On shell renormalization scheme

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. If these conditions are not fulfilled, one can turn to other schemes, like the minimal subtraction scheme (MS scheme).
Fermion propagator in the interacting theory.
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 theory where the only field is the Dirac field, the Feynman propagator reads
where formula_2 is the time-ordering operator, formula_3 the vacuum in the non interacting theory, formula_4 and formula_5 the Dirac field and its Dirac adjoint, and where the left-hand side of the equation is the two-point correlation function of the Dirac field.
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, formula_6. The general form of the propagator should remain unchanged, meaning that if formula_7 now represents the vacuum in the interacting theory, the two-point correlation function would now read
Two new quantities have been introduced. First the renormalized mass formula_9 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). The quantity formula_10 represents the new strength of the Dirac field. As the interaction is turned down to zero by letting formula_11, these new parameters should tend to a value so as to recover the propagator of the free fermion, namely formula_12 and formula_13.
This means that formula_9 and formula_10 can be defined as a series in formula_6 if this parameter is small enough (in the unit system where formula_17, formula_18, where formula_19 is the fine-structure constant). 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 formula_6 using Feynman diagrams. These modifications are summed up in the fermion self energy formula_23
These corrections are often divergent because they contain loops.
By identifying the two expressions of the correlation function up to a certain order in formula_6, the counterterms can be defined, and they are going to absorb the divergent contributions of the corrections to the fermion propagator. Thus, the renormalized quantities, such as formula_9, will remain finite, and will be the quantities measured in experiments.
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 formula_6 in the interacting theory. The photon self energy is noted formula_28 and the metric tensor formula_29 (here taking the +--- convention)
The behaviour of the counterterm formula_31 is independent of the momentum of the incoming photon formula_32. To fix it, the behaviour of QED at large distances (which should help recover classical electrodynamics), i.e. when formula_33, is used :
Thus the counterterm formula_35 is fixed with the value of formula_36.
Vertex function.
A similar reasoning using the vertex function leads to the renormalization of the electric charge formula_37. 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 formula_38, which is, in fact, equal to formula_39 because of the Ward–Takahashi identity. It is this calculation that account for the anomalous magnetic dipole moment of fermions.
Rescaling of the QED Lagrangian.
We have considered some proportionality factors (like the formula_40) 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
where formula_42 is the field strength tensor, formula_43 is the Dirac spinor (the relativistic equivalent of the wavefunction), and formula_44 the electromagnetic four-potential. The parameters of the theory are formula_43, formula_44, formula_47 and formula_6. 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 formula_50 are called counterterms (some other definitions of them are possible). They are supposed to be small in the parameter formula_6. 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 formula_43 and formula_44 when they do not interact (which corresponds to removing the term formula_55 in the Lagrangian).