We compute next-to-next-to-leading order QCD corrections to the gluon-induced production cross section of Higgs boson pairs in the large top quark mass limit using the soft-virtual approximation. In the limit of infinitely-heavy top quark we confirm the results in the literature. We add two more expansion terms in the inverse top quark mass to the Mt→∞ result. Since the 1/Mt expansion converges poorly, we try to improve on it by factorizing the exact leading order cross section. We discuss two ways of doing that and conclude that the finite top quark mass effects shift the cross section at most by about 10% at next-to-leading order and by about 5% at next-to-next-to-leading order.
Grigo, J., Hoff, J., & Steinhauser, M. (2015). Higgs boson pair production: Top quark mass effects at NLO and NNLO. Nuclear Physics B, 900, 412–430. https://doi.org/10.1016/j.nuclphysb.2015.09.012