Dispersive construction of two-loop P->3\pi (P=K,\eta) amplitudes

TL;DR

This paper develops a unified, relativistic, model-independent method based on fundamental principles to construct two-loop amplitudes for P->3π decays, aiding the analysis of quark mass ratios and pion scattering lengths.

Contribution

It introduces a novel approach combining unitarity, analyticity, crossing symmetry, and chiral counting to derive two-loop amplitudes for P->3π decays in a model-independent way.

Findings

Provides a general procedure for two-loop amplitude construction.

Illustrates the method with e9 decay amplitude at leading isospin breaking.

Enables improved analysis of decay processes for fundamental parameters.

Abstract

The branching ratio of the \eta->3\pi decay is an important source of information on the value of the quark mass ratio 1/R=(m_d-m_u)/(m_s-\hat m). Furthermore, isospin breaking effects in the decays K->3\pi provide information on the pion scattering lengths. The cusp effect in the K->3\pi decays is presently being analyzed by the NA48 and KTeV experiments. From the theoretical point of view, these processes have been studied by different methods. We propose a unified and relativistic treatment relying on very general principles, unitarity, analyticity and crossing symmetry, combined with chiral counting, in order to construct model-independent representations of the corresponding amplitudes that are valid at two loops. A general description of the procedure is given and is illustrated in the case of the \eta decay amplitude in the leading order in the isospin breaking.