On the computation of the spectral density of two-point functions: complex masses, cut rules and beyond

TL;DR

This paper develops a method to compute the spectral density of two-point functions with complex masses in Euclidean space, comparing traditional cut rules with a Stieltjes inversion approach, relevant for theories with non-standard propagators.

Contribution

It introduces an alternative Stieltjes inversion method for spectral density calculation, applicable when cut rules are invalid due to complex masses or non-standard propagators.

Findings

Results agree with cut rule calculations for complex masses.

Stieltjes inversion is useful when cut rules are not applicable.

Method applicable to gauge theories with Gribov-like propagators.

Abstract

We present a steepest descent calculation of the Kallen-Lehmann spectral density of two-point functions involving complex conjugate masses in Euclidean space. This problem occurs in studies of (gauge) theories with Gribov-like propagators. As the presence of complex masses and the use of Euclidean space brings the theory outside of the strict validity of the Cutkosky cut rules, we discuss an alternative method based on the Widder inversion operator of the Stieltjes transformation. It turns out that the results coincide with those obtained by naively applying the cut rules. We also point out the potential usefulness of the Stieltjes (inversion) formalism when non-standard propagators are used, in which case cut rules are not available at all.