Stars as Laboratories for Fundamental Physics - MPP Theory Group

328 Chapter 9
9.3.6 Small Mixing Angle
In practice the mixing angle is usually small, allowing for substantial
further simplifications. In this limit the approach to flavor equilibrium
is much slower than that to kinetic equilibrium for each flavor separately
(ν x is taken to be one of the active flavors ν µ or ν τ ). Therefore, each
flavor is characterized by a Fermi-Dirac distribution so that it is enough
to specify the total number density n νx rather than the occupation
numbers of individual modes. Integrating Eq. (9.37) over all modes,
using detailed balance to lowest order in s 2 p, and with t p = s p one finds
for the evolution of the ν x number density
∫
ṅ νx = 1 dp dp ′ W
4
P P ′[
(gx s p − g e s p ′) 2 fp(1 e − fp x ′)
− (g e s p − g x s p ′) 2 f x p(1 − f e p ′)] , (9.41)
and a similar equation for ṅ νe . Together with the condition of β equilibrium,
µ n − µ p = µ e − µ νe , that of charge neutrality, n p = n e , and the
conservation of the trapped lepton number, d(n e + n νe + n νx )/dt = 0,
these equations represent differential equations for the chemical potentials
µ νx (t) and µ νe (t) if the temperature is fixed.
9.3.7 Flavor Conversion by Neutral Currents
Next I turn to the conceptually interesting question whether flavor
conversion (or the damping of neutrino oscillations) is possible by NC
collisions alone. Considering only standard flavors the matrix of coupling
constants G is then proportional to the unit matrix. In this case
Stodolsky’s damping formula in the form Eq. (9.29) gives ˙ρ p,coll = 0.
This formula applies in the limit when the neutrino energies do not
change in collisions (a medium of “heavy” fermions). If one lifts this
restriction the situation is more complicated, but it simplifies again
for weak damping and a small mixing angle. Then one may apply
Eq. (9.41) with g e = g x = 1,
∫
ṅ νx = 1 dp dp ′ W
4
P P ′ (s p − s p ′) 2[ fp(1 e − fp x ′) − f p(1 x − fp e ′)] .(9.42)
If in a collision |p| = |p ′ | and thus s p = s p ′ one recovers the previous
result ṅ νx = 0. However, if the mixing angle is a function of the neutrino
momentum, NC collisions do lead to flavor conversion and thus to the
damping of oscillations.
Of course, if only true NC interactions existed, the mixing angle
in the medium would be fixed at its vacuum value and so no flavor