Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 312 / 2 / c

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- Solution c

Solution

The variation of the product is

Δ [ϕ (k) ϕ (k^{'})] = c (2 π)^{3} [δ^{(3)} (k) ϕ (k^{'}) + δ^{(3)} (k^{'}) ϕ (k)] .

Its expectation value vanishes because the background has

⟨ ϕ ⟩ = 0

. The Ward identity and part b therefore give the shift-symmetry soft theorem for a scalar bispectrum

q \to 0 lim \frac{B _{ϕ} ( q , k , ∣ k + q ∣ )}{P _{ϕ} ( q )} = 0 .

Thus an exact internal shift creates no leading soft response of the two hard modes.

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