Four-dimensional unsubtraction from the loop-tree duality

TL;DR

This paper introduces a four-dimensional algorithm based on loop-tree duality for calculating NLO corrections to cross-sections, enabling local infrared divergence cancellation without subtraction terms.

Contribution

It presents a novel four-dimensional method for NLO calculations using LTD, simplifying divergence handling and extending to multi-leg processes.

Findings

Local cancellation of infrared divergences achieved at integrand level

Algorithm successfully applied to scalar three-point function and gamma* to quark-antiquark process

Potential extension to NNLO briefly discussed

Abstract

We present a new algorithm to construct a purely four dimensional representation of higher-order perturbative corrections to physical cross-sections at next-to-leading order (NLO). The algorithm is based on the loop-tree duality (LTD), and it is implemented by introducing a suitable mapping between the external and loop momenta of the virtual scattering amplitudes, and the external momenta of the real emission corrections. In this way, the sum over degenerate infrared states is performed at integrand level and the cancellation of infrared divergences occurs locally without introducing subtraction counter-terms to deal with soft and final-state collinear singularities. The dual representation of ultraviolet counter-terms is also discussed in detail, in particular for self-energy contributions. The method is first illustrated with the scalar three-point function, before proceeding with the calculation of the physical cross-section for $\gamma^* \to q \bar{q}(g)$, and its generalisation to multi-leg processes. The extension to next-to-next-to-leading order (NNLO) is briefly commented.