Analysis of averaged multichannel delay times

TL;DR

This paper critically examines various formalisms of time delay in scattering theory, clarifies their relationships, and extends the understanding of multichannel delay times, highlighting differences and consistencies among them.

Contribution

It clarifies the relationships between different delay time formalisms and extends Nussenzveig's angular delay to multichannel scattering, correcting previous misinterpretations.

Findings

Smith's delay time is consistent with a Hermitian lifetime matrix.

Eisenbud's lifetime matrix violates time reversal invariance.

Multichannel average delay reduces to Smith's delay after proper averaging.

Abstract

The physical significances and the pros and cons involved in the usage of different time delay formalisms are discussed. The delay time matrix introduced by Eisenbud, where only s-waves participate in a reaction, is in general related to the definition of an angular time delay which is shown not to be equivalent to the so-called phase time delay of Eisenbud and Wigner even for single channel scattering. Whereas the expression due to Smith which is derived from a time delayed radial wave packet is consistent with a lifetime matrix which is Hermitian, this is not true for any Eisenbud-type lifetime matrix which violates time reversal invariance. Extending the angular time delay of Nussenzveig to multiple channels, we show that if one performs an average over the directions and subtracts the forward angle contribution containing an interference of the incident and scattered waves, the multichannel angle dependent average time delay reduces to the one given by Smith. The present work also rectifies a recently misinterpreted misnomer of the relation due to Smith.