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We calculate the two-loop QCD corrections to $gg \to ZZ$ involving a closedtop-quark loop. We present a new method to systematically construct linearcombinations of Feynman integrals with a convergent parametric representation,where we also allow for irreducible numerators, higher powers of propagators,dimensionally shifted integrals, and subsector integrals. The amplitude isexpressed in terms of such finite integrals by employing syzygies derived withlinear algebra and finite field techniques. Evaluating the amplitude usingnumerical integration, we find agreement with previous expansions in asymptoticlimits and provide ab initio results also for intermediate partonic energiesand non-central scattering at higher energies.

Two-loop helicity amplitudes for $gg \to ZZ$ with full top-quark mass effects