Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
where θV is the vacuum mixing angle and U the lepton mixing matrix, which is
a rotation matrix of angle θV . Using Eq.(1.5), the transition probability can be
written as:
P(νe → νµ; t) = |Uµ1 e −iE1t U ∗ e1 + Uµ2 e −iE2t U ∗ e2 | 2 ,
= cos 2 θV sin 2 θV |e i(E 2 −E 1 )
2 t − e −i (E 2 −E 1 )
2 t | 2 ,
= sin 2 2θV sin 2
(E2 − E1)
t . (1.7)
2
Since we are considering relativistic neutrinos of momentum p the following approximation
can be used 3 :
Ei =
p2 + m2 i ≃ p + m2i 2p ≃ p + m2i , (1.8)
2E
and therefore defining ∆m2 = m2 2 − m2 1, we have E2 − E1 = ∆m2
2E .
Finally, the transition probabilities are
P(νe → νµ; t) = P(νµ → νe; t) = sin 2 2θV sin 2
2 ∆m
4E t
. (1.9)
Note here that the T-symmetry is conserved since the probabilities for the two
processes νe → νµ and νµ → νe are equal. Since the Hamiltonian is hermitian,
and the wave functions normalized to 1, one has:
P(νe → νµ; t) + P(νe → νe; t) = 1 (1.10)
Physically, this means that in two flavors the electron neutrino can only give either
an electron neutrino or a muon neutrino. Therefore, the survival probabilities are
P(νe → νe; t) = P(νµ → νµ; t) = 1 − sin 2 2θV sin 2
2 ∆m
4E t
. (1.11)
It is convenient to rewrite the transition probability in terms of the distance L
travelled by neutrinos. For relativistic neutrinos L ≃ t, and one has
P(νe → νµ; L) = sin 2 2θV sin 2
π L
, (1.12)
losc
where losc is the oscillation length defined as losc = (4πE)/∆m 2 .
It is equal to the distance between any two closest minima or maxima of the
transition probability (see fig. 1.1).
3 assuming that neutrinos are emitted with a fixed and equal momentum.
11