The analytic structure of three-point functions from contour deformations

TL;DR

This paper develops a contour deformation method to analyze the analytic structure of three-point functions, enabling continuation from spacelike to timelike regimes and identifying physical thresholds through Landau analysis.

Contribution

It introduces a novel contour deformation approach for three-point functions, linking Landau conditions to the emergence of singularities and cuts in complex integrals.

Findings

Contour deformation effectively reveals the cut structure of three-point functions.

Landau conditions can be interpreted through contour deformation to identify singularities.

Numerical application to $ ext{phi}^3$ theory demonstrates the method's practical utility.

Abstract

We explore the analytic structure of three-point functions using contour deformations. This method allows continuing calculations analytically from the spacelike to the timelike regime. We first elucidate the case of two-point functions with explicit explanations how to deform the integration contour and the cuts in the integrand to obtain the known cut structure of the integral. This is then applied to one-loop three-point integrals. We explicate individual conditions of the corresponding Landau analysis in terms of contour deformations. In particular, the emergence and position of singular points in the complex integration plane are relevant to determine the physical thresholds. As an exploratory demonstration of this method's numerical implementation we apply it to a coupled system of functional equations for the propagator and the three-point vertex of $\phi^3$ theory. We demonstrate that under generic circumstances the three-point vertex function displays cuts which can be determined from modified Landau conditions.