Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 44 / 3 / c / Solution

The divergent part follows from the elementary integrals ${\int }_0^{1}xdx=1/2$ and ${\int }_0^{1}dx=1$:Thus a wave-function counterterm and a mass counterterm are required. Write their contribution asIn the insertion convention of part (a), the inverse Dirac propagator receives $\delta Z_2/p-\delta m_{\mathrm{coeff}}$. With $\kappa =e^2/(8{\pi }^2ϵ)$, cancellation requiresTo distinguish the coefficient counterterm from the multiplicative mass renormalization, write $m_0=Z_{m}m$ and ${\psi }_0=Z_2^{1/2}\psi$. Then $\delta m_{\mathrm{coeff}}=m(\delta Z_2+\delta Z_m)$ at this order, giving $\delta Z_m=-3\kappa$. In modified minimal subtraction, replace $\kappa$ by $e^2{\Delta }_{\overline{\mathrm{MS}}}/(16{\pi }^2)$. No new derivative structure is needed for this two-point divergence; charge and photon counterterms are determined from other functions.