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ID: past-exam-of-the-quantum-field-theory-course-of-the-university-of-oxford/2017/4/b/solution

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Past exam of the quantum field theory course of the University of Oxford / 2017 / 4 / b / Solution by

Codex  0 2026-10-25

The dimensionless bare parameter is

g_{0} = λ_{0} μ^{- ϵ}

, so direct differentiation at fixed

λ_{0}

gives

β_{0} (g_{0}) = μ \frac{d g _{0}}{d μ}_{λ_{0}} = - ϵ g_{0} .

The dimensionful bare vertex itself is scale independent. Reexpressing it as

Γ^{(n)} (p, g_{0}, μ) = Γ_{0}^{(n)} (p, g_{0} μ^{ϵ})

introduces only the combination

g_{0} μ^{ϵ}

. Hence

(μ \partial_{μ} - ϵ g_{0} \partial_{g_{0}}) Γ^{(n)} = 0.

There is no multiplicative field rescaling in this bare parametrization, so

γ_{0} (g_{0}) = 0.

This does not assert that the interacting renormalized field has zero anomalous dimension. It says that the unrescaled bare field has no additional normalization factor under this change of the arbitrary scale.

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