Requiring $\int d^dx\,(\nabla\phi)^2$ to be dimensionless gives the <engineering dimensions>
$$
[\phi]=\frac{d-2}{2},
\qquad [\mu_0]=1,
$$
and
$$
[\lambda_0]=d-3[\phi]=\frac{6-d}{2},
$$
$$
[g_0]=d-4[\phi]=4-d,
$$
$$
[\gamma_0]=d-6[\phi]=6-2d.
$$
Thus $\mu_0$ is relevant. The cubic coupling is relevant, marginal, or irrelevant for $d<6$, $d=6$, or $d>6$; the quartic coupling has the same classification around $d=4$; and the sextic coupling has it around $d=3$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 303 / 2 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-303/2/b/solution