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

13. Neutrino mixing 197
exp[i(Δm2 j1 L/(2p)] ∼ = 1, j =2, ..., (n − 1). Under these conditions we
obtain from Eq. (13.13) and Eq. (13.14), keeping only the oscillating terms
involving Δm2 n1 : P (νl(l ′ ) → νl ′ (l) ) ∼ = P (¯ν l(l ′ ) → ¯ν l ′ (l) ),
P (νl(l ′ ) → νl ′ (l) ) ∼ �
2
= δll ′ − 4|Uln| δll ′ −|Ul ′ n | 2�
sin 2 Δm2n1 4p
L. (13.20)
It follows from the neutrino oscillation data that in the case of 3-neutrino
mixing, one of the two independent neutrino mass squared differences,
say Δm2 21 , is much smaller in absolute value than the second one, Δm231 :
|Δm2 21 |/|Δm231 | ∼ = 0.032, |Δm2 31 | ∼ = 2.4 × 10−3 eV2 .Eq.(13.20) with
n = 3, describes with a relatively good precision the oscillations of i)
reactor ¯νe ( l, l ′ = e) onadistanceL∼1km, corresponding to the
CHOOZ and the Double Chooz, Daya Bay and RENO experiments, and of
ii) the accelerator νμ (l, l ′ = μ), seen in the K2K and MINOS experiments.
The νμ → ντ oscillations, which the OPERA experiment is aiming to
detect, can be described in the case of 3-neutrino mixing by Eq. (13.20)
with n =3andl = μ, l ′ = τ.
In certain cases the dimensions of the neutrino source, ΔL, and/or
the energy resolution of the detector, ΔE, have to be included in
the analysis of the neutrino oscillation data. If [29] 2πΔL/Lv jk ≫ 1,
and/or 2π(L/Lv jk )(ΔE/E) ≫ 1,theinterferencetermsinP (νl → νl ′)and
P (¯ν l ′ → ¯ν l) will be strongly suppressed and the neutrino flavour conversion
will be determined by the average probabilities: ¯ P (νl → νl ′)= ¯ P (¯ν l →
¯ν l ′) ∼ = �
j |Ul ′ j |2 |Ulj| 2 . Suppose next that in the case of 3-neutrino mixing,
|Δm2 21 | L/(2p) ∼ 1, while |Δm2 31(32) | L/(2p) ≫ 1, and the oscillations due
to Δm2 31(32) are strongly suppressed (averaged out) due to integration
over the region of neutrino production, etc. In this case we get for the νe
and ¯νe survival probabilities: P (νe → νe) =P (¯νe → ¯νe) ≡ Pee,
Pee ∼ = |Ue3| 4 �
+ 1 −|Ue3| 2� 2 �
1 − sin 2 2θ12 sin 2 Δm221 4p L
�
(13.26)
with θ12 determined by cos2 θ12 = |Ue1| 2 /(1 −|Ue3| 2 ), sin2 θ12 =
|Ue2| 2 /(1 −|Ue3| 2 ). Eq. (13.26) describes the effects of reactor ¯νe
oscillations observed by the KamLAND experiment (L ∼ 180 km).
The data of ν-oscillations experiments is often analyzed assuming
2-neutrino mixing: |νl〉 = |ν1〉 cos θ + |ν2〉 sin θ, |νx〉 = −|ν1〉 sin θ +
|ν2〉 cos θ, where θ is the neutrino mixing angle in vacuum and νx is
another flavour neutrino or sterile (anti-) neutrino, x = l ′ �= l or νx ≡ ¯νs.
In this case we have [41]: Δm2 = m2 2 − m21 > 0,
P 2ν (νl → νl)=1− sin 2 2θ sin 2 π L
Lv , Lv =4πp/Δm 2 , (13.30)
P 2ν (νl → νx) =1− P 2ν (νl → νl). Eq. (13.30) with l = μ, x = τ was used,
e.g., in the atmospheric neutrino data analysis [13], in which the first
compelling evidence for neutrino oscillations was obtained.
III. Matter effects in neutrino oscillations. When neutrinos
propagate in matter (e.g., in the Earth, Sun or a supernova), their
coherent forward-scattering from the particles present in matter can
change drastically the pattern of neutrino oscillations [25,26,52]. Thus,
the probabilities of neutrino transitions in matter can differ significantly
from the corresponding vacuum oscillation probabilities.
In the case of, e.g., solar νe transitions in the Sun and 3-neutrino
mixing, the oscillations due to Δm2 31 are strongly suppressed by the