Possible explanations of the observed Λ_c resonances

Singly-charmed baryons contain a charm quark and two light quarks, which provide a good opportunity to study the dynamics of quark confinement [1, 2]. Singly-charmed baryon spectroscopy has always been a prominent research topic. To date, especially in the last six years, considerable progress has been achieved experimentally, and many new excited singly-charmed baryons have been discovered [3]. To decode the inner structures of these newly observed states, much effort has been made in both experiment and theory.

The $ \Lambda_c $ baryon spectrum is an important member of the singly-charmed baryons, and valuable data have been obtained experimentally. According to the PDG 2024 [4], there are eight $ \Lambda_c $ baryons: $ \Lambda_c^+ $, $ \Lambda_c(2595)^+/\Lambda_c(2625)^+ $, $ \Lambda_c(2765)^+ $(or $ \Sigma_c(2765)^+ $), $ \Lambda_c(2860)^+/ \Lambda_c(2880)^+ $, $ \Lambda_c(2910)^+ $, and $ \Lambda_c(2940)^+ $. The $ \Lambda_c^+ $ ground state is the lowest-lying charmed baryon and was first observed by Fermilab in 1976 [5]. $ \Lambda_c(2595)^+ $ and $ \Lambda_c(2625)^+ $ are the $ \Lambda_c^+ $ orbital excitations. They were first reported by the CLEO Collaboration in 1995 [6] and ARGUS Collaboration in 1993 [7], respectively, and soon confirmed by subsequent experiments [6, 8−10]. The spin-parity of $ \Lambda_c(2595)^+ $ is almost certainly $ 1/2^- $, and that of $ \Lambda_c(2625)^+ $ is expected to be $ 3/2^- $ [4]. $ \Lambda_c(2765)^+ $ is a rather broad structure first reported in the $ \Lambda_c^+\pi^+\pi^- $ channel by the CLEO Collaboration in 2001 [11] and later also observed in the $ \Sigma_c^{++/0}\pi^{\mp} $ decay by the Belle Collaboration in 2007 [12]. However, nothing at all is known about its quantum numbers, including whether it is a $ \Lambda_c^+ $ or $ \Sigma_c^+ $state. In 2017, the Belle Collaboration determined its isospin to be zero and suggested this particle to be a $ \Lambda_c^+ $ state [13]. The $ \Lambda_c(2860)^+ $ resonance of spin-parity $ 3/2^+ $ was reported by the LHCb Collaboration in the $ D^0p $ amplitude in 2017 [14] and is expected to be confirmed by other experiments in the future. Another orbital excitation $ \Lambda_c(2880)^+ $ of spin-parity $ 5/2^+ $ was first observed by the CLEO Collaboration [11]. The mass, decay width, and quantum number were further determined by later experiments [12, 14−16]. In addition, the BaBar Collaboration reported a new structure $ \Lambda_c(2940)^+ $ in the same paper [15], and soon after the Belle Collaboration confirmed the existence of this state and reported the first observation of $ \Lambda_c(2940)^+\rightarrow \Sigma_c(2455)^{0,++}\pi^{+,-} $ decay [12]. Furthermore, the LHCb Collaboration confirmed this structure in the $ D^0p $ amplitude [14] and suggested its spin-parity to be $ 3/2^- $. More recently, the Belle Collaboration measured the branching fractions of $ \Lambda_c(2940)^+ $ decaying to $ \Lambda_c^+\eta $ and $ D^0p $ relative to $ \Sigma_c\pi $ [16], which provided a more accurate reference for the theory. The new resonance $ \Lambda_c(2910)^+ $ was observed by the Belle Collaboration in $ \bar{B}^0\rightarrow \Sigma_c(2455)^{0,++}\pi^{\pm}\bar{p} $ decays in 2023 [17], but its spin-parity has not been determined yet.

Meanwhile, there exist many theoretical calculations to decode the inner structures of those observed $ \Lambda_c^+ $ baryons via mass spectrum [18−33] and decay properties [19, 33−47]. Besides the $ \Lambda_c^+ $ ground state, $ \Lambda_c(2595)^+ $ and $ \Lambda_c(2625)^+ $ can be well interpreted as the P-wave $ \Lambda_c $ states of $ J^P=1/2^- $ and $ J^P=3/2^- $ [18−20, 34−39], respectively. Meanwhile, for $ \Lambda_c(2860)^+ $ and $ \Lambda_c(2880)^+ $, most of the literature suggests that they together form the D-wave $ \Lambda_c $ doublet of $ J^P=3/2^+ $ and $ J^P=5/2^+ $, respectively [23, 25−27, 41, 42]. However, some literature does not support $ \Lambda_c(2880)^+ $ as the $ J^P=5/2^+ $ state [27, 41] and maintains that this state is an F-wave $ \Lambda_c $ state of $ J^P=5/2^- $ [41]. Hence, the relation between these two states still needs to be carefully examined in future experimental and theoretical studies. Regarding $ \Lambda_c(2940)^+ $, its internal structure is controversial. In addition to being interpreted as a traditional hadron state of $ J^P=1/2^- $ [2], $ 3/2^{\pm} $ [22, 24, 41, 43], $ 5/2^{\mp} $ [43, 45], or $ 7/2^+ $ [44], it has been interpreted as a $ D^*N $ molecular state with $ J^P=3/2^- $ [48, 49]. Meanwhile, some papers [50, 51] discuss the properties of $ \Lambda_c(2940) $ to help us clarify its nature. Fortunately, the latest experimental measurements by the Belle Collaboration [16] on the partial decay width ratios of $ \Lambda_c(2940)^+ $ provide a stronger basis for decoding its inner structure. Compared to $ \Lambda_c(2940)^+ $, the properties of $ \Lambda_c(2765)^+ $ (or $ \Sigma_c(2765)^+ $) and $ \Lambda_c(2910)^+ $ are more controversial, and it is not even certain whether they are $ \Lambda_c $ or $ \Sigma_c $ states. At present, theoretical explanations suggest that $ \Lambda_c(2765)^+ $ (or $ \Sigma_c(2765)^+ $) may be $ \Lambda_c(2S)1/2^+ $ [21, 22], $ \Lambda_c(1P)1/2^- $ [22], or $ \Sigma_c(1P)3/2^- $ [21] resonance. For $ \Lambda_c(2910)^+ $, it can be assigned as $ \Lambda_c $ resonance with spin-parity $ 1/2^+ $ [22] or $ 1/2^- $ [24]. Moreover, the assignment of the two states as $ D^{(*)}N $ molecular states also exists [49, 52].

To decode the inner structures of these undetermined $ \Lambda_c $resonances, more theoretical and experimental efforts are essential. Meanwhile, studies on the strong decay properties of ρ-mode excitations are scarce. Hence, in this paper, we present a systematic analysis of 1P-, 1D-, 2S-, and 2P-wave $ \Lambda_c $ states for both ρ- and λ-mode excitations within the quark pair creation model. On the one hand, we explain the properties of controversial states, and on the other hand, we predict the decays for unobserved $ \Lambda_c $ states. The predicted masses and possible decay channels within the quark pair creation model are listed in Table 1.

Notation	Quantum Number								Mass					Decay channel
$ \Lambda_c $$ {\left|{J^P,j}\right\rangle}_{\lambda(\rho)} $	$ n_{\lambda} $	$ n_{\rho} $	$ l_{\lambda} $	$ l_{\rho} $	L	$ s_{\rho} $	j	$ J^P $	RQM [28]	RQM [30]	RFT [31]	NCQM [25]	QM [32]
$ \Lambda_c $$ {\left|{J^P={1}/{2}^-,1}\right\rangle}_\lambda $	0	0	1	0	1	0	1	$ {1}/{2}^{-} $	2598	2630	2591	2614	2625	$ \Sigma_c\pi $
$ \Lambda_c $$ {\left|{J^P={3}/{2}^-,1}\right\rangle}_\lambda $	0	0	1	0	1	0	1	$ {3}/{2}^{-} $	2627	2640	2629	2639	2636
$ \Lambda_c $$ {\left|{J^P={1}/{2}^-,0}\right\rangle}_\rho $	0	0	0	1	1	1	0	$ {1}/{2}^{-} $		2780			2816	$ \Sigma^{(*)}_c\pi $
$ \Lambda_c $$ {\left|{J^P={1}/{2}^-,1}\right\rangle}_\rho $	0	0	0	1	1	1	1	$ {1}/{2}^{-} $		2830			2816
$ \Lambda_c $$ {\left|{J^P={3}/{2}^-,1}\right\rangle}_\rho $	0	0	0	1	1	1	1	$ {3}/{2}^{-} $		2840			2830
$ \Lambda_c $$ {\left|{J^P={3}/{2}^-,2}\right\rangle}_\rho $	0	0	0	1	1	1	2	$ {3}/{2}^{-} $		2885			2830
$ \Lambda_c $$ {\left|{J^P={5}/{2}^-,2}\right\rangle}_\rho $	0	0	0	1	1	1	2	$ {5}/{2}^{-} $		2900			2872
$ \Lambda_c $$ {\left|{J^P={3}/{2}^+,2}\right\rangle}_{\lambda\lambda} $	0	0	2	0	2	0	2	$ {3}/{2}^{+} $	2874	2910	2857	2843	2887	$ \Sigma^{(*)}_c\pi $,$ DN $
$ \Lambda_c $$ {\left|{J^P={5}/{2}^+,2}\right\rangle}_{\lambda\lambda} $	0	0	2	0	2	0	2	$ {5}/{2}^{+} $	2880	2910	2879	2851	2887
$ \Lambda_c $$ {\left|{J^P={3}/{2}^+,2}\right\rangle}_{\rho\rho} $	0	0	0	2	2	0	2	$ {3}/{2}^{+} $		$ \sim3035 $			3073	$ \Lambda^{+}_c\omega^0 $,$ \Sigma^{(*)}_c\pi $,$ \Xi^{'}_cK $,$ \Sigma_c(1P_{\lambda})\pi $
$ \Lambda_c $$ {\left|{J^P={5}/{2}^+,2}\right\rangle}_{\rho\rho} $	0	0	0	2	2	0	2	$ {5}/{2}^{+} $		$ \sim3140 $			3092
$ \Lambda_{c1} $$ {\left|{J^P={1}/{2}^+,0}\right\rangle}_\lambda $	1	0	0	0	0	0	0	$ {1}/{2}^{+} $	2769	2775	2766	2772	2791	$ \Sigma^{(*)}_c\pi $,$ DN $
$ \Lambda_{c1} $$ {\left|{J^P={1}/{2}^+,0}\right\rangle}_\rho $	0	1	0	0	0	0	0	$ {1}/{2}^{+} $		2970				$ \Sigma^{(*)}_c\pi $,$ D^{(*)}N $,$ \Sigma_c(1P_{\lambda})\pi $
$ \Lambda_{c1} $$ {\left|{J^P={1}/{2}^-,1}\right\rangle}_\lambda $	1	0	1	0	1	0	1	$ {1}/{2}^{-} $	2983	3030	2989	2980		$ \Sigma^{(*)}_c\pi $,$ D^{(*)}N $,$ \Sigma_c(1P_{\lambda})\pi $
$ \Lambda_{c1} $$ {\left|{J^P={3}/{2}^-,1}\right\rangle}_\lambda $	1	0	1	0	1	0	1	$ {3}/{2}^{-} $	3005	3035	3000	3004
$ \Lambda_{c1} $$ {\left|{J^P={1}/{2}^-,0}\right\rangle}_\rho $	0	1	0	1	1	1	0	$ {1}/{2}^{-} $		3200				$ \Lambda^{+}_c\omega^0/\eta $,$ \Sigma^{(*)}_c\pi $,$ \Xi_cK $,$ \Xi_c'^{(*)}K $, $ \Sigma_c(1P_{\lambda/\rho})\pi $, $ \Lambda^{+}_c(1P_{\lambda})\eta $
$ \Lambda_{c1} $$ {\left|{J^P={1}/{2}^-,1}\right\rangle}_\rho $	0	1	0	1	1	1	1	$ {1}/{2}^{-} $		3240
$ \Lambda_{c1} $$ {\left|{J^P={3}/{2}^-,1}\right\rangle}_\rho $	0	1	0	1	1	1	1	$ {3}/{2}^{-} $		3240
$ \Lambda_{c1} $$ {\left|{J^P={3}/{2}^-,2}\right\rangle}_\rho $	0	1	0	1	1	1	2	$ {3}/{2}^{-} $		3255
$ \Lambda_{c1} $$ {\left|{J^P={5}/{2}^-,2}\right\rangle}_\rho $	0	1	0	1	1	1	2	$ {5}/{2}^{-} $		3130

Table 1.  Predicted masses of $ \Lambda_c $ states (1P-, 1D-, 2S-, and 2P-wave) in various quark models and quark pair creation modes.

The remainder of this paper is structured as follows. In Sec. II, we briefly introduce the quark pair creation model. Then, we present our theoretical results and discussions in Sec. III. Finally, a summary is given in Sec. IV.

As a phenomenological method, the quark pair creation model [53−57] has been successfully employed in the description of the OZI-allowed two-body strong decays. The main idea of this model is that the quark-antiquark pair with $ 0^{++} $ is created from the vacuum and then regroups with the quarks from the initial hadron to produce two outing hadrons. Hence, for the $ \Lambda_c $ system, there are three decay processes, as shown in Fig. 1.

Figure 1.  (color online) Possible manners of decays for the $ \Lambda_c $ system within the quark pair creation model.

In the framework of the quark pair creation model, the transition operator for a two-body decay ($ A\rightarrow B+C $) in the nonrelativistic limit reads

The pair creation strength γ is a dimensionless parameter and fixed by fitting experimental data. $ {\bf{p}}_i(i=4,5) $ denotes the three-vector momentum of the ith quark of the created quark pair. $ \omega^{45}_0 $=$ \delta_{ij} $ and $ \chi^{45}_{1,-m} $ represent the color singlet and spin triplet of the quark pair, respectively. $ \phi^{45}_0 =(u\bar{u}+d\bar{d}+s\bar{s})$/$ \sqrt{3} $ represents the flavor function. The solid harmonic polynomial $ {\cal{Y}}^m_1 $=$ |{\bf{p}}| {\rm{Y}}^m_1 $($ \theta_p\phi_p $) corresponds to the momentum-space distribution. The creation operator $ a^{\cal{Y}}_{4i}d^{\cal{Y}}_{5j} $ represents quark pair-creation in vacuum.

According to the definition of the mock state [58], the wave functions of the baryon (denoted as $ |A\rangle $) and meson (denoted as $ |C\rangle $) are respectively given by

$ {\bf{p}}_i $ ($ i = 1, 2, 3 $ and $ a, b $) represents the momentum of quarks in baryon $ |A\rangle $ and meson $ |C\rangle $. $ {\bf{p}}_A $($ {\bf{p}}_C $) denotes the momentum of the hadron $ |A\rangle $($ |C\rangle $). The spatial wave functions of hadrons are described with simple harmonic oscillator wave functions. For a baryon without the radial excitation,

and for a baryon with one radial excitation ($ n_{\lambda/\rho}=1 $),

$ {\bf{p}}_{\rho} $ represents the relative momentum within the light diquark, and $ {\bf{p}}_{\lambda} $ denotes the relative momentum between the light diquark and heavy quark. The spatial wave function for a ground meson $ |C\rangle $ is

where $ {\bf{p}}_{ab} $ is the relative momentum between the quark and antiquark in the meson.

Then, in the center of mass frame, we can obtain the partial decay amplitude as

$ I^{M_{L_A},m}_{M_{L_B},M_{L_C}}({\bf{p}}) $ denotes the spatial integration, and the Clebsch-Gorden coefficient $ \prod_{A,B,C} $ reads

Considering that the vertex given by the quark pair creation model is too complicated at high momenta, we modify the vertex by adopting a form factor $ {\rm e}^{-\frac{{\bf{p}}^2}{2\Lambda^2}} $ as in previous literature [59−61], which gives the quark-pair-creation vertex a finite-size rather than point-like behavior. It reads

In this equation, we fix the cut-off parameter $ \Lambda=780 $ MeV, which is the same value used in Ref. [62]. $ {\bf{p}} $ represents the momentum of the daughter baryon B in the center of mass frame of the parent baryon A and reads

Finally, the decay width $ \Gamma[A\rightarrow BC] $ can be calculated by the following formula:

In this work, we adopt $ m_u $=$ m_d $=330 MeV, $ m_s $=450 MeV, and $ m_c $=1700 MeV for the constituent quark mass. The masses of the final baryons and mesons involved in our calculations are listed in Table 2. The harmonic oscillator strength $ R=2.5 $ GeV^–1 for light flavor mesons $ \pi/K^{(*)}/\omega/\eta $, $ R=1.67 $ GeV^–1 for the D meson, and $ R=1.94 $ GeV^–1 for the $ D^* $ meson [64]. The parameter of the ρ-mode excitation between the two light quarks is taken as $ \alpha_{\rho} $=0.4 GeV. The other parameter $ \alpha_\lambda $ is obtained by the relation [45]

State	Mass		State	Mass		State	Mass
p	938.27		$ \pi^0 $	134.98		$ \eta' $	957.78
n	939.57		$ \pi^{\pm} $	139.57		ρ	775.26
$ \Lambda_c^+ $	2286.46		η	547.862		$ \Lambda_c|J^{P}={1}/{2}^-,1\rangle_{\lambda} $	2592.25
$ \Sigma_c^0 $	2453.75		ω	782.66		$ \Lambda_c|J^{P}={3}/{2}^-,1\rangle_{\lambda} $	2628.00
$ \Sigma_c^+ $	2452.65		$ K^0 $	497.611		$ \Sigma_c|J^{P}={1}/{2}^-,0\rangle_{\lambda} $	2823
$ \Sigma_c^{++} $	2453.97		$ K^+ $	493.677		$ \Sigma_c|J^{P}={1}/{2}^-,1\rangle_{\lambda} $	2809
$ \Xi_c^+ $	2467.71		$ K^{0*} $	895.55		$ \Sigma_c|J^{P}={3}/{2}^-,1\rangle_{\lambda} $	2829
$ \Xi_c^0 $	2470.44		$ K^{+*} $	891.67		$ \Sigma_c|J^{P}={3}/{2}^-,2\rangle_{\lambda} $	2802
$ \Xi_c'^{+} $	2578.2		$ D^0 $	1864.84		$ \Sigma_c|J^{P}={5}/{2}^-,2\rangle_{\lambda} $	2835
$ \Xi_c'^{0} $	2578.7		$ D^+ $	1869.66		$ \Sigma_c|J^{P}={1}/{2}^-,1\rangle_{\rho} $	2909
$ \Xi_c'^{+*} $	2645.1		$ D^{0*} $	2006.85		$ \Sigma_c|J^{P}={3}/{2}^-,1\rangle_{\rho} $	2910
$ \Xi_c'^{0*} $	2646.16		$ D^{+*} $	2010.26

Table 2.  Masses (MeV) of the final baryons and mesons [4, 29, 63].

The value of vacuum pair-production strength γ is determined by fitting the well measured decay $ \Sigma_c^{++}(2520)\rightarrow \Lambda_c^+\pi^+ $ and is obtained as $ \gamma= $11.51.

The two-body strong decays of 1P-, 1D-, 2S-, and 2P-wave excited $ \Lambda_c $ baryons within the j-j coupling scheme are systematically investigated by the quark pair creation model. Both λ-mode and ρ-mode excitations are considered in this calculation. We attempt to decode the inner structures of the observed controversial $ \Lambda_c $excitations while providing strong decay predictions for the missing $ \Lambda_c $ states, which may be helpful for the observations in forthcoming experiments.

For the 1P-wave λ-mode $ \Lambda_c $ baryons, there are two states according to the quark model classification: $ \Lambda_c|J^P=1/2^-,1\rangle_{\lambda} $ and $ \Lambda_c|J^P=3/2^-,1\rangle_{\lambda} $. They correspond to the well determined states $ \Lambda_c(2595)^+ $ and $ \Lambda_c(2625)^+ $, respectively. Hence, we fix the masses of the two states at the physical masses and list their decay properties in Table 3.

Decay width	$ \Lambda_{c}{\left|{J^{P}={1}/{2}^{-},1}\right\rangle}_{\lambda} $	$ \Lambda_{c}{\left|{J^{P}={3}/{2}^{-},1}\right\rangle}_{\lambda} $
	$ \Lambda_c(2595)^+ $	$ \Lambda_c(2625)^+ $
$ \Gamma[\Sigma_c^0\pi^+] $	–	0.02
$ \Gamma[\Sigma_c^+\pi^0] $	7.07	0.01
$ \Gamma[\Sigma_c^{++}\pi^-] $	–	0.01
$ \Gamma_{\rm{Total}} $	7.07	0.04
Expt.	$ 2.59\pm0.30\pm0.47 $	<0.52

Table 3.  Strong decay properties of the λ-mode 1P-wave $ \Lambda_c $ states, which are taken as $ \Lambda_c(2595)^+ $ and $ \Lambda_c(2625)^+ $, respectively. $ \Gamma_{\rm{Total}} $ represents the total decay width, and Expt. denotes the experimental value. The units are MeV.

Considering the uncertainties for the experimental data and theoretical calculations, the theoretical value is roughly consistent with the observations, which proves the applicability of the quark pair creation model. In addition, it should be noted that the mass of $ \Lambda_c(2595)^+ $ is very close to the threshold of $ \Sigma_c\pi $, and the partial decay widths are highly sensitive to the precision of mass.

For the 1P-wave ρ-mode $ \Lambda_c $ baryons, there are five states: $ \Lambda_{c}{\left|{J^{P}=1/2^{-},0}\right\rangle}_{\rho} $, $ \Lambda_{c}{\left|{J^{P}=1/2^{-},1}\right\rangle}_{\rho} $, $ \Lambda_{c}{\big|{J}^{P}= {3/2^{-},1}\big\rangle}_{\rho} $, $ \Lambda_{c}{\left|{J^{P}=3/2^{-},2}\right\rangle}_{\rho} $, and $ \Lambda_{c}{\left|{J^{P}=5/2^{-},2}\right\rangle}_{\rho} $. According to the theoretical predictions by various methods, the mass of the 1P-wave ρ-mode $ \Lambda_c $baryons is approximately $ M\sim $2.85 GeV. Meanwhile, we notice that their masses are above the threshold of $ ND $, while their strong decays are forbidden due to the orthogonality of spatial wave functions. This is true for all of the ρ-mode excitations. Hence, we mainly focus on their strong decays into $ \Sigma_c\pi $ and $ \Sigma_c^*\pi $. Fixing the masses of 1P-wave ρ-mode $ \Lambda_c $ baryons at the predictions in Ref. [30], we study their strong decay properties, as listed in Table 4.

Decay width	$ \Lambda_{c}{\left|{J^{P}={1}/{2}^{-},0}\right\rangle}_{\rho} $	$ \Lambda_{c}{\left|{J^{P}={1}/{2}^{-},1}\right\rangle}_{\rho} $	$ \Lambda_{c}{\left|{J^{P}={3}/{2}^{-},1}\right\rangle}_{\rho} $	$ \Lambda_{c}{\left|{J^{P}={3}/{2}^{-},2}\right\rangle}_{\rho} $		$ \Lambda_{c}{\left|{J^{P}={5}/{2}^{-},2}\right\rangle}_{\rho} $
	M=2780	M=2830	M=2840	M=2885	$ \Lambda_c(2910)^+ $	M=2900	$ \Lambda_c(2910)^+ $
$ \Gamma[\Sigma_c\pi] $	0.00	782.44	10.33	31.53	41.96	16.34	16.65
$ \Gamma[\Sigma_c^*\pi] $	0.00	6.42	768.63	14.37	21.12	27.50	32.85
$ \Gamma_{\rm{Total}} $	0.00	788.86	778.96	45.90	63.08	43.84	49.50
Expt.	-	-	-	$ 51.8\pm20.0\pm18.8 $
Decay width	$ \Lambda_{c1}{\left|{J^{P}={1}/{2}^{-},0}\right\rangle}_{\rho} $	$ \Lambda_{c1}{\left|{J^{P}={1}/{2}^{-},1}\right\rangle}_{\rho} $	$ \Lambda_{c1}{\left|{J^{P}={3}/{2}^{-},1}\right\rangle}_{\rho} $	$ \Lambda_{c1}{\left|{J^{P}={3}/{2}^{-},2}\right\rangle}_{\rho} $		$ \Lambda_{c1}{\left|{J^{P}={5}/{2}^{-},2}\right\rangle}_{\rho} $
	M=3200	M=3240	M=3240	M=3255		M=3130
$ \Gamma[\Sigma_c\pi] $	0.00	19.02	7.10	12.15		6.78
$ \Gamma[\Sigma_c^*\pi] $	0.00	16.47	21.08	14.47		22.55
$ \Gamma[\Lambda_c\omega] $	0.00	7.30	7.30	12.58		2.10
$ \Gamma[\Lambda_c\eta] $	3.61	0.00	0.00	3.38		4.15
$ \Gamma[\Lambda_c\eta'] $	-	-	-	3.38		-
$ \Gamma[\Sigma_c\rho] $	-	0.23	0.11	176.79		-
$ \Gamma[\Xi_cK] $	0.13	0.00	0.00	2.65		1.36
$ \Gamma[\Xi_c'K] $	0.00	1.86	0.58	1.21		0.05
$ \Gamma[\Xi_c^*K] $	0.00	0.44	6.05	0.53		-
$ \Gamma[\Lambda_c|J^P=1/2^-,1\rangle_{\lambda}\eta] $	0.00	0.04	0.01	1.95		-
$ \Gamma[\Lambda_c|J^P=3/2^-,1\rangle_{\lambda}\eta] $	0.00	0.01	0.03	0.25		-
$ \Gamma[\Sigma_c|J^P=1/2^-,0\rangle_{\lambda}\pi] $	0.00	2.58	2.58	0.00		0.00
$ \Gamma[\Sigma_c|J^P=1/2^-,1\rangle_{\lambda}\pi] $	0.15	6.64	1.66	13.51		0.02
$ \Gamma[\Sigma_c|J^P=3/2^-,1\rangle_{\lambda}\pi] $	0.28	2.77	6.92	1.41		6.84
$ \Gamma[\Sigma_c|J^P=3/2^-,2\rangle_{\lambda}\pi] $	0.00	0.24	0.41	3.76		0.14
$ \Gamma[\Sigma_c|J^P=5/2^-,2\rangle_{\lambda}\pi] $	0.00	0.40	0.05	3.29		0.90
$ \Gamma[\Sigma_c|J^P=1/2^-,1\rangle_{\rho}\pi] $	57.44	10.96	2.74	33.77		0.01
$ \Gamma[\Sigma_c|J^P=3/2^-,1\rangle_{\rho}\pi] $	113.60	5.44	14.27	7.94		6.94
$ \Gamma_{\rm{Total}} $	175.21	74.40	70.89	293.02		51.84

Table 4.  Strong decay properties of the ρ-mode 1P- and 2P-wave $ \Lambda_c $ states within the quark pair creation model, where masses are taken from the predictions in Ref. [30]. $ \Gamma_{\rm{Total}} $ represents the total decay width, and Expt. denotes the experimental value. The units are MeV.

Within the $ j-j $ coupling scheme, the total decay width of $ \Lambda_c|J^P=1/2^-,0\rangle_{\rho} $ is most likely to be near zero. We know that the states in the $ j-j $ coupling scheme can be expressed by a linear combination of the configurations in the $ L-S $ coupling scheme, which reads

In this expression, $ l_{\rho} $ and $ l_{\lambda} $ are the ρ- and λ-mode quantum numbers of the orbital angular momentum, respectively. The total orbital angular momentum L = $ |l_{\rho}-l_{\lambda}|,\cdots , l_{\rho}+l_{\lambda} $. $ s_{\rho} $ is the quantum number of the total spin of the two light quarks, and $ s_Q $ is the spin of the heavy quark. The total spin angular momentum S = $ |s_{\rho}-s_Q|,\cdots , s_{\rho}+s_Q $. J is the total angular momentum. This means the states in the $ j-j $ coupling scheme have a mixing angle θ. Considering the heavy quark symmetry not being strictly true and slightly breaking in the $ \Lambda_c $ system, the mixing angle θ fluctuates around the center value ($ \theta\simeq35^{\circ} $). To investigate this effect, we plot the strong decay widths of $ \Lambda_c|J^P=1/2^-,0\rangle_{\rho} $ as a function of the mixing angle in Fig 2. We obtain that $ \Lambda_c|J^P=1/2^-,0\rangle_{\rho} $ is still a very narrow state, and the $ \Sigma_c\pi $ decay channel almost saturates its total decay widths.

Figure 2.  (color online) Partial and total strong decay widths of $ \Lambda_c|J^P=1/2^-,0\rangle_{\rho} $ as a function of mixing angle.

The two states $ \Lambda_c|J^P=1/2^-,1\rangle_{\rho} $ and $ \Lambda_c|J^P=3/2^-,1\rangle_{\rho} $ are probably broad states with a total decay width around $ \Gamma_{\rm{Total}}\sim780 $ MeV. $ \Lambda_c|J^P=1/2^-,1\rangle_{\lambda} $ dominantly decays into the $ \Sigma_c\pi $ channel, while $ \Lambda_c|J^P=3/2^-,1\rangle_{\rho} $ mainly decays into the $ \Sigma_c^*\pi $ channel. In this case, $ \Lambda_c|J^P=1/2^-,1\rangle_{\rho} $ and $ \Lambda_c|J^P=3/2^-,1\rangle_{\rho} $ might be too broad to be observed experimentally.

The states $ \Lambda_c|J^P=3/2^-,2\rangle_{\rho} $ and $ \Lambda_c|J^P=5/2^-,2\rangle_{\rho} $ may be moderate states with a total width of several tens of MeV, and their strong decays are governed by the $ \Sigma_c\pi $ and $ \Sigma_c^*\pi $ channels. We note that the new state $ \Lambda_c(2910)^+ $ was observed in the $ \Sigma_c\pi $ channel by the Belle Collaboration [17]. Its mass and width were measured to be $ M= 2913.8\pm 5.6\pm3.8 $ MeV and $ \Gamma=51.8\pm20.0\pm18.8 $ MeV, respectively. Combining the predicted masses and decay properties, $ \Lambda_c|J^P=3/2^-,2\rangle_{\rho} $ and $ \Lambda_c|J^P=5/2^-,2\rangle_{\rho} $ may be candidates of $ \Lambda_c(2910)^+ $. Hence, we fix the masses of the two states as $ M=2914 $ MeV, and their decays are listed in Table 4. It is found that the total decay width of $ \Lambda_c|J^P=3/2^-,2\rangle_{\rho} $ is approximately

which is consistent with the observation. Meanwhile, the main decay channel is $ \Sigma_c\pi $, and the predicted partial decay width ratio is

This calculation is consistent with the fact that $ \Lambda_c(2910)^+ $ was first observed in the $ \Sigma_c\pi $ invariant mass distribution by the Belle Collaboration [17].

Regarding $ \Lambda_c|J^P=5/2^-,2\rangle_{\rho} $, its total decay width is approximately

which is in good agreement with the observation. The dominant decay channel is $ \Sigma_c^*\pi $, and the corresponding branching fraction is

Meanwhile, the predicted branching ratio of the $ \Sigma_c\pi $ channel is

which is sufficiently large to be observed experimentally. To further decode the nature of $ \Lambda_c(2910)^+ $and determine whether it is $ \Lambda_c|J^P=3/2^-,2\rangle_{\rho} $ or $ \Lambda_c|J^P=5/2^-,2\rangle_{\rho} $, the partial decay width ratio between $ \Sigma_c\pi $ and $ \Sigma_c^*\pi $ may be a good criterion.

Nevertheless, considering that the current experimental data are limited and bare large errors, there are other possible explanations, such as the $ D^*N $ molecular state [49, 52] or $ \Lambda_c $ resonances with different spin-parity [22, 24]. To further clarify the nature of $ \Lambda_c(2910)^+ $, more discussions are necessary.

According to the symmetry of wave functions, there are two λ-mode 1D-wave $ \Lambda_c $ states (see Table 1): $ \Lambda_c|J^P= 3/2^+,2\rangle_{\lambda\lambda} $ and $ \Lambda_c|J^P=5/2^+,2\rangle_{\lambda\lambda} $. As listed in Table 1, the masses of the two 1D λ-mode $ \Lambda_c $ states fluctuate around $ \sim2.85 $ GeV. Based on the predicted masses, $ \Lambda_c|J^P= 3/2^+,2\rangle_{\lambda\lambda} $ and $ \Lambda_c|J^P=5/2^+,2\rangle_{\lambda\lambda} $ are probably assignments of observed states $ \Lambda_c(2860)^+ $ and $ \Lambda_c(2880)^+ $, respectively. Hence, we fix the masses of $ \Lambda_c|J^P=3/2^+,2\rangle_{\lambda\lambda} $ and $ \Lambda_c|J^P=5/2^+,2\rangle_{\lambda\lambda} $ at $ M=2856 $ MeV and $ M=2882 $ MeV, respectively, and list their decays in Table 5.

Decay width	$ \Lambda_c|J^P=3/2^+,2\rangle_{\lambda\lambda} $	$ \Lambda_c|J^P=5/2^+,2\rangle_{\lambda\lambda} $
	$ \Lambda_c(2860)^+ $	M=2882
$ \Gamma[\Sigma_c\pi] $	24.32	1.74
$ \Gamma[\Sigma_c^*\pi] $	3.86	24.28
$ \Gamma[pD^0] $	21.61	0.35
$ \Gamma[nD^+] $	18.77	0.27
$ \Gamma_{\rm{Total}} $	68.56	26.64
Expt.	$ 67.6^{+10.1}_{-8.1}\pm1.4^{+5.9}_{-20.0} $	$ 5.6^{+0.8}_{-0.6} $

Table 5.  Partial decay widths of $ \Lambda_c(2860)^+ $ and $ \Lambda_c(2880)^+ $ assigned as λ-mode 1D-wave $ \Lambda_c $ states $ \Lambda_c|J^P= $$ 3/2^+,2\rangle_{\lambda\lambda} $ and $ \Lambda_c|J^P=5/2^+,2\rangle_{\lambda\lambda} $, respectively. The units are MeV.

The total width of $ \Lambda_c|J^P=3/2^+,2\rangle_{\lambda\lambda} $ is obtained as

which is in good agreement with the experimental central value. Furthermore, $ pD^0 $ is one of the main decay modes, and the predicted branching fraction is

This result is consistent with the fact that $ \Lambda_c(2860)^+ $ was observed in the $ pD^0 $ invariant mass distribution [14]. In addition, we obtain

If the observed state $ \Lambda_c(2860)^+ $ corresponds to $ \Lambda_c|J^P= 3/2^+,2\rangle_{\lambda\lambda} $, besides the $ pD^0 $ channel, $ \Sigma_c\pi $ and $ nD^+ $ may be two interesting channels for the observation of $ \Lambda_c(2860)^+ $ in future experiments. The $ \Lambda_c(2860)^+ $ resonance should be observed in the $ \Lambda_c\pi\pi $ and $ nD^+ $ final states as well.

For the state $ \Lambda_c|J^P=5/2^+,2\rangle_{\lambda\lambda} $(see Table 5), fixing its mass to M=2882 MeV, the total decay width

is approximately five times that of the observation for $ \Lambda_c(2880)^+ $. Meanwhile, the predicted partial decay width ratio between $ pD^0 $ and $ \Sigma_c\pi $ is

This value is much smaller than the ratio measured ($ 0.75\pm0.03\pm0.07 $) by the Belle Collaboration [16]. Meanwhile, our theoretical calculation indicates that the $ \Sigma_c^*\pi $ decay channel almost saturates the total decay widths. The partial decay width ratio between $ \Sigma_c^*\pi $ and $ \Sigma_c\pi $ is

which is inconsistent with the analysis from the CLEO Collaboration [11]. Hence, according to our investigation, the experimental widths and some partial decay width ratios cannot be reproduced. To further clarify the properties of the $ \Lambda_c(2880)^+ $ resonance, more experimental and theoretical investigations may be necessary.

Within the quark model, there are two ρ-mode 1D-wave $ \Lambda_c $ states: $ \Lambda_c|J^P=3/2^+,2\rangle_{\rho\rho} $ and $ \Lambda_c|J^P=5/2^+,2\rangle_{\rho\rho} $. According to the mass predictions listed in Table 1, their masses are approximately $ M\sim3.10 $ GeV. First, we fix the masses at the predictions within a relativized quark potential model in Ref. [30] and list the decay properties in Table 6.

Decay width	$ \Lambda_c|J^P=3/2^+,2\rangle_{\rho\rho} $	$ \Lambda_c|J^P=5/2^+,2\rangle_{\rho\rho} $
	M=3035	M=3140
$ \Gamma[\Sigma_c\pi] $	82.20	33.38
$ \Gamma[\Sigma_c^*\pi] $	25.66	107.24
$ \Gamma[\Lambda_c\omega] $	-	31.25
$ \Gamma[\Xi_c'K] $	-	0.02
$ \Gamma[\Sigma_c|J^P=1/2^-,0\rangle_\lambda\pi] $	0.01	0.29
$ \Gamma[\Sigma_c|J^P=1/2^-,1\rangle_\lambda\pi] $	0.13	0.79
$ \Gamma[\Sigma_c|J^P=3/2^-,1\rangle_\lambda\pi] $	0.06	1.91
$ \Gamma[\Sigma_c|J^P=3/2^-,2\rangle_\lambda\pi] $	11.81	0.11
$ \Gamma[\Sigma_c|J^P=5/2^-,2\rangle_\lambda\pi] $	0.00	2085
$ \Gamma_{\rm{Total}} $	119.87	195.84

Table 6.  Partial decay widths of the two ρ-mode 1D-wave $ \Lambda_c $ states, with masses taken from the predictions in Ref. [30]. $ \Gamma_{\rm{Total}} $ represents the total decay width, and units are MeV.

The $ \Lambda_c|J^P=3/2^+,2\rangle_{\rho\rho} $ state may be a moderate state with a width of $ \Gamma_{\rm{Total}}\simeq120 $ MeV and mainly decays into $ \Sigma_c\pi $. The predicted branching fraction is

Hence, the $ \Lambda_c|J^P=3/2^+,2\rangle_{\rho\rho} $ state is likely to be observed in the $ \Lambda_c\pi\pi $ final state via the decay chain $ \Lambda_c|J^P= 3/2^+,2\rangle_{\rho\rho}\rightarrow \Sigma_c\pi\rightarrow \Lambda_c\pi\pi $.

Meanwhile, the partial decay width of $ \Sigma_c^*\pi $ is sizable, and the branching fraction is

Thus, $ \Lambda_c|J^P=3/2^+,2\rangle_{\rho\rho}\rightarrow \Sigma_c^*\pi\rightarrow \Lambda_c\pi\pi $ may be another interesting decay chain for experimental exploration.

The other ρ-mode 1D-wave state $ \Lambda_c|J^P=5/2^+,2\rangle_{\rho\rho} $ has a width of $ \Gamma_{\rm{Total}}\simeq196 $ MeV and mainly decays into $ \Sigma_c^*\pi $ with a branching fraction

Furthermore, $ \Lambda_c|J^P=5/2^+,2\rangle_{\rho\rho} $ may have a considerable decay rate into $ \Sigma_c\pi $ and $ \Lambda_c\omega $. The predicted branching fractions are

However, this state may be too broad to be observed experimentally.

Then, accounting for the uncertainty of the predicted masses, which may bring uncertainties to the theoretical results, we plot the decay properties of the 1D-wave ρ-mode $ \Lambda_c $ as functions of masses within the range of $ M=(3.00-3.15) $ GeV, as shown in Fig. 3. We can find that the total decay widths vary within the scope of $ \Gamma_{\rm{Total}}<200 $ MeV as the mass increases. In addition, when the masses of the 1D-wave ρ-mode $ \Lambda_c $ states are above the threshold of $ \Lambda_c\omega $, the partial decay width of this channel becomes sizable and increases dramatically with mass.

Figure 3.  (color online) Partial and total strong decay widths of the two ρ-mode 1D-wave $ \Lambda_c $ states as functions of the masses. Some decay channels are too small to show in the figure.

The 2S-wave states are the first radial excited states, and the radial quantum number $ n_{\lambda}=1 $ or $ n_{\rho}=1 $. Hence, according to the symmetry of the wave function, there are two 2S-wave $ \Lambda_c $ states: $ \Lambda_{c1}|J^P=1/2^+,0\rangle_{\lambda} $($ n_{\lambda}=1 $) and $ \Lambda_{c1}|J^P=1/2^+,0\rangle_{\rho} $($ n_{\rho}=1 $). As listed in Table 1, the mass of the $ \Lambda_{c1}|J^P=1/2^+,0\rangle_{\lambda} $ state is approximately $ M\sim (2.76- 2.79) $ MeV, while the mass of the $ \Lambda_{c1}|J^P= 1/2^+,0\rangle_{\rho} $ state is slightly higher at approximately $ M\sim2.97 $ MeV. Fixing the masses at the predictions in Ref. [30], we analyze the decay properties of the two 2S-wave $ \Lambda_c $ states and list their partial strong decay widths in Table 7.

Decay width	$ \Lambda_{c1}|J^P=1/2^+,0\rangle_{\lambda} $		$ \Lambda_{c1}|J^P=1/2^+,0\rangle_{\rho} $
	M=2775	$ \Lambda_c(2765)^+ $	M=2970
$ \Gamma[\Sigma_c\pi] $	11.63	10.90	81.48
$ \Gamma[\Sigma_c^*\pi] $	11.57	10.24	149.88
$ \Gamma[\Sigma_c|J^P=1/2^-,0\rangle_\lambda\pi] $	−	−	13.15
$ \Gamma[\Sigma_c|J^P=1/2^-,1\rangle_\lambda\pi] $	−	−	0.00
$ \Gamma[\Sigma_c|J^P=3/2^-,1\rangle_\lambda\pi] $	−	−	0.00
$ \Gamma[\Sigma_c|J^P=3/2^-,2\rangle_\lambda\pi] $	−	−	0.01
$ \Gamma_{\rm{Total}} $	23.20	21.14	244.52
Expt.	50		−

Table 7.  Partial decay widths of the two 2S-wave $ \Lambda_c $ states, with masses taken from the predictions in Ref. [30]. $ \Gamma_{\rm{Total}} $ represents the total decay width, and the units are MeV.

The $ \Lambda_{c1}|J^P=1/2^+,0\rangle_{\lambda} $ state probably has a narrow width of several tens of MeV and mainly decays into the $ \Sigma_c\pi $ and $ \Sigma_c^*\pi $ channels. In this case, this state has a good potential to be observed in the $ \Lambda_c\pi\pi $ final state by the intermediate channels $ \Sigma_c\pi $ and $ \Sigma_c^*\pi $. Combining the predicted mass and our calculations, the $ \Lambda_{c1}|J^P=1/2^+,0\rangle_{\lambda} $ state may be an assignment of the observed state $ \Lambda_c(2765)^+ $. Hence, we further take this state as $ \Lambda_c(2765)^+ $ and list its decay properties in Table 7. With the mass of $ \Lambda_{c1}|J^P=1/2^+,0\rangle_{\lambda} $fixed at $ M=2767 $ MeV, the total decay width

is approximately half of the experimental value ($ \Gamma_{\rm{Expt.}}= $ 50 MeV). The predicted partial decay width ratio between the dominant modes $ \Sigma_c\pi $ and $ \Sigma_c^*\pi $ is

This result is consistent with the fact that $ \Lambda_c(2765)^+ $ was first observed in the $ \Sigma_c\pi $ and $ \Sigma_c^*\pi $ channels by the CLEO Collaboration [11]. Thus, the state $ \Lambda_{c1}|J^P=1/2^+,0\rangle_{\lambda} $ may be a candidate of $ \Lambda_c(2765)^+ $.

For the other 2S-wave state $ \Lambda_{c1}|J^P=1/2^+,0\rangle_{\rho} $, the main decay channels are $ \Sigma_c\pi $ and $ \Sigma_c^*\pi $ as well, and the partial width ratio is

This state is most likely to be a broad state with a width of approximately $ \Gamma_{\rm{Total}}\simeq245 $ MeV. Thus, it is difficult to observe the $ \Lambda_{c1}|J^P=1/2^+,0\rangle_{\rho} $ state in experiments due to its broad decay width.

Considering the uncertainty of the masses of the 2S-wave $ \Lambda_c $ states, we further investigate the strong decay widths as a function of the mass in Fig. 4. It is shown that the decay properties of the 2S-wave $ \Lambda_c $ excitations are sensitive to their masses varying in the considered region. Furthermore, if the mass of $ \Lambda_{c1}|J^P=1/2^+,0\rangle_{\lambda} $ is above the threshold of $ ND $, the corresponding partial decay width of $ ND $ increases dramatically with mass and is crucial in the strong decay.

Figure 4.  (color online) Partial and total strong decay widths of the two 2S-wave $ \Lambda_c $ states as functions of the masses. Some decay channels are too small to show in the figure.

In these calculations, the 2P-wave λ-mode $ \Lambda_c $ excitations correspond to the radial quantum number $ n_{\lambda}=1 $ and orbital quantum number $ l_{\lambda}=1 $. Hence, based on the quark model classification, there are two 2P-wave λ-mode $ \Lambda_c $ states: $ \Lambda_{c1}|J^P=1/2^-,1\rangle_{\lambda} $ and $ \Lambda_{c1}|J^P=3/2^-,1\rangle_{\lambda} $. Their theoretical masses and possible two-body decay channels are listed in Table 1.

From the table, it is known that the predicted masses of the two 2P-wave λ-mode $ \Lambda_c $baryons are approximately $ M\sim3.00 $ GeV, which is close to the measured mass of the observed state $ \Lambda_c(2940)^+ $. As a possible assignment, it is crucial to investigate the decay behaviors of the two 2P-wave λ-mode $ \Lambda_c $ baryons. Considering the uncertainties of the predicted masses of $ \Lambda_{c1}|J^P=1/2^-,1\rangle_{\lambda} $ and $ \Lambda_{c1}|J^P=3/2^-,1\rangle_{\lambda} $, we plot the decay width as a function of the mass in the range of $ M=(2.90-3.05) $ GeV in Fig. 5.

Figure 5.  (color online) Partial and total strong decay widths of the two λ-mode 2P-wave $ \Lambda_c $ states as functions of the masses. Some decay channels are too small to show in the figure.

From Fig. 5, the total decay widths of the two 2P-wave λ-mode $ \Lambda_c $ baryons are approximately dozens of MeV within the mass range that we considered. When their masses lie below the threshold of $ ND^* $, their strong decays are dominated by the $ ND $ channel. However, if their masses are above the threshold of $ ND^* $, the dominant decay channel should be $ ND^* $, and their total decay widths show a strong dependence on the masses.

In addition, we notice that the measured mass and total decay width of $ \Lambda_c(2940)^+ $ are consistent with the properties the two 2P-wave λ-mode $ \Lambda_c $ baryons. Hence, we further fix the masses of $ \Lambda_{c1}|J^P=1/2^-,1\rangle_{\lambda} $ and $ \Lambda_{c1}|J^P=3/2^-,1\rangle_{\lambda} $ at $ M=2940 $ MeV and list their decay properties in Table 8.

Decay width	$ \Lambda_{c1}|J^P=1/2^-,1\rangle_{\lambda} $	$ \Lambda_{c1}|J^P=3/2^-,1\rangle_{\lambda} $
	M=2940	$ \Lambda_c(2940)^+ $
$ \Gamma[\Sigma_c\pi] $	0.68	2.28
$ \Gamma[\Sigma_c^*\pi] $	3.18	1.61
$ \Gamma[pD^0] $	5.72	10.21
$ \Gamma[nD^+] $	6.89	9.55
$ \Gamma_{\rm{Total}} $	16.47	23.65
Expt.	$ 20^{+6}_{-5} $

Table 8.  Partial decay widths of $ \Lambda_c(2940)^+ $ assigned as the two 2P-wave λ-mode $ \Lambda_c $ states $ \Lambda_{c1}|J^P=1/2^-,1\rangle_{\lambda} $ and $ \Lambda_{c1}|J^P= $$ 3/2^-,1\rangle_{\lambda} $.

It can be seen that the total decay width of $ \Lambda_{c1}|J^P= 1/2^-,1\rangle_{\lambda} $ is

which is in agreement with the experimental observation. The dominant decay channels are $ pD^0 $ and $ nD^+ $ with predicted branching fractions of

Meanwhile, the $ \Sigma_c^*\pi $ decay channel occupies a sizable branching fraction, and the corresponding branching fraction is approximately

However, we notice that the partial decay width of $ \Lambda_{c1}|J^P=1/2^-,1\rangle_{\lambda}\rightarrow \Sigma_c\pi $ seems slightly small, which does not accord with the fact that $ \Lambda_c(2940) $ was observed in the $ \Sigma_c\pi $ channel. Meanwhile, the partial decay width ratio between $ pD^0 $ and $ \Sigma_c\pi $

is significantly greater than the latest experimental value ($ 3.59\pm0.21\pm0.56 $) measured by the LHCb Collaboration [16].

The total decay width of $ \Lambda_{c1}|J^P=3/2^-,1\rangle_{\lambda} $

agrees with the experimental value as well. Meanwhile, the decays are governed by $ pD^0 $ and $ nD^+ $, and the predicted branching fractions are

In addition, $ \Lambda_{c1}|J^P=3/2^-,1\rangle_{\lambda} $ has a sizable decay width in $ \Sigma_c\pi $. The predicted partial decay width ratio between $ pD^0 $ and $ \Sigma_c\pi $ is

which is close to the upper limit of the measurement [16].

In conclusion, our calculation indicates that the strong decay properties of $ \Lambda_{c1}|J^P=3/2^-,1\rangle_{\lambda} $ are in good agreement with the nature of $ \Lambda_c(2940) $, and $ \Lambda_{c1}|J^P=3/2^-,1\rangle_{\lambda} $is a good candidate. It should be pointed out that the threshold of the main decay channel $ DN $ is close to the mass of $ \Lambda_c(2940) $, which may suggest the importance of the coupled-channel effects for understanding the $ \Lambda_c(2940)^+ $ state.

In the present study, the 2P-wave ρ-mode excitations correspond to the radial quantum number $ n_{\rho} $=1 and orbital quantum number $ l_{\rho} $=1. According to the quark model, there are five 2P-wave ρ-mode $ \Lambda_c $ baryons: $ \Lambda_{c1}|J^P= 1/2^-,0\rangle_{\rho} $, $ \Lambda_{c1}|J^P=1/2^-,1\rangle_{\rho} $, $ \Lambda_{c1}|J^P=3/2^-,1\rangle_{\rho} $, $ \Lambda_{c1}|J^P= 3/2^-,2\rangle_{\rho} $, and $ \Lambda_{c1}|J^P=5/2^-,2\rangle_{\rho} $. For their masses, there are a few discussions in theoretical references, and we list them in Table 1. From the table, the masses of the 2P-wave ρ-mode $ \Lambda_c $excitations are approximately $ M\sim3.20 $ GeV. Fixing the masses of the 2P-wave ρ-mode $ \Lambda_c $ excitations on the predicted masses from Ref. [30], we discuss their decay properties and list the results in Table 4.

For the $ \Lambda_{c1}|J^P=1/2^-,0\rangle_{\rho} $ state, the total decay width is approximately $ \Gamma_{\rm{Total}}\simeq $175 MeV. The dominant decay modes are $ \Sigma_c|J^P=1/2^-,1\rangle_{\rho}\pi $ and $ \Sigma_c|J^P=3/2^-,1\rangle_{\rho}\pi $ with the partial decay ratio

Hence, this state may be observed in the $ \Lambda_c\pi\pi\pi $ final state via the decay chains $ \Lambda_{c1}|J^P=1/2^-,0\rangle_{\rho}\rightarrow \Sigma_c|J^P= 1/2^-,1\rangle_{\rho}\pi\rightarrow \Sigma_c\pi\pi\rightarrow \Lambda_c\pi\pi\pi $ and $ \Lambda_{c1}|J^P=1/2^-,0\rangle_{\rho}\rightarrow \Sigma_c|J^P=3/2^-,1\rangle_{\rho}\pi\rightarrow \Sigma_c^*\pi\pi\rightarrow \Lambda_c\pi\pi\pi $.

Meanwhile, the partial decay width of $ \Gamma[\Lambda_{c1}|J^P= 1/2^-,0\rangle_{\rho}\rightarrow \Lambda_c\eta] $ is considerable. The branching fraction is

The $ \Lambda_c\eta $ channel may also be a notable decay mode for future exploration of the $ \bar{\Lambda}_c|J^P=1/2^-,0\rangle_{\rho} $ state.

The states $ \Lambda_{c1}|J^P=1/2^-,1\rangle_{\rho} $ and $ \Lambda_{c1}|J^P=3/2^-,1\rangle_{\rho} $ are most likely to be the moderate states with a total decay width of $ \Gamma_{\rm{Total}}\sim70 $ MeV. Meanwhile, their dominant decay channels are different. $ \Lambda_{c1}|J^P=1/2^-,1\rangle_{\rho} $ mainly decays via the $ \Sigma_c\pi $ and $ \Sigma_c^*\pi $ channels, and the branching fractions are

In addition, the $ \Lambda_{c1}|J^P=1/2^-,1\rangle_{\rho} $ state has sizable partial widths decaying into $ \Sigma_c|J^P=1/2^-,1\rangle_{\rho}\pi $, $ \Lambda_c\omega $, and $ \Sigma_c|J^P= 1/2^-,1\rangle_{\lambda}\pi $. The predicted branching fractions of these channels are approximately $ 9\%-15\% $.

As for $ \Lambda_{c1}|J^P=3/2^-,1\rangle_{\rho} $, it decays mainly through the $ \Sigma_c^*\pi $ channel. The predicted branching fraction is

Meanwhile, the partial decay widths of $ \Sigma_c|J^P=3/2^-,1\rangle_{\rho}\pi $, $ \Sigma_c|J^P=3/2^-,1\rangle_{\lambda}\pi $, $ \Lambda_c\omega $, and $ \Sigma_c\pi $ are considerable, and the corresponding branching fractions are approximately $ (10\%-20\%) $.

The decay width of the state $ \Lambda_{c1}|J^P=3/2^-,2\rangle_{\rho} $ is approximately $ \Gamma\simeq293 $ MeV. Its strong decays are governed by the $ \Sigma_c\rho $ channel with the branching fraction around ~60%. Meanwhile, this state has a sizable decay rate, decaying into the $ \Sigma_c|J^P=1/2^-,1\rangle_{\rho}\pi $ channel, and the predicted branching ratio is approximately ~12%. However, $ \Lambda_{c1}|J^P= 3/2^-,2\rangle_{\rho} $ might be too broad to be observed experimentally.

The state $ \Lambda_{c1}|J^P=5/2^-,2\rangle_{\rho} $ may be a narrow state with a total decay width around $ \Gamma\simeq52 $ MeV, and it mainly decays into the $ \Sigma_c^*\pi $ channel. The predicted branching fraction is

Thus, this state is most likely to be observed in the $ \Lambda_c\pi\pi $ final state via the decay chain $ \Lambda_{c1}|J^P= 5/2^-, 2\rangle_{\rho}\rightarrow \Sigma_c^*\pi\rightarrow \Lambda_c\pi\pi $. In addition, the partial widths of $ \Sigma_c\pi $ and $ \Sigma_c|J^P=3/2^-,1\rangle_{\lambda/\rho}\pi $ are sizable as well, and all of the branching fractions are approximately 13%. Hence, the $ \Lambda_c\pi\pi $ and $ \Lambda_c\pi\pi\pi $ final states via the decay chains $ \Lambda_{c1}|J^P=5/2^-,2\rangle_{\rho}\rightarrow \Sigma_c\pi\rightarrow \Lambda_c\pi\pi $ and $ \Lambda_{c1}|J^P=5/2^-, 2\rangle_{\rho}\rightarrow \Sigma_c|J^P=3/2^-, 1\rangle_{\lambda/\rho}\pi \rightarrow \Sigma_c^*\pi\pi\rightarrow \Lambda_c\pi\pi\pi $ may also be interesting decay channels for experimental observations.

Similarly, we also plot the partial decay widths of the 2P-wave ρ-mode $ \Lambda_c $baryons as a function of mass in the region of $ M=(3.10-3.30) $ GeV. The sensitivities of the decay properties of these states to their masses are shown in Fig. 6. From the figure, the partial widths of dominant decay channels for most of the states vary slightly with increasing mass. In addition, if the 2P-wave ρ-mode $ \Lambda_c $ baryons are above the threshold of $ \Sigma_c\rho $, most of the states can decay via $ \Sigma_c\rho $ with a partial width of several MeV or more.

Figure 6.  (color online) Partial and total strong decay widths of the ρ-mode 2P-wave $ \Lambda_c $ states as functions of mass. Some decay channels are too small to show in the figure.

In the present study, we systematically investigated the strong decay properties of the low-lying 1P-, 1D-, 2S-, and 2P-wave $ \Lambda_c $ baryons in the framework of the quark pair creation model within the $ j-j $coupling scheme. Our main results are summarized as follows.

The two 1P-wave λ-mode $ \Lambda_c $ baryons $ \Lambda_c|J^P= 1/2^-,1\rangle_{\lambda} $ and $ \Lambda_c|J^P=3/2^-,1\rangle_{\lambda} $ correspond to the well determined states $ \Lambda_c(2595)^+ $ and $ \Lambda_c(2625)^+ $, respectively, and we can reproduce the experimental data well in theory. Meanwhile, we notice that the mass of $ \Lambda_c(2595)^+ $ is very close to the threshold of $ \Sigma_c\pi $, and this causes the decay widths to be highly sensitive to its mass precision.

For the 1P-wave ρ-mode $ \Lambda_c $ baryons, $ \Lambda_c|J^P= 1/2^-,0\rangle_{\rho} $ is most likely to be a very narrow state, and the $ \Sigma_c\pi $ decay mode almost saturates its total decay widths. Hence, the $ \Lambda_c\pi\pi $ final state may be an ideal decay channel to explore $ \Lambda_c|J^P=1/2^-,0\rangle_{\rho} $ in future experiments. The states $ \Lambda_c|J^P=1/2^-,1\rangle_{\rho} $ and $ \Lambda_c|J^P=3/2^-,1\rangle_{\rho} $ are probably relatively broad states with a width of approximately $ \Gamma_{\rm{Total}}\sim780 $ MeV. Their dominant decay channels are $ \Sigma_c\pi $ and $ \Sigma_c^*\pi $, respectively. Considering that the decay widths are too broad, the two states may be difficult to observe experimentally. The total decay widths of $ \Lambda_c|J^P= 3/2^-,2\rangle_{\rho} $ and $ \Lambda_c|J^P= 5/2^-,2\rangle_{\rho} $ are several tens of MeV, and their strong decays are governed by the $ \Sigma_c\pi $ and $ \Sigma_c^*\pi $ channels. Combining the mass and decay properties of the newly observed state $ \Lambda_c(2910)^+ $, both $ \Lambda_c|J^P= 3/2^-,2\rangle_{\rho} $ and $ \Lambda_c|J^P= 5/2^-,2\rangle_{\rho} $ may be good candidates. To further determine which of the two is the most suitable one, the partial decay width ratio between $ \Sigma_c\pi $ and $ \Sigma_c^*\pi $ may be a good criterion.

As for the 1D-wave λ-mode $ \Lambda_c $ excitations, $ \Lambda_c|J^P= 3/2^+,2\rangle_{\lambda\lambda} $ is probably a good assignment of the observed state $ \Lambda_c(2860)^+ $. In addition, if the observed state $ \Lambda_c(2860)^+ $ corresponds to $ \Sigma_c|J^P=3/2^+,2\rangle_{\lambda\lambda} $, besides the $ pD^0 $ channel, $ \Sigma_c\pi $ and $ nD^+ $ may be two interesting channels for future experimental observation. The state $ \Lambda_c|J^P=5/2^+,2\rangle_{\lambda\lambda} $ may be a moderate state with a width of approximately $ (20-30) $ MeV and mainly decays via the $ \Sigma_c^*\pi $ channel. According to our investigation, if we take the observed state $ \Lambda_c(2880)^+ $ as $ \Lambda_c|J^P=5/2^+,2\rangle_{\lambda\lambda} $, the measured total decay widths and some partial decay width ratios cannot be reproduced. To further clarify the inner structure of the $ \Lambda_c(2880)^+ $ resonance, more experimental and theoretical efforts are needed.

For the 1D-wave ρ-mode $ \Lambda_c $ excitations, $ \Lambda_c|J^P= 3/2^+,2\rangle_{\rho\rho} $ may be a moderate state with a width of $ \Gamma_{\rm{Total}}\sim $ 120 MeV, and it mainly decays into $ \Sigma_c\pi $. Meanwhile, the partial decay width of $ \Sigma_c^*\pi $ is sizable. Hence, $ \Lambda_c|J^P= 3/2^+,2\rangle_{\rho\rho} $ has the possibility to be observed in the $ \Lambda_c\pi\pi $ final state via the decay chains $ \Sigma_c|J^P= 3/2^+,2\rangle_{\rho\rho}\rightarrow \Sigma_c^{(*)}\pi\rightarrow \Lambda_c\pi\pi $. The other 1D-wave ρ-mode state $ \Lambda_c|J^P= 5/2^+,2\rangle_{\rho\rho} $ has a width of $ \Gamma_{\rm{Total}}\sim196 $ MeV and dominantly decays into $ \Sigma_c^*\pi $. Moreover, the decay rates into $ \Sigma_c\pi $ and $ \Lambda_c\omega $ are considerable. However, this state may be too broad to be observed experimentally.

The 2S-wave state $ \Lambda_{c1}|J^P=1/2^+,0\rangle_{\lambda} $ probably has a width of dozens of MeV and mainly decays into the $ \Sigma_c\pi $ and $ \Sigma_c^*\pi $ channels. Combining the predicted mass and our calculations, the possibility of $ \Lambda_{c1}|J^P=1/2^+,0\rangle_{\lambda} $ being an assignment of the observed state $ \Lambda_c(2765)^+ $ cannot be ruled out entirely. For the other 2S-wave state $ \Lambda_{c1}|J^P= 1/2^+,0\rangle_{\rho} $, the main decay channels are $ \Sigma_c\pi $ and $ \Sigma_c^*\pi $. However, this state is most likely to be a broad state with a width of approximately $ \Gamma_{\rm{Total}}\sim245 $MeV. Thus, it is highly challenging to observe the $ \Lambda_{c1}|J^P=1/2^+,0\rangle_{\rho} $ state in experiments for its broad decay width.

The total decay widths of the two 2P-wave λ-mode $ \Lambda_c $ states $ \Lambda_{c1}|J^P=1/2^-,1\rangle_{\lambda} $ and $ \Lambda_{c1}|J^P=3/2^-,1\rangle_{\lambda} $ are dozens of MeV, and the $ ND $ decay channel almost saturates their decay widths. Comparing the masses and total decay widths, $ \Lambda_{c1}|J^P=1/2^-,1\rangle_{\lambda} $ and $ \Lambda_{c1}|J^P= 3/2^-,1\rangle_{\lambda} $ are allowed as good assignments of $ \Lambda_c(2940)^+ $. Meanwhile, the partial decay width ratio between $ pD^0 $ and $ \Sigma_c\pi $ for $ \Lambda_{c1}|J^P=3/2^-,1\rangle_{\lambda} $ is predicted to be $ 4.48 $, which is close to the upper limit of the newest measured value by the Belle Collaboration. In this case, $ \Lambda_{c1}|J^P=3/2^-,1\rangle_{\lambda} $ is more favorable.

For the 2P-wave ρ-mode $ \Lambda_c $ states, $ \Lambda_{c1}|J^P=1/2^-,0\rangle_{\rho} $ probably has a width of $ \Gamma_{\rm{Total}}\sim175 $ MeV and dominantly decays into $ \Sigma_c|J^P=1/2^-,1\rangle_{\rho}\pi $ and $ \Sigma_c|J^P=3/2^-,1\rangle_{\rho}\pi $. The states $ \Lambda_{c1}|J^P=1/2^-,1\rangle_{\rho} $ and $ \Lambda_{c1}|J^P=3/2^-,1\rangle_{\rho} $ are most likely to be moderate states with a width of approximately $ \Gamma_{\rm{Total}}\sim70 $ MeV. Besides the main decay channels $ \Sigma_c\pi $ and $ \Sigma_c^*\pi $, the partial decay widths of the final states containing a P-wave baryon are considerable. The state $ \Lambda_{c1}|J^P=3/2^-,2\rangle_{\rho} $ may be a broad state with a width of $ \Gamma_{\rm{Total}}\sim293 $ MeV, and it mainly decays into $ \Sigma_c^*\pi $ and $ \Sigma_c\pi $. Meanwhile, if its mass lies above the threshold of $ \Sigma_c\rho $, the strong decays are governed by the $ \Sigma_c\rho $ channel. As for $ \Lambda_{c1}|J^P=5/2^-,2\rangle_{\rho} $, it may be a relatively narrow state with a total width of $ \Gamma_{\rm{Total}}\sim52 $ MeV, and it mainly decays into $ \Sigma_c^*\pi $. Thus, this state is most likely to be observed in the $ \Lambda_c\pi\pi $ final state via the decay chain $ \Lambda_{c1}|J^P=5/2^-,2\rangle_{\rho}\rightarrow \Sigma_c^*\pi\rightarrow \Lambda_c\pi\pi $.