Possible assignments of highly excited $\varLambda _c(2860)^+$, $\varLambda _c(2880)^+$ and $\varLambda _c(2940)^+$

Abstract

Possible assignments of highly excited $\varLambda _c(2860)^+$, $\varLambda _c(2880)^+$ and $\varLambda _c(2940)^+$ are explored in a $^3P_0$ strong decay model. Decay widths, branching fraction ratios $R={\varGamma (\varSigma _c(2520)\pi )\over \varGamma (\varSigma _c(2455)\pi )}$ and the branching fractions of DN channels of theses assignments are computed. $D^0p$ channel is a very important channel to provide information on the inner excitation and structure of these highly excited $\varLambda _c$. In our analysis, $\varLambda _c(2860)^+$ may be a 1D-wave excited $\varLambda _c$ with $J^P={3\over 2}^+$, which has dominant DN decay channels with a branching fraction ${\mathcal {B}}(\varLambda _c(2860)^+\rightarrow DN)=75\%$ and a branching ratio $R={\varGamma (\varSigma _c(2520)\pi )\over \varGamma (\varSigma _c(2455)\pi )}=0.12$. $\varLambda _c(2880)^+$ is very possibly a 1F-wave excited $\varLambda _c$ with $J^P=\frac{5}{2}^-$; In this assignment, the predicted total decay width ($\varGamma \approx 4.49$ MeV) is comparable to the measured $\varGamma =5.6^{+0.8}_{-0.6}$ MeV, and the predicted $R={\varGamma (\varSigma _c(2520)\pi )\over \varGamma (\varSigma _c(2455)\pi )}=0.12$ is consistent with the measured $R=0.225\pm 0.062\pm 0.025$; The DN channels are its dominant strong decay channels with a branching fraction ${\mathcal {B}}(\varLambda _c(2880)^+\rightarrow DN)=94\%$. $\varLambda _c(2880)^+$ seems impossibly a 1D-wave excited $\varLambda _c$ with $J^P=\frac{5}{2}^+$ once the presently measured $R={\varGamma (\varSigma _c(2520)\pi )\over \varGamma (\varSigma _c(2455)\pi )}$ is confirmed. $\varLambda _c(2940)^+$ may be a 2P-wave excited $\varLambda ^{1,0}_{c1,1}(\frac{3}{2}^-,2P)$. In this case, $\varLambda _c(2940)^+$ has a total decay width $\varGamma =17.56$ MeV, a branching ratio $R={\varGamma (\varSigma _c(2520)\pi )\over \varGamma (\varSigma _c(2455)\pi )}=0.89$ and the DN decay channels with a branching fraction ${\mathcal {B}}(\varLambda _c(2940)^+\rightarrow DN)=43\%$. In order to understand the inner excitation and structure of these highly excited $\varLambda _c$, measurements of those predicted quantities are required in the future.

1 Introduction

Charmed baryons with single charmed quark provide ideal windows to study the baryon structure and quark dynamics. The heavy-quark symmetry works approximately in singly-charmed baryons, and the quarks inside may correlate and exhibit their structure through their strong decays. There are now 36 established singly-charmed baryons [1], but their $J^P$ numbers have seldom been measured in experiments.

$\varLambda _c$ baryons are composed of one u quark, one d quark and one charmed quark. $\varLambda _c(2286)^+$ is believed the ground state with $J^P={1\over 2}^+$, $\varLambda _c(2593)^+$ and $\varLambda _c(2625)^+$ are believed the P-wave excited states with $J^P={1\over 2}^-$ and $J^P={3\over 2}^-$, respectively. $\varLambda _c(2765)^+$ or $\varSigma _c(2765)^+$ [2] was seen in $\varLambda ^+_c\pi ^+\pi ^-$ with mass difference $m(\varLambda _c(2765)^+)-m(\varLambda _c)=480.1\pm 2.4$ MeV, but nothing is known about its $J^P$ and isospin quantum numbers as indicated in PDG2020 (This state was reported as a $\varLambda _c$ with zero isospin in HADRON 2019 [3]).

$\varLambda _c(2880)^+$ was first observed by CLEO collaboration [2], and its quantum numbers have been constrained by Belle and LHCb collaborations [4, 5] with $J^P={5\over 2}^+$. The spin hypothesis $J={5\over 2}$ is favored from an angular analysis in the experiment [4] and the positive parity is assumed through the predictions of the heavy quark symmetry [4, 6,7,8]. $\varLambda _c(2940)$ was first observed by BaBar collaboration [9] and $\varLambda _c(2860)^+$ was first observed by LHCb collaboration [5], the quantum numbers $J^P$ of $\varLambda _c(2860)^+$ and $\varLambda _c$ $(2940)^+$ have not been measured. The masses, decay modes and decay widths of $\varLambda _c(2860)^+$, $\varLambda _c(2880)^+$ and $\varLambda _c(2940)^+$ were reported in these experiments.

In normal baryon interpretations, the quantum numbers and possible internal excitation of $\varLambda _c(2860)^+$, $\varLambda _c(2880)^+$ and $\varLambda _c(2940)^+$ have been studied in many models. The measured mass of $\varLambda _c(2860)^+$ is consistent with the theoretical predictions of the orbital 1D-wave $\varLambda _c$ excitation with quantum numbers ${3\over 2}^+$ [10, 11], so $\varLambda _c(2860)^+$ is supposed with the quantum numbers $J^P={3\over 2}^+$. $\varLambda _c(2880)^+$ was supposed with quantum numbers $J^P=\frac{5}{2}^+$ in a framework of heavy hadron chiral perturbation theory [7, 8], a constituent quark model [12, 13], a relativistic flux tube model [10], and a $^3P_0$ strong decay model [14, 15], etc. $\varLambda _c(2880)^+$ was also assumed with quantum numbers $J^P=\frac{3}{2}^+$ in a chiral quark model [16]. The assignments of $\varLambda _c(2940)^+$ is much more contradictory. $\varLambda _c(2940)^+$ was assumed with quantum numbers $J^P=\frac{1}{2}^-$, $J^P=\frac{3}{2}^+$, $J^P=\frac{3}{2}^-$, $J^P=\frac{5}{2}^-$ or $J^P=\frac{5}{2}^+$ in different models [7, 8, 12,13,14,15,16]. Obviously, the quantum numbers of $\varLambda _c(2860)^+$, $\varLambda _c(2880)^+$ and $\varLambda _c(2940)^+$ have not been fixed, their internal structures are not clear either.

The quarks in baryons may make complex structures and have complex excitations. In order to study the internal structures and excitations, the three-quark baryons are usually described by Jacobi coordinates: a relative coordinate $\rho$ between any two quarks, and a relative coordinate $\lambda$ between the center of mass of the two quarks and the other quark. As known, a diquark may be an important correlation and cluster in hadrons with more than two quarks, and the diquark has been introduced to interpret the light scalar mesons, the missing nucleons, the charmonium-like X,  Y,  Z, and so on. The diquark has also been employed to describe singly-charmed baryons in many models [10, 12, 13, 17,18,19,20,21]. However, there is no evidence for the existence of diquark in baryons. In this paper, the strong decay properties of $\varLambda _c$ baryons with different $\rho$ or $\lambda$ mode excitations will be studied, and the relation between the excitations and the diquark correlation is explored.

In Ref. [15], all the observed $\varLambda _c$ states except for the ground $\varLambda _c(2286)^+$ were systematically examined as the 1P-wave, 1D-wave, or 2S-wave $\varLambda _c$ baryons from their strong decay properties in the $^3P_0$ model, and their possible assignments were suggested. In this paper, we continue the examination of $\varLambda _c(2860)^+$, $\varLambda _c(2880)^+$ and $\varLambda _c(2940)^+$ with the highly excited 1D, 1F and 2P orbital or radial excitations assignments in detail.

The paper is organized as follows. A simple introduction of $^3P_0$ strong decay model and analyses of the strong decay properties of $\varLambda _c(2860)^+$, $\varLambda _c(2880)^+$ and $\varLambda _c(2940)^+$ are given in Sect. 2. Conclusions and discussions are reserved in Sect. 3.

2 1D-wave, 1F-wave and 2P-wave possibilities of $\varLambda _c(2860)^+$, $\varLambda _c(2880)^+$ and $\varLambda _c(2940)^+$

In order to fix the quantum numbers and to understand the internal structure of $\varLambda _c(2860)^+$, $\varLambda _c(2880)^+$ and $\varLambda _c(2940)^+$, the $^3P_0$ strong decay model is employed. As well known, the $^3P_0$ model is usually known as the quark pair creation model. It was proposed by Micu [22] and developed by Le Yaouanc et al. [23,24,25,26,27,28]. This model has been employed to compute the Okubo-Zweig-Iizuka-allowed (OZI) strong decays widths with two final states and obtained good agreements with experiments.

Following Refs. [15, 29,30,31,32,33], the strong decay width for an initial baryon A decaying into two final hadrons B and C in the $^3P_0$ model is

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

(1)

where ${{\mathcal {M}}^{M_{J_A}M_{J_B}M_{J_C}}}$ is the helicity amplitude. The explicit expression of the helicity amplitude, the flavor matrix, the space integral and some relevant notations could be found in detail in Ref. [15].

As indicated in Ref. [15], $\rho$ is the relative coordinate between the two light quarks (quarks 1 and 2), and $\lambda$ is the relative coordinate between the center of mass of the two light quarks and the charmed quark. In a constituent quark model, the internal structure of a baryon is also described by a set of quantum numbers $n_\rho$, $n_\lambda$, $L_\rho$, $L_\lambda$ and $S_\rho$. $n_\rho$ and $n_\lambda$ denote the nodal quantum numbers of the $\rho$ and $\lambda$ coordinates, respectively. $L_\rho$ and $L_\lambda$ denote the orbital angular momentum between the two light quarks and the orbital angular momentum between the charm quark and the two-light-quark system. $S_\rho$ denotes the total spin of the two light quarks. The total orbital angular momentum $L=L_\rho +L_\lambda$ and the total angular momentum of the baryons $J=J_l+{1\over 2}$ with $J_l=L+S_\rho$.

Therefore, in the constituent quark model with the heavy-quark symmetry [15, 34], there are one 1S-wave, seven 1P-wave, seventeen 1D-wave, and thirty-one 1F-wave $\varLambda _c$ baryons. For the first radial excitations, the corresponding states doubled. That is to say, there are two 2S-wave, fourteen 2P-wave, thirty-four 2D-wave, and sixty-two 2F-wave $\varLambda _c$ baryons. Internal quantum numbers of the 1D-wave excited $\varLambda _c$ were given in Ref. [15], quantum numbers of the 1F-wave and 2P-wave excited $\varLambda _c$ are given in the appendix.

Table 1 Possible $\varLambda _c$ decaying into DN final states

Some parameters are chosen as those in Refs. [15, 32, 35]. The dimensionless pair-creation $\gamma =13.4$. The $\beta _{\lambda ,\rho }=600$ MeV in 1S-wave baryon wave function, $\beta =400$ MeV in the wave function of $\pi$ and K mesons, $\beta =600$ MeV for the D meson [15, 35]. $\beta _{\rho ,\lambda }=400$ MeV is for the excited $\varLambda _c(2860)^+$, $\varLambda _c(2880)^+$ and $\varLambda _c(2940)^+$. The masses of relevant hadrons are chosen from Particle Data Group [1].

$D^0p$ mode is an important channel for $\varLambda _c$ baryons in the $^3P_0$ model. In this channel, the heavy charmed quark in initial baryon enters the final D meson and other two light quarks enter the final p baryon. Therefore, this channel may provide some information on the inner excitation and structure of $\varLambda _c$.

In theory, the helicity amplitudes of many high-lying $\varLambda _c$ decaying into $D^0p$ channel vanish. Therefore, many possible assignments of $\varLambda _c(2860)^+$, $\varLambda _c(2880)^+$ and $\varLambda _c(2940)^+$ can be excluded through the observed $D^0p$ final states. So does the $D^+n$ channel. Possible high-lying $\varLambda _c$ which can decay into $D^0p$ channel are given in Table 1. In these $\varLambda _c$ excitations, $\varLambda ^{1,0}_{c1,1}(\frac{1}{2}^-,2P)$ and $\varLambda ^{1,0}_{c1,1}(\frac{3}{2}^-,2P)$ have radial $\rho$ mode excitation, while others have only $\lambda$ mode excitation.

2.1 1D-wave excitations

Among the seventeen 1D-wave $\varLambda _c$ states, there are only two $\lambda$ mode excited states $\varLambda ^{2,0}_{c2,0}(\frac{3}{2}^+,1D)$ and $\varLambda ^{2,0}_{c2,0}(\frac{5}{2}^+,1D)$ with $D^0p$ decay channel. The masses of $\varLambda _c(2860)^+$ and $\varLambda _c(2880)^+$ are comparable to the predicted spectrum of D-wave excited $\varLambda _c$ [10, 11], and they could be the 1D-wave excitations. These two 1D-wave excitations are examined through their strong decay properties in this section.

In the framework of $^3P_0$ model, the decay widths of possible assignments of $\varLambda _c(2860)^+$, $\varLambda _c(2880)^+$ and $\varLambda _c(2940)^+$ are computed. The numerical results of $\varLambda _c(2860)^+$, $\varLambda _c(2880)^+$ and $\varLambda _c(2940)^+$ are presented in Tables 2,  3 and  4, respectively. Their total strong decay widths are also given. The results in these three tables are similar except for a new $\varSigma _c(2800)\pi$ channel for $\varLambda _c(2940)^+$. In our computation, $\varSigma _c(2800)$ is regarded as a 1P-wave $\varSigma _c$ with $J^P={3\over 2}^-$ [36].

Table 2 Possible decay widths (MeV), branching fraction of DN channels and $R={\varGamma (\varSigma _c(2520)\pi )\over \varGamma (\varSigma _c(2455)\pi )}$ of $\varLambda _c(2860)^+$ as $\varLambda ^{2,0}_{c2,0}(\frac{3}{2}^+,1D)$ and $\varLambda ^{2,0}_{c2,0}(\frac{5}{2}^+,1D)$

Table 3 Possible decay widths (MeV), branching fraction of DN channels and $R={\varGamma (\varSigma _c(2520)\pi )\over \varGamma (\varSigma _c(2455)\pi )}$ of $\varLambda _c(2880)^+$ as $\varLambda ^{2,0}_{c2,0}(\frac{3}{2}^+,1D)$ and $\varLambda ^{2,0}_{c2,0}(\frac{5}{2}^+,1D)$

Table 4 Possible decay widths (MeV), branching fraction of DN channels and $R={\varGamma (\varSigma _c(2520)\pi )\over \varGamma (\varSigma _c(2455)\pi )}$ of $\varLambda _c(2940)^+$ as $\varLambda ^{2,0}_{c2,0}(\frac{3}{2}^+,1D)$ and $\varLambda ^{2,0}_{c2,0}(\frac{5}{2}^+,1D)$

In each table, the main differences for the assignments of $\varLambda ^{2,0}_{c2,0}(\frac{3}{2}^+,1D)$ and $\varLambda ^{2,0}_{c2,0}(\frac{5}{2}^+,1D)$ are the total decay widths, branching fraction of DN channels and the branching fraction ratios $R={\varGamma (\varSigma _c(2520)\pi )\over \varGamma (\varSigma _c(2455)\pi )}$. $\varLambda _c(2860)^+$ or $\varLambda _c(2880)^+$ in the $\varLambda ^{2,0}_{c2,0}(\frac{3}{2}^+,1D)$ assignment has a larger total width, a smaller R and a dominant DN channel in comparison with the $\varLambda ^{2,0}_{c2,0}(\frac{5}{2}^+,1D)$ assignment.

In experiments, $\varLambda _c(2880)^+$ has a small total decay width ($\varGamma =5.6^{+0.8}_{-0.6}$ MeV) and a branching ratio $R=0.225\pm 0.062\pm 0.025$ [1], which was doubted by an influence from its nearby state $\varLambda _c(2860)^+$ in Ref. [37]. $\varLambda _c(2860)^+$ and $\varLambda _c(2940)^+$ have total decay widths $\varGamma =67.6^{+10.1}_{-8.1}\pm 1.4^{+5.9}_{-20.0}$ MeV and $\varGamma =20^{+6}_{-5}$ MeV, respectively, but no branching fraction has been measured.

From the total strong decay width, $\varLambda _c(2860)^+$ is very possibly the 1D-wave excitation $\varLambda ^{2,0}_{c2,0}(\frac{3}{2}^+,1D)$ while impossibly the 1D-wave excitation $\varLambda ^{2,0}_{c2,0}(\frac{5}{2}^+,1D)$. In this assignment, DN are the main two body strong decay channels with branching fraction ${\mathcal {B}}(\varLambda _c(2860)^+\rightarrow DN)=75\%$, and the ratio $R={\varGamma (\varSigma _c(2520)\pi )\over \varGamma (\varSigma _c(2455)\pi )}=0.12$. Their measurement in the future will provide more information on $\varLambda _c(2860)^+$.

From the total strong decay width, $\varLambda _c(2880)^+$ is impossible the the 1D-wave excitation $\varLambda ^{2,0}_{c2,0}(\frac{3}{2}^+,1D)$, but may be a 1D-wave excitation $\varLambda ^{2,0}_{c2,0}(\frac{5}{2}^+,1D)$. In the $\varLambda ^{2,0}_{c2,0}(\frac{5}{2}^+,1D)$ assignment, $\varSigma _c(2520)\pi$ are the dominant two body strong decay channels with branching fraction ${\mathcal {B}}(\varLambda _c(2880)^+\rightarrow \varSigma _c(2520)\pi )=94\%$. DN channels have branching fraction ${\mathcal {B}}(\varLambda _c(2880)^+\rightarrow DN)=3\%$. However, the predicted branching ratio $R={\varGamma (\varSigma _c(2520)\pi )\over \varGamma (\varSigma _c(2455)\pi )}=44.29$ is much larger than the observed $R=0.225\pm 0.062\pm 0.025$. Similarly large R was predicted in Refs. [37, 38]. Even though the theoretical uncertainties in the $^3P_0$ model have been taken into account, it is difficult to assign the $\varLambda _c(2880)^+$ with the 1D-wave excitation $\varLambda ^{2,0}_{c2,0}(\frac{5}{2}^+,1D)$ through its strong decay properties.

Through the strong decay widths only, $\varLambda _c(2940)^+$ could be the 1D-wave excitation $\varLambda ^{2,0}_{c2,0}(\frac{5}{2}^+,1D)$ and is impossibly the 1D-wave excitation $\varLambda ^{2,0}_{c2,0}(\frac{3}{2}^+,1D)$. Taking into account the fact that $\varLambda _c(2940)^+$ has a higher mass than the predicted 1D-wave excited $\varLambda _c$, it can not be the 1D-wave excitation.

2.2 1F-wave excitations

Among the thirty-one 1F-wave $\varLambda _c$ states, there are also only two $\lambda$ mode excited states $\varLambda ^{3,0}_{c3,0}(\frac{5}{2}^-,1F)$ and $\varLambda ^{3,0}_{c3,0}(\frac{7}{2}^-,1F)$ with $D^0p$ decay channel. As a 1F-wave excitation candidate, $\varLambda _c(2940)^+$ has numerical results similar to $\varLambda _c(2860)^+$ and $\varLambda _c(2880)^+$ except for the $\varSigma _c(2800)\pi$ channel. The decay widths of $\varLambda _c(2860)^+$, $\varLambda _c(2880)^+$ and $\varLambda _c(2940)^+$ are computed. To avoid tedious duplicate tables, only the results of $\varLambda _c(2880)^+$ are presented in Table 5.

Table 5 Possible decay widths (MeV), branching fraction of DN channels and $R={\varGamma (\varSigma _c(2520)\pi )\over \varGamma (\varSigma _c(2455)\pi )}$ of $\varLambda _c(2880)^+$ as $\varLambda ^{3,0}_{c3,0}(\frac{5}{2}^-,1F)$ and $\varLambda ^{3,0}_{c3,0}(\frac{7}{2}^-,1F)$

Obviously, as a $\varLambda ^{3,0}_{c3,0}(\frac{5}{2}^-,1F)$, the predicted total decay widths $\varGamma =4.49$ MeV and the branching ratio $R={\varGamma (\varSigma _c(2520)\pi )\over \varGamma (\varSigma _c(2455)\pi )}=0.12$ of $\varLambda _c(2880)^+$ are consistent with the observed total decay width $\varGamma =5.6^{+0.8}_{-0.6}$ MeV and branching ratio $R=0.225\pm 0.062\pm 0.025$. $\varLambda _c(2880)^+$ is very possibly the 1F-wave excitation $\varLambda ^{3,0}_{c3,0}(\frac{5}{2}^-,1F)$. In this assignment, the channels DN are the dominant two-body decay channels with branching fraction ${\mathcal {B}}(\varLambda _c(2880)^+\rightarrow DN)=94\%$.

$\varLambda _c(2860)^+$ seems impossibly the $\varLambda ^{3,0}_{c3,0}(\frac{5}{2}^-,1F)$ for its much larger decay width in comparison to the predicted one. Since no branching ratio has been measured, it is difficult to assign $\varLambda _c(2940)^+$ with the $\varLambda ^{3,0}_{c3,0}(\frac{5}{2}^-,1F)$ only from its total decay width.

Table 6 Possible decay widths (MeV), branching fraction of DN channels and $R={\varGamma (\varSigma _c(2520)\pi )\over \varGamma (\varSigma _c(2455)\pi )}$ of $\varLambda _c(2940)^+$ as four 2P-wave excitations

As a $\varLambda ^{3,0}_{c3,0}(\frac{7}{2}^-,1F)$, the predicted total decay widths of $\varLambda _c(2860)^+$, $\varLambda _c(2880)^+$ and $\varLambda _c(2940)^+$ are much smaller than the observed ones, so $\varLambda _c(2860)^+$, $\varLambda _c(2880)^+$ and $\varLambda _c(2940)^+$ are impossibly the 1F-wave excitation $\varLambda ^{3,0}_{c3,0}(\frac{7}{2}^-,1F)$.

2.3 2P-wave excitations

There are fourteen 2P-wave excited $\varLambda _c$, among which there are four excitations with $D^0p$ decay channels. As indicated in Table 1, these excitations are $\varLambda ^{1,0}_{c1,0}(\frac{1}{2}^-,2P)$, $\varLambda ^{1,0}_{c1,0}(\frac{3}{2}^-,2P)$, $\varLambda ^{1,0}_{c1,1}(\frac{1}{2}^-,2P)$ and $\varLambda ^{1,0}_{c1,1}(\frac{3}{2}^-,2P)$. From the appendix, $\varLambda ^{1,0}_{c1,0}$ $(\frac{1}{2}^-,2P)$ and $\varLambda ^{1,0}_{c1,0}(\frac{3}{2}^-,2P)$ are $\lambda$ mode radial excitations, and $\varLambda ^{1,0}_{c1,1}(\frac{1}{2}^-,2P)$ and $\varLambda ^{1,0}_{c1,1}(\frac{3}{2}^-,2P)$ are $\rho$ mode radial excitations.

The decay widths of possible assignments of $\varLambda _c(2940)^+$ are computed when it is regarded as $\varLambda ^{1,0}_{c1,0}(\frac{1}{2}^-,2P)$, $\varLambda ^{1,0}_{c1,0}(\frac{3}{2}^-,2P)$, $\varLambda ^{1,0}_{c1,1}(\frac{1}{2}^-,2P)$ or $\varLambda ^{1,0}_{c1,1}(\frac{3}{2}^-,2P)$. The numerical results are given in Table 6. $\varLambda _c(2860)^+$ and $\varLambda _c(2880)^+$ have similar numerical results which have not been presented explicitly.

Once the strong decay widths are taken into account only, $\varLambda _c(2860)^+$ could be the $\varLambda ^{1,0}_{c1,1}(\frac{1}{2}^-,2P)$ under large uncertainty and can not be any other 2P-wave excited $\varLambda _c$. Taking into account the fact that $\varLambda _c(2860)^+$ has a lower mass in comparison to theoretical prediction of 2P-wave, $\varLambda _c(2860)^+$ can not be a 2P-wave excited $\varLambda _c$.

When the predicted total decay widths are compared with the observed ones of $\varLambda _c(2880)^+$ and $\varLambda _c(2940)^+$, $\varLambda _c(2880)^+$ is impossibly a 2P-wave excitation for its small total decay width, and $\varLambda _c(2940)^+$ is possibly the $\varLambda ^{1,0}_{c1,1}(\frac{3}{2}^-,2P)$. As a $\varLambda ^{1,0}_{c1,1}(\frac{3}{2}^-,2P)$ excitation, the total decay width $\varGamma =17.56$ MeV, the branching ratio $R={\varGamma (\varSigma _c(2520)\pi )\over \varGamma (\varSigma _c(2455)\pi )}=0.89$ and the branching fraction ${\mathcal {B}}(\varLambda _c(2940)^+\rightarrow DN)=43\%$ are predicted for $\varLambda _c(2940)^+$.

In comparison with the D-wave and F-wave excited $\varLambda _c$ with $\lambda$ mode excitation only, the 2P-wave excited $\varLambda _c$ with $\rho$ mode excitation has a much lower branching fraction of DN channels.

3 Conclusions and discussions

The 1D-wave, 1F-wave and 2P-wave assignments of the high-lying $\varLambda _c(2860)^+$, $\varLambda _c(2880)^+$ and $\varLambda _c(2940)^+$ are examined from their strong decay properties in the $^3P_0$ model.

Table 7 2P-wave excited $\varLambda _c$

Table 8 1F-wave excited $\varLambda _c$

Based on experimental results, some possible or favored assignments of these excited $\varLambda _c$ are suggested to them, and some impossible assignments are pointed out.

$\varLambda _c(2860)^+$ may be the 1D-wave excited $\varLambda ^{2,0}_{c2,0}(\frac{3}{2}^+,1D)$, it is impossibly the 1D-wave excited $\varLambda ^{2,0}_{c2,0}(\frac{5}{2}^+,1D)$, 1F-wave excitation or 2P-wave excited $\varLambda _c$. The $D^0p$ mode is the dominant decay channel with branching fraction ${\mathcal {B}}(\varLambda _c(2860)^+\rightarrow DN)=75\%$, and the branching ratio $R={\varGamma (\varSigma _c(2520)\pi )\over \varGamma (\varSigma _c(2455)\pi )}=0.12$. In experiment, only the $D^0p$ channel has been observed, the observation of other decay channels such as $\varSigma _c\pi$ or $\varSigma _c(2520)\pi$ and measurement of their branching fractions are required to the understand its inner excitation and structure.

As reported in Ref. [4], an analysis of angular distribution in $\varLambda _c(2880)^+\rightarrow \varSigma _c(2455)^{0,++}\pi ^{+,-}$ strongly favors the $\varLambda _c(2880)^+$ with spin ${5\over 2}$. In their analysis, the measured $R={\varGamma (\varSigma _c(2520)\pi )\over \varGamma (\varSigma _c(2455)\pi )}=0.225\pm 0.062\pm 0.025$ is found around the prediction of heavy quark symmetry $R=0.23-0.36$ for the ${5\over 2}^+$ state [6,7,8], so the positive parity was assumed to the $\varLambda _c(2880)^+$ [4]. In our analysis, the $\varLambda _c(2880)^+$ is very possibly the 1F-wave excited $\varLambda ^{3,0}_{c3,0}(\frac{5}{2}^-,1F)$ with negative parity. As a $\varLambda ^{3,0}_{c3,0}(\frac{5}{2}^-,1F)$, our prediction of the total decay width and the branching ratio agrees well with experiments. Furthermore, the dominant decay channel $D^0p$ with branching fraction ${\mathcal {B}}(\varLambda _c(2880)^+\rightarrow DN)=94\%$ is also predicted. $\varLambda _c(2880)^+$ is impossibly the 1D-wave excitation, the 1F-wave excited $\varLambda ^{3,0}_{c3,0}(\frac{7}{2}^-,1F)$ or the 2P-wave excitation. Accordingly, the J, P quantum numbers of $\varLambda _c(2880)^+$ can not be $J^P={3\over 2}^+$. In experiment, the channels $\varSigma _c(2455)\pi$, $\varSigma _c(2520)\pi$, $\varLambda _c\pi \pi$ and $D^0p$ have been observed, the measurement of all the branching fractions of these channels is very important for the understanding of this state.

In [5], the most likely spin-parity assignment for $\varLambda _c(2940)^+$ was suggested with $J^P={3\over 2}^-$. However, other solutions with spins ${1\over 2}$ to ${7\over 2}$ have not been excluded. In our analysis, $\varLambda _c(2940)^+$ could be the 2P-wave excited $\varLambda ^{1,0}_{c1,1}(\frac{3}{2}^-,2P)$. It is impossibly the 1D-wave excited $\varLambda _c$, 1F-wave excited $\varLambda ^{3,0}_{c3,0}(\frac{7}{2}^-,1F)$ or any other 2P-wave excitations. In the $\varLambda ^{1,0}_{c1,1}(\frac{3}{2}^-,2P)$ assignment, $\varLambda _c(2940)^+$ has a total decay width $\varGamma =17.56$ MeV, the branching ratio $R={\varGamma (\varSigma _c(2520)\pi )\over \varGamma (\varSigma _c(2455)\pi )}=0.89$ and the DN channels with branching fraction ${\mathcal {B}}(\varLambda _c(2940)^+\rightarrow DN)=43\%$.

So far, $D^0p$ channel has been observed in all highly excited $\varLambda _c$ above their threshold, which may imply that the two light quarks in initial baryons enters the final baryon in the strong decay process. In the same time, the DN channels are dominant and the two internal light quarks in initial baryons coupling with a total spin $S_\rho =0$ in all possible assignments of $\varLambda _c(2860)^+$, $\varLambda _c(2880)^+$ and $\varLambda _c(2940)^+$, which may imply that the two light quarks in initial $\varLambda _c$ make a good diquark. Furthermore, the 2P-wave excited $\varLambda _c$ with $\rho$ mode excitation has a much lower branching fraction of DN channel in comparison with the 1D-wave and 1F-wave excited $\varLambda _c$ with $\lambda$ mode excitation only. The existence and properties of diquark require more exploration.

In addition to the normal uncertainties, three-body decay cannot be computed in the $^3P_0$ model. In our analyses, the parameters $\beta$ are chosen the same for $\rho$ mode and $\lambda$ mode for simplicity though the parameters $\beta$ (represent the inverse root mean square radius) of $\rho$ and $\lambda$ mode excitation may be different. More highly excited possibilities to these $\varLambda _c$ have not yet analyzed. In order to identify these highly-excited $\varLambda _c$ baryons and to understand their inner structure and dynamics, measurements of the J, P quantum numbers and branching fractions of the main decay channels of these highly excited $\varLambda _c$ are required, more theoretical analyses in different models are also required.

Appendix

In the constituent quark model with the heavy-quark symmetry, internal quantum numbers of 1F-wave and 2P-wave excited $\varLambda _c$ and their name are given in the following two tables (Tables 7 and 8).