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

TL;DR

This paper develops a dispersive framework to construct two-loop level amplitudes for $P o \pi\pi\pi$ decays, improving theoretical understanding of these processes at low energies.

Contribution

It introduces a novel dispersive method for two-loop amplitude calculations in $P o \pi\pi\pi$ decays, including all pion mass configurations.

Findings

Constructed two-loop amplitudes for various pion mass configurations.

Analyzed analyticity properties of the amplitudes.

Provided detailed case studies for specific decay channels.

Abstract

We present and develop a general dispersive framework allowing us to construct representations of the amplitudes for the processes $P\pi\to\pi\pi$, $P=K,\eta$, valid at the two-loop level in the low-energy expansion. The construction proceeds through a two-step iteration, starting from the tree-level amplitudes and their S and P partial-wave projections. The one-loop amplitudes are obtained for all possible configurations of pion masses. The second iteration is presented in detail in the cases where either all masses of charged and neutral pions are equal, or for the decay into three neutral pions. Issues related to analyticity properties of the amplitudes and of their lowest partial-wave projections are given particular attention. This study is introduced by a brief survey of the situation, for both experimental and theoretical aspects, of the decay modes into three pions of charged and neutral kaons and of the eta meson.