Interactions as conformal intertwiners in 4D QFT

TL;DR

This paper explores how conformal integrals in 4D quantum field theory can be understood through topological field theory and representation theory, revealing new structures in one-loop correlators and their relation to equivariant maps.

Contribution

It extends previous work by linking 1-loop conformal integrals to projectors in so(4,2) representations and relates quantum equations of motion to indecomposable representations, offering new insights into multiplet recombination.

Findings

The log term coefficient acts as a projector in tensor product representations.

The 1-loop 4-point integral decomposes into four terms related to external legs.

Quantum equations of motion connect to indecomposable representations of so(4,2).

Abstract

In a recent paper we showed that the correlators of free scalar field theory in four dimensions can be constructed from a two dimensional topological field theory based on so(4,2) equivariant maps (intertwiners). The free field result, along with results of Frenkel and Libine on equivariance properties of Feynman integrals, are developed further in this paper. We show that the coefficient of the log term in the 1-loop 4-point conformal integral is a projector in the tensor product of so(4,2) representations. We also show that the 1-loop 4-point integral can be written as a sum of four terms, each associated with the quantum equation of motion for one of the four external legs. The quantum equation of motion is shown to be related to equivariant maps involving indecomposable representations of so(4,2), a phenomenon which illuminates multiplet recombination. The harmonic expansion method for Feynman integrals is a powerful tool for arriving at these results. The generalization to other interactions and higher loops is discussed.