Towards multiloop local renormalization within Causal Loop-Tree Duality

TL;DR

This paper develops an integrand-level renormalization method based on Loop-Tree Duality, enabling local counter-terms for multiloop Feynman diagrams, facilitating numerical calculations in quantum field theory.

Contribution

It introduces a novel integrand-level renormalization formalism using causal Loop-Tree Duality, suitable for semi-numerical and numerical approaches in quantum field theory.

Findings

Successfully tested on diagrams up to three-loops

Achieved smooth cancellation of divergences

Enabled local counter-terms in four dimensions

Abstract

Renormalization is a well-known technique to get rid of ultraviolet (UV) singularities. When relying on Dimensional Regularization (DREG), these become manifest as $\epsilon$-poles, allowing to define counter-terms with useful recursive properties. However, this procedure requires to work at \emph{integral-level} and poses difficulties to achieve a smooth combination with semi-numerical approaches. This article is devoted to the development of an integrand-level renormalization formalism, better suited for semi or fully numerical calculations. Starting from the Loop-Tree Duality (LTD), we keep the causal representations of the integrands of multiloop Feynman diagrams and explore their UV behaviour. Then, we propose a strategy that allows to build local counter-terms, capable of rendering the expressions integrable in the high-energy limit and in four space-time dimensions. Our procedure was tested on diagrams up to three-loops, and we found a remarkably smooth cancellation of divergences. The results of this work constitute a powerful step towards a fully local renormalization framework in QFT.