$P$-wave charmed baryons of the $SU(3)$ flavor $\mathbf{6}_F$

We use QCD sum rules to study mass spectra of $P$-wave charmed baryons of the $SU(3)$ flavor $\mathbf{6}_F$. We also use light-cone sum rules to study their $S$- and $D$-wave decays into ground-state charmed baryons together with light pseudoscalar and vector mesons. We work within the framework of heavy quark effective theory, and we also consider the mixing effect. Our results can explain many excited charmed baryons as a whole, including the $\Sigma_c(2800)^0$, $\Xi_c(2923)^0$, $\Xi_c(2939)^0$, $\Xi_{c}(2965)^{0}$, $\Omega_c(3000)^0$, $\Omega_c(3050)^0$, $\Omega_c(3066)^0$, $\Omega_c(3090)^0$, and $\Omega_c(3119)^0$. Their masses, mass splittings within the same multiplets, and decay properties are extracted for future experimental searches.


arXiv:2106.15488v3 [hep-ph] 10 Aug 2021
In this paper we shall systematically investigate Pwave charmed baryons of the SU (3) flavor 6 F . In Refs. [85,86] we have studied mass spectra of P -wave bottom baryons using the method of QCD sum rules [87,88], and in the present study we shall replace the bottom quark by the charm quark, and reanalyse those results. In Ref. [89] we have studied decay properties of Pwave bottom baryons using the method of light-cone sum rules [90][91][92][93][94], and in the present study we shall apply the same method to study P -wave charmed baryons of the SU (3) flavor 6 F . We shall study their S-and D-wave decays into ground-state charmed baryons together with pseudoscalar mesons π/K and vector mesons ρ/K * . We shall work within the framework of the heavy quark effective theory (HQET) [95][96][97], and we shall also consider the mixing effect between two different HQET multiplets.
This paper is organized as follows. In Sec. II we briefly introduce our notations, and use the method of QCD sum rule to study mass spectra of P -wave charmed baryons of the SU (3) flavor 6 F . The obtained results are further used in Sec. III to study their S-and D-wave decays into ground-state charmed baryons together with light pseudoscalar and vector mesons. The mixing effect between different HQET multiplets is investigated in Sec. IV, and the obtained results are summarized in Sec. V, where we conclude this paper.

II. MASS SPECTRA THROUGH QCD SUM RULES
In this section we follow Ref. [27] and classify Pwave charmed baryons. A singly charmed baryon consists of one charm quark and two light up/down/strange quarks, and its internal symmetries are: • The color structure of the two light quarks is antisymmetric (3 C ).
• The flavor structure of the two light quarks is either symmetric (6 F ) or antisymmetric (3 F ).
• The spin structure of the two light quarks is either symmetric (s l ≡ s qq = 1) or antisymmetric (s l = 0).
• The orbital structure of the two light quarks is either symmetric or antisymmetric. We call the former λ-type with l ρ = 0 and l λ = 1, and the latter ρ-type with l ρ = 1 and l λ = 0. Here l ρ denotes the orbital angular momentum between the two light quarks, and l λ denotes the orbital angular momentum between the charm quark and the two-lightquark system.
According to the Pauli principle, the total symmetry of the two light quarks is antisymmetric, so that we can categorize P -wave charmed baryons into eight multiplets. Four of them belong to the SU (3) flavor 6 F representation, as shown in Fig. 1. We denote them as [F (flavor), j l , s l , ρ/λ], where j l = l λ ⊗ l ρ ⊗ s l is the total angular momentum of the light components. There are one or two charmed baryons contained in each multiplet, with the total angular momenta j = j l ⊗ s b = |j l ± 1/2|. We have systematically studied mass spectra of P -wave bottom baryons in Refs. [85,86]. In the present study we just need to replace the bottom quark by the charm quark, and reanalyse those results. The newly obtained results for charmed baryons are summarized in Table I. In the calculation we have used the following QCD parameters at the renormalization scale 1 GeV [4,62,[98][99][100][101][102][103]: g ss σGs = M 2 0 × ss , M 2 0 = 0.8 GeV 2 , g 2 s GG = (0.48 ± 0.14) GeV 4 . Besides, we have used the PDG value m c = 1.275 ± 0.025 GeV [4] for the charm quark mass in the MS scheme.
To better understand P -wave charmed baryons, we shall further investigate their decay properties in the next section. The parameters given in Table I will be used as inputs. To better describe P -wave charmed baryons, we select the following mass values when calculating their decay widths: 20 , which are calculated through:

III. DECAY PROPERTIES THROUGH LIGHT-CONE SUM RULES
We have systematically studied decay properties of Pwave bottom baryons of the SU (3) flavor 6 F in Ref. [89] using the method of light-cone sum rules within HQET. In this paper we apply the same method to study P -wave charmed baryons of the SU (3) flavor 6 F . We shall study their S-and D-wave decays into ground-state charmed baryons together with pseudoscalar mesons π/K and vector mesons ρ/K * , including: In the above expressions, the superscripts S and D denote S-and D-wave decays, respectively; X and V µ denote P -wave charmed baryons, ground-state charmed baryons, light pseudoscalar mesons, and light vector mesons, respectively. We shall use Ω 0 c (3/2 − ) belonging to [6 F , 2, 1, λ] as an example, and study its D-wave decay into Ξ + c (1/2 + ) and K − (0 − ) in Sec. III A. Then we shall apply the same method to systematically investigate the four charmed baryon multiplets [6 F ], separately in the following subsections.
Finally, we use the amplitude, to evaluate its partial decay width to be: In the following subsections we shall similarly investigate the four charmed baryon multiplets [6 F , 1, 0, ρ], We study their S-and D-wave decays into ground-state charmed baryons together with light pseudoscalar and vector mesons. We derive the following nonzero coupling constants: Some of these coupling constants are shown in Fig. 2 as functions of the Borel mass T . We further use these coupling constants to derive the following decay channels that are kinematically allowed: We summarize the above results in Table II.
The [6 F , 0, 1, λ] doublet contains altogether three charmed baryons: We study their S-and D-wave decays into ground-state charmed baryons together with light pseudoscalar and vector mesons. We derive the following non-zero coupling constants: Some of these coupling constants are shown in Fig. 3 as functions of the Borel mass T . We further use these coupling constants to derive the following decay channels that are kinematically allowed: We summarize the above results in Table III.
The [6 F , 1, 1, λ] doublet contains altogether six charmed baryons:    c [ 1 ground-state charmed baryons together with light pseudoscalar and vector mesons. We derive the following nonzero coupling constants: Some of these coupling constants are shown in Fig. 4 as functions of the Borel mass T . We further use these coupling constants to derive the following decay channels that are kinematically allowed: We summarize the above results in Table IV. We study their S-and D-wave decays into ground-state charmed baryons together with light pseudoscalar and vector mesons. We derive the following nonzero coupling constants: Some of these coupling constants are shown in Fig. 5 as functions of the Borel mass T . We further use these coupling constants to derive the following decay channels that are kinematically allowed: We summarize the above results in Table V.
To explain these discrepancies, we recall that the HQET is an effective theory, which works quite well for bottom baryons [89], but not so perfect for charmed baryons [105]. Therefore, the three J P = 1/2 − charmed baryons can mix, and the three J P = 3/2 − charmed baryons can also mix. Accordingly, we just need a tiny mixing angle θ 1 ≈ 0 • to make it possible to observe all the P -wave Ξ c baryons in the Λ c K decay channel.

V. SUMMARY AND DISCUSSIONS
In this paper we perform a rather complete study on P -wave charmed baryons of to the SU (3) flavor 6 F . We use the method of QCD sum rules to study their mass spectra. We also use the method of light-cone sum rules to study their decay properties, including their S-and Dwave decays into ground-state charmed baryons together with pseudoscalar mesons π/K and vector mesons ρ/K * . We work within the framework of heavy quark effective theory, and we also consider the mixing effect between different HQET multiplets.
Accordingly to the heavy quark effective theory, we categorize P -wave charmed baryons of the SU (3) We find it possible to interpret the Ω c (3000) 0 as the P -wave Ω c baryon of either J P = 1/2 − or 3/2 − , belonging to this doublet. However, total widths of Σ c ( with the mixing angle fine-tuned to be θ 2 = 37±5 • . Our results suggest: a) the Ξ c (2923) 0 and Ω c (3050) 0 can be interpreted as the P -wave Ξ c and Ω c baryons of J P = 1/2 − , belonging to [6 F , 1, 1, λ]; b) the Ω c (3119) 0 can be interpreted the P -wave Ω c baryon of J P = 5/2 − , belonging to [6 F , 2, 1, λ]; c) the Ξ c (2939) 0 , Ξ c (2965) 0 , Ω c (3066) 0 , and Ω c (3090) 0 can be interpreted the P -wave Ξ c and Ω c baryons of J P = 3/2 − , belonging to the mixing of the [6 F , 1, 1, λ] and [6 F , 2, 1, λ] doublets.   To arrive at our interpretations, we need to pay attention to: there exist considerable uncertainties in our results for absolute values of charmed baryon masses due to their dependence on the charm quark mass [85,86]; however, mass splittings within the same doublets do not depend much on this, and are calculated with much less uncertainties; moreover, we can extract more useful information from decay properties of charmed baryons.
Summarizing the above results, the present sum rule Σc(2800) 0 Ωc ( Table VI. We suggest the Belle-II, CMS, and LHCb Collaborations to further study them to verify our interpretations. Especially, we propose to further study the Σ c (2800) 0 to examine whether it can be separated into several excited charmed baryons. For convenience, we show their total widths and branching ratios in Fig. 6 using pie charts.
The sum rule equation for the Ω 0 c [ 5