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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 $\log ({\Lambda }^2/F)=16{\pi }^{2}I(0)$.

For a bubble carrying momentum $p$, a Feynman parameter and a shift of the loop momentum give, after Wick rotation and cutoff regularization,where $F(p^2)=m^2-x(1-x)p^2-i0$. The quantum effective action therefore containsThe renormalization condition ${\Gamma }_4(M^2,M^2,M^2)=\lambda$ fixesSubstitution cancels both the cutoff and the scheme-dependent constant and leavesas required.