Unitarity and unstable-particle scattering amplitudes

TL;DR

This paper derives unitarity equations for scattering amplitudes involving unstable particles by extending the principles of unitarity and analyticity, revealing positivity constraints similar to the optical theorem.

Contribution

It introduces a method to formulate unitarity equations for unstable-particle scattering amplitudes using residues at complex poles, aligning with stable-particle frameworks.

Findings

Derived unitarity equations for unstable particles

Established positivity constraints analogous to the optical theorem

Connected unstable-particle amplitudes to stable-particle analyticity

Abstract

Unitarity is the fundamental property of the S-matrix while its usage for a scattering of unstable particles has been subtle as unstable particles do not appear in the asymptotic states. Defining unstable-particle amplitudes as residues of a higher-point amplitude at an appropriate complex pole, we find unitarity equations for the 2-to-2 unstable-particle amplitudes from unitarity and analyticity of stable-particle scattering amplitudes. The unstable-particle unitarity equations take a form analogous to those of the stable-particle amplitudes when the in and out states are chosen to be complex-conjugate positions. In particular, as in the optical theorem, we find a positivity constraint on a discontinuity of the amplitudes in a positive region of the momentum transfer variable.