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

Repeated insertions of the fermion self-energy form a geometric Dyson resummation:The physical fermion mass is the pole mass: after analytic continuation, $m_{\mathrm{phys}}$ is determined by the zero of the exact inverse propagator at $p^2=-m_{\mathrm{phys}}^2$. Ifthen to all orders the pole obeyswith the signs fixed by the displayed Dyson convention.

The one-loop graph is a fermion line that emits and reabsorbs one internal photon:

The QED Feynman rules assign $(-ie{\gamma }^{\mu })(-ie{\gamma }^{\nu })$ to the two vertices, the Feynman-gauge photon propagator contracts $\mu$ and $\nu$ and contributes ${\delta }_{\mu \nu }/(p-k{)}^2$, and the internal Dirac propagator is $(-i/k+m)/(k^2+m^2)$. Hence

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.

Use a Feynman parameter with $A=k^2+m^2$ and $B=(p-k{)}^2$. ThenThe identities for gamma matrices giveAfter the shift $\ell =k-(1-x)p$, the term odd in $\ell$ integrates to zero. The rotationally symmetric loop integral isConsequentlyThus

Restore the dimensional-regularization factor ${\mu }^{ϵ}$ and defineUsing $\Gamma (ϵ/2)=2/ϵ-\gamma +O(ϵ)$ givesThe first term is the ultraviolet divergence. In the modified minimal subtraction scheme, subtraction of $2/ϵ-\gamma +\log 4\pi$ leavesOn the tree-level mass shell, $p^2=-m^2$ and $i/p=-m$, so $\Delta =x^2{m}^2$. The mass counterterm isThe pole condition $m_{\mathrm{phys}}=m-{\Sigma }_{\overline{\mathrm{MS}}}{|}_{i/p=-m}+O(e^4)$ therefore givesEvaluating the elementary parameter integral yields the equivalent expression