The Dual Conformal Box Integral in Minkowski Space

TL;DR

This paper investigates the dual conformal box integral in Minkowski space, revealing its branch structure and how dual conformal transformations can change its value, thus uncovering subtle symmetry breaking effects.

Contribution

It classifies conformally equivalent configurations of four points in Minkowski space and demonstrates how dual conformal transformations affect the integral's branch structure.

Findings

The integral's value depends on the kinematic region and can jump between branches.

Most configurations with real parameters can be mapped to a region where the integral is directly calculable.

Up to four branches of the integral can be reached via dual conformal transformations.

Abstract

The dual conformal box integral in Minkowski space is not fully determined by the conformal invariants $z$ and $\bar{z}$. Depending on the kinematic region its value is on a 'branch' of the Bloch-Wigner function which occurs in the Euclidean case. Dual special conformal transformations in Minkowski space can change the kinematic region in such a way that the value of the integral jumps to another branch of this function, encoding a subtle breaking of dual conformal invariance for the integral. We classify conformally equivalent configurations of four points in compactified Minkowski space. We show that starting with any configuration, one can reach up to four branches of the integral using dual special conformal transformations. We also show that most configurations with real $z$ and $\bar{z}$ can be conformally mapped to a configuration in the same kinematic region with two points at infinity, where the box integral can be calculated directly in Minkowski space using only the residue theorem.