Cutkosky's Theorem for Massive One-Loop Feynman Integrals -- Part 1

TL;DR

This paper extends Cutkosky's Theorem to massive one-loop Feynman integrals, providing a method to analyze their discontinuities and Landau surfaces without solving Landau equations explicitly.

Contribution

It formulates and proves Cutkosky's Theorem for massive one-loop integrals using intersection theory and develops an algorithm to compute Landau surfaces efficiently.

Findings

Derived explicit formulas for discontinuities of massive one-loop integrals.

Provided an algorithm to determine Landau surfaces without solving Landau equations.

Analyzed specific cases like bubble, triangle, and box graphs.

Abstract

We formulate and prove Cutkosky's Theorem regarding the discontinuity of Feynman integrals in the massive one-loop case up to the involved intersection index. This is done by applying the techniques to treat singular integrals developed in \cite{app-iso}. We write one-loop integrals as an integral of a holomorphic family of holomorphic forms over a compact cycle. Then, we determine at which points simple pinches occur and explicitly compute a representative of the corresponding vanishing sphere. This also yields an algorithm to compute the Landau surface of a one-loop graph without explicitly solving the Landau equations. We also discuss the bubble, triangle and box graph in detail.