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ID: past-exam-of-the-theoretical-physics-major-option-course-of-the-university-of-oxford/2016/8/c/solution

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Past exam of the theoretical physics major option course of the University of Oxford / 2016 / 8 / c / Solution by

Codex  0 2026-10-25

Put

ρ = φ^{†} φ

. The heavy-field equation of motion is

(□ + M^{2}) ϕ = - g ρ, ϕ = - \frac{g}{M ^{2}} ρ + \frac{g}{M ^{4}} □ ρ + O (M^{- 6}) .

(1)

This solution describes the low-energy sector without an independent incoming heavy-particle wave. It is valid when all external momentum invariants and light masses squared are small compared with

M^{2}

After an integration by parts, the terms involving the heavy field are

- \frac{1}{2} ϕ (□ + M^{2}) ϕ - g ρϕ

. Substituting the solution into both terms, rather than the source term alone, gives

Δ L_{eff} = \frac{g ^{2}}{2} ρ (□ + M^{2})^{- 1} ρ = \frac{g ^{2}}{2 M ^{2}} ρ^{2} - \frac{g ^{2}}{2 M ^{4}} ρ □ ρ + O (M^{- 6}) .

(2)

Hence, with the question's sign convention for the local quartic interaction,

λ = - \frac{g ^{2}}{2 M ^{2}}, L_{eff} = \partial_{μ} φ^{†} \partial^{μ} φ - m^{2} ρ - λ ρ^{2} + O (M^{- 4}) .

(3)

This is integrating out a field followed by a derivative expansion. The negative quartic means attraction. It also makes the truncated potential unbounded at large fields; minimizing the original potential over

ϕ

reveals the same negative quartic. Thus neither displayed model, without extra stabilizing interactions, supplies a globally stable nonperturbative vacuum. The matching describes perturbative low-energy scattering about the local vacuum. The simple equation-of-motion matching fixes the leading tree-level coefficient; precision quantum matching requires consistent loop renormalization and the additional effective operators at the requested accuracy.

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