 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-354/1/c/i/solution

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

For three identical operators, conformal invariance gives

⟨ O (x) O (0) O (y)⟩ = \frac{C _{3}}{∣ x ∣ ^{Δ} ∣ y ∣ ^{Δ} ∣ y - x ∣ ^{Δ}} .

When

∣ x ∣ ≪ ∣ y ∣

∣ y - x ∣ = ∣ y ∣ [1 + O (∣ x ∣/∣ y ∣)]

, so

⟨ O (x) O (0) O (y)⟩ \sim \frac{C _{3}}{∣ x ∣ ^{Δ} ∣ y ∣ ^{2Δ}} .

Inserting the operator product expansion, the identity term has zero expectation with

O (y)

, while the

O (0)

term gives

\frac{C _{OOO}^{OPE}}{∣ x ∣ ^{Δ}} ⟨ O (0) O (y)⟩ = \frac{C _{OOO}^{OPE} C _{2}}{∣ x ∣ ^{Δ} ∣ y ∣ ^{2Δ}},

which has precisely the required position dependence.

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