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

After Wick rotation, the zero-momentum bubble integral with cutoff regularization isThe three channels contribute $3{\lambda }^{2}I(0)/2$. 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 byand hence $F=e{m}^2$. Keeping the exact cutoff expression merely replaces $F$ by the finite cutoff-dependent quantity defined by $\log ({\Lambda }^2/F)=16{\pi }^{2}I(0)$.