Single heavy baryons with chiral partner structure in a three-flavor chiral model

We construct an effective hadronic model including single heavy baryons (SHBs) belonging to the $(\mathbf{3},\mathbf{3})$ representation under $\mbox{SU}(3)_L \times \mbox{SU}(3)_R$ symmetry, respecting the chiral symmetry and heavy-qaurk spin-flavor symmetry. When the chiral symmetry is spontaneously broken, the SHBs are divided into the baryons with negative parity of $\bar{\mathbf 3}$ representation under $\mbox{SU}(3)$ flavor symmetry which is the chiral partners to the ones with positive parity of ${\mathbf 6}$ representation. We determine the model parameters from the available experimental data for the masses and strong decay widths of $\Sigma_c^{(\ast)}$, $\Lambda_c (2595)$, $\Xi_c (2790)$, and $\Xi_c (2815)$. Then, we predict the masses and strong decay widths of other baryons including $\Xi_b$ with negative parity. We also study radiative decays of SHBs including $\Omega_c^\ast$ and $\Omega_b^\ast$ with positive parity.


I. INTRODUCTION
The spontaneous chiral symmetry breaking, which is one of the most essential properties of QCD, is expected to generate a part of hadron masses and causes the mass difference between chiral partners. Investigation of chiral partner structure will provide some clues to understand the chiral symmetry. In particular, study of the chiral partner structure of hadrons including heavy quarks gives information which are not obtained from the hadrons including only light quarks.
In Ref.

II. EFFECTIVE LAGRANGIAN
In this section, we construct an effective model of single heavy baryons (SHBs) by extending the two-flavor model provided in the previous work [12], to three-flavor case.
We introduce a set of fields, S µ Q (Q = b, c), for SHBs in which the light-quark cloud carries the spin 1 and belongs to (3,3) representation under SU(3) L × SU(3) R symmetry. The field transforms as where g L,R ∈ SU(3) L,R . When the chiral symmetry is spontaneously broken, S µ Q is divided into two parts. One is for the positive parity SHBs belonging to 6 representation under SU(3) flavor symmetry,B 6µ Q , and another for the negative parity SHBs to3,B3 µ Q : We would like to stress thatB 6µ Q andB3 µ Q are chiral partners to each other in the present model. The physical states are embedded aŝ We introduce a 3 × 3 matrix field M for scalar and pseudoscalar mesons made from a light quark and a light anti-quark, which belongs to the (3,3) representation under the chiral SU(3) L × SU(3) R symmetry. The transformation properties of M under the chiral symmetry and the parity are given by We assume that the potential terms for M in the model are constructed in such a way that the M has a vacuum expectation value (VEV) which breaks the chiral symmetry spontaneously: where f π is the pion decay constant and σ s is written as σ s = 2f K − f π with the Kaon decay constant f K . 1 In the following, for studying the decays of the SHBs with emitting pions, we parameterize the field M as with π being the 3 × 3 matrix field including pions as In addition, we introduce two fields, one belonging to (3, 1) representation under SU(3) L × SU(3) R symmetry and another to (1,3) representation. It is convenient to use anti-symmetric 3 × 3 matrix fields which transform as where S QLL and S QRR denote the fields of (3, 1) and (1,3) representations, respectively. They are related to each others by parity transformation as We introduce the parity eigenstates as where A3 Q and C3 Q carry the negative and positive parities, respectively. They include the flavor anti-symmetric fields aŝ These A3 Q and C3 Q express spin-1/2 fields respectively. Since the particles which are expressed by A3 Q are still undiscovered, we neglect A3 Q in the following discussion. Now, let us write down an effective Lagrangian including the baryon fields S µ Q , S QLL , and S QRR together with the meson field M , based on the heavy-quark spin-flavor symmetry and the chiral symmetry. We do not consider the terms including more than square of M field or more than two derivatives. A possible Lagrangian is given by where m ΛQ (Q = c, b) are the masses of Λ c (2286) and Λ b in the ground state, ∆ provides the difference between the chiral invariant mass of S µ Q and that of S QLL and S QRR .
, and h 2 are dimensionless coupling constants. We note that we include g v 2 -term to incorporate the heavy-flavor violation needed for explaining the mass differences of charm and bottom sectors (See Ref. [12].). Although we can add heavy-quark flavor violation terms corresponding to g 1term, such contributions are absorbed into the definition of ∆. We expect that heavy-quark flavor violating contributions to terms other than g v 2 term are small. Since thresholds of B 6 * Q → B 6 Q π are not open, the related terms are not included here. We note that the above chiral partner 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 2 , g v 2 , and κ 2 from masses of relevant SHBs, and g 3 from Σ whereκ i = κ im , and g Q 2 is defined as We determine the fraction of strange quark mass m s and up or down quark massm from the masses of the pion and kaon as m s /m = 25.9 using In the present analysis, we assign the following physical states to the flavor3 representation: and are the chiral partner to the flavor 6 representation: In the bottom sector,3 includes We list experimental data of their masses and full decay widths [39] in Table I. Here, we cannot determine the values of ∆, g 1 , and κ 1 , separately. Instead, we introducē to rewrite mass formulas as We estimate the values of mass parameters and coupling constants in charm sector from experimental data in a way explained in Ref. [12]: We calculate the spinaveraged mass of SHBs in a heavy-quark multiplet with including errors to include the masses of members belonging to the multiplet as shown in Table II.
To include the heavy quark flavor symmetry violation, we determined the value of g b 2 from the mass difference between spin-averaged masses of Λ to determine the coupling constant g 3 as done in Ref. [12]. We show the estimated values of model parameters in Table III. Using the estimated value of g 3 , we predict the decay widths of Σ ( * ) Table IV. These predictions are consistent with experimental data because light flavor symmetry violation and heavy quark symmetry violation are small for the g 3 -term.
We can estimate the masses of bottom baryons included in our model using the parameters in Table III. In the present analysis, we assume heavy quark spin symmetry, so that we predict the spin-averaged masses which are shown 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 values predicted for the members in a multiplet in Ref. [30]. This table shows that our predictions are consistent with those in Ref. [30,36].
We can see that our predictions for Σ are consistent with the spin-averaged masses of experimentally observed masses. For Ω b , only the mass of the spin-1/2 member is known experimentally. Although our prediction of the spin-average is slightly larger than the observed mass of the spin-1/2 member, we expect that the spin-3/2 member is slightly heavier which makes the spin-averaged larger and consistent with our prediction. Future experimental observation of spin-3/2 member as well as Ξ ( * ) b1 will be a test of the present model. We note that Ξ ( * ) b1 in the present analysis are unlikely to make a multiplet including Ξ b (6227) reported in Ref. [37], since the predicted mass of Ξ ( * ) b1 is about 100 MeV smaller than the observed mass of Ξ b (6227).

IV. PION DECAYS OF SINGLE HEAVY BARYONS WITH NEGATIVE PARITY
In this section, we consider decays of B3 In Ref. [12], we used the two-pion decay width of Λ c (2595) to determine the values of derivative coupling constants, h I 1 and h 2 . Here, we also include the decay widths of Ξ c (2790) and Ξ c (2815). There exists violation of the heavy quark spin symmetry between the decay widths of Ξ c (2790) and Ξ c (2815). Instead of treating this violation precisely, we include the violation as systematic errors of the model. Therefore, we use values of a decay width of Λ c (2595) and, a spin averaged decay width between Ξ c (2790) and Ξ c (2815) as inputs to determine h I 1 and h 2 . The region colored by dark purple in Fig. 1 shows the allowed values of h I 1 and h 2 determined from the decay width of Λ c (2595) where the errors of g c 2 , g 3 and the total width with Λ(2595) are taken into account. The region by light purple are obtained from the spin averaged width of Ξ c (2790) and Ξ c (2815) with the errors of model parameters included. In the following analysis, we use the values of h I 1 and h 2 in the overlapped region of two colors in Fig. 1 to make predictions of the decay widths of excited SHBs with negative pariry. We show the results of total decay widths in Table VI, where we list predictions by the quark model in Refs. [25,36] for comparison. [MeV] Λc(2595) Λcπ + π − 0.562-1.09 Λcπ 0 π 0 1.23-2.31 sum 1.82-3.36 (input) · · · · · · 2.59 ± 0.30 ± 0.47 Λc(2625) Λcπ + π − 0.0618-0.507 Λcπ 0 π 0 0.0431-0.226 sum 0.106-0.733 · · · · · · < 0.97 We note that we use the predicted masses of Ξ b with negative parity shown in Table V with including their errors. So the predicted decay widths take a wide range of values, which includes predictions in Refs. [25,36]. In particular, since the minimum value shown in Table V is very close to the threshold of the relevant decays, the minimum values of the predictions of one-pion decays of Ξ ( * ) b1 in Table VI are Table V, and Ξ ′ * b to be the central mass values in Table I. We note that, unlikely to the decays of Λ b (5912) and Λ b (5920), the decays of Ξ * 0 b1 and Ξ * − b1 are not dominated by the non-resonant contribution.

V. RADIATIVE DECAYS
In this section, we study radiative decays of the SHBs. The relevant Lagrangian is given by where F µν is the field strength of the photon andF µν is its dual tensor:F µν = (1/2)ǫ µνρσ F ρσ , r i (i = 1, ..., 4) are dimensionless constants, and F is a constant with dimension one. In this analysis, we take F = 350 MeV following Ref. [17]. We note that the values of the constants r i are of order one based on quark models [17]. Let us first study the electromagnetic intramultiplet transitions governed by the r 1 -term in Eq. (32). Let B * denotes the decaying baryon with spin-3/2 (B * = Λ * Q1 , Ξ * Q1 , Σ * Q , Ξ ′ * Q , Ω * Q ), and B, the daughter baryon with spin-1/2 (Λ Q1 , Ξ Q1 , Σ Q , Ξ ′ Q , Ω Q ). Then the radiative decay width is given by 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 Here we would like to stress that the radiative decay widths of positive parity SHBs (Σ * Q , Ξ * Q , Ω * Q ) and those of negative parity SHBs (Λ * Q1 , Ξ * Q1 ) are determined by just one coupling constant r 1 , reflecting the chiral partner structure. We think that checking the relation among these radiative decays will be one of the crucial test of the chiral partner structure. In Table VIII and IX, we show our predictions comparing with those in Ref. [18,32]. In radiative decays, the chiral loop considered in Ref. [18] might have contribution. However, our predictions are consistent with those in Ref. [18] for r 1 ∼ 1, which implies the contribution from the chiral loop is small in this radiative decay. On the other hand, our results are consistent with the lattice results if r 1 ∼ 0.2.
We note that Ω * Q does not have any strong decays and the main mode must be Ω * Q → Ω Q γ. We expect the coupling constant r 1 will be determined by the decay of Ω * Q in future experiments, and other radiative decay widths related to r 1 -type interaction will be estimated.  IX. Predicted widths of radiative decays between heavy quark multiplets of bottom baryons. We also show the predictions in Ref. [18] for comparison.
decay mode predicted width [18] [keV] [keV] We next study the radiative decays between the SHBs with negative parity in the flavor 3 representations and the SHBs with positive parity in the flavor 6 representations, which concern the r 2 -term. The decay widths are expressed as In Table X and XI, we show our predictions comparing with those in Ref. [17,20]. Our results are consistent with those in Ref. [17] when r 2 ∼ c RS / √ 2, and with those in Ref. [20] when r 2 ∼ 1/2.
The r 3 -term generates the radiative decays between the negative parity SHBs in the flavor 3 representations and the positive parity SHBs in the flavor 3 representa-TABLE X. Predicted widths of radiative decays between negative parity charm baryons in the flavor 3 representations and positive parity charm baryons in the flavor 6 representations. We also show the predictions in Ref. [17,20] for comparison. decay mode predicted width [17] [20] [keV] [keV] [keV] tions, the widths of which are expressed as In Tabel XII and XIII, we show our predictions together with the ones in Ref. [17,20]. We think that the differences between our predictions and those in Ref. [20] are from the value of σ s : we use σ s = 2f K − f π while σ s = f π is used in Ref. [20].
The widths of radiative decays between the positive parity SHBs in the flavor 6 representations and the positive parity SHBs in the flavor 3 representations via the   decay mode predicted width r 4 -term are given by and the predicted values are shown in Table XIV and XV with the ones in Ref. [18,31]. Our results are consistent with the lattice results in Ref. [31] if r 4 ∼ 0.1. On the 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 regarding negative parity 3 representations as chiral partners to positive parity 6 representations, based on the chiral symmetry and heavy-quark spinflavor symmetry. We determine the model parameters from the experimental data for relevant masses and decay widths of Σ c (2455, 1/2 + ), Σ c (2520, 3/2 + ), Λ c (2595, 1/2 − ), Ξ c (2790, 1/2 − ), and Ξ c (2815, 1/2 − ). XIV. Predicted widths of radiative decays between positive parity charm baryons in the flavor 6 representations and positive parity charm baryons in the flavor 3 representations. We also show the predictions in Ref. [18,31]. decay mode predicted width [18] [31] [keV] [keV] [keV] Σ + c → Λ + c γ 42.9r 2 are close to the threshold of hadronic decays, the radiative decay widths can be comparable with the strong decay widths depending on the precise values of the masses. We summarize the decays of bottom SHBs with negative parity in Table XVI.
We expect that experimental study of these radiative decays will provide a clue to understand the chiral partner structure. In addition, we predict the Ω ( * ) Q → Ω Q γ decay which is the sole decay mode of Ω ( * ) Q . Experimental observation of this in future will be a check of the present framework based on the effective model respecting the chiral symmetry and the heavy-quark spin-flavor symmetry. In addition, we expect that the future lattice simulations for the radiative decay of negative parity SHBs also provide some clues to the chiral partner structure.