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

The Quantum effective action contains the following diagrams. For ${\Gamma }_2$, there are the tree inverse propagator, the one-loop tadpole at order $\lambda$, and at order ${\lambda }^2$ the two-loop sunset and the two-loop tadpole with a tadpole insertion. For ${\Gamma }_4$, there are the tree vertex $-i\lambda$ and, at order ${\lambda }^2$, the three one-loop bubble diagrams in the $s$, $t$, and $u$ channels. Disconnected and one-particle-reducible graphs do not occur in the quantum effective action.

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

Introducing a Feynman parameter and subtracting at zero external momentum giveThe bare coupling ${\lambda }_0=\lambda +{\delta }_{\lambda }$ is independent of the renormalization scale. Differentiating its logarithmic counterterm at fixed ${\lambda }_0$ gives the leading beta function