Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
A.3 The Majorana case
We have discussed neutrino oscillations in the case of Dirac neutrinos. What
happens if neutrinos have a Majorana mass term rather than the Dirac one? Eq.
(A.2) now has to be modified: the term (mD)ij ν f
+ h.c. has to be replaced
by (mM)ij νc f f
iL νjR + h.c. = (mM)ij ν f
iL
T
iLνf jR
C ν f
jR + h.c.. This mass term breaks
not only the individual lepton flavours but also the total lepton number. The
symmetric Majorana mass matrix (mM)ij is diagonalized by the transformation
U T L mM UL = (mM)diag, so one can again use the field transformations (A.3).
Therefore the structure of the charged current interactions is the same as in
the case of the Dirac neutrinos, and the diagonalization of the neutrino mass
matrix in the case of the N fermion generations again gives N mass eigenstates.
Thus the oscillation probabilities in the case of the Majorana mass term are the
same as in the case of the Dirac mass term. This, in particular, means that one
cannot distinguish between Dirac and Majorana neutrinos by studying neutrino
oscillations. Essentially this is because the total lepton number is not violated
by neutrino oscillations.
Concerning the MNSP matrix in the Majorana case, there is less freedom to
rephase the fields because of the form of the Majorana mass terms and so the
phases of neutrino fields cannot be absorbed. Therefore only N phases can be
removed, leaving N(N + 1)/2 − N = N(N − 1)/2 physical phases. Out of these
phases, (N − 1)(N − 2)/2 are the usual, Dirac-type phases while the remaining
N − 1 are specific for the Majorana case, so called Majorana phases. The MNSP
UM matrix in the Majorana case then becomes
UM = U ∗ D = U ∗ diag(e −iϕ1 , 1, e −iϕ2 ) (A.23)
Since Majorana phases do not lead to any observable effects for neutrino oscillations,
we shall not consider them here.
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