Chiral dynamics and s-wave exotic hadrons

Based on chiral dynamics, existence of exotic hadrons is discussed in the SU(3) symmetric limit. The low energy s-wave interaction of the Nambu-Goldstone boson with a hadron is known to be determined model-independently, which has been used to generate some hadron resonances in nonexotic channels. We show that this interaction in any exotic channels is not strong enough to generate a bound state. PACS. 14.20.–c Baryons (including antiparticles) – 11.30.Rd Chiral symmetries – 11.30.Hv Flavor symmetries Experimentally, more than hundred of hadrons have been discovered so far, whose properties are summarized by the Particle Data Group (PDG) [1]. Most of the hadrons can be, in principle, described in terms of q̄q or qqq, and the only state with exotic flavor quantum numbers in PDG is the S = +1 baryon Θ [2]. Thus, exotic hadrons are indeed “exotic” from an experimental point of view, while there is no clear theoretical explanation of the nonobservation (or non-existence) of the exotic hadrons. This is certainly a non-trivial issue to be explained theoretically, irrespective to the existence of the Θ. Here we report our recent work on this issue [3,4], in which we have studied the existence of the exotic hadrons in s-wave scattering of a hadron and the Nambu-Goldstone (NG) boson. We utilize the theoretical framework based on the S-matrix theory. In 60’s, the framework was used to describe hadron resonances such as Λ(1405) [5,6] with the effective vector meson exchange interaction. Recently, the interaction has been founded by chiral symmetry [7, 8,9,10], leading to a successful description of the hadron resonances in chiral unitary approaches [11,12,13]. In recent applications of the chiral unitary approach, it was shown that some resonances obtained in the coupled channel dynamics with SU(3) breaking became bound states of a single channel in the flavor SU(3) limit [11,14,15,16, 17,18]. Therefore, we expect that the origin of the physical resonances may be clarified by studying the bound states in the SU(3) limit. We focus on the s-wave scatterings, since the low energy interaction of the NG boson with any hadrons in s-wave are uniquely determined by chiral symmetry. In this framework, we examine the possibility to generate the exotic hadron as a bound state of the NG boson and a hadron. The low energy s-wave interaction of the NG boson (Ad) with a target hadron (T ) is model-independently given by Fig. 1. (a) : Notation of the representations α, Ad and T for the WT term. (b) : The bound state pole diagram after unitarization of the amplitude. the Weinberg-Tomozawa theorem [19,20] as Vα = − ω 2f2 Cα,T , (1) with the decay constant (f) and energy (ω) of the NG boson. The group theoretical factor Cα,T is determined by specifying the flavor representations of the target T and the scattering system α ∈ T⊗Ad (see Fig. 1): Cα,T = −〈2FT · FAd〉α = C2(T )− C2(α) + 3, (2) where C2(R) is the quadratic Casimir of SU(3) for the representation R, and we use C2(Ad) = 3 for the adjoint representation of the NG boson. For the target hadron with an arbitrary SU(3) representation T = [p, q], possible representations α for the scattering channels are obtained as [p, q]⊗ [1, 1] = [p+ 1, q + 1] ⊕[p+ 2, q − 1]⊕ [p− 1, q + 2]⊕ [p, q]⊕ [p, q] ⊕[p+ 1, q − 2]⊕ [p− 2, q + 1]⊕ [p− 1, q − 1], where the labels of representations [a, b] should satisfy a, b ≥ 0, and one of the two [p, q] representations on the right hand side should satisfy p ≥ 1 and the other q ≥ 1. Using Eq. (2), we evaluate the coupling strengths Cα,T for 2 Tetsuo Hyodo et al.: Chiral dynamics and s-wave exotic hadrons Table 1. Properties of the WT interaction in the channel α of the NG boson scattering on the target hadron with the T = [p, q] representation. The coupling strengths of the WT term is denoted as Cα,T , ∆E is the differences of the exoticness E between the channel α and the target hadron T , and Cα,T (Nc) denotes the coupling strengths for arbitrary Nc. α Cα,T ∆E Cα,T (Nc) [p+ 1, q + 1] −p− q 1 or 0 3−Nc 2 − p− q [p+ 2, q − 1] 1− p 1 or 0 1− p [p− 1, q + 2] 1− q 1 or 0 5−Nc