Stars as Laboratories for Fundamental Physics - MPP Theory Group

Particle Dispersion and Decays in Media 197
z = 0 which contains n scattering centers per unit volume, and which is
infinite in the x- and y-directions. At a distance z from the slab, large
compared with k −1 , the asymptotic form of the parent plus scattered
wave is, ignoring the temporal variation e −iωt ,
∫ ∞
Φ(z) ∝ e iωz e ik(ρ2 +z 2 ) 1/2
+ nδa
f(ω, θ) 2πρ dρ , (6.2)
0 (ρ 2 + z 2 )
1/2
where ρ ≡ (x 2 + y 2 ) 1/2 and θ = arctan(ρ/z). Moreover, it was assumed
that in vacuum the wave propagates relativistically so that k = ω.
The integral in Eq. (6.2) is ill defined because the integrand oscillates
with a finite amplitude even for large values of ρ. It is made
convergent by substituting k → k + iκ with κ > 0 an infinitely small
real parameter. Integration by parts then yields
[
Φ(z) ∝ e iωz 1 + i 2πnδa ]
f 0 (ω) , (6.3)
ω
where a term of order (ωz) −1 was neglected which becomes small for
large z. Here, f 0 (ω) ≡ f(ω, 0) is the forward scattering amplitude.
Turn next to a slab of finite thickness a. The phase change of the
transmitted wave is obtained by compounding infinitesimal ones with
δa = a/j and taking the limit j → ∞,
lim
j→∞
[
1 + i 2πna
jω
f 0(ω)
] j
= e i(2πn/ω)f 0a . (6.4)
Inserting this result in Eq. (6.3) reveals that over a distance a in the
medium the wave accumulates a phase e in refrωa where
n refr = 1 + 2π
ω 2 n f 0(ω) (6.5)
is recognized as the index of refraction.
If the relativistic approximation |n refr −1| ≪ 1 is not valid one must
treat the wave self-consistently in the medium and distinguish carefully
between frequency and wavenumber. In this case one finds (Foldy 1945)
n 2 refr = 1 + 4π
k 2 n f 0(k), (6.6)
where the forward scattering amplitude must be calculated taking the
modified dispersion relation into account.
For the propagation of a field with several spin or flavor components
the same result applies if one remembers that “forward scattering” not