Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 303 / 3 / d

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- Solution d

Solution

Let

s = lo g ζ

. For an infinitesimal momentum shell,

\frac{d I _{d}}{d s} = \frac{Ω _{d - 1}}{( 2 π ) ^{d}} Λ^{d - 2} .

Differentiating the inverse-coupling relation and using

d (1/ g) / d s = - g^{- 2} d g / d s

gives the one-loop beta function

\frac{d g}{d s} = - (d - 2) g + \frac{Ω _{d - 1}}{( 2 π ) ^{d}} Λ^{d - 2} (N - 2) g^{2} .

For

d = 2 + ϵ

,

Ω_{d - 1} / (2 π)^{d} = 1/ (2 π) + O (ϵ)

, so

\frac{d g}{d s} ≃ - ϵ g + (N - 2) Λ^{ϵ} \frac{g ^{2}}{2 π} .

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