Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
Appendix B
The adiabaticity notion
In general, for oscillations in a matter of an arbitrary non-uniform density, the
evolution equation (1.44) does not allow an analytic solution and has to be solved
numerically. However, there is an important particular case in which one can get
an illuminating approximate analytic solution. An adiabatic evolution is the case
of a slowly varying matter density. Mathematically, it means that the difference
between the off-diagonal terms in Eq.(1.47) is very small compared with the difference
between the eigenvalues of the matter Hamiltonian. Physically, it means
that the transitions between the instantaneous matter eigenstates are suppressed.
In astrophysical objects like supernovae (without considering shockwaves) or the
Sun, neutrinos are produced at high density and propagate through the star
seeing a monotonically decreasing density. Our goal here is to derive general
oscillation probability equations which include the possible transitions between
the matter eigenstates νm1 and νm2 due to the violation of the adiabaticity.
B.1 An analytic approximate formula
The idea here is to calculate the average probability for a νe created nigh the
Sun’s core to exit in vacuum as a νe. We first focus on the Sun as a dense matter
environment, since such resonance phenomenon first found its application in the
solar neutrino problem. Defining |ν(t)〉 as the quantum state of the considered
neutrino, at creation we have:
|ν(t = ti)〉 = |νe〉 = cosθm,i|νm1〉 + sin θm,i|νm2〉 (B.1)
Before reaching the resonance at time t = t − r , one can consider an adiabatic
evolution, the matter states just pick up a phase (equal to the integral over the
traveled distance of the respective eigen-energies),
|ν(t − r )〉 = cosθm,i e −i t − r
t i Em 1 (t)dt |νm1〉 + sin θm,i e −i t − r
t i Em 2 (t)dt |νm2〉 (B.2)
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