Bootstrapping one-loop QCD amplitudes with general helicities

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 A6;1(1^-,2^-,3^-,4⁺,5⁺,6⁺), as well as numerical results for A7;1(1^-,2^-,3^-,4⁺,5⁺,6⁺,7⁺), A8;1(1^-,2^-,3^-,4⁺,5⁺,6⁺,7⁺,8⁺), and A8;1(1^-,2^-,3^-,4^-,5⁺,6⁺,7⁺,8⁺). We expect the on-shell bootstrap approach to have widespread applications to phenomenological studies at colliders.