Toward multiloop local renormalization within causal loop-tree duality

Renormalization is a well-known technique to get rid of ultraviolet (UV) singularities. When relying on dimensional regularization, these become manifest as <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>ε</a:mi></a:mrow></a:math> poles, allowing one to define counterterms with useful recursive properties. However, this procedure requires one to work at “integral level” and poses difficulties to achieve a smooth combination with seminumerical 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, we keep the causal representations of the integrands of multiloop Feynman diagrams and explore their UV behavior. Then, we propose a strategy that allows one to build local counterterms, 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 toward a fully local renormalization framework in quantum field theory.