Renormalisation

TL;DR

This work surveys the renormalisation framework used in quantum field theory, beginning with the foundational treatment of UV divergences, regularisation, and the renormalisation prescription that yields finite, predictive Green functions via $Z$-factors and a subtraction scale $\mu$. It then develops the renormalisation group in QCD, detailing the beta-function $\beta(\alpha_s)$, the running coupling $\alpha_s(\mu)$, and the mass anomalous dimension $\gamma_m$, including high-order results and the concept of the conformal window. The text discusses scheme dependence, the role of the $\Lambda$ parameter, and how extended renormalisation group transformations relate different schemes, with emphasis on matching across quark thresholds and scheme transitions. Finally, it addresses the practical renormalisation scale setting problem in perturbative QCD, reviewing optimization strategies—PMS, FAC, and PMC—and highlighting their potential to reduce residual scale-scheme ambiguities, thereby enhancing precision for SM tests and future collider phenomenology. Throughout, the treatment uses $D=4-2\varepsilon$ dimensional regularisation and standard schemes (e.g., $\overline{\text{MS}}$), ensuring RG invariance and consistent scale evolution across processes.

Abstract

We give an introduction to renormalisation, focusing first on a pedagogical description of fundamental concepts of the procedure and its features, then we introduce the renormalisation group and its equations. We discuss then the case of gauge theories such as QCD summarising the current state of the art. We introduce the renormalisation scale setting problem in QCD and we give an illustration of the possible optimisation procedures currently in use.

Figures (10)

• Figure 1: a)1PI-diagram contributions to the Feynman propagator; b) summed geometric series with A the propagator at lowest order and B the1PI insertion; c) exact scalar propagator in momentum space.
• Figure 2: Structure of the 4-points connected Green’s function $G^{(4)}$, decomposed into exact propagators and amputated $\Gamma^{(4)}$ Green’s function.
• Figure 3: Feynman diagrams for the counterterms in the renormalized QED lagrangian for: 1. the photon propagator, 2. the fermion propagator and 3. the vertex, respectively.
• Figure 4: Feynman diagrams for the 1PI Green’s functions: 1. the photon propagator, 2. the fermion propagator and 3. the vertex, respectively.
• Figure 5: Feynman diagram of two fermion fields coupled to a vector current.