Vector boson pair production at one loop: analytic results for the process qqℓℓℓ^'ℓ^'g

We present compact analytic results for the one-loop amplitude for the process 0 → qqℓℓℓ^'ℓ^'g , relevant for both the production of a pair of Z and W -bosons in association with a jet. We focus on the gauge-invariant contribution mediated by a loop of quarks. We explicitly include all effects of the loop-quark mass m , appropriate for the production of a pair of Z -bosons. In the limit m → 0, our results are also applicable to the production of W -boson pairs, mediated by a loop of massless quarks. Implemented in a numerical code, the results are fast. The calculation uses novel advancements in spinor-helicity simplification techniques, for the first time applied beyond five-point massless kinematics. We make use of primary decompositions from algebraic-geometry, which now involve non-radical ideals, and p -adic numbers from number theory. We show how to infer whether numerator polynomials belong to symbolic powers of non-radical ideals through numerical evaluations.