Essentially Two-Particle Recoil Corrections

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- 

Eides et al. Theory of Light Hydrogenic Bound States, STMP 222, 81—98 (2007) 

Complete formal analysis of the recoil corrections in the framework of the relativistic two-particle equations, with derivation of all relevant kernels, perturbation theory contributions, and necessary subtraction terms may be performed along the same lines as was done for hyperfine splitting in [3]. However, these results may also be understood without a cumbersome formalism by starting with the simple scattering approximation. We will discuss recoil corrections below using this less rigorous but more physically transparent approach. 

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The book is organized as follows. In the introductory part we briefly discuss the main theoretical approaches to the physics of weakly bound two-particle systems. A detailed discussion then follows of the nuclear spin independent corrections to the energy levels. First, we discuss corrections which can be calculated in the external field approximation. Second, we turn to the essentially two-particle recoil and radiative-recoil corrections. Consideration of the spin-independent corrections is completed with discussion of the nuclear size and structure contributions. A special section is devoted to the spin-independent... 

A weakly bound state is necessarily nonrelativistic, v Za (see discussion of the electron in the field of a Coulomb center above). Hence, there are two small parameters in a weakly bound state, namely, the fine structure constant a. and nonrelativistic velocity v Za. In the leading approximation weakly bound states are essentially quantum mechanical systems, and do not require quantum field theory for their description. But a nonrelativistic quantum mechanical description does not provide an unambiguous way for calculation of higher order corrections, when recoil and many particle effects become important. On the other hand the Bethe-Salpeter equation provides an explicit quantum field theory framework for discussion of bound states, both weakly and strongly bound. Just due to generality of the Bethe-Salpeter formalism separation of the basic nonrelativistic dynamics for weakly bound states becomes difficult, and systematic extraction of high order corrections over a and V Za becomes prohibitively complicated. 

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