Rational QCD loop amplitudes and quantum theories on twistor space

TL;DR

This paper reveals linear relations among one-loop gluon amplitudes in QCD derived from twistor space anomalies, providing a new understanding of amplitude dependencies and their combinatorial structure.

Contribution

It demonstrates how anomalies in twistor uplift lead to specific linear relations among gluon amplitudes and determines their independence using combinatorial mathematics.

Findings

Linear relations among one-loop gluon amplitudes are established.

Number of independent amplitudes is expressed via Stirling numbers.

No additional linear dependencies exist beyond those identified.

Abstract

We show how curing an anomaly of the twistor uplift of self-dual Yang-Mills theory implies linear relations among one-loop, $n$-gluon, color-ordered subamplitudes in QCD, when all $n$ gluon helicities are positive, or when exactly one is negative. We compute the number of linearly independent subamplitudes as determined by these relations, in terms of unsigned Stirling numbers. Then we use a momentum-twistor parametrization to show that there are no further linear dependencies.