Define the two shell <integrals>
$$
I_a(\zeta)=\int_{\Lambda/\zeta}^{\Lambda}
\frac{d^dq}{(2\pi)^d}\frac1{q^2+\mu_{0,a}^2}.
$$
For an external $\phi_1$, the $\phi_1^4$ part of the interaction gives the scalar tadpole coefficient $12g_0I_1$, while $2g_0\phi_1^2\phi_2^2$ gives $4g_0I_2$. Interchanging the components gives
$$
\boxed{
\mu_1^2(\zeta)=\zeta^2
\left[\mu_{0,1}^2+4g_0(3I_1+I_2)\right]},
$$
$$
\boxed{
\mu_2^2(\zeta)=\zeta^2
\left[\mu_{0,2}^2+4g_0(I_1+3I_2)\right]}.
$$
When the masses agree, $I_1=I_2$ and both formulas reduce to $4(N+2)g_0I_1$ with $N=2$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 303 / 3 / b / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-303/3/b/ii/solution