Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 344 / 1 / c

Table of contents
- Solution c

Solution

The terms involving

p

can be written by completing the square:

\frac{ν}{2} ∣ p ∣^{2} + λ p \cdot \nabla ϕ = \frac{ν}{2} p + \frac{λ}{ν} \nabla ϕ^{2} - \frac{λ ^{2}}{2 ν} ∣\nabla ϕ ∣^{2} .

Translation of the integration variable in the Gaussian functional integral over

p

contributes only a

ϕ

-independent determinant. The effective free energy is therefore

F [ϕ] = \int [\frac{a}{2} ϕ^{2} + \frac{b}{4} ϕ^{4} + \frac{κ}{2} ∣\nabla ϕ ∣^{2} + \frac{γ}{2} (\nabla^{2} ϕ)^{2}] d r,

where

κ = κ - \frac{λ ^{2}}{ν} .

This is the surfactant renormalization of the square-gradient coefficient.

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