Resummed Higgs cross section at N$^3$LL

We present accurate predictions for the inclusive production of a Higgs boson in proton-proton collisions, via gluon-gluon fusion. Our calculation includes next-to-next-to-leading order (NNLO) corrections in perturbative QCD, as well as the resummation of threshold-enhanced contributions to next-to-next-to-next-to-leading logarithmic (N$^3$LL) accuracy, with the inclusion of the recently-determined three-loop constant coefficient (sometimes referred to as N$^3$LL' accuracy). Our result correctly accounts for finite top, bottom and charm masses at leading order (LO) and next-to-leading order (NLO), and includes top mass dependence at NNLO. At the resummed level the dependence on top, bottom and charm mass is accounted for at NLL, while only the top mass at NNLL. The all-order calculation is improved by a suitable choice of the soft terms, dictated by analyticity conditions and by the inclusion of subleading corrections of collinear origin, which improve the accuracy of the resummation away from the threshold region. We present results for different collider energies and we study perturbative uncertainties by varying renormalization and factorization scales. We find that, at current LHC energies, the resummation corrects the NNLO result by as much as 20 % at $\mu_R=\mu_F=m_H$, while the correction is much smaller, 5.5 %, at $\mu_R=\mu_F=m_H /\,2$. While the central value of NNLO+N$^3$LL result depends very mildly on the scale choice, we argue that a more reliable estimate of the theoretical uncertainty is found if the perturbative scales are canonically varied about $m_H$.