Expand the quintic interaction after the slow-fast split. Its term with three slow fields and two fast fields is
$$
\gamma_0\binom52\phi_-^3\phi_+^2.
$$
The first term of the <cumulant expansion> therefore contains
$$
10\gamma_0\int d^dx\,\phi_-^3(x)\langle\phi_+^2(x)\rangle_+.
$$
Since the coincident fast-mode propagator is
$$
\langle\phi_+^2(x)\rangle_+
=\int_{\Lambda/\zeta<|q|<\Lambda}\frac{d^dq}{(2\pi)^d}\frac1{q^2+\mu_0^2},
$$
the lowest-order correction is
$$
\boxed{\delta\alpha_0
=10\gamma_0\int_{\Lambda/\zeta<|q|<\Lambda}\frac{d^dq}{(2\pi)^d}\frac1{q^2+\mu_0^2}}.
$$
Its <Feynman diagram> is one five-valent vertex with three external slow-field legs and the remaining two legs contracted into a <tadpole diagram>.

Solution

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

 Solution

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