Stars as Laboratories for Fundamental Physics - MPP Theory Group

Two-Photon Coupling of Low-Mass Bosons 179
5.4 Axion-Photon Oscillations
5.4.1 Mixing Equations
The “axion haloscope” discussed in the previous section was based on
the Primakoff conversion between axions and photons in a macroscopic
magnetic or electric field. The same idea can be applied to other axion
fluxes such as that expected from the Sun (“axion helioscope,” Sikivie
1983). Typical energies of solar axions are in the keV range (Sect. 5.2.4)
so that a macroscopic laboratory magnetic field is entirely homogeneous
on the scale of the axion wavelength.
In this case the conversion process is best formulated in a way analogous
to neutrino oscillations (Anselm 1988; Raffelt and Stodolsky 1988).
This approach may seem surprising as the axion has spin zero while
the photon is a spin-1 particle. States of different spin-parity can mix,
however, if the mixing agent (here the external magnetic field) matches
the missing quantum numbers. Therefore, only a transverse magnetic
or electric field can mix a photon with an axion; a longitudinal field
respects azimuthal symmetry whence it cannot mediate transitions between
states of different angular momentum components in the field
direction.
The starting point for the magnetically induced mixing between
axions and photons is the classical equation of motion for the system
of electromagnetic fields and axions in the presence of the interaction
Eq. (5.1). In terms of the electromagnetic field-strength tensor F and
its dual ˜F one finds, apart from the constraint ∂ µ ˜F µν = 0,
∂ µ F µν = J ν + g aγ ˜F µν ∂ µ a,
( + m 2 a) a = − 1 4 g aγ F µν ˜F µν , (5.24)
where J is the electromagnetic current density.
In a physical situation with a strong external field plus radiation one
may approximate g ˜F µν µν
µν
aγ ∂ µ a → g aγ ˜F ext∂ µ a because a term g aγ ˜F rad ∂ µa
is of second order in the weak radiation fields. Moreover, if only an external
magnetic field is present, the wave equation for the time-varying
part of the vector potential A and for the axion field are
A = g aγ B T ∂ t a,
( − m 2 a) a = −g aγ B T · ∂ t A, (5.25)
where B T is the transverse external magnetic field. If one specializes to
a wave of frequency ω propagating in the z-direction, and denoting the