Stars as Laboratories for Fundamental Physics - MPP Theory Group

Particle Dispersion and Decays in Media 247
ing electric neutrality it is (Botella, Lim, and Marciano 1987; see also
Semikoz 1992 and Horvat 1993)
V ντ − V νµ = 3G2 Fm 2 [ ( )
τ m
2
n
2π 2 B ln W
− 1 + Y ]
n
, (6.116)
3
with a sign change for V ντ − V νµ . The shift of the “effective mass” m 2 eff
is numerically
m 2 τ
2ω(V ντ − V νµ ) = ( 2.06×10 −6 eV ) 2
ρ
g cm −3
ω
MeV
6.61 + Y n /3
,
7
(6.117)
much smaller than the corresponding difference between ν e and ν µ,τ .
6.7.3 The Sun a Neutrino Lens
The most important consequence of neutrino refraction in media is its
impact on neutrino oscillations because different flavors experience a
different index of refraction. In optics, the most notable consequence
of refraction is the possibility to deflect light and thus to use lenses and
other optical instruments. In principle, the same is also possible for
neutrinos. The Sun, for example, could act as a gigantic neutrino lens.
One may easily calculate the deflection caused by a given body. If
s is a unit vector along the direction of a propagating wave, and if s
is a coordinate along the beam, the deflection is given by (Sommerfeld
1958)
|ds/ds| = n −1
refr |s × ∇n refr|, (6.118)
where n refr is the refractive index. One may equally write
|dα/ds| = n −1
refr |∇ ⊥n refr |, (6.119)
where α is the angle relative to the local tangential vector, i.e. dα/ds
is the local curvature of the beam, and ∇ ⊥ is the transverse gradient.
The total angle of deflection is
|∆α| =
∫ +∞
−∞
ds |∇ ⊥ n refr | (6.120)
if the curvature is small which is the case for |n refr − 1| ≪ 1.
If the beam hits a spherically symmetric body (radius R) at an
impact parameter b < R its angle against the radial direction at a
radius r from the center is sin β = b/r so that ∇ ⊥ n refr = (b/r) ∂ r n refr .