Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 304 / 3 / b

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- Solution b

Solution

The disjoint Fourier supports make the quadratic cross term

\int \partial ϕ \cdot \partial χ

vanish. Therefore

S [ϕ + χ] = S [ϕ] + S_{0, χ} [χ] + S_{int} [ϕ, χ],

where

S_{int} [ϕ, χ] = \frac{g}{3 !} \int d^{d} x (3 ϕ^{2} χ + 3 ϕ χ^{2} + χ^{3}) .

Define the high-mode free generating functional with source convention

Z_{χ} [J] = \int D χ e^{- S_{0, χ} [χ] - \int J χ} .

Then inserting

χ = - δ / δ J

reproduces every high-field factor, and

e^{- W [ϕ]} = e^{- S [ϕ]} e^{- S_{int} [ϕ, - δ / δ J]} Z_{χ} [J]_{J = 0} .

Expanding the interaction exponential and applying Wick theorem evaluates the Wilsonian effective action as a sum of connected diagrams whose internal lines are restricted to the high-momentum shell.

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