Towards three-body unitarity in $D^+ \to K^- \pi^+ \pi^+$

TL;DR

This paper develops a unitarity-consistent model for the three-body decay $D^+ o K^- \pi^+\pi^+$, incorporating final state interactions based on unitarized chiral perturbation theory and Faddeev-inspired integral equations, and compares results with experimental data.

Contribution

It introduces a novel approach combining two- and three-body unitarity in charm meson decays using a Faddeev-inspired formalism and unitarized $K\pi$ amplitudes.

Findings

Proper three-body effects are significant at threshold.

The model reproduces qualitative features of the decay amplitude.

Results agree well with experimental phase data.

Abstract

We assess the importance of final state interactions in $D^+ \rar K^- \p^+ \p^+$, stressing the consistency between two- and three-body interactions. The basic building block in the calculation is a $K\pi$ amplitude based on unitarized chiral perturbation theory and with parameters determined by a fit to elastic LASS data. Its analytic extension to the second sheet allows the determination of two poles, associated with the $\k$ and the $K^*(1430)$, and a representation of the amplitude based on them is constructed. The problem of unitarity in the three-body system is formulated in terms of an integral equation, inspired in the Faddeev formalism, which implements a convolution between the weak vertex and the final state hadronic interaction. Three different topologies are considered for the former and, subsequently, the decay amplitude is expressed as a perturbation series. Each term in this series is systematically related to the previous one and a re-summation was performed. Remaining effects owing to single and double rescattering processes were then added and results compared to FOCUS data. We found that proper three-body effects are important at threshold and fade away rapidly at higher energies. Our model, based on a vector weak vertex, can describe qualitative features of the modulus of the decay amplitude and agrees well with its phase in the elastic region.