Stars as Laboratories for Fundamental Physics - MPP Theory Group

Two-Photon Coupling of Low-Mass Bosons 183
5.4.4 Vacuum Birefringence
Besides the conversion between photons and axions there is a more
subtle effect that can serve to search for the two-photon vertex of lowmass
pseudoscalars. In an external transverse E or B field the mixing
between the A ∥ component with a leads to a backreaction on A ∥
(Fig. 5.8b) which amounts to a retardation of its phase. Put another
way, the ∥ and ⊥ polarization states have different refractive indices in
vacuum with an E or B field (vacuum Cotton-Mouton effect). Therefore,
if one shines a light beam which is linearly polarized at 45 ◦ relative
to a transverse B field, the beam will develop a small degree of elliptic
polarization (Maiani, Petronzio, and Zavattini 1986).
A vacuum Cotton-Mouton effect is expected even in the absence of
axions from the QED amplitude shown in Fig. 5.8a; in the nonforward
direction it describes Delbrück scattering on a charged particle. More
generally, an electron loop mediates an effective γγ interaction which
for low energies can be described by the Euler-Heisenberg Lagrangian,
L γγ = 2α2 [
(E 2 − B 2 ) 2 + 7(E · B) 2] (5.30)
45m 4 e
(Heisenberg and Euler 1936; see also Itzykson and Zuber 1983).
Fig. 5.8. Vacuum birefringence in the presence of external fields. (a) QED
contribution according to the Euler-Heisenberg γγ interaction. (b) γa or
γπ ◦ oscillations in an external E or B field. (c) Photon birefringence in an
external axion field (axionic domain walls, cosmic axion field). (d) Axionmediated
contribution in a strong E · B field, e.g. near a pulsar.