Exact isospin symmetry would make the proton and neutron degenerate. Their observed mass difference is generated by explicit isospin breaking from $m_d-m_u$ together with electromagnetic effects. The approximate symmetry explains why the splitting is small compared with the nucleon mass, while <chiral perturbation theory> parametrizes it through symmetry-breaking operators and low-energy constants; symmetry alone does not predict its numerical value.

Solution

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

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.

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

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Chiral perturbation theory is the low-energy effective field theory of pseudo-Goldstone bosons from chiral symmetry breaking. It expands amplitudes in derivatives and light-quark masses over the chiral scale.


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