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. These logarithms include the distributions $((1-z)^{-1} \log^i(1-z))_+$ resulting from soft plus virtual (SV) and the logarithms $\log^i(1-z)$ from next to SV contributions. We use collinear factorisation and renormalisation group invariance to achieve this. We find that the resummed result is a solution to Sudakov type differential equation and hence it can predict soft plus virtual contributions as well as next to SV contributions to all orders in strong coupling constant to the partonic coefficient function in terms of infrared anomalous dimensions and process independent functions. The $z$ space resummed result is shown to have integral representation which allows us to resum the large logarithms of the form $\log^i(N)$ retaining $1/N$ corrections resulting from next to SV terms. We show that in $N$ space, tower of logarithms $a_s^n/N^\alpha \log^{2n-\alpha} (N), a_s^n/N^\alpha \log^{2n-1-\alpha}(N) \cdots $ etc for $\alpha =0,1$ are summed to all orders in $a_s$.