# Pad\'e approximants and analytic continuation of Euclidean Phi-derivable approximations

## TL;DR

This paper explores the effectiveness of Padé approximants for analytically continuing Euclidean propagator data to extract spectral functions and physical information, tested on benchmarks and applied to the O(4) model.

## Contribution

It demonstrates the use of Padé approximants for analytic continuation in Phi-derivable approximations and compares zero-momentum and pole masses in a nonperturbative context.

## Key findings

- Padé approximants successfully access spectral functions from Euclidean data.
- The method reveals differences between zero-momentum and pole masses.
- Application to the O(4) model shows potential for nonperturbative analysis.

## Abstract

We investigate the Pad\'e approximation method for the analytic continuation of numerical data and its ability to access, from the Euclidean propagator, both the spectral function and part of the physical information hidden in the second Riemann sheet. We test this method using various benchmarks at zero temperature: a simple perturbative approximation as well as the two-loop Phi-derivable approximation. The analytic continuation method is then applied to Euclidean data previously obtained in the O(4) symmetric model (within a given renormalization scheme) to assess the difference between zero-momentum and pole masses, which is in general a difficult question to answer within nonperturbative approaches such as the Phi-derivable expansion scheme.