Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
matter states are the instantaneous matter eigenstates which allow a instantaneous
diagonalization of the effective Hamiltonian written in the flavour basis
Hfv as in Eq.(1.44):
Um(t) † Hfv(t)Um(t) = Hd(t) = diag(Em1(t), Em2(t)) . (1.46)
with Em1(t) and Em2(t) are instantaneous eigenvalues of Hfv(t). The evolution
equation in the basis of the instantaneous eigenstates can therefore be written as
i(d/dt)ν = [Hd − iU † m (dUm/dt)]ν, or
i d
νm1
dt νm2
=
Em1(t) −i ˙ θm(t)
i ˙ θm(t) Em2(t)
νm1
νm2
, (1.47)
where ˙ θm ≡ dθm/dt. Notice that the effective matter Hamiltonian in this basis is
not diagonal since the mixing angle θm(t) is not constant, i.e. the matter eigenstate
basis changes with time. To obtain the oscillation probability equations
with the same simple form as in vacuum, we can study the case where the matter
density (and chemical composition) is taken constant (i.e Ne = const). Therefore,
we obtain a diagonal effective Hamiltonian for the matter eigenstates since
dUm/dt = 0 just like the vacuum Hamiltonian is diagonal in the mass basis. To
derive an explicit oscillation probability equation one has to express the matter
mixing angle as a function of density and the vacuum mixing angle, by linking
the matter basis to the flavour basis. Starting from Eq.(1.47) with no off-diagonal
terms, and rotating in the favour basis, one obtains, after removing the diagonal
(Em1(t) + Em2(t))/2:
i d
νe
dt νµ
= (Em1(t) − Em2(t))
2
cos 2θm(t) sin 2θm(t)
sin 2θm(t) − cos 2θm(t)
νe
νµ
. (1.48)
By comparing this form of the effective Hamiltonian in the flavour basis to the
first one given in Eq.(1.44), one obtains the two following relations:
sin 2θm =
sin 2θV
Em1(t) − Em2(t)
∆m 2
2E
(1.49)
cos 2θm = −∆m2
2E cos 2θV + √ 2GF Ne
. (1.50)
Em1(t) − Em2(t)
Em1(t) and Em2(t) are easily found by diagonalizing the effective Hamiltonian of
Eq.(1.44) and their difference is :
Em1(t) − Em2(t) =
∆m 2
2E cos 2θV − √ 2GF Ne
21
2
∆m2 2
+ sin
2E
2 2θV (1.51)