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

For $q=(u,d)^T$, massless two-flavour <Quantum chromodynamics> has
$$
\mathcal L=-\frac12\operatorname{tr}(G_{\mu\nu}G^{\mu\nu})
+i\bar q\not Dq.
$$
Classically its continuous global symmetry is $SU(2)_L\times SU(2)_R\times U(1)_V\times U(1)_A$, up to finite central quotients. The <Adler-Bell-Jackiw anomaly> destroys the continuous axial $U(1)_A$ in the quantum theory, leaving $SU(2)_L\times SU(2)_R\times U(1)_V$, together with an anomaly-preserved discrete axial subgroup.

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 305 / 2 / i / Solution

{c}
{title2=QCD}
{wiki}

Quantum chromodynamics is the $SU(3)_c$ gauge theory of quarks and gluons. At low energies it confines coloured fields into gauge-singlet hadrons.


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 Incoming links: Quantum chromodynamics