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

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- Solution ii

Solution

Define the two shell integrals

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

For an external

ϕ_{1}

, the

ϕ_{1}^{4}

part of the interaction gives the scalar tadpole coefficient

12 g_{0} I_{1}

, while

2 g_{0} ϕ_{1}^{2} ϕ_{2}^{2}

gives

4 g_{0} I_{2}

. Interchanging the components gives

μ_{1}^{2} (ζ) = ζ^{2} [μ_{0, 1}^{2} + 4 g_{0} (3 I_{1} + I_{2})],

μ_{2}^{2} (ζ) = ζ^{2} [μ_{0, 2}^{2} + 4 g_{0} (I_{1} + 3 I_{2})] .

When the masses agree,

I_{1} = I_{2}

and both formulas reduce to

4 (N + 2) g_{0} I_{1}

with

N = 2

.

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