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

196 13. Neutrino mixing
between the states of νj and νk issubjecttoanumberofconditions,the
localization condition and the condition of overlapping of the wave packets
of νj and νk at the detection point being the most important (see, e.g.,
[33,35,37]) .
For the νl → νl ′ and ¯ν l → ¯ν l ′ oscillation probabilities we get from
Eq. (13.8), Eq. (13.9),andEq.(13.11):
P (νl → νl ′)= �
R
j
l′ l
jj +2� |R
j>k
l′ l
jk | cos(Δm2 jk
2p L − φl′ l
jk ), (13.13)
P (¯ν l → ¯ν l ′)= �
R
j
l′ l
jj +2� |R
j>k
l′ l
jk | cos(Δm2 jk
2p L + φl′ l
jk ), (13.14)
where l, l ′ = e, μ, τ, Rl′ l
jk = Ul ′ j U ∗ lj Ulk U ∗ l ′ k and φl′ l
jk =arg
�
Rl′ �
l
jk .
It follows from Eq. (13.8) - Eq. (13.10) that in order for neutrino
oscillations to occur, at least two neutrinos νj should not be degenerate
in mass and lepton mixing should take place, U �= 1. The oscillations
effects can be large if at least for one Δm2 jk we have |Δm2 jk |L/(2p) =
2πL/Lv jk � 1, i.e. the oscillation length Lv jk is of the order of, or smaller,
than source-detector distance L (otherwise the oscillations will not have
time to develop before neutrinos reach the detector).
We see from Eq. (13.13) and Eq. (13.14) that P (νl → νl ′)=P (¯ν l ′ → ¯ν l).
This is a consequence of CPT invariance. The conditions of CP invariance
read [30,42,43]: P (νl → νl ′)=P (¯ν l → ¯ν l ′), l, l ′ = e, μ, τ. In the case of
CPT invariance, which we will assume to hold, we get for the survival
probabilities: P (νl → νl)=P (¯ν l → ¯ν l), l, l ′ = e, μ, τ. Thus, the study
of the “disappearance” of νl and ¯ν l, caused by oscillations in vacuum,
cannot be used to test the CP invariance in the lepton sector. It follows
from Eq. (13.13) - Eq. (13.14) that we can have CP violation effects in
neutrino oscillations only if U is not real. Eq. (13.2) and Eq. (13.13) -
Eq. (13.14) imply that P (νl → νl ′)andP (¯ν l → ¯ν l ′) do not depend on
the Majorana phases in the neutrino mixing matrix U [30]. Thus, i) in
the case of oscillations in vacuum, only the Dirac phase(s) in U can cause
CP violating effects leading to P (νl → νl ′) �= P (¯ν l → ¯ν l ′), l �= l ′ ,and
ii) the experiments investigating the νl → νl ′ and ¯ν l → ¯ν l ′ oscillations
cannot provide information on the nature - Dirac or Majorana, of massive
neutrinos [30,44].
As a measure of CP violation in neutrino oscillations we can consider
the asymmetry: A (l′ l)
CP = P (νl → νl ′) − P (¯ν l → ¯ν l ′)=−A (ll′ )
CP . In the case
of 3-neutrino mixing one has [45]: A (μe)
CP = −A(τe)
CP = A(τμ)
CP ,
A (μe)
CP =4J �
CP sin Δm232 2p L +sinΔm2 21
2p L +sinΔm2 13
2p L
�
, (13.18)
�
where JCP =Im Uμ3U ∗ e3Ue2U ∗ �
μ2 is analogous to the rephasing invariant
associated with the CP violation in the quark mixing [46]. Thus, JCP controls the magnitude of CP violation effects in neutrino oscillations in
the case of 3-neutrino mixing. Even if JCP �= 0, we will have A (l′ l)
CP =0
unless all three sin(Δm2 ij /(2p))L �= 0inEq.(13.18).
Consider next neutrino oscillations in the case of one neutrino
mass squared difference “dominance”: suppose that |Δm2 j1 |≪|Δm2n1 |,
j =2, ..., (n − 1), |Δm2 n1 | L/(2p) � 1and|Δm2 j1 | L/(2p) ≪ 1, so that