# Scattering amplitudes and contour deformations

## TL;DR

This paper demonstrates how contour deformations can be used to compute scattering amplitudes in Minkowski space within a scalar model, highlighting challenges in locating resonance poles and proposing direct second sheet calculations.

## Contribution

It introduces a method for calculating scattering amplitudes on the second Riemann sheet using contour deformations and unitarity, addressing limitations of analytic continuation methods.

## Key findings

- Contour deformations enable access to Minkowski space scattering regions.
- The scalar model shows virtual states instead of resonance poles.
- Direct second sheet calculations can overcome analytic continuation difficulties.

## Abstract

We employ a scalar model to exemplify the use of contour deformations when solving Lorentz-invariant integral equations for scattering amplitudes. In particular, we calculate the onshell 2 -> 2 scattering amplitude for the scalar system. The integrals produce branch cuts in the complex plane of the integrand which prohibit a naive Euclidean integration path. By employing contour deformations, we can also access the kinematical regions associated with the scattering amplitude in Minkowski space. We show that in principle a homogeneous Bethe-Salpeter equation, together with analytic continuation methods such as the Resonances-via-Pad\'e method, is sufficient to determine the resonance pole locations on the second Riemann sheet. However, the scalar model investigated here does not produce resonance poles above threshold but instead virtual states on the real axis of the second sheet, which pose difficulties for analytic continuation methods. To address this, we calculate the scattering amplitude on the second sheet directly using the two-body unitarity relation which follows from the scattering equation.