A semi-analytical x-space solution for parton evolution — Application to non-singlet and singlet DGLAP equation

We present a novel semi-analytical method for parton evolution. It is based on constructing a family of analytic functions spanning x-space which is closed under the considered evolution equation. Using these functions as a basis, the original integro-differential evolution equation transforms into a system of coupled ordinary differential equations, which can be solved numerically by restriction to a suitably chosen finite subsystem. The evolved distributions are obtained as analytic functions in x with numerically obtained coefficients, providing insight into the analytic behavior of the evolved parton distributions. As a proof-of-principle, we apply our method to the leading order non-singlet and singlet DGLAP equation. Comparing our results to traditional Mellin-space methods, we find good agreement. The method is implemented in the code POMPOM in Mathematica as well as in Python.