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

Fix the sign convention by taking the proper two-point insertion to be $i\Sigma (/p)$. This convention matches the loop expression and mass conversion printed later. Dyson resummation of successive fermion self-energy insertions givesThe subscript denotes the renormalized self-energy. If the insertion is instead named $-i{\Sigma }_{\mathrm{usual}}$, then ${\Sigma }_{\mathrm{usual}}=-{\Sigma }_R$ and the denominator is written $/p-m-{\Sigma }_{\mathrm{usual}}$. These are the same physical convention.

Lorentz covariance allows ${\Sigma }_R=\mathcal{A}(p^2)/p+\mathcal{B}(p^2)m$. The physical mass-shell condition isIt locates the mass-shell singularity of the Dirac propagator. In infrared-regulated perturbation theory this is the pole mass; near a simple pole the quantum field theory propagator has the form $i{Z}_{\mathrm{pole}}(/p+m_{\mathrm{phys}})/(p^2-m_{\mathrm{phys}}^2+i0)$. The mass and pole residue are different quantities: the former fixes the singularity's location, the latter the field normalization. The calculation below uses this standard perturbative pole-mass definition.