Two-loop helicity amplitudes for $gg\to W^{+}{W}^{-}$ using high-energy and forward expansions

We calculate next-to-leading order QCD corrections for the process $gg\to W^{+}{W}^{-}$ with two off-shell $W$ bosons, accounting for virtual top quarks in the loop diagrams. Our approach relies on combining expansions valid in different regions of phase space. Specifically, we perform a high-energy expansion in which the Mandelstam variables are much larger than the top-quark mass. Furthermore, we expand around the forward-scattering limit, where the Mandelstam variable $t$ is small. We also provide an expansion for large top-quark mass, which serves as an important cross-check. We provide technical details regarding the various expansions and demonstrate that precise results are obtained for all helicity amplitudes across the entire phase space. Our results exhibit an explicit dependence on all input parameters.