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We perform the all orders resummation of threshold enhanced contributions forthe Higgs boson pair production cross section via gluon fusion, includingfinite top quark mass ($M_t$) effects. We present results for the total crosssection and Higgs pair invariant mass ($M_{hh}$) distribution. We obtainresults at next-to-leading logarithmic accuracy (NLL) which retain the full$M_t$ dependence, and are matched to the full next-to-leading order (NLO)prediction. Our NLL+NLO results represent the most advanced prediction withfull $M_t$ dependence for this process, and produce an increase of about 4% inthe total cross section with respect to the NLO result for LHC energies, andfor a central scale $\mu_0 = M_{hh}/2$. We also consistently combine the fullNLL with the next-to-next-to-leading logarithmically (NNLL) accurateresummation computed in the Born-improved large-$M_t$ limit, and match it tothe next-to-next-to-leading order approximation of Ref.\cite{Grazzini:2018bsd}, so called NNLO_FTa. We find that the resummationeffects are very small at NNLL for $\mu_0 = M_{hh}/2$, in particular below 1%at 13 TeV, indicating that the perturbative expansion is under control. In allcases the resummation effects are found to be substantially larger for thecentral scale $\mu_0 = M_{hh}$, resulting in a more stable cross section withrespect to scale variations than the fixed order calculation.

Soft gluon resummation for Higgs boson pair production including finite Mt effects