Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 303 / 3 / a / v

Table of contents
- Solution v

Solution

Let

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

For a radial integrand

h (q)

, differentiating the thin shell at

ζ = e^{s}

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 the bare parameters by running ones 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}} .

 Ancestors (11)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version