Define the shell integrals
$$
I_1(\zeta)=\int_{\Lambda/\zeta<|q|<\Lambda}
\frac{d^dq}{(2\pi)^d}\frac1{q^2+\mu_0^2},
$$
$$
I_2(\zeta)=\int_{\Lambda/\zeta<|q|<\Lambda}
\frac{d^dq}{(2\pi)^d}\frac1{(q^2+\mu_0^2)^2}.
$$
The index multiplicities described in part a and the canonical rescaling give
$$
\boxed{
\mu^2(\zeta)=\zeta^2
\left[\mu_0^2+4(N+2)g_0I_1(\zeta)\right],}
$$
$$
\boxed{
g(\zeta)=\zeta^{4-d}
\left[g_0-4(N+8)g_0^2I_2(\zeta)\right].}
$$
For $N=1$, these reduce to the stated Ising coefficients $12$ and $36$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 303 / 3 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-303/3/b/solution