Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
case, and see the consequence of the hierarchy. We then show the influence of the
matter on the pendulum behaviour. Finally, we study the system when a more
realistic case is taken, namely with varying neutrino density and with initial
neutrino/anti-neutrino asymmetry. We follow the analytical derivations given in
[72].
The pendulum model
We consider here the simplest bipolar system, where initially equal densities of
pure νe and ¯νe. In addition, we take all neutrinos to have the same energy, so
every neutrino behave in the same way. We take the exact same notations that
the ones used in the previous section.
∂tP = +ωB + µ P − ¯ P × P ,
∂t ¯ P = −ωB + µ P − ¯ P × ¯ P, (5.27)
where ¯ P corresponds to the anti-neutrino polarization vector. We then define
two new variables D and S from P and ¯ P:
and
they are the solution of the new equations of motion:
D = P − ¯ P (5.28)
S = P + ¯ P. (5.29)
˙S = ωB × D + µD × S ,
˙D = ωB × S . (5.30)
In order to have simpler E.O.M, instead of using S we use
which finally yield:
Q = S − ω
µ B . (5.31)
˙Q = µD × Q ,
˙D = ωB × Q . (5.32)
since ˙ S = ˙ Q and B × Q = B × S.
Multiplying the E.O.M for Q in Eqs.(5.32), one can see that the squared
modulus of Q is constant and therefore the length of Q is conserved. Using the
initial conditions, namely P(0) = ¯ P(0) = (0, 0, 1), we have5 :
Q = |Q| =
4 +
2 ω
+ 4
µ
ω
µ cos 2θV
1/2
5 We have used |B| 2 = 1, and the initial values |S| 2 = 4 and B · S = −2 cos2θ0.
99
, (5.33)