Stars as Laboratories for Fundamental Physics - MPP Theory Group

184 Chapter 5
The vacuum refractive index in an external magnetic field which
results from this interaction was studied by a number of authors—
see Tsai and Erber (1975, 1976) for references to the early literature.
Adler (1971) provided a comprehensive study and derived the correct
expression for the related photon splitting rate γ → γγ in an external
field. 28 The refractive indices for the ∥ and ⊥ polarization states are 29
n ∥ = 1 + 7 2α2
45
B 2
m 4 e
and
n ⊥ = 1 + 4 2α2
45
B 2
m 4 e
, (5.31)
where
(2α 2 /45) B 2 /m 4 e = 1.32×10 −32 (B/Gauss) 2 . (5.32)
Note that 1 Gauss corresponds to 10 −4 Tesla, and to 1.95×10 −2 eV 2 in
natural units (Appendix A).
Clearly one needs very strong magnetic fields for vacuum birefringence
effects to become important. So far, no positive experimental
measurement exists. A proposal to measure the acquired elliptic polarization
of a laser beam was put forth by Iacopini and Zavattini (1979).
More recently Cantatore et al. (1991) proposed to use polarized light
scattered off an electron beam, a method which allows one to obtain
polarized GeV photons. As the relative phase shift is (n ∥ − n ⊥ )ωl for
a distance of travel l in the magnetic field, high-energy photons show
a much stronger effect for otherwise equal conditions.
An experiment to search for the axion contribution of Fig. 5.8b
was recently performed (Semertzidis et al. 1990; Cameron et al. 1993).
Note that there are two axion-induced effects on a laser beam trapped
in an optical cavity. One is the birefringence effect analogous to the
QED effect which leads to a small amount of elliptical polarization.
Another is the loss of ∥ photons into the axion channel which depletes
the amplitude of the ∥ mode relative to the ⊥ one, which in turn leads
to a rotation of the plane of polarization. Both effects are of the same
order in the coupling constant; experimentally, the rotation effect led to
28 The photon-splitting box graph with one external field and three real photons
attached to an electron loop does not contribute. The lowest-order amplitude is
with the external field attached three times, and three real photons (hexagon diagram).
For references to the early literature and a discussion of the astrophysical
implications of the photon-splitting process see Baring (1991).
29 ∥, ⊥ refer to the electric field of the wave relative to the external transverse B
field while Adler (1971) refers with ∥, ⊥ to the magnetic field of the wave. Note
also that I use rationalized units where α = e 2 /4π = 1/137 while in the literature
on photon refraction unrationalized units with α = e 2 = 1/137 are often employed.