ar X iv : h ep - p h / 07 01 05 8 v 1 9 J an 2 00 7 Reply to Takamatsu ’ s paper on the kappa

The essential elements in my way of fitting the κ are reviewed. A number of incorrect claims made by Takamatsu are clarified and corrected. The 'sigma' group takes the κ and σ as conventional Breit-Wigner resonances with widths linearly proportional to phase space ρ for decay. These resonances alone do not fit the Kπ (and ππ) elastic phase shifts. An s-dependent background is added from a repulsive core so as to achieve agreement with experiment. The idea is described fully in Takamatsu's review. This is one way of fitting the data, although the background needs to be parametrised empirically. The approach led to early clarification of the existence of the κ [2] and σ resonances [3,4]. Since then, things have moved on. My approach [5], and that of many others, is instead to treat the κ as a resonance with s-dependent width: f el = N(s) D(s) = MΓ Kπ (s) M 2 − s − iMΓ total (s). (1) No background is required. In elastic ππ and Kπ scattering, it is well known that the Adler self-consistency condition, supplemented by linearity of amplitudes within the Mandelstam triangle, leads to zeros at s = m 2 π /2 for the ππ S-wave amplitude and at s = m 2 K − m 2 π /2 for Kπ. It accounts naturally for the fact that my Γ(s) near threshold is closely proportional to (s − s A)ρ. For the σ, data are more extensive and the required s-dependence is somewhat more complicated: see Ref. [6] for full details. The denominator D(s) in Eq. (1) originates from the right-hand cut and should be common to elastic scattering and production. Phases for the σ, derived from BES data for J/ψ → ωπ + π − , are indeed consistent with the same s-dependence as ππ elastic scattering from 450 to 950 MeV within experimental errors of ∼ 3.5 • [7]. Likewise κ phases, derived from BES II data [8], are consistent with the s-dependence of LASS data for Kπ elastic scattering [9] within similar errors. These are important consistency checks which could fail if the initial assumptions in Eq. (1) are wrong. There is however more to the story. The numerator N(s) originates from the left-hand cut. The 'sigma' group makes no use of this information. Neither did the first fits to production data. This situation is now being remedied. Caprini et al …