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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-305/2/ii/solution