Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 303 / 3 / c

Table of contents
- Solution c

Solution

Put

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

For

ζ = e^{s}

, differentiating a thin radial shell at

s = 0

gives

\frac{d}{d s} \int_{Λ e^{- s}}^{Λ} \frac{d ^{d} q}{( 2 π ) ^{d}} h (q)_{s = 0} = K_{d} Λ^{d} h (Λ) .

Replacing bare parameters by running parameters after each infinitesimal step yields the beta functions

\frac{d μ ^{2}}{d s} = 2 μ^{2} + 4 (N + 2) g K_{d} \frac{Λ ^{d}}{Λ ^{2} + μ ^{2}},

\frac{d g}{d s} = (4 - d) g - 4 (N + 8) g^{2} K_{d} \frac{Λ ^{d}}{( Λ ^{2} + μ ^{2} ) ^{2}} .

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