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We compute the two-loop massless QCD corrections to the helicity amplitudesfor the production of two massive vector bosons in quark-antiquarkannihilation, allowing for an arbitrary virtuality of the vector bosons: $q\bar q' \to V_1V_2$. Combining with the leptonic decay currents, we obtain thefull two-loop QCD description of the corresponding electroweak four-leptonproduction processes. The calculation is performed by projecting the two-loopdiagrams onto an appropriate basis of Lorentz structures. All two-loop Feynmanintegrals are reduced to a basis of master integrals, which are then computedusing the differential equations method and optimised for numericalperformance. We provide a public C++ code which allows for fast and precisenumerical evaluations of the amplitudes.

The two-loop helicity amplitudes for $q \bar q' \to V_1 V_2 \to 4~\mathrm{leptons}$