Precise determination of the f0(500) and f0(980) parameters in dispersive analysis of the ππ data

Use of the new and precise dispersive equations with imposed crossing symmetry condition to solve the long-standing puzzle in the parameters of the f0(500), as well as the f0(980) is presented. This puzzle is finally being settled thanks to analyzes carried out during the last years [J. Beringer et al. (Particle Data Group), Phys. Rev. D86, (2012) 010001]. In this report we show how our very recent dispersive data analysis allowed for a precise and model independent determination of the amplitudes for the S, P, D and F waves [R. Garcia-Martin, R. Kaminski, J. R. Pelaez, J. Ruiz de Elvira and F.J. Yndurain, Phys. Rev. D83, (2011) 074004; R. Garcia-Martin, R. Kamiński, J.R. Pelaez and J. Ruiz de Elvira, Phys. Rev. Lett. 107, (2011) 072001; R. Kamiński, Phys. Rev. D83, (2011) 076008]. Especially we present that the analytic continuation of once subtracted dispersion relations for the S 0 wave to the complex energy plane leads to very precise results for the f0(500) pole: spole=457−13+14−i279−7+11 MeV and for the f0(980) pole: spole=996±7−i25−6+10MeV. We also mention on first (or one of the first) practical application of presented dispersion relations in refitting and in significant improving of the ππ S-wave amplitudes below 1000 MeV.