Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 304 / 3 / c / Solution

The kinetic terms determine the engineering dimensions in $d$ dimensions:Requiring $e\bar{\psi }{\gamma }^{\mu }{A}_{\mu }\psi$ to have dimension $d$ givesThus the electric charge is dimensionless only in four dimensions. In dimensional regularization one writes the bare interaction with $e_0={\mu }^{ϵ/2}{Z}_{e}e$, where $e$ is dimensionless and the renormalization scale $\mu$ supplies the missing dimension.