The free energy is dimensionless, while $d^dx$ has engineering dimension $-d$ and each derivative has dimension one. Requiring
$$
\int d^dx\,(\nabla\phi)^2
$$
to be dimensionless gives $2+2[\phi]-d=0$. Thus the <engineering dimension> of the field is
$$
\boxed{[\phi]_{\rm eng}=\frac{d-2}{2}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 303 / 2 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-303/2/b/solution