For opposite contour branches the two vertex integrals factorize. Since
$$
\int_{-\infty(1-i\epsilon)}^0\eta^2e^{iE\eta}\,d\eta=\frac{2i}{E^3},
$$
their product contributes $4/(E_L^3E_R^3)$. The two assignments $LR$ and $RL$ then give
$$
\boxed{B_{4,s}^{(lr+rl)}
=\frac{\lambda^2H^8s}{4k_1k_2k_3k_4}
\frac1{(E_RE_L)^3}}.
$$
Consequently
$$
\boxed{C_4=\frac{\lambda^2H^8}{4},
\qquad \alpha_4=0,
\qquad \beta_4=3}.
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 312 / 1 / iv / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-312/1/iv/solution