Dispersion relations for unphysical particles

TL;DR

This paper develops generalized dispersion relations for unphysical particles, such as confined degrees of freedom, and identifies the rational function component crucial for reconstructing their propagators, with applications to Yang-Mills theory.

Contribution

It introduces a method to determine the missing rational part of unphysical particle propagators, demonstrated through explicit expressions in Yang-Mills theory using the massive expansion.

Findings

The rational part of the propagator can be explicitly identified.

Multi-particle spectral contributions are negligible.

The rational part yields an approximate propagator matching phenomenological models.

Abstract

Generalized dispersion relations are discussed for unphysical particles, e.g. confined degrees of freedom that are not present in the physical spectra but can give rise to observable bound states. While in general the propagator of the unphysical particles can have complex poles and cannot be reconstructed from the knowledge of the imaginary part, under reasonable assumptions the missing piece of information is shown to be in the rational function that contains the poles and must be added to the integral representation. For pure Yang-Mills theory, the rational part and the spectral term are identified in the explicit analytical expressions provided by the massive expansion of the gluon propagator. The multi particle spectral term turns out to be very small and the simple rational part provides, from first principles, an approximate propagator that is equivalent to the tree-level result of simple phenomenological models like the refined Gribov-Zwanziger model.