Analytic structure of three-point functions from contour deformations

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.

Automated summary

We use contour deformations to investigate the analytic structure of three-point functions, allowing calculations to continue analytically from the spacelike to the timelike regime. We demonstrate how to deform the integration contour and cuts in the integrand to obtain the known cut structure of two-point functions. This method is then applied to one-loop three-point integrals, revealing the relevance of singular points in determining physical thresholds.