## Methods for the renormalization of the top quark

##### Summary

The (automated) multiloop renormalization of the top quark mass and field in quantum chromodynamics is discussed, which involves calculating fermion propagator type Feynman diagrams. First a theoretical description is given of renormalization at multiloop order, after which the building blocks are discussed of an automated setup which calculates Feynman diagrams contributing to the quark propagator. Large sets of Feynman integrals are reduced to a small set of so-called master integrals using integration-by-parts identities.
The alpha-parametrization is introduced as a powerful way of turning Feynman integrals into scalar integrals. Mathematical techniques are studied to evaluate alpha-parametrized integrals either analytically or as Laurent series around d=4 in terms of epsilon (d=4-2*epsilon). To derive a Laurent series of a divergent integral one first has to do a finite integral expansion to rewrite the divergent integral in terms of finite integrals, which is possible for on-shell integrals. The poles of the divergent integral are captured in the coefficients in front of the finite integrals. Care has to be taken that the finite integral expansion does not contain 'spurious poles', which are poles that drop out due to non-trivial cancellations of the coefficients of the power series in epsilon of the finite integrals.
A new method called the 'projective trick' is introduced for deriving finite integral expansions, and rules of thumb are developed to derive finite integral expansions without spurious poles for the 2- and 3-loop on-shell integrals with one mass scale considered in this thesis. These finite integrals are then expanded as power series up to the desired order in epsilon, and the series coefficients are evaluated. It was possible for almost all integrals to evaluate the coefficients exactly using the Maple package HyperInt. Alternatively, the coefficients can be calculated with a numerical integrator.