# Spectral representation of lattice gluon and ghost propagators at zero temperature

## TL;DR

This paper develops a regularized method for analytically continuing lattice QCD propagator data to Minkowski space using spectral density extraction, addressing the ill-posed nature of the problem and applying it to gluon and ghost propagators.

## Contribution

It introduces a Tikhonov regularization approach with the Morozov discrepancy principle for spectral density extraction from lattice data, including handling IR singularities and comparing equivalent integral forms.

## Key findings

- Successful application to toy models demonstrating method validity
- Effective regularization of ill-posed spectral density extraction
- Insights into differences between equivalent spectral integral forms

## Abstract

We consider the analytic continuation of Euclidean propagator data obtained from 4D simulations to Minkowski space. In order to perform this continuation, the common approach is to first extract the K\"all\'en-Lehmann spectral density of the field. Once this is known, it can be extended to Minkowski space to yield the Minkowski propagator. However, obtaining the K\"all\'en-Lehmann spectral density from propagator data is a well known ill-posed numerical problem. To regularize this problem we implement an appropriate version of Tikhonov regularization supplemented with the Morozov discrepancy principle. We will then apply this to various toy model data to demonstrate the conditions of validity for this method, and finally to zero temperature gluon and ghost lattice QCD data. We carefully explain how to deal with the IR singularity of the massless ghost propagator. We also uncover the numerically different performance when using two ---mathematically equivalent--- versions of the K\"all\'en-Lehmann spectral integral.