Abstract
The evaluation method of the one-loop self-energy of the electron bound in a Coulomb field is described. The method combines the relativistic multipole expansion with the free-particle approximation in the virtual states without breaking any integrations of pieces.
The relativistic multipole expansion is based on a single assumption: except for the part of the time component of the electron four-momentum corresponding to the electron rest mass, the exchange of four-momentum between the virtual electron and photon can be treated perturbatively. This assumption holds very well, except for the electron virtual states with very high three-momentum.
It is shown that for such virtual states one can always rearrange the pertinent expression in a way that allows the electron to be treated as free. The fraction of the free-particle approximation contained in the relativistic multipole expansion carried out to a given order is precisely determined.
Furthermore, it is pointed out that in the virtual states with very large wave numbers the electron ceases to feel the Coulomb force from the nucleus for arbitrarily strong fields. This results in a simple scaling behavior of the integrals over the large electron wave numbers.
This in turn enables us to avoid a decomposition of convergent integrals into the sum of divergent integrals encountered earlier. By taking the method up to the ninth order and estimating the remainder of the series, the result obtained for the ground state of the hydrogen atom differs from the other result of comparable accuracy by two parts in 10(9).
This amounts to the difference of 18 Hz for 2s-1s transition in hydrogen. This is by four orders smaller than the uncertainty in determination of the proton radius.
With an increasing nuclear charge Z, the rate of convergence of the expansion slows down. Nonetheless, the obtained results are in very good agreement with the results obtained by partial wave expansion up to Z = 90.