Strong Decays of observed $\Lambda_c$ Baryons in the $^3P_0$ Model

The strong decay widths and some important branching ratios of possible Okubo-Zweig-Iizuka(OZI)-allowed strong decay channels of $\Lambda_c(2595)^+$, $\Lambda_c(2625)^+$, $\Lambda_c(2765)^+$ ($\Sigma_c(2765)^+$), $\Lambda_c(2860)^+$, $\Lambda_c(2880)^+$ and $\Lambda_c(2940)^+$ are computed in a $^{3}P_{0}$ model, and possible assignments of these $\Lambda_c$ are given. (1), $\Lambda_c(2595)^+$ and $\Lambda_c(2625)^+$ are possibly the $1P$-wave charmed baryons $\Lambda_{c1}(\frac{1}{2}^-)$ and $\Lambda_{c1}(\frac{3}{2}^-)$, respectively. (2), $\Lambda_c(2765)^+$ ($\Sigma_c(2765)^+$) seems impossibly the $1P$-wave $\Lambda_{c}$, it could be the $2S$-wave or $1D$-wave charmed baryon. So far, the experimental information has not been sufficient for its identification. (3), $\Lambda_c(2860)^+$ seems impossibly $2S$-wave charmed baryon, it may be the $P$-wave $\tilde\Lambda_{c2}^{ }(\frac{3}{2}^-)$ or $\tilde\Lambda_{c2}^{ }(\frac{5}{2}^-)$, it could also be the $D$-wave $\check\Lambda_{c1}^{2}(\frac{1}{2}^+)$ or $\check\Lambda_{c1}^{2}(\frac{3}{2}^+)$. If the hypothesis that $\Lambda_c(2860)^+$ has $J^P={3\over 2}^+$ is true, $\Lambda_c(2860)^+$ is possibly the $D$-wave $\check\Lambda_{c1}^{2}(\frac{3}{2}^+)$ which has a predicted branching ratio $R=\Gamma(\Sigma_c(2520)\pi)/\Gamma(\Sigma_c(2455)\pi)=2.8$. (4), $\Lambda_c(2880)^+$ is impossibly a $1P$-wave or $2S$-wave charmed baryon, it may be a $D$-wave $\check\Lambda_{c3}^{2}(\frac{5}{2}^+)$ with $\Gamma_{total}=1.3$ MeV. The predicted branching ratio $R=\Gamma(\Sigma_c(2520)\pi)/\Gamma(\Sigma_c(2455)\pi)=0.35$, which is consistent with experiment. (5), $\Lambda_c(2940)^+$ is the $P$-wave $\tilde\Lambda_{c2}^{ }(\frac{3}{2}^-)$ or $\tilde\Lambda_{c2}^{ }(\frac{5}{2}^-)$, it is also possibly the $D$-wave $\check\Lambda_{c3}^{2}(\frac{5}{2}^+)$ or $\check\Lambda_{c3}^{2}(\frac{7}{2}^+)$.


I. INTRODUCTION
In the past years, in addition to established ground states, more and more highly excited charmed baryons have been observed by Belle, BABAR, CLEO and LHCb et al [1]. Λ c baryons have two light u, d quarks and one heavy c quark inside. The two light quarks couple with isospin zero. The heavy quark symmetry works approximately in Λ c baryons, and the light quarks in Λ c baryons may correlate and make a diquark. The Λ c states provide an excellent window to explore the baryon structure and quark dynamics in baryons.
So far, in the review of particle physics [1], Λ c , Λ c (2595) + , Λ c (2625) + , Λ c (2765) + (or Σ c (2765) + ), Λ c (2860) + , Λ c (2880) + , Λ c (2940) + have been listed. The masses, total decay widths and possible decay channels of these Λ c are presented in Table. I. The spins and parities of these Λ c states have not been measured by experiments. In order to identify these states, it is important to determine their J P quantum numbers and to learn their internal dynamics in every model.
As known, a study of the strong decays of Λ c baryons is an important way to determine their J P quantum numbers. As a phenomenological method, the 3 P 0 model was proposed to compute the OZI-allowed hadronic decay widths of hadrons [34][35][36][37]. There are also some attempts to make a bridge between the phenomenological 3 P 0 model and QCD [38][39][40]. The 3 P 0 model has been employed to study the strong decays of Λ c baryons [10,[41][42][43][44]. In addition to the computation of strong decay widths, the dynamics and structure of the Λ c baryons have also been explored in these references. However, the studies aim at the separate analysis of one Λ c baryon or few observed Λ c baryons. The Λ c baryons have not been systematically analyzed in the 3 P 0 model.
In this work, all the observed Λ c except for Λ c (2286) + will be systematically examined as the 1P -wave, 1D-wave or 2S-wave Λ c baryons from their strong decay properties in the 3 P 0 model. In particular, their internal structure (especially the ρ-mode and λ-mode excitations ) will be paid attention to. The paper is organized as follows. In Sec.II, the 3 P 0 model is briefly introduced, some notations of heavy baryons and related parameters are indicated. We present our numerical results and analyses in Sec.III. In the last section, we give our conclusions and discussions.
II. 3 P0 MODEL, SOME NOTATIONS AND PARAMETERS 3 P 0 model is also known as a quark pair creation (QPC) model. It was first proposed by Micu [34] and further developed by Yaouanc et al, [35][36][37]. The basic idea of this model assumes: a pair of qq are firstly created from the QCD vacuum with quantum numbers J P C = 0 ++ ; Subsequently, the created quark and antiquark recombine with the quarks from the initial hadron A to form two daughter hadrons B and C [34]. The decays follow the OZI rule. For baryon decays, one quark of the initial baryon regroups with the created antiquark to form a meson, and the other two quarks regroup with the created quark to form a daughter baryon. There are three ways for the processes of recombination as follows, A(q 1 q 2 q 3 ) + P (q 4 q 5 ) → B(q 1 q 4 q 2 ) + C(q 3 q 5 ), (1) A(q 1 q 2 q 3 ) + P (q 4 q 5 ) → B(q 1 q 4 q 3 ) + C(q 2 q 5 ), (2) A(q 1 q 2 q 3 ) + P (q 4 q 5 ) → B(q 4 q 2 q 3 ) + C(q 1 q 5 ), (3) which are shown in Fig. 1, where each quark was numbered for a convenience. The two-body hadronic decay width Γ for a baryon A into B and C final states follows as in the 3 P 0 model [37,[41][42][43][44][45], The partial wave amplitude M JL is related to the helicity amplitude M MJ A MJ B MJ C via a Jacob-Wick formula [46]. In the equation, p is the momentum of the daughter baryon in A's center of mass frame, m A and J A are the mass and total angular momentum of the initial baryon A, respectively. m B and m C are the masses of the final hadrons. The helicity amplitude M MJ A MJ B MJ C reads [10,41,42,44] The conservation of the total angular momentum and the angular momentum of the light quarks freedom is indicated explicitly in the equation. F is a factor equal to 2 when each one of the two quarks in C has isospin 1 2 , and F = 1 when one of the two quarks in C has isospin 0.
In last equation, the matrix ϕ 1,4,3 B ϕ 2,5 C |ϕ 1,2,3 A ϕ 4,5 0 of the flavor wave functions ϕ i (i = A, B, C, 0) can also be presented in terms of C-G coefficients of the isospin as follows [37,41,45] with where f = ( The space integral follows as with a simple harmonic oscillator(SHO) wave functions for the baryons [8,41,48] where N represents a normalization coefficient of the total wave function. Explicitly, where L L+1/2 n p 2 β 2 denotes the Laguerre polynomial function, and Y LML (Ω p ) is a spherical harmonic function. The relation between the solid harmonica polynomial In order to describe three-body systems, a center of mass motion and a two-body systems of internal relative motion in the Jacobi coordinate [47] are usually employed. As displayed in Fig. 2, ρ is the relative coordinate between the two light quarks (quark 1 and 2), and λ is the relative coordinate between the center of mass of the two light quarks and the charmed quark.
In these tables, L ρ denotes the orbital angular momentum between the two light quarks, L λ denotes the orbital angular momentum between the charm quark and the two light quark system, S ρ denotes the total spin of the two light quarks. L is the total orbital angular momentum of L ρ and L λ (L =L ρ + L λ ), and J l is the total angular momentum of L and S ρ (J l = L + S ρ ). J is the total angular momentum of the baryons (J = J l + 1 2 ). InΛ L cJ l (Σ L cJ l ), a superscript L denotes the total angular orbital momentum, a tilde indicates L ρ = 1, and the one without a tilde indicates L ρ = 0. More details about the notations could be found in Refs. [10,43,44,48] In the 3 P 0 model, the qq quark pair created from the vacuum may be uū, dd or ss. So far, there is no sign of an ss creation in observed strong decay channels of Λ c states. In addition to masses, decay widths, experimentally observed strong decay channels, theoretically predicted strong decay channels of all the Λ c states are also given in Table I. Masses of relevant mesons and baryons involved in our calculation are presented in Table V [1]. The parameters are chosen as follows. The dimensionless pair-creation strength γ = 13.4. The β λ,ρ = 600 MeV in the 1S-wave baryon wave functions are chosen, the β λ,ρ = 500 MeV in the P -wave baryon wave functions are chosen, and the β λ,ρ = 400 MeV in the 2S-wave and D-wave baryon wave functions are chosen. These β λ,ρ are consistent with those in Refs. [10,41,[49][50][51]. The R = 2.5 GeV −1 in the harmonic oscillator wave functions of π/K meson and R = 1.67 GeV −1 for D meson [10,41,[49][50][51].
Their J P are supposed 1 2 − and 3 2 − , respectively [1]. In our analyses, all the hypothesises that Λ c (2595) + and Λ c (2625) + are the low-lying 1P -wave, 2S-wave, and 1Dwave charmed baryons are examined. In Table VI, the numerical results of the decay widths of Λ c (2595) + as the 1P -wave and 2S-wave states are given. Similar numerical results for Λ c (2625) + are presented in Table VII. In Table VIII and Table IX, the numerical results of the decay widths of Λ c (2595) + and Λ c (2625) + as D-wave charmed baryons are given, respectively. In these tables, some branching ratios are also given.
a vanishing (denoted with "0" in the table) total decay width or approximately vanishing total decay width (denoted with "≈ 0" in the table). It is impossibly theΛ c1 ( 1 2 − ) either for a large total decay width.
, for a much lower predicted branching frac- The predicted total decay width is much smaller either in comparison with experimental data.
From Table VIII, neither the branching ratios nor the total decay widths are consistent with experimental measurements. Therefore, Λ c (2595) + is impossibly a D-wave excitation of Λ c . Account for the branching fractions B = Γ(Σ (++) c π (−) )/Γ total and the total decay width, for a large predicted decay width or a vanishing Σ ++ c π − mode. For Λ c (2625) + , Σ c (2455)π are the only two-body decay modes of this state, and the branching fraction of the direct threebody decay mode Λ + c ππ is large, so it is impossible to learn this state only from the branching fraction of these two-body strong decay modes. However, the predicted citations and the 1D-wave excitations are much higher than that of Λ c (2625) + [14,18,21]. Account for this fact, Λ c (2625) + seems impossibly these charmed baryons. In In the given configurations of Λ c (2595) + and Λ c (2625) + , there is a λ-mode excitation while there is not a ρ-mode excitation. The two light quarks inside couple with total spin (2765)) Λ c (2765) + (or Σ c (2765) + ) is a broad state first observed in Λ + c π − π + channel by CLEO Collaboration [55]. However, nothing is known about its J P . One even does not know whether it is a Λ c or a Σ c . Λ c (2765) + (or Σ c (2765) + ) was suggested as a first orbital excitation of Λ c with J P = 1  [56].
In this subsection, all the possibilities of Λ c (2765) (or Σ c (2765)) as the 1P -wave, 2S-wave and 1D-wave charmed baryon with isospin I = 0 are examined. When Λ c (2765) (or Σ c (2765)) is assigned in these configurations, the relevant hadronic decay widths are calculated in the 3 P 0 model and are shown in Table X.
From Table X, account for the fact that Λ c (2595) + and Λ c (2625) + have been assigned with the Λ c1 ( 1 2 − ) and Λ c1 ( 3 2 − ), respectively, Λ c (2765) + (or Σ c (2765)) seems impossibly a P -wave Λ c . Otherwise, Λ c (2765) + (or Σ c (2765)) has an extremely small or extremely large decay width. Except for the total decay width, the strong     When Λ c (2765) (or Σ c (2765)) is assumed with a 1Dwave baryon with isospin I = 0, the relevant hadronic decay widths are calculated and presented in Table XI. From this table, the predicted total decay widths are around the measured one in several configurations. That is to say, Λ c (2765) (or Σ c (2765)) is possibly a Dwave charmed baryons. However, one has no accurate measurement of the total decay width of Λ c (2765) (or Σ c (2765)), and has no measurement of any branching fraction or branching ratio on its decay channel. In fact, it is not suitable to draw a confirmative conclusion in terms of such less information of Λ c (2765) (or Σ c (2765)). Λ c (2860) + as a newly reported Λ c baryon was first observed by the LHCb Collaboration in the D 0 p channel [33]. The mass and width of Λ c (2860) + were measured. The mass of Λ c (2860) + is consistent with the predictions for an orbital D-wave Λ c excitation with J P = 3 2 + [12,18]. In particular, quantum numbers of Λ c (2860) + were found to be J P = 3 2 + , the other quantum numbers were excluded with a significance of more than 6 standard deviations [33]. Λ c (2880) + was first observed by the CLEO Collaboration in Λ + c π − π + [55] and confirmed by the BaBar Collaboration in the D 0 p channel [59]. From an analysis of angular distributions in Λ c (2880) + → Σ c (2455) 0,++ π +,− decays and the measured R = Γ(Σ c (2520)π)/Γ(Σ c (2455)π) = 0.225 ± 0.062 ± 0.0255, the preferred quantum numbers of Λ c (2880) + state were constrained to J P = 5 2 + by Belle Collaboration [61]. Recently, the LHCb Collaboration studied the spectrum of excited Λ c states that decay into D 0 p channel and measured the mass and width of Λ c (2880) + . The preferred spin of Λ c (2880) + is found to be 5 2 , and the spin assignments 1 2 and 3 2 were excluded [33]. Λ c (2940) + was first observed by the BaBar Collaboration in D 0 p invariant mass distribution [59]. The spin-parity of Λ c (2940) + was constrained to J P .08 × 10 −3 1.08 × 10 −2 7.43 × 10 −2 9.42 × 10 −2 9.64% , it was also assigned as a D-wave state with J P = 3 2 + [19,20]. In most references [9-14, 16-18, 21, 60], Λ c (2880) + was conjectured as an excited charmed baryon with J P = 5 2 + though its structure may be different in these references.