Strong decay of $\Lambda _c(2940)$ as a 2P state in the $\Lambda _c$ family

Abstract

Considering the mass, parity and $D^0 p$ decay mode, we tentatively assign the $\Lambda _c(2940)$ as the $P-$wave states with one radial excitation. Then, via studying the strong decay behavior of the $\Lambda _c(2940)$ within the $^3P_0$ model, we obtain that the total decay widths of the $\Lambda _{c1}(\frac{1}{2}^-,2P)$ and $\Lambda _{c1}(\frac{3}{2}^-,2P)$ states are 16.27 and 25.39 MeV, respectively. Compared with the experimental total width $27.7^{+8.2}_{-6.0}\pm 0.9^{+5.2}_{-10.4}~\mathrm {MeV}$ measured by LHCb Collaboration, both assignments are allowed, and the $J^P=\frac{3}{2}^-$ assignment is more favorable. Other $\lambda$-mode $\Sigma _c(2P)$ states are also investigated, which are most likely to be narrow states and have good potential to be observed in future experiments.

1 Introduction

The singly charmed baryons are composed of one charm quark and two light quarks. Constraints on the nonstrange light quarks, they can be further categorized into the $\Lambda _c$ and $\Sigma _c$ families, which belong to the antisymmetric flavor structure $\bar{3}_F$ and symmetric flavor structure $6_F$, respectively. Establishing the spectrum of these charmed baryons has attracted lots of theoretical and experimental attentions [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36]. From the Particle Data Group book, there exist nine $\Lambda _c$ and $\Sigma _c$ baryons, $\Lambda _c(2286)$, $\Lambda _c(2593)$, $\Lambda _c(2625)$, $\Lambda _c(2765)$, $\Lambda _c(2880)$, $\Lambda _c(2940)$, $\Sigma (2455)$, $\Sigma (2520)$, and $\Sigma (2800)$ [37]. $\Lambda _c(2286)$, $\Sigma (2455)$, and $\Sigma (2520)$ are the $S-$wave ground states, and $\Lambda _c(2593)$ and $\Lambda _c(2625)$ can be well understood as the $P-$wave $\Lambda _c$ states in the conventional quark model. In the cqq configuration, $\Lambda _c(2765)$ and $\Lambda _c(2880)$ might be classified into the 2S and 1D $\Lambda _c$ states, respectively, while $\Sigma _c(2800)$ is possibly a 1P $\Sigma _c$ state. Other conventional or exotic interpretations are also suggested for the $\Lambda _c(2765)$, $\Lambda _c(2880)$, and $\Sigma _c(2800)$ states. Detailed discussions of various assignments and properties can be found in Refs. [32,33,34].

The $\Lambda (2940)$ was firstly observed in the $D^0p$ mass distribution by the BaBar Collaboration [38], and then seen in the $\Lambda _c \pi ^+ \pi ^-$ channel by the Belle Collaboration [39]. In 2017, the LHCb Collaboration performed an amplitude analysis of the $\Lambda _b^0 \rightarrow D^0p\pi ^-$ decay process in the $D^0p$ channel, and observed three $\Lambda _c$ resonances, $\Lambda _c(2860)$, $\Lambda _c(2880)$, and $\Lambda _c(2940)$ [40]. Their masses and decay widths were measured as follows,

$$\begin{aligned} m[\Lambda _c(2860)^+]= & {} 2856.1^{+2.0}_{-1.7}\pm 0.5^{+1.1}_{-5.6}~\mathrm {MeV}, \end{aligned}$$

(1)

$$\begin{aligned} \Gamma [\Lambda _c(2860)^+]= & {} 67.6^{+10.1}_{-8.1}\pm 1.4^{+5.9}_{-20.0}~\mathrm {MeV}, \end{aligned}$$

(2)

$$\begin{aligned} m[\Lambda _c(2880)^+]= & {} 2881.75\pm 0.29\pm 0.07^{+0.14}_{-0.20}~\mathrm {MeV}, \end{aligned}$$

(3)

$$\begin{aligned} \Gamma [\Lambda _c(2880)^+]= & {} 5.43^{+0.77}_{-0.71}\pm 0.29^{+0.75}_{-0.00}~\mathrm {MeV}, \end{aligned}$$

(4)

$$\begin{aligned} m[\Lambda _c(2940)^+]= & {} 2944.8^{+3.5}_{-2.5}\pm 0.4^{+0.1}_{-4.6}~\mathrm {MeV}, \end{aligned}$$

(5)

$$\begin{aligned} \Gamma [\Lambda _c(2940)^+]= & {} 27.7^{+8.2}_{-6.0}\pm 0.9^{+5.2}_{-10.4}~\mathrm {MeV}. \end{aligned}$$

(6)

The quantum numbers of $\Lambda _c(2860)$ and $\Lambda _c(2880)$ were determined to be $J^P=\frac{3}{2}^+$ and $J^P=\frac{5}{2}^+$, respectively. The measured information indicates that they may be good candidates of the 1D-wave $\Lambda _c$ resonances. The spin and parity of the $\Lambda _c(2940)$ state were constrained. The most likely spin-parity quantum numbers of $\Lambda _c(2940)$ are $J^P=\frac{3}{2}^-$, while other possibilities cannot be excluded completely [40]. With the favorable $J^P=\frac{3}{2}^-$ assignment, the $\Lambda _c(2940)$ may correspond to a conventional 2P-wave $\Lambda _c$ resonance in the quark model.

In the past years, from the point view of the mass of $\Lambda _c(2940)$, its nature was attempted to be explained within various quark models. For example, some people studied the $\Lambda _c$ spectrum in the consistent quark model, and found $\Lambda _c(2940)$ could be an excited $\Lambda _c$ state with $J^P=3/2^+$ [3, 41]. Within the diquark picture, $\Lambda _c(2940)$ can be interpreted as the 2P-wave $\Lambda _c$ resonance with $J^P=1/2^-$ or the 2S-wave state with $J^P=3/2^+$ in the relativistic quark model [4], the 2P-wave $\Lambda _c$ resonance with $J^P=1/2^-$ state in the relativized quark model [42], and the $J^P=5/2^-$ 1D-wave state or the 2P -wave $\Lambda _c$ resonances in flux tube model [5, 30]. Meanwhile, the $D^*N$ molecular state interpretations were suggested in some works [6,7,8,9,10], where with the $S-$wave $1/2^-$ or $3/2^-$ assignment, the near threshold behavior of $\Lambda _c(2940)$ can be naturally explained.

Besides the mass spectrum, the $\Lambda _c(2940)$ resonance was also investigated via its decay and production processes. For example, the strong decays of $\Lambda _c(2940)$ were studied within the chiral perturbation theory, one found that the spin-parity numbers might be $3/2^+$ or $5/2^-$ [1]. Within the quark model, the strong decays indicated $\Lambda _c(2940)$ can be described as the $D-$wave $\Lambda _c$ state with spin-parity numbers $5/2^+$ [12] or $7/2^+$ [16]. Meanwhile, the decay behaviors of the $J^P= 1/2^-$, $3/2^-$, $1/2^+$ $D^*N$ molecule states were investigated [6,7,8, 17], and no definitive conclusion was obtained. Furthermore, the productions of $\Lambda _c(2940)$ in the $\bar{p} p$, $\pi ^-p$, $\gamma n$, and $K^-p$ processes were studied within effective Lagrangian approaches [43,44,45,46,47], which provide helpful references for future PANDA and COMPASS experiments.

It is shown that the theoretical works perform lots of interpretations on $\Lambda _c(2940)$, while the quantum numbers $J^P = \frac{3}{2}^-$ determined by LHCb Collaboration favor the conventional 2P $\Lambda _c$ resonance or the exotic $D^*N$ molecule description. Although there are many discussions of $\Lambda _c(2940)$ in the literature as mentioned before, less discussions of the decay behaviors as the conventional 2P $\Lambda _c$ states can be found. Hence, in this work, we study the strong decays of the 2P charmed baryons within the $^3P_0$ quark pair creation model. Our results indicate that $\Lambda _c(2940)$ as the $\lambda$-mode $\Lambda _{c1}(\frac{1}{2}^-,2P)$ and $\Lambda _{c1}(\frac{3}{2}^-,2P)$ states are both allowed, and the $J^P=3/2^-$ state $\Lambda _{c1}(\frac{3}{2}^-,2P)$ is more favorable.

This paper is organized as follows. The $^3P_0$ model is briefly introduced in Sect. 2. The strong decays of the 2P $\Lambda _c$ and $\Sigma _c$ charmed baryons are estimated in Sect. 3. A short summary is presented in the last section.

2 $^3P_0$ model

In this work, we adopt the $^3P_0$ model to calculate the Okubo–Zweig–Iizuka-allowed two-body strong decays of the 2P $\Lambda _c$ and $\Sigma _c$ states. The $^3P_0$ model, also known as the quark pair creation model, has been extensively employed to study the strong decays with considerable successes [2, 18, 48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65]. In this model, the hadrons decay occurs through a quark-antiquark pair with the vacuum quantum number $J^{PC}=0^{++}$ [55]. Here we perform a brief review of the $^3P_0$ model. In the nonrelativistic limit, the transition operator T of the decay $A\rightarrow BC$ in the $^3P_0$ model can be assumed as [2, 60]

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

(7)

where $\gamma$ is a dimensionless $q_4\bar{q}_5$ pair-production strength, and $\varvec{p}_4$ and $\varvec{p}_5$ are the momenta of the created quark $q_4$ and antiquark $\bar{q}_5$, respectively. The i and j are the color indices of the created quark and antiquark. $\phi ^{45}_{0}=(u\bar{u} + d\bar{d} +s\bar{s})/\sqrt{3}$, $\omega ^{45}=\delta _{ij}$, and $\chi _{{1,-m}}^{45}$ are the flavor singlet, color singlet, and spin triplet wave functions of the $q_4\bar{q}_5$, respectively. The solid harmonic polynomial $\mathcal{{Y}}^m_1(\varvec{p})\equiv |p|Y^m_1(\theta _p, \phi _p)$ reflects the P-wave momentum-space distribution of the $q_4\bar{q}_5$ quark pair.

For the initial baryon A, we adopt the definition of the mock states [66]

$$\begin{aligned}&|A(n^{2S_A+1}_AL_{A}\,_{J_A M_{J_A}})(\varvec{P}_A)\rangle \nonumber \\&\quad \equiv \sqrt{2E_A}\sum _{M_{L_A},M_{S_A}}\langle L_A M_{L_A} S_A M_{S_A}|J_A M_{J_A}\rangle \nonumber \\&\qquad \times \int d^3\varvec{p}_1d^3\varvec{p}_2d^3\varvec{p}_3\nonumber \\&\qquad \times \ \delta ^3(\varvec{p}_1+\varvec{p}_2+\varvec{p}_3-\varvec{P}_A)\psi _{n_AL_AM_{L_A}}(\varvec{p}_1,\varvec{p}_2,\varvec{p}_3)\nonumber \\&\qquad \times \ \chi ^{123}_{S_AM_{S_A}} \phi ^{123}_A\omega ^{123}_A \times \ |q_1(\varvec{p}_1)q_2(\varvec{p}_2)q_3(\varvec{p}_3) \rangle , \end{aligned}$$

(8)

which satisfies the normalization condition

$$\begin{aligned} \langle A(\varvec{P}_A)|A(\varvec{P}^\prime _A)\rangle =2E_A\delta ^3(\varvec{P}_A-\varvec{P}^\prime _A). \end{aligned}$$

(9)

The $\varvec{p}_1$, $\varvec{p}_2$, and $\varvec{p}_3$ are the momenta of the quarks $q_1$, $q_2$, and $q_3$, respectively. $\varvec{P}_A$ denotes the momentum of the initial state A. $\chi ^{123}_{S_AM_{S_A}}$, $\phi ^{123}_A$, $\omega ^{123}_A$, $\psi _{n_AL_AM_{L_A}}(\varvec{p}_1,\varvec{p}_2,\varvec{p}_3)$ are the spin, flavor, color, and space wave functions of the baryon A composed of $q_1q_2q_3$ with total energy $E_A$, respectively. The definitions of the mock states B and C are similar to that of initial state A, and can be find in Ref. [2].

For the decay of the charmed baryon A, three possible rearrangements exist,

$$\begin{aligned}&A(q_1,q_2,c_3)+P(q_4,\bar{q}_5)\rightarrow B(q_2,q_4,c_3)\nonumber \\&\quad + \ C(q_1,\bar{q}_5), \end{aligned}$$

(10)

$$\begin{aligned}&A(q_1,q_2,c_3)+P(q_4,\bar{q}_5)\rightarrow B(q_1,q_4,c_3)\nonumber \\&\quad +\ C(q_2,\bar{q}_5), \end{aligned}$$

(11)

$$\begin{aligned}&A(q_1,q_2,c_3)+P(q_4,\bar{q}_5)\rightarrow B(q_1,q_2,q_4)\nonumber \\&\quad +\ C(c_3,\bar{q}_5), \end{aligned}$$

(12)

where the $q_i$ and $c_3$ denote the light quark and charm quark, respectively. These three ways of recouplings are also shown in Fig. 1.

Fig. 1

The baryon decay process $A\rightarrow B+C$ in the $^3P_0$ model

The S matrix can be defined as

$$\begin{aligned} \langle f|S|i\rangle =I-i2\pi \delta (E_f-E_i)\mathcal{{M}}^{M_{J_A}M_{J_B}M_{J_C}}, \end{aligned}$$

(13)

where the $\mathcal{{M}}^{M_{J_A}M_{J_B}M_{J_C}}$ is the helicity amplitude of the decay process $A\rightarrow B+C$. Taken the process $A(q_1,q_2,c_3)+P(q_4,\bar{q}_5)\rightarrow B(q_1,q_4,c_3)+C(q_2,\bar{q}_5)$ shown in Fig. 1b as an example, the helicity amplitude $\mathcal{{M}}^{M_{J_A}M_{J_B}M_{J_C}}$ reads [2, 57, 58],

$$\begin{aligned}&\delta ^3(\varvec{p}_B+\varvec{p}_C-\varvec{p}_A)\mathcal{{M}}^{M_{J_A}M_{J_B}M_{J_C}} \nonumber \\&\quad = - \ \gamma \sqrt{8E_AE_BE_C} \sum _{M_{\rho _A}} \sum _{M_{L_A}} \sum _{M_{\rho _B}} \sum _{M_{L_B}} \sum _{M_{S_1}, M_{S_3}, M_{S_4}, m}\nonumber \\&\qquad \times \ \langle J_{l_A} M_{J_{l_A}}S_3M_{S_3}|J_AM_{J_A}\rangle \langle L_{\rho _A} M_{L_{\rho _A}}L_{\Lambda _A}M_{L_{\Lambda _A}}|L_AM_{L_A} \rangle \nonumber \\&\qquad \times \ \langle L_A M_{L_A}S_{12}M_{S_{12}}|J_{l_A}M_{J_{l_A}}\rangle \langle S_1M_{S_1}S_2M_{S_2}|S_{12}M_{S_{12}} \rangle \nonumber \\&\qquad \times \ \langle J_{l_B} M_{J_{l_B}}S_3M_{S_3}|J_BM_{J_B}\rangle \langle L_{\rho _B} M_{L_{\rho _B}}L_{\Lambda _B}M_{L_{\Lambda _B}}|L_BM_{L_B}\rangle \nonumber \\&\qquad \times \ \langle L_B M_{L_B}S_{14}M_{S_{14}}|J_{l_B}M_{J_{l_B}}\rangle \langle S_1M_{S_1}S_4M_{S_4}|S_{14}M_{S_{14}}\nonumber \\&\qquad \times \ \langle 1m 1-m|00\rangle \langle S_4M_{S_4}S_5M_{S_5}|1-m \rangle \nonumber \\&\qquad \times \ \langle L_C M_{L_C}S_CM_{S_C}|J_CM_{J_C}\rangle \langle S_2M_{S_2}S_5M_{S_5}|S_CM_{S_C}\rangle \nonumber \\&\qquad \times \ \langle \phi _B^{143} \phi _C^{25}|\phi _A^{123}\phi _0^{45}\rangle I^{M_{L_A}m}_{M_{L_B}M_{L_C}}(\varvec{p}), \end{aligned}$$

(14)

where $\langle \phi _B^{143} \phi _C^{25}|\phi _A^{123}\phi _0^{45}\rangle$ are the overlap of the flavor wavefunctions. The $I^{M_{L_A}m}_{M_{L_B}M_{L_C}}(\varvec{p})$ are the spatial overlaps of the initial and final states, which can be written as

$$\begin{aligned} I^{M_{L_A}m}_{M_{L_B}M_{L_C}}(\varvec{p})= & {} \int d^3\varvec{p}_1d^3\varvec{p}_2d^3\varvec{p}_3d^3\varvec{p}_4d^3\varvec{p}_5 \nonumber \\&\times \ \delta ^3(\varvec{p}_1+\varvec{p}_2+\varvec{p}_3-\varvec{P}_A)\delta ^3(\varvec{p}_4+\varvec{p}_5)\nonumber \\&\times \ \delta ^3(\varvec{p}_1+\varvec{p}_4+\varvec{p}_3-\varvec{P}_B)\delta ^3(\varvec{p}_2+\varvec{p}_5-\varvec{P}_C) \nonumber \\&\times \ \psi ^*_B(\varvec{p}_1,\varvec{p}_4,\varvec{p}_3) \psi ^*_C(\varvec{p}_2,\varvec{p}_5)\nonumber \\&\times \ \psi _A(\varvec{p}_1,\varvec{p}_2,\varvec{p}_3)\mathcal{{Y}}^m_1\left( \frac{\varvec{p}_4-\varvec{p}_5}{2}\right) . \end{aligned}$$

(15)

In this issue, we employ the simplest vertex which assumes a spatially constant pair production strength $\gamma$ [55], the relativistic phase space, and the simple harmonic oscillator wave functions. With the relativistic phase space, the decay width $\Gamma (A\rightarrow BC)$ can be expressed as follows

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

(16)

where $p=|\varvec{p}|=\frac{\sqrt{[M^2_A-(M_B+M_C)^2][M^2_A-(M_B-M_C)^2]}}{2M_A}$, and $M_A$, $M_B$, and $M_C$ are the masses of the hadrons A, B, and C, respectively. $s=1/(1+\delta _{BC})$ is a statistical factor which is needed if B and C are identical particles. Due to B and C correspond to baryon and meson, respectively, the s always equals to one in this work.

3 Strong decay

3.1 Notations and parameters

In our calculation, we adopt the same notations of $\Lambda _c$, $\Sigma _c$ and $\Xi _c$ baryons as those in Refs. [2, 32]. For the spatial 2P excited states, the symbol 2P are added. In Table 1, The $n_\rho$ and $L_\rho$ stand the nodal and orbital angular momentum between the two light quarks, while $n_\lambda$ and $L_\lambda$ denote the nodal and angular momentum between the two light quark system and the charm quark. L is the total orbital angular momentum, $S_\rho$ is the total spin of the two light quarks, $J_l$ is total angular momentum of L and $S_\rho$, and J is the total angular momentum.

Table 1 Notations and quantum numbers of the relevant $\Lambda _c$, $\Sigma _c$ and $\Xi _c$ baryons

For the masses of the two $\Lambda _{c1}(2P)$ states, we adopt the mass of $\Lambda (2940)$ from LHCb experimental data. Masses of the other 2P states are taken from theoretical predictions. For the final ground states, their masses are adopted from the Particle Data Group [37]. For the harmonic oscillator parameters of mesons, we use the effective values obtained by relativized quark model, i.e., $R= 2.5~\mathrm {GeV^{-1}}$ for $\pi /\rho /\omega /K/\eta$ meson, $R= 1.67~\mathrm {GeV^{-1}}$ for D meson, $R= 1.94~\mathrm {GeV^{-1}}$ for $D^*$ meson, and $R= 1.54~\mathrm {GeV^{-1}}$ for $D_s$ meson [64]. For the baryon parameters, we use $\alpha _\rho =400~\mathrm {MeV}$ and

$$\begin{aligned} \alpha _\lambda =\Bigg (\frac{3m_Q}{2m_q+m_Q} \Bigg )^\frac{1}{4} \alpha _\rho , \end{aligned}$$

(17)

where the $m_Q$ and $m_q$ are the heavy and light quark masses, respectively [12]. The $m_{u/d}=220~\mathrm {MeV}$, $m_s=419~\mathrm {MeV}$, and $m_c=1628~\mathrm {MeV}$ are introduced to explicitly break the SU(4) symmetry [64, 67, 68]. There is an overall parameter $\gamma$, which is determined by the well determined width of the $\Sigma _c(2520)^{++} \rightarrow \Lambda _c \pi ^+$ process. The $\gamma =9.83$ is obtained by reproducing the width, $\Gamma [\Sigma _c(2520)^{++} \rightarrow \Lambda _c \pi ^+]=14.78~\mathrm {MeV}$ [37].

3.2 $\Lambda _c(2940)$

In the constituent quark model, there are two $\lambda -$type 2P states in the $\Lambda _c$ family. The predicted masses in the literature are presented in Table 2, which suggests that the $\Lambda _c(2940)$ is a good candidate of the $\Lambda _c(2P)$ states. The strong decays of $\Lambda _c(2940)$ as the $\Lambda _{c1}(\frac{1}{2}^-,2P)$ and $\Lambda _{c1}(\frac{3}{2}^-,2P)$ assignments are calculated. The results are listed in Table 3. It is shown that the total decay widths of the $\Lambda _{c1}(\frac{1}{2}^-,2P)$ and $\Lambda _{c1}(\frac{3}{2}^-,2P)$ states are 16.27 and 25.39 MeV, respectively. Compared with the experimental total width $27.7^{+8.2}_{-6.0}\pm 0.9^{+5.2}_{-10.4}~\mathrm {MeV}$ measured by LHCb Collaboration, both assignments are allowed. However, the $J^P=\frac{3}{2}^-$ assignment is more favorable. The main decay mode is the DN channel, and the partial decay widths of the $\Sigma _c \pi$ and $\Sigma ^*_c \pi$ channels are rather small, which is consistent with the fact that $\Lambda _c(2940)$ was observed in $D^0p$ invariant mass distribution. The partial decay width ratios of the $J^P=\frac{1}{2}^-$ state are predicted to be

$$\begin{aligned} \Gamma [\Sigma _c \pi ]{:}\Gamma [\Sigma _c^* \pi ]{:}\Gamma [DN] = 1{:}0.52{:}6.91, \end{aligned}$$

(18)

and the partial decay width ratios of the $J^P=\frac{3}{2}^-$ state are predicted to be

$$\begin{aligned} \Gamma [\Sigma _c \pi ]{:}\Gamma [\Sigma _c^* \pi ]{:}\Gamma [DN] = 1{:}3.22{:}22.03. \end{aligned}$$

(19)

These ratios are independent with the overall parameter $\gamma$ in the $^3P_0$ model, and the divergence of these two set of quantum number assignments can be tested in future experimental data.

Table 2 Predicted masses for the $\lambda$-mode $\Lambda _c(2P)$ states in the literature. The units are in MeV

Table 3 Decay widths of the $\Lambda _c(2940)$ as the $\Lambda _{c1}\left( \frac{1}{2}^-,2P\right)$ and $\Lambda _{c1}\left( \frac{3}{2}^-,2P\right)$ in MeV

Fig. 2

The decay widths of the $\Lambda _{c1}\left( \frac{1}{2}^-,2P\right)$ and $\Lambda _{c1}\left( \frac{3}{2}^-,2P\right)$ states as functions of the initial state mass. The blue line with a yellow band denotes the LHCb experimental data

Fig. 3

The dependence of the decay widths on the harmonic oscillator parameters $\alpha _\rho$

In Fig. 2, we plot the variation of the decay widths as a function of the initial baryon mass. It is seen that the partial width of the DN channel decreases for the $1/2^-$ state, while increases for the $3/2^-$ state. The $\Sigma _c \pi$ and $\Sigma _c^* \pi$ decay modes are small enough in this mass region. When the mass lies above the $D^*N$ threshold, the $D^*N$ channel also performs significant contributions to the total decay widths in both cases. Since the mass splitting of $\Lambda _c(1P)$ is

$$\begin{aligned} m[\Lambda _c(2625)]-m[\Lambda _c(2595)] = 36~\mathrm {MeV}, \end{aligned}$$

(20)

the mass splitting of the two $\Lambda _c(2P)$ states is smaller than $36~\mathrm {MeV}$. Considering $\Lambda _c(2940)$ as the $\Lambda _{c1}(\frac{3}{2}^-,2P)$ state, the mass of the $\Lambda _{c1}(\frac{1}{2}^-,2P)$ state should lie in $2909\sim 2945 ~\mathrm {MeV}$. From Fig. 2, the $\Lambda _{c1}(\frac{1}{2}^-,2P)$ state has a width of $16\sim 33 ~\mathrm {MeV}$, which can be searched in the DN final state in future experiments.

The dependence on the harmonic oscillator parameter $\alpha _\rho$ is also investigated in Fig. 3. When the $\alpha _\rho$ increases, the total decay width also increases for the $1/2^-$ state. While, the total decay width of the $3/2^-$ state is almost unchanged with the $\alpha _\rho$ variation. Within this reasonable range of the parameter $\alpha _\rho$, our conclusions remain.

Table 4 Predicted masses for the $\lambda$-mode $\Sigma _c(2P)$ states in the literature. The units are in MeV

Table 5 The strong decay behaviors of the five $\lambda$-mode $\Sigma _c(2P)$ states. The masses of the initial baryons are taken from Ref. [68]. The units are in MeV

Fig. 4

The decay widths of the five $\lambda$-mode $\Sigma _c(2P)$ states as functions of their masses. The partial decay widths of $\Lambda _c \pi$, $\Lambda _c \rho$, $\Sigma _c \pi$, $\Sigma _c^* \pi$, $\Sigma _c \eta$, $\Sigma _c^*\eta$, $\Sigma _c \rho$, $\Sigma _c \omega$, and $D_s \Sigma$ channels are relatively small, which are not presented here

3.3 $\Sigma _c(2P)$

There are five $\lambda$-mode $\Sigma _c(2P)$ states, denoted as $\Sigma _{c0}(\frac{1}{2}^-,2P)$, $\Sigma _{c1}(\frac{1}{2}^-,2P)$, $\Sigma _{c1}(\frac{3}{2}^-,2P)$, $\Sigma _{c2}(\frac{3}{2}^-,2P)$, and $\Sigma _{c2}(\frac{5}{2}^-,2P)$, respectively. Although no information exists for these states in the experiments, some theoretical works have investigated their masses [4, 18, 68,69,70]. In Table 4, we collect the predicted masses of $\lambda$-mode $\Sigma _c(2P)$ states in the literature. Here, we employ the masses predicted by the relativized quark model [68] to calculate their strong decays, and the results are listed in Table 5. The total decay widths of these five states are about $28\sim 69~\mathrm {MeV}$, which are relatively narrow. The main decay modes are light baryon plus heavy meson channels, while the heavy baryon plus light meson channels are rather small. The narrow total decay widths and large $D^{(*)}N$ branching ratios suggest that these states have good potential to be observed in future experiments. Moreover, the decay widths as functions of their initial masses are plotted in Fig. 4 for reference.

There are also $\rho$-mode excited 2P states, where a symbol $``\sim$” are added to distinguish them from the $\lambda$-mode states in Table 1. The theoretical predictions of these states are scarce. In the singly heavy baryon sector, exciting the $\lambda$-mode is much easier than the $\rho$-mode, hence, the $\rho$-mode excited 2P states should be much higher than the $\lambda$-mode states. With the higher masses, more strong decay channels will be open. Due to the lack of mass information and the uncertainties of many decay channels, it seems untimely to study their properties in present work.

4 Summary

In this work, we study the strong decays of the $\Lambda _c(2940)$ baryon within the $^3P_0$ model. Considering the mass, parity and $D^0 p$ decay mode, we tentatively assign $\Lambda _c(2940)$ as the $\lambda$-mode $\Lambda _c(2P)$ states. The main decay mode is DN channel for both $1/2^-$ and $3/2^-$ states. The total decay width of the $\Lambda _{c1}(\frac{1}{2}^-,2P)$ and $\Lambda _{c1}(\frac{3}{2}^-,2P)$ states are 16.27 and 25.39 MeV, respectively. Compared with the total width measured by LHCb Collaboration, both assignments are allowed, and the $J^P=\frac{3}{2}^-$ assignment is more favorable. Other $\lambda -$mode $\Sigma _c(2P)$ states are also investigated. The relatively narrow total decay widths and large $D^{(*)}N$ branching ratios can be tested in future experimental searches.