Bootstrapping one-loop QCD amplitudes with general helicities

Abstract

The recently developed on-shell bootstrap for computing one-loop amplitudes in nonsupersymmetric theories such as QCD combines the unitarity method with loop-level on-shell recursion. For generic helicity configurations, the recursion relations may involve undetermined contributions from nonstandard complex singularities or from large values of the shift parameter. Here we develop a strategy for sidestepping difficulties through the use of pairs of recursion relations. To illustrate the strategy, we present sets of recursion relations needed for obtaining $n$-gluon amplitudes in QCD. We give a recursive solution for the one-loop $n$-gluon QCD amplitudes with three or four color-adjacent gluons of negative helicity and the remaining ones of positive helicity. We provide an explicit analytic formula for the QCD amplitude ${A}_{6;1}({1}^{\ensuremath{-}},{2}^{\ensuremath{-}},{3}^{\ensuremath{-}},{4}^{+},{5}^{+},{6}^{+})$, as well as numerical results for ${A}_{7;1}({1}^{\ensuremath{-}},{2}^{\ensuremath{-}},{3}^{\ensuremath{-}},{4}^{+},{5}^{+},{6}^{+},{7}^{+})$, ${A}_{8;1}({1}^{\ensuremath{-}},{2}^{\ensuremath{-}},{3}^{\ensuremath{-}},{4}^{+},{5}^{+},{6}^{+},{7}^{+},{8}^{+})$, and ${A}_{8;1}({1}^{\ensuremath{-}},{2}^{\ensuremath{-}},{3}^{\ensuremath{-}},{4}^{\ensuremath{-}},{5}^{+},{6}^{+},{7}^{+},{8}^{+})$. We expect the on-shell bootstrap approach to have widespread applications to phenomenological studies at colliders.