Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
The amplitude of the process νe → νe is:
A(νe, ti → νe, tf) = < νe|ν(tf) > (B.11)
= A1(ti) cos θV e −i t f
t + r
Em 1 (t)dt
+ A2(ti) sin θV e −i t f
t + r
Em 2 (t)dt
and therefore one obtains for the survival probability:
P(νe, ti → νe, tf) = 1 cos 2θV
+ |A1(ti)|
2 2
2 − |A2(ti)| 2
(B.12)
tf
+ |A1(ti)A2(ti)| sin2θV cos (Em1(t) − Em2(t))dt + Ω
where Ω = arg(A1(ti)A ∗ 2 (ti)). Averaging over the final time, the probability of
detecting this neutrino is given by:
with
1 cos 2θV
< P(νe, ti → νe, tf) >tf = +
2 2
t + r
|A1(ti)| 2 − |A2(ti)| 2
(B.13)
|A1(ti)| 2 = |a1| 2 cos 2 θm,i + |a2| 2 sin 2 θm,i (B.14)
− |a1a2| sin 2θm,i cos
t − r
ti
(Em1(t) − Em2(t))dt + ω
|A2(ti)| 2 = |a2| 2 cos 2 θm,i + |a1| 2 sin 2 θm,i (B.15)
+ |a1a2| sin 2θm,i cos
t − r
ti
(Em1(t) − Em2(t))dt + ω
where ω = arg(a1a2). The last term in |A1(ti)| 2 or |A2(ti)| 2 shows that the phase
of the neutrino oscillation at the point the neutrino enters resonance can substantially
affect this probability. Since we must also average over the production
position to obtain the fully averaged probability of detecting an electron neutrino
as:
P(νe → νe) = 1
) cos 2θV cos 2θθm,i (B.16)
2 + (|a1| 2 − |a2| 2
2
As said earlier |a1| 2 + |a2| 2 = 1 and |a2| 2 represent the probability to go from
νm1 to νm2 at resonance, i.e the hopping probability between matter eigenstates
written Phop. Finally, we have:
P(νe → νe) = 1 1
+
2 2 (1 − 2Phop) cos 2θV cos 2θm,i. (B.17)
Logically, in the case of very small mixing angles 2 one has:
P(νe → νe) = Phop. (B.18)
2 From Eq.(1.50), one can see this is the case for θm since the matter is very dense in the
Sun where the electron neutrino is created. We now know that θV is not very small, but we
take it so for illustrative purpose.
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