Momentum Space Landau Equations Via Isotopy Techniques

TL;DR

This paper derives and generalizes Landau equations in momentum space for Feynman integrals using isotopy techniques, addressing singularities and regularization, and discusses implications for renormalization.

Contribution

It introduces a novel isotopy-based method to derive and extend Landau equations for Feynman integrals, including second-type singularities.

Findings

Derived Landau equations in momentum space from isotopy techniques

Generalized equations to include second-type singularities

Discussed behavior on the principal branch and compatibility with renormalization

Abstract

We investigate the analytic structure of functions defined by integrals with integrands singular on a finite union of quadrics. The main motivation comes from Feynman integrals which belong to this class. Using isotopy techniques we derive the Landau equations in momentum space from the theory of Feynman integrals and generalize these equations to naturally include singularities of the second type. For this purpose we introduce a regularization of analytic families of quadratic forms rendering the isotopy techniques applicable. In the case of Feynman integrals we comment on what is known about the behavior on the principal branch where only specific solutions of the Landau equations contribute to non-analytic points. Finally we discuss compatibility with renormalization.