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

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- Solution e

Solution

Expand the quintic interaction after the slow-fast split. Its term with three slow fields and two fast fields is

γ_{0} (2 5) ϕ_{-}^{3} ϕ_{+}^{2} .

The first term of the cumulant expansion therefore contains

10 γ_{0} \int d^{d} x ϕ_{-}^{3} (x) ⟨ ϕ_{+}^{2} (x) ⟩_{+} .

Since the coincident fast-mode propagator is

⟨ ϕ_{+}^{2} (x) ⟩_{+} = \int_{Λ/ ζ < ∣ q ∣ < Λ} \frac{d ^{d} q}{( 2 π ) ^{d}} \frac{1}{q ^{2} + μ _{0}^{2}},

the lowest-order correction is

δ α_{0} = 10 γ_{0} \int_{Λ/ ζ < ∣ q ∣ < Λ} \frac{d ^{d} q}{( 2 π ) ^{d}} \frac{1}{q ^{2} + μ _{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.

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