Crossing symmetry including non planar diagrams in perturbative QFT

TL;DR

This paper extends the proof of crossing symmetry in perturbative quantum field theory to include non-planar diagrams, classifying them into trivial and non-trivial cases, and provides explicit examples and arguments for the general case.

Contribution

It provides the first proof of crossing symmetry for non-planar diagrams in perturbative QFT, including explicit examples at 3-loop order and a general argument for higher loops.

Findings

Proof of crossing symmetry for non-planar diagrams

Explicit 3-loop non-trivial case example

General argument for higher loops and non-planar edges

Abstract

We venture a proof of crossing symmetry for non-planar diagrams in perturbative QFT. For the planar diagrams a proof of crossing is available in the literature and our method closely follows the one depicted in that case. We classify the non-planar diagrams broadly into two types. For one of these types the proof is pretty straightforward and hence the result extends to all point all loop on-shell amplitudes. These are called the "trivial" cases while for the other type we find certain cases called the "non trivial" cases for which the proof is much more subtle. We present an explicit example of such a "non trivial" case at 3-loop order and argue how the proof of crossing symmetry holds true when all subtleties are taken into consideration. Based on this simple example we argue how the proof works out in general for these "non-trivial" cases at higher loop and with arbitrary number of non-planar edges.