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

We perform the all orders resummation of threshold enhanced contributions for the Higgs boson pair production cross section via gluon fusion, including finite top quark mass ($M_t$) effects. We present results for the total cross section and Higgs pair invariant mass ($M_{hh}$) distribution. We obtain results 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 with full $M_t$ dependence for this process, and produce an increase of about 4% in the total cross section with respect to the NLO result for LHC energies, and for a central scale $\mu_0 = M_{hh}/2$. We also consistently combine the full NLL with the next-to-next-to-leading logarithmically (NNLL) accurate resummation computed in the Born-improved large-$M_t$ limit, and match it to the next-to-next-to-leading order approximation of Ref. \cite{Grazzini:2018bsd}, so called NNLO_FTa. We find that the resummation effects 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 all cases the resummation effects are found to be substantially larger for the central scale $\mu_0 = M_{hh}$, resulting in a more stable cross section with respect to scale variations than the fixed order calculation.