Consider a hypothetical universe described by a conformal fi...
Prompt
Consider a hypothetical universe described by a conformal field theory (CFT) in 𝑑 = 4 d=4 dimensions, with a scalar primary sector 𝜙 ϕ of scaling dimension Δ Δ. Write the general expression for the two-point function ⟨ 𝜙 ( 𝑥 1 ) 𝜙 ( 𝑥 2 ) ⟩ ⟨ϕ(x 1 )ϕ(x 2 )⟩ in Euclidean space, fixing the normalization to 1. Suppose there exists a composite operator 𝑂 = : 𝜙 2 : O=:ϕ 2 : that is also primary, with scaling dimension Δ 𝑂 Δ O . Using only conformal symmetry and dimensional analysis, determine the form of the three-point function ⟨ 𝜙 ( 𝑥 1 ) 𝜙 ( 𝑥 2 ) 𝑂 ( 𝑥 3 ) ⟩ ⟨ϕ(x 1 )ϕ(x 2 )O(x 3 )⟩ up to a structure constant 𝐶 𝜙 𝜙 𝑂 C ϕϕO . Impose the following physical constraint: in the limit 𝑥 1 → 𝑥 2 x 1 →x 2 , the three-point function must reduce, via the OPE expansion, to the two-point function multiplied by the OPE coefficient 𝐶 𝜙 𝜙 𝑂 C ϕϕO and the appropriate power of ∣ 𝑥 1 − 𝑥 2 ∣ ∣x 1 −x 2 ∣. Use this constraint to express Δ 𝑂 Δ O in terms of Δ Δ, and to fix the explicit dependence on ∣ 𝑥 1 − 𝑥 2 ∣ ∣x 1 −x 2 ∣, ∣ 𝑥 1 − 𝑥 3 ∣ ∣x 1 −x 3 ∣, ∣ 𝑥 2 − 𝑥 3 ∣ ∣x 2 −x 3 ∣ in the three-point function. Finally, briefly discuss whether and how this result would change if the theory were defined on Minkowski spacetime instead of Euclidean space, focusing only on the conformal structure (ignore issues of causality and vacuum choice). Answer in a self-consistent way, showing the key steps and explicitly checking that scaling dimensions are consistent in every expression.
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