The Dπ S-wave

A pole in the Dπ S-wave analogous to the σ and κ is predicted at M− iΓ/2 = 2098± 40− i(260 ± 40) MeV. The main objective of this paper is to provide formulae for fitting it to data. PACS numbers: 12.39.Ki, 13.25.Gv, 14.40.Lb. There are well-known σ and κ poles in ππ and Kπ. The objective of this paper is to predict analogous poles in Dπ and Bπ S-waves and provide appropriate parametrisations of them. It has not been observed experimentally yet. There are earlier predictions of the poles in Dπ and Bπ S-waves by van Beveren and Rupp [1]. Their result for Dπ agrees fairly closely with that found here. For Bπ they predict the P0 bn̄ state close to the Bπ threshold. Guo et al. give similar predictions for poles in Dπ and Bπ but with widths roughly a factor 2 smaller than those found here [2]. Another recent paper by Guo with other authors discusses the Dπ and Bπ scattering lengths and related pole positions [3]. The existence of these poles and their parameters are of interest to understanding Chiral Symmetry Breaking and its possible relation to Confinement. These could be separate phase transitions, i.e. condensates. On the other hand it is possible that they are two facets of a single phase transition. What will emerge from the discussion here is that the predicted pole near the Dπ threshold overlaps significantly with the lowest P0 cn̄ state. How or whether they mix and possibly coalesce is of fundamental interest to understanding how confinement works. Let us begin by reviewing briefly the facts about σ and κ. Caprini et al. predict the σ pole at 441 −8 − i272 −12.5 MeV [4]. It appears as a peak in E791 data for D → 3π [5] at ∼ 478 MeV and, more prominently, in BES2 data on J/Ψ → ωππ at ∼ 500 MeV [6]. Pole positions from these and other data depend somewhat on formulae used to fit them. Descotes-Genon et al. predict a similar κ pole inKπ elastic scattering at 658±13−i(279±12) MeV [7] The corresponding peak is clearly visible in D → Kππ data of E791 [8] [9] and less clearly in BES2 data for J/Ψ → KKππ [10] [11]. Again experimental values are somewhat higher in mass, but strongly dependent on fitting formulae. Most analyses of data do not include the s-dependence required by Chiral Symmetry Breaking, hence confusing the experimental situation. This needs to be remedied and a prime objective of the present paper is to provide simple formulae and discuss their foundations. Algebra will be presented for the Dπ S-wave and the resulting pole will be called ∂ (dabba) as a shorthand. Formulae carry over trivially to ππ, Kπ and Bπ. For a resonance, the elastic scattering amplitude is T (s) = MΓ(s) M − s− iMΓ(s) (1) T(s) = M − s MΓ(s) − i = cot δ − i (2) email: [email protected]