Optical theorem at second Born order ($\mathrm{Im}{f}^{(2)}(0)=k{\sigma }_{\mathrm{total}}^{(1)}/(4\pi )$)

For a real elastic scattering potential the first Born forward amplitude is real, while the leading total scattering cross-section is quadratic in the potential. The second Born amplitude supplies an imaginary part from its outgoing Green function. Symmetrizing the two interaction positions makes its imaginary part equal to $k/(4\pi )$ times the integrated squared first Born amplitude. The identity therefore compares quantities at the same perturbative order; a real first Born amplitude alone cannot test it against a nonzero scattering cross-section.