Holonomies of gauge fields in twistor space 4: functional MHV rules and one-loop amplitudes

TL;DR

This paper extends the CSW rules to one-loop amplitudes in N=4 super Yang-Mills theory using a holonomy formalism in twistor space, providing new analytic expressions and a novel regularization scheme.

Contribution

It introduces a quantum-level S-matrix functional implementing CSW rules, reproduces known one-loop MHV amplitudes, and proposes a new regularization for higher amplitudes.

Findings

Successfully reproduces BST one-loop MHV amplitudes

Derives new analytic expressions for one-loop NMHV and N$^2$MHV amplitudes

Proposes a novel regularization scheme using polylogarithm iterated integrals

Abstract

We consider generalization of the Cachazo-Svrcek-Witten (CSW) rules to one-loop amplitudes of N=4 super Yang-Mills theory in a recently developed holonomy formalism in twistor space. We first reconsider off-shell continuation of the Lorentz-invariant Nair measure for the incorporation of loop integrals. We then formulate an S-matrix functional for general amplitudes such that it implements the CSW rules at quantum level. For one-loop MHV amplitudes, the S-matrix functional correctly reproduces the analytic expressions obtained in the Brandhuber-Spence-Travaglini (BST) method. Motivated by this result, we propose a novel regularization scheme by use of an iterated-integral representation of polylogarithms and obtain a set of new analytic expressions for one-loop NMHV and N$^2$MHV amplitudes in a conjectural form. We also briefly sketch how the extension to one-loop non-MHV amplitudes in general can be carried out.