Magic Identities for the Conformal Four-Point Integrals; the Minkowski Metric Case

TL;DR

This paper proves the 'magic identities' for four-point conformal integrals in Minkowski space, clarifying the cycle choices needed, and builds on previous mathematical interpretations involving Lie groups and quaternionic analysis.

Contribution

It extends the proof of magic identities from Euclidean to Minkowski metrics, specifying the integration cycles and using quaternionic analysis and Lie group representations.

Findings

Proof of magic identities in Minkowski space

Specification of integration cycles for correctness

Connection to quaternionic analysis and Lie group representations

Abstract

The original "magic identities" are due to J.M.Drummond, J.Henn, V.A.Smirnov and E.Sokatchev; they assert that all n-loop box integrals for four scalar massless particles are equal to each other [DHSS]. The authors give a proof of the magic identities for the Euclidean metric case only and claim that the result is also true in the Minkowski metric. However, the Minkowski case is much more subtle and requires specification of the relative positions of cycles of integration to make these identities correct. In this article we prove the magic identities in the Minkowski metric case and, in particular, specify the cycles of integration. Our proof of magic identities relies on previous results from [L1, L2], where we give a mathematical interpretation of the n-loop box integrals in the context of representations of a Lie group U(2,2) and quaternionic analysis. The main result of [L1, L2] is a (weaker) operator version of the "magic identities". No prior knowledge of physics or Feynman diagrams is assumed from the reader. We provide a summary of all relevant results from quaternionic analysis to make the article self-contained.