Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
We can express the second vector of the r.h.s of Eq.(D.26) as a vector product:
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
˙αPvu 0u Puu
⎝ − ˙αPuv ⎠ = − ˙α ⎝ 0v ⎠ × ⎝ Pvv ⎠ . (D.27)
0w 1w Pww
The adiabaticy notion within the polarization vector formalism
In this subsection, we come back on the adiabaticity notion: It is also possible
to measure adiabaticity like it was previously done when we studied the MSW
resonance, by defining the adiabaticity parameter has the ratio of the off diagonal
term of the instantly diagonalized Hamiltonian over the diagonal terms of the
same Hamiltonian. Recalling that the off diagonal terms of a matrix correspond
to the transverse term in the polarization vector formalism, one can define the
medium mixing angle by 5
cosθω ≡ Hω⊥/Hω
(D.28)
Thus, the speed for the Hω evolution in the co-rotating plane is dθω/dt, while Pω
precesses with speed Hω. The evolution is adiabatic if the adiabaticity parameter
γω ≡ |dθω/dt| H−1 ω ≪ 1.
Let us now look to the link between the adiabaticity notion and the consequences
on the relevant variables. If we insert Eq.(5.60), in the E.O.M. given by Eq.(5.59)
we obtain
∂tHω(µ) = ∂t(µD) = 0 , (D.29)
which is satisfied exactly if ∂tµ = 0 and ∂tD = 0. That is, the medium has
constant neutrino density and the difference vector does not change. The fact
that µ should be independent of time is coherent with the previous assumption
which considered a µ changing slowly with time. The same reasoning applies
for the second condition which says that D should be independent of time. In
the adiabatic approximation we supposed a slow variation for Hω and a fortiori
for the Pω which follow Hω. We see therefore a perfect coherence between the
different assumptions taken and the solutions yielded by the E.O.M. with such
assumptions.
5 To obtain such formula we considered that the flavour basis almost coincide with the
mass basis by taking a small vacuum mixing angle (which is quite correct for θ13), and since
the in-medium mixing angle near a SN core is small, the mass basis almost coincide with
the interaction/medium basis. Consequently, we considered that the flavour basis and the
interaction basis almost coincide too. That is why we can define the in-medium mixing angle
with Hω written in the flavour basis.
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