# The Bethe-Salpeter approach to bound states: from Euclidean to Minkowski space

## TL;DR

This paper introduces a practical method to transform Euclidean Bethe-Salpeter amplitudes into Minkowski space, revealing their analytic structure and enabling applications like parton density extraction.

## Contribution

The paper presents a novel un-Wick rotation technique for Bethe-Salpeter equations, bridging Euclidean and Minkowski formulations for bound state analysis.

## Key findings

- Revealed the analytic structure of the Bethe-Salpeter amplitude in Minkowski space.
- Demonstrated the method's effectiveness for two-boson bound states.
- Suggested potential for extracting parton distribution functions.

## Abstract

The challenge to obtain from the Euclidean Bethe--Salpeter amplitude the amplitude in Minkowski is solved by resorting to un-Wick rotating the Euclidean homogeneous integral equation. The results obtained with this new practical method for the amputated Bethe--Salpeter amplitude for a two-boson bound state reveals a rich analytic structure of this amplitude, which can be traced back to the Minkowski space Bethe--Salpeter equation using the Nakanishi integral representation. The method can be extended to small rotation angles bringing the Euclidean solution closer to the Minkowski one and could allow in principle the extraction of the longitudinal parton density functions and momentum distribution amplitude, for example.