# Universal four-dimensional representation of $H \to \gamma \gamma$ at two loops through the Loop-Tree Duality

## TL;DR

This paper extends the Loop-Tree Duality formalism to two-loop level for the $H\to\gamma\gamma$ process, demonstrating universality, providing a local renormalisation algorithm, and enabling fully four-dimensional numerical computations.

## Contribution

It introduces a two-loop extension of the Loop-Tree Duality method with a local renormalisation algorithm, maintaining universality and facilitating 4D numerical calculations.

## Key findings

- Perfect numerical agreement with existing analytic results.
- Successful local renormalisation at the integrand level.
- Foundation for a 4D framework for higher-order computations.

## Abstract

We extend useful properties of the $H\to\gamma\gamma$ unintegrated dual amplitudes from one- to two-loop level, using the Loop-Tree Duality formalism. In particular, we show that the universality of the functional form -- regardless of the nature of the internal particle -- still holds at this order. We also present an algorithmic way to renormalise two-loop amplitudes, by locally cancelling the ultraviolet singularities at integrand level, thus allowing a full four-dimensional numerical implementation of the method. Our results are compared with analytic expressions already available in the literature, finding a perfect numerical agreement. The success of this computation plays a crucial role for the development of a fully local four-dimensional framework to compute physical observables at Next-to-Next-to Leading order and beyond.