Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
Indeed, initially we consider a time where the temperature is between 100 MeV
and 10 MeV, and neutrinos are in thermal and chemical equilibrium. The equations
of motion for the density matrices in an expanding universe are :
UM
i(∂t−Hp∂p)ρ(p, t) =
2U †
2p − 8√2GFp 3m2
E +
W
√
2GF(ρ − ρ), ρ(p, t) +C[ρ(p, t)],
(8.19)
where H = ˙a(t)/a(t) is the Hubble parameter (a(t) the expansion parameter),
GF is the Fermi constant and mW the W boson mass. On the l.h.s. of Eq.(8.19),
we substituted ∂t → ∂t − Hp∂p with H the cosmic expansion parameter because
the Universe is expanding.
The vacuum term
The first term in the commutator [·, ·] UM2U †
is the vacuum oscillation term where
2p
M2 = diag(m2 1 , m22 , m23 ) and U the unitary Maki-Nakagawa-Sakata-Pontecorvo
matrix as described in appendix A.
The matter term
The second term in the commutator
− 8√ 2GFp
3m 2 W
E ≡ − 8√2GFp 3m2 (〈Eℓ−〉nℓ− + 〈Eℓ +〉nℓ +) (8.20)
W
represents the energy densities of charged leptons and corresponds to the refractive
effects of the medium that neutrinos experience. Actually, this term is due
to the presence of thermally populated charged leptons in the plasma, which induces
a thermal potential from finite-temperature modification of the neutrino
mass [108, 91]. E is the 3 × 3 flavor matrix of charged-lepton energy densities:
where
⎛
E = ⎝
Eαα = gα
2π 2
Eee + Eµµ 0 0
0 Eµµ 0
0 0 0
∞
dpp 2
0
E
⎞
1 + exp E
T
⎠ . (8.21)
(8.22)
with E = p2 + mµ ±2 and gα is the number of spin states of the species, for
electrons for instance ge = 2×2 = 4, since the spin can be + 1 or −1 and we have
2 2
to count the particle and its anti-particle, here the positron. For electrons and
positrons, since T, p >> me ± we consider that those particles are ultra-relativistic
then E ≃ p. Therefore, we have:
4 2T
Eee =
π2 ∞
u
du
3
4
= −2T
1 + exp u π2 Li4(−1)Γ(4) (8.23)
0
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