Vacuum Polarization in a Strong Coulomb Field

Point-particle effective field theory III: relativistic fermions and the Dirac equation

“…A standard contribution to atomic energy levels that also can be captured using S p is the contribution (or parts of the contribution) due to vacuum polarization. It is well-known that the effects of vacuum polarization on the field of a point charge, Ze, can be described by the Uehling potential [54][55][56][57], of the form 17) in which m is the mass of the particle circulating within the loop. Since the range of this interaction is of order R ∼ m −1 the electron and muon vacuum polarizations fall into the category of physical effects acting over much smaller distances than typical sizes of orbits in ordinary atoms.…”

Higher order vacuum polarization for finite radius nuclei

“…(10) as (2.2) where SF satisfies (2.3) For time independent potentials ~,SF(x,x') depends on time only through t -t', and consequently, ...... (2.6) This relation, then, is the basis of the Wichmann-Kroll formalism (8) for the calculation of P vp to all orders in Za. Note that the Green's function in this relation must be properly regulated to insure that the limit x, ~ x exists and that the integral over z con-'I'¥.…”

“…where k = ±(j + 1/2) for a given total angular momentum j. From the following relation (8,12) Tr G( x, x'; z)1 ... ..…”

“…(3.1) for the case of a pure Coulomb potential (R = 0). These solutions are well known (8,12). Letting 1VtC denote the solution regular at r = 0 and ~ denote the solution regular at: r = 00, then To obtain the interior solutions, the nuclear charge density must be specified.…”

“…In section 2, the Wichmann-KrOll formalism (8) for the calculation of the VP density P vp is reviewed. The modifications necessary for very large Z nuclei are discussed in detail, and formal relations between P vp and the Green's function for the Dirac equation are established.…”