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Dimensional transmutation replaces a dimensionless coupling by a dimensionful renormalization-group invariant scale. In an asymptotically free theory, specifying the coupling at one scale is equivalent to specifying its <strong-coupling scale>.

Dimensional transmutation

The <strong-coupling scale> $\Lambda$ is defined as the scale at which the one-loop inverse coupling vanishes. Setting $g^{-2}(\Lambda)=0$ gives
$$
\boxed{\Lambda=\Lambda_{UV}
\exp\!\left[-\frac{3(4\pi)^2}{2b_0g_0^2}\right]}.
$$
Equivalently, in terms of the coupling measured at any scale $\mu$,
$$
\boxed{\Lambda=\mu
\exp\!\left[-\frac{3(4\pi)^2}{2b_0g^2(\mu)}\right]}.
$$
This <dimensional transmutation> defines $\Lambda_{\rm QCD}$ for <Quantum chromodynamics>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 305 / 2 / ii / Solution

{title2=$\Lambda$}

The strong-coupling scale of an asymptotically free gauge theory is the renormalization-group invariant scale at which the one-loop running coupling formally diverges. It replaces a dimensionless ultraviolet coupling through <dimensional transmutation>.

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 Incoming links: Strong-coupling scale