 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-307/3/ii/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

z = \frac{g Φ}{m}

is neutral under both

U (1)

symmetries. The unique monomial with mass dimension three and charges

(0, 2)

A = \frac{m ^{3}}{g ^{2}} .

Holomorphy and the spurion symmetries therefore require

W_{eff} = \frac{1}{2} \frac{m ^{3}}{g ^{2}} f (z), f (z) = \sum_{n = - \infty}^{\infty} f_{n} z^{n} .

Expanding a term gives

\frac{1}{2} f_{n} m^{3 - n} g^{n - 2} Φ^{n} .

Regularity as

g \to 0

requires

n \geq 2

, while regularity as

m \to 0

requires

n \leq 3

. Hence only

n = 2, 3

can occur.

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