Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
At time t = tr, the resonance occurs and mixes the matter states, the new matter
states are therefore:
′ ν m1
ν ′
a1 a2
=
m2 −a∗ 2 a∗
νm1 , (B.3)
1 νm2
The norm of the total states must remain constant therefore we have the following
condition on a1 and a2:
|a1| 2 + |a2| 2 = 1 (B.4)
One can easily interpret a1 and a2: their squared moduli represent the probability
for the matter states to remain as they were the before resonance Prem = |a1| 2 , and
the probability for the matter states to be interchanged Pint = |a2| 2 respectively.
Just after the resonance (t = tf), the state is:
|ν(t + r )〉 = cosθm,i e −i t + r
t i Em 1 (t)dt |ν ′ m1 〉 + sin θm,i e −i t + r
t i Em 2 (t)dt |ν ′ m2 〉 (B.5)
Here we can notice that we made an approximation by considering an instantaneous
mixing between the matter states : the matter states do not evolve temporally
during the resonance. Rewriting Eq.(B.5) with the initial matter states,
we obtain:
|ν(t + r )〉 = A1(ti)|νm1〉 + A2(ti)|νm2〉 (B.6)
where
A1(ti) = a1 cosθm,i e −i t + r
t i Em 1 (t)dt − a ∗ 2 sin θm,i e −i t + r
t i Em 2 (t)dt
A2(ti) = a2 cosθm,i e −i t + r
t i Em 1 (t)dt + a ∗ 1 sin θm,i e −i t + r
t i Em 2 (t)dt
(B.7)
Eventually, considering an adiabatic evolution after the resonance occured 1 , the
final state where the neutrino exits the Sun at time t = tf will be:
|ν(tf)〉 = A1(ti) e −i t f
t + r
Em 1 (t)dt
|νm1〉 + A2(ti) e −i t f
t + r
Em 2 (t)dt
In vacuum the matter basis obviously coincides with the mass basis:
therefore the νe state can be written as:
|νm2〉 (B.8)
|ν1m〉 = |ν1〉 and |ν2m〉 = |ν2〉, (B.9)
|νe〉 = cosθV |νm1〉 + sin θV |νm2〉 (B.10)
1 This is the same approximation than before the resonance, it only means that when the
difference between the matter state eigenvalues is not minimal it dominates over the off-diagonal
terms which implies that the eigenstates just pick up a phase when propagating in the matter.
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