Khuri\u2013Treiman equations for pi pi scattering

Joint Physics Analysis Center ,M Albaladejo,V Mathieu,A Pilloni,C Fernández-Ramírez,A Jackura,N Sherrill,J Nys,A P Szczepaniak,M Mikhasenko

doi:10.1140/epjc/s10052-018-6045-0

Abstract

The Khuri–Treiman formalism models the partial-wave expansion of a scattering amplitude as a sum of three individual truncated series, capturing the low-energy dynamics of the direct and cross channels. We cast this formalism into dispersive equations to study pi pi scattering, and compare their expressions and numerical output to the Roy and GKPY equations. We prove that the Khuri–Treiman equations and Roy equations coincide when both are truncated to include only S- and P-waves. When higher partial waves are included, we find an excellent agreement between the Khuri–Treiman and the GKPY results. This lends credence to the notion that the Khuri–Treiman formalism is a reliable low-energy tool for studying hadronic reaction amplitudes.

Highlights

S-matrix theory, such as analyticity, unitarity, and crossing symmetry for the analysis of three-particle final states are mandatory.One of the main issues posed by the presence of hadrons in any reaction is their final-state interactions, which are formally expressed in terms of the unitarity of the T -matrix
In the case of π π scattering, the results obtained for the σ meson with the Roy equations are very similar to those obtained with approaches that neglect altogether the left-hand cut, or take it into account perturbatively
We explored the various aspects of the KT formalism within the context of π π scattering

Summary

Introduction

S-matrix theory, such as analyticity, unitarity, and crossing symmetry for the analysis of three-particle final states are mandatory. [7,8,9] and references therein) Despite crossing symmetry, this is certainly not the case for a 1 → 3 decay, where ideally one wants to take into account unitarity in the three possible two-hadron channels in the final state. It is known that this model fails to properly account for asymptotic behavior at high energies, it is expected to accurately describe the physics at low energies For this reason, it has often been used to study meson decays, where the energy range is limited by the decay kinematics. 4 we numerically study the results obtained with the KT equations considering D- and F-waves, using as an input the parameterization of the π π scattering shift in Ref. In the Appendix we discuss the contributions of the LHC to the partial waves

Amplitudes

Analytical comparison of KT and Roy equations for S- and P-waves

Numerical results

Discussion and conclusions