Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 304 / 1 / c

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- Solution c

Solution

Introducing a Feynman parameter and subtracting at zero external momentum give

Γ_{4} (s, t, u) = λ + \frac{λ ^{2}}{32 π ^{2}} \sum_{q^{2} = s, t, u} \int_{0}^{1} d x lo g \frac{m ^{2}}{m ^{2} - x ( 1 - x ) q ^{2} - i ϵ} + O (λ^{3}) .

The bare coupling

λ_{0} = λ + δ_{λ}

is independent of the renormalization scale. Differentiating its logarithmic counterterm at fixed

λ_{0}

gives the leading beta function

β_{λ} = μ \frac{d λ}{d μ} = \frac{3 λ ^{2}}{16 π ^{2}} + O (λ^{3}) .

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