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).
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
\langle0|T(\psi(x)\bar{\psi}(0))|0\rangle=iSF(x)=\int
d4p | |
(2\pi)4 |
ie-ip ⋅ | |
p/-m+i\epsilon |
where
T
|0\rangle
\psi(x)
\bar{\psi}(x)
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,
e
|\Omega\rangle
\langle\Omega|T(\psi(x)\bar{\psi}(0))|\Omega\rangle=\int
d4p | |
(2\pi)4 |
iZ2e-i | |
p/-mr+i\epsilon |
Two new quantities have been introduced. First the renormalized mass
mr
Z2
e → 0
mr → m
Z2 → 1
This means that
mr
Z2
e
\hbar=c=1
e=\sqrt{4\pi\alpha}\simeq0.3
\alpha
Z2=1+\delta2
mr=m+\deltam
On the other hand, the modification to the propagator can be calculated up to a certain order in
e
\Sigma(p)
\langle\Omega|T(\psi(x)\bar{\psi}(0))|\Omega\rangle=\int
d4p | |
(2\pi)4 |
ie-i | |
p/-m-\Sigma(p)+i\epsilon |
These corrections are often divergent because they contain loops.By identifying the two expressions of the correlation function up to a certain order in
e
mr
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
e
\Pi(q2)
η\mu\nu
\langle\Omega|T(A\mu(x)A\nu(0))|\Omega\rangle=\int
d4q | |
(2\pi)4 |
-iη\mu\nue-i | |
q2(1-\Pi(q2))+i\epsilon |
=\int
d4q | |
(2\pi)4 |
-iZ3η\mu\nue-i | |
q2+i\epsilon |
The behaviour of the counterterm
\delta3=Z3-1
q
q2 → 0
-iη\mu\nue-i | \sim | |
q2(1-\Pi(q2))+i\epsilon |
-iη\mu\nue-i | |
q2 |
Thus the counterterm
\delta3
\Pi(0)
A similar reasoning using the vertex function leads to the renormalization of the electric charge
er
\delta1
\delta2
We have considered some proportionality factors (like the
Zi
lL=-
1 | |
4 |
F\muF\mu+\bar{\psi}(i\partial/-m)\psi+e\bar{\psi}\gamma\mu\psiA\mu
where
F\mu
\psi
A
\psi
A
m
e
\psi=\sqrt{Z2}\psir A=\sqrt{Z3}Ar m=mr+\deltam e=
Z1 | |
Z2\sqrt{Z3 |
The
\deltai
e
lL=-
1 | |
4 |
Z3F\mu
\mu\nu | |
F | |
r |
+Z2\bar{\psi}r(i\partial/-mr)\psir-\bar{\psi}r\deltam\psir+Z1er\bar{\psi}r\gamma\mu\psirA\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
\psi
A
e\bar{\psi}\gamma\mu\psiA\mu