Stars as Laboratories for Fundamental Physics - MPP Theory Group

180 Chapter 5
components of A parallel and perpendicular to B T with A ∥ and A ⊥ ,
respectively, one finds (Raffelt and Stodolsky 1988)
⎡
⎛
⎞⎤
⎛ ⎞
n ⊥ − 1 n R 0 A ⊥
⎢
⎣ω 2 + ∂z 2 + 2ω 2 ⎜
⎟⎥
⎜ ⎟
⎝ n R n ∥ − 1 g aγ B T /2ω ⎠⎦
⎝ A ∥ ⎠ = 0,
0 g aγ B T /2ω −m 2 a/2ω 2 a
(5.26)
where the off-diagonal terms were made real by a suitable global transformation
of the fields. Further, a photon index of refraction was included
because in practice one never has a perfect vacuum.
The refractive index is generally different for the two linear polarization
states parallel and perpendicular to B T (Cotton-Mouton effect).
Also, there may be mixing between the A ⊥ and A ∥ fields, i.e. the plane
of polarization may rotate in optically active media, an effect characterized
by n R . In general, any medium becomes optically active if there
is a magnetic field component along the direction of propagation (Faraday
effect). Therefore, in general the refractive indices n ⊥,∥ depend on
the transverse, the index n R on the longitudinal magnetic field.
Equation (5.26) is made linear by an approach that will be discussed
in more detail for neutrinos in Sect. 8.2. For propagation in the positive
z-direction and for very relativistic axions and photons one may expand
(ω 2 + ∂z) 2 = (ω + i∂ z )(ω − i∂ z ) → 2ω (ω − i∂ z ). Then one obtains the
usual “Schrödinger equation”
⎡
⎢
⎣ω +
⎛
⎞ ⎤ ⎛
∆ ⊥ ∆ R 0
⎜
⎟ ⎥ ⎜
⎝ ∆ R ∆ ∥ ∆ aγ ⎠ + i∂ z ⎦ ⎝
0 ∆ aγ ∆ a
⎞
A ⊥
⎟
A ∥ ⎠ = 0, (5.27)
a
where ∆ ∥,⊥ = (n ∥,⊥ − 1) ω, ∆ R = n R ω, ∆ a = −m 2 a/2ω, and ∆ aγ =
1
2 g aγB T .
If one ignores a possible optical activity or Faraday effect (n R = 0),
the lower part of this equation represents a 2 × 2 mixing problem. The
matrix is made diagonal by a rotation about an angle
1
tan 2θ = ∆ aγ
=
2
∆ ∥ − ∆ a
g aγ B T ω
. (5.28)
(n ∥ − 1)2ω 2 + m 2 a
In analogy to neutrino oscillations (Sect. 8.2.2) the probability for an
axion to convert into a photon after travelling a distance l in a transverse
magnetic field is
prob(a → γ) = sin 2 (2θ) sin 2 ( 1 2 ∆ oscl), (5.29)
where ∆ 2 osc = (∆ ∥ − ∆ a ) 2 + ∆ 2 aγ so that the oscillation length is l osc =
2π/∆ osc .