Considerations on the Schmid theorem for triangle singularities

TL;DR

This paper examines the validity of the Schmid theorem in particle decay processes, showing that finite resonance widths can enhance triangle singularity effects, challenging the theorem's assumptions and implications.

Contribution

It analyzes the limits of the Schmid theorem's applicability, especially considering realistic resonance widths, and highlights conditions where triangle singularities significantly impact cross sections.

Findings

The theorem holds strictly when the resonance width approaches zero.

Finite widths can cause triangle singularities to dominate or match tree level contributions.

Interference patterns reveal residual effects of the Schmid theorem at realistic widths.

Abstract

We investigate the Schmid theorem, which states that if one has a tree level mechanism with a particle decaying to two particles and one of them decaying posteriorly to two other particles, the possible triangle singularity developed by the mechanism of elastic rescattering of two of the three decay particles does not change the cross section provided by the tree level. We investigate the process in terms of the width of the unstable particle produced in the first decay and determine the limits of validity and violation of the theorem. One of the conclusions is that the theorem holds in the strict limit of zero width of that resonance, in which case the strength of the triangle diagram becomes negligible compared to the tree level. Another conclusion, on the practical side, is that for realistic values of the width, the triangle singularity can provide a strength comparable or even bigger than the tree level, which indicates that invoking the Schmid theorem to neglect the triangle diagram stemming from elastic rescattering of the tree level should not be done. Even then, we observe that the realistic case keeps some memory of the Schmid theorem, which is visible in a peculiar interference pattern with the tree level.