Single heavy baryons with chiral partner structure in a three-flavor chiral model
aa r X i v : . [ h e p - ph ] A p r Single heavy baryons with chiral partner structure in a three-flavor chiral model
Yohei Kawakami ∗ and Masayasu Harada † Department of Physics, Nagoya University, Nagoya, 464-8602, Japan (Dated: April 8, 2019)We construct an effective hadronic model including single heavy baryons (SHBs) belonging tothe ( , ) representation under SU(3) L × SU(3) R symmetry, respecting the chiral symmetry andheavy-qaurk spin-flavor symmetry. When the chiral symmetry is spontaneously broken, the SHBsare divided into the baryons with negative parity of ¯ representation under SU(3) flavor symmetrywhich is the chiral partners to the ones with positive parity of representation. We determine themodel parameters from the available experimental data for the masses and strong decay widths ofΣ ( ∗ ) c , Λ c (2595), Ξ c (2790), and Ξ c (2815). Then, we predict the masses and strong decay widths ofother baryons including Ξ b with negative parity. We also study radiative decays of SHBs includingΩ ∗ c and Ω ∗ b with positive parity. I. INTRODUCTION
The spontaneous chiral symmetry breaking, which isone of the most essential properties of QCD, is expectedto generate a part of hadron masses and causes the massdifference between chiral partners. Investigation of chiralpartner structure will provide some clues to understandthe chiral symmetry. In particular, study of the chiralpartner structure of hadrons including heavy quarks givesinformation which are not obtained from the hadrons in-cluding only light quarks.There are several studies of hadrons including heavyquarks based on the chiral partner structure. The chi-ral partner structure of heavy-light mesons is studiedin e.g., Refs. [1–5], that of doubly heavy baryons is ine.g., Refs. [6–8], and the single heavy baryons (SHBs)are studied in e.g., Refs. [9–13].In Ref. Ref. [12], we proposed a new chiral partnerstructure for SHBs including a heavy quark and twolight quarks. There, we considered the chiral part-ners of Σ Q ( Q = c, b ) baryons with positive parityas Λ Q baryons with negative parity: a heavy quarkdoublet of (cid:0) Λ c (2595; J P = 1 / − ) , Λ c (2625; 3 / − ) (cid:1) is regarded as the chiral partners to thedoublet of (Σ c (2455; 1 / + ) , Σ c (2520; 3 / + )),and (Λ b (5912; 1 / − ) , Λ b (5920; 3 / − )) to(Σ b (1 / + ) , Σ ∗ b (3 / + )). Based on this structure, wepredicted the pionic and photonic decay widths of theseexcited SHBs. The results show that, although thedecay of Λ c (2595) is dominated by the resonant contri-bution through Σ c (2455), nonresonant contributions areimportant for Λ c (2625), Λ b (5912), and Λ b (5920), whichreflects the chiral partner structure. In Ref. [13], itsexperimental verification was proposed.In the present work, we extend the chiral partner modelin Ref. [12], which is based on SU(2) L × SU(2) R symme-try, to SU(3) L × SU(3) R symmetry. We consider theSHBs with negative parity which belongs to ¯3 represen- ∗ [email protected] † [email protected] tation under SU(3) flavor symmetry are the chiral part-ner to the SHBs with positive parity to representation.We introduce a chiral field belonging to ( , ) represen-tation under SU(3) L × SU(3) R symmetry to construct aneffective Lagrangian including the interactions to lightpseudo-scalar mesons. Determining the model parame-ters from existing experimental data, we give predictionson the masses and pionic decay widths which are not ex-perimentally determined. We also study radiative decaysby introducing the interactions with photon field in a chi-ral invariant way. The single heavy baryons have beenstudied experimentally (e.g., Refs. [14–16]), and theoret-ically based on chiral models (e.g., Refs. [17–19]), quarkmodels (e.g., Refs. [20–26]), the sum rule (e.g., Refs. [27–29]), the Regge theory (e.g., Ref. [30]), lattice simulations(e.g., Refs. [31, 32]), molecule models (e.g., Refs. [33, 34])(See for a review, e.g., Ref. [35] and references therein.).In this paper, we make comparisons of our predictionswith those in chiral effective models [17, 18], quark mod-els [20, 25, 30], and lattice simulations [31, 32].This paper is organized as follows: We construct aneffective Lagrangian in section II. Sections III and IV aredevoted to study the masses and the hadronic decays ofSHBs. We also study the radiative decays of SHBs insection V. Finally, we give a summary and discussions insection VI. II. EFFECTIVE LAGRANGIAN
In this section, we construct an effective model of singleheavy baryons (SHBs) by extending the two-flavor modelprovided in the previous work [12], to three-flavor case.We introduce a set of fields, S µQ ( Q = b, c ), for SHBs inwhich the light-quark cloud carries the spin 1 and belongsto ( , ) representation under SU(3) L × SU(3) R symme-try. The field transforms as S µQ Ch . → g R S µQ g TL , ( Q = c, b ) , (1)where g L,R ∈ SU(3)
L,R . When the chiral symmetry isspontaneously broken, S µQ is divided into two parts. Oneis for the positive parity SHBs belonging to represen-tation under SU(3) flavor symmetry, ˆ B µQ , and another forthe negative parity SHBs to ¯ , ˆ B ¯3 µQ : S µQ = ˆ B µQ + ˆ B ¯3 µQ . (2)We would like to stress that ˆ B µQ and ˆ B ¯3 µQ are chiral part-ners to each other in the present model. The physicalstates are embedded asˆ B µQ = Σ I =1 µQ √ Σ I =0 µQ √ Ξ ′ I = µQ √ Σ I =0 µQ Σ I = − µQ √ Ξ ′ I = − µQ √ Ξ ′ I = µQ √ Ξ ′ I = − µQ Ω µQ , ˆ B ¯3 µQ = 1 √ µQ Ξ I = µQ − Λ µQ I = − µQ − Ξ I = µQ − Ξ I = − µQ . (3)These B µQ and B ¯3 µQ are decomposed into spin-3/2 baryonfields and spin-1/2 fields as B µQ = B ∗ µQ − √ γ µ + v µ ) γ B Q ,B ¯3 µQ = B ¯3 ∗ µQ − √ γ µ + v µ ) γ B ¯3 Q , (4)where B ∗ µQ and B ¯3 ∗ µQ denote the spin-3/2 baryon fields,and B Q and B ¯3 Q the spin-1/2 fields, respectively. We notethat the parity transformation of the S µQ field is given by S µQ P → − γ S TQµ , (5)where T denotes the transposition of the 3 × S µQ = S µ † Q γ . (6)We introduce a 3 × M for scalar andpseudoscalar mesons made from a light quark and a lightanti-quark, which belongs to the ( , ¯ ) representation un-der the chiral SU(3) L × SU(3) R symmetry. The transfor-mation properties of M under the chiral symmetry andthe parity are given by M Ch . → g L M g † R , (7) M P → M † . (8)We assume that the potential terms for M in the modelare constructed in such a way that the M has a vacuumexpectation value (VEV) which breaks the chiral sym-metry spontaneously: h M i = f π f π
00 0 σ s , (9)where f π is the pion decay constant and σ s is written as σ s = 2 f K − f π with the Kaon decay constant f K . In the following, for studying the decays of the SHBs withemitting pions, we parameterize the field M as M = ξ f π f π
00 0 σ s ξ (10)where ξ = e iπ/f π , (11)with π being the 3 × π = 12 π √ π + √ π − − π
00 0 0 . (12)In addition, we introduce two fields, one belonging to(¯ , ) representation under SU(3) L × SU(3) R symmetryand another to ( , ¯ ) representation. It is convenient touse anti-symmetric 3 × S QLL Ch . → g L S QLL g TL , S QRR Ch . → g R S QRR g TR , (13)where S QLL and S QRR denote the fields of (¯ , ) and( , ¯ ) representations, respectively. They are related toeach others by parity transformation as S QLL P → − γ S QRR . (14)We introduce the parity eigenstates as S QLL = ˆ A ¯3 Q − ˆ C ¯3 Q , S QRR = ˆ A ¯3 Q + ˆ C ¯3 Q , (15)where A ¯3 Q and C ¯3 Q carry the negative and positive pari-ties, respectively. They include the flavor anti-symmetricfields asˆ A ¯3 Q = µQ Ξ I = µQ − Λ µQ I = − µQ − Ξ I = µQ − Ξ I = − µQ , (16)ˆ C ¯3 Q = µQ Ξ I = µQ − Λ µQ I = − µQ − Ξ I = µQ − Ξ I = − µQ . (17)These A ¯3 Q and C ¯3 Q express spin-1/2 fields respectively.Since the particles which are expressed by A ¯3 Q are stillundiscovered, we neglect A ¯3 Q in the following discussion.Now, let us write down an effective Lagrangian includ-ing the baryon fields S µQ , S QLL , and S QRR together withthe meson field M , based on the heavy-quark spin-flavorsymmetry and the chiral symmetry. We do not considerthe terms including more than square of M field or morethan two derivatives. A possible Lagrangian is given by L Q = − tr ¯ S µQ ( v · iD − ∆) S Qµ + ¯ S QLL ( v · iD ) S QLL + ¯ S QRR ( v · iD ) S QRR + g f π tr (cid:16) ¯ S µQ M † M S Qµ + ¯ S TQµ
M M † S µTQ (cid:17) − g f π tr ¯ S µQ M † S TQµ M T − g v m Λ Q tr ¯ S µQ M † S TQµ M T + κ f π tr (cid:16) ¯ S µQ M † M S Qµ + ¯ S µQ M † M S Qµ + ¯ S TQµ M M † S µTQ + ¯ S TQµ M M † S µTQ (cid:17) − κ f π tr (cid:16) ¯ S µQ M † S TQµ M T + ¯ S µQ M † S TQµ M T (cid:17) − i h I − ih R f π tr (cid:16) ¯ S µQ M † v · ∂M S Qµ + ¯ S µTQ M v · ∂M † S TQµ (cid:17) − i − h I − ih R f π tr (cid:16) ¯ S µQ v · ∂M † M S Qµ + ¯ S µTQ v · ∂M M † S TQµ (cid:17) + h f π tr (cid:16) ¯ S µQ v · ∂M † S TQµ M T + ¯ S µTQ v · ∂M S Qµ M ∗ (cid:17) − g f π tr (cid:16) ¯ S QLL ∂ µ M S Qµ + ¯ S µQ ∂ µ M † S QLL + ¯ S QRR ∂ µ M † S TQµ + ¯ S µTQ ∂ µ M S
QRR (cid:17) , (18)where m Λ Q ( Q = c, b ) are the masses of Λ c (2286) and Λ b in the ground state, ∆ provides the difference betweenthe chiral invariant mass of S µQ and that of S QLL and S QRR . g i ( i = 1 , , g v , κ i ( i = 1 , h I , h R , and h are dimensionless coupling constants. We note that weinclude g v -term to incorporate the heavy-flavor violationneeded for explaining the mass differences of charm andbottom sectors (See Ref. [12].). Although we can addheavy-quark flavor violation terms corresponding to g -term, such contributions are absorbed into the definitionof ∆. We expect that heavy-quark flavor violating con-tributions to terms other than g v term are small. Sincethresholds of B ∗ Q → B Q π are not open, the related termsare not included here. We note that the above chiral part-ner may not be necessarily a three-quark state but can be also a molecular state such as the one in Ref. [34]. III. MASSES AND ONE-PION DECAYS
In this section, we determine the coupling constants g , g v , and κ from masses of relevant SHBs, and g fromΣ ( ∗ ) c → Λ c π decays. Then we make predictions of theone-pion decay widths of other members of the flavor representation.When the chiral symmetry is spotaneously broken,the light meson field M acquires its vacuum expecta-tion value as in Eq. (9). Then the masses of the particlesincluded in the model are expressed as M (Σ Q ) = M Λ Q + ∆ + g f π − g Q f π + ¯ κ − ¯ κ (19) M (Ξ ′ Q ) = M Λ Q + ∆ + g f π + σ s f π − g Q σ s + ¯ κ f π + σ s m s ¯ m f π − ¯ κ f π m s ¯ m + σ s f π (20) M (Ω Q ) = M Λ Q + ∆ + g σ s f π − g Q σ s f π + ¯ κ m s ¯ m σ s f π − ¯ κ m s ¯ m σ s f π (21) M (Λ Q ) = M Λ Q + ∆ + g f π + g Q f π + ¯ κ + ¯ κ (22) M (Ξ Q ) = M Λ Q + ∆ + g f π + σ s f π + g Q σ s + ¯ κ f π + σ s m s ¯ m f π + ¯ κ f π m s ¯ m + σ s f π . (23) Here we adopt the normalization of f π = 92 . f K = 1 . f π where ¯ κ i = κ i ¯ m , and g Q is defined as g Q = g + g v f π m Λ Q . (24)We determine the fraction of strange quark mass m s andup or down quark mass ¯ m from the masses of the pionand kaon as m s / ¯ m = 25 . m K m π = m s + ¯ m m . (25)In the present analysis, we assign the following physicalstates to the flavor ¯3 representation:(Λ c , Λ ∗ c ) = (cid:0) Λ c (2595; J P = 1 / − ) Λ c (2625; 3 / − ) (cid:1) , (Ξ c , Ξ ∗ c ) = (cid:0) Ξ c (2790; J P = 1 / − ) , Λ c (2815; 3 / − ) (cid:1) (26)and are the chiral partner to the flavor representation:(Σ c , Σ ∗ c ) = (cid:0) Σ c (2455; 1 / + ) , Σ c (2520; 3 / + ) (cid:1) , (Ξ ′ c , Ξ ′∗ c ) = (cid:0) Ξ ′ c (1 / + ) , Ξ ′ c (3 / + ) (cid:1) , (Ω c , Ω ∗ c ) = (cid:0) Ω c (1 / + ) , Ω c (2770; 3 / + ) (cid:1) . (27)In the bottom sector, ¯3 includes(Λ b , Λ ∗ b ) = (cid:0) Λ c (5912; J P = 1 / − ) , Λ c (5920; 3 / − ) (cid:1) , (Ξ b , Ξ ∗ b ) = (cid:0) Ξ b (1 / − ) , Ξ b (3 / − ) (cid:1) , (28)and includes(Σ b , Σ ∗ b ) = (cid:0) Σ b (1 / + ) , Σ b (3 / + ) (cid:1) , (Ξ ′ b , Ξ ′∗ b ) = (cid:0) Ξ ′ b (5935; 1 / + ) , Ξ c (5945 , / + ) (cid:1) , (Ω b , Ω ∗ b ) = (cid:0) Ω b (1 / + ) , Ω b (3 / + ) (cid:1) . (29)We list experimental data of their masses and full decaywidths [39] in Table I.Here, we cannot determine the values of ∆, g , and κ ,separately. Instead, we introduce¯∆ = ∆ + g f π + ¯ κ ∆ s = ∆ + g f π + σ s f π + ¯ κ f π + σ s m s ¯ m f π ∆ Ω = ∆ + g σ s f π + ¯ κ m s ¯ m σ s f π , (30)to rewrite mass formulas as M (Σ Q ) = M Λ Q + ¯∆ − g Q f π − ¯ κ ,M (Ξ ′ Q ) = M Λ Q + ∆ s − g Q σ s − ¯ κ f π m s ¯ m + σ s f π ,M (Ω Q ) = M Λ Q + ∆ Ω − g Q σ s f π − ¯ κ m s ¯ m σ s f π ,M (Λ Q ) = M Λ Q + ¯∆ + g Q f π + ¯ κ ,M (Ξ Q ) = M Λ Q + ∆ s + g Q σ s + ¯ κ f π m s ¯ m + σ s f π . (31) TABLE I. Experimental data of masses and decay widths ofheavy baryons included in the present analysis.particle J P mass[MeV] full width[MeV]Λ c / + . ± .
14 no strong decaysΞ + c / + . ± .
30 no strong decaysΞ c / + . +0 . − . no strong decaysΣ ++ c (2455) 1 / + . ± .
14 1 . +0 . − . Σ + c (2455) 1 / + . ± . < . c (2455) 1 / + . ± .
14 1 . +0 . − . Σ ++ c (2520) 3 / + . +0 . − . . +0 . − . Σ + c (2520) 3 / + . ± . < c (2520) 3 / + . ± .
20 15 . +0 . − . Ξ ′ + c / + . ± . ′ c / + . ± . + c (2645) 3 / + . ± .
31 2 . ± . c (2645) 3 / + . ± .
31 2 . ± . ± . c / + . ± . c (2770) 3 / + . ± . c (2595) 1 / − . ± .
28 2 . ± . ± . c (2625) 3 / − . ± . < . + c (2790) 1 / − . ± . . ± . ± . c (2790) 1 / − . ± . . ± . ± . + c (2815) 3 / − . ± .
31 2 . ± . ± . c (2815) 3 / − . ± .
32 2 . ± . ± . b / + . ± .
17 no strong decaysΞ b / + . ± . − b / + . ± . + b / + . +0 . − . ± . . +3 . − . . − . Σ b / + · · · · · · Σ − b / + . +0 . − . ± . . +3 . − . ± . ∗ + b / + . ± . +1 . − . . +2 . − . . − . Σ ∗ b / + · · · · · · Σ ∗− b / + . ± . +1 . − . . +2 . − . . − . Ξ ′ b / + · · · · · · Ξ ′− b (5935) 1 / + . ± . ± . < . b (5945) 3 / + . ± . . ± . ± . − b (5955) 3 / + . ± . ± .
05 1 . ± . ± . b / + . ± . ∗ b / + · · · · · · Λ b (5912) 1 / − . ± . ± . < . b (5920) 3 / − . ± . < . b / − · · · · · · Ξ − b / − · · · · · · Ξ b / − · · · · · · Ξ − b / − · · · · · · We estimate the values of mass parameters and cou-pling constants in charm sector from experimental datain a way explained in Ref. [12]: We calculate the spin-averaged mass of SHBs in a heavy-quark multiplet withincluding errors to include the masses of members be-longing to the multiplet as shown in Table II.To include the heavy quark flavor symmetry viola-tion, we determined the value of g b from the mass dif-ference between spin-averaged masses of Λ ( ∗ ) b and Σ ( ∗ ) b .In addition, we use the weighted average of Σ ++ c → TABLE II. Spin averaged masses and widths used as inputsto determine the model parameters.input value (MeV) M (Λ c ) 2286 . M (cid:16) Σ ( ∗ ) c (cid:17) . +21 . − . M (cid:16) Ξ ( ∗ ) c (cid:17) . +22 . − . M (cid:16) Ω ( ∗ ) c (cid:17) . +23 . − . M (cid:16) Λ ( ∗ ) c (cid:17) . +10 . − . M (cid:16) Ξ ( ∗ ) c (cid:17) . +8 . − . Γ (cid:16) Σ ( ∗ ) c → Λ c π (cid:17) . +4 . − . M (Λ b ) 5619 . M (cid:16) Λ ( ∗ ) b (cid:17) − M (cid:16) Σ ( ∗ ) b (cid:17) . +8 . − . Λ + c π + , Σ c → Λ + c π − , Σ ∗ ++ c → Λ + c π + and Σ ∗ c → Λ + c π − to determine the coupling constant g as done in Ref. [12].We show the estimated values of model parameters in Ta-ble III. TABLE III. Estimated values of model paramtersparameter value¯∆ 270 +17 − MeV∆ s +16 − MeV∆ Ω +13 − MeV g c . +0 . − . g b . +0 . − . ¯ κ . +0 . − . g . +0 . − . Using the estimated value of g , we predict the de-cay widths of Σ ( ∗ ) Q → Λ Q π and Ξ ′ ( ∗ ) Q → Ξ Q π as shownin Table IV. These predictions are consistent with ex-perimental data because light flavor symmetry violationand heavy quark symmetry violation are small for the g -term. TABLE IV. Decay widths Σ ( ∗ ) Q → Λ Q π predicted in ourmodel.decay modes our model [MeV] expt. [MeV]Σ ++ c → Λ + c π + . +0 . − . . +0 . − . Σ + c → Λ + c π . +0 . − . < . c → Λ + c π − . +0 . − . . +0 . − . Σ ∗ ++ c → Λ + c π + . +0 . − . . +0 . − . Σ ∗ + c → Λ + c π . +0 . − . < ∗ c → Λ − c π . +0 . − . . +0 . − . Σ + b → Λ b π + . +0 . − . . +3 . − . . − . Σ b → Λ b π . +0 . − . · · · Σ − b → Λ b π − . +0 . − . . +3 . − . ± . ∗ + b → Λ b π + . +0 . − . . +2 . − . . − . Σ ∗ b → Λ b π . +0 . − . · · · Σ ∗− b → Λ b π − . +0 . − . . +2 . − . . − . Ξ ′ + c → Ξ + c π · · · no strong decaysΞ ′ + c → Ξ c π + Ξ ′ c → Ξ + c π − · · · no strong decaysΞ ′ c → Ξ c π Ξ ′∗ + c → Ξ c π . +0 . − . . ± . ′∗ + c → Ξ + c π . +0 . − . · · · Ξ ′∗ + c → Ξ c π + . +0 . − . · · · Ξ ′∗ c → Ξ c π . +0 . − . . ± . ± . ′∗ c → Ξ c π . +0 . − . · · · Ξ ′∗ c → Ξ + c π − . +0 . − . · · · Ξ ′ b → Ξ b π . +0 . − . · · · Ξ ′ b → Ξ b π . +0 . − . · · · Ξ ′ b → Ξ − b π + . +0 . − . · · · Ξ ′− b → Ξ b π . +0 . − . < . ′− b → Ξ − b π . +0 . − . · · · Ξ ′− b → Ξ b π − . +0 . − . · · · Ξ ′∗ b → Ξ b π . +0 . − . . ± . ± . ′∗ b → Ξ b π . +0 . − . · · · Ξ ′∗ b → Ξ − b π + . +0 . − . · · · Ξ ′∗− b → Ξ b π . +0 . − . . ± . ± . ′∗− b → Ξ − b π . +0 . − . · · · Ξ ′∗− b → Ξ b π − . +0 . − . · · · We can estimate the masses of bottom baryons in-cluded in our model using the parameters in Table III. Inthe present analysis, we assume heavy quark spin symme-try, so that we predict the spin-averaged masses which areshown in Table V. Here, we show the result in Ref. [30, 36]and experimental values for comparison. We note that,in Table V, we just put the minimum and maximum val-ues predicted for the members in a multiplet in Ref. [30].This table shows that our predictions are consistent withthose in Ref. [30, 36].We can see that our predictions for Σ ( ∗ ) b , Ξ ′ ( ∗ ) b and Λ ( ∗ ) b are consistent with the spin-averaged masses of experi-mentally observed masses. For Ω b , only the mass of thespin-1 / / / / TABLE V. Predicted values of the spin-averaged masses ofbottom baryons. For comparison we list the spin-averages ofexperimentally observed masses and the predicted values inRef. [30, 36].particle our model [30] [36] expt.(spin averaged)Σ ( ∗ ) b +20 − − · · · . ′ ( ∗ ) b +18 − · · · · · · . ( ∗ ) b +15 − − · · · . ( ∗ ) b +20 − − · · · . ( ∗ ) b +20 − − · · · well as Ξ ( ∗ ) b will be a test of the present model. We notethat Ξ ( ∗ ) b in the present analysis are unlikely to make amultiplet including Ξ b (6227) reported in Ref. [37], sincethe predicted mass of Ξ ( ∗ ) b is about 100 MeV smaller thanthe observed mass of Ξ b (6227). IV. PION DECAYS OF SINGLE HEAVYBARYONS WITH NEGATIVE PARITY
In this section, we consider decays of B ¯3( ∗ ) Q , the nega-tive parity excited SHBs belonging to the flavor repre-sentation. The main modes of Λ ( ∗ ) Q are three body decay,Λ Q → Λ Q ππ because Λ ( ∗ ) Q → Σ ( ∗ ) Q π decay thresholdsare closed in most cases. In the decays of Ξ ( ∗ ) c , the decaythresholds of Ξ ( ∗ ) c → Ξ ′ ( ∗ ) c π are completely open, so themain mode is the two body decay.In Ref. [12], we used the two-pion decay width ofΛ c (2595) to determine the values of derivative coupling constants, h I and h . Here, we also include the decaywidths of Ξ c (2790) and Ξ c (2815). There exists violationof the heavy quark spin symmetry between the decaywidths of Ξ c (2790) and Ξ c (2815). Instead of treatingthis violation precisely, we include the violation as sys-tematic errors of the model. Therefore, we use values ofa decay width of Λ c (2595) and, a spin averaged decaywidth between Ξ c (2790) and Ξ c (2815) as inputs to de-termine h I and h . The region colored by dark purple inFig. 1 shows the allowed values of h I and h determinedfrom the decay width of Λ c (2595) where the errors of g c , g and the total width with Λ(2595) are taken into ac-count. The region by light purple are obtained from thespin averaged width of Ξ c (2790) and Ξ c (2815) with theerrors of model parameters included. FIG. 1. Allowed range of h I and h . In the following analysis, we use the values of h I and h in the overlapped region of two colors in Fig. 1 tomake predictions of the decay widths of excited SHBswith negative pariry. We show the results of total decaywidths in Table VI, where we list predictions by the quarkmodel in Refs. [25, 36] for comparison. TABLE VI. Predicted widths of excited SHBs. We usedthe spin and isospin averaged value of the decay widths ofΞ + c (2790), Ξ c (2790), Ξ + c (2815) and Ξ c (2815) in addition tothe decay width of Λ c (2595) as inputs.initial mode Our model [36] [25] expt.[MeV] [MeV] [MeV] [MeV]Λ c (2595) Λ c π + π − . . c π π . . . .
36 (input) · · · · · · . ± . ± . c (2625) Λ c π + π − . . c π π . . . . · · · · · · < . b (5912) Λ b π + π − (0 . . × − Λ b π π (1 . . × − sum (2 . × − · · · · · · < . b (5920) Λ b π + π − (0 . × − Λ b π π (2 . × − sum (3 . × − · · · · · · < . + c (2790) Ξ ′ + c π Ξ ′ c π + sum input · · · .
61 8 . ± . ± . c (2790) Ξ ′ c π Ξ ′ + c π − sum input · · · .
61 10 . ± . ± . + c (2815) Ξ + c (2645) π Ξ c (2645) π + sum input · · · .
80 2 . ± . ± . c (2815) Ξ c (2645) π Ξ + c (2645) π − sum input · · · .
80 2 . ± . ± . b Ξ ′ b π . − . ′− b π + . − . . − . . . · · · Ξ − b Ξ ′− b π . − . ′ b π − . − . . − . . . · · · Ξ ∗ b Ξ ∗ b π . − . ∗− b π + < . . − . . . · · · Ξ ∗− b Ξ ∗− b π . − . ∗ b π − < . . − . . . · · · We note that we use the predicted masses of Ξ b withnegative parity shown in Table V with including theirerrors. So the predicted decay widths take a wide rangeof values, which includes predictions in Refs. [25, 36]. Inparticular, since the minimum value shown in Table Vis very close to the threshold of the relevant decays, theminimum values of the predictions of one-pion decays ofΞ ( ∗ ) b in Table VI are very small, and three-body decayssuch as Ξ b → Ξ b ππ become dominant. Here, we studythe contributions of possible intermediate states of three-body decays and show the results in Table VII, where weset the parameters as g b = 0 . , ¯ κ = 0 . , g = 0 . , h I = − .
40, and h = 0, and set the masses ofΞ ∗ b to be their minimum of the predicted values shown inTable V, and Ξ ′∗ b to be the central mass values in Table I.We note that, unlikely to the decays of Λ b (5912) andΛ b (5920), the decays of Ξ ∗ b and Ξ ∗− b are not dominatedby the non-resonant contribution. TABLE VII. Estimated values of the decay widths of bottombaryons. initial decay intermediate widthstate mode state [keV]Ξ ∗ b Ξ b π π non-resonant 0 . ′∗ b ′∗ b . b π + π − NR 0 . ′∗− b . ′∗− b . − b π + π Ξ ′∗ b . × Ξ ′∗− b . ′∗ b & Ξ ′∗− b . ∗− b Ξ b π π NR 0 . ′∗− b . ′∗− b . − b π + π − NR 0 . ′∗ b ′∗ b . b π − π Ξ ′∗− b ′∗ b . × Ξ ′∗− b & Ξ ′∗ b . V. RADIATIVE DECAYS
In this section, we study radiative decays of the SHBs.The relevant Lagrangian is given by L rad = r F tr (cid:16) ¯ S µQ Q light S νQ + ¯ S µTQ Q light S νTQ (cid:17) F µν + r F tr (cid:16) ¯ S µQ Q light S νQ − ¯ S µTQ Q light S νTQ (cid:17) ˜ F µν + r F tr (cid:16) ¯ S QLL
M S µQ Q light v ν + ¯ S QRR M † S µQ Q light v ν (cid:17) F µν + h . c . + r F tr (cid:16) ¯ S QLL
M S µQ Q light v ν + ¯ S QRR M † S µQ Q light v ν (cid:17) ˜ F µν + h . c . , (32)where F µν is the field strength of the photon and ˜ F µν isits dual tensor: ˜ F µν = (1 / ǫ µνρσ F ρσ , r i ( i = 1 , ...,
4) aredimensionless constants, and F is a constant with dimen-sion one. In this analysis, we take F = 350 MeV followingRef. [17]. We note that the values of the constants r i areof order one based on quark models [17].Let us first study the electromagnetic intramultiplettransitions governed by the r -term in Eq. (32). Let B ∗ denotes the decaying baryon with spin-3/2 ( B ∗ =Λ ∗ Q , Ξ ∗ Q , Σ ∗ Q , Ξ ′∗ Q , Ω ∗ Q ), and B , the daughter baryonwith spin-1/2 (Λ Q , Ξ Q , Σ Q , Ξ ′ Q , Ω Q ). Then the ra-diative decay width is given byΓ B ∗ → Bγ = C B ∗ Bγ αr F m B m B ∗ E γ (33)where α is the electromagnetic fine structure constant, E γ is the photon energy and C B ∗ Bγ is the Clebsh-Gordon constants given by C Σ ∗ ++ c Σ ++ c γ = C Σ ∗ + b Σ + b γ = 23 ,C Σ ∗ + c Σ + c γ = C Σ ∗ b Σ b γ = 16 ,C Σ ∗ c Σ c γ = C Σ ∗− b Σ − b γ = − ,C Ξ ∗ + c Ξ + c γ = C Ξ ∗ b Ξ b γ = 16 ,C Ξ ∗ c Ξ c γ = C Ξ ∗− b Ξ − b γ = − ,C Ω ∗ c Ω c γ = C Ω ∗− b Ω − b γ = − ,C Λ ∗ + c Λ + c γ = C Λ ∗ b Λ b γ = − ,C Ξ ∗ + c Ξ + c γ = C Ξ ∗ b Ξ b γ = − ,C Ξ ∗ c Ξ c γ = C Ξ ∗− b Ξ − b γ = 13 . (34)Here we would like to stress that the radiative decaywidths of positive parity SHBs (Σ ∗ Q , Ξ ∗ Q , Ω ∗ Q ) and thoseof negative parity SHBs (Λ ∗ Q , Ξ ∗ Q ) are determined byjust one coupling constant r , reflecting the chiral part-ner structure. We think that checking the relation amongthese radiative decays will be one of the crucial test of thechiral partner structure. In Table VIII and IX, we showour predictions comparing with those in Ref. [18, 32]. Inradiative decays, the chiral loop considered in Ref. [18]might have contribution. However, our predictions areconsistent with those in Ref. [18] for r ∼
1, which im-plies the contribution from the chiral loop is small in thisradiative decay. On the other hand, our results are con-sistent with the lattice results if r ∼ . ∗ Q does not have any strong decays andthe main mode must be Ω ∗ Q → Ω Q γ . We expect thecoupling constant r will be determined by the decayof Ω ∗ Q in future experiments, and other radiative decaywidths related to r -type interaction will be estimated. TABLE VIII. Predicted widths of radiative decays betweenheavy quark multiplets of charm baryons. We also show thepredictions in Ref. [18, 32] for comparison.decay mode predicted width [18] [32][keV] [keV] [keV]Σ ∗ ++ c → Σ ++ c γ . r . · · · Σ ∗ + c → Σ + c γ . r . · · · Σ ∗ c → Σ c γ . r . · · · Ξ ′∗ + c → Ξ ′ + c γ . r . · · · Ξ ′∗ c → Ξ ′ c γ . r . · · · Ω ∗ c → Ω c γ . r .
82 0 . ∗ + c → Λ + c γ . r · · · · · · Ξ ∗ + c → Ξ + c γ . r · · · · · · Ξ ∗ c → Ξ c γ . r · · · · · · TABLE IX. Predicted widths of radiative decays betweenheavy quark multiplets of bottom baryons. We also show thepredictions in Ref. [18] for comparison.decay mode predicted width [18][keV] [keV]Σ ∗ + b → Σ + b γ . r . ∗ b → Σ b γ . r . ∗− b → Σ − b γ . r . ′∗ b → Ξ ′ b γ . r · · · Ξ ′∗− b → Ξ ′− b γ . r · · · Ω ∗− b → Ω − b γ . − . r · · · Λ ∗ b → Λ b γ . r · · · Ξ ∗ b → Ξ b γ < . r · · · Ξ ∗− b → Ξ − b γ < . r · · · We next study the radiative decays between the SHBswith negative parity in the flavor representations andthe SHBs with positive parity in the flavor representa-tions, which concern the r -term. The decay widths areexpressed as Γ Λ Q → Σ Q γ = 16 αr F m Σ Q m Λ Q E γ , Γ Λ Q → Σ ∗ Q γ = 8 αr F m Σ ∗ Q m Λ Q E γ , Γ Λ ∗ Q → Σ Q γ = 4 r F m Σ Q m Λ ∗ Q E γ , Γ Λ ∗ Q → Σ ∗ Q γ = 20 αr F m Σ ∗ Q m Λ ∗ Q E γ Γ Ξ + Q → Ξ ′ + Q γ = 16 αr F m Ξ ′ + Q m Ξ + Q E γ , Γ Ξ + Q → Ξ ′∗ + Q γ = 8 αr F m Ξ ′∗ + Q m Ξ + Q E γ , Γ Ξ ∗ + Q → Ξ ′ + Q γ = 4 r F m Ξ ′ + Q m Ξ ∗ + Q E γ , Γ Ξ ∗ + Q → Ξ ′∗ + Q γ = 20 αr F m Ξ ′∗ + Q m Ξ ∗ + Q E γ . (35)In Table X and XI, we show our predictions comparingwith those in Ref. [17, 20]. Our results are consistent withthose in Ref. [17] when r ∼ c RS / √
2, and with those inRef. [20] when r ∼ / r -term generates the radiative decays betweenthe negative parity SHBs in the flavor representationsand the positive parity SHBs in the flavor representa- TABLE X. Predicted widths of radiative decays between neg-ative parity charm baryons in the flavor representations andpositive parity charm baryons in the flavor representations.We also show the predictions in Ref. [17, 20] for comparison.decay mode predicted width [17] [20][keV] [keV] [keV]Λ + c → Σ + c γ r c RS ± + c → Σ ∗ + c γ . r c RS ± . ∗ + c → Σ + c γ r c RS ± . ∗ + c → Σ ∗ + c γ r c RS ± . + c → Ξ ′ + c γ r · · · · · · Ξ + c → Ξ ′∗ + c γ r · · · · · · Ξ ∗ + c → Ξ ′ + c γ r · · · · · · Ξ ∗ + c → Ξ ′∗ + c γ r · · · · · · TABLE XI. Predicted widths of radiative decays between neg-ative parity bottom baryons in the flavor representationsand positive parity bottom baryons in the flavor represen-tations. decay mode predicted width[keV]Λ b → Σ b γ . r Λ b → Σ ∗ b γ . r Λ ∗ b → Σ b γ . r Λ ∗ b → Σ ∗ b γ . r Ξ b → Ξ ′ b γ − r Ξ b → Ξ ′∗ b γ − r Ξ ∗ b → Ξ ′ b γ . − r Ξ ∗ b → Ξ ′∗ b γ − r tions, the widths of which are expressed asΓ Λ ( ∗ ) Q → Λ Q γ = 8 αr f π F m Λ Q m Λ ( ∗ ) Q E γ , Γ Ξ ( ∗ )+ Q → Ξ + Q γ = 8 αr ( f π − σ s ) F m Ξ + Q m Ξ ( ∗ )+ Q E γ , Γ Ξ ( ∗ )0 Q → Ξ Q γ = 8 αr ( f π + σ s ) F m Ξ Q m Ξ ( ∗ )0 Q E γ . (36)In Tabel XII and XIII, we show our predictions togetherwith the ones in Ref. [17, 20]. We think that the differ-ences between our predictions and those in Ref. [20] arefrom the value of σ s : we use σ s = 2 f K − f π while σ s = f π is used in Ref. [20].The widths of radiative decays between the positiveparity SHBs in the flavor representations and the pos-itive parity SHBs in the flavor representations via the0 TABLE XII. Predicted widths of radiative decays betweennegative parity charm baryons in the flavor representationsand positive parity charm baryons in the flavor representa-tions. We also show the predictions in Ref. [17, 20].decay mode predicted width [17] [20][keV] [keV] [keV]Λ c → Λ c γ . r c sRT ± ∗ c → Λ c γ . r c RT ± + c → Ξ + c γ . r · · · · · · Ξ ∗ + c → Ξ + c γ r · · · ± c → Ξ c γ r · · · · · · Ξ ∗ c → Ξ c γ r · · · ± representationsand positive parity bottom baryons in the flavor represen-tations. decay mode predicted width[keV]Λ b → Λ b γ . r Λ ∗ b → Λ b γ . r Ξ b → Ξ b γ . − r Ξ ∗ b → Ξ b γ . − r Ξ − b → Ξ − b γ − r Ξ ∗− b → Ξ − b γ − r r -term are given byΓ Σ ( ∗ ) Q → Λ Q γ = 8 αr f π F m Λ Q m Σ ( ∗ ) Q E γ , Γ Ξ ′ ( ∗ )+ Q → Ξ + Q γ = 8 αr ( f π + 2 σ s ) F m Ξ + Q m Ξ ′ ( ∗ )+ Q E γ , Γ Ξ ′ ( ∗ )0 Q → Ξ Q γ = 8 αr ( f π − σ s ) F m Ξ Q m Ξ ′ ( ∗ )0 Q E γ , (37)and the predicted values are shown in Table XIV andXV with the ones in Ref. [18, 31]. Our results are consis-tent with the lattice results in Ref. [31] if r ∼ .
1. Onthe other hand, comparison with the results in Ref. [18]indicates that the chiral loop may be important.
VI. A SUMMARY AND DISCUSSIONS
We constructed an effective hadronic model re-garding negative parity representations as chi-ral partners to positive parity representations,based on the chiral symmetry and heavy-quark spin-flavor symmetry. We determine the model parame-ters from the experimental data for relevant massesand decay widths of Σ c (2455 , / + ), Σ c (2520 , / + ),Λ c (2595 , / − ), Ξ c (2790 , / − ), and Ξ c (2815 , / − ). TABLE XIV. Predicted widths of radiative decays betweenpositive parity charm baryons in the flavor representationsand positive parity charm baryons in the flavor representa-tions. We also show the predictions in Ref. [18, 31].decay mode predicted width [18] [31][keV] [keV] [keV]Σ + c → Λ + c γ . r · · · Σ ∗ + c → Λ + c γ r · · · Ξ ′ + c → Ξ + c γ . r . . . ′∗ + c → Ξ + c γ . r · · · Ξ ′ c → Ξ c γ . r .
02 0 . ′∗ c → Ξ c γ . r . · · · TABLE XV. Predicted widths of radiative decays betweenpositive parity bottom baryons in the flavor representationsand positive parity bottom baryons in the flavor represen-tations. We also show the predictions in Ref. [18].decay mode predicted width [18][keV] [keV]Σ b → Λ b γ . r ∗ b → Λ b γ . r ′ b → Ξ b γ . r · · · Ξ ′∗ b → Ξ b γ . r ′− b → Ξ − b γ . r · · · Ξ ′∗− b → Ξ − b γ . r . Then, we studied the decay widths of Λ c (2625), Λ b (5912),Λ b (5920), and negative parity Ξ ( ∗ ) b which have not beenyet discovered in any experiments We think that Ξ ( ∗ ) b hereis unlikely to be Ξ b (6227) reported in Ref. [37], which maybe explained as e.g., a molecule state in Ref. [38]. Usingthe model parameters, we predict the values for massesand decay widths of negative parity excited Ξ b .As shown in our previous work Ref. [12], the chiralpartner structure is reflected in the direct decay processesin three-body decays of negative parity representations.Our results for the three-body decays of Ξ ∗ b → Ξ b ππ are dominated by the resonant decay modes unlikely tothe decays of Λ c (2625), Λ b (5912) and Λ b (5920) shown inRef. [12]. However, the Dalitz analysis, which was per-formed in Ref. [13] for the decays of Λ c s and Λ b s, maygive an information of the direct decays of Ξs. There-fore, we would like to stress that future investigationsof detailed three-body decay processes of negative paritySHBs will provide some clues to understand the chiralpartner structure.We also studied the radiative decays of the SHBs in-cluded in the present model using the effective interactionLagrangians in Eq. (32). We showed that there is a rela-tion among the radiative decay widths of positive paritySHBs (Σ ∗ Q , Ξ ∗ Q , Ω ∗ Q ) and those of negative parity SHBs(Λ ∗ Q , Ξ ∗ Q ), reflecting the chiral partner structure. Sincethe masses of negative parity SHBs in the bottom sector1are close to the threshold of hadronic decays, the radia-tive decay widths can be comparable with the strong de-cay widths depending on the precise values of the masses.We summarize the decays of bottom SHBs with negativeparity in Table XVI. TABLE XVI. Pionic and radiative decays of bottom SHBswith negative parity.SHB J P decay Our model exp.modes [MeV] [MeV]Λ b / − Λ b π + π − (0 . . × − < . b π π (1 . . × − Σ b γ . r Σ ∗ b γ . r Λ b γ . r Λ ∗ b / − Λ b π + π − (0 . × − < . b π π (2 . × − Λ b γ . r × − Σ b γ . r Σ ∗ b γ . r Λ b γ . r Ξ b / − Ξ ′ b π . − . · · · Ξ ′ b γ . − . r Ξ ′∗ b γ . − . r Ξ b γ . − . r Ξ − b / − Ξ ′ b π . − . · · · Ξ ′− b γ · · · Ξ ′∗− b γ · · · Ξ − b γ . − . r Ξ ∗ b / − Ξ ′ b π . − . · · · Ξ b γ < . r × − Ξ ′ b γ . − . r Ξ ′∗ b γ . − . r Ξ b γ . − . r Ξ ∗− b / − Ξ ′ b π . − . · · · Ξ − b γ < . r × − Ξ ′− b γ · · · Ξ ′∗− b γ · · · Ξ − b γ . − . r We expect that experimental study of these radiativedecays will provide a clue to understand the chiral part-ner structure. In addition, we predict the Ω ( ∗ ) Q → Ω Q γ decay which is the sole decay mode of Ω ( ∗ ) Q . Experimen-tal observation of this in future will be a check of thepresent framework based on the effective model respect-ing the chiral symmetry and the heavy-quark spin-flavorsymmetry. In addition, we expect that the future lat-tice simulations for the radiative decay of negative par-ity SHBs also provide some clues to the chiral partnerstructure.While we are writing this manuscript, we are informedof Ref. [40], in which the chiral partner structure of SHBsis studied based on the mirror assignment of parity dou-blet structure including three chiral representations, i.e.,( , ), (¯ , ) + ( , ¯ ) and ( , ) + ( , ). Acknowledgments
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