Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
the adiabaticity parameter is consequently:
γ = |Em1 − Em2|
|2 ˙ θm(t)|
(B.19)
A similar problem is actually well known in atomic physics. In such a context,
the hopping probability has been calculated by Landau and Zener in 1932 and in
their approximation one obtains:
π
−
Phop ≃ e 2 γr (B.20)
where γr, the adiabaticity evaluated at the resonance point, is:
γr = sin2 2θV ∆m
cos 2θV
2
d lnNe(t) −1
2E dt
B.3 The Landau-Zener formula
t=tr
(B.21)
Considering the crossing of a polar and homopolar state of a molecule Landau
and Zener, in 1932, independently derived, with some approximations, an analytical
formula for non-adiabatic crossing probability. About some 50 years later
people realised the importance of this formula in neutrino physics. We follow
an approach similar to Zener derivation to demonstrate the formula, adapted for
neutrinos. Let us rewrite the effective matter Hamiltonian in the flavour basis:
Hfl =
=
−∆m2 4E cos 2θV + √ 2
2 GFNe
∆m2 sin 2θV 4E
H1 H12
H12 H2
∆m 2
4E
sin 2θV
∆m 2
4E cos 2θV − √ 2
2 GFNe
(B.22)
To derive such formula we consider that the region of transition we are interested
in is so small that H1 − H2 = √ 2GF Ne = −α t can be approximated to a linear
function. This assumption is the same that making a Taylor development to the
first order of Ne(t) at the time of resonance t = tr:
Ne(t) ≃ Ne(tr) + dNe(t)
|t=tr(t − tr) (B.23)
dt
In its paper Zener was considering another approximation ˙ H12 = 0, but in our
case, this is true without any approximation since θV is constant. Considering a
neutrino state |ν(t)〉 = Ce(t)|νe〉 + Cµ(t)|νµ〉, the evolution equation we have to
solve is:
i d
dt
Ce
Cµ
=
− ∆m2
4E cos 2θV + √ 2
2 GFNe
∆m2 sin 2θV 4E
163
∆m 2
4E
sin 2θV
∆m 2
4E cos 2θV − √ 2
2 GFNe
Ce
Cµ
(B.24)
.