Results and techniques of multi-loop calculations

In this review some recent multi-loop results obtained in the framework of perturbative quantum chromodynamics (QCD) and quantum electrodynamics (QED) are discussed. After reviewing the most advanced techniques used for the computation of renormalization group functions, we consider the decoupling of heavy quarks. In particular, an effective method for the evaluation of the decoupling constants is presented and explicit results are given. Furthermore, the connection to observables involving a scalar Higgs boson is worked out in detail. An all-order low-energy theorem is derived which establishes a relation between the coefficient functions in the hadronic Higgs decay and the decoupling constants. We review the radiative corrections of a Higgs boson into gluons and quarks and present explicit results up to order α_s⁴ and α_s³, respectively. In this review special emphasis is put on the applications of asymptotic expansions. A method is described which combines expansion terms of different kinematical regions with the help of conformal mapping and Padé approximation. This method allows us to proceed beyond the present scope of exact multi-loop calculations. As far as physical processes are concerned, we review the computation of three-loop current correlators in QCD taking into account the full mass dependence. In particular, we concentrate on the evaluation of the total cross section for the production of hadrons in e⁺e⁻ annihilation. The knowledge of the complete mass dependence at order α_s² has triggered a bunch of theory-driven analyses of the hadronic contribution to the electromagnetic coupling evaluated at high energy scales. The status is summarized in this review. In a further application four-loop diagrams are considered which contribute to the order α² QED corrections to the μ decay. Its relevance for the determination of the Fermi constant G_F is discussed. Finally, the calculation of the three-loop relation between the $\text{MS}$ and on-shell quark mass definitions is presented and physical applications are given. To complete the presentation, some technical details are presented in the appendix, where also explicit analytical results are listed.