Photon exchange

Coupling the motion of the mosaic cell (TLS and boson peak) to phonons is necesssary to explain thermal conductivity therefore the interaction effects discussed later follow from our identification of the origin of amorphous state excitations. The emission of a phonon followed by its absorption by another cell will give an effective interaction, in the same way that photon exchange leads to... 

Leading recoil corrections in Za (of order (Za) (m/M)") still may be taken into account with the help of the effective Dirac equation in the external field since these corrections are induced by the one-photon exchange. This is impossible for the higher order recoil terms which reflect the truly relativistic two-body nature of the bound state problem. Technically, respective contributions are induced by the Bethe-Salpeter kernels with at least two-photon exchanges and the whole machinery of relativistic QFT is necessary for their calculation. Calculation of the recoil corrections is simplified by the absence of ultraviolet divergences, connected with the purely radiative loops. 

An explicit expression for the Breit potential was derived in [2] from the one-photon exchange amplitude with the help of the Foldy-Wouthuysen transformation ... 

Recoil corrections depending on odd powers of Za are also missing in (3.5), since as was explained above all corrections generated by the one-photon exchange necessarily depend on the even powers of Za. Hence, to calculate recoil corrections of order Za) one has to consider the nontrivial contribution of the box diagram. We postpone discussion of these corrections until Sect. 4.1. 

The mass dependence of the correction of order a Za) beyond the reduced mass factor is properly described by the expression in (3.7) as was proved in [11, 12]. In the same way as for the case of the leading relativistic correction in (3.4), the result in (3.7) is exact in the small mass ratio m/M, since in the framework of the effective Dirac equation all corrections of order Za) are generated by the kernels with one-photon exchange. In some earlier papers the reduced mass factors in (3.7) were expanded up to first order in the small mass ratio m/M. Nowadays it is important to preserve an exact mass dependence in (3.7) because current experiments may be able to detect quadratic mass corrections (about 2 kHz for the IS level in hydrogen) to the leading nonrecoil Lamb shift contribution. 

The external field approximation is clearly inadequate for calculation of the recoil corrections and, in principle, one needs the machinery of the relativistic two-particle equations to deal with such contributions to the energy levels. The first nontrivial recoil corrections are generated by kernels with two-photon exchanges. Naively one might expect that all corrections of order Za) m/M)m are generated only by the two-photon exchanges in Fig. 4.1. However, the situation is more complicated. More detailed consideration shows that the two-photon kernels are not sufficient and irreducible kernels in Fig. 4.2 with arbitrary number of the exchanged Coulomb pho-... 

Let us emphasize that the total contribution of the double transverse exchange is given by the matrix element of the two-photon exchanges between... 

Let us start systematic discussion of such corrections with the recoil corrections to the leading contribution to the Lamb shift. The most important observation here is that the mass dependence of all corrections of order a." Za.Y obtained above is exact, as was proved in [1, 2], and there is no additional mass dependence beyond the one already present in (3.7)-(3.24). This conclusion resembles the similar conclusion about the exact mass dependence of the contributions to the energy levels of order (Za) m discussed above, and it is valid essentially for the same reason. The high frequency part of these corrections is generated only by the one photon exchanges, for which we know the exact mass dependence, and the only mass scale in the low frequency part, which depends also on multiphoton exchanges, is the reduced mass. 

Corrections of relative order (Za) connected with the nonelementarity of the nucleus are generated by the diagrams with two-photon exchanges. As usual all corrections of order (Za), originate from high (on the atomic scale) intermediate momenta. Due to the composite nature of the nucleus, besides intermediate elastic nuclear states, we also have to consider the contribution of the diagrams with inelastic intermediate states. 

The main feature of the polarizability contribution to the energy shift is its logarithmic enhancement [26, 30]. The appearance of the large logarithm may easily be understood with the help of the skeleton integral. The heavy particle factor in the two-photon exchange diagrams is now described by the photon-nucleus inelastic forward Compton amplitude [31]... 

Fig. 7.6. One-photon exchange with one-loop polarization insertion...

Due to the analogy between contributions of the diagrams with muon and hadron vacuum polarizations, it is easy to see that insertion of hadron vacuum polarization in one of the exchanged photons in the skeleton diagrams with two-photon exchanges generates a correction of order (x Zotf (see Fig. 7.12). Calculation of this correction is straightforward. One may even take into account the composite nature of the proton and include the proton form factors in photon-proton vertices. Such a calculation was performed in [51, 52] and produced a very small contribution... 

This effective Hamiltonian for the interaction of two magnetic moments may also easily be derived from the one photon exchange diagram in Fig. 8.2. 

I. Lindgren, H. Persson, Sten Salomonson, L. Labzowsky, Full QED calculations of two-photon exchange for heliumlike systems Analysis in the Coulomb and Feynman gauges, Phys. Rev. A 51 (2) (1995) 1167. 

Conventionally, the evaluation of bound-state QED corrections is made tractable by including the vacuum fluctuations in several steps. The corrections thus calculated are called radiative corrections, and their evaluation can be made by making two expansions. The first is in powers of (a/7r) and denotes the number of photon propagator loops present. The second is in the number of photon exchanges with the nucleus and is in powers of (Zot), where Z is the nuclear charge. The expression in equation 1 shows the first two terms of the (o/tt) expansion for the case of hydrogenic S-states, i.e. up to two photon loops. In this expression the second expansion has not yet been made and the (Za) dependence is still contained within the functions F and H. 

The relativistic correction to the fermion kinetic energy is represented as a potential. The Breit-Fermi interaction includes the effects of transverse photon exchange as well as relativistic corrections to Coulomb photon exchange. The potentials are given with the assumption that the states acted on are S states with total spin 1. 

---