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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/iii/paper-49/2/ii/solution

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Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 49 / 2 / ii / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

The Derrick scaling of the Abelian Higgs energy must rescale the gauge potential as well as the Higgs field. Set

Φ_{λ} (x) = Φ (λ x), A_{j, λ} (x) = λ A_{j} (λ x) .

Then

B_{λ} (x) = λ^{2} B (λ x)

and

(\partial_{j} - i A_{j, λ}) Φ_{λ} (x) = λ (D_{j} Φ) (λ x) .

Let

E_{B} = \int B^{2}

,

E_{D} = \int ∣ D Φ ∣^{2}

and

E_{P} = \frac{1}{4} \int (1 - ∣Φ ∣^{2})^{2}

, with all integrals over the plane. Changing variables gives

V (λ) = λ^{2} E_{B} + E_{D} + λ^{- 2} E_{P} .

At a static solution,

V^{'} (1) = 2 E_{B} - 2 E_{P} = 0

. Consequently

\int_{R^{2}} B^{2} d^{2} x = \frac{1}{4} \int_{R^{2}} (1 - ∣Φ ∣^{2})^{2} d^{2} x .

The covariant-gradient contribution is scale invariant, while the magnetic and potential terms balance. This is why the scalar-only nonexistence conclusion does not extend to the Abelian Higgs model.

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