Crossing symmetry in the ππ D - and F - wave scattering amplitudes and new precise results for the S - wave amplitude ∗

Recently presented new-one subtracted dispersion relations with imposed crossing symmetry condition for the ππ Sand P-wave scattering amplitudes and the well known Roy’s equations with two subtractions have led to a set of many partial wave amplitudes in very wide energy range [1]. They allow e.g. for a very precise and unambiguous determination of scattering lengths and parameters of the f0(600) (often called σ ) and f0(980) resonances in the S wave. Similarly, one subtracted dispersion relations for the D and F waves have also been recently derived and presented [2]. Here, general structure of these equations with imposed crossing symmetry condition and results of their first practical application in the testing of the input amplitudes obtained in [1] is presented. It can be seen that these equations are very demanding i.e. produce D and F wave output amplitudes with very small errors. This significantly increases the accuracy of determined amplitudes and indirectly can further improve the precision of parameters in the other waves, such as S and P. It is worthy noting that although the presented amplitudes of the D and F waves were not fitted directly to dispersion relations in [1], they fulfill crossing symmetry quite well up to ∼ 800 MeV (some work is still needed). Recently, new and very precise dispersive analysis of the S and P wave amplitudes appeared [3]. Using similar to presented here once subtracted dispersion equations (plus other dispersion relations) and very recent Kl4 experimental results we have calculated set of ππ partial wave amplitudes fulfilling, inter alia, crossing symmetry condition. Making analytic continuation to the complex plane we have found the f0(600) pole at (457+14 −13− i279 +11 −7 ) MeV and f0(980) pole at (996± 7− i25+10 −6 ) MeV. Those new dispersion relations, for the D and F waves, together with the previous ones (called GKPY see [1]) for the S and P waves form a complementary set of theoretical constraints that imposed on the experimental amplitude can define them clearly and precisely. The analysis is