Low-Lying Charmed and Charmed-Strange Baryon States

Eur. Phys. J. C (2017) 77:154 DOI 10.1140/epjc/s10052-017-4708-x

Regular Article - Theoretical Physics

Low-lying charmed and charmed-strange baryon states

Bing Chen1,3,a, Ke-Wei Wei1,b, Xiang Liu2,3,c, Takayuki Matsuki4,5,d 1 Department of Physics, Anyang Normal University, Anyang 455000, China 2 School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China 3 Research Center for Hadron and CSR Physics, Institute of Modern Physics of CAS & Lanzhou University, Lanzhou 730000, China 4 Tokyo Kasei University, 1-18-1 Kaga, Itabashi, Tokyo 173-8602, Japan 5 Theoretical Research Division, Nishina Center, RIKEN, Saitama 351-0198, Japan

Received: 15 December 2016 / Accepted: 19 February 2017 © The Author(s) 2017. This article is published with open access at Springerlink.com

( )0,+,++ ( )0,+ ( )0,+ ( )0,+ Abstract In this work, we systematically study the mass c 2520 , c 2470 , c 2580 , c 2645 , 0,+ 0,+ () 0 0,+ spectra and strong decays of 1P and 2S charmed and c(2790) , c(2815) , c (2930) , c(2980) , 0,+ 0,+ () + charmed-strange baryons in the framework of non-relativistic c(3055) , c(3080) , and c (3123) . For brevity, constituent quark models. With the light quark cluster–heavy we call these charmed and charmed-strange baryons just quark picture, the masses are simply calculated by a poten- charmed baryons in the following. Among these observed tial model. The strong decays are studied by the Eichten– states, some new measurements have been performed by Hill–Quigg decay formula. Masses and decay properties experiments in the past several years. The masses and 0,+,++ 0,+,++ + of the well-established 1S and 1P states can be repro- widths of c(2455) , c(2520) , c(2595) , ( )0,+,++ + duced by our method. c 2800 can be assigned as and c(2625) have been measured with significantly small − − a c2(3/2 ) or c2(5/2 ) state. We prefer to interpret the uncertainties with the efforts of CDF [2] and Belle [3]. Very 0 + signal c(2850) as a 2S(1/2 ) state although at present recently, the Belle Collaboration updated the measurements ( )0,+ ( )0,+ ( )0,+ ( )0,+ we cannot thoroughly exclude the possibility that this is of c 2580 , c 2645 , c 2790 , c 2815 , ( )0 ( )+ ( )+ 0,+ the same state as c 2800 . c 2765 or c 2765 and c(2980) [4]. On the other hand, new decay modes +( ) + ( / −) could be explained as the c 2S state or c1 1 2 state, for the higher excited charmed baryon states have been found respectively. We propose to measure the branching ratio by experiment. For instance, the decay channel of D+ was B( ( )π)/B( ( )π) + + of c 2455 c 2520 in the future, which may first found for c(3055) and c(3080) , and the follow- disentangle the puzzle of this state. Our results support ing ratios of branching fractions were first reported by Belle 0,+ 0,+ c(2980) as the first radial excited state of c(2470) several months ago [5]: P = / + ( )0 with J 1 2 . The assignment of c 2930 is analogous + + 0,+,++ − − B(c(3055) → D ) to c(2800) , i.e., a (3/2 ) or (5/2 ) state. In = . ± . ± . , c2 c2 B( ( )+ → ( )++ −) 5 09 1 01 0 76 addition, we predict some typical ratios among partial decay c 3055 c 2455 K widths, which are valuable for experimental search for these B( ( )+ → +) missing charmed and charmed-strange baryons. c 3080 D = . ± . ± . , + ++ − 1 29 0 30 0 15 B(c(3080) → c(2455) K ) and 1 Introduction B( ( )+ → ( )++ −) c 3080 c 2520 K = . ± . ± . , B( ( )+ → ( )++ −) 1 07 0 27 0 01 At present, the particle data group (PDG) lists nine charmed c 3055 c 2455 K + and ten charmed-strange baryons [1]. They are c(2286) , where the uncertainties are statistical and systematic. Obvi- + + + + c(2595) , c(2625) , c(2765) (or c(2765) ), ously, these new measurements are very useful to understand + + 0,+,++ 0,+,++ c(2880) , c(2940) , c(2800) , c(2455) , the nature of these excited charmed baryon states. Theoretically, the charmed baryons which contain one a e-mail: [email protected] heavy quark and two light quarks occupy a particular posi- b e-mail: [email protected] tion in the baryon physics. Since the chiral symmetry and c e-mail: [email protected] heavy quark symmetry (HQS) can provide some qualitative d e-mail: [email protected] insight into the dynamics of charmed baryons, the investiga- 123 154 Page 2 of 17 Eur. Phys. J. C (2017) 77:154 tion of charmed baryons should be more helpful for improv- of low-lying 1P and 2S charmed baryons. We pay attention ing our understanding of the confinement mechanism. The to only the charmed baryons inside of which degrees of free- spectroscopy of charmed baryons has been investigated in dom of two light quarks are frozen. It means that two light various models. So far, the several kinds of quark potential quarks are not considered to be excited, neither radially nor models [6–10], the relativistic flux tube (RFT) model [11,12], orbitally. As illustrated in Ref. [44], these kinds of charmed the coupled channel model [13],theQCDsumrule[14–16], excitations carry lower-excited energies, which means these and the Regge phenomenology [17] have been applied to excited charmed baryons may more likely be detected by study the mass spectra of excited charmed baryons, and so did experiments. Fortunately, our results indicate that most of the lattice QCD [18,19]. The strong decay behaviors of charmed observed charmed baryons can be accommodated in this way. baryons have been studied by several methods, such as the The paper is arranged as follows. In Sect. 2, the masses heavy hadron chiral perturbation theory (HHChPT) [20,21], of low-excited charmed baryons are calculated by the non- 3 the chiral quark model [22,23], the P0 model [24], and a relativistic quark potential model. In Sect. 3, the Eichten– non-relativistic quark model [25]. The decays of 1P c and Hill–Quigg (EHQ) decay formula, which is employed to c baryons have also been investigated by a light front quark study the strong decays of excited charmed baryons, is intro- model [26,27], a relativistic three-quark model [28], and the duced. The properties of low-lying charmed baryon states QCD sum rule [29]. are fully discussed in Sect. 4. Finally, the paper ends with a Although many experimental and theoretical efforts have conclusion and an outlook. Some detailed calculations and been made for the research of charmed baryons, most of definitions are collected in the appendices. the 1P and 2S charmed baryons are not yet established. + 0,+,++ Several candidates, including c(2765) , c(2800) , 0 0,+ c(2930) , and c(2980) are still in controversy. 2 The deduction of mass spectra + c(2765) was first observed by the CLEO Collaboration ( )+ → +π +π − in the decay channel of c 2765 c [30], and 2.1 Treating charmed baryon system as a two-body confirmed by Belle in the c(2455)π mode [31,32]. Because problem + + +π +π − both c and c excitations can decay through c and c(2455)π, we even do not know whether the observed To study the baryon dynamics, one crucial question which + + charmed baryon signal around 2765 MeV is the c or c should be answered is “What are the relevant degrees of free- + − state, or their overlapping structure [33]. In the e e anni- dom in a baryon?” [45]. In constituent quark models, a baryon 0,+,++ hilation process, an isotriplet state, c(2800) ,was system consists of three confined quarks. Thus the dynamics +π observed by Belle in the channel of c , and it was ten- of a baryon resonance is surely more complex than a meson. P − tatively identified as the c2 state with J = 3/2 [34]. Due to the HQS, however, the dynamics of charmed baryons Interestingly, another neutral resonance was later found by could be greatly simplified. The HQS suggests that the cou- − → ∗0 ¯ → +π − ¯ BaBar in the process of B c p c p with the plings between a c quark and two light quarks are weak [46]. ± ± +33 mass 2846 8 10 MeV and decay width 86−22 MeV [35]. Therefore, two light quarks in a charmed baryon could first The higher mass and the weak evidence of J = 1/2 indi- couple with each other to form a light quark cluster.1 Then cate that the signal observed by BaBar might be different the light quark cluster couples with a charm quark, and a from the Belle observation. In this paper, we denote the sig- charmed baryon resonance forms. With this assumption, two 0 0 nal discovered by BaBar by c(2850) . c(2930) , which light quarks have the same status to a c quark, including the + − was only seen by BaBar in the decay mode c K [36], still average distances to a c quark. In the light cluster–heavy 0,+ needs more confirmation. c(2980) was first reported by quark picture, the dynamics of heavy baryon can be sim- + −π + + 0π − Belle in the channels c K and c KS [37], and plified. In the non-relativistic constituent quark model, the was later confirmed by Belle [4,38] and BaBar [39]inthe spin-independent part of the Hamiltonian is ( )π ( )π ( ) channels c 2580 , c 2645 and c 2455 K , respec- 3 2 α tively. However, the decay widths reported by Refs. [4,37– pi 2 b H = + mi + − + rij , (1) 39] were quite different from each other. More experimental 2mi 3rij 2 i=1 i< j information as regards the charmed baryons can be found in Refs. [40–43]. where the Cornell potential [47] is used as a phenomenolog- Obviously, a systematic study of masses and decays is ical confining term. The Jacobi coordinates are usually taken required for these unestablished charmed baryons. More to deal with the three-body problem for baryons. ρ, λ, and R importantly, most of 2S and 1P charmed baryons have not yet are related to the quark positions by been detected by any experiments. Such a research can also help the future experiments to find them. In the present work, 1 In some work, the light quark cluster may also be named a light we will focus on both the mass spectra and the strong decays diquark. 123 Eur. Phys. J. C (2017) 77:154 Page 3 of 17 154

ρ =r1 −r2, color–spin interaction is proportional to the inverse of the + m1r1 m2r2 quark masses, two light quarks in the heavy-baryon system λ = −rQ, m1 + m2 are expected to strongly couple to each other [49]. Thus, they + + may develop into a quark cluster. In fact, the existence of a = m1r1 m2r2 m QrQ , R light quark cluster correlations was partly confirmed by the m1 + m2 + m Q lattice QCD [50] and the Bethe–Salpeter equation [51]. where the indices, 1, 2, and Q, are for two light quarks and Thus, we expect the light quark cluster to be an effective a heavy quark, respectively. The momenta pρ, pλ and pR, degree of freedom for a charmed baryon. Since the ρ mode of which are conjugate to the Jacobi coordinates above, can a cluster is not considered here, the Schrödinger equation (3) be defined easily. Now the spin-independent Hamiltonian is simplified as becomes 2 2 pρ p2 p α = + λ + R + + − 2 + b ∇2 α H M r12 − λ − 4 + τ + ψ = ψ. 2mρ 2mλ 2M 3r12 2 b C E (4) 2mλ 3τ 2α b 2α b + − + r1Q + − + r2Q , (2) 3r1Q 2 3r2Q 2 2.2 Adopted effective potentials where mρ = m1m2/(m1 + m2), mλ = (m1 + m2)m Q/M, and M = m1 +m2 +m Q. According to the definitions above, the relative motion between two light quarks is usually called As a whole, the light quark cluster which occupies an the ρ-mode, while the one between the center of mass of the antitriplet color structure interacts with the c quark. Then λ two light quarks and the heavy quark is called the λ-mode. we would like to substitute (distance between light cluster τ As emphasized above, the average distances to c quark and c quark) for (distance between light quark and c quark). should be equal for two light quarks in a cluster, i.e., τ ≡ To this end, the effective potential [52] r1Q = r2Q (see Fig. 1). In practice, we should solve the following Schrödinger equation for the mass of a heavy baryon: 4 αs ν conf (λ) =− + λ − HQ−cl. b CQqq (5) 2 3 λ ∇ρ ∇2 bρ 2α 4α − − λ + − + bτ − + C 2mρ 2mλ 2 3ρ 3τ describes the interaction between the cluster and c quark, × ψ = Eψ. (3) where ν is an adjustable parameter. This approximation can greatly decrease the computational complexity. As shown Since the excited mode between two light quarks is not con- in Tables 2 and 3, the mass spectra given in this way are sidered in this paper, the light quark cluster can be treated reasonable for the low-lying excited charmed baryons. as a block with the antitriplet color structure and peculiar As a two-body problem, we treat the masses of differ- size. Specifically, a color singlet baryon system should be ¯ ¯ ent kinds of clusters as parameters and first fix them before formed as 3q ⊗ 3q ⊗ 3Q 3cl. ⊗ 3Q 1Q−cl.. In this way, 1 2 calculating the masses of the low-lying charmed baryons. a heavy baryon could be treated as a quasi-two-body sys- According to the flavor and spin, the light cluster can be clas- tem. In the light quark cluster–heavy quark picture, Isgur has sified into two kinds: one is the “scalar” cluster, and another discussed the similarity of dynamics between heavy baryons is the “vector” cluster. Constrained by the Pauli exclusion and heavy–light mesons [48]. It should be stressed that the principle, the total wave function of the light quark clus- scenario of a light cluster–heavy quark picture is not con- ter should be antisymmetric in the exchange of two quarks. tradictory to the one-gluon exchange interaction. Since the Because the spatial and color parts of this light quark cluster are always symmetric and antisymmetric, respectively, the function |flavor ×|spin should be symmetric. There- fore, the scalar light quark cluster [qq] (S = 0) is always flavor antisymmetric, and the axial-vector light quark cluster {qq} (S = 1) is flavor symmetric. In Jaffe’s terminology [49], the “scalar” and “vector” quark clusters are named the “good” and “bad” quark clusters, respectively. The masses of the “good” light quark cluster [qq] and [qs] are taken from our previous work where m[qq] and m[qs] were fixed as 450 and 630 MeV by the RFT model [11], respectively. Following Fig. 1 A single heavy baryon in the light cluster–heavy quark picture. With the SU(3) flavor symmetry, the relation of τ = λ2 + ρ2/4is Jaffe’s method [49], the bad light quark cluster masses can obvious be evaluated by the following relationships: 123 154 Page 4 of 17 Eur. Phys. J. C (2017) 77:154

4 × (2520) + 2 × (2455) c c where S denotes the spin of a baryon, S = SQ + S .. Another − c(2286) ≈ 210 MeV, cl 6 spin–orbit interaction is the Thomas-precession term, × ∗( ) + × ( ) 4 c 2645 2 c 2580 − ( ) ≈ . c 2470 150 MeV conf 6 ( ) 1 ∂ H − . S · L S . · L H SO tp =− Q cl Q + cl . (11) Q−cl. λ ∂λ 2 2 Evidently, the masses of {qq} and {qs} are about 660 MeV 2 m Q mcl. + and 780 MeV, respectively. Henceforth, we will call the c 0,+ 0,+,++ To reflect the importance of the heavy quark symmetry, and c baryons the G-type baryons, and c and 0,+ we rewrite the spin-dependent interactions as c the B-type baryons for convenience. Due to an antitriplet color structure, the spin-dependent = · + · + · + ˆ . interactions between light cluster and c quark are expected HS Vss SQ Scl. V1 Scl. L V2 SQ jcl. Vt S12 to be the same as the meson systems. In a constituent quark (12) model [53], the spin-dependent interaction is written as The degrees of freedom of the light quark cluster are char- = cont + ten + SO . HS HQ−cl. HQ−cl. HQ−cl. (6) acterized by its total angular momentum jcl., i.e., jcl. = S . + L. Obviously, the orbital angular momentum L of cont cl The color contact interaction HQ−cl. is usually given by the a charmed baryon in the present picture is defined by the following form: angular momentum between light quark cluster and c quark, ˆ π α i.e., L = Lλ. The tensor operator is defined as S12 = cont 32 s ˜ H = δσ (λ)S · S ., (7) Q−cl. Q cl 3 S · λ S . · λ /λ2 − S · S .. 9 m Qmcl. Q cl Q cl With the confining term of Eq. (5), the coefficients Vss, where SQ and Scl. refer to the heavy quark and light clus- V1, V2, and Vt in Eq. (12) are defined by √ 2 2 ter spins. A Gaussian-smeared function (σ/ π)3e−σ λ is ˜ α α ν normally used for δσ (λ) [54]. If the SU(3) flavor symmetry 1 32 s 3 −σ 2λ2 1 2 s b ν−2 Vss = √ σ e − − λ m π m 3λ3 2 is kept for the charmed baryons, we may modify the color Q 9 Q contact interaction as α −4 s 1 , λ3 α σ 3 3 mcl. cont 32 s −σ 2λ2 H = √ e S · S ., (8) α ν α Q−cl. π Q cl 1 1 2 s b ν−2 4 s 1 9 m Q V1 = − λ + , m . m . 3λ3 2 3λ3 m cl cl Q where the mass of a light quark cluster is just replaced by a α ν α 1 1 2 s b ν−2 4 s 1 V2 = − λ + , unit. This assumption is supported by the mass differences m m 3λ3 2 3λ3 m . B Q Q cl of the 1S -type charmed baryons, 4α 1 V = s . (13) + . t 3 ( )++ − ( )++ = . 0 25 , 3λ m Qmcl. c 2520 c 2455 64 44−0.24 MeV ( )+ − ( )+ = . ± . , c 2645 c 2580 70 2 3 0MeV 0 0 2.3 Getting masses of charmed baryons c(2770) − c(2695) = 70.7 ± 2.6MeV.

The mass differences shown above are mainly due to the color In our calculation, the following Schrödinger equation is contact interaction in the quark potential model. Clearly, solved for the nS state: these values are almost independent of the light quark cluster ∇2 masses. The color tensor interaction in Eq. (6)is − λ + conf + cont = . HQ−cl. HQ−cl. E (14) ⎛ ⎞ 2mλ α 3 SQ · λ Scl. · λ ten 4 s 1 ⎝ ⎠ H = −S · S . . The conﬁning and contact terms have been given by Eqs. (5) Q−cl. λ3 λ2 Q cl 3 m Qmcl. and (8). For the orbital excitations, all spin-dependent inter- (9) actions are treated as the leading-order perturbations. Our calculation indicates that the color contact interaction can be SO Finally, HQ−cl. denotes the spin–orbit interaction which con- ignored for the orbital excitations. tains two terms. One is the color magnetic interaction which Two bases are employed to extract the mass matrix ele- arises from one-gluon exchange, ments. One is deﬁned by the eigenstates |Scl., L, jcl., SQ, J ( jj coupling scheme) and the other is by |Scl., SQ, S, L, J α · · · SO(cm) 4 s S L SQ L Scl. L (LScoupling scheme). The relation between these two bases H − . = + + , (10) Q cl λ3 . 2 2 3 m Qmcl m Q mcl. is 123 Eur. Phys. J. C (2017) 77:154 Page 5 of 17 154

1 2 1P Table 1 jcl. 0 J 1 2, j 0 Values of the parameters of the non-relativistic quark potential cl. ν+1 J 1 2, j 1 model. The unit of b is GeV which varies depending on each value cl. 1 2 1P ν jcl. 1 of J 3 2, jcl. 1 3 2 1P m 1.68 GeV b 0.145 C 0.233 GeV L 1 V 1 V 2 V ss V t c C J 3 2, jcl. 2 α m[qq] 0.45 GeV 0.45 CC 0.100 GeV jcl. 2 ν m[qs] 0.63 GeV [c,c] 0.84 CC 0.156 GeV J 5 2, jcl. 2 5 2 1P m{qq} 0.66 GeV ν[ , ] 0.70 C 0.060 GeV 3 2 1P c c C { } σ physical states m qs 0.78 GeV 1.00 GeV

Fig. 2 A schematic diagram for the splittings of p-wave c and c. j = L + j . h | = − / , P Here l cl and subindices and of the last column mean low | = jl J 1 2 J . and high states in mass after including V˜ and V˜ interactions J P ss t | jl = J + 1/2, J + + + The mass matrix of 1D states can also be obtained by the |[ ., ] , = (− )Scl. SQ L J Scl L jcl. SQ J 1 similar procedure. As shown above, there are seven parame- S√ × ( + )( + ) ters in the non-relativistic quark potential model, which are 2Jcl. 1 2S 1 (15) m Q, mcl., b, α, γ , ν, and CQqq . All values of parameters are × scl. Ljcl. |[ , ] , . Scl. SQ S L J listed in Table 1. If the SU(3) flavor symmetry is taken into JSQ S account for the charmed and charmed-strange baryons, the = · + Due to Scl. 0, only V2 SQ jcl. contributes to the masses dynamics of c states should be like c. The case of c and G γ of -type charmed baryons. With a bad light quark cluster, c is alike. Accordingly, the same value of is selected for however, B-type charmed baryons have more complicated the G-type charmed baryons, as well as the case of B-type. splitting structures. Within the framework of the heavy quark We have adopted the typical values in the quark potential effective theory, the spin of an axial-vector light quark cluster, models for mc, b, α, and ν (see Table 1). It is an effective Scl., first couples with the orbital angular momentum L.As method to investigate charmed baryons in the heavy quark– illustrated in Fig. 2, in the heavy quark limit mc →∞, light quark cluster picture. We do not expect the values of there are only three states which are characterized by jcl. ν to be the same for G-type and B-type baryons. Here, ν of +/ / for 1P charmed baryons. When the heavy quark spin SQ c c is slightly larger than c c. The predicted masses couples with jcl., the degeneracy is resolved and the five of the low-excited charmed baryons are collected in Tables − − states appear. These are two J P = 1/2 ,twoJ P = 3/2 , 2 and 3. − and one J P = 5/2 states. Lastly, the states with the same As mentioned earlier, the nonzero off-diagonal elements P · J mix with each other by the interactions of Vss SQ Scl. in mass matrices of 1/2 | HS | 1/2 and 3/2 | HS | ˆ and Vt S12, and physical states are formed. 3/2 cause the mixing between two states with the same P We now turn to a calculation of the mass matrix in the jj J but different jcl.. However, the mechanism of mixing − coupling scheme. For 1P states with J P = 1/2 ,themass effects in hadron physics is still unclear. In principle, a phys- matrix is given by ical hadron state with a specific J P comprises all possible Fock states with the same total spin and parity. As the − − − Vss√ 4Vt 2V1 4Vt most famous member of the XYZ family, X(3872) may be / | H | / = 2 . 1 2 S 1 2 Vss√−4Vt Vss −V − V − − 2Vt 2 1 2 2 explained as a mixture of charmonium and the molecular PC ++ P state with J = 1 [55]. Here we take the | jcl., J P − Similarly, for two states with J = 3/2 , basis to describe the mixing for the B-type baryons. Then | | two physical states characterized by different masses can be 3/2 HS 3/2 + − + V2 + Vss + 5Vss√16Vt denoted V1 2 4 4Vt = 4 5 . P P 5Vss+√16Vt 3V2 3Vss 4Vt |High, J cos φ sin φ |J − 1/2, J V1 − − + = . 4 5 2 4 5 |Low, J P − sin φ cos φ |J + 1/2, J P − For the J P = 5/2 state, (16) = , P = / − | | = , P = / − P − jl 2 J 5 2 HS jl 2 J 5 2 For example, two 1P c states with J = 1/2 can be 1 6 represented as = V1 + V2 + Vss − Vt . 2 5 ( ) . ◦ . ◦ c 2765 = cos 125 4 sin 125 4 | , , , , | , P ( ) − . ◦ . ◦ In the following, we denote Scl. L jcl. SQ J as jcl. J c 2702 sin 125 4 cos 125 4 − (17) for brevity. Then the notations | / and | / appearing | , / 1 2 3 2 × 0 1 2 . above are defined by |1, 1/2− 123 154 Page 6 of 17 Eur. Phys. J. C (2017) 77:154

+ Table 2 Predicted masses for c and c states of ours and other approaches in Refs. [10,11,56,57] compared to experimental data [1](inMeV) + States c baryons c baryons PDG [1] Prediction Ref. [10]Ref.[11] Refs. [56]PDG[1] Prediction Ref. [10]Ref.[11]Ref.[57]

| 1S, 1/2+ 2286.86 2286 2286 2286 2265 2470.88 2470 2476 2467 2466 | 2S, 1/2+ 2766.6 2772 2769 2766 2775 2968.0 2940 2959 2959 2924 | 3S, 1/2+ 3116 3130 3112 3170 3265 3323 3325 | 1P, 1/2− 2592.3 2614 2598 2591 2630 2791.8 2793 2792 2779 2773 | 1P, 3/2− 2628.1 2639 2627 2629 2640 2819.6 2820 2819 2814 2783 | 1D, 3/2+ 2843 2874 2857 2910 3054.2 3033 3059 3055 3012 | 1D, 5/2+ 2881.53 2851 2880 2879 2910 3079.9 3040 3076 3076 3004 | 2P, 1/2− 2939.3 2980 2983 2989 3030 3122.9 3140 3179 3195 | 2P, 3/2− 3004 3005 3000 3035 3164 3201 3204

Table 3 Predicted masses for c and c states of ours and other approaches in Refs. [9,10,56,58] compared to experimental data [1](inMeV) States c baryons c baryons PDG [1] Prediction Ref. [9]Ref.[10] Refs. [56]Ref.[58]PDG[1] Prediction Ref. [9]Ref.[10]

| 1S, 1/2+ 2452.9 2456 2439 2443 2440 2452 2575.6 2579 2579 2579 | 1S, 3/2+ 2517.5 2515 2518 2519 2495 2501 2645.9 2649 2654 2649 | 2S, 1/2+ 2846a 2850 2864 2901 2890 2961 2977 2984 2983 | 2S, 3/2+ 2876 2912 2936 2985 2996 3007 3035 3026 | 3S, 1/2+ 3091 3271 3035 3381 3215 3323 | 3S, 3/2+ 3109 3293 3200 3403 3236 3396 − | 1P, 1/2 l 2702 2795 2713 2765 2832 2839 2928 2854 − | 1P, 1/2 h 2766.6 2765 2805 2799 2770 2841 2900 2934 2936 − | 1P, 3/2 l 2785 2761 2773 2770 2812 2931 2921 2900 2912 − | 1P, 3/2 h 2801 2798 2798 2798 2805 2822 2932 2931 2935 | 1P, 5/2− 2790 2799 2789 2815 2796 2927 2921 2929 | 1D, 1/2+ 2949 3014 3041 3005 3075 3132 3163 + | 1D, 3/2 l 2952 3005 3040 3060 3089 3127 3160 + | 1D, 3/2 h 2964 3010 3043 3065 3081 3131 3167 + | 1D, 5/2 l 2942 2960 3023 3065 3091 3087 3153 + | 1D, 5/2 h 2962 3001 3038 3080 3077 3123 3166 | 1D, 7/2+ 2943 3015 3013 3090 3078 3136 3147 − | 2P, 1/2 l 2971 3176 3125 3185 3245 3094 3294 3267 − | 2P, 1/2 h 3018 3186 3172 3195 3256 3144 3300 3313 − | 2P, 3/2 l 3036 3147 3151 3195 3223 3172 3269 3293 − | 2P, 3/2 h 3044 3180 3172 3210 3233 3165 3296 3311 | 2P, 5/2− 3040 3167 3161 3220 3203 3170 3282 3303 a The mass value for the | 2S, 1/2+ state is taken from the measurement of BaBar [35]

− − Here we have denoted the physical states by their masses (see inated by |2, 1/2 , and the heavy c(2798) by |1, 1/2 . Table 3). The mixing angles for other states in Table 3 with The mixing of 2P states is similar to the 1P states. For the the same J P are listed in Table 4. 1D states, one notices that both 3/2+ and 5/2+, with heavier Our results of the mixing angles in Table 4 indicate that the masses, are dominated by smaller jcl. components. − − heavier 1/2 state, c(2765), is dominated by a |1, 1/2 The uncertainty may exist in the mixing angles. Firstly, the − component, while c(2702) is dominated by a |0, 1/2 loop corrections to the spin-dependent one-gluon-exchange − component. For two 3/2 states, the light c(2785) is dom- potential may be important for the heavy–light hadrons. As

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Table 4 The mixing angles for ( / −) ( / −) ( / −) ( / −) ( / +) ( / +) / 1P 1 2 1P 3 2 2P 1 2 2P 3 2 1D 3 2 1D 5 2 the 1P,2P,and1D c c states ◦ ◦ ◦ ◦ ◦ ◦ c 125.4 −156.8 124.8 −151.4 172.2 −175.6 . ◦ − . ◦ . ◦ − . ◦ . ◦ − . ◦ c 125 0 153 6 124 3 145 1 168 9 173 8

± an example, the lower mass of Ds(2317) compared with contain a light flavor meson, a charmed meson, a light flavor the old calculations [53] can be well explained by the cor- baryon, e.g., π, K, D, p, and .Fortheβ of these hadrons, rected spin-dependent potential [59,60]. If we use this type of the following potential will be used: potential in our calculation, of course, the mixing angle will 2 2 α 3 3 32α σ 3e−σ r change. Secondly, the mixing angles depend on the parame- V (r) = F · F s − br + C + √s S · S , q1 q2 π q1 q2 ters. Thirdly, there are other mechanisms, e.g., hadron loop r 4 4 9 mq mq effects [61], which may contribute to the mixing phenomenon (18) in hadron physics. Anyway, we expect that the mixing angles in Table 4 reflect the main features of the mixing states. Due where to the uncertainties of the mixing angles, however, we ignore the mixing effects as the first step to the study of the decays of charmed excitations in the next subsection. Obviously, it is a good approximation only when the mixing effects are not large. Fortunately, this crude procedure is partially sup- α ported by the former analysis of charmed mesons [62–64]. If Here, the parameters s and b are taken as 0.45 and 0.145 ν+1 the decay properties obtained in this way describe the prin- GeV as in Table 1, respectively. To reproduce the masses cipal characteristics of the mixing states, the angles obtained of the light quark clusters in Table 1, the masses of u/d, s are by the potential model may be overestimated. fixed as 0.195 and 0.380 GeV. While σ and C are treated as adjustable parameters, the masses of the π/ρ, K/K ∗, D/D∗, p/, and families are fitted with experimental data. Mean- 2.4 Simple harmonic oscillator (SHO) β values while, the values of β for the corresponding states are also obtained; they are collected in Table 6. In the next section, the Okubo–Zweig–Iizuka (OZI) allowed Before ending this section, we briefly summarize the com- decays will be calculated for the 1P and 2S charmed baryons plicated deduction presented here. Firstly, the dynamics of where the SHO wave functions are used to evaluate the tran- heavy baryon is simplified as a two-body system when the sition factors via the 3 P model. We will also discuss the 0 symmetric configuration is considered. Secondly, the mass mixing effects for the decays of the relevant states. Following matrices were calculated in the jjcoupling scheme. By solv- the method of Ref. [65], all values of the SHO wave func- ing the Schrödinger equation, we obtain the mass spectra and tion scale, denoted β in the following, are calculated (see mixing angles for the relevant states. For estimating the two- Table 5). The values of β reflect the distances between the body strong decays in the next section, finally, we also present light quark cluster and the c quark. the values of the SHO wave function scale for all initial and In our calculation of strong decays, we will consider the final states. structures of light diquarks. What is more important is that the possible final states of an excited charmed baryon may

3 Strong decays Table 5 The meson effective β values in GeV In this section, we will use the formula provided by Eichten, States + c c c c Hill, and Quigg (EHQ) [66] to extract the decay widths of 1S excited charmed baryons. Since the dynamical behavior of 1/2+ 0.291 0.331 0.335 0.362 the heavy–light hadrons is governed by the light degrees 3/2+ 0.296 0.315 of freedom in the limit of heavy quark symmetry, a dou- 2S 0.145 0.162 0.144 0.152 blet formed by two states with the same jcl. but different J 3S 0.102 0.113 0.098 0.103 shall have similar decay properties. More speciﬁcally, the 1P 0.184 0.205 0.182 0.192 transitions between two doublets should be determined by 2P 0.117 0.130 0.112 0.118 a single amplitude which is proportional to the products of 1D 0.142 0.156 0.136 0.143 four Clebsch–Gordan coefﬁcients [46]. Some typical ratios of excited charmed baryons with negative parity were pre- 123 154 Page 8 of 17 Eur. Phys. J. C (2017) 77:154

Table 6 The effective β values in GeV for the light quark cluster and various hadrons (the second row). The values of σ and C are given in the square brackets for various hadron structures (the third row) [qq]{qq}[qs]{qs} πρ KK∗ DD∗ p

0.201 0.143 0.207 0.159 0.298 0.179 0.291 0.201 0.250 0.230 0.189 0.226 [1.17, 0.39] [1.57, 0.38] [0.73, 0.63] [0.83, 0.48] [1.20, 0.63] [−, 0.38] [−, 0.26] dicted by this law [46]. Later, a more concise formula (the 4 Discussion EHQ formula) was proposed for the widths of heavy–light mesons [66]. The EHQ formula has been applied systemat- 4.1 Experimentally well-established 1S and 1P states ically to the decays of excited open-charm mesons [62–64]. Recently, the EHQ formula has been extended to the study At present, all the ground states and 1P G-type charmed states of the decay properties of 1D c and c states [11]. have been experimentally established [1]. These states have For the charmed baryons, the EHQ formula can be written been observed, at least, by two different collaborations, and as their properties including masses and decays have been well determined. With good precision, the strong decays of these s , j ,J 2 , 2 2 ˜2 A→BC = ξ C Q B B M jA jB (q) q2+1 e−q /β , jC , jC , jA,JA jC , states provide a crucial test of our method. (20) Among the 1S charmed baryons, the measurements of c(2455) and c(2520) have been largely improved [2,3] where ξ is the flavor factor given in Table 13 in Appendix B. (see Table 7). In our calculation, the mass and decay width ++ q =|q| denotes the three-momentum of a final state in the of c(2520) measured by CDF will be taken as input 3 rest frame of an initial state. A and B represent the initial and data to fix the constant γ peculiar to the P0 model. With the final heavy–light hadrons, respectively. C denotes the light transition factor for the process c(2520) → c(2286) + π flavor hadron (see Fig. 3). The explicit expression of β˜ is [see Eq. (A.12) in Appendix A], the value of γ is fixed as sQ , jB ,JB . 2 given in Eq. (A.11) in Appendix A. In addition, C is 1 296. jC , jA,JA a normalized coefficient given by the following equation: As shown in Tables 8 and 9, the predicted widths of other G √ ground and 1P -type charmed baryons are well consistent s , j ,J + + + C Q B B = (−1)JA jB jC sQ (2 j + 1)(2J + 1) with experiments. Our results of mass spectra and decay jC , jA,JA A B ( )+ ( )+ ( )0,+ s j J (21) widths indicate that c 2595 , c 2625 , c 2790 , × Q B B , ,+ ( )0 P G jC JA jA and c 2815 can be accommodated with the 1 - + 0,+ type charmed baryons. c(2595) and c(2790) can / − ( )+ where jC ≡sC + . The symbols sC and represent the be classified into the 1 2 states while c 2625 and 0,+ − spin of the light hadron C and the orbital angular momentum c(2815) into the 3/2 states. The predicted mass split- , − − relative to B, respectively. The transition factors M jA jB (q) tings between the 1P 1/2 and 3/2 states are 25 MeV jC , involved in the concrete dynamics can only be calculated by and 27 MeV for the c and c baryons, respectively, which various phenomenological models. For the decays of heavy– are also consistent with the experiments. The assignments of + + 0,+ 0,+ light mesons, transition factors have been calculated by the c(2595) , c(2625) , c(2790) , and c(2815) are 3 relativistic chiral quark model [67] and the P0 model [62, also supported by other work [9–12] in which the light quark 3 64,68]. In our work, we will employ the P0 model [69–71] cluster scenario was also employed. In addition, the mass to obtain the transition factors. More details of an estimate spectra obtained by different types of the quark potential of the transition factors are given in Appendix A. models in the three-body picture also support these assignments [7,8,56–58]. However, the investigations by QCD sum rules indicate that these 1P candidates may have more complicated structures [14–16]. Especially, the work by Chen + + et al. suggested that c(2595) and c(2625) form the ˜ − − heavy doublet c1(1/2 , 3/2 ) (the same assignments as 0,+ 0,+ the case of c(2790) and c(2815) )[16], which is

2 For the different conventions to extract the color and ﬂavor factors, Fig. 3 The two topological diagrams for an excited charmed baryon the value of γ here is different from those in Refs. [24,73,74]. The A decaying into the ﬁnal states B and C.Thebrown line 3 denotes a deviation, of course, does not affect the predictions since γ is regarded 3 charm quark as an adjustable parameter in the P0 model. 123 Eur. Phys. J. C (2017) 77:154 Page 9 of 17 154

Table 7 The masses and widths ( )++ ± ± ± (in units of MeV) of c 2455 2453.90 0.13 0.14 2.34 0.47 CDF (2455)++ and (2520)++ ± ± ± ± +0.07 c c 2453.97 0.01 0.02 0.14 1.84 0.04−0.20 Belle measured by CDF [2]and ++ (2520) 2517.19 ± 0.46 ± 0.14 15.03 ± 2.52 CDF Belle [3] c ± ± ± ± +0.18 2518.45 0.10 0.02 0.14 14.77 0.25−0.30 Belle

Table 8 Open-flavor strong decay widths of 1S c and c in MeV According to the predicted masses in Table 3, ( )0,+,++ | , / − c 2800 could be assigned to either the 1P 3 2 l , 1S c and c − − or the |1P, 3/2 h,orthe|1P, 5/2 states. When we con- + + 1/2 3/2 sider the decay properties of these three states (see Table ( )++ ( )+ ( )++ ( )+ − c 2455 c 2580 c 2520 c 2645 10), the possibility of assignment to the |1P, 3/2 l state +π + +π + 0π + c 1.53 c Input c 1.54 can be excluded since the Belle Collaboration observed this +π 0 +π 3 c 1.01 state in the c mode. At present, the Belle Collabora- 0,+,++ 1.53 − Input 2.55 tion tentatively identified c(2800) as members of the + . − . 0 09 − . ± . . ± . c2(3/2 ) isospin triplet, which agrees with our results of 1 89−0.18 [1] 14 9 1 5[1]26 0 6[72] both mass spectrum and strong decays. When the measured 0 − mass of c(2800) (2806 MeV) is used for the c2(3/2 ) Table 9 Open-flavor strong decay widths of 1P c and c in MeV state, the predicted width is about 40.1 MeV, which is com-

1P c and c parable with the experiment [34]. However, we notice that the quantum number J P of (2800)0,+,++ has not yet been / − / − c 1 2 3 2 measured. Then the possibility of this state as the (5/2−) ( )+ ( )+ ( )+ ( )+ c2 c 2595 c 2790 c 2625 c 2815 candidate cannot be excluded by our results since the decay π π π π + c 2.78 c 6.01 c 0.04 c 0.15 π mode of c is dominant for this state. In addition, the pre- ∗π 4.09 − c dicted mass and total width of the c2(5/2 ) state are also 2.78 6.01 0.04 4.24 0,+,++ compatible with experimental data of the c(2800) 2.6 ± 0.6[1]8.9 ± 1.4[4] <0.97 [1]2.43 ± 0.37 [4] baryon. Therefore, we would like to point out that the signal 0,+,++ of c(2800) found by Belle might be their overlapping structure. We hope the future experiments measure the different from our conclusion. Since the quantum numbers following branching ratios to disentangle this state. P − of J have not yet been determined for these 1P charmed For the c2(3/2 ) state, states, more experiments are required in the future. B( ( / −) → ( )π) c2 3 2 c 2455 = . ; − 1 90 (22) 0,+,++ B(c2(3/2 ) → c(2520)π) 4.2 1P c states − B(c2(3/2 ) → c(2286)π) As shown in Tables 2 and 3,themassesof1P c states are = 4.07. (23) + B( ( / −) → ( )π) predicted in the range of 2700–2800 MeV. Then c(2765) c2 3 2 c 2455 0,+,++ and c(2800) can be grouped into the candidates of ( / −) − For the c2 5 2 state, 1P c family. The predicted mass of the |1P, 1/2 h state is B( (5/2−) → (2455)π) about 2765 MeV, which is in good agreement with the mea- c2 c = 0.58; (24) + B( ( / −) → ( )π) sured mass of c(2765) . In addition, the theoretical result c2 5 2 c 2520 − for the decay width of the c1(1/2 ) state in Table 10 is B( ( / −) → ( )π) about 63.52 MeV, which is also in agreement with the mea- c2 5 2 c 2286 = . . + − 9 85 (25) surements [30–32]. Furthermore, the signal of c(2765) B(c2(5/2 ) → c(2455)π) has been observed in the (2455)π intermediate state, while c As mentioned earlier, the signal (2850)0 discovered by the there is no clear evidence for the decay of (2765)+ through c c BaBar Collaboration may be a J = 1/2 state. If (2850)0 is (2520)π [31,32]. This is also consistent with our results c c the 1/2+(2S) state, the corresponding ratios (see Sect. 4.5) of the |1P, 1/2− . Based on the combined analysis of the h are different from Eqs. (22)–(25). So the measurements of mass spectrum and strong decays, we therefore conclude + that c(2765) could be regarded as a good candidate of 3 ( / −) ( / −) ( / −) Even if the possible mixing between c1 3 2 and c2 3 2 c1 1 2 . Considering the uncertainties of the quark poten- +π is considered, the partial width of c is only 3.87 MeV for the tial models, the masses obtained by Refs. [9,10,56] are not |1P, 3/2− state where the mixing angle obtained in Table 4 has been + l contradictory to our assignment to c(2765) . used. 123 154 Page 10 of 17 Eur. Phys. J. C (2017) 77:154

Table 10 The partial and total − − − Decay modes 1/2 (1P) 3/2 (1P) 5/2 (1P) decay widths of 1P c states in MeV c0(2702)c1(2765)c1(2798)c2(2785)c2(2790)

cπ 3.64 ××24.06 24.63

c(2455)π × 58.94 3.48 5.22 2.50

c(2520)π × 1.70 63.72 2.47 4.34

c(2595)π 2.88 2.31 1.93 0.03

c(2625)π 3.12 0.07 0.63 Theory 3.64 63.52 72.63 33.75 32.13 ≈ +22 Expt. [1] 50 72−15

0,+,++ these ratios of branching fractions can help us understand and c(2800) is about 130 MeV, which is compa- 0,+,++ 0 the nature of c(2800) and c(2850) . rable with the mass differences among sextet states of the − − Although, at present, the c0(1/2 ) and c1(3/2 ) ground charmed baryons [21]. With a chiral quark model, ( )0 states are still missing in the experiments, our results Liu et al. also analyzed the c 2930 by the two-body ( / −) ( )0 indicate that the c0 1 2 state may be a narrow res- strong decays [23]. Their results support c 2930 as the +π |2 , / − |4 , / − onance and its predominant decay channel c is only c Pλ 1 2 or c Pλ 1 2 state. Since the heavy quark about 3.64 MeV (see Table 10). Since the decay mode of symmetry was not considered in Ref. [23], the notations of − c(2520)πis the largest for the c1(3/2 ) state, we sug- charm-strange baryons in Ref [23] are different from our ( / −) ( / −) gest to search this channel for this state in future experi- c0 1 2 and c1 1 2 . Although the results in Table 11 + ments. In the heavy quark limit, the branching ratio for the indicate that the c K decay mode dominates the decay of ( / −) ( / −) c1 3 2 state, c0 1 2 state, the mass of this state is predicted to be about 2840 MeV, which is much smaller than 0(2930). In addi- B( (3/2−) → (2455)π) c c1 c = . , tion, the + K decay mode is forbidden for the (1/2−) B( ( / −) → ( )π) 0 04 (26) c c1 c1 3 2 c 2520 0( ) state. Thus, according to our results, c 2930 is unlikely − P = / − is much smaller than c2(3/2 ) (Eq. 22). to be a 1P state with J 1 2 . 0,+ Another charm-strange baryon, c(2980) , is slightly 0,+ 4.3 1P c states higher than the predicted mass range of 1P c states. ( ) ( )π This state has been observed in c 2455 K , c 2580 , ( )π + ¯ π As shown in Table 3, the predicted mass of 1P is in the c 2645 , and nonresonant c K decay channels. How- c + ¯ range from 2840 to 2930 MeV. Then the resonance struc- ever, it was not seen in the decay modes of c K and − π ture observed by BaBar [36] in the decay channel B → c [37–39]. Comparing the mass and decay properties of ,+ ( )0¯ − → + −¯ − (2980)0 with our results, the possibility of a 1P c 2930 c c K c with an invariant mass of 2.93 c c state might be excluded. As shown in the next subsection, GeV could be a good candidate of 1P c members. The ( )0,+ results of the decays in Table 11 favor (2930)0 as the c 2980 could be a good 2S c candidate. Based on c − − − 0 ( / ) (3/2 ) or (5/2 ) state. Then (2930) might be our results on strong decays, we ﬁnd that the c1 1 2 and c2 c2 c − 0,+,++ ( / ) regarded as the strange partner of c(2800) by our c1 3 2 are quite narrow (see Table 11). ( )0 results. Interestingly, the mass difference between c 2930

Table 11 The partial and total / − ( ) / − ( ) / − ( ) Decay modes 1 2 1P 3 2 1P 5 2 1P decay widths of 1P c states in MeV ( ) ( ) ( ) ( ) ( ) c0 2839 c1 2900 c1 2932 c2 2921 c2 2927

c K 46.59 ××11.59 12.43

cπ 4.39 ×× 7.42 7.75 ( )π × c 2580 9.44 0.76 1.20 0.57 ( )π × c 2645 0.52 3.23 0.75 1.31 c(2790)π 0.01 Theory 50.98 9.96 4.00 20.96 22.06 Expt. 36 ± 7 ± 11 [36]

123 Eur. Phys. J. C (2017) 77:154 Page 11 of 17 154

+ +,0 Table 12 The partial and total decay widths of 2S c and c states in MeV 1/2+ (2S) 1/2+ (2S) 3/2+ (2S) ( )+ ( )( )0 ( )( )0 ( ) c 2765 c 2980 c 2850 c 3000 c 2880 c 3030 ( )π ( ) +π + +π + c 2455 26.23 c 2455 K 3.14 c 35.11 c K 17.42 c 34.96 c K 18.37 ( )π ( )π ( )π ( )π ( )π ( )π c 2520 19.28 c 2580 11.47 c 2455 57.16 c 2580 12.56 c 2455 15.98 c 2580 3.50 ( )π ( )π ( )π ( )π ( )π c 2645 12.83 c 2520 17.54 c 2645 4.13 c 2520 54.52 c 2645 12.92 c(2595)π 6.92 c(2790)π 5.89 c(2595)π 1.07 c(2790)π 0.40

c(2625)π 1.57 c(2815)π 0.13 c(2625)π 7.62 c(2815)π 6.22 0 0 D n 0.03 c(2455)K 15.34 D n 3.03 c(2455)K 6.49 D0 0.01 45.51 27.44 118.33 55.47 117.18 47.91 ≈ ± . +1.0 +33 50 [1] 28.1 2 4−5.0 [4]86−22 [35]

+ 0,+ ( ) 4.4 2S c and c states c 2470 , the branching ratio B( ( ) → ( )π) c 2980 c 2580 = . According to the mass spectrum (see Table 2), c/ 0 89 (29) + B(c(2980) → c(2645)π) c(2765) can also be regarded as the first radial (2S) exci- + P + ( ) tation of the c(2286) with J = 1/2 . Interestingly, the is predicted for c 2980 , which can be tested by future results of strong decays in Table 12 do not contradict this experiments. Recently, the ratio of branching fractions + 0 + assignment. Our calculation indicates that the decay chan- B(c(2980) → (2580) π ) ( )π +( ) c ≈ 75% nel c 2455 is a dominant decay channel for the c 2S + 0 + + − + B(c(2815) → c(2645) π → c π π ) state. This is in line with the observations by Belle [31,32]. At (30) / +( )+ / −( )+ present, both 1 2 2S c and 1 2 1P c are possible + for the assignment of c/c(2765) . However, there is the has been estimated by the Belle Collaboration [4]. Com- very important feature for experiments to distinguish these bining this with the predicted partial widths of c(2815) two assignments in future. Specifically, we suggest to search and c(2645) in Tables 8 and 9, the branching fraction + B( ( )+ → ( )0π +) c/c(2765) in the channel of c(2520)π.Asshownin c 2980 c 2580 is evaluated to be about . Table 12, the channel c(2520)π is large enough to find 40%, which is consistent with our direct result of 41 8%. +( ) the c 2S state. On the other hand, this mode seems to be − 0,+,++ 0,+ too small to be detected for the c1(1/2 ) (see Table 10). 4.5 2S c and c states Explaining the criteria concretely, we give the following + + branching ratios for these two states. In Table 3,massesofthe2S c(1/2 , 3/2 ) states are For the c(2S) state, predicted as 2850 and 2876 MeV, respectively. The neu- 0 tral c(2850) found by the BaBar Collaboration in the − + − B(c(2765) → c(2520)π) → ( )0 ¯ → π ¯ = 0.74. (27) decay channel B c 2850 p c p [35] can B( ( ) → ( )π) P + c 2765 c 2455 be regarded as the 2S c state with J = 1/2 .Themass and width of the neutral (2800)0 and (2850)0 are col- ( / −) c c For the c1 1 2 state, lected here: + + B(c(2765) → c(2520)π) (2800)0 : m = 2806 5 MeV,= 72 22 MeV; = 0.03. (28) c −7 −15 B( (2765) → (2455)π) ( )0 : = ± ± ,= +33 . c c c 2850 m 2846 8 10 MeV 86−22 MeV

The branching ratio of B(c(2520)π)/B(c(2455)π) for For lack of experimental information, at present, PDG − 0 0,+,++ the c1(1/2 ) state is roughly an order of magnitude smaller treated c(2850) and c(2800) as the same state [1]. than c(2S).Ifc(2765) is the 2S excitation, c(2980) As pointed out by the BaBar Collaboration [35], however, could be a good candidate as its charm-strange analog [21]as there are indications that these two signals detected by ( ) ∗ seen in Table 12. The mass difference between c 2765 and Belle [34] and BaBar [35] are two different c states. The c(2980) is about 200 MeV, which nearly equals the mass main reasons are listed as follows: difference between c(2287) and c(2470). The predicted 0,+,++ 0 width of c(2980) is 27.44 MeV which is in good agreement 1. Although the widths of c(2800) and c(2850) with the experimental data [4,39]. As the 2S excitation of are consistent with each other, their masses are 3σ apart. 123 154 Page 12 of 17 Eur. Phys. J. C (2017) 77:154

B( (3030) → (2580)π) 2. The Belle Collaboration tentatively identified the c c = 0.27. (35) 0,+,++ B( ( ) → ( )π) c(2800) as the J = 3/2 isospin triple, while the c 3030 c 2645 0 BaBar Collaboration found weak evidence of c(2850) as a J = 1/2 state. 5 Summary and outlook 0,+,++ Our results also indicate that c(2800) and 0 ρ λ c(2850) are the different c excited states. One notices In principle, both and modes can be excited in a baryon + ρ that the predicted mass of 1/2 (2S)c state in this work system. For charmed baryons, the excitation energies of the and in Ref. [9] are around 2850 MeV. Even the results in and λ modes are different due to the heavier mass of a c quark. Refs. [10,56] are only about 50 MeV larger than the mea- For the ordinary confining potential, such as the linear or har- surements. Due to the intrinsic uncertainties of the quark monic form, the excited energy of the ρ mode is larger than 0 λ potential model, it is appropriate to assign c(2850) as a the mode [44]. Hence the low-excited charmed baryons 2S 1/2+ state. More importantly, the predicted decay width may be dominated by the λ mode excitations. Recently, the + of c(1/2 , 2S) state is 118.36 MeV, which is comparable investigation by Yoshida et al. confirmed this point [75]. Fur- with the measurement by BaBar [35]. The partial width of thermore, they find that the ρ and λ modes are well separated 0 cπ is 35.11 MeV, which can explain why c(2850) was for the charmed and bottom baryons, which means the com- first found in this channel. We find that the decay modes of ponent of the ρ mode can be ignored for the low-excited c(2455)π and c(2520)π are also large. Finally, we give charmed baryons. Interestingly, Refs. [9–11]havealsoshown the following branching ratios: that the masses of existing charmed baryons can be explained by the λ mode. Hence, our study of strong decays of the low- B( (2850) → (2455)π) c c = . , excited charmed baryons is an important complement to this B( ( ) → ( )π) 3 26 (31) c 2850 c 2520 work [9–11,75].

B(c(2850) → c(2286)π) Up to now, several candidates of the 1P and 2S charm = 0.61, (32) and charm-strange baryons have been found by experiments, B(c(2850) → c(2455)π) and some of them are still open to debate. To better under- 0 which can be tested by future experiments. If c(2850) stand these low-excited charmed baryons, in this paper, we + is the 1/2 (2S) state, the mass of its doublet partner carry out a systematical study of the mass spectra and strong in the heavy quark effective theory is predicted as 2876 decays for the 1P and 2S charmed baryon states in the frame- MeV (denoted c(2880)). According to the predicted decay work of the non-relativistic constituent quark model. The +π widths in Table 12, this state might also be broad. c , masses have been calculated in the potential model where c(2455)π, and c(2520)π are also dominant for the decay the charmed baryons are simply treated as a quasi-two-body of c(2880). The ratio of (c(2455)π)/ (c(2520)π) system in a light quark cluster picture. The strong decays are for c(2880) is different from c(2850), whose numerical computed by the EHQ decay formula where the transition 3 value is given by factors are determined by the P0 model. When calculating the decays, the inner structure of a light quark cluster has B(c(2880) → c(2455)π) = 0.29. (33) also been considered. Except for the unique parameter γ of B( (2880) → (2520)π) c c the QPC model, the parameters in the potential model and in Even though the strange partners of c(2850) and c(2880) the EHQ decay formula have the same values. have not been found by any experiments, their decay proper- The well-established ground and 1P G-type charmed ties are calculated and presented in Table 12. Our results indi- baryons provide a good test to our method. The experimental + ( )π ( ) cate that c K , c 2580 , and c 2455 K are the domi- properties including both masses and widths for these states ( ) P = / + nant decay modes of the c 3000 state with J 1 2 , can be well explained by our results. This success has made + ( )π ( ) 4 while c K and c 2645 are those of the c 3030 . us more conﬁdent of our predictions for other 1P and 2S Besides the masses and decay widths, the following branch- states. Our main conclusions are as follows. + + ing ratios may also be valuable for future experiments: The broad state c(2765) (or c(2765) ) which is still / +( )+ B( (3000) → (2580)π) ambiguous could be assigned to the 1 2 2S c ,orthe c c = 3.04, (34) / −( )+ B( ( )π)/ B( ( ) → ( )π) 1 2 1P c1 state. The branching ratio c 2455 c 3000 c 2645 B(c(2520)π) is found to be different for these two assignments, which may help us understand the nature of this state. 0,+,++ c(2800) observed by the Belle Collaboration 4 ( ) ( ) If c 2980 is the ﬁrst radial excited state of c 2470 . Then our in e+e− annihilation processes [34] can be regarded as predicted masses for 2S charm-strange baryons may be about 20–30 P = / − / − MeV lower than the experiments. To compensate for this difference, a negative-parity state with J 3 2 ,or52 ,or we increase by about 25 MeV for the 2S c states in this case. their overlapping structure. We suggest to measure the 123 Eur. Phys. J. C (2017) 77:154 Page 13 of 17 154

B(c(2455)π)/B(c(2520)π) in the future. Another neu- of excited charmed baryons. As an example, the process 0 − tral state, c(2850) , which was found in the B meson c(2520) → c(2280)π will be constructed and the transi- decay [35] could be a good candidate for the first radial tion factor for the EHQ formula will be extracted. excited state of c(2455). With the above assignments, the As pointed out in Sect. 3, there are two possible decay ratios of B(c(2287)π)/B(c(2455)π) shall be very dif- processes for an excited charmed baryon state (see Fig. 3). 0,+,++ 0 ferent for c(2800) and c(2850) , i.e., 4.07 for The final states of the left figure contain a charmed baryon 0,+,++ 0 c(2800) and 0.61 for c(2850) . The puzzle of and a light meson. The right one contains a charmed meson 0,+,++ 0 3 c(2800) and c(2850) may be disentangled if these and a light baryon. If a baryon decays via the so-called P0 branching ratios are measured in the future. In addition, the mechanism, a quark–antiquark pair is created from the vac- ratio of branching fractions B(c(2455)π)/B(c(2520)π) uum and then regroups two outgoing hadrons by a quark 0 for c(2850) is predicted to be 3.26. rearrangement process. In the non-relativistic limit, the tran- ˆ 3 The analysis of the mass and decay properties supports that sition operator T of the P0 model is given by 0,+ c(2980) is the 2S excitation (the first radial excited state 0 Tˆ =− γ , ; , − | , 3 3 δ3( + ) of c(2470)). The existence of c(2930) is still in dispute. 3 1 m 1 m 0 0 d k4d k5 k4 k5 ( / −) ( / −) m If it exists, the assignments of c2 3 2 and c2 5 2 are − possible. In other words, it could be regarded as a strange part- ×Ym(k4 k5 )ω(4,5)ϕ(4,5)χ (4,5) †( ) †( ), 0,+,++ 1 ,− d k4 d k5 (A.1) ner of c(2800) . Some useful ratios of partial decay 2 0 1 m 4 5 widths are also presented for (2980)0,+ and (2930)0. c c ω(4,5) ϕ(4,5) Although both the masses and the strong decays have been where and 0 are the color and flavor wave func- ¯ ω(4,5) = explained in the heavy quark–light quark cluster picture for tions of the q4q5 pair created√ from the vacuum. Thus, √ ( ¯ + ¯ + ¯ )/ ϕ(4,5) = ( ¯ + ¯ + ¯)/ the observed 2S and 1P candidates, it is not the end of the RR GG B B 3 and 0 uu dd ss 3 are color and flavor singlets. The pair is also assumed to story of the study of the excited charmed baryon states. Inves- ++ tigation of the ρ mode excited states with higher energies are carry the quantum number of 0 , suggesting that they 3 χ (4,5) also important to identify the effective degrees of freedom of areina P0 state. The 1,−m represents the pair produc- charmed baryons. However, this topic needs much laborious tion in a spin triplet state. The solid harmonic polynomial Ym() ≡||Ym(θ ,φ ) work and is beyond the scope of the present work. In addition, 1 k k 1 k k reflects the momentum-space dis- ¯ γ the quark model employed here neglects the effect of virtual tribution of the q4q5. is a dimensionless constant which hadronic loops. In the future, a more reasonable scheme for expresses the strength of the quark–antiquark pair created γ studying the properties of heavy baryons will be obtained by from the vacuum. The value of is usually fixed by fitting the unquenched quark model. Another topic which is left as the well-measured partial decay widths. a future task is to calculate the sum rules among the branch- When the mock state [77] is adopted to describe the spatial wave function of a meson, the helicity amplitude ing fractions of charmed baryons by applying the technique , , M jA jB jC ( ) found in Ref. [76]. q can easily be constructed in the LSbasis [71]. The mock state for an A meson is Acknowledgements Bing Chen thanks Franz F. Schöberl for the pack- 2SA+1 JA jA A n A L PA age which is very useful to solve the Schrödinger equation. This project A is supported by the National Natural Science Foundation of China under ≡ ω123φ123 3 3 3 δ3 + + − Grant Nos. 11305003, 11222547, 11175073, 11447604, and U1204115. A A d k1d k2d k3 k1 k2 k3 PA Xiang Liu is also supported by the National Program for Support of A Top-notch Young Professionals. × L AlA k , k , k q k q k q k . n A 1 2 3 1 1 2 2 3 3 (A.2) Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm As for the left decay process in Fig. 3, the wave function of ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, a B baryon can be constructed in the same way. The wave and reproduction in any medium, provided you give appropriate credit function of a C meson is to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. + J j C n 2SC 1 L C C P Funded by SCOAP3. C C C ≡ ω15φ15 3 3 δ3 + − C C d k1d k5 k1 k5 PC C j , j Appendix A: Transition factor M A B (q) in the QPC × LC lC k , k q k q¯ k . jC , nC 1 5 1 1 5 5 model (A.3) ( = , ) In the following, we will show how to obtain the partial wave Here, the symbols of i i A B, and C represent the 3 amplitudes by the P0 strong decay model for the decays Clebsch–Gordan coefficients for the initial and final hadrons, 123 154 Page 14 of 17 Eur. Phys. J. C (2017) 77:154 which arise from the couplings among the orbital, spin, and functions are employed to describe the spatial wave function total angular momentum and their projection of lz and sz to of a hadron. In the momentum space, the SHO radial wave ( = , ) ψn ( ) jz. More specifically, i i A B, and C are given by function, Lm q , is given by = , , ψn (q) s1m1 s2m2 s12m12 s12m12 s3m3 SAsA Lm A q2 (− )n ( − )! L − + / 2 × , , 1 2 2n 1 q 2β2 L 1 2 q L AlA SAsA JA jA = / e L Y (q), β3 2 (n+L+ 1 ) β n−1 β2 Lm 2 = s2m2, s5m5 s25m25 s25m25, s3m3 SBsB B (A.8) × L BlB, SB sB JB jB , with q = (mi k j − m j ki )/(mi + m j ) and YLm(q) = L+1/2 = s1m1, s4m4 SC sC LClC , SC sC JC jC , | |L ( ) ( 2/β2) C q YLm q . Ln−1 q is an associated Laguerre polynomial. The values of the SHO wave function scale respectively. , , parameter β have been given in Tables 5 and 6. In the light The helicity amplitude M jA jB jC (q) is defined by quark cluster picture, the wave function of a charmed baryon ˆ 3 jA, jB , jC BC|T |A =δ (PA − PB − PC )M (q), (A.4) can easily be constructed. Taking the A baryon as an example, the wave functions corresponding to the 1S,2S, and 1P where q represents the momentum of an outgoing meson in states are given as follows, respectively: the rest frame of a meson A. For comparison with experi- 2 2 − ( + ) − ( + ) ments, one obtains the partial wave amplitudes M (q) via / 1 m1k2 m2k1 1 m1 m2 k3 mQ k1 k2 LS 3 4 − + − + + 3 β2 m1 m2 β2 m1 m2 mQ 0 = 2 dA 2 A ; the Jacob–Wick formula [78], 00 / / e π 3/2β3 2β3 2 √ dA A + 2L 1 2 2 − ( + ) − ( + ) M ( ) = | 1 m1k2 m2k1 1 m1 m2 k3 mQ k1 k2 LS q L0JjA JA jA / − + − + + J + 1 33 4 2β2 m1 m2 2β2 m1 m2 mQ 2 A 1 =−√ / / e dA A jB , jC 00 π3/2β3 2β3 2 6 dAA jA, jB , jC 2 ×J j , J j |Jj M (q). (A.5) (m1+m2)k3−m Q (k1+k2) B B C C A × 3 − 2 ; β2 m1+m2+m Q ( → ) A Then the decay width A BC is derived analytically 2 2 − ( + ) − ( + ) √ 1 m1k2 m2k1 1 m1 m2 p3 m Q k1 k2 in terms of the partial wave amplitudes in the A rest frame, 3/4 − + − + + 3 ×2 2/3 β2 m1 m2 β2 m1 m2 m Q 0 = 2 dA 2 A 1m 3/2 5/2 e EB EC 2 πβ β (A → BC) = 2π q , |MLS(q)| . (A.6) dA A MA L S ( + ) − ( + ) ×Y m1 m2 k3 m Q k1 k2 . 1m m1+m2+m Q Finally, the full expression of MLS(q) in the rest frame of the baryon A is With the help of Eq. (A.7), the transition amplitude can be ( ) → MLS(q) =−3γ L0; Jj|JA jA JB jB; JC jC |Jj obtained. In the following, we take the process c 2520 ( )+π li ,m j c 2280 as an example. The wave functions of initial and final states are ×s1m1; s2m2|sdAm12 sdAm12; s3m3|SAsA 2 2 × ; | ; | − ( + ) − ( + ) SAsA L AlA JA jA s2m2 s m sdBm 1 m1k2 m2k1 1 m1 m2 k3 m3 k1 k2 5 5 25 / − + − + + 33 4 2β2 m1 m2 2β2 m1 m2 m3 A = / / e dA A ; ×s m ; s m |S s S s ; L l |J j π3/2β3 2β3 2 dB 25 3 3 B B B B B B B B dA A 2 2 − ( + ) − ( + ) 1 m5k2 m2k5 1 m2 m5 k3 m Q p2 k5 ×s m ; s m |S s / − + − + + 1 1 4 4 C C 33 4 2β2 m2 m5 2β2 m2 m3 m5 = / / e dB B ; B π3/2β3 2β3 2 × ; | ; | − dB B SC sC LClC JC jC s4m4 s5m5 1 m 2 − 1 m1k4 m4k1 − + ×1, m; 1, −m|0, 0 1 2β2 m1 m2 ψ =− / e C . C π3/4β3 2 ×ϕ235ϕ14|ϕ45ϕ123 ω235ω14|ω45ω123 ··· 3 ··· C B C 0 A B C 0 A d k1 BasedonEq.(A.7), we obtain the amplitude: ×d3k δ3(k +k +k )δ3(q−k −k )δ3(q + k +k + k ) 5 1 2 3 1 4 2 3 5 − 2 3g − 4 fg g q2 3 ∗ M ( ) =− 4 f , ×δ (k + k ) (k , k , k ) (k , k , k ) 1 1 q / / / / / pe 4 5 A 1 2 3 B 1 2 4 2 8π 5/4 f 5/2λ3/2β3 2β3 2β3 2β3 2β3 2 A dA B dB C ∗ k − k ×ψ (k , k )Ym 3 4 , (A.7) (A.9) C 3 5 1 2 where where i = A, B, C and j = 1, 2,...,5. The color matrix 2 235 14 45 123 1 1 1 μ element ω ω |ω ω is a constant which can be f = + + − ; B C 0 A β2 β2 β2 λ absorbed into the parameter γ . The flavor matrix element 2 dA 2 dB 2 M 4 ξ =ϕ235ϕ14|ϕ45ϕ123 will be presented in the next sub- 1 ε ε μν B C 0 A g = + 3 + 4 − ; section. To obtain the analytical amplitudes, the SHO wave β2 β2 β2 λ dA dB M 2 123 Eur. Phys. J. C (2017) 77:154 Page 15 of 17 154

2 2 2 2 1 ε ε ε ν Table 13 The ﬂavor matrix element ξ for different decay channels of h = + 2 + 3 + 4 − ; β2 β2 β2 β2 λ the charmed baryons 2 dA 2 B 2 dB 2 M 4 1 1 ε2 ε2 Initial Final states λ = + + 1 + 3 ; state β2 β2 β2 β2 2 A 2 B 2 dA 2 dB + ++,+,0π −,0,+ + / 0 2 c c D n D p ε1 ε3 ε1 ε3 ε √ √ μ = + ; ν = + + 3 ; / / β2 β2 β2 β2 β2 1 3 1 3 dA dB dA B dB ++ ++,+ 0,+ + + + c π π D p m1 m3 c √ √c √ ε = ; ε = ; / / / 1 + 2 + + 1 3 1 3 1 6 m1 m2 m1 m3 m5 ++, + 0π −,+ +π 0 D+n/D0 p ε = m5 ; ε = m4 , c √c √c √ 3 4 1/3 − 1/3 1/12 m2 + m5 m1 + m4 0 +π −/0π 0 +π − D0n and c √c c √c √ 1/3 1/3 1/6 [ 2 − ( + )2][ 2 − ( − )2] ()+ + () + ()+ ++ − + MA MB MC MA MB MC K 0 0π π 0 K K 0 q = . c √c √c √c √c √c 2MA 1/6 1/6 1/12 1/3 1/6 () ()+ () (A.10) 0 + K − π − 0π 0 + K − 0 K 0 c √c √c √c √c √c 1/6 1/6 1/12 1/6 1/3 Here, MA, MB, and MC are the masses of hadrons A, B, and C, respectively. Then the β˜ in Eq. (20) is given by

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