Past exam of the quantum field theory course of the University of Oxford / 2017 / 4 / d / Solution

At fixed regulator abbreviateThe coupling expansion is $g=g_0+A{g}_0^2+O(g_0^3)$, whose inverse is $g_0=g-A{g}^2+O(g^3)$. Using part c before removing the regulator,Therefore the regulator-free coefficients at the orders determined by the supplied expansions areThe canonical term $-ϵg$ vanishes, and the factors of $ϵ$ cancel the simple poles. The finite coefficients $a_2,z_2$ and higher positive powers of $ϵ$ do not enter these leading regulator-free terms. Inverting the coupling is essential: substituting $g_0=g$ too early would incorrectly give $-2a_1{g}^2$.

The calculation establishes finiteness only at the displayed perturbative orders. Higher-order pole cancellations require the higher-order coefficients, as the next part makes explicit.