In four dimensions the action is dimensionless and $[\partial_\mu]=1$. The kinetic terms give
$$
2[\Phi]+2=4,
\qquad
2[\Psi]+1=4,
$$
so $[\Phi]=1$ and $[\Psi]=3/2$.

Up to a constant and total derivatives, the most general renormalizable Lagrangian invariant under
$$
\Phi\mapsto e^{2i\alpha}\Phi,
\qquad
\Psi\mapsto e^{i\alpha}\Psi
$$
is
$$
\begin{aligned}
\mathcal L={}&
\partial_\mu\Phi^*\partial^\mu\Phi
-m_\Phi^2|\Phi|^2-\lambda|\Phi|^4
+\overline\Psi(i\not\partial-m_\Psi)\Psi\\
&-\frac12\left(y\Phi^*\Psi^{\mathsf T}C\Psi
+y^*\Phi\,\overline\Psi C\overline\Psi^{\mathsf T}\right).
\end{aligned}
$$
Here $\lambda>0$ is required for stability at large field. The last line is the allowed charge-conserving scalar-fermion Yukawa interaction; its precise two-component expression depends on the chosen spinor notation.


Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 305 / 3 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-305/3/a/solution