# Manifestly Dual-Conformal Loop Integration

## TL;DR

This paper introduces a systematic method to regulate and evaluate dual-conformal loop integrals in planar supersymmetric Yang-Mills theory, preserving symmetry and enabling direct computation of finite quantities at two loops.

## Contribution

It presents a new regularization scheme for dual-conformal integrals that maintains symmetry and allows direct evaluation of finite loop integrals in planar theories.

## Key findings

- Successfully applied to two-loop ratio and remainder functions for six particles.
- Reproduced known results using the new regularization method.
- Identified novel features in regularization at two loops not seen at one loop.

## Abstract

Local, manifestly dual-conformally invariant loop integrands are now known for all finite quantities associated with observables in planar, maximally supersymmetric Yang-Mills theory through three loops. These representations, however, are not infrared-finite term by term and therefore require regularization; and even using a regulator consistent with dual-conformal invariance, ordinary methods of loop integration would naively obscure this symmetry. In this work, we show how any planar loop integral through at least two loops can be systematically regulated and evaluated directly in terms of strictly finite, manifestly dual-conformal Feynman-parameter integrals. We apply these methods to the case of the two-loop ratio and remainder functions for six particles, reproducing the known results in terms of individually regulated local loop integrals, and we comment on some of the novelties that arise for this regularization scheme not previously seen at one loop.