On shell renormalization scheme

Photon propagator

Just like what has been done with the fermion propagator, the form of the photon propagator inspired by the free photon field will be compared to the photon propagator calculated up to a certain order in ${\displaystyle e}$ in the interacting theory. The photon self energy is noted ${\displaystyle \mathrm{\Pi }(q^2)}$ and the metric tensor ${\displaystyle {\eta }^{\mu \nu }}$ (here taking the +--- convention)

${\displaystyle ⟨\mathrm{\Omega }|T(A^{\mu }(x)A^{\nu }(0))|\mathrm{\Omega }⟩=\int \frac{d^{4}q}{(2\pi {)}^4}\frac{-i{\eta }^{\mu \nu }{e}^{-ip\cdot x}}{q^2(1-\mathrm{\Pi }(q^2))+iϵ}=\int \frac{d^{4}q}{(2\pi {)}^4}\frac{-i{Z}_3{\eta }^{\mu \nu }{e}^{-ip\cdot x}}{q^2+iϵ}}$

The behaviour of the counterterm ${\displaystyle {\delta }_3=Z_3-1}$ is independent of the momentum of the incoming photon ${\displaystyle q}$. To fix it, the behaviour of QED at large distances (which should help recover classical electrodynamics), i.e. when ${\displaystyle q^2\to 0}$, is used :

${\displaystyle \frac{-i{\eta }^{\mu \nu }{e}^{-ip\cdot x}}{q^2(1-\mathrm{\Pi }(q^2))+iϵ}\sim \frac{-i{\eta }^{\mu \nu }{e}^{-ip\cdot x}}{q^2}}$

Thus the counterterm ${\displaystyle {\delta }_3}$ is fixed with the value of ${\displaystyle \mathrm{\Pi }(0)}$.

Vertex function

A similar reasoning using the vertex function leads to the renormalization of the electric charge ${\displaystyle e_r}$. This renormalization, and the fixing of renormalization terms is done using what is known from classical electrodynamics at large space scales. This leads to the value of the counterterm ${\displaystyle {\delta }_1}$, which is, in fact, equal to ${\displaystyle {\delta }_2}$ because of the Ward–Takahashi identity. It is this calculation that accounts for the anomalous magnetic dipole moment of fermions.

Rescaling of the QED Lagrangian

We have considered some proportionality factors (like the ${\displaystyle Z_i}$) that have been defined from the form of the propagator. However they can also be defined from the QED Lagrangian, which will be done in this section, and these definitions are equivalent. The Lagrangian that describes the physics of quantum electrodynamics is

${\displaystyle \mathcal{L}=-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }+\overline{\psi }(i\mathrm{\partial }/-m)\psi +e\overline{\psi }{\gamma }^{\mu }\psi A_{\mu }}$

where ${\displaystyle F_{\mu \nu }}$ is the field strength tensor, ${\displaystyle \psi }$ is the Dirac spinor (the relativistic equivalent of the wavefunction), and ${\displaystyle A}$ the electromagnetic four-potential. The parameters of the theory are ${\displaystyle \psi }$, ${\displaystyle A}$, ${\displaystyle m}$ and ${\displaystyle e}$. These quantities happen to be infinite due to loop corrections (see below). One can define the renormalized quantities (which will be finite and observable):

${\displaystyle \psi =\sqrt{Z_2}{\psi }_{r}A=\sqrt{Z_3}{A}_{r}m=m_r+\delta me=\frac{Z_1}{Z_2\sqrt{Z_3}}{e}_r\text{with}{Z}_i=1+{\delta }_i}$

The ${\displaystyle {\delta }_i}$ are called counterterms (some other definitions of them are possible). They are supposed to be small in the parameter ${\displaystyle e}$. The Lagrangian now reads in terms of renormalized quantities (to first order in the counterterms):

${\displaystyle \mathcal{L}=-\frac{1}{4}{Z}_3{F}_{\mu \nu ,r}{F}_r^{\mu \nu }+Z_2{\overline{\psi }}_r(i\mathrm{\partial }/-m_r){\psi }_r-{\overline{\psi }}_r\delta m{\psi }_r+Z_1{e}_r{\overline{\psi }}_r{\gamma }^{\mu }{\psi }_r{A}_{\mu ,r}}$

A renormalization prescription is a set of rules that describes what part of the divergences should be in the renormalized quantities and what parts should be in the counterterms. The prescription is often based on the theory of free fields, that is of the behaviour of ${\displaystyle \psi }$ and ${\displaystyle A}$ when they do not interact (which corresponds to removing the term ${\displaystyle e\overline{\psi }{\gamma }^{\mu }\psi A_{\mu }}$ in the Lagrangian).