Charmed Baryons Circa 2015

Front. Phys. 10(6), 101406 (2015) DOI 10.1007/s11467-015-0483-z REVIEW ARTICLE Charmed baryons circa 2015

Hai-Yang Cheng

Institute of Physics, Academia Sinica, Taipei E-mail: [email protected] Received August 27, 2015; accepted September 8, 2015

This is essentially an update of Ref. [1] [H. Y. Cheng, Int. J. Mod. Phys. A 24 (Suppl. 1), 593 (2009)], a review of charmed baryon physics around 2007. Topics covered in this review include the spectroscopy, strong decays, lifetimes, nonleptonic and semileptonic weak decays, and electromagnetic decays of charmed baryons. Keywords charmed baryons, spectroscopy, decays of baryons, chiral Lagrangian PACS numb ers 14.20.Lq, 13.30.-a, 12.39.Fe

Contents 1 Introduction 1 Introduction 1 2 Spectroscopy 2 Charm baryon spectroscopy provides an excellent ground 2.1 Singly charmed baryons 2 for studying the dynamics of light quarks in the envi- 2.1.1 Λc states 4 ronment of a heavy quark. In the past decade, many 2.1.2 Σc states 6 new excited charmed baryon states have been discov- 2.1.3 Ξc states 6 ered by BaBar, Belle, CLEO and LHCb. B decays and + − 2.1.4 Ωc states 6 the e e → cc¯ continuum are both very rich sources 2.2 Doubly charmed baryons 8 of charmed baryons. Many efforts have been made to 3 Strong decays 8 identify the quantum numbers of these new states and 3.1 Strong decays of s-wave charmed baryons 9 understand their properties. ∗ 3.2 Strong decays of p-wave charmed baryons 10 Consider the strong decays ΣQ → ΛQπ and ΞQ → 4 Lifetimes 12 ΞQπ,whereQ = c, b. The mass differences between Σc ∗ 4.1 Singly charmed baryons 12 and Λc and between Ξc and Ξc in the charmed baryon 4.2 Doubly charmed baryons 13 sector were found to be large enough that the strong ∗ 5 Hadronic weak decays 14 decays of Σc and Ξc are kinematically allowed. Conse- 5.1 Nonleptonic decays 14 quently, the charmed baryon system offers a unique and 5.2 Discussion 15 excellent laboratory for testing the ideas and predictions + 5.2.1 Λc decays 15 of heavy quark symmetry of the heavy quark and chi- + 5.2.2 Ξc decays 15 ral symmetry of the light quarks. These tests will have 0 5.2.3 Ξc decays 16 interesting implications for the low-energy dynamics of 0 5.2.4 Ωc decays 16 heavy baryons interacting with Goldstone bosons. 5.3 Charm-flavor-conserving nonleptonic decays 16 Theoretical interest in hadronic weak decays of 5.4 Semileptonic decays 17 charmed baryons peaked around the early 1990s and 6 Electromagnetic and weak radiative decays 17 then faded away. To date, we still do not have a good 6.1 Electromagnetic decays 17 phenomenological model, not mentioning the quantum 6.2 Weak radiative decays 21 chromodynamics (QCD)-inspired approach as in meson 7 Conclusions 22 decays, to describe the complicated physics of baryon Acknowledgements 22 decays. We need cooperative efforts from both experi- References 22 mentalists and theorists to make progress in this arena. This review is essentially an update of Ref. [1], which ∗Special Topic: Potential Physics at a Super τ-Charm Factory (Ed. described charmed baryon physics around 2007. The out- Hai-Bo Li). line of the content is the same as that of Ref. [1] except

c The Author(s) 2015. This article is published with open access at www.springer.com/11467 and journal.hep.com.cn/fop REVIEW ARTICLE that we add discussions of the spectroscopy and lifetimes of doubly charmed baryons. Several excellent review articles on charmed baryons can be found in Refs. [2–7].

2Spectroscopy

2.1 Singly charmed baryons Fig. 1 Singly charmed baryon where Lρ describes relative orbital excitation of the two light quarks and Lλ the orbital excitation The singly charmed baryon is composed of a charmed of the center of the mass of the two light quarks relative to the quark and two light quarks. Each light quark is a triplet charmed quark. of flavor SU(3). There are two different SU(3) multiplets of charmed baryons: a symmetric sextet 6 and an tations of the two light quarks, and the λ-orbital mo- ¯ + + 0 antisymmetric antitriplet 3.TheΛc ,Ξc and Ξc form mentum Lλ describes orbital excitations of the center of ¯ 0 a 3 representation and they all decay weakly. The Ωc, mass of the two light quarks relative to the heavy quark + 0 ++,+,0 Ξc ,Ξc and Σc form a 6 representation; among (see Fig. 1). The p-wave heavy baryon can be in either 0 them, only Ωc decays weakly. We follow the Particle Data the (Lρ =0,Lλ =1)λ-state or the (Lρ =1,Lλ =0)ρ- Group’s (PDGs) convention [8] of using a prime to dis- state. The orbital λ-state (ρ-state) is clearly symmetric ¯ tinguish the Ξc in the 6 from that in the 3. (antisymmetric) under the interchange of p1 and p2.In B P B˜ P In the quark model, the orbital angular momentum of the following, we use the notation cJ (J )( cJ (J )) the light diquark can be decomposed into L = Lρ + Lλ, to denote the states that are symmetric (antisymmetric) where Lρ is the orbital angular momentum between the in the orbital wave functions under the exchange of two two light quarks, and Lλ is the orbital angular momen- light quarks. The lowest-lying orbitally excited baryon tum between the diquark and the charmed quark. Al- states are the p-wave charmed baryons, the quantum though the separate spin angular momentum S and or- numbers of which are listed in Table 1. bital angular momentum L of the light degrees of free- The next orbitally excited states are the positive- dom are not well defined, they are included for guid- parity excitations with Lρ + Lλ = 2. There exist mul- ance from the quark model. In the heavy quark limit, tiplets (e.g., Λc2 and Λˆ c2) with the symmetric orbital the spin of the charmed quark Sc, and the total angu- wave function, corresponding to Lλ =2,Lρ =0and lar momentum of the two light quarks J = S + L, Lλ =0,Lρ = 2 (see Table 2). We use a hat to distinguish are separately conserved. The total angular momentum them. Because the orbital Lλ = Lρ = 1 states are an- is given by J = Sc + J.ItisconvenienttouseS, tisymmetric under the interchange of two light quarks, L,andJ to enumerate the spectrum of states. More- we use a tilde to denote them. Moreover, the notation B˜L P 3¯ over, one can define two independent relative momenta, cJ (J ) is reserved for tilde states in the ,asthequan- 1 1 p √ p − p p √ p p − p tum number L is needed to distinguish different states. ρ = 2 ( 1 2)and λ = 6 ( 1 + 2 2 c), from the two light quark momenta p1, p2 and the heavy quark The observed mass spectra and decay widths of momentum pc. Denoting the quantum numbers Lρ and charmed baryons are summarized in Table 3 (see also 2 2 Lλ as the eigenvalues of Lρ and Lλ, respectively, the Fig. 2). Note that except for the parity of the lightest + + P ρ-orbital momentum Lρ describes relative orbital exci- Λc and the heavier one Λc(2880) , none of the other J

B P B˜ P Table 1 The p-wave charmed baryons denoted by cJ (J )and cJ (J )whereJ is the total angular momentum of the two light quarks. The orbital ρ-states with Lρ =1andLλ = 0 have odd orbital wave functions under the permutation of the two light quarks and are denoted by a tilde. P P State SU(3) S L(Lρ,Lλ) J State SU(3) S L(Lρ,Lλ) J 1 − 3 − ¯ − 1 − − Λc1( 2 , 2 ) 3 01(0,1)1 Σc0( 2 ) 6 11(0,1)0 ˜ 1 − ¯ − 1 − 3 − − Λc0( 2 ) 3 11(1,0)0Σc1( 2 , 2 ) 6 11(0,1)1 ˜ 1 − 3 − ¯ − 3 − 5 − − Λc1( 2 , 2 ) 3 11(1,0)1Σc2( 2 , 2 ) 6 11(0,1)2 ˜ 3 − 5 − ¯ − ˜ 1 − 3 − − Λc2( 2 , 2 ) 3 11(1,0)2Σc1( 2 , 2 ) 6 01(1,0)1 1 − 3 − ¯ − 1 − − Ξc1( 2 , 2 ) 3 01(0,1)1 Ξc0( 2 ) 6 11(0,1)0 ˜ 1 − ¯ − 1 − 3 − − Ξc0( 2 ) 3 11(1,0)0Ξc1( 2 , 2 ) 6 11(0,1)1 ˜ 1 − 3 − ¯ − 3 − 5 − − Ξc1( 2 , 2 ) 3 11(1,0)1Ξc2( 2 , 2 ) 6 11(0,1)2 ˜ 3 − 5 − ¯ − ˜ 1 − 3 − − Ξc2( 2 , 2 ) 3 11(1,0)2Ξc1( 2 , 2 ) 6 01(1,0)1

101406-2 Hai-Yang Cheng, Front. Phys. 10(6), 101406 (2015) REVIEW ARTICLE

P P L P Table 2 The ﬁrst positive-parity excitations of charmed baryons denoted by BcJ (J ), BˆcJ (J )andB˜ (J ). Orbital Lρ = Lλ =1 cJ states with antisymmetric orbital wave functions are denoted by a tilde. States with the symmetric orbital wave functions Lρ =2and Lλ = 0 are denoted by a hat. For convenience, we drop the superscript L for tilde states in the sextet.

P P State SU(3) S L(Lρ,Lλ) J State SU(3) S L(Lρ,Lλ) J 3 + 5 + ¯ + 1 + 3 + + Λc2( 2 , 2 ) 3 02(0,2)2Σc1( 2 , 2 ) 6 12(0,2)1 ˆ 3 + 5 + ¯ + 3 + 5 + + Λc2( 2 , 2 ) 3 02(2,0)2Σc2( 2 , 2 ) 6 12(0,2)2 ˜ 1 + 3 + ¯ + 5 + 7 + + Λc1( 2 , 2 ) 3 10(1,1)1Σc3( 2 , 2 ) 6 12(0,2)3 ˜ 1 1 + ¯ + ˆ 1 + 3 + + Λc0( 2 ) 3 11(1,1)0Σc1( 2 , 2 ) 6 12(2,0)1 ˜ 1 1 + 3 + ¯ + ˆ 3 + 5 + + Λc1( 2 , 2 ) 3 11(1,1)1Σc2( 2 , 2 ) 6 12(2,0)2 ˜ 1 3 + 5 + ¯ + ˆ 5 + 7 + + Λc2( 2 , 2 ) 3 11(1,1)2Σc3( 2 , 2 ) 6 12(2,0)3 ˜ 2 1 + 3 + ¯ + ˜ 1 + + Λc1( 2 , 2 ) 3 12(1,1)1 Σc0( 2 ) 6 00(1,1)0 ˜ 2 3 + 5 + ¯ + ˜ 1 + 3 + + Λc2( 2 , 2 ) 3 12(1,1)2Σc1( 2 , 2 ) 6 01(1,1)1 ˜ 2 5 + 7 + ¯ + ˜ 3 + 5 + + Λc3( 2 , 2 ) 3 12(1,1)3Σc2( 2 , 2 ) 6 02(1,1)2 3 + 5 + ¯ + 1 + 3 + + Ξc2( 2 , 2 ) 3 02(0,2)2Ξc1( 2 , 2 ) 6 12(0,2)1 ˆ 3 + 5 + ¯ + 3 + 5 + + Ξc2( 2 , 2 ) 3 02(2,0)2Ξc2( 2 , 2 ) 6 12(0,2)2 ˜ 1 + 3 + ¯ + 5 + 7 + + Ξc1( 2 , 2 ) 3 10(1,1)1Ξc3( 2 , 2 ) 6 12(0,2)3 ˜1 1 + ¯ + ˆ 1 + 3 + + Ξc0( 2 ) 3 11(1,1)0Ξc1( 2 , 2 ) 6 12(2,0)1 ˜1 1 + 3 + ¯ + ˆ 3 + 5 + + Ξc1( 2 , 2 ) 3 11(1,1)1Ξc2( 2 , 2 ) 6 12(2,0)2 ˜1 3 + 5 + ¯ + ˆ 5 + 7 + + Ξc2( 2 , 2 ) 3 11(1,1)2Ξc3( 2 , 2 ) 6 12(2,0)3 ˜2 1 + 3 + ¯ + ˜ 1 + + Ξc1( 2 , 2 ) 3 12(1,1)1 Ξc0( 2 ) 6 00(1,1)0 ˜2 3 + 5 + ¯ + ˜ 1 + 3 + + Ξc2( 2 , 2 ) 3 12(1,1)2Ξc1( 2 , 2 ) 6 01(1,1)1 ˜2 5 + 7 + ¯ + ˜ 3 + 5 + + Ξc3( 2 , 2 ) 3 12(1,1)3Ξc2( 2 , 2 ) 6 02(1,1)2

Fig. 2 Charmed baryons and their excitations [8].

Hai-Yang Cheng, Front. Phys. 10(6), 101406 (2015) 101406-3 REVIEW ARTICLE

Table 3 Mass spectra and widths (in units of MeV) of charmed baryons. Experimental values are taken from the Particle Data Group 0/++ 0/++ [8]. For the widths of the Σc(2455) and Σc(2520) baryons, we have taken into account the recent Belle measurement [11] for + 0 average. The width of Ξc(2645) is taken from Ref. [12]. For Ξc(3055) , we quote the preliminary result from Belle [13].

P P State J S L J Mass Width Decay modes + 1 + + ± Λc 2 000 2286.46 0.14 weak + 1 − − ± ± Λc(2595) 2 011 2592.25 0.28 2.59 0.56 Λcππ, Σcπ + 3 − − ± Λc(2625) 2 011 2628.11 0.19 < 0.97 Λcππ, Σcπ + ? Λc(2765) ? ? ? ? 2766.6 ± 2.450 Σcπ, Λcππ + 5 + ± ± (∗) 0 Λc(2880) 2 ? ? ? 2881.53 0.35 5.8 1.1Σc π, Λcππ, D p + ? +1.4 +8 (∗) 0 Λc(2940) ? ??? 2939.3−1.5 17−6 Σc π, Λcππ, D p ++ 1 + + ± +0.08 Σc(2455) 2 101 2453.98 0.16 1.94−0.16 Λcπ + 1 + + ± Σc(2455) 2 101 2452.9 0.4 < 4.6Λcπ 0 1 + + ± +0.09 Σc(2455) 2 101 2453.74 0.16 1.87−0.17 Λcπ ++ 3 + + ± +0.3 Σc(2520) 2 101 2517.9 0.614.8−0.4 Λcπ + 3 + + ± Σc(2520) 2 101 2517.5 2.3 < 17 Λcπ 0 3 + + ± +0.3 Σc(2520) 2 101 2518.8 0.615.3−0.4 Λcπ ++ ? +4 +22 (∗) Σc(2800) ? ??? 2801−6 75−17 Λcπ, Σc π, Λcππ + ? +14 +60 (∗) Σc(2800) ? ??? 2792− 5 62−40 Λcπ, Σc π, Λcππ 0 ? +5 +22 (∗) Σc(2800) ? ??? 2806−7 72−15 Λcπ, Σc π, Λcππ + 1 + + +0.4 Ξc 2 000 2467.8−0.6 weak 0 1 + + +0.34 Ξc 2 000 2470.88−0.80 weak + 1 + + ± Ξc 2 101 2575.6 3.1Ξcγ 0 1 + + ± Ξc 2 101 2577.9 2.9Ξcγ + 3 + + +0.5 ± Ξc(2645) 2 101 2645.9−0.6 2.6 0.5Ξcπ 0 3 + + ± Ξc(2645) 2 101 2645.9 0.9 < 5.5Ξcπ + 1 − − ± Ξc(2790) 2 011 2789.9 3.2 < 15 Ξcπ 0 1 − − ± Ξc(2790) 2 011 2791.8 3.3 < 12 Ξcπ + 3 − − ± ∗ Ξc(2815) 2 011 2816.6 0.9 < 3.5Ξc π, Ξcππ, Ξcπ 0 3 − − ± ∗ Ξc(2815) 2 011 2819.6 1.2 < 6.5Ξc π, Ξcππ, Ξcπ 0 ? Ξc(2930) ? ??? 2931 ± 636± 13 ΛcK + ? Ξc(2980) ? ? ? ? 2971.4 ± 3.326± 7ΣcK,ΛcKπ,Ξcππ 0 ? Ξc(2980) ? ? ? ? 2968.0 ± 2.620± 7ΣcK,ΛcKπ,Ξcππ + ? Ξc(3055) ? ? ? ? 3054.2 ± 1.317± 13 ΣcK,ΛcKπ,DΛ 0 ? Ξc(3055) ? ? ? ? 3059.7 ± 0.87.4 ± 3.9ΣcK,ΛcKπ,DΛ + ? Ξc(3080) ? ? ? ? 3077.0 ± 0.45.8 ± 1.0ΣcK,ΛcKπ,DΛ 0 ? Ξc(3080) ? ? ? ? 3079.9 ± 1.45.6 ± 2.2ΣcK,ΛcKπ,DΛ + ? ∗ Ξc(3123) ? ? ? ? 3122.9 ± 1.34.4 ± 3.8Σc K,ΛcKπ 0 1 + + ± Ωc 2 101 2695.2 1.7weak 0 3 + + ± Ωc(2770) 2 101 2765.9 2.0Ωcγ

− ∗ − quantum numbers given in Table 3 has been measured. Λc1(1/2 ) → [Σcπ]S, [Σc π]D and Λc1(3/2 ) → ∗ One has to rely on the quark model to determine the [Σcπ]D, [Σc π]S,D, [Λcππ]P . This explains why the width + + spin-parity assignments. of Λc(2625) is narrower than that of Λc(2595) .Be- + In the following, we discuss some of the excited cause of isospin conservation in strong decays, Λc1 is not + 0 charmed baryon states: allowedtodecayintoΛc π . + + + − Λc(2765) is a broad state ﬁrst seen in Λc π π by 2.1.1 Λc states CLEO [15]. However, it is still not known whether it is + + aΛc or a Σc and whether the width might be due to ˜ 1 − The lowest-lying p-wave Λc states are Λc0( 2 ), overlapping states. The Skyrme model [16] and the quark 1 − 3 − ˜ 1 − 3 − ˜ 3 − 5 − P 1 + Λc1( 2 , 2 ), Λc1( 2 , 2 )andΛc2( 2 , 2 ). The model [17, 18] suggest a J = 2 Λc state with a mass of 1 − 3 − + + c doublet Λc1( 2 , 2 )isformedbyΛc(2595) and 2742 and 2775 MeV, respectively. Therefore, Λ (2765) + Λc(2625) [14]. The allowed strong decays are could be the ﬁrst positive-parity excitation of Λc.Inthe

101406-4 Hai-Yang Cheng, Front. Phys. 10(6), 101406 (2015) REVIEW ARTICLE diquark model, it has also been proposed to be either It is worth mentioning that the Peking group [22] has P 1 − the first radial (2S) excitation of the Λc with J = 2 studied the strong decays of charmed baryons on the 3 containing the light scalar diquark or the first orbital ex- basis of the so-called P0 recombination model. For the P 3 − citation (1P )oftheΣc with J = 2 containing the Λc(2880), the Peking group found that (i) Λc(2880) can- light axial vector diquark [19]. not be a radial excitation, as its decay into D0p is pro- + 3 The state Λc(2880) ,firstobservedbyCLEO[15] hibited in the P0 model if Λc(2880) is the first radial + + − 0 5 + ˜ 1 5 + in Λc π π , was also seen by BaBar in the D p spec- excitation of Λc, and (ii) the states Λc2( 2 ), Λc2( 2 ) 5 + 0 trum [20]. Belle studied the experimental constraint on and Λˆ c2( ) are excluded, as they do not decay to D p P + 2 the J quantum numbers of Λc(2880) [21] and found 3 + according to the P0 model. Moreover, the predicted ra- P 5 ∗ that J = 2 is favored by the angular analysis of tios Σc π/Σcπ are either too large or too small compared + 0,++ ± ∗ Λc(2880) → Σc π together with the ratio Σ π/Σπ, to experiment, for example, which was measured to be + ∗ c2 → ∗ ± Γ(Λ (5/2 ) Σc π) Γ(Λc(2880) → Σc π ) +1.1 + =89, ≡ ± Γ(Λc2(5/2 ) → Σcπ) R ± =(24.1 6.4−4.5)%. (2.1) Γ(Λc(2880) → Σcπ ) + ∗ Γ Λˆ c2(5/2 ) → Σc π In the quark model, the candidates for the parity-even + + + + =0.75 . (2.4) 5 5 ˆ 5 ˜ 1 5 ˜ 2 5 + spin- 2 state are Λc2( 2 ), Λc2( 2 ), Λc2( 2 ), Λc2( 2 ), Γ Λˆ c2(5/2 ) → Σcπ ˜ 2 5 + and Λc3( 2 ) (see Table 2). The first four candidates, ∗ ˆ with J = 2, decay to Σcπ in an F wave and to Σc π in Both symmetric states Λc2 and Λc2, are thus ruled out. ˜ 2 5 + F and P waves. Neglecting the P -wave contribution for Hence, it appears that Λc3( 2 ) dictates the inner struc- 1) the moment, ture of Λc(2880). However, there are several problems + ∗ 7 ∗ with this assignment: (i) the quark model indicates a Γ(Λc2(5/2 ) → [Σc π]F ) 4 pπ(Λc(2880) → Σc π) 5 + Λc2( ) state around 2910 MeV, which is close to the + → = 7 → 2 Γ(Λc2(5/2 ) [Σcπ]F ) 5 pπ(Λc(2880) Σcπ) ˜ 2 5 + mass of Λc(2880), whereas the mass of Λc3( 2 )iseven 4 2 5 + = × 0.29 = 0.23 , (2.2) higher [17, 18], (ii) Λ˜ ( ) can decay to an F -wave Λcπ, 5 c3 2 and this has not been seen by BaBar and Belle, and (iii) where the factor of 4/5 follows from heavy quark symme- the calculated width, 28.8 MeV, is too large compared try. At first glance, this appears to be in good agreement ± ∗ to the measured one, 5.8 1.1 MeV. One may argue that 3 with experiment. However, the Σc π channel is available 0 4 the P model’s prediction can easily differ from the ex- via a P -wave and is enhanced by a factor of 1/pπ relative perimental measurement by a factor of 2–3 owing to its to the F -wave one. However, heavy quark symmetry can- inherent uncertainties [22]. ∗ + not be applied to calculate the contribution of the [Σc π]F Interestingly, the quantum numbers J P = 5 for the channel to the ratio R, as the reduced matrix elements 2 Λc(2880) were correctly predicted on the basis of the for the P -wave and F -wave modes differ. In this case, one diquark concept in Refs. [24, 25] before the Belle exper- has to rely on a phenomenological model to compute the 2 5 + ∗ iment. ratio R.AsforΛ˜ c3( ), it decays to Σc π,Σcπ and Λcπ + 2 The highest state Λc(2940) was first discovered by all in F waves. Further, 0 BaBar in the D p decay mode [20] and confirmed by 0 + ++ − 2 + ∗ ˜ → F 7 ∗ Belle in the decays Σcπ , Σc π , which subsequently Γ Λc3(5/2 ) [Σc π] 5 p (Λc(2880) → Σ π) + + − π c π π = 7 decay into Λc [21]. Its spin-parity assignment is 2 + 4 pπ(Λc(2880) → Σcπ) Γ Λ˜ c3(5/2 ) → [Σcπ]F quite diverse. For example, it has been argued that + P 5 Λc(2940) is the radial excitation of Λc(2595) with J = = × 0.29 = 0.36 . (2.3) 1 − 4 2 , but the predicted mass is too large by the order of 40 MeV. Alternatively, it could be the first radial exci- Although this deviates from the experimental measure- P + tation of Σc (not Λc!) with J =3/2 [26]. The latter ment (2.1) by 1σ, it is a robust prediction. This has mo- assignment has the advantage that the predicted mass is tivated us to conjecture that the first positive-parity ex- + in better agreement with experiment. Because the mass cited charmed baryon Λc(2880) couldbeanadmixture + ∗0 + + + of Λc(2940) is barely below the threshold of D p,this 5 ˆ 5 ˜ 2 5 of Λc2( 2 ), Λc2( 2 )andΛc3( 2 )[10]. observation motivated the authors of Ref. [27] to suggest

1) | 2S+1 P | 2 3 + It has been argued in Ref. [23] that in the chiral quark model Λc(2880) favors to be the state Λc LσJ = Λc Dλλ 2 with | 2 5 + 0 Lρ =0andLλ = 2 rather than Λc DA 2 with Lρ = Lλ = 1 as the latter cannot decay into D p. However, this is not our case ˜ 2 5 + 0 as Λc3( 2 ) does decay to D p and can reproduce the measured value of R. Hai-Yang Cheng, Front. Phys. 10(6), 101406 (2015) 101406-5 REVIEW ARTICLE an exotic molecular state of D∗0 and p with a binding en- antitriplet. P 1 − + ergy of the order of 6 MeV and J = 2 for Λc(2940) . In the relativistic quark–diquark model [26], Ξc(2980) 5 − P 1 + The quark potential model predicts a 2 Λc state at 2900 is a sextet J = 2 state. According to Table 2, possible 3 + 1 + ˆ 1 + ˜ 1 + ˜ 1 + MeV and a 2 Λc state at 2910 MeV [17, 18]. A similar candidates are Ξc1( 2 ), Ξc1( 2 ), Ξc0( 2 ), and Ξc1( 2 ). 3 + result of 2906 MeV for Λc was also obtained in the As pointed out in Ref. [32], strong decays of these four 2 3 relativistic quark model [28]. states, which were studied in Ref. [22] using the P0 ˜ 1 + model show that Ξc1( 2 ) does not decay to Ξcπ and 2.1.2 Σc states ΛcK, and has a width of 28 MeV, consistent with experiment. Therefore, the favored candidate for Ξc(2980) is ++,+,0 ˜ 1 + The highest isotriplet charmed baryons, Σc(2800) , Ξc1( 2 ), which has J = L =1. which decay to Λ+π, were first measured by Belle [29] P 5 + c The possible quark states for the J = 2 Ξc(3080) with widths of the order of 70 MeV. The possible 5 + 5 + 1 5 + baryon in an antitriplet are Ξc2( ), Ξˆc2( ), Ξ˜ ( ), 1 − 1 − 3 − ˜ 1 − 3 − 2 2 c2 2 quark states are Σc0( 2 ), Σc1( 2 , 2 ), Σc1( 2 , 2 ), 2 5 + 2 5 + Ξ˜ ( ), and Ξ˜ ( ) (see Table 2). Because Ξc(3080) is 3 − 5 − ˜ c2 2 c3 2 and Σc2( 2 , 2 ). The states Σc1 and Σc1 are ruled out + above the DΛ threshold, the two-body mode DΛ should because their decays to Λc π are prohibited in the heavy 3 − 5 − exist although it has not been searched for in the DΛ 0 quark limit. Now the Σc2( 2 , 2 ) baryon decays primar- c − spectrum. Recall that the neutral Ξ (3055) was ob- 1 0 ily into the Λcπ system in a D-wave, whereas Σc0( 2 ) served recently by Belle in the D Λ spectrum [13]. Ac- 3 decays into Λcπ in an S-wave. Because heavy hadron chi- cordingtothe P0 model, the first four states are ex- ral perturbation theory (HHChPT) implies a very broad cluded as they do not decay into DΛ[22].Theonly + Σc0 with a width of the order of 885 MeV (see Section ˜2 5 remaining possibility is Ξc3( 2 ). This is the analog of 3.2 below), this p-wave state is also excluded. There- 2 5 + Λ˜ ( )forΛc(2880). Nevertheless, the identification of ++,+,0 3 − c3 2 fore, Σc(2800) are likely to be either Σc2( )or 2 5 + 2 ˜ c 5 − Ξc3( 2 )withΞ(3080) encounters two potential prob- Σc2( ) or a mixture of the two. In the quark-diquark + 2 lems: (i) its width is dominated by the Ξcπ and Λc K model [26], both of them have very close masses com- modes, which have not been seen experimentally, and patible with experiment. Given that for light strange (ii) the predicted width of the order of 47 MeV [22] is baryons, the first orbital excitation of the Σ also has P − too large compared to the measured one, which is of the the quantum numbers J =3/2 (see Fig. 2), we will order of 5.7 MeV. − advocate a Σc2(3/2 ) state for Σc(2800).

2.1.4 Ωc states 2.1.3 Ξc states Only two ground states have been observed thus far: The states Ξc(2790) and Ξc(2815) form the doublet + 0 + 0 1/2 Ωc and 3/2 Ωc(2770) . The latter was seen by 1 − 3 − − → + Ξc1( 2 , 2 ). Because the diquark transition 1 0 +π BaBar in the electromagnetic decay Ωc(2770) → Ωcγ 1 − 3 − is prohibited, Ξc1( 2 , 2 ) cannot decay to Ξcπ.The [39]. 1 − dominant decay modes are [Ξcπ]S for Ξc1( 2 )and ∗ 3 − Molecular picture S c1 [Ξc π] for Ξ ( 2 ). + c c c Because Λ (2940) and Σ (2800) are barely below the Many excited charmed baryon states Ξ (2980), ∗0 c c c D p and DN thresholds, respectively, it is tempting to Ξ (3055), Ξ (3080), and Ξ (3123) have been seen at B ∗0 0 factories [12, 30, 31]. Another state Ξc(2930) ,which conjecture an exotic molecular structure of D and p is omitted from the PDG summary table, has been for the former and a molecular state of DN for the lat- + − ter [27, 40–44]. Likewise, Ξc(2980) could be a molecular seen only by BaBar in the Λc K mass projection of − + − − state of DΛ. B → Λc Λ¯ c K [33]. However, as we shall see below, it may form a sextet with Σc(2800) and Ωc(3050). A coupled-channel calculation of the baryon–meson The states Ξc(2980), Ξc(3055), Ξc(3080), and Ξc(3123) ND system has been performed to look for an isospin– c spin channel that is attractive enough to form a molec- could be the ﬁrst positive-parity excitations of the Ξ . P 1 − ular state [42, 45]. (I)J =(0)2 was found to be the The study of Regge phenomenology is very useful for − P P 3 the J assignment of charmed baryons [26, 34]. The most attractive one, followed by (I)J =(1)2 .This P + Regge analysis suggests that J =3/2 for Ξc(3055) suggests that Σc(2800) might be an s-wave DN molec- + P 1 − and 5/2 for Ξc(3080) [26]. From Table 5 below, we shall ular state with (I)J =(0)2 and Λc(2940) an s-wave P + ∗ P 3 − see that Ξc(3080) and Λc(2880) form a nice J =5/2 D N molecular state with (I)J =(1)2 (see Fig. 3

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P 1 + 1 − 3 − ¯ of [45]). Another possibility is a DN molecular state are established: the J = 2 , 2 ,and2 3 states, P 3 − ∗ + + 0 + + 0 with (I)J =(1)2 for Σc(2800) and a D N state (Λc ,Ξc , Ξc), (Λc(2595) ,Ξc(2790) , Ξc(2790) ), and − + + 0 P 1 (Λc(2625) ,Ξc(2815) , Ξc(2815) ), respectively, and the with (I)J =(0)2 for Λc(2940). Because Σc(2800) has P 1 + 3 + 6 ∗ ∗ ∗ isospin 1, and moreover, as we have noted in passing, J = 2 and 2 states, (Ωc, Σc, Ξc)and(Ωc , Σc , Ξc ), P − − Σc(2800) will be too broad if it is assigned to J =1/2 , respectively. The mass difference mΞc mΛc in the an- we conclude that the second possibility is preferable (see titriplet states clearly lies between 180 and 200 MeV. also [44]). We note in passing that Ξc(3080) should carry the quan- P + The possible spin-parity quantum numbers of the tum numbers J =5/2 . From Table 5, we see that P + higher excited charmed baryon resonances that have Ξc(3080) and Λc(2880) form a nice J =5/2 antitriplet been suggested in the literature are partially summarized as the mass difference between Ξc(3080) and Λc(2880) in Table 4. Some of the predictions are already ruled out is consistent with that observed in other antitriplets. P 5 + P − by experiment. For example, Λc(2880) has J = 2 ,as Likewise, the mass differences in the J =3/2 sextet seen by Belle. More experimental studies are certainly to (Ωc(3050), Ξc(2930), Σc(2800)) predicted by the quark– pin down the quantum numbers. diquark model are consistent with that measured in P + + Charmed baryon spectroscopy has been studied exten- J =1/2 and 3/2 sextets. Note that there is no P 1 − sively in various models. It appears that the spectroscopy J = 2 sextet as the Σc(2800) with these spin-parity is well described by the heavy quark–light diquark pic- quantum numbers will be too broad to be observed. ture elaborated by Ebert, Faustov, and Galkin (EFG) On the basis of the QCD sum rules, many charmed ¯ [26] (see also Ref. [35]). As noted in passing, the quantum baryon multiplets classified according to [6F (or 3F ), 5 + P numbers J = 2 of Λc(2880) were correctly predicted J ,S ,ρ/λ)] were recently studied in Ref. [36]. Three sex- in a model based on the diquark idea before the Belle experiment [24, 25]. Moreover, EFG have shown that all the available experimental data on heavy baryons fit the linear Regge trajectories nicely, namely, the trajectories 2 2 in the (J, M )and(nr,M ) planes for orbitally and ra- dially excited heavy baryons, respectively, 2 2 J = αM + α0,nr = βM + β0, (2.5) where nr is the radial excitation quantum number, α and β are the slopes, and α0 and β0 are intercepts. The linearity, parallelism, and equidistance of the Regge trajectories were verified. The predictions of the spin-parity quantum numbers of charmed baryons and their masses in Ref. [26] can be regarded as a theoretical benchmark (see Fig. 3).

Antitriplet and sextet states The antitriplet and sextet states of charmed baryons Fig. 3 Singly charmed baryon states where the spin-parity quan- are listed in Table 5. To date, the following states tum numbers in red are taken from Ref. [26].

Table 4 Possible spin-parity quantum numbers for excited charmed baryon resonances.

Λc(2765) Λc(2880) Λc(2940) Σc(2800) Ξc(2930) Ξc(2980) Ξc(3055) Ξc(3080) Ξc(3123) 1 + 3 + 5 − 3 − 5 − Capstick et al. [17, 18] 2 2 , 2 2 , 2 1 + 5 + 1 − 1 + 3 + 5 + 1 − B. Chen et al. [35] 2 (2S) 2 (1D) 2 (2P ) 2 (2S) 2 (1D) 2 (1D) 2 (2P ) 1 + 1 − 1 − 3 − 1 − 3 − 5 − H. Chen et al.[36] 2 , 2 2 , 2 2 , 2 2 3 − 1 + 5 + Cheng et al. [10] 2 2 2 1 + 5 + 1 − 3 + 1 − 3 − 1 − 3 − 5 − 1 + 3 + 5 + 7 + Ebert et al. [26] 2 (2S) 2 (1D) 2 (2P ), 2 (2S) 2 , 2 (1P ) 2 , 2 , 2 2 (2S) 2 (1D) 2 (1D) 2 (1D) 1 + 1 − 3 − 3 + 1 − 3 − Garcilazo et al. [28] 2 2 , 2 2 2 , 2 5 − 1 − 5 − Gerasyuata et al. [37] 2 2 2 1 − 1 − 3 − 3 + 1 + 3 + 5 + Liu et al.[38] 2 (1P ) 2 , 2 (1P ) 2 (1D) 2 (2S) 2 , 2 (1D) 5 + Wilczek et al. [24, 25] 2 1 − 3 + 5 + 1 − 5 − Zhong et al. [23] 2 (1P ) 2 (1D) 2 (1D) 2 , 2 (1P )

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P Table 5 Antitriplet and sextet states of charmed baryons. The J quantum numbers of Ξc(3080), Ξc(2930), Σc(2800) are not yet − c ≡ − ≡ − ≡ established and the Ω (3/2 ) state has not been observed. Mass diﬀerences ΔmΞcΛc mΞc mΛc ,ΔmΞcΣc mΞc mΣc ,ΔmΩcΞc − mΩc mΞc are in units of MeV. J P States Mass diﬀerence Status + + + 0 3¯ 1/2 Λc(2287) ,Ξc(2470) , Ξc(2470) ΔmΞcΛc = 183 estab − + + 0 1/2 Λc(2595) ,Ξc(2790) , Ξc(2790) ΔmΞcΛc = 198 estab − + + 0 3/2 Λc(2625) ,Ξc(2815) , Ξc(2815) ΔmΞcΛc = 190 estab + + + 0 5/2 Λc(2880) ,Ξc(3080) , Ξc(3080) ΔmΞcΛc = 196 [26] + 0 +,0 ++,+,0 c c 6 1/2 Ω (2695) ,Ξc(2575) , Σ (2455) ΔmΞcΣc = 124, ΔmΩcΞc = 119 estab + 0 +,0 ++,+,0 c c 3/2 Ω (2770) ,Ξc(2645) , Σ (2520) ΔmΞcΣc = 128, ΔmΩcΞc = 120 estab − 0 +,0 ++,+,0 c c 3/2 Ω (3050) ,Ξc(2930) , Σ (2800) ΔmΞcΣc = 131, ΔmΩcΞc = 119 [26]

tets were proposed in this work: (Ωc(3250), Ξc(2980), lation. The predicted doubly charmed baryon masses cal- P − − Σc(2800)) for J =1/2 , 3/2 and (Ωc(3320), Ξc(3080), culated in various models are tabulated in Refs. [51, 52]. P − Σc(2890)) for J =5/2 .NotethatΞc(2980) and For recent QCD sum rule calculations, see e.g. [53–55]. Ξc(3080) were treated as p-wave baryons rather than the Chiral corrections to the masses of doubly heavy baryons first positive-parity excitations, as we have discussed be- up to N3LO were presented in Ref. [56]. fore. The results for the multiplet [6F , 1, 0,ρ]ledtheau- Figure 4 shows the results of recent lattice studies of thors of Ref. [36] to suggest that there are two Σc(2800), doubly and triply charmed baryon spectra by different P − Ξc(2980), and Ωc(3250) states with J =1/2 and groups: RQCD [57], HSC [58], Brown et al.[59],ETMC J P =3/2−. The mass splittings are 14 ± 7, 12 ± 7, [60], ILGTI [61], PACS-CS [62], Durr et al. [63], Briceno and 10 ± 6 MeV, respectively. The predicted mass of et al.[64],Liuet al. [65], and Na et al. [66]. A new lat- − − (∗) Ωc(1/2 , 3/2 ) is around 3250 ± 200 MeV. Using the tice calculation of Ωcc and Ωccc was available in Ref. central value of the predicted masses to label the states [67]. The various lattice results are consistent with each in the multiplet [6F , 1, 0,ρ] (see Table I of Ref. [36]), one other and they fall into the ranges P − obtains (Ωc(3250), Ξc(2960), Σc(2730)) for J =1/2 P − M(Ξcc)=3.54–3.68 GeV, and (Ωc(3260), Ξc(2980), Σc(2750)) for J =3/2 .One ∗ ± can check that ΔmΞcΣc = 230 234 MeV, and ΔmΩcΞc M(Ξcc)=3.61–3.72 GeV, is of the order of 285±250 MeV. Owing to the large theo- M(Ωcc)=3.57–3.76 GeV, retical uncertainties in the masses, it is not clear whether ∗ the QCD sum rule calculations are compatible with the M(Ωcc)=3.68–3.85 GeV, (2.6) P + + mass differences measured in the J =1/2 and 3/2 and sextets. In any event, it will be interesting to test these two different model predictions for J P =3/2− and 1/2− M(Ωccc)=4.70–4.84 MeV. (2.7) sextets in the future. Although lattice studies suggest that the mass of the low-lying Ξcc exceeds 3519 MeV, it is interesting to note 2.2 Doubly charmed baryons that the authors of [51] calculated the masses of doubly and triply charmed baryons on the basis of the Regge Evidence of doubly charmed baryon states has been re- + +40.6 + + − + phenomenology and found M(Ξcc) = 3520.2−39.8 MeV, ported by SELEX in Ξcc(3520) → Λc K π [46]. Fur- + + − in good agreement with SELEX. ther observation of Ξcc → pD K was also announced by SELEX [47]. However, none of the doubly charmed states discovered by SELEX has been confirmed by FO- 3Strongdecays CUS [48], BaBar [49], Belle [12] and LHCb [50], although 6 10 Λc events are produced in B factories, for example, Owing to the rich mass spectrum and relatively narrow versus 1630 Λc events observed at SELEX. widths of the excited states, the charmed baryon sys- (∗)++ (∗)+ (∗)+ The doubly charmed baryons Ξcc , Ξcc , Ωcc tem offers an excellent ground for testing the ideas and with the quark contents ccu, ccd, ccs form an SU(3) predictions of heavy quark symmetry and light flavor triplet. They have been studied extensively using many SU(3) symmetry. The pseudoscalar mesons involved in different approaches: the quark model, light quark– the strong decays of charmed baryons such as Σc → Λcπ heavy diquark model, QCD sum rules, and lattice simu- are soft. Therefore, heavy quark symmetry of the heavy

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Fig. 4 Doubly charmed low-lying baryon spectra taken from Ref. [57].

2 h mΣ quark and chiral symmetry of the light quarks will have − → 2 c 2 Γ(Λc1(1/2 ) Σcπ)= 2 Eπpπ, interesting implications for the low-energy dynamics of 2πfπ mΛc1 2 heavy baryons interacting with Goldstone bosons. h mΛ − → 3 c 2 The strong decays of charmed baryons are most con- Γ(Σc0(1/2 ) Λcπ)= 2 Eπpπ, 2πfπ mΣc0 veniently described by the HHChPT, into which heavy 2 2h mΣ − → 8 c 5 quark symmetry and chiral symmetry are incorporated Γ(Λc1(3/2 ) Σcπ)= 2 pπ, 9πf mΛ (3/2) [68–70]. Heavy baryon chiral Lagrangians were ﬁrst con- π c1 2 − h mΣ(∗) structed in Ref. [68] for strong decays of s-wave charmed → (∗) 9 c 5 Γ Σc1(3/2 ) Σc π = 2 pπ, baryons and in Refs. [9, 14] for p-wave ones. Previous 9πfπ mΣc1(3/2) phenomenological studies of the strong decays of p-wave 2 − 4h mΛ → 10 c 5 charmed baryons based on HHChPT can be found in Γ Σc2(3/2 ) Λcπ = 2 pπ, 15πfπ mΣc2 Refs. [9, 10, 14, 71, 72]. The chiral Lagrangian involves 2 − h mΣ(∗) → (∗) 11 c 5 two coupling constants, g1 and g2,forP -wave transitions Γ Σc2(3/2 ) Σc π = 2 pπ, 10πf mΣ between s-wave and s-wave baryons [68]; six couplings, π c2 2 − 2h mΣ h2 − h7,fortheS-wave transitions between s-wave and → 11 c 5 Γ Σc2(5/2 ) Σcπ = 2 pπ, p-wave baryons; and eight couplings; h8 − h15,forthe 45πfπ mΣc2 2 − 7h mΣ∗ D-wave transitions between s-wave and p-wave baryons → ∗ 11 c 5 Γ Σc2(5/2 ) Σc π = 2 pπ, (3.1) [9]. The general chiral Lagrangian for heavy baryons cou- 45πfπ mΣc2 pled to pseudoscalar mesons can be expressed compactly where fπ = 132 MeV. The dependence on the pion mo- in terms of superﬁelds. We will not write the relevant 3 5 mentum is proportional to pπ, pπ and pπ for S-wave, Lagrangians here; instead the reader is referred to Eqs. P -wave and D-wave transitions, respectively. It is obvi- (3.1) and (3.3) of Ref. [9]. The partial widths relevant for ous that the couplings g1,g2,h2, ···,h7 are dimension- −1 our purposes are [9] less, whereas h8, ···,h15 have canonical dimension E . 2 ∗ g1 mΣc 3 Γ(Σc → Σcπ)= pπ, 2 ∗ 3.1 Strong decays of s-wave charmed baryons 2πfπ mΣc 2 ∗ g mΛ → → 2 c 3 Because the strong decay Σc Σcπ is kinematically Γ(Σc Λcπ)= 2 pπ, 2πfπ mΣc prohibited, the coupling g1 cannot be extracted directly

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∗ Table 6 Decay widths (in units of MeV) of s-wave charmed is also clear that the Σc width is smaller than that of Σc baryons where the measured rates are taken from 2006 PDG [73]. by a factor of ∼ 7, although they will become the same Decay Expt. HHChPT in the limit of heavy quark symmetry. ++ + + Σc → Λc π 2.23 ± 0.30 input + + 0 Σc → Λc π < 4.62.6 ± 0.4 3.2 Strong decays of p-wave charmed baryons 0 + − Σc → Λc π 2.2 ± 0.42.2 ± 0.3 ++ + + + + Σc(2520) → Λc π 14.9 ± 1.916.7 ± 2.3 Because Λc(2595) and Λc(2625) form the doublet − − + → + 0 ± 1 3 Σc(2520) Λc π < 17 17.4 2.3 Λc1( 2 , 2 ), it appears from Eq. (3.1) that the cou- 0 + − Σc(2520) → Λc π 16.1 ± 2.116.6 ± 2.2 plings h2 and h8 caninprinciplebeextractedfrom + → 0,+ +,0 ± Ξc(2645) Ξc π < 3.12.8 0.4 Λc(2595) → Σcπ and from Λc(2625) → Σcπ, respec- 0 → +,0 −,0 ± Ξc(2645) Ξc π < 5.52.9 0.4 tively. Likewise, the information on the couplings h10 and h11 can be inferred from the strong decays of Σc(2800) from the strong decays of heavy baryons. In the frame- − identified with Σc2(3/2 ). Couplings other than h2, h8, work of HHChPT, one can use some measurements as and h10 can be related to each other via the quark model input to fix the coupling g2, which, in turn, can be used [9]. to predict the rates of other strong decays. Among the (∗) ++ + + Although the coupling h2 can be inferred from the two- strong decays Σc → Λcπ,Σc → Λc π is the most body decay Λc(2595) → Σcπ, this method is less accurate well-measured. Hence, we shall use this mode to extract ++ because this decay is kinematically barely allowed or even the coupling g2. Using the 2006 data [73] Γ(Σc )= + ++ + + prohibited, depending on the mass of Λc(2595) .Forthe Γ(Σc → Λc π )=2.23 ± 0.30 MeV, the coupling g2 old mass measurement m(Λc(2595)) = 2595.4 ± 0.6MeV is extracted as Table 8 Decay widths (in units of MeV) of p-wave charmed | | +0.039 g2 2006 =0.605−0.043 . (3.2) baryons where the measured rates are taken from 2006 PDG [73]. The predicted rates of other modes are shown in Ta- Decay Expt. HHChPT ble 6. The agreement between theory and experiment [73] [10] + → + +1.56 is clearly excellent, except that the predicted width for Λc(2595) (Λc ππ)R 2.63−1.09 input + → ++ − +0.41 +0.43 ∗++ → + + Λc(2595) Σc π 0.65−0.31 0.72−0.30 Σc Λc π is slightly too large. + 0 + +0.41 +0.46 Λc(2593) → Σc π 0.67−0.31 0.77−0.32 Using the new data from the 2014 Particle Data Group + + 0 +0.93 Λc(2593) → Σc π 1.57−0.65 [8] in conjunction with the new measurements of the Σc + → ++ − ∗ Λc(2625) Σc π < 0.10 0.029 and Σc widths by Belle [11], we obtain the new aver- + → 0 + ++ + + +0.08 Λc(2625) Σc π < 0.09 0.029 age Γ(Σ → Λ π )=1.94 MeV (see Table 3). + + 0 c c −0.16 Λc(2625) → Σc π 0.041 2 + + Therefore, the coupling g is reduced to Λc(2625) → Λc ππ < 1.90.21 ++ (∗) +22 +0.011 Σc(2800) → Λcπ, Σc π 75 input |g2| =0.565 . (3.3) −17 2015 −0.024 + (∗) +60 Σc(2800) → Λcπ, Σc π 62−40 input 0 (∗) +28 From Table 7 we see that the agreement between the- Σc(2800) → Λcπ, Σc π 61−18 input + 0,+ +,0 +4.7 ory and experiment is further improved: The predicted Ξc(2790) → Ξc π < 15 8.0−3.3 + 0 +,0 −,0 +5.0 Ξc(2645) width is consistent with the first new mea- Ξc(2790) → Ξc π < 12 8.5−3.5 + → ∗+,0 0,+ +2.0 surement by Belle [12], and the new calculated width for Ξc(2815) Ξc π < 3.53.4−1.4 ∗++ + + 0 → ∗+,0 −,0 +2.1 Σc → Λc π is now in agreement with experiment. It Ξc(2815) Ξc π < 6.53.6−1.5

Table 7 Decay widths (in units of MeV) of s-wave charmed baryons. Data are taken from 2014 PDG [8] together with the new ∗ + measurements of Σc,Σc [11] and Ξc(2645) widths [12]. Theoretical predictions of [74] are taken from Table IV of [75]. Decay Expt. HHChPT Tawﬁq Ivanov Huang Albertus [8] et al. [74] et al.[75] et al. [76] et al. [78] ++ + + +0.08 Σc → Λc π 1.94−0.16 input 1.51 ± 0.17 2.85 ± 0.19 2.5 2.41 ± 0.07 + + 0 +0.1 Σc → Λc π < 4.62.3−0.2 1.56 ± 0.17 3.63 ± 0.27 3.2 2.79 ± 0.08 0 + − +0.1 +0.1 Σc → Λc π 1.9−0.2 1.9−0.2 1.44 ± 0.16 2.65 ± 0.19 2.4 2.37 ± 0.07 ++ + + +0.3 +0.5 Σc(2520) → Λc π 14.8−0.4 14.5−0.8 11.77 ± 1.27 21.99 ± 0.87 8.2 17.52 ± 0.75 + + 0 +0.6 Σc(2520) → Λc π < 17 15.2−1.3 8.6 17.31 ± 0.74 0 + − +0.4 +0.6 Σc(2520) → Λc π 15.3−0.5 14.7−1.2 11.37 ± 1.22 21.21 ± 0.81 8.2 16.90 ± 0.72 + 0,+ +,0 +0.1 Ξc(2645) → Ξc π 2.6 ± 0.52.4−0.2 1.76 ± 0.14 3.04 ± 0.37 3.18 ± 0.10 0 +,0 −,0 +0.1 Ξc(2645) → Ξc π < 5.52.5−0.2 1.83 ± 0.06 3.12 ± 0.33 3.03 ± 0.10

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Table 9 Decay widths (in units of MeV) of p-wave charmed baryons. Data are taken from 2014 PDG [8] together with the new ∗ + measurements of Σc,Σc [11] and Ξc(2645) widths [12]. Theoretical predictions of [74] are taken from Table IV of [75]. Decay Expt. HHChPT Tawﬁq Ivanov Huang Zhu [8] et al. [74] et al. [75] et al. [76] et al. [22] + + Λc(2595) → (Λc ππ)R 2.59 ± 0.56 input 2.5 + ++ − +1.3 Λc(2595) → Σc π 1.47 ± 0.57 0.79 ± 0.09 0.55−0.55 0.64 + 0 + Λc(2595) → Σc π 1.78 ± 0.70 0.83 ± 0.09 0.89 ± 0.86 0.86 + + 0 +0.57 Λc(2595) → Σc π 2.74−0.60 1.18 ± 0.46 0.98 ± 0.12 1.7 ± 0.49 1.2 + ++ − Λc(2625) → Σc π < 0.10 ∼< 0.028 0.44 ± 0.23 0.076 ± 0.009 0.013 0.011 + 0 + Λc(2625) → Σc π < 0.09 ∼< 0.040 0.47 ± 0.25 0.080 ± 0.009 0.013 0.011 + + 0 Λc(2625) → Σc π ∼< 0.029 0.42 ± 0.22 0.095 ± 0.012 0.013 0.011 + + Λc(2625) → Λc ππ < 0.97 ∼< 0.35 0.11 ++ (∗) +22 Σc(2800) → Λcπ, Σc π 75−17 input + (∗) +60 Σc(2800) → Λcπ, Σc π 62−40 input 0 (∗) +22 Σc(2800) → Λcπ, Σc π 72−15 input + 0,+ +,0 +3.6 Ξc(2790) → Ξc π < 15 16.7−3.6 0 +,0 −,0 +2.9 Ξc(2790) → Ξc π < 12 17.7−3.8 + ∗+,0 0,+ +1.5 Ξc(2815) → Ξc π < 3.57.1−1.5 2.35 ± 0.93 0.70 ± 0.04 0 ∗+,0 −,0 +1.7 Ξc(2815) → Ξc π < 6.57.7−1.7

+ ++ − 0 + + [73], Λc(2595) → Σc π , Σcπ and Λc(2595) → It has been noted [72] that the proximity of the + 0 + Σ π are kinematically barely allowed. However, for the Λc(2595) mass to the sum of the masses of its decay new measurement by the CDF, m(Λc(2595)) = 2592.25± products will lead to an important threshold effect that + 0.28 MeV [79], only the last mode is allowed. Moreover, will lower the Λc(2595) mass by 2 − 3 MeV compared the finite width effect of the intermediate resonant states to the observed mass. A more sophisticated treatment + + + − could become important [71]. of the mass lineshape of Λc(2595) → Λc π π by the + We next turn to the three-body decays Λc ππ of CDF yields m(Λc(2595)) = 2592.25 ± 0.28 MeV [79], + + Λc(2595) and Λc(2625) to extract h2 and h8.Asshown which is 3.1 MeV smaller than the 2006 world average. +2.0 in Ref. [10], the 2006 data Γ(Λc(2595)) = 3.6−1.3 MeV Therefore, the strong decay Λc(2595) → Λcππ is very [73] and for the Λc(2595) mass lead to the resonant rate close to the threshold. With the new measurement of [10] m(Λc(2595)), we have (in units of MeV) [32] + → + +1.56 + + Γ(Λc(2593) Λc ππ)R =(2.63−1.09)MeV, (3.4) Γ(Λc(2595) → Λc ππ)R 2 2 2 − as shown in Table 9. Assuming the pole contributions to = g2(20.45h2 +43.92h8 8.95h2h8), + → + + + Λc(2595) Λc ππ due to the intermediate states Σc Γ(Λc(2625) → Λc ππ)R ∗ + → and Σc , the resonant rate for the process Λc1 (2595) 2 2 6 2 + + − = g2(1.78h2 +4.557 × 10 h8 − 79.75h2h8). (3.7) Λc π π can be calculated in the framework of HHChPT [9]. Numerically, we found By fitting to the measured M(pK−π+π+) − − + 2 + + M(pK π ) mass difference distributions and using g2 = Γ(Λc(2595) → Λ ππ)R c 2 ± | | ± 2 2 0.365, CDF obtained h2 =0.36 0.08 or h2 =0.60 0.07 =13.82h +26.28h − 2.97h2h8, + 2 8 [79]. This corresponds to a decay width Γ(Λc(2595) )= + + + Γ(Λc(2625) → Λc ππ)R 2.59 ± 0.30 ± 0.47 MeV [79]. For the width of Λc(2625) , 2 6 2 =0.617h2 +0.136 × 10 h8 − 27h2h8, (3.5) CDF observed a value consistent with zero and therefore calculated an upper limit of 0.97 MeV using a Bayesian + + + − + 0 0 + where Λc ππ =Λc π π +Λc π π . It is clear that the approach. From the CDF measurements Γ(Λc(2595) )= + limit on Γ(Λc(2625)) gives an upper bound on h8 of the 2.59 ± 0.56 MeV and Γ(Λc(2625) ) < 0.97 MeV, we ob- −3 −1 order of 10 (in units of MeV ), whereas the decay tain width of Λc(2595) is entirely governed by the coupling | | ± h2. Specifically, we have [10] h2 2015 =0.63 0.07 , | | × −3 −1 | | +0.114 h8 2015 < 2.32 10 MeV . (3.8) h2 2006 =0.437−0.102 , | | × −3 −1 h8 2006 < 3.65 10 MeV . (3.6) Hence, the magnitude of the coupling h2 is greatly en-

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hanced from 0.437 to 0.63 . Our h2 differs slightly from The quark model relation |h8| = |h10| then leads to the value of 0.60 obtained by CDF. This is because the +0.11 −3 −1 |h8|≈(0.85−0.08) × 10 MeV , (3.10) CDF used |g2| =0.604 to calculate the mass dependence + of Γ(Λc ππ), whereas we used |g2| =0.565. which improves the previous limit (3.8) by a factor of + The large difference between the values of the cou- 3. The calculated partial widths of Λc(2625) shown in pling h2 obtained in 2006 and 2015 is ascribed to Table 9 are consistent with experimental limits. + the fact that the mass of Λc(2595) is 3.1 MeV lower than the previous world average because of the threshold effect. To illustrate this, we consider the 4 Lifetimes + + − 2 + 0 0 2 dependence of Γ(Λc π π )/h2 and Γ(Λc π π )/h2 on + + ΔM(Λc(2595)) ≡ M(Λc(2595) ) − M(Λc )asdepicted 4.1 Singly charmed baryons + 2 in Fig. 5. Γ(Λc ππ)/h2 at ΔM(Λc(2595)) = 305.79 MeV is clearly smaller than that at 308.9 MeV. This explains Among singly charmed baryons, the antitriplet states + + 0 0 why h2 should become larger when ΔM(Λc(2595)) be- Λc , Ξc , Ξc ,andtheΩc baryon in the sextet decay comes smaller. weakly. In 2006, the world averages of their lifetimes were The Ξc(2790) and Ξc(2815) baryons form the doublet [73] 1 − 3 − Ξc1( , ). The Ξc(2790) and Ξc(2815) widths pre- + −15 2 2 τ(Λc ) = (200 ± 6) × 10 s, dicted using the coupling h2 obtained from Eq. (3.8) + ± × −15 and assuming SU(3) flavor symmetry are shown in Ta- τ(Ξc ) = (442 26) 10 s, 0 0 +13 −15 ble 9. The predicted two-body decay rates of Ξc(2790) τ(Ξc ) = (112−10) × 10 s, + and Ξc(2815) clearly exceed the current experimental 0 −15 τ(Ωc )=(69± 12) × 10 s. (4.1) limits because of the enhancement of h2 (see Table 9). Hence, there is a tension for the coupling h2,asits These results remain the same even in 2014 [8]. As we + + + + value extracted from from Λc(2595) → Λc ππ will imply shall see below, the lifetime hierarchy τ(Ξc ) >τ(Λc ) > 0 + ∗ 0 0 Ξc(2790) → Ξcπ and Ξc(2815) → Ξc π rates slightly τ(Ξc ) >τ(Ωc ) is understood qualitatively but not quan- above current limits. It is conceivable that SU(3) flavor titatively in the operator product expansion (OPE) ap- symmetry breaking can help account for the discrepancy. proach. Some information on the coupling h10 can be inferred On the basis of the OPE approach to analysis of in- from the strong decays of Σc(2800).√ From Eq. (3.1) and clusive weak decays, the inclusive rate of the charmed the quark model relation |h3| = 3|h2| from Ref. [9], baryon is schematically represented by ++ → + + ≈ we obtain, for example, Γ(Σc0 Λc π ) 885 MeV. G2 m5 − B → F c Hence, Σc(2800) cannot be identified with Σc0(1/2 ). Γ( c f)= 3 VCKM 2 2 192π Using the quark model relation h11 =2h10 and the mea- ++,+,0 · A2 A3 O 1 sured widths of Σc(2800) (Table 3), we obtain A0 + 2 + 3 + ( 4 ) , (4.2) mc mc mc +0.11 −3 −1 |h10| =(0.85 ) × 10 MeV . (3.9) −0.08 where VCKM is the relevant Cabibbo-Kobayashi- Maskawa matrix element. The A0 term comes from the c quark decay and is common to all charmed hadrons. There are no linear 1/mQ corrections to the inclusive decay rate owing to the lack of gauge-invariant dimension- four operators [81–84], a consequence known as Luke’s theorem [85]. Nonperturbative corrections start at order 2 1/mQ and are model-independent. Spectator effects in inclusive decays due to the Pauli interference and W - 3 exchange contributions account for the 1/mc corrections and they have two noteworthy features: First, the es- timate of the spectator effects is model dependent; the hadronic four-quark matrix elements are usually eval-

+ 0 0 2 uated by assuming the factorization approximation for Fig. 5 Calculated dependence of Γ(Λc π π )/h2 (full curve) + + − 2 + + and Γ(Λc π π )/h2 (dashed curve)onm(Λc(2595) ) − m(Λc ), mesons and the quark model for baryons. Second, there 2 wherewehaveusedtheparametersg2 =0.565, h2 =0.63, and is a two-body phase-space enhancement factor of 16π for −3 −1 h8 =0.85 × 10 MeV . spectator eﬀects relative to the three-body phase space

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Table 10 Various contributions to the decay rates (in units of 10−12 GeV) of singly charmed baryons [86]. Experimental values are taken from Ref. [8]. dec ann int int tot −13 −13 Γ Γ Γ− Γ+ ΓSL Γ τ(10 s) τexpt(10 s) + Λc 1.006 1.342 −0.196 0.323 2.492 2.64 2.00 ± 0.06 + Ξc 1.006 0.071 −0.203 0.364 0.547 1.785 3.68 4.42 ± 0.26 0 +0.13 Ξc 1.006 1.466 0.385 0.547 3.404 1.93 1.12−0.10 0 Ωc 1.132 0.439 1.241 1.039 3.851 1.71 0.69 ± 0.12 for heavy quark decay. This implies that spectator ef- contains two s quarks, it is natural to expect that 3 int 0 int fects, which are of the order of 1/mc, are comparable to Γ+ (Ωc ) Γ+ (Ξc). The W -exchange contribution (Fig. 2 0 + and even exceed the 1/mc terms. 6(b)) occurs only for Ξc and Λc at the same Cabibbo- The total width of the charmed baryon Bc gener- allowed level. In the heavy quark expansion approach, ally receives contributions from inclusive nonleptonic and the above-mentioned spectator eﬀects can be described semileptonic decays: Γ(Bc)=ΓNL(Bc)+ΓSL(Bc). The in terms of the matrix elements of local four-quark op- nonleptonic contribution can be decomposed into erators. dec ann int int The inclusive nonleptonic rates of charmed baryons ΓNL(Bc)=Γ (Bc)+Γ (Bc)+Γ (Bc)+Γ (Bc), − + in the valence quark approximation and in the limit (4.3) ms/mc = 0 are expressed as [86]: + dec + 2 ann int 2 int corresponding to the c-quark decay, W -exchange contri- ΓNL(Λc )=Γ (Λc )+cosθC Γ +Γ− +sinθC Γ+ , bution, and destructive and constructive Pauli interfer- + dec + 2 ann int 2 int ΓNL(Ξc )=Γ (Ξc )+sinθC Γ +Γ− +cosθC Γ+ , ence. The inclusive decay rate is known to be governed 0 dec 0 ann int ΓNL(Ξ )=Γ (Ξ )+Γ +Γ , by the imaginary part of an eﬀective nonlocal forward c c + dec 0 dec 0 2 ann 10 2 int transition operator T . Therefore, Γ corresponds to the ΓNL(Ωc)=Γ (Ωc )+6sinθC Γ + cos θC Γ+ , imaginary part of Fig. 6(a) sandwiched between the same 3 dec (4.4) Bc states. At the Cabibbo-allowed level, Γ represents → ¯ ann the decay rate of c sud,andΓ denotes the con- where θC is the Cabibbo angle. tribution from the W -exchange diagram cd → us.The The results of a model calculation in Ref. [86] are int int interference Γ− (Γ+ ) arises from destructive (construc- shown in Table 10. The lifetime pattern tive) interference between the u (s) quark produced in c- + + 0 0 quark decay and the spectator u (s) quark in the charmed τ(Ξc ) >τ(Λc ) >τ(Ξc ) >τ(Ωc ) (4.5) B baryon c. Note that the constructive Pauli interference clearly agrees with experiment. This lifetime hierarchy is unique to the charmed baryon sector, as it does not + is qualitatively understandable. The Ξc baryon is the occur in the bottom sector. From the quark content of longest-lived among charmed baryons because of the the charmed baryons, it is clear that at the Cabibbo- smallness of W -exchange and partial cancellation be- + allowed level, destructive interference occurs in Λc and tween constructive and destructive Pauli interference, + + 0 0 Ξc decays (Fig. 6(c)), whereas Ξc , Ξc and Ωc can have whereas Ωc is the shortest-lived owing to the presence int 0 constructive interference Γ+ (Fig. 6(d)). Because Ωc of two s quarks in the Ωc, which greatly enhances the int int int contribution of Γ+ . Because Γ+ is always positive, Γ− is negative, and the constructive interference has a larger magnitude than the destructive interference, this ex- + + plains why τ(Ξc ) >τ(Λc ). It is also clear from Table 10 that, although the qualitative feature of the lifetime pattern is comprehensive, the quantitative estimates of charmed baryon lifetimes and their ratios are still rather poor.

4.2 Doubly charmed baryons Fig. 6 Contributions to nonleptonic decay rates of charmed baryons from four-quark operators: (a) c-quark decay, (b) W - The inclusive nonleptonic rates of doubly charmed exchange, (c) destructive Pauli interference, and (d) constructive baryons in the valence quark approximation and in the interference.

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Table 11 Predicted lifetimes of doubly charmed baryons in units dropped, the discrepancy between theory and experi- −13 of 10 s. ment is greatly reduced [91–93]. This leads to the so- Kiselev et al.Guberinaet al. Chang et al. Karliner et al. called large-Nc approach to describing hadronic D decays [87, 88] [89] [90] [52] [94]. Theoretically, explicit calculations based on QCD ++ Ξcc 4.6 ± 0.5 10.5 6.7 1.85 sum-rule analysis [95–97] indicate that the Fierz terms + ± Ξcc 1.6 0.5 2.0 2.5 0.53 are indeed largely compensated by the nonfactorizable + ± Ωcc 2.7 0.63.02.1 corrections. c limit ms/mc = 0 are expressed as As the 1/N expansion method greatly reduces the discrepancy between theory and experiment for charmed + dec + 2 ann 2 int ΓNL(Ξcc)=Γ (Ξcc)+cosθCΓ +sinθC Γ+ , meson decays, it is natural to ask if this scenario also ++ dec ++ int works in the baryon sector. This issue can be settled by ΓNL(Ξcc )=Γ (Ξcc )+Γ− , + dec + 2 ann 2 int experimental measurement of the Cabibbo-suppressed ΓNL(Ω )=Γ (Ω )+sinθ Γ +cosθ Γ . (4.6) + cc cc C C + mode Λc → pφ, which receives contributions only from int int the factorizable diagrams. As pointed out in Ref. [98], Because Γ+ is positive and Γ− is negative, it is obvi- theratepredictedbythelarge-Nc approach is in good ous that Ξ++ is longest-lived, whereas Ξ+ (Ω+ )isthe cc cc cc agreement with the measured value. In contrast, its de- shortest-lived if Γint > Γann (Γint < Γann). In general, we + + cay rate predicted by the naive factorization approxima- have tion will be too small by a factor of 15. Therefore, the ++ + + 1/Nc approach also works for the factorizable amplitude τ(Ξcc ) τ(Ωcc) ∼ τ(Ξcc). (4.7) of charmed baryon decays. This also implies that the in- The predictions available in the literature are summa- clusion of nonfactorizable contributions is inevitable and + rized in Table 11. Note that the lifetime of Ξcc was mea- necessary. If nonfactorizable effects amount to a redefini- + −13 sured by SELEX to be τ(Ξcc) < 0.33 × 10 s[46]. tion of the effective parameters a1, a2, and are universal ∗ Because the mass splitting between Ξcc and Ξcc and (i.e., channel-independent) in charm decays, then we still ∗ between Ωcc and Ωcc is less than 100 MeV (see also Eq. have a new factorization scheme with the universal pa- (2.6) for the lattice calculations), rameters a1,a2 to be determined from experiment. It is known that for heavy mesons, the nonfactoriz- ∗ − ∗ − ≈ mΞcc mΞcc = mΣc mΣc 65 MeV, able contributions will render color suppression of in-

∗ − ∗ − ≈ ternal W -emission ineffective. However, W -exchange in mΩcc mΩcc = mΩc mΩc 71 MeV, (4.8) baryon decays is not subject to color suppression even it is clear that only electromagnetic decays are allowed in the absence of nonfactorizable terms. A simple way to ∗ ∗ for Ωcc and Ξcc. see this is to consider the large-Nc limit. Although the W -exchange diagram is down by a factor of 1/Nc relative to the external W -emission one, this difference is com- 5Hadronicweakdecays pensated by the fact that the baryon contains Nc quarks in the limit of large Nc, thus allowing Nc different possi- 5.1 Nonleptonic decays bilities for W exchange between heavy and light quarks [100]. That is, the pole contribution can be as impor- In contrast to the significant advances made over the tant as the factorizable one. Experimental measurement last 10 years or so in the study of hadronic weak de- + 0 + ++ − of the decay modes Λc → Ξ K , Δ K , which pro- cays in the bottom baryon sector, progress in the arena ceed only through the W -exchange contributions, indi- of charmed baryons, both theoretical and experimental, cates that W -exchange indeed plays an essential role in has been very slow. charmed baryon decays. In the naive factorization approach, the coefficients a1 Various theoretical approaches to weak decays of for the external W -emission amplitude and a2 for inter- heavy baryons have been investigated, including the cur- c2 c1 nal W -emission are given by (c1 + )and(c2 + ), Nc Nc rent algebra approach [101–117], factorization scheme, respectively. However, we have learned from charmed pole model technique [98, 118–122], relativistic quark meson decays that the naive factorization approach model [100, 123], and quark diagram scheme [124, 125]. never works for the decay rate of color-suppressed de- Model predictions of the branching fractions and decay cay modes, although it usually works for color-allowed asymmetries can be found in Tables VI–VII of [1] for decays. Empirically, it was learned in the 1980s that if Bc →B+ P decays, Table VIII for Bc →B+ V decays the Fierz-transformed terms characterized by 1/Nc are 3 + and Table IX for Bc →B( 2 )+P (V ) decays.

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+ Table 12 Branching fractions of the Cabibbo-allowed two-body decays of Λc in units of %. Data are taken from PDG [8] except that + − + +0.21 the absolute branching fraction B(Λc → pK π )=(5.0 ± 1.3)% is replaced by the new measurement of (6.84 ± 0.24−0.27)% by Belle [126]. Decay B Decay B Decay B + + + + + ++ − Λc → Λπ 1.46± 0.13 Λc → Λρ < 6.5 Λc → Δ K 1.16 ± 0.07 + 0 + + 0 + + ∗0 + Λc → Σ π 1.44± 0.14 Λc → Σ ρ Λc → Σ π + + 0 + + 0 + ∗+ 0 Λc → Σ π 1.37± 0.30 Λc → Σ ρ < 1.9 Λc → Σ π + + + + + ∗+ Λc → Σ η 0.75± 0.11 Λc → Σ ω 3.7±1.0 Λc → Σ η 1.16 ± 0.35 + + + + + ∗+ Λc → Σ η Λc → Σ φ 0.42± 0.07 Λc → Σ η + 0 + + 0 ∗+ + ∗0 + Λc → Ξ K 0.53± 0.13 Λc → Ξ K 0.53± 0.19 Λc → Ξ K 0.36 ± 0.10 + 0 + ∗0 + + 0 Λc → pK¯ 3.2± 0.3 Λc → pK¯ 2.1± 0.3 Λc → Δ K¯ 1.36± 0.44

5.2 Discussion BaBar [130], respectively. Their branching fractions are of the order of 10−3 −10−4. The first measured Cabibbo- + + → 5.2.1 Λc decays suppressed mode Λc pφ, is of particular interest because it receives contributions only from the factorizable Experimentally, nearly all the branching fractions of the diagram and is expected to be color suppressed in the + − + Λc are measured relative to the pK π mode. On the naive factorization approach. A calculation in Refs. [131, basis of ARGUS and CLEO data, the PDG has made a 132] yields model-dependent determination of the absolute branch- + −3 2 + − + B(Λc → pφ)=2.26 × 10 a2, ing fraction, B(Λc → pK π )=(5.0 ± 1.3)% [8]. Re- +0.21 + → − cently, Belle reported a value of (6.84±0.24−0.27)% [126] α(Λc pφ)= 0.10 . (5.1) ∗ + from the reconstruction of D pπ recoiling against the Λc + + − From the experimental measurement B(Λc → pφ)= production in e e annihilation. Hence, the uncertain- −4 2) (11.2 ± 2.3) × 10 [8] , it follows that ties are much smaller, and, most importantly, this measurement is model independent! More recently, BESIII |a2|expt =0.70 ± 0.07 . (5.2) has also measured this mode directly with the prelimi- + − + This is consistent with the result obtained by the 1/Nc nary result B(Λc → pK π )=(5.84 ± 0.27 ± 0.23)% approach, a2 = c2(mc) ≈−0.59 . [127]. Its precision is comparable to that of Belle’s re- All the models except that of Ref. [121] predict a pos- sult. Another approach is to exploit a particular decay, + + 0 + + + −− itive decay asymmetry α for the decay Λc → Σ π (see B → pπ π Σc , and its charge conjugate to measure + − + Table VII of Ref. [1]). Therefore, the measurement of B(Λc → pK π ), also in a model independent manner α = −0.45 ± 0.31 ± 0.06 by CLEO [133] is very surpris- [128]. ing. If the negative sign of α is confirmed in the future, The branching fractions of the Cabibbo-allowed two- + this will imply that the s-wave and p-wave amplitudes for body decays of Λc are listed in Table 12. The central this decay have opposite signs, contrary to the model ex- values of data taken from the PDG [8] are scaled up pectation. The implication of this has been discussed in by a factor of 1.37 because of the new measurement of + − + detail in Refs. [98, 99]. Because the error of the previous B(Λc → pK π ) by Belle [126]. BESIII recently mea- + CLEO measurement is very large, it is crucial to more ac- sured 2-body, 3-body, and 4-body decay modes of Λc curately measure the decay asymmetry for Λ+ → Σ+π0. with significantly improved precision [127]. For example, c the result B(Λ+ → Λπ+)=(1.24±0.07±0.03)% obtained c 5.2.2 + decays by BESIII has much better precision than the value of Ξc (1.07 ± 0.28)% quoted by the PDG [8]. + + + − + No absolute branching fractions have been measured. Many of the Λc decay modes such as Σ K K ,Σ φ, (∗) (∗)+ ++ − The branching ratios listed in Tables VI and VIII of Ξ K ,andΔ K can proceed only through W - + − + + Ref. [1] are those relative to Ξc → Ξ π π . Several exchange. Experimental measurement of them implies Cabibbo-suppressed decay modes such as pK¯ ∗0,Σ+φ, the importance of W -exchange, which is not subject to Σ+π+π−,Σ−π+π+, and Ξ(1690)K+ have been observed color suppression in charmed baryon decays. [8]. Some Cabibbo-suppressed modes such as Λ+ → ΛK+ c The Cabibbo-allowed decays Ξ+ →B(3/2+)+P have and Λ+ → Σ0K+ have been measured by Belle [129] and c c been studied and are believed to be forbidden, as they

2) We have scaled up the PDG number (8.7 ± 2.7) × 10−4 [8] by a factor of 1.37 for its central value.

Hai-Yang Cheng, Front. Phys. 10(6), 101406 (2015) 101406-15 REVIEW ARTICLE do not receive factorizable and 1/2± pole contributions pate in weak interactions; that is, while the two light [100, 120]. However, the Σ∗+K¯ 0 mode was seen earlier by quarks undergo weak transitions, the heavy quark be- FOCUS [134], and this may indicate the importance of haves as a “spectator”. As the emitted light mesons are pole contributions beyond low-lying 1/2± intermediate soft, the ΔS = 1 weak interactions among light quarks states. can be handled by the well known short-distance effective Hamiltonian. This special class of weak decays can 0 5.2.3 Ξc decays usually be calculated more reliably than the conventional charmed baryon weak decays. The synthesis of the heavy No absolute branching fractions have been measured to quark and chiral symmetries provides a natural setting date. However, there are several measurements of the ra- for investigating these reactions [136]. The weak decays tios of branching fractions, for example [8], ΞQ → ΛQπ with Q = c, b were also studied in Refs. [137–139]. Γ(Ξ0 → ΛK0 ) c S ± ± The combined symmetries of heavy and light quarks R1 = 0 → − + =0.21 0.02 0.02, Γ(Ξc Ξ π ) severely restrict the weak interactions allowed. In the 0 → − + B −B B∗ −B Γ(Ξc Ω K ) ± symmetry limit, it is found that 3¯ 6 and 6 6 R2 = 0 − + =0.297 0.024 . (5.3) Γ(Ξc → Ξ π ) nonleptonic weak transitions [136] cannot occur. Sym- B¯ −B¯ 0 − + + − metries alone permit three types of transitions: 3 3, → ∗ The decay modes Ξc Ω K and Σ K and B6 −B6 B −B6 + − and 6 transitions. However, in both Σ π proceed only through W -exchange. The measured B¯ −B¯ − + − + the MIT bag and diquark models, only 3 3 tran- branching ratio of Ω K relative to Ξ π implies that sitions have nonzero amplitudes. The general amplitude the W -exchange mechanism plays a significant role. The for Bi →Bf + P is given by model of Körner and Krämer [100] predicts R2 =0.33 (see Table IX of Ref. [1]), in agreement with experiment, M(Bi →Bf + P )=iu¯f (A − Bγ5)ui, (5.4) but its prediction R1 =0.06 is too small compared to where A and B are the S-andP -wave amplitudes, re- the data. spectively. The S-wave amplitude can be evaluated using current algebra in terms of the parity-violating com- 5.2.4 0 decays Ωc mutator term. For example, the S-wave amplitude of + + 0 0 0 Ξc → Λc π is given by A unique feature of Ωc decays is that the decay Ωc → − + Ω π proceeds only via external W -emission, whereas + → + 0 −√1 + ↑|HPC| + ↑ 0 ∗0 0 A(Ξc Λc π )= Λc eff Ξc , (5.5) Ωc → Ξ K¯ proceeds via the factorizable internal W - 2fπ emission diagram. Various model predictions of Cabibbo- whereas the P -wave amplitude arises from the ground- 0 →B + allowed Ωc (3/2 )+P (V ) are listed in Table IX of state baryon poles [136]: [1] with the unknown parameters a1 and a2.Fromthe + + + 0 decay Λc → pφ we learn that |a2| =0.70 ± 0.07. The B(Ξc → Λc π ) 0 hadronic weak decays of the Ωc were recently studied in g2 mΞc + mΞc + PC + = Λc ↑|Heff |Ξc ↑ sin φ, (5.6) great detail in Ref. [135], where most of the decay chan- 2fπ mΛc − mΞc2 0 nels in Ωc decays were found to proceed only through the where φ is the mixing angle of Ξc,andΞ and Ξc1,Ξc2 W -exchange diagram; moreover, the W -exchange contri- c are their mass eigenstates. The matrix element Λ+ ↑ butions dominate in the rest of the processes, with some c |HPC|Ξ+ ↑ was evaluated in Ref. [136] using two dif- exceptions. Observation of such decays would shed light eff c ferent models: the MIT bag model [140, 141] and the on the mechanism of W -exchange effects in these decay diquark model. modes. The predicted rates are [136] 0 + − −15 5.3 Charm-flavor-conserving nonleptonic decays Γ(Ξc → Λc π )=1.7 × 10 GeV, Γ(Ξ+ → Λ+π0)=1.0 × 10−15 GeV, There is a special class of weak decays of charmed c c 0 + − −17 baryons that can be studied reliably, namely, heavy- Γ(Ωc → Ξc π )=4.3 × 10 GeV, (5.7) flavor-conserving nonleptonic decays. Some examples are and the corresponding branching fractions are the singly Cabibbo-suppressed decays Ξc → Λcπ and 0 + − −4 Ωc → Ξcπ. The idea is simple: In these decays, only B(Ξc → Λc π )=2.9 × 10 , the light quarks inside the heavy baryon will partici- + + 0 −4 B(Ξc → Λc π )=6.7 × 10 ,

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0 + − −6 + 0 + B(Ωc → Ξc π )=4.5 × 10 . (5.8) and Ξc → Ξ e ν are predicted to lie in the ranges 0 + − (0.8 − 2.0)% and (3.3 − 8.1)%, respectively, except that As stated above, the B6 −B6 transition Ω → Ξ π c c the QCD sum rule calculation in Ref. [150] predicts a vanishes in the chiral limit. It receives a finite factor- + much larger rate for Ξc → Ξe νe. Experimentally, only izable contribution as a result of a symmetry-breaking the ratios of the branching fractions are available to date effect. At any rate, the predicted branching fractions 0 + − [8]: for the charm-flavor-conserving decays Ξc → Λc π and + → + 0 −3 − −4 + → 0 + Ξc Λc π are of the order of 10 10 and should Γ(Ξc Ξ e ν) ± +0.3 + − + + =2.3 0.6−0.6, be readily accessible in the near future. Γ(Ξc → Ξ π π ) 0 → − + Γ(Ξc Ξ e ν) ± +0.3 0 − + =3.1 1.0−0.5 . (5.12) 5.4 Semileptonic decays Γ(Ξc → Ξ π ) Exclusive semileptonic decays of charmed baryons: There have been active studies of semileptonic decays + + + + 0 + 0 − + Λc → Λe (μ )νe,Ξc → Ξ e νe and Ξc → Ξ e νe of doubly charmed baryons. The interested reader can have been observed experimentally. Their rates depend consult [155–159] for further references. 2 2 on the Bc →Bform factors fi(q )andgi(q )(i =1, 2, 3), Just as with the hadronic decays discussed in the which are defined as last subsection, there are also heavy-flavor-conserving 0 + + − semileptonic processes, for example, Ξc → Λc (Σc )e ν¯e Bf (pf )|Vμ − Aμ|Bc(pi) 0 + − and Ωc → Ξc e ν¯e. In these decays only the light quarks 2 2 ν 2 =¯uf (pf )[f1(q )γμ + if2(q )σμν q + f3(q )qμ inside the heavy baryon will participate in weak interactions, while the heavy quark behaves as a spectator. − 2 2 ν 2 (g1(q )γμ + ig2(q )σμν q + g3(q )qμ)γ5]ui(pi). (5.9) This topic was recently investigated in Ref. [139]. Ow- These form factors have been evaluated using the non- ing to the severe phase-space suppression, the branching −6 relativistic quark model [131, 132, 142–144, 151], MIT fractions are of order 10 in the best cases, and typically −7 −8 bag model [142, 143], relativistic quark model [145, 149], 10 to 10 . light-front quark model [146], and QCD sum rules [147, 148, 150]. Experimentally, the only information 6 Electromagnetic and weak radiative decays available to date is the form-factor ratio measured in → the semileptonic decay Λc Λeν¯. In the heavy quark Although radiative decays are well measured in the limit, the six Λc → Λ form factors are reduced to two: ∗ ∗+ + charmed meson sector, e.g., D → Dγ and Ds → Ds γ, ΛcΛ Λ(p)|sγ¯ μ(1 − γ5)c|Λc(v) =¯u (F (v · p) only three of the radiative modes in the charmed baryon Λ 1 0 0 + sector have been seen, namely, Ξc → Ξc γ,Ξc → ΛcΛ · − +v/F2 (v p))γμ(1 γ5)uΛ . (5.10) + ∗0 0 c Ξc γ and Ωc → Ωcγ. This is understandable because 2 − ≈ ∗ − ≈ Assuming that the form factors exhibit dipole q behav- mΞc mΞc 108 MeV, and mΩc mΩc 71 MeV. ΛcΛ ΛcΛ ∗ ior, the ratio R = F˜2 /F˜1 is measured by CLEO to Hence, Ξc and Ωc are governed by the electromagnetic be [152] decays. However, it will be difficult to measure the rates of these decays because these states are too narrow to − ± ± R = 0.31 0.05 0.04 . (5.11) be experimentally resolvable. Nevertheless, we shall sys- Various model predictions of the charmed baryon tematically study the two-body electromagnetic decays semileptonic decay rates and decay asymmetries are of charmed baryons and also weak radiative decays. shown in Table 13. Dipole q2 dependence of the form factors is assumed whenever the form factor momentum 6.1 Electromagnetic decays dependence is not available in the model. Four differ- + + In the charmed baryon sector, the following two-body ent sets of predictions for Λc → ne νe, which are not listedinTable13,werepresentedinthesumrulecal- electromagnetic decays are of interest: c culations of Ref. [153]. The semileptonic decays of Ω B6 →B3 + γ :Σc → Λc + γ, Ξc → Ξc + γ, were treated in Ref. [154] within the framework of a ∗ ∗ ∗ B →B + γ :Σ → Λc + γ, Ξ → Ξc + γ, constituent quark model. From Table 13, we see that 6 3 c c + + B∗ →B ∗ → ∗ → the computed branching fractions of Λc → Λe ν,which 6 6 + γ :Σc Σc + γ, Ξc Ξc + γ, ∼ ∗ fall in the range 1.4% 2.6% are slightly smaller than Ωc → Ωc + γ, (6.1) the experimental values, (2.9 ± 0.5)% [(2.1 ± 0.6)%, of 1 0 − + B6 B the PDG [8]]. The branching fractions of Ξc → Ξ e ν wherewedenotethespin 2 baryons as and 3 for

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Table 13 Predicted semileptonic decay rates (in units of 1010s−1) and decay asymmetries (second entry) in various models. The + − + +0.21 absolute branching fraction B(Λc → pK π )=(5.0 ± 1.3)% is replaced by the new measurement of (6.84 ± 0.24−0.27)% by Belle [126] + 0 + for the data of Γ(Λc → Λ ν) taken from the PDG [8]. Predictions of [142, 143] are obtained in the non-relativistic quark model and the MIT bag model (in parentheses). Process [131, 132] [142, 143] [144] [145] [146] [147] [148] [149] [150] [151] Expt. [8] + 0 + Λc → Λ e νe 7.1 11.2 (7.7) 9.8 7.22 7.0 13.2 ± 1.810.9 ± 3.014.4 ± 2.6 −0.812 −1 −0.88 ± 0.03 −0.86 ± 0.04 + 0 + Λc → Λ μ νe 7.1 11.2 (7.7) 9.8 7.22 7.0 13.2 ± 1.810.9 ± 3.013.3 ± 2.8 + + Λc → ne νe 1.32 1.01 0.96, 1.37 + Ξc → Ξe νe 7.4 18.1 (12.5) 8.5 8.16 9.7 64.8 ± 22.6 seen + Ξc → Σe νe 3.3 ± 1.7

6 3¯ μ ∗μ 1 μ μ the symmetric sextet and antisymmetric antitriplet , T = B3¯,S= B − √ (γ + v )γ5B6 . (6.3) 3 ∗ 6 respectively, and the spin 2 baryon by B6. 3 An ideal theoretical framework for studying the above- It follows that [161, 162] mentioned electromagnetic decays is provided by the for- μ → ν malism in which the heavy quark symmetry and the chi- A[Sij (v) Sij + γ(ε, k)] ral symmetry of light quarks are combined [68–70]. When 3 ν μ μ = i a1U (Qii +Qjj)(kν εμ −kμεν )U −i6a Q U k/ε/Uμ, supplemented by the nonrelativistic quark model, the 2 1 μ formalism determines completely the low energy dynam- A[Sij (v) → Tij + γ(ε, k)] ics of heavy hadrons. The electromagnetic interactions − ν α β − U μ of heavy hadrons consist of two distinct contributions: = 2 3/2 a2 μναβ u¯3¯v k ε (Qii Qjj) (istrong interaction Lagrangians for heavy mesons and where kμ is the photon 4-momentum, and εμ is the polar- baryons given in Refs. [68–70], and the other from the ization 4-vector. As stressed in Ref. [160], SU(3) breaking anomalous magnetic moment couplings of the heavy par- effects due to light-quark mass differences can be incor- ticles. The heavy quark symmetry reduces the number porated by replacing the charge matrix Q with of free parameters needed to describe the magnetic couplings to the photon. There are two undetermined pa- 2 α β Q → Q˜ =diag , − , − (6.5) rameters for the ground-state heavy baryons. All of these 3 3 3 parameters are related simply to the magnetic moments where α = Mu/Md and β = Mu/Ms. To avoid any con- of the light quarks in the nonrelativistic quark model. fusion with the current quark mass mq,wehaveused However, the charmed quark is not particularly heavy 2 capital letters to denote the constituent quark masses. (mc 1.6 GeV), and it carries a charge of e.Conse- 3 In the quark model, the coefficients a1 and a2 are simply quently, the contribution from its magnetic moment can- related to the Dirac magnetic moments of light quarks, not be neglected. The chiral and electromagnetic gauge- e 1 e 1 invariant Lagrangian for heavy baryons can be found in a1 = − ,a2 = √ , (6.6) Eqs. (3.8) and (3.9) of Ref. [160], and are denoted by 3 Mu 2 6 Mu L(1) L(2) B and B , respectively. whereas a1 is connected to those of heavy quarks. Ex- The most general gauge-invariant Lagrangian Eq. (3.9) plicitly, a1 is fixed by heavy quark symmetry as of Ref. [160] for magnetic transitions of heavy baryons e 1 can be recast in terms of superfields [161, 162]: a1 = . (6.7) 12 MQ (2) μ ν L = −i3a1tr(S¯ QFμν S ) B √ Within the framework of HHChPT, the authors of μ ν αβ + 3a2 μναβ tr(S¯ Qv F T )+h.c. [163] proceeded to construct chiral Lagrangians at the O 2 O 3 μ 3 level (p )and (p ) and then calculated the electro- +3a1tr(S¯ Q σ · FSμ) − a1tr(TQ¯ σ · FT), (6.2) 2 magnetic decay amplitudes of charmed baryons up to 3 2 μν O(p ). It is not clear whether their O(p ) Lagrangian where σ · F ≡ σμν F , Q =diag(2/3, −1/3, −1/3) is (see Eq. (12) of Ref. [163]), characterized by the four the charge matrix for the light u, d, and s quarks, and couplings f2,f3, f˜3 and f4, is equivalent to the first two Q = eQ is the charge of the heavy quark. In the above terms of the O(p) Lagrangian given by Eq. (6.2). The equation,

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unknown couplings there were also estimated using the masses, Mu = 338 MeV, Md = 322 MeV, Ms = 510 quark model. MeV [8], and Mc =1.6 GeV, are summarized in the sec- The general amplitudes of electromagnetic decays are ond column of Table 14. Some other model predictions given by [160] are also listed there for comparison. μ ν Radiative decays of s-wave charmed baryons are con- A(B6 →B3¯ + γ)=iη1u¯3¯σμν k ε u6, sidered in Ref. [165] in the quark model, and the predic- ∗ ν α β μ A(B6 →B3¯ + γ)=iη2 μναβ u¯3¯γ k ε u , tions are similar to ours. A similar procedure is followed ∗ ν α β μ A(B6 →B6 + γ)=iη3 μναβ u¯6γ k ε u . (6.8) in Ref. [166] where the heavy quark symmetry is supplemented with light-diquark symmetries to calculate the + + ∗ The corresponding decay rates are [160] widths of Σc → Λc γ and Σc → Σcγ. The authors of Ref. 3 [75] apply the relativistic quark model to predict vari- 2 k Γ(B6 →B3¯ + γ)=η , 1 π ous electromagnetic decays of charmed baryons. In addi- 3 2 2 tion to the magnetic dipole (M1) transition, the author ∗ 2 k 3mi + mf Γ(B6 →B3¯ + γ)=η2 2 , of Ref. [167] also considered and estimated the electric 3π 4mi ∗+ + quadrupole (E2) amplitude for Σc → Λc γ arising from 3 2 2 ∗ 2 k 3mi + mf the chiral loop correction. The E2 contributions were an- Γ(B6 →B6 + γ)=η3 2 , (6.9) 3π 4mi alyzed in detail in Ref. [168]. The E2 amplitudes appear at different higher orders for the three types of decays: where mi (mf ) is the mass of the parent (daughter) 2 ∗ 2 ∗ O(1/Λχ)forB6 →B6 + γ, O(1/mQΛχ)forB6 →B3¯ + γ baryon. The coupling constants ηi can be calculated us- 3 2 and O(1/mQΛχ)forB6 →B3¯ +γ. Therefore, the E2 con- ing the quark model for a1, a2,anda1 [160, 164]: tribution to B6 →B3¯ + γ is completely negligible. The + + e 2 1 electromagnetic decays were calculated in Refs. [169–171] η1(Σc → Λc )= √ + , 6 3 Mu Md using the QCD sum rule method, and they were studied within the framework of the modified bag model in Ref. + → + √e 2 1 η1(Ξc Ξc )= + , [172]. 6 3 Mu Ms It is evident from Table 14 that the predictions in Refs. 0 0 e 1 1 η1(Ξc → Ξc)= √ − , [160, 164] and [163] all based on HHChPT are quite dif- 6 3 Ms Md ∗++ ++ ferent for the following three modes: Σc → Σc γ, ∗+ + ∗+ + ∗+ + e 2 1 Σc → Λc γ and Ξc → Ξc γ. Indeed, the results η2(Σc → Λc )= √ + , 3 6 Mu Md for the last two modes in Ref. [163] are larger than ∗+ + e 2 1 all the other existing predictions by one order of mag- η2(Ξc → Ξc )= √ + , 3 6 Mu Ms nitude! It is naively expected that all HHChPT approaches should agree with each other to the lowest or- ∗0 0 e 1 1 η2(Ξc → Ξc)= √ − + , der of chiral expansion provided that the coefficients are 3 6 Md Ms √ inferred from the nonrelativistic quark model. The low- ∗+ + ∗++ ++ 2 2e 1 1 est order predictions Γ(Σc → Λc γ) = 756 keV and η3(Σc → Σc )= − , 9 Mu Mc Γ(Ξ∗+ → Ξ+γ) = 403 keV obtained in Ref. [163] are √ c c ∗0 0 2 2e 1 1 still very large. Note that a recent lattice calculation in → − − ∗ η3(Σc Σc)= , → c ± d c Ref. [174] yields Γ(Ωc Ω γ)=0.074 0.008 keV which 9 2M M ∗ √ is much smaller than the value of Γ(Ωc )=4.82 keV pre- ∗+ + 2e 1 1 2 η3(Σc → Σc )= − − , dicted in Ref. [163]. 9 Mu 2Md Mc √ Chiral loop corrections to the M1 electromagnetic de- ∗0 0 2 2e 1 1 cays and to the strong decays of heavy baryons were η3(Ωc → Ωc)= − − , 9 2Ms Mc computed at the one loop order in Refs. [161, 162]. The √ leading chiral-loop effects we found are nonanalytic in ∗+ + 2e 1 1 2 2 2 2 2 1/2 η3(Ξc → Ξc )= − − , the forms of m/Λχ and (m /Λχ)ln(Λ /m )(ormq and 9 Mu 2Ms Mc √ mq ln mq,wheremq is the light quark mass). Some re- ∗0 0 2e 1 1 2 sults are [161, 162] η3(Ξc → Ξc )= − − − . 9 2Md 2Ms Mc + + Γ(Σc → Λc γ) = 112 keV, (6.10) + + Γ(Ξ c → Ξc γ)= 29keV, The results calculated using the constituent quark 0 0 Γ(Ξ c → Ξc γ)= 0.15 keV, (6.11)

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Table 14 Electromagnetic decay rates (in units of keV) of s-wave charmed baryons. Among the four different results listed in Refs. (0) [165] and [173], we quote those denoted by Γγ and “Present (ecqm)”, respectively. Decay HHChPT HHChPT Dey Ivanov Tawfiq Bañuls Aliev Wang Bernotas Majethiya [160, 164] [163] et al. [165] et al. [75] et al. [166] et al. [168] et al. [169] [170, 171] et al. [172] et al. [173] + + Σc → Λc γ 91.5 164.16 120 60.7 ± 1.5 87 46.1 60.55 ∗+ + Σc → Λc γ 150.3 892.97 310 151 ± 4 130 ± 45 126 154.48 ∗++ ++ +6.79 Σc → Σc γ 1.3 11.60 1.6 3.04 2.65 ± 1.20 6.36−3.31 0.826 1.15 ∗+ + +0.43 −4 Σc → Σc γ 0.002 0.85 0.001 0.14 ± 0.004 0.19 0.40 ± 0.16 0.40−0.21 0.004 < 10 ∗0 0 +1.68 Σc → Σc γ 1.2 2.92 1.2 0.76 0.08 ± 0.03 1.58−0.82 1.08 1.12 + + Ξc → Ξc γ 19.7 54.31 14 12.7 ± 1.5 10.2 ∗+ + Ξc → Ξc γ 63.5 502.11 71 54 ± 352± 25 44.3 63.32 ∗+ + +1.47 Ξc → Ξc γ 0.06 1.10 0.10 0.96−0.67 0.011 0 0 Ξc → Ξcγ 0.4 0.02 0.33 0.17 ± 0.02 1.2 ± 0.7 0.0015 ∗0 0 Ξc → Ξcγ 1.1 0.36 1.7 0.68 ± 0.04 5.1 ± 2.70.66 ± 0.32 0.908 0.30 ∗0 0 +0.80 Ξc → Ξc γ 1.0 3.83 1.6 1.26−0.46 1.03 ∗0 0 +1.12 Ωc → Ωc γ 0.9 4.82 0.71 1.16−0.54 1.07 2.02

Table 15 Electromagnetic decay rates (in units of keV) of p-wave charmed baryons. Decay Ivanov Tawﬁq Aziza Baccouche Zhu Chow Gamermann et al.[75] et al. [166] et al. [176] [177] [178] et al. [179] 1/2− → 1/2+(3/2+)γ + + Λc(2595) → Λc γ 115 ± 1 25 36 16 274 ± 52 + + Λc(2595) → Σc γ 77 ± 171 11 2.1 ± 0.4 + ∗+ Λc(2595) → Σc γ 6 ± 0.111 1 3/2− → 1/2+(3/2+)γ + + Λc(2625) → Λc γ 151 ± 24821 + + Λc(2625) → Σc γ 35 ± 0.5 130 5 + ∗+ Λc(2625) → Σc γ 46 ± 0.632 6 + + Ξc(2815) → Ξc γ 190 ± 5 0 0 Ξc(2815) → Ξc γ 497 ± 14 which should be compared with the corresponding quark- details on the theoretical side. Some predictions are col- model results: 92, 20, and 0.4 keV, respectively (Table lected in Table 15, and they are more diversiﬁed than 14). the s-wave case. For electromagnetic decays of doubly ∗0 0 0 The electromagnetic decays Ξc → Ξc γ and Ξc → charmed baryons, see, e.g., Refs. [172, 181]. 0 Ξcγ are of special interest. It was advocated in Ref. The electromagnetic decays considered to date do not [175] that a measurement of their branching fractions critically test the heavy quark symmetry or chiral sym- will allow us to determine one of the coupling constants metry. The results follow simply from the quark model. in HHChPT, namely, g1. They are forbidden at tree level There are examples in which both the heavy quark sym- in the SU(3) limit [see Eq. (6.10)]. In heavy baryon chiral metry and chiral symmetry enter in a crucial way. These perturbation theory, this radiative decay is induced via are the radiative decays of heavy baryons involving an chiral loops where SU(3) symmetry is broken by the light emitted pion. Some examples that are kinematically al- current quark masses. By identifying the chiral loop con- lowed are tribution to Ξ∗0 → Ξ0γ with the quark model prediction c c → ∗ → ∗ → ∗ → given in Eq. (6.10), it was found in Ref. [164] that one of Σc Λcπγ, Σc Λcπγ, Σc Σcπγ, Ξc Ξcπγ. the two possible solutions agrees with the quark model (6.12) expectation for g1. L(1) For the electromagnetic decays of p-wave charmed The contact interaction dictated by the Lagrangian B + + 0 → + − baryons, the search for Λc(2593) → Λc γ and can be nicely tested by the decay Σc Λc π γ,whereas + + L(2) Λc(2625) → Λc γ has not yet succeeded. The interested a test of the chiral structure of B is provided by the + + 0 reader is referred to Refs. [14, 75, 166, 175–180] for more process Σc → Λc π γ; see Ref. [160] for the analysis.

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6.2 Weak radiative decays Nonpenguin weak radiative decays of charmed baryons, such as those in Eq. (6.13), are characterized by At the quark level, three different types of processes emission of a hard photon and the presence of a highly can contribute to the weak radiative decays of heavy virtual intermediate quark between the electromagnetic hadrons, namely, single-, two-, and three-quark transi- and weak vertices. It was shown in Ref. [185] that these tions [183]. The single-quark transition mechanism arises features should make it possible to analyze these pro- from the so-called electromagnetic penguin diagram. Un- cesses by perturbative QCD; that is, these processes can fortunately, the penguin process c → uγ is highly sup- be described by an effective local and gauge invariant pressed, so it plays no role in radiative decays of charmed Lagrangian: hadrons. There are two contributions from the two-quark GF ∗ F F transitions: one from the W -exchange diagram accom- Heff (cu¯ → sdγ¯ )= √ VcsVud(c+O+ + c−O−), (6.16) panied by photon emission from the external quark, and 2 2 the other from the same W -exchange diagram but with with a photon radiated from the W boson. The latter is typi- 2 F ¯ e mf mi cally suppressed by a factor of mqk/MW (where k is the O±(cu¯ → sdγ)= 2 2 es + eu m − m ms mu photon energy) compared to the former bremsstrahlung i f process [182]. For charmed baryons, the Cabibbo-allowed μν mf mi × F˜μν + iFμν O± − ed + ec decay modes via cu¯ → sdγ¯ (Fig. 7) or cd → usγ are md mc

+ + 0 0 Λc → Σ γ, Ξc → Ξ γ. (6.13) μν × F˜μν − iFμν O∓ , (6.17) Finally, the three-quark transition involving W -exchange between two quarks and photon emission by the third 1 αβ where mi = mc +mu, mf = ms +md, F˜μν ≡ μναβF quark is quite strongly suppressed because of the very 2 and small probability of finding three quarks that are ad- μν μ ν equately kinematically matched with the baryons [183, O± =¯sγ (1 − γ5)cuγ¯ (1 − γ5)d 184]. μ ν ±sγ¯ (1 − γ5)duγ¯ (1 − γ5)c. (6.18) The general amplitude of weak radiative decays of baryons is given by For radiative decays of charmed baryons, one needs to μν μ ν evaluate the matrix element Bf |O± |Bi . Because the A(Bi →Bf γ)= iu¯f (a + bγ5)σμν ε k ui, (6.14) quark-model wave functions most resemble the hadronic where a and b are the parity-conserving and -violating states in the frame where both baryons are static, the amplitudes, respectively. The corresponding decay rate static MIT bag model was adopted in Ref. [185] for the is calculation. The predictions are3) 3 2 − 2 + + −5 1 mi mf 2 2 B(Λc → Σ γ)= 4.9 × 10 , Γ(Bi →Bf γ)= (|a| + |b| ). (6.15) 8π mi + + α(Λc → Σ γ)=−0.86 , 0 0 −5 B(Ξc → Ξ γ)=3.5 × 10 , 0 0 α(Ξc → Ξ γ)=−0.86 . (6.19) A different analysis of the same decays in Ref. [186] yielded the following results: + + −4 B(Λc → Σ γ)= 2.8 × 10 , + + α(Λc → Σ γ)=0.02 , 0 0 −4 B(Ξc → Ξ γ)=1.5 × 10 , 0 0 α(Ξc → Ξ γ)=−0.01 . (6.20) Evidently, these predictions (especially the decay asymmetry) differ greatly from those obtained in Ref. [185]. Fig. 7 W -exchange diagrams contributing to the quark-quark bremsstrahlung process c +¯u → s + d¯ + γ.TheW -annihilation Finally, it is worth remarking that, analogous to type diagrams are not shown here. the heavy-flavor-conserving nonleptonic weak decays dis-

3) 0 0 0 The branching fraction of Ξc → Ξ γ has been updated using the current lifetime of Ξc. Hai-Yang Cheng, Front. Phys. 10(6), 101406 (2015) 101406-21 REVIEW ARTICLE

cussed in Section 6.3, there is a special class of weak to attend the International Workshop on Physics at Future High radiative decays in which heavy flavor is conserved, for Intensity Collider @ 2–7 GeV in China held at University of Sci- ence and Technology of China on January 13–16, 2015. He urged example, Ξc → Λcγ and Ωc → Ξcγ. In these decays, weak me to write up the review on charmed baryons which will appear radiative transitions arise from the light quark sector of in the special topic of Frontiers of Physics, “Potential Physics at a the heavy baryon, whereas the heavy quark behaves as Super Tau-Charm Factory”. This research was supported in part a spectator. However, the dynamics of these radiative by the Ministry of Science and Technology of R.O.C. under Grant No. 104-2112-M-001-022. decays is more complicated than that of their counter- → part in nonleptonic weak decays, e.g., Ξc Λcπ.Inany Open Access This article is distributed under the terms of the event, it merits an investigation. Creative Commons Attribution License which permits any use, dis- tribution, and reproduction in any medium, provided the original author(s) and the source are credited. 7 Conclusions References In this report, we began with a brief overview of the spectroscopy of charmed baryons and discussed their possi- 1. H. Y. Cheng, Charm baryon production and decays, Int. J. ble structure and spin-parity assignments in the quark Mod. Phys. A 24, Suppl. 1, 593 (2009) [arXiv: 0809.1869 3 − model. For p-wave baryons, we have assigned Σc2( 2 ) [hep-ex]], Chap. 24 to Σc(2800). As for first positive-parity excitations, with 2. J. G. Körner, M. Krämer, and D. 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