Relations of loop partial amplitudes in gauge theory by Unitarity cut method

TL;DR

This paper explores relations among gauge theory loop partial amplitudes using unitarity cut methods, revealing new linear relations at tree and multi-loop levels, and highlighting differences between one-loop and higher-loop structures.

Contribution

It demonstrates that double trace partial amplitudes at one-loop are linear combinations of single trace amplitudes, and uncovers new Kleiss-Kuijf relations for subleading-color amplitudes at six and seven points.

Findings

Double trace amplitudes are linear combinations of single trace at one-loop.

New Kleiss-Kuijf relations for six and seven-point subleading amplitudes.

Modified relations for eight-point subleading amplitudes.

Abstract

It is well known that under the color-decomposition, one-loop amplitude of gluons contains partial amplitudes of single and double trace structures, and particularly all partial amplitudes of double trace structure can be expressed as a linear combination of partial amplitudes of single trace structure. Using unitarity cut method, we prove that this result is the natural consequence of tree-level Kleiss-Kuijf relation. Generalizing the unitarity cut method to two-loop (triple cut in this case), we show that, unlike the one-loop case, partial amplitudes of double and triple trace structures can not be expressed as a linear combination of partial amplitudes of leading-color single trace structure. For partial amplitudes of subleading-color single trace structure, we have shown a very nontrivial Kleiss-Kuijf relation for six and seven-point amplitudes, which is one new result of our paper and can not be obtained by U(1)-decoupling method. Mysteriously, when we consider the case of eight points, Kleiss-Kuijf relation must be modified for subleading-color single trace partial amplitudes.