On next to soft corrections to Drell-Yan and Higgs Boson productions

We present a framework that resums threshold enhanced large logarithms to all orders in perturbation theory for the production of a pair of leptons in Drell-Yan process and of Higgs boson in gluon fusion as well as in bottom quark annihilation. We restrict ourselves to contributions from diagonal partonic channels. These logarithms include the distributions $((1-z)^{-1} \ln^i(1-z))_+$ resulting from soft plus virtual (SV) and the logarithms $\ln^i(1-z)$ from next-to-SV (NSV) contributions. We use collinear factorisation and renormalisation group invariance to achieve this. The former allows one to define a Soft-Collinear (SC) function which encapsulates soft and collinear dynamics of the perturbative results to all orders in strong coupling constant. The logarithmic structure of these results are governed by universal infrared anomalous dimensions and process dependent functions of Sudakov differential equation that the SC satisfies. The solution to the differential equation is obtained by proposing an all-order ansatz in dimensional regularisation, owing to several state-of-the-art perturbative results available to third order. The $z$ space solutions thus obtained provide an integral representation to sum up large logarithms originating from both soft and collinear configurations, conveniently in Mellin $N$ space. We show that in $N$ space, tower of logarithms $a_s^n/N^\alpha \ln^{2n-\alpha} (N), a_s^n/N^\alpha \ln^{2n-1-\alpha}(N) \cdots$ etc for $\alpha =0,1$ are summed to all orders in $a_s$.