After <Wick rotation>, the zero-momentum bubble integral with <cutoff regularization> is
$$
I(0)=\int_{|k|<\Lambda}\frac{d^4k}{(2\pi)^4}\frac1{(k^2+m^2)^2}
=\frac1{16\pi^2}\left[\log\frac{\Lambda^2+m^2}{m^2}+\frac{m^2}{\Lambda^2+m^2}-1\right].
$$
The three channels contribute $3\lambda^2I(0)/2$. For $\Lambda\gg m$, this is $\frac{3\lambda^2}{32\pi^2}[\log(\Lambda^2/m^2)-1]$, so the zero-momentum <renormalization condition> is enforced by
$$
\delta_\lambda=-\frac{3\lambda^2}{32\pi^2}\log\frac{\Lambda^2}{e m^2},
$$
and hence $F=e m^2$. Keeping the exact cutoff expression merely replaces $F$ by the finite cutoff-dependent quantity defined by $\log(\Lambda^2/F)=16\pi^2I(0)$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 304 / 1 / b / Solution

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A Wick rotation deforms a time or energy contour from real to imaginary values, converting suitable Minkowski integrals into Euclidean integrals while respecting pole prescriptions.


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 Incoming links: Wick rotation