A Reduced Lagrangian for Photon-Photon Interactions in Vacuum*

Electromagnetic waves travelling through vacuum excite virtual electron-positron pairs that can modify their propagation. Nonlinear wave equations describing this propagation can be derived from the Euler-Heisenberg Lagrangian density, which captures vacuum polarization effects up to the one-loop contribution. Here, we introduce a reduced action integral approach that facilitates modeling of processes arising from the Euler-Heisenberg Lagrangian, including photon-photon scattering and vacuum birefringence between two spectrally distinct light beams. The reduced Lagrangian derived from this action describes the evolution of familiar light-beam parameters, such as the centroid, spot size, phase, polarization, and phase-front curvature. To demonstrate the approach, the reduced action is applied to the scattering of an X-ray beam from a counter-propagating ultrahigh-intensity optical beam.