Probing the vacuum as a chiral medium

We study the experienced by linearly polarized photons colliding with a circularly polarized background creating a vacuum of definite chirality (handedness). For this scenario the standard Heisenberg-Euler approach fails and must be supplemented by derivative corrections which we match to known Hilbert series. Choosing a plane wave background, we find equivalence in the low-energy limit between three approaches: (i) adding derivative corrections to the Heisenberg-Euler Lagrangian, (ii) improving the locally constant field approximation to the one-loop polarization tensor, and (iii) performing a low-energy expansion of the direct <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mn>2</a:mn> <a:mo stretchy="false">→</a:mo> <a:mn>2</a:mn> </a:math> QED photon-photon scattering amplitude. Going beyond plane-wave backgrounds, we analyze an example of a circularly polarized standing wave sensitive to derivative corrections. We find a parameter regime where these corrections could be probed in experiments.