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

Table of contents
- Solution b

Solution

Define the shell integrals

I_{1} (ζ) = \int_{Λ/ ζ < ∣ q ∣ < Λ} \frac{d ^{d} q}{( 2 π ) ^{d}} \frac{1}{q ^{2} + μ _{0}^{2}},

I_{2} (ζ) = \int_{Λ/ ζ < ∣ q ∣ < Λ} \frac{d ^{d} q}{( 2 π ) ^{d}} \frac{1}{( q ^{2} + μ _{0}^{2} ) ^{2}} .

The index multiplicities described in part a and the canonical rescaling give

μ^{2} (ζ) = ζ^{2} [μ_{0}^{2} + 4 (N + 2) g_{0} I_{1} (ζ)],

g (ζ) = ζ^{4 - d} [g_{0} - 4 (N + 8) g_{0}^{2} I_{2} (ζ)] .

For

N = 1

, these reduce to the stated Ising coefficients

12

and

36

.

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