Stars as Laboratories for Fundamental Physics - MPP Theory Group

244 Chapter 6
to understand the absolute sign of neutrino-neutrino scattering. It is
a case where identically “charged” fermions scatter by the exchange of
a vector boson. The structure of this process is analogous to Coulomb
scattering of like-charged particles which experience a repulsive force.
(The exchange of a spin-0 or spin-2 boson leads to an attractive force.)
Thus a neutrino of given momentum in a region of space filled with
other neutrinos will have a positive potential energy V in addition to
its kinetic energy. The correct absolute sign was first pointed out by
Langacker, Leveille, and Sheiman (1983).
The deviation from relativistic propagation described by Eq. (6.107)
can be expressed as an effective refractive index which includes the
vacuum neutrino mass,
n refr =
[ (
1 − V ω
) 2
− m2
ω 2 ] 1/2
→ 1 − V ω − m2
2ω 2 (6.111)
(relativistic limit). Therefore, the effect of a medium can be expressed
as an effective mass
m 2 eff = m 2 + 2ωV. (6.112)
A numerical comparison with the vacuum mass is achieved by
( )
( √ ) 1/2 ( ) 1/2 ρ ω 1/2
2ω 2GF n B = 3.91×10 −4 eV
.
g cm −3 MeV
(6.113)
Of course, the term “effective mass” is a misnomer because m eff depends
on the energy ω, and m 2 eff can be negative depending on the vacuum
mass, the medium composition, the flavor of the neutrino, and whether
it is ν or ν.
The phase velocity v phase = ω/k = nrefr −1 can be larger or less than
the speed of light, depending on those parameters. However, the group
velocity v group = dω/dk remains at its vacuum value (1 + m 2 /k 2 ) −1/2
for a given momentum k because ω is shifted by a constant amount V ,
independently of k.
Because normal media contain about equal numbers of protons and
neutrons Y n ≈ Y e ≈ 1 and so V 2 ν e
≈ 4√ 1 2GF n B ≈ −V νµ,τ . Therefore, ν e
and ν µ,τ are shifted by almost exactly opposite amounts. An exception
is the proton-rich material of normal stars which initially contain about
75% hydrogen. Another exception is the neutron-rich matter of neutron
stars where even V νe is negative.