Pion-pion scattering amplitude. III. Improving the analysis with forward dispersion relations and Roy equations

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We complete and improve the fits to experimental π π scattering amplitudes, both at low and high energies, that we performed in the previous papers of this series. We then verify that the corresponding amplitudes satisfy analyticity requirements, in the form of partial wave analyticity at low energies, forward dispersion relations (FDR) at all energies, and Roy equations below K̅ K threshold; the first by construction, the last two, inside experimental errors. Then we repeat the fits including as constraints FDR and Roy equations. The ensuing central values of the various scattering amplitudes verify very accurately FDR and, especially, Roy equations, and change very little from what we found by just fitting data, with the exception of the D2 wave phase shift, for which one parameter moves by 1.5 σ. These improved parametrizations therefore provide a reliable representation of pion- pion amplitudes with which one can test various physical relations. We also present a list of low energy parameters and other observables. In particular, we find a_(0)^((0)) = 0.223 ± 0.009M_(π)^(-1), a_(0) ^(2) = -0.0444 ± 0.0045M -(π)^(-1), and δ_((0))^(0) (m_(K)^(2)) – δ_((2))^(0)(m_(K)^(2)) = 50.9 ± 1.2º
©2008 The American Physical Society. F. J. Y.’s work was supported in part by the Spanish DGI of the MEC under Contract No. FPA2003-04597. J. R. P.’s research is partially funded by Spanish CICYT Contracts No. FPA2005-02327 and No. BFM2003-00856 as well as Banco Santander/Complutense Contract No. PR27/05-13955-BSCH, and is part of the EU integrated infrastructure initiative HADRONPHYSICS PROJECT, under Contract No. RII3-CT-2004-506078. We are grateful to H. Leutwyler for his remarks about the KK threshold rise of the D0 wave and R. García Martín for information concerning the location of the pole. R. K. thanks the Universidad Autónoma de Madrid and Universidad Complutense de Madrid, where part of this research was carried out.
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