# Quark propagator in Minkowski space

## TL;DR

This paper introduces a spectral representation method to analyze the quark propagator in Minkowski space, effectively handling complex singularities and enabling nonperturbative calculations.

## Contribution

It presents a novel computational approach using spectral functions to solve Dyson-Schwinger equations in Minkowski space, avoiding traditional numerical complications.

## Key findings

- Spectral functions can be used to handle poles and branch cuts exactly.
- Derived model-independent integral equations for spectral functions.
- Implemented a numerical algorithm with explicit calculations in a schematic model.

## Abstract

The analytic structure of the quark propagator in Minkowski space is more complex than in Euclidean space due to the possible existence of poles and branch cuts at timelike momenta. These singularities impose enormous complications on the numerical treatment of the nonperturbative Dyson-Schwinger equation for the quark propagator. Here we discuss a computational method that avoids most of these complications. The method makes use of the spectral representation of the propagator and of its inverse. The use of spectral functions allows one to handle in exact manner poles and branch cuts in momentum integrals. We obtain model-independent integral equations for the spectral functions and perform their renormalization by employing a momentum-subtraction scheme. We discuss an algorithm for solving numerically the integral equations and present explicit calculations in a schematic model for the quark-gluon scattering kernel.