Particle Physics Booklet - Particle Data Group - Lawrence Berkeley ...

194 13. Neutrino mixing
(L(¯νj) =− 1). The neutrinos νj can be Majorana particles if no lepton
charge is conserved (see, e.g., Ref. 29). A massive Majorana particle χj is
identical with its antiparticle ¯χj: χj ≡ ¯χj. On the basis of the existing
neutrino data it is impossible to determine whether the massive neutrinos
are Dirac or Majorana fermions.
In the case of n neutrino flavours and n massive neutrinos, the n × n
unitary neutrino mixing matrix U can be parametrised by n(n − 1)/2
Euler angles and n(n +1)/2phases. If the massive neutrinos νj are Dirac
particles, only (n − 1)(n − 2)/2 phases are physical and can be responsible
for CP violation in the lepton sector. In this respect the neutrino mixing
with Dirac massive neutrinos is similar to the quark mixing. For n =3
there is one CP violating phase in U, “the Dirac CP violating phase.” CP
invariance holds if U is real, U ∗ = U.
If, however, the massive neutrinos are Majorana fermions, νj ≡ χj, the
neutrino mixing matrix U contains n(n − 1)/2 CP violation phases [30,31],
i.e., by(n−1) phases more than in the Dirac neutrino case: in contrast to
Dirac fields, the massive Majorana neutrino fields cannot “absorb” phases.
In this case U can be cast in the form [30]
U = VP (13.2)
where the matrix V contains the (n − 1)(n − 2)/2 Dirac CP violation
phases, while P is a diagonal matrix with the additional (n − 1) Majorana
CP violation phases α21, α31,..., αn1,
�
P = diag 1,e i α21 2 ,e i α31 2 , ..., e i α �
n1
2 . (13.3)
The Majorana phases will conserve CP if [32] αj1 = πqj, qj =0, 1, 2,
j =2, 3, ..., n. Inthiscaseexp[i(αj1−αk1)] = ±1 is the relative CP-parity
of Majorana neutrinos χj and χk. The condition of CP invariance of the
leptonic CC weak interaction reads [29]:
U ∗ lj = Ulj ρj , ρj = − iηCP(χj) =±1 , (13.4)
where ηCP(χj) is the CP parity of the Majorana neutrino χj [32].
In the case of n = 3 there are 3 CP violation phases - one Dirac and
two Majorana. Even in the mixing involving only 2 massive Majorana
neutrinos there is one physical CP violation Majorana phase.
II. Neutrino oscillations in vacuum. Neutrino oscillations are a
quantum mechanical consequence of the existence of nonzero neutrino
masses and neutrino (lepton) mixing, Eq. (13.1), and of the relatively
small splitting between the neutrino masses.
Suppose the flavour neutrino νl is produced in a CC weak interaction
process and is observed by a neutrino detector capable of detecting also
neutrinos νl ′, l ′ �= l. If lepton mixing, Eq. (13.1), takes place and the
masses mj of all neutrinos νj are sufficiently small, the state of the
neutrino νl, |νl〉, will be a coherent superposition of the states |νj〉 of
neutrinos νj:
|νl〉 = �
U ∗ lj |νj; ˜pj〉, l = e, μ, τ , (13.5)
j
˜pj being the 4-momentum of νj. For the state vector of the flavour
antineutrino ¯ν l, produced in a weak interaction process, we get:
|¯ν l〉 = �
Ulj |¯νj; ˜pj〉 ,l = e, μ, τ . (13.7)
j
Note the presence of U in Eq. (13.5) and U ∗ in Eq. (13.7).