicted total decay widths of the $J^P=5/2^-$ $\Lambda _b$ and $\Lambda _c$ states are relatively small, and the main decay channels are the $\Sigma _b^{(*)} \pi$ and $\Sigma _c^{(*)} \pi$, respectively. These two states have good potentials to be observed in future experiments, which may help us to distinguish the three-quark model and diquark model.

Until now, there exist abundant $\lambda$-mode states in the heavy baryon systems, while no $\rho$-mode excitation has been confirmed both theoretically and experimentally. Under our assignment, the $\Lambda _b(6072)$ should correspond to the $\rho$-mode excitation for the singly heavy baryons. We hope these theoretical predictions on ${\tilde{\Lambda }}_{b(c)}(1P)$ and ${\tilde{\Sigma }}_{b(c)}(1P)$ states can provide valuable information for searching more $\rho$-mode excitations in future experiments.