Probing the vacuum as a chiral medium

TL;DR

The paper addresses vacuum circular birefringence in quantum electrodynamics, showing that a standard Heisenberg-Euler treatment is insufficient for backgrounds with definite chirality and that derivative corrections (organized via Hilbert-series counting) are essential. It demonstrates the equivalence of three approaches—the derivative expansion of the LCFA, derivative corrections to the HE Lagrangian, and the low-energy limit of photon-photon scattering amplitudes—in plane-wave backgrounds, and extends the analysis to rotating standing-wave backgrounds. The key result is that NLO derivative terms can dominate the CP-induced birefringence signal, enabling a feasible experimental probe of higher-dimensional QED operators. The work provides both a rigorous theoretical framework and practical guidance for experimental tests of vacuum chirality using current and upcoming high-intensity light sources.

Abstract

We study the circular birefringence experienced by linearly polarised photons colliding with a circularly polarised 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 between three approaches: (i) adding derivative corrections to the Heisenberg-Euler Lagrangian; (ii) improving the locally constant field approximation to the one-loop polarisation tensor; (iii) performing a low-energy expansion of the direct $2\to 2$ QED photon-photon scattering amplitude. Going beyond plane-wave backgrounds, we analyse an example of a circularly polarised standing wave sensitive to derivative corrections. We find a parameter regime where these corrections could be probed in experiments.

Figures (5)

• Figure 1: Microscopic view of the circular birefringence scenario: There is a non-vanishing probability for the linear polarisation of probe photons to flip upon scattering off the CP plane-wave background (momentum and polarisation labels shown).
• Figure 2: Poincaré sphere with polarisation changes for linear and circular birefringence [cases (i) and (ii) of Table \ref{['tab:birefs']}]. The input state is $|D\rangle = (1/\sqrt{2}) (|1\rangle - |2\rangle)$. For linear birefringence, this is mapped to an elliptically polarised state $|E\rangle$ (red arrows), for circular polarisation (a chiral background), $|D\rangle$ is mapped to a rotated LP state, $|L\rangle$ (green arrows).
• Figure 3: Poincaré sphere with polarisation changes for a CP probe [cases (iii) and (iv) of Table \ref{['tab:birefs']}. The input state is $|+\rangle$ which is a fixed point, hence mapped to itself, for a CP background (green arrows). For an LP background, $|+\rangle$ is mapped to an elliptically polarised state $|E\rangle$ (red arrows), unless the phase difference is fine tuned to $\pi/2$ (as for a quarter waveplate) for which one reaches the equator, hence an LP configuration.
• Figure 4: a) the LO term in field strength of the HE Lagrangian ($0 \to 0$ process, i.e. a dressed vacuum loop), with the probe and background as classical fields (dashed lines); b) the LO expansion in field strength of the $1\to 1$ process of photon scattering in a plane-wave background (polarisation tensor); c) $2\to 2$ photon-photon scattering with one incoming and one outgoing photon (wavy lines) replaced by a classical plane wave background field.
• Figure 5: Sketch of the circular standing-wave scenario: The linear polarisation of probe photons flips due to scattering off the standing CP plane-wave background (with momentum and polarisation labels). The main signal has unchanged momentum $\ell$ (forward scattering), while the photons on the side peaks have altered momenta $\ell_{r,l}$ and polarisations $\neq \varepsilon_{1,2}$.