All one-loop maximally helicity violating gluonic amplitudes in QCD

Abstract

We use on-shell recursion relations to compute analytically the one-loop corrections to maximally helicity-violating $n$-gluon amplitudes in QCD. The cut-containing parts have been computed previously; our work supplies the remaining rational parts for these amplitudes, which contain two gluons of negative helicity and the rest positive, in an arbitrary color ordering. We also present formulae specific to the six-gluon cases with helicities ($\ensuremath{-}+\ensuremath{-}+++$) and ($\ensuremath{-}++\ensuremath{-}++$), as well as numerical results for six, seven, and eight gluons. Our construction of the $n$-gluon amplitudes illustrates the relatively modest growth in complexity of the on-shell-recursive calculation as the number of external legs increases. These amplitudes add to the growing body of one-loop amplitudes known for all $n$, which are useful for studies of general properties of amplitudes, including their twistor-space structure.