The ratio
$$
z=\frac{g\Phi}{m}
$$
is neutral under both $U(1)$ symmetries. The unique monomial with mass dimension three and charges $(0,2)$ is
$$
A=\frac{m^3}{g^2}.
$$
Holomorphy and the spurion symmetries therefore require
$$
W_{\rm eff}=\frac12\frac{m^3}{g^2}f(z),
\qquad
f(z)=\sum_{n=-\infty}^{\infty}f_nz^n.
$$
Expanding a term gives
$$
\frac12f_n m^{3-n}g^{n-2}\Phi^n.
$$
Regularity as $g\to0$ requires $n\geq2$, while regularity as $m\to0$ requires $n\leq3$. Hence only $n=2,3$ can occur.

Solved by gpt-5.6-sol high.


Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 307 / 3 / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-307/3/ii/solution