Higher-Order Corrections in QCD Evolution Equations and Tools for Their Calculation

TL;DR

This thesis introduces a modified regularization method for NLO QCD calculations that improves Monte Carlo simulations of parton showers, and provides a new computational tool for these complex calculations.

Contribution

It proposes the NPV prescription for regularizing singularities in QCD evolution equations, enabling more effective Monte Carlo simulations and introduces the Axiloop package for NLO axial gauge calculations.

Findings

The NPV prescription simplifies the cancellation of divergences in NLO calculations.

The NLO non-singlet splitting function was successfully calculated using the new method.

The Axiloop package facilitates NLO calculations in the axial gauge.

Abstract

In this thesis we calculate the NLO one-loop virtual contributions to the QCD DGLAP splitting functions in a form suitable for Monte Carlo simulations. We use the standard technique based on the factorization properties of mass singularities in the light-cone axial gauge \cite{EGMPR79,CFP80} but we propose a modification to this approach by introducing a New Principal Value prescription \cite{GJKS14} in which we use the PV prescription for regularization of {\em all} singularities in the light-cone plus components of the four-momenta. The main advantage of the NPV prescription is that exclusive splitting functions calculated with its help can be used for construction of the Monte-Carlo parton showers. The reason for this is that in the NPV prescription some of the higher order poles in dimensional $\epsilon$ parameter are replaced by the logarithms of a cut-off parameter $\delta$ which has a geometrical meaning in four dimensions. As a consequence, cancellation of the higher order poles between real and virtual components is reduced. On the other hand, at the inclusive level the NPV results agree with the results in the standard PV prescription which shows compatibility of both approaches. With the help of the NPV prescription, we calculate virtual one-loop contributions to the NLO non-singlet splitting function $P_{q\to q}^{NLO}$ and selected contributions to the singlet $P_{g\to g}^{NLO}$ one. We also discuss the dependence of the results on the choice of the integration variable related to the evolution time in Monte Carlo parton showers. Finally, we present the {\tt Axiloop} package written in Wolfram Mathematica language, that is dedicated to perform NLO calculations in the axial gauge. Results presented in this thesis were obtained with the help of the {\tt Axiloop} package.