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 Incoming links: Integration-by-parts reduction of scalar operators

Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2025 / 1 / e / Solution 2026-10-25

Write

X ≃ Y

when the two densities differ by a total derivative. Under the boundary conditions used in the action, they have the same integral. The following identities establish the integration-by-parts reduction of scalar operators:

\partial_{μ} (ϕ^{3} \partial^{μ} ϕ) \partial_{μ} (ϕ \partial^{μ} □ ϕ) \partial_{μ} (\partial^{μ} ϕ □ ϕ) \partial_{μ} (\partial_{ν} ϕ \partial^{μ} \partial^{ν} ϕ) = 3 ϕ^{2} (\partial ϕ)^{2} + ϕ^{3} □ ϕ, = \partial_{μ} ϕ \partial^{μ} □ ϕ + ϕ □^{2} ϕ, = (□ ϕ)^{2} + \partial_{μ} ϕ \partial^{μ} □ ϕ, = (\partial_{μ} \partial_{ν} ϕ)^{2} + \partial_{ν} ϕ \partial^{ν} □ ϕ, O_{2} O_{4} O_{5} O_{7} ≃ - \frac{1}{3} O_{3}, ≃ - O_{5}, ≃ - O_{6}, ≃ - O_{5} .

Consequently

{ϕ^{6}, ϕ^{3} □ ϕ, (□ ϕ)^{2}}

spans the operator basis in effective field theory before equation-of-motion reduction of scalar operators. These four independent relations reduce the seven representatives to three; the three sectors have different field or derivative content, so integration by parts cannot identify them further.

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