Coulomb and screening effects in Delbrück scattering, the scattering of light by an electromagnetic field, are a subject of ongoing theoretical interest. Experimental results for (large angle) scattering from high-Z atomic targets at 1.33 MeV [1,2] appear to be in agreement with theory if the available Born approximation results are used to describe the Delbrück amplitude (they are not in agreement if the Delbrück amplitude is neglected altogether). However results at 2.754 MeV are not in agreement with theory even when the Delbrück amplitude is included in the Born approximation, and it has been argued this indicates the importance of Coulomb effects in the Delbrück amplitude there [3,4]. There is still no complete calculation of the Delbrück scattering amplitude, including effects beyond Born approximation, available to clarify these questions. (There are calculations which involve limiting approximations, e. g. assumptions of small angles and high energies [5,6].) For a detailed discussion of the current situation we refer the reader to the recent review article [7], which builds on an earlier review [8].
For forward angle Delbrück scattering one can
use the optical theorem (in an expansion in the fine
structure constant 
)
and a dispersion relation to
obtain the scattering amplitude from knowledge of the lower
order total cross sections for pair production. Though a
discussion of forward angle Delbrück scattering takes us
away from the situations of present experimental interest,
it still serves as a useful check for a future more general
calculation.
Rohrlich and Gluckstern [9] calculated the forward Delbrück scattering amplitude in the Born approximation with two methods: (1) performing the evaluation of the corresponding Feynman graphs, though restricted to the case of forward angle in which the calculation simplifies (later Papatzacos and Mork [10] and Cheng et al. [11] did the full Born calculation, not restricted to forward angle), and (2) using the optical theorem to relate the imaginary part of the amplitude to the total cross section for pair production (also evaluated in the Born approximation), and using a dispersion relation to then obtain the real part. In this way the same result for the forward Delbrück scattering amplitude in the Born approximation was obtained with two different methods.
Extending method (1) to include Coulomb and screening effects would involve calculating the relevant Furry diagrams, where the electron-positron propagators are no longer free propagators but the full propagators in the atomic field. Analogous S-matrix calculations have been done numerically for other processes, such as Rayleigh scattering [12,13,14], Compton scattering [15] and bremsstrahlung [16], but the corresponding calculation for Delbrück scattering is less developed, though a formalism has been given [17], (based on the formalism of Wichmann and Kroll [18] for vacuum polarization), and some limited numerical results and partial calculations have been reported [19] within this formalism. This calculation is complicated both numerically and in principle since the amplitude describing this process is divergent and requires performing (external field) renormalization.
Extending method (2) is much simpler and free of renormalization issues (but it is restricted to describe forward scattering only). One simply replaces the results for the pair production cross section in the Born approximation with better estimates for the pair production cross section, including Coulomb and screening effects. This directly gives the imaginary part of the forward Delbrück amplitude, including Coulomb and screening effects, and the dispersion relation gives the real part. Solberg et al. [20] obtained the corrections to the Born result for the forward Delbrück amplitude due to Coulomb and screening effects in the ordinary (electron in continuum) pair production total cross section using this procedure (though, as will be discussed, their screening corrections were not accurate for photon energies near the pair production threshold).
However the Born approximation completely neglects bound-electron pair production, which should also be considered. Furthermore, given the usual partitioning of the elastic scattering amplitude (see next section), one should consider the total cross section for bound-electron pair production into all bound states, regardless of occupation (and not just the physically accessible unoccupied bound states). We will show that these corrections can be comparable with the corrections due to Coulomb and screening effects in ordinary pair production in the energy regime around the pair production threshold and below, while they are becoming unimportant at higher energies.
In Sec. 2 the elastic amplitude is partitioned into Rayleigh and Delbrück amplitudes in the single-electron formalism, describing elastic scattering off the bound atomic electrons and off virtual electron-positron pairs, respectively, together with amplitudes describing elastic scattering off the nucleus. The optical theorem and dispersion relation are written for each amplitude separately, corresponding to this partitioning. In Sec. 3 numerical results are given for the contribution of the bound-electron pair production total cross section to the forward Delbrück amplitude in the case of scattering from neutral ground-state uranium (Z=92). This contribution is compared with the corrections due to Coulomb and screening effects in the ordinary pair production cross section. Conclusions are presented in Sec. 4.