Stars as Laboratories for Fundamental Physics - MPP Theory Group

332 Chapter 9
9.5 Flavor Conversion in a SN Core
9.5.1 Rate Equation
As a first application of the formalism developed in the previous sections
consider two-flavor mixing between ν e and another active neutrino
species ν x = ν µ or ν τ (vacuum mixing angle θ 0 ). In a SN core immediately
after collapse electron lepton number is trapped and so the ν e ’s
have a large chemical potential on the order of 200 MeV. The trapped
energy and lepton number diffuses out of the SN core and is radiated
away within a few seconds. Will ν x achieve equilibrium with ν e on this
time scale and thus share the large chemical potential For the µ flavor
this would also imply the production of muons by the subsequent
charged-current absorption of ν µ so that the lepton number would be
shared between e, µ, ν e , and ν µ .
If the mixing angle in the medium were not small, flavor conversion
would occur about as fast as it takes to establish β equilibrium. In
this case a detailed calculation is not necessary so that one may focus
on the limit of small mixing angles. In addition the oscillations are
fast which allows one to use Eqs. (9.41) and (9.50). Moreover, the
medium properties are assumed to be isotropic so that the production
and absorption rates P e and A e of ν e ’s depend only on their energy E.
Also, the transition rate W P P ′ for the scattering of a neutrino with four
momentum P to one with P ′ may be replaced by an angular average
which depends only on the energies E and E ′ . Altogether one finds a
rate of change for the ν x density of
∫
]
ṅ νx = 1 dp s 2 4 p[
(1 − f
x
p ) PE e − fp x A e E
∑
∫
[
dp dp ′ WEE a (g
a ′ x s p − ge a s p ′) 2 fp(1 e − fp x ′)
+ 1 4
a
− (g a e s p − g a xs p ′) 2 f x p(1 − f e p ′)] , (9.51)
where f e p and f x p are the occupation numbers of ν e and ν x which are
given by Fermi-Dirac distributions because kinetic equilibrium was assumed
for both flavors. Also, 1 4 s2 p = 1 4 sin2 2θ E = θ 2 E for small mixing
angles.
A summation over different species a of medium fermions was restored;
g a e and g a x are dimensionless effective NC coupling constants of
ν e and ν x to fermion species a. For a = n or p these constants are
the same for all active neutrino species. Electrons as scattering targets
are very relativistic so that they may be classified into a l.h. and a