All-Order Coulomb Corrections to Delbrück Scattering above the Pair-Production Threshold

We report calculations of Delbrück scattering that include all-order Coulomb corrections for photon energies above the threshold of electron-positron pair creation. Our approach is based on the application of the Dirac-Coulomb Green’s function and accounts for the interaction between the virtual electron-positron pair and the nucleus to all orders in the nuclear binding strength parameter $\alpha Z$. Practical calculations are performed for the scattering of 2.754 MeV photons off plutonium atoms. We find that including the Coulomb corrections enhances the scattering cross section by up to 50% in this case. The obtained results resolve the long-standing discrepancy between experimental data and theoretical predictions and demonstrate that an accurate treatment of the Coulomb corrections is crucial for the interpretation of existing and guidance of future Delbrück scattering experiments on heavy atoms.

Figure 1

Feynman diagram for Delbrück scattering to leading order in $\alpha$ and all orders in $\alpha Z$.

Figure 2

Original (upper panel) and Wick rotated (lower panel) contour for the $z^{\prime }$ integration in the Delbrück amplitude (1). The modified contour consists of paths along the imaginary axis ($C_1$) and around the branch cuts ($C_2$) as well as contributions from the residue ($C_3$). Moreover, the branch cuts (black zigzag lines) and a finite subset of the countably infinite singularities (black crosses) of the Green’s functions in Eq. (1) are shown for the case of above threshold photon energies.

Figure 3

Analytic structure of the integrand and the integration contour after the redefinition of the vacuum to include the lowest-lying bound state. The corresponding pole in the lower left quadrant moves up and is not encircled anymore, whereas the pole in the lower right quadrant is also moving up and gets encircled by the integration contour.

Figure 4

Real (upper panels) and imaginary (lower panels) parts of the amplitude for Delbrück scattering (1) of 2.754 MeV photons by bare neon (left panels) and plutonium (right panels) nuclei. Calculations have been performed for linear polarization of the incoming and outgoing photons parallel (black solid line) as well as perpendicular (red dashed line) to the scattering plane. Moreover, the lowest-order Born predictions from Ref. [31] are shown (diamonds). The amplitudes are given in units $(\alpha Z{)}^2{r}_0$, where $r_0=2.818\mathrm{fm}$ is the classical electron radius.

Figure 5

Differential cross section for elastic scattering of 2.754 MeV unpolarized photons by plutonium atoms. The black dots display the experimental data from Ref. [15], the black solid line indicates the theoretical results based on all order in $\alpha Z$ Delbrück calculations while the shaded region shows the theoretical error. Theoretical predictions using the lowest-order Born approximation for Delbrück scattering are displayed with the red dashed line.