Dimensional transmutation

In particle physics, dimensional transmutation is a physical mechanism that transforms a dimensionless parameter into a dimensionful parameter.

In classical field theory such as gauge theory in four-dimensional spacetime, the coupling constant is a dimensionless constant. However, logarithmic divergences in one-loop diagrams imply that this "constant" actually depends on the typical energy scale of the processes under considerations, called the renormalization group (RG) scale. The "running" is determined by the beta-function and renormalization group.

Consequently, the interaction may be characterised by a dimensionful parameter Λ, namely the value of the RG scale at which the coupling constant reaches the value 1. In the case of quantum chromodynamics, this energy scale Λ is called the QCD scale and its value 220 MeV "replaces" the original dimensionless coupling constant. Note that perturbation theory is only valid for a (dimensionless) coupling constant g << 1. In the case of QCD, the energy scale Λ is an infra-red cutoff such that μ >> Λ implies g << 1, with μ the RG scale. On the other hand, in the case of theories such as QED, Λ is an ultra-violet cutoff such that μ << Λ implies g << 1.

This is also a way of saying that the conformal symmetry of the classical theory is anomalously broken, thereby setting up a mass scale. See conformal anomaly.