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

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 MeV and 25.39 MeV, respectively. Compared with the experimental total width $27.7^{+8.2}_{-6.0}\pm0.9^{+5.2}_{-10.4}~\rm{MeV}$ 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.


I. 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 Λ c and Σ c families, which belong to the antisymmetric flavor struc-ture3 F and symmetric flavor structure 6 F , respectively. Establishing the spectrum of these charmed baryons has attracted lots of theoretical and experimental attentions . From the Particle Data Group book, there exist nine Λ c and Σ c baryons, Λ c (2286), Λ c (2593), Λ c (2625), Λ c (2765), Λ c (2880), Λ c (2940), Σ(2455), Σ(2520), and Σ(2800) [37]. Λ c (2286), Σ(2455), and Σ(2520) are the S −wave ground states, and Λ c (2593) and Λ c (2625) can be well understood as the P−wave Λ c states in the conventional quark model. In the cqq configuration, Λ c (2765) and Λ c (2880) might be classified into the 2S and 1D Λ c states, respectively, while Σ c (2800) is possibly a 1P Σ c state. Other conventional or exotic interpretations are also suggested for the Λ c (2765), Λ c (2880), and Σ c (2800) states. Detailed discussions of various assignments and properties can be found in Refs. [32][33][34].
In 2017, the LHCb Collaboration performed an amplitude analysis of the Λ 0 b → D 0 pπ − decay process in the D 0 p channel, and observed three Λ c resonances, Λ c (2860), Λ c (2880), and Λ c (2940) [38]. Their masses and decay widths were measured as follows, The quantum numbers of Λ c (2860) and Λ c (2880) were determined to be J P = 3 2 + and J P = 5 2 + , respectively. The measured information indicates that they may be good candidates of the 1D-wave Λ c resonances. The spin and parity of the Λ c (2940) state were constrained. The most likely spin-parity quantum numbers of Λ c (2940) are J P = 3 2 − , while other possibilities cannot be excluded completely [38]. With the favorable J P = 3 2 − assignment, the Λ c (2940) may correspond to a conventional 2P-wave Λ c resonance in the quark model. In the past years, from the point view of the mass spectrum the properties of Λ c (2940) were attempted to be understood within various quark models. For example, some people studied the Λ c spectrum in the consistent quark model, and found Λ c (2940) could be an excited Λ c state with J P = 3/2 + [3,39]. Within the diquark picture, Λ c (2940) can be interpreted as the 2P-wave Λ c resonance with J P = 1/2 − or the 2S -wave state with J P = 3/2 + in the relativistic quark model [4], the 2Pwave Λ c resonance with J P = 1/2 − state in the relativized quark model [40], and the J P = 5/2 − 1D-wave state or the 2P -wave Λ 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 Λ c (2940) can be naturally explained.
It is shown that the theoretical works perform lots of interpretations on Λ c (2940), while the quantum numbers J P = 3 2 − determined by LHCb Collaboration favor the conventional 2P Λ c resonance or the exotic D * N molecule description. Although there are many discussions of Λ c (2940) in the literature as mentioned before, less discussions of the decay behaviors as the conventional 2P Λ c states can be found. Hence, in this work, we study the strong decays of the 2P charmed baryons within the 3 P 0 quark pair creation model. Our results indicate that Λ c (2940) as the λ−mode Λ c1 ( 1 2 − , 2P) and This paper is organized as follows. The 3 P 0 model is briefly introduced in Sec. II. The strong decays of the 2P Λ c and Σ c charmed baryons are estimated in Sec. III. A short summary is presented in the last section.

II. 3 P 0 MODEL
In this work, we adopt the 3 P 0 model to calculate the Okubo-Zweig-Iizuka-allowed two-body strong decays of the 2P Λ c and Σ c states. The 3 P 0 model, also known as the quark pair creation model, has been extensively employed to study the strong decays with considerable successes [2,18,[46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63]. In this model, the hadrons decay occurs through a quark-antiquark pair with the vacuum quantum number J PC = 0 ++ [53]. Here we perform a brief review of the 3 P 0 model. In the nonrelativistic limit, the transition operator T of the decay A → BC in the 3 P 0 model can be assumed as [2,58] where γ is a dimensionless q 4q5 pair-production strength, and p 4 and p 5 are the momenta of the created quark q 4 and antiquarkq 5 , respectively. The i and j are the color indices of the created quark and antiquark. φ 45 0 = (uū + dd + ss)/ √ 3, ω 45 = δ i j , and χ 45 1,−m are the flavor singlet, color singlet, and spin triplet wave functions of the q 4q5 , respectively. The solid harmonic polynomial Y m 1 (p) ≡ |p|Y m 1 (θ p , φ p ) reflects the P−wave momentum-space distribution of the q 4q5 quark pair.
For the initial baryon A, we adopt the definition of the mock states [64] which satisfies the normalization condition The p 1 , p 2 , and p 3 are the momenta of the quarks q 1 , q 2 , and q 3 , respectively. P A denotes the momentum of the initial state A. χ 123 are the spin, flavor, color, and space wave functions of the baryon A composed of q 1 q 2 q 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, (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 Figure 1.
where the M M J A M J B M J C is the helicity amplitude of the decay process A → B + C. Taken the process Fig. 1(b) as an example, the helicity amplitude M M J A M J B M J C reads [2,55,56], where φ 143 B φ 25 C |φ 123 In this issue, we employ the simplest vertex which assumes a spatially constant pair production strength γ [53], the relativistic phase space, and the simple harmonic oscillator wave functions. With the relativistic phase space, the decay width Γ(A → BC) can be expressed as follows where , and M A , M B , and M C are the masses of the hadrons A, B, and C, respectively. s = 1/(1 + δ 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.

A. Notations and parameters
In our calculation, we adopt the same notations of Λ c , Σ c and Ξ c baryons as those in Ref. [2,32]. For the spatial 2P excited states, the symbol 2P are added. In Table I, The n ρ and L ρ stand the nodal and orbital angular momentum between the two light quarks, while n λ and L λ denote the nodal and angular momentum between the two light quark system and the charm quark. L is the total orbital angular momentum, S ρ is the total spin of the two light quarks, J l is total angular momentum of L and S ρ , and J is the total angular momentum.
For the masses of the two Λ c1 (2P) states, we adopt the mass of Λ(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 GeV −1 for π/ρ/ω/K/η meson, R = 1.67 GeV −1 for D meson, R = 1.94 GeV −1 for D * meson, and R = 1.54 GeV −1 for D s meson [62]. For the baryon parameters, we use α ρ = 400 MeV and where the m Q and m q are the heavy and light quark masses, respectively [12]. The m u/d = 220 MeV, m s = 419 MeV, State and m c = 1628 MeV are introduced to explicitly break the SU(4) symmetry [62,65,66]. There is an overall parameter γ, which is determined by the well determined width of the Σ c (2520) ++ → Λ c π + process. The γ = 9.83 is obtained by reproducing the width, Γ[Σ c (2520) ++ → Λ c π + ] = 14.78 MeV [37].  assignment is more favorable. The main decay mode is the DN channel, and the partial decay widths of the Σ c π and Σ * c π channels are rather small, which is consistent with the fact These ratios are independent with the overall parameter γ in the 3 P 0 model, and the divergence of these two set of quantum number assignments can be tested in future experimental data. In Figure. 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 Σ c π and Σ * c π 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 Λ c (1P) is

Mode
Λ c (2940)    The dependence on the harmonic oscillator parameter α ρ is also investigated in Fig. 3. When the α ρ 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 α ρ variation. Within this reasonable range of the parameter α ρ , our conclusions remain.  [4,18,[66][67][68]. In Tab. III, we collect the predicted masses of λ-mode Σ c (2P) states in the literature. Here, we employ the masses predicted by the relativized quark model [66] to calculate their strong decays, and the results are listed in Tab. IV. The total decay widths of these five states are about 28 ∼ 69 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 ρ-mode excited 2P states, where a symbol " ∼ " are added to distinguish them from the λ-mode states in Tab I. The theoretical predictions of these states are scarce. In the singly heavy baryon sector, exciting the λ-mode is much easier than the ρ-mode, hence, the ρ-mode excited 2P states should be much higher than the λ-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.

IV. SUMMARY
In this work, we study the strong decays of the Λ c (2940) baryon within the 3 P 0 model. Considering the mass, parity and D 0 p decay mode, we tentatively assign Λ c (2940) as the λ-mode Λ c (2P) states. The main decay mode is DN channel for both 1/2 − and 3/2 − states. The total decay width of the MeV, respectively. Compared with the total width measured by LHCb Collaboration, both assignments are allowed, and the J P = 3 2 − assignment is more favorable. Other λ−mode Σ 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.