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We present the two-loop leading colour QCD helicity amplitudes for theprocess $pp\to W(\to l\nu)\gamma+j$. We implement a complete reduction of theamplitudes, including the leptonic decay of the $W$-boson, using finite fieldarithmetic, and extract the analytic finite remainders using a recentlyidentified basis of special functions. Simplified analytic expressions areobtained after considering permutations of a rational kinematic parametrisationand multivariate partial fractioning. We demonstrate efficient numericalevaluation of the two-loop colour and helicity summed finite remainders forphysical kinematics, and hence the suitability for phenomenologicalapplications.

Two-loop leading colour helicity amplitudes for $W^\pm\gamma+j$ production at the LHC