Results and Techniques of Multi-Loop Calculations

TL;DR

This review surveys cutting-edge multi-loop calculations in perturbative QCD and QED, emphasizing four-loop renormalization group functions, heavy-quark decoupling, and the use of dimension-four scalar operators in Higgs physics. A central theme is the development and application of asymptotic expansions and Padé approximants, often in conjunction with conformal mapping, to extract high-order results when exact multi-scale integrals are intractable. Key contributions include explicit four-loop β and γ_m results, MS-bar to on-shell mass and coupling relations, and the construction of effective Lagrangians with universal decoupling constants that enable precise predictions for observables like R(s), α(M_Z), and Higgs decay widths. The methods and results enable high-precision tests of the Standard Model, particularly in precision electroweak and Higgs phenomenology, and provide practical tools for running couplings and masses across heavy-quark thresholds.

Abstract

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. We review the radiative corrections of a Higgs boson into gluons and quarks and present explicit results up to order $α_s^4$ and $α_s^3$, 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 a further application four-loop diagrams are considered which contribute to the order $α^2$ QED corrections to the $μ$ decay. Finally the calculation of the three-loop relation between the $\bar{\rm MS}$ and on-shell quark mass definitions is presented. To complete the presentation, some technical details are presented in the Appendix, where also explicit analytical results are listed.

Figures (34)

• Figure 2.1: Scalar two-loop diagram.
• Figure 2.2: Sample diagrams contributing to the fermion propagator.
• Figure 2.3: (a) Graphical representation of Eq. (\ref{['eq:Zem_a']}). In the case of the fermion propagator the analogous diagram looks like the one in (b). In both cases an artificial mass is introduced in the propagator of the fermion line attached to the left vertex.
• Figure 2.4: One- and two-loop diagrams contributing the vector current correlator.
• Figure 3.1: Feynman diagram contribution to the process $e^+e^-\to b\bar{b}$ which gives rise to top-quark-dependent terms.