Stars as Laboratories for Fundamental Physics - MPP Theory Group

Neutrino Oscillations 287
Fig. 8.2. Flavor oscillation as a “spin precession.” (After Stodolsky 1987.)
8.2.3 Distribution of Sources and Energies
If the neutrino source region is not point-like relative to the oscillation
length, one has to average the appearance or survival probabilities
accordingly. If the source locations z 0 are distributed according to a
normalized function f(z 0 ) the ν µ appearance probability is
∫
prob (ν e → ν µ ) = sin 2 2θ
dz 0 f(z 0 ) sin 2 π (z − z 0)
l osc
. (8.20)
For example, consider a Gaussian distribution f(z 0 ) = e −z2 0 /2s2 /s √ 2π
of size s for which
prob (ν e → ν µ ) = 1 2 sin2 2θ [ 1 − e −2π2 (s/l osc) 2 cos(2πz/l osc ) ] .(8.21)
For s = 1 l 5 osc this result is shown in Fig. 8.3. For s = 0 Eq. (8.21) is
identical with Eq. (8.17) while for s ≫ l osc it is 1 2 sin2 2θ which reflects
that the beam is an incoherent mixture: the relative phases between
different flavor components have been averaged to zero.
No source is exactly monochromatic; usually the neutrino energies
are broadly distributed. With a point source and a normalized distribution
g(ω) one finds
∫
prob (ν e → ν µ ) = sin 2 2θ dω g(ω) sin 2 (m2 2 − m 2 1) z
. (8.22)
4ω
As an example let g(ω) such that ∆ = 2π/l osc follows a Gaussian
distribution e −(∆−∆ 0) 2 /2δ 2 /δ √ 2π of width δ and with ∆ 0 = 2π/l 0 . Then
prob (ν e → ν µ ) = 1 2 sin2 2θ [ 1 − e −δ2 z 2 /2 cos(2πz/l 0 ) ] (8.23)
which is shown in Fig. 8.4 for δ = 1
10 ∆ 0. For δ = 0 Eq. (8.23) reproduces
Eq. (8.17) while for z ≫ δ −1 it approaches 1 2 sin2 2θ.