Solving integral equations in $\eta\to 3\pi$

TL;DR

This paper introduces a new numerical method for solving integral equations in the dispersive analysis of eta to 3 pi decays, overcoming previous technical challenges related to complex plane path integrals and singularities.

Contribution

A novel deformation technique for integration paths simplifies the numerical solution of dispersive integral equations in eta to 3 pi decay analysis.

Findings

The method effectively handles singularities and complex path integrals.

Numerical solutions are obtained more straightforwardly than previous approaches.

The approach is expected to be applicable to omega to 3 pi decays.

Abstract

A dispersive analysis of $\eta\to 3\pi$ decays has been performed in the past by many authors. The numerical analysis of the pertinent integral equations is hampered by two technical difficulties: i) The angular averages of the amplitudes need to be performed along a complicated path in the complex plane. ii) The averaged amplitudes develop singularities along the path of integration in the dispersive representation of the full amplitudes. It is a delicate affair to handle these singularities properly, and independent checks of the obtained solutions are demanding and time consuming. In the present article, we propose a solution method that avoids these difficulties. It is based on a simple deformation of the path of integration in the dispersive representation (not in the angular average). Numerical solutions are then obtained rather straightforwardly. We expect that the method also works for $\omega\to 3\pi$.