Heavy sterile neutrinos - MPP Theory Group

566 A.D. Dolgov et al. / Nuclear Physics B 590 (2000) 562–574
and measure all masses in units of m 0 . In terms of these variables the Boltzmann equations
describing the evolution of the sterile neutrino, ν s , and the active neutrinos, ν a , can be
written as
where
(
ν
∂ x f νs (x, y) = s
− f νs ) 1.48 x 10.75
τ νs /s (T x) 2 g ∗ (T )
[
Ms
× + 3 × 27 T 3 ( 3ζ(3)
E νs Ms
3 + 7π 4 ( ))]
Eνs T
4 144 Ms
2 + p2 ν s
T
3E νs Ms
2 , (10)
∂ x δf νa = D a (x,y,M s ) + S a (x, y), (11)
(f
eq
) 1/2
f eq
ν s
= ( e E/T + 1 ) −1 , δfνa = f νa − ( e y + 1 ) −1 , Eνs =
√
M 2 s + (y/x)2
and p νs = y/x are the energy and momentum of ν s .Further,g ∗ (T ) = ρ tot /(π 2 Tγ 4/30)
is the effective number of massless species in the plasma determined as the ratio of
the total energy density to the equilibrium energy density of one bosonic species with
temperature T γ .
The scattering term in Eq. (11) comes from interactions of active neutrinos between
themselves and from their interaction with electrons and positrons. For ν e it has the form
( ) 10.75 1/2
(
S νe (x, y) = 0.26
1 + g
2
g L + g 2 )(
R y/x
4 )
∗
{
× −δf νe + 2 e −y [ (
1 + 0.75 g
2
15 1 + gL 2 + L + gR)]
2
g2 R
[ ∫
× dy 2 y2 3 δf ν e
(x, y 2 ) + 1 ∫
dy 2 y 3 (
2 δfνµ (x, y 2 ) + δf ντ (x, y 2 ) )]
8
+ 3 5 (T x − 1) gL 2 + }
g2 R
1 + gL 2 + e −y (11y/12 − 1) , (12)
g2 R
where g L = sin 2 θ W + 1/2, while g R = sin 2 θ W . The corresponding term for ν µ and ν τ is
given by Eq. (12) with the exchange g L →˜g L = sin 2 θ W − 1/2.
The decay term in Eq. (11) comes from the decay of heavy sterile neutrino according to
reactions Eq. (2). This term depends on the mixing channel. For ν τ ↔ ν s mixings we have
D νe ,ν µ
(x, y) = 467
M 3 s τ ν s
x 2 ( 10.75
g ∗
) 1/2 (
1 − 16y
9M s x
D ντ (x, y) = 935
M 3 s τ ν s
x 2 ( 10.75
g ∗
) 1/2 [
1 − 16y
9M s x + 2 3
) (nνs
− n eq
ν s
)
θ(Ms x/2 − y), (13)
(
)(
1 +˜g L 2 + g2 R 1 − 4y )]
3M s x
× ( n νs − n eq
ν s
)
θ(Ms x/2 − y), (14)
where n νs (x) is the number density of ν s and θ(y) is the step function which ensures
energy conservation in the decay. For ν µ ↔ ν s mixing these terms come from Eq. (13) and
Eq. (14) by exchange of ν µ ↔ ν τ .
For ν e ↔ ν s mixing those terms are