Abstract
Meson-meson amplitudes are important on their own and also play key roles in analyses of heavy-meson and tau decays. In this work we propose a new phenomenological model suited to all $SU(3)$ mesonic two-body final-state interactions up to energies around 2 GeV. It is aimed at replacing those entering the old isobar model, produced in the 1960s, long before the development of QCD. The only similarity between our new proposal and amplitudes used in the isobar model concerns vector resonances in the elastic regime. In other situations, especially those involving scalar resonances and coupled channels, the isobar model is not compatible with post-QCD dynamics. In order to support these claims convincingly and to motivate our approach, we consider applications to the $\ensuremath{\pi}\ensuremath{\pi}$ amplitude and compare our version with the isobar model in several different instances. We also show that the new model provides a clear indication of the mechanism responsible for the sharp rise observed in the $\ensuremath{\pi}\ensuremath{\pi}$ phase around 1 GeV. The phenomenological amplitudes proposed here are suited to any number of resonances in a given channel and rely just on masses and coupling constants as free parameters. Concerning theory, they incorporate chiral symmetry at low energies, include coupled channels, and respect unitarity whenever appropriate.
Highlights
In this work we propose a new phenomenological model suited to all SUð3Þ mesonic twobody final-state interactions up to energies around 2 GeV
Final-state interactions incorporated into the decay amplitude T include both proper three-body interactions and a wide range of elastic and inelastic two-body subamplitudes A involving resonances and coupled channels, as we review in Sec
VI we present our model for the real part of two-meson propagators which allows one to go beyond the K-matrix approximation
Summary
A considerable amount of precise data has been produced from BABAR, Belle, BES, LHCb experiments on nonleptonic three-body decays of D and B mesons as well as on tau decays into pseudoscalars. (1) Even if one overlooks the problem of ascribing physical meanings to parameters extracted from the isobar model, there is another issue Speaking, their numerical values depend on the particular assumptions underlying how Eq (1) is used, namely, the nonresonant term and the number and isospins of resonances employed. Guess functions for fitting heavy-meson decay data departing from Lagrangians deal with the same free parameters as employed in scattering amplitudes. This makes the mutual comparison of their values meaningful. II we review how heavy-meson decay amplitudes are related to weak vertices, scattering amplitudes, and form factors This is intended to provide a broad conceptual framework for criticisms of the isobar model.
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