Dispersion Relation Bounds for pi pi Scattering

TL;DR

This paper uses fundamental axiomatic principles to derive bounds on pi pi scattering amplitudes, compares these bounds with previous results, and computes the linear sigma model amplitude to analyze consistency with these bounds.

Contribution

It introduces new positivity bounds on pi pi scattering parameters and computes the linear sigma model amplitude in the MS-bar scheme, highlighting violations at certain mass ratios.

Findings

Positivity bounds constrain ar{l}_1 and ar{l}_2.

Linear sigma model results violate bounds at small m_sigma/m_pi.

The bounds are consistent with axiomatic principles despite violations.

Abstract

Axiomatic principles such as analyticity, unitarity and crossing symmetry constrain the second derivative of the pi pi scattering amplitudes in some channels to be positive in a region of the Mandelstam plane. Since this region lies in the domain of validity of chiral perturbation theory, we can use these positivity conditions to bound linear combinations of \bar{l}_1 and \bar{l}_2. We compare our predictions with those derived previously in the literature using similar methods. We compute the one-loop pi pi scattering amplitude in the linear sigma model (LSM) using the MS-bar scheme, a result hitherto absent in the literature. The LSM values for \bar{l}_1 and \bar{l}_2 violate the bounds for small values of m_sigma/m_pi. We show how this can occur, while still being consistent with the axiomatic principles.