Decays of Roper-like singly heavy baryons in a chiral model

We present a chiral model describing decays of the Roper-like $\Lambda_c(2765)$ and $\Xi_c(2970)$ as well as those of the heavy quark spin-doublet $\{\Sigma_c(2455),\Sigma_c(2520)\}$ and $\{\Xi_c',\Xi_c(2645)\}$, by focusing on chiral representations of diquarks inside the heavy baryons. Based on our chiral model together with the heavy quark spin symmetry, we derive a relation satisfied by axial charges of the heavy baryons that controls magnitude of their one pion (kaon) decays. Besides, it is shown that the large decay widths of the Roper-like $\Lambda_c(2765)$ and $\Xi_c(2970)$, which cannot be implemented by conventional non-relativistic quark models, are explained by reasonable values of the axial charges. In addition, we apply the chiral model to bottom baryons and predict decay widths of the undiscovered Roper-like $\Xi_b$. We expect that our present investigation leads to elucidation of dynamical properties of the excited heavy baryons from the viewpoint of chiral symmetry.


Introduction
Recent development of hadron experiments at, e.g., KEK, LHC, and SLAC has been providing us with clues to understand properties of singly heavy baryons. Since those baryons include one heavy (c or b) quark acting as a spectator for the remaining light quarks, in particular, we can examine a role of diquarks inside the hadrons from them.
Among singly heavy baryons, the Roper-like baryons such as Λ c (2765) and Ξ c (2967) which are heavy quark spinsinglet and J P = (1/2 + ) [1,2] are attracting attention. Their masses are larger than the masses of the corresponding ground-state Λ c (2286) and Ξ c (2470) by about 500 MeV [3], which is similar to the Roper resonance N (1440) for the nucleon sector [4]. Besides, the decay widths of those baryons are tens of MeV, and it is known that such large widths cannot be explained by the conventional nonrelativistic quark model [5]. i This discrepancy is also similar to the nucleon sector [8].
Chiral models based on the chiral representation of the light quarks are useful in studying hadron properties. Thus far, investigation of hadrons by a chiral model has been broadly done not only for light hadrons but also for the heavylight mesons [9,10]. Moreover, recently the model is applied to singly heavy baryons by focusing on the chiral representation of the diquarks inside them [11][12][13][14][15]. In the present work, we extend the chiral model for singly heavy baryons to include the Roper-like baryons Λ c (2765) and Ξ c (2967) in addition to the ground-state ones Λ c (2286) and Ξ c (2470). In particular, in order to treat both the baryons, we introduce the mirror diquark which can be understood as a tetra-diquark state (qqqq) as well as the conventional diquark (qq) [16].
This write-up is organized as follows. In Sec. 2, we introduce the mirror and conventional diquarks, and construct an effective Lagrangian for the singly heavy baryons based on their chiral representations. In this section we also explain input parameters. In Sec. 3 we present the resultant mass spectrum of the heavy baryons. Our present analysis is based on the linear representation of chiral symmetry, so the parity-partner structure of the positive-parity and negativeparity heavy baryons is expected to be realized. Thus, in Sec. 4 we demonstrate mass degeneracies of the parity partners at the chiral restoration point. And in Sec. 5 we conclude the present study.

Model
In this section, we construct an effective model describing the singly heavy baryons which are heavy quark spin-singlet and flavor-singlet, based on chiral symmetry of diquarks. For this purpose, we introduce the following two types of diquarks [16]: where the superscripts (a, b, · · · ) and subscripts (i, j, · · · ) represent color and flavor indices, respectively, and q R(L) = (1 ± γ 5 /2)q is the left-handed (right-handed) quark. In Eq (1) as

the chiral transformation laws of those baryons read
It should be noted that B R(L) corresponds to the three-quark state while B R(L) the pentaquark state. From the chiral transformation laws in Eq. (2), a chiral Lagrangian which is invariant under SU (3) L ×SU )(3) R chiral and parity transformations is obtained as within the heavy baryon effective theory. In Eq. (3), v is a velocity of the heavy baryons and Σ is a light meson nonet. Besides, µ 1 , µ 2 and µ 3 are responsible for parts of the heavy baryon masses while g 1 , g 2 and g 3 are couplings between the baryons and light mesons. Under the spontaneous breakdown of chiral symmetry, the light meson nonet is replaced by its vacuum expectation value (VEV) Σ = diag(σ q , σ q , σ s ) with σ q = 93 MeV. We note that the value of σ s is different from σ q due to the small violation of SU (3) L+R , and σ s will be determined later. In Eq. (3) the heavy baryon fields are expressed in terms of the chiral indices L and R. The parity eigenstates are given by [13], however, still these B ± and B ± are not mass eigenstates due to mixings between them. In order to diagonalize the mixings, we define mass eigenstates B L ±,i and B H ±,i by introducing mixing angles θ B±,i as The corresponding mass eigenstates are given by In our model (3), we have seven parameters: µ 1 , µ 2 , µ 3 , g 1 , g 2 , g 3 , and σ s . To fix them, we use masses of the observed baryons Λ c (2286), Ξ c (2470), Λ c (2765) and Ξ c (2967) as inputs [3]: In addition to them, we also employ masses of the conventional diquarks measured by the lattice simulation [17] together with the chiral model [13] as additional inputs: where the subscripts ± and i refer to the parity eigenvalue and flavor index of the diquarks, respectively. Those diquark masses were measured without corrections from the mirror diquarks, hence, the masses are related to those of the heavy baryons of three-quark states B ± with no corrections from the pentaquark ones B ± . In our model, the masses of such uncorrelated three-quark states are read by switching off the pentaquark states in Eq. That is, we have which fixes σ s and g 1 . The above procedure enables us to fix six parameters, however, one parameter remains undetermined. In order to fix it, we further employ a mass of the negative-parity and heavy quark spin-singlet Λ c baryon estimated by a quark model as an input: M (B L −,3 ) = 2890 MeV [18].

Mass spectrum
In Sec. 2 we have constructed our chiral model. In this section, based on the model we show the resultant mass spectrum of positive-parity and negative-parity Λ c 's and Ξ c 's which are heavy quark spin-singlet.  Since the mass formulae in Eq. (5) include square roots, we can get several parameter sets. In particular, we have obtained two physically distinct parameter sets, as summarized in Table I. We note that the arbitrary mass parameter m B is fixed to be m B = 2868 MeV as mentioned before.
The resultant mass spectrum of Λ c s and Ξ c s for J P = 1/2 + (red) and J P = 1/2 − (blue) are shown in Fig. 1. In this figure the ratio indicated below the bars stands for Qqq : Qqqqq of each heavy baryon, where the upper and lower ones are corresponding to the parameter set (I) and set (II) in Table I, respectively. Besides, the asterisk ( * ) means the input mass. The figure shows that the Roper-like Λ c (2765) and Ξ c (2967) are mostly dominated by the pentaquark state composed of the mirror diquark (qqqq), while the ground-state Λ c (2286) and Ξ c (2470) by the three-quark state composed of the conventional diquark (qq) for both the parameter sets. On the other hand, the ratio of negative-parity baryons are largely dependent on the parameter set.
Since our analysis is based on chiral symmetry of the diquarks, we can obtain an extended Goldberger-Treiman relation that the couplings among the heavy baryons and light pseudo-scalar mesons satisfy:

Mass degeneracy of parity partners
One of the peculiarities of chiral models is a mass degeneracy of the parity partners at the chiral restoration point. The mirror diquarks proposed in the present study can be regarded as an analogue of the mirror nucleons for negativeparity nucleon N * (1535) in the parity doublet model [19,20]. Within this model, it has been suggested that the ground-state N (940) and the excited negative-parity N * (1535) tend to degenerate as chiral symmetry restores, showing the paritypartner structure [21][22][23][24][25]. Thus, in the present model for the singly heavy baryons as well, we can expect that mass degeneracies of the parity partners occur in a similar way. In this section, we show such mass degeneracies by simply changing the VEV's σ q and σ s in Eq. (5) to demonstrate the paritypartner structure of the heavy baryons. The resultant mass changes of Λ c 's (left) and Ξ c 's (right) with respect to σ q and σ s are shown in Fig 2. It should be noted that σ s = 212 MeV and σ q = 93 MeV correspond to the vacuum while σ s = σ q = 0 the chiral restoration point. The figure clearly shows that holds at the chiral restoration point. That is, the parity-partner Although our analysis shows the partner structure in the simplest way, in more realistic situation the chiral restoration occurs at finite temperature, density, and volume where other corrections contribute. Hence, investigation including such effects together with the changes of σ q and σ s on the masses of Λ c 's and Ξ c 's are inevitable, and we leave such a study for future work. From such an investigation, a better understanding of the relation between the mirror diquark and the conventional diquark from the viewpoint of chiral symmetry is expected.

Conclusions
In the present work we have studied the Roper-like Λ c (2765) and Ξ c (2967) together with the ground-state Λ c (2286) and Ξ c (2470) in a chiral model, by introducing the mirror diquark (qqqq) in addition to the conventional diquark (qq). As a result, we have found that the Roper-like Λ c (2765) and Ξ c (2967) are mostly pentaquark state (Qqqqq) while the ground-state Λ c (2286) and Ξ c (2470) are three-quark state (Qqq). Besides, chiral representation of the diquarks has yielded the mass spectrum of negative-parity Λ c 's and Ξ c 's in addition to the positive-parity ones. Furthermore, a sum rule of the heavy baryon masses and an extended Goldberger-Treiman relation have been derived, as a consequence of field-theoretical treatment based on chiral symmetry. In addition to those findings, we have demonstrated the paritypartner structure of the heavy baryons by showing mass degeneracies at the chiral restoration point.
We expect that our present investigation leads to a better understanding of the diquarks inside hadrons from the viewpoint of chiral symmetry. We also expect that our findings and predictions provide future experiments with useful information on unobserved negative-parity Λ c and Ξ c .
i. The relativistic corrections in the quark model was found to improve the discrepancy [6]. Also, the 3 P0 model can reproduce the decay width reasonably [7].