At energies well below $\Lambda_{\mathrm{QCD}}$, collect the pion fields into
$$
U(x)=\exp\left(\frac{i\pi^a(x)\tau^a}{f_\pi}\right),
\qquad
U\mapsto LUR^\dagger.
$$
The leading <chiral perturbation theory> Lagrangian is
$$
\mathcal L_\chi
=\frac{f_\pi^2}{4}\operatorname{Tr}
(\partial_\mu U^\dagger\partial^\mu U)
+\frac{f_\pi^2B_0}{2}
\operatorname{Tr}(MU^\dagger+UM^\dagger)
+\cdots,
$$
where $M=\operatorname{diag}(m_u,m_d)$ is treated as a symmetry-breaking spurion. The derivative and quark-mass expansion organizes the omitted terms.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 305 / 4 / c / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-305/4/c/ii/solution