Stars as Laboratories for Fundamental Physics - MPP Theory Group

Neutrino Oscillations 285
two-flavor scenario. Of course, if one were to observe the appearance
of a certain flavor—rather than the disappearance of ν e as for solar
neutrinos or of ν µ in the atmospheric case—one would have to consider
the possibility that they arise from sequential transitions of the sort
ν e → ν µ → ν τ . Note that in the bottom-left entry of the approximate
expression for the CKM matrix Eq. (7.6) the term S 12 S 23 had to be
kept because it is larger than the direct term S 13 . The neutrino mixing
angles could show a similar hierarchy.
With these caveats in mind we turn to the two-flavor mixing case
where U has the 2×2 Cabbibo form Eq. (7.5),
U = cos θ I + i sin θ σ 2 , (8.12)
with the mixing angle θ, the 2×2 unit matrix I, and the Pauli matrix 47
σ 2 . The mass matrix may be written in the form
M 2 /2ω = b 0 − 1 B · σ, (8.13)
2
where b 0 = (m 2 1 + m 2 2)/4ω. In the weak-interaction basis
⎛ ⎞
B = 2π sin 2θ
⎜ ⎟
⎝ 0 ⎠ , (8.14)
l osc
cos 2θ
a vector which is tilted with regard to the 3-axis by twice the mixing
angle (Fig. 8.2). Further,
l osc ≡
4π ω
(8.15)
m 2 2 − m 2 1
is the oscillation length. Its meaning will presently become clear.
In this representation it is straightforward to work out the spatial
behavior of a stationary neutrino beam. From K = ω − b 0 + 1 B · σ 2
one finds
W = e i(ω−b 0)z
[ ( ) ( πz πz
cos − i sin
l osc
) ( − cos 2θ
l osc sin 2θ
sin 2θ
cos 2θ
)]
.
(8.16)
Assuming that the oscillations are among the first two families, the
appearance probability for a ν µ and the ν e survival probability are for
an initial ν e
prob (ν e → ν µ ) = |W eµ | 2 = sin 2 (2θ) sin 2 (πz/l osc ),
prob (ν e → ν e ) = |W ee | 2 = 1 − prob (ν e → ν µ ). (8.17)
The oscillation behavior is shown in Fig. 8.1.
)
, σ 2 =
47 The Pauli matrices are σ 1 =
(
0 1
1 0
(
0 −i
i 0
) ( )
1 0
, and σ 3 = .
0 −1