This diagram is convergent.
One-loop divergent diagrams
Diagram Effective degree
0 2 2 0
0 3 =0 Furry’s theorem
0 4 0 finite
2 0 1 0
2 1 0 0
Therefore, we have only the following one-loop divergent diagrams:
Self energy Vacuum polarization
We regularize the corresponding integrals by a parameter . They diverge as .
Calculations by perturbation theory give all physical quantities (e.g., , , …) in terms of the bare electron mass , the bare electron charge , and the regularization parameter ( and determine the electron mass and charge, if the interaction is switched off):
When the interaction is switched on, and are not more physical quantities. The observed (physical) electron mass and charge can be calculated by perturbation theory
The physical values of and are known from experiment. Using the smallness of , we can invert the equations for and :
Substituting these expressions for and into the equation for , we obtain
It can be shown that is finite (in every order in ) as
Let us consider the self-energy (SE) diagram for a free electron
A direct evaluation of this diagram yields
From the other side,
because is not changed due to this interaction. We have
So, the self-energy diagram changes the electron mass
In the free-electron theory, this effect can be simply accounted for by replacing , where is the physical electron mass, and by omitting the self-energy diagram. However, if the electron is not free, the SE diagram has further effects. To account this effect in calculations of other physical quantities, we should put in our equations
It results in occurring the term
in the interaction Hamiltonian and an additional vertex in the Feynman rules:
SE mass counterterm
The calculation of the SE diagrams for a bound electron together with the counterterm results in the Lamb shift of atomic levels.
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