NNLO logarithmic expansions and exact solutions of the DGLAP equations from x-space: New algorithms for precision studies at the LHC

A NNLO analysis of certain logarithmic expansions, developed for precision studies of the evolution of the QCD parton distributions (pdf) at the large hadron collider, is presented. We elaborate on their relations to all the solutions of the DGLAP equations that have been hitherto obtained from Mellin space, to which are equivalent. Exact expansions, equivalent to exact solutions of the equations, are constructed in the non-singlet sector. The algorithmic features of our approach are also emphasized, since this method allows to obtain numerical solutions of the evolution equations with the same accuracy of other methods, based on Mellin space, and of brute force methods, which solve the equations by finite differences. The implementation of our analysis allows to compare with existing benchmarks for the evolution of the pdf's, useful for applications at the LHC, and to extend them significantly in a systematic fashion, especially when solutions that retain logarithmic corrections only of a certain accuracy are searched for.