In quantum field theory, and especially in quantum electrodynamics, the interacting theory leads to infinite quantities that have to be absorbed in a renormalization procedure, in order to be able to predict measurable quantities.
The renormalization scheme can depend on the type of particles that are being considered.
For particles that can travel asymptotically large distances, or for low energy processes, the on-shell scheme, also known as the physical scheme, is appropriate.
Knowing the different propagators is the basis for being able to calculate Feynman diagrams which are useful tools to predict, for example, the result of scattering experiments.
In a new theory, the Dirac field can interact with another field, for example with the electromagnetic field in quantum electrodynamics, and the strength of the interaction is measured by a parameter, in the case of QED it is the bare electron charge,
The general form of the propagator should remain unchanged, meaning that if
now represents the vacuum in the interacting theory, the two-point correlation function would now read Two new quantities have been introduced.
has been defined as the pole in the Fourier transform of the Feynman propagator.
This is the main prescription of the on-shell renormalization scheme (there is then no need to introduce other mass scales like in the minimal subtraction scheme).
represents the new strength of the Dirac field.
, these new parameters should tend to a value so as to recover the propagator of the free fermion, namely
if this parameter is small enough (in the unit system where
Thus these parameters can be expressed as On the other hand, the modification to the propagator can be calculated up to a certain order in
These modifications are summed up in the fermion self energy
By identifying the two expressions of the correlation function up to a certain order in
, the counterterms can be defined, and they are going to absorb the divergent contributions of the corrections to the fermion propagator.
, will remain finite, and will be the quantities measured in experiments.
(here taking the +--- convention) The behaviour of the counterterm
is independent of the momentum of the incoming photon
To fix it, the behaviour of QED at large distances (which should help recover classical electrodynamics), i.e. when
A similar reasoning using the vertex function leads to the renormalization of the electric charge
It is this calculation that accounts for the anomalous magnetic dipole moment of fermions.
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 where
is the Dirac spinor (the relativistic equivalent of the wavefunction), and
These quantities happen to be infinite due to loop corrections (see below).
One can define the renormalized quantities (which will be finite and observable): The
The Lagrangian now reads in terms of renormalized quantities (to first order in the counterterms): 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
when they do not interact (which corresponds to removing the term