Improvements to the Method of Dispersion Relations for B Nonleptonic Decays

TL;DR

This paper enhances the dispersion relations method for analyzing final state interactions in B nonleptonic decays by adding an extra term, refining spectral function properties, and including comprehensive Regge contributions, with applications to specific decay modes.

Contribution

It introduces an additional term in the dispersion representation, refines the treatment of spectral functions, and incorporates both t and u-channel Regge contributions for better analysis of B decays.

Findings

Improved dispersion relation method with an extra term from equal-time commutator.

Enhanced spectral function analysis considering reality properties.

Inclusion of both t and u-channel Regge contributions in decay analysis.

Abstract

We bring some clarifications and improvements to the method of dispersion relations in the external masses variables, that we proposed recently for investigating the final state interactions in the B nonleptonic decays. We first present arguments for the existence of an additional term in the dispersion representation, which arises from an equal-time commutator in the LSZ formalism and can be approximated by the conventional factorized amplitude. The reality properties of the spectral function and the Goldberger-Treiman procedure to perform the hadronic unitarity sum are analyzed in more detail. We also improve the treatment of the strong interaction part by including the contributions of both t and u-channel trajectories in the Regge amplitudes. Applications to the $B^0\to \pi^+\pi^-$ and $B^+\to \pi^0 K^+$ decays are presented.