Stars as Laboratories for Fundamental Physics - MPP Theory Group

294 Chapter 8
The dispersion relation for a single flavor in a medium was given
by Eq. (6.107). For three flavors which mix according to Eq. (8.3) the
Klein-Gordon equation in Fourier space is
⎧⎡
⎛
⎞⎤2
⎪⎨
⎢
⎣ω − G 3Y e − 1 0 0
√
Fn B ⎜
⎟⎥
⎝ 0 Y e − 1 0 ⎠⎦
⎪⎩ 2
0 0 Y e − 1
⎛
m 2 ⎞ ⎫ ⎛
−k 2 1 0 0 ⎪⎬
⎜
− U ⎝ 0 m 2 ⎟
2 0 ⎠ U † ⎜
⎝
0 0 m 2 ⎪ ⎭
3
⎞
ν e
⎟
ν µ
ν τ
⎠ = 0, (8.26)
where a possible neutrino background was ignored and Y n = 1 − Y p =
1−Y e was used. Also, the higher-order difference between ν µ and ν τ was
ignored. This equation has nonzero solutions only if det{. . .} vanishes,
a condition that gives us the dispersion relation for the three normal
modes. For unmixed neutrinos where U is the unit matrix one recovers
Eq. (6.107) for each flavor.
For all practical cases the neutrinos are highly relativistic so that
one may linearize this equation,
(ω − k − M 2 eff/2k) Ψ = 0 (8.27)
where it is easy to read the matrix M 2 eff from Eq. (8.26). Because to
lowest order ω = k one may equally use M 2 eff/2ω in order to derive the
dispersion relation, depending on whether one wishes to write ω as a
function of k or vice versa.
For two-flavor mixing between ν e and ν µ or ν τ (mixing angle θ 0 ) the
effective mass matrix may be written in the same form as in vacuum
Meff/2ω 2 = b 0 − 1 B · σ, (8.28)
2
where b 0 = (m 2 1 + m 2 2)/4k + √ 2 G F n B (Y e − 1 2 ) and
⎛ ⎞
B = 2π sin 2θ
⎜
⎝ 0
l osc
cos 2θ
⎟
⎠ = m2 2 − m 2 1
2ω
⎛
⎜
⎝
⎞
sin 2θ 0
⎟
0
cos 2θ 0
⎛ ⎞
⎠ − √ 0
⎜ ⎟
2 G F n e ⎝ 0 ⎠ .
1
(8.29)
This equation defines implicitly the mixing angle θ as well as the oscillation
length l osc in the medium in terms of the masses, the vacuum
mixing angle θ 0 , and the electron density n e .