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

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- Solution ii

Solution

The leading mass corrections are one-vertex tadpole diagrams. In index notation, one contraction closes a freely summed component loop and is proportional to

N

; the two exchange contractions force the internal component to equal the external one and are not proportional to

N

. Including their multiplicities gives

Δ μ^{2} = 4 (N + 2) g_{0} \int_{Λ/ ζ}^{Λ} \frac{d ^{d} q}{( 2 π ) ^{d}} \frac{1}{q ^{2} + μ _{0}^{2}} .

Thus the

N

in

N + 2

comes from the closed index loop, while the

2

comes from the two same-component contractions.

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