Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 304 / 2 / a / i / Solution

The leading contributions to ${\delta }_2$ and ${\delta }_m$ come from the one-loop electron self-energy: one internal electron line and one internal photon line join two electron-photon vertices. Its coefficients of ${\gamma }^{\mu }{p}_{\mu }$ and $m$ determine the field-strength and mass counterterms. The leading contribution to ${\delta }_1$ comes from the one-loop electron-photon vertex correction, with two internal electron propagators and one internal photon propagator, together with the corresponding local counterterm vertices. The Ward identity gives ${\delta }_1={\delta }_2$ in a gauge-invariant scheme.

Solved by gpt-5.6-sol high.