Next-to soft virtual corrections to Inclusive cross-sections at the colliders
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We present a framework that resums threshold enhanced large logarithms to all orders in perturbation theory for the production of a color singlet final state in the hadronic collisions. We restrict ourselves to contributions from diagonal partonic channels. 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 (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 the strong coupling constant. The logarithmic structure of these results is governed by universal infrared anomalous dimensions and process-dependent functions of the 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^α log^(2n−α) (N ), a_s^n /N^α log^(2n−1−α) (N ) · · ·, etc for α = 0, 1 are summed to all orders in a_s. We apply this formalism to study the phenomenological aspects of the NSV resummation for the case of the Drell-Yan process as well as the Higgs productions process in gluon fusion.