Neutrino oscillation is a quantum mechanical
phenomenon whereby a neutrino created with
a specific lepton family number ("lepton flavor":
electron, muon, or tau) can later be measured
to have a different lepton family number.
The probability of measuring a particular
flavor for a neutrino varies between 3 known
states, as it propagates through space.First
predicted by Bruno Pontecorvo in 1957, neutrino
oscillation has since been observed by a multitude
of experiments in several different contexts.
Notably, the existence of neutrino oscillation
resolved the long-standing solar neutrino
problem.
Neutrino oscillation is of great theoretical
and experimental interest, as the precise
properties of the process can shed light on
several properties of the neutrino. In particular,
it implies that the neutrino has a non-zero
mass, which requires a modification to the
Standard Model of particle physics. The experimental
discovery of neutrino oscillation, and thus
neutrino mass, by the Super-Kamiokande Observatory
and the Sudbury Neutrino Observatories was
recognized with the 2015 Nobel Prize for Physics.
== Observations ==
A great deal of evidence for neutrino oscillation
has been collected from many sources, over
a wide range of neutrino energies and with
many different detector technologies. The
2015 Nobel Prize in Physics was shared by
Takaaki Kajita and Arthur B. McDonald for
their early pioneering observations of these
oscillations.
Neutrino oscillation is a function of the
ratio L/E, where L is the distance traveled
and E is the neutrino's energy. (Details in
§ Propagation and interference below.) Neutrino
sources and detectors are far too large to
move, but all available sources produce a
range of energies, and oscillation can be
measured with a fixed distance and neutrinos
of varying energy. The preferred distance
depends on the most common energy, but the
exact distance is not critical as long as
it is known. The limiting factor in measurements
is the accuracy with which the energy of each
observed neutrino can be measured. Because
current detectors have energy uncertainties
of a few percent, it is satisfactory to know
the distance to within 1%.
=== Solar neutrino oscillation ===
The first experiment that detected the effects
of neutrino oscillation was Ray Davis's Homestake
experiment in the late 1960s, in which he
observed a deficit in the flux of solar neutrinos
with respect to the prediction of the Standard
Solar Model, using a chlorine-based detector.
This gave rise to the Solar neutrino problem.
Many subsequent radiochemical and water Cherenkov
detectors confirmed the deficit, but neutrino
oscillation was not conclusively identified
as the source of the deficit until the Sudbury
Neutrino Observatory provided clear evidence
of neutrino flavor change in 2001.Solar neutrinos
have energies below 20 MeV. At energies above
5 MeV, solar neutrino oscillation actually
takes place in the Sun through a resonance
known as the MSW effect, a different process
from the vacuum oscillation described later
in this article.
=== Atmospheric neutrino oscillation ===
Following the theories that were proposed
in the 1970s suggesting unification of weak,
strong and Electromagnetic forces, a few experiments
on proton decay followed in the 80's. Large
detectors such as IMB, MACRO, and Kamiokande
II have observed a deficit in the ratio of
the flux of muon to electron flavor atmospheric
neutrinos (see muon decay). The Super-Kamiokande
experiment provided a very precise measurement
of neutrino oscillation in an energy range
of hundreds of MeV to a few TeV, and with
a baseline of the diameter of the Earth; the
first experimental evidence for atmospheric
neutrino oscillations was announced in 1998.
=== Reactor neutrino oscillation ===
Many experiments have searched for oscillation
of electron anti-neutrinos produced at nuclear
reactors. No oscillations were found until
the detector was installed at a distance 1–2
km. Such oscillations give the value of the
parameter θ13. Neutrinos produced in nuclear
reactors have energies similar to solar neutrinos,
of around a few MeV. The baselines of these
experiments have ranged from tens of meters
to over 100 km (parameter θ12). Mikaelyan
and Sinev proposed to use two identical detectors
to cancel systematic uncertainties in reactor
experiment to measure the parameter θ13.
In December, 2011 the Double Chooz firstly
found that θ13 ≠ 0 and in 2012 the Daya
Bay experiment announced a discovery that
θ13 ≠ 0 with a significance of 5.2σ; these
results have since been confirmed by RENO.
=== Beam neutrino oscillation ===
Neutrino beams produced at a particle accelerator
offer the greatest control over the neutrinos
being studied. Many experiments have taken
place that study the same oscillations as
in atmospheric neutrino oscillation using
neutrinos with a few GeV of energy and several-hundred-km
baselines. The MINOS, K2K, and Super-K experiments
have all independently observed muon neutrino
disappearance over such long baselines.Data
from the LSND experiment appear to be in conflict
with the oscillation parameters measured in
other experiments. Results from the MiniBooNE
appeared in Spring 2007 and contradicted the
results from LSND, although they could support
the existence of a fourth neutrino type, the
sterile neutrino.In 2010, the INFN and CERN
announced the observation of a tau particle
in a muon neutrino beam in the OPERA detector
located at Gran Sasso, 730 km away from the
source in Geneva.T2K, using a neutrino beam
directed through 295 km of earth and the Super-Kamiokande
detector, measured a non-zero value for the
parameter θ13 in a neutrino beam. NOνA,
using the same beam as MINOS with a baseline
of 810 km, is sensitive to the same.
== Theory ==
Neutrino oscillation arises from mixing between
the flavor and mass eigenstates of neutrinos.
That is, the three neutrino states that interact
with the charged leptons in weak interactions
are each a different superposition of the
three (propagating) neutrino states of definite
mass. Neutrinos are created (and absorbed)
in weak processes in their flavor eigenstates.As
a neutrino superposition propagates through
space, the quantum mechanical phases of the
three mass states advance at slightly different
rates, due to the slight differences in their
respective neutrino masses. This results in
a changing superposition mixture of mass eigenstates
as the neutrino travels; but a different mixture
of mass eigenstates corresponds to a different
mixture of flavor states. So a neutrino born
as, say, an electron neutrino will be some
mixture of electron, mu, and tau neutrino
after traveling some distance. Since the quantum
mechanical phase advances in a periodic fashion,
after some distance the state will nearly
return to the original mixture, and the neutrino
will be again mostly electron neutrino. The
electron flavor content of the neutrino will
then continue to oscillate – as long as
the quantum mechanical state maintains coherence.
Since mass differences between neutrino flavors
are small in comparison with long coherence
lengths for neutrino oscillations, this microscopic
quantum effect becomes observable over macroscopic
distances.
In contrast, due to their larger masses, the
charged leptons (electrons, muons, and tau
leptons) have never been observed to oscillate.
In nuclear beta decay, muon decay, pion decay,
and kaon decay, when a neutrino and a charged
lepton are emitted, the charged lepton, by
virtue of its large mass, is emitted incoherently
in mass eigenstates, such as |e−〉. Weak-force
couplings force the corresponding neutrino
to be emitted in a "charged-lepton-centric"
superposition such as |νe〉, which is an
eigenstate for a "flavor" that is pegged to
the electron's mass eigenstate, and not by
one of the neutrino's own mass eigenstates.
Because the neutrino is in a coherent superposition
that is not a mass eigenstate, its state detectably
oscillates as it travels. No analogous mechanism
exists in the Standard Model that would make
charged leptons oscillate meaningfully. In
the four decays given above, where the charged
lepton is emitted in a mass eigenstate, the
charged lepton will not oscillate, as mass
eigenstates propagate without oscillation.
The case of (real) W boson decay is more complex:
W boson decay is sufficiently energetic to
generate a charged lepton that is not in a
mass eigenstate; however, the charged lepton
would lose coherence, if it had any, over
interatomic distances (0.1 nm) and would thus
quickly cease any meaningful oscillation.
More importantly, no mechanism in the Standard
Model is capable in the first place of pinning
down a charged lepton into a coherent state
that is not a mass eigenstate; instead, while
the charged lepton from the W boson decay
is not initially in a mass eigenstate, neither
is it in any "neutrino-centric" eigenstate,
nor in any other coherent state. It cannot
meaningfully be said that such a featureless
charged lepton oscillates or that it does
not oscillate, as any "oscillation" transformation
would just leave it the same generic state
that it was before the oscillation. Therefore,
detection of a charged lepton oscillation
from W boson decay is infeasible on multiple
levels.
=== Pontecorvo–Maki–Nakagawa–Sakata
matrix ===
The idea of neutrino oscillation was first
put forward in 1957 by Bruno Pontecorvo, who
proposed that neutrino–antineutrino transitions
may occur in analogy with neutral kaon mixing.
Although such matter–antimatter oscillation
has not been observed, this idea formed the
conceptual foundation for the quantitative
theory of neutrino flavor oscillation, which
was first developed by Maki, Nakagawa, and
Sakata in 1962 and further elaborated by Pontecorvo
in 1967. One year later the solar neutrino
deficit was first observed, and that was followed
by the famous article by Gribov and Pontecorvo
published in 1969 titled "Neutrino astronomy
and lepton charge".The concept of neutrino
mixing is a natural outcome of gauge theories
with massive neutrinos, and its structure
can be characterized in general.
In its simplest form it is expressed as a
unitary transformation relating the flavor
and mass eigenbasis and can be written as
|
ν
α
⟩
=
∑
i
U
α
i
∗
|
ν
i
⟩
,
{\displaystyle \left|\nu _{\alpha }\right\rangle
=\sum _{i}U_{\alpha i}^{*}\left|\nu _{i}\right\rangle
,}
|
ν
i
⟩
=
∑
α
U
α
i
|
ν
α
⟩
,
{\displaystyle \left|\nu _{i}\right\rangle
=\sum _{\alpha }U_{\alpha i}\left|\nu _{\alpha
}\right\rangle ,}
where
|
ν
α
⟩
{\displaystyle \left|\nu _{\alpha }\right\rangle
}
is a neutrino with definite flavor α = e
(electron), μ (muon) or τ (tauon),
|
ν
i
⟩
{\displaystyle \left|\nu _{i}\right\rangle
}
is a neutrino with definite mass
m
i
{\displaystyle m_{i}}
,
i
=
1
,
2
,
3
{\displaystyle i=1,2,3}
,
the asterisk (
∗
{\displaystyle ^{*}}
) represents a complex conjugate; for antineutrinos,
the complex conjugate should be dropped from
the first equation and added to the second.
U
α
i
{\displaystyle U_{\alpha i}}
represents the Pontecorvo–Maki–Nakagawa–Sakata
matrix (also called the PMNS matrix, lepton
mixing matrix, or sometimes simply the MNS
matrix). It is the analogue of the CKM matrix
describing the analogous mixing of quarks.
If this matrix were the identity matrix, then
the flavor eigenstates would be the same as
the mass eigenstates. However, experiment
shows that it is not.
When the standard three-neutrino theory is
considered, the matrix is 3×3. If only two
neutrinos are considered, a 2×2 matrix is
used. If one or more sterile neutrinos are
added (see later), it is 4×4 or larger. In
the 3×3 form, it is given by
U
=
[
U
e
1
U
e
2
U
e
3
U
μ
1
U
μ
2
U
μ
3
U
τ
1
U
τ
2
U
τ
3
]
=
[
1
0
0
0
c
23
s
23
0
−
s
23
c
23
]
[
c
13
0
s
13
e
−
i
δ
0
1
0
−
s
13
e
i
δ
0
c
13
]
[
c
12
s
12
0
−
s
12
c
12
0
0
0
1
]
[
e
i
α
1
/
2
0
0
0
e
i
α
2
/
2
0
0
0
1
]
=
[
c
12
c
13
s
12
c
13
s
13
e
−
i
δ
−
s
12
c
23
−
c
12
s
23
s
13
e
i
δ
c
12
c
23
−
s
12
s
23
s
13
e
i
δ
s
23
c
13
s
12
s
23
−
c
12
c
23
s
13
e
i
δ
−
c
12
s
23
−
s
12
c
23
s
13
e
i
δ
c
23
c
13
]
[
e
i
α
1
/
2
0
0
0
e
i
α
2
/
2
0
0
0
1
]
,
{\displaystyle {\begin{aligned}U&={\begin{bmatrix}U_{e1}&U_{e2}&U_{e3}\\U_{\mu
1}&U_{\mu 2}&U_{\mu 3}\\U_{\tau 1}&U_{\tau
2}&U_{\tau 3}\end{bmatrix}}\\&={\begin{bmatrix}1&0&0\\0&c_{23}&s_{23}\\0&-s_{23}&c_{23}\end{bmatrix}}{\begin{bmatrix}c_{13}&0&s_{13}e^{-i\delta
}\\0&1&0\\-s_{13}e^{i\delta }&0&c_{13}\end{bmatrix}}{\begin{bmatrix}c_{12}&s_{12}&0\\-s_{12}&c_{12}&0\\0&0&1\end{bmatrix}}{\begin{bmatrix}e^{i\alpha
_{1}/2}&0&0\\0&e^{i\alpha _{2}/2}&0\\0&0&1\\\end{bmatrix}}\\&={\begin{bmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta
}\\-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta
}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta
}&s_{23}c_{13}\\s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta
}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta
}&c_{23}c_{13}\end{bmatrix}}{\begin{bmatrix}e^{i\alpha
_{1}/2}&0&0\\0&e^{i\alpha _{2}/2}&0\\0&0&1\\\end{bmatrix}},\end{aligned}}}
where cij = cos θij, and sij = sin θij.
The phase factors α1 and α2 are physically
meaningful only if neutrinos are Majorana
particles — i.e. if the neutrino is identical
to its antineutrino (whether or not they are
is unknown) — and do not enter into oscillation
phenomena regardless. If neutrinoless double
beta decay occurs, these factors influence
its rate. The phase factor δ is non-zero
only if neutrino oscillation violates CP symmetry;
this has not yet been observed experimentally.
If experiment shows this 3×3 matrix to be
not unitary, a sterile neutrino or some other
new physics is required.
=== Propagation and interference ===
Since
|
ν
i
⟩
{\displaystyle \left|\nu _{i}\right\rangle
}
are mass eigenstates, their propagation can
be described by plane wave solutions of the
form
|
ν
i
(
t
)
⟩
=
e
−
i
(
E
i
t
−
p
→
i
⋅
x
→
)
∣
ν
i
(
0
)
⟩
,
{\displaystyle |\nu _{i}(t)\rangle =e^{-i(E_{i}t-{\vec
{p}}_{i}\cdot {\vec {x}})}\mid \nu _{i}(0)\rangle
,}
where
quantities are expressed in natural units
(
c
=
1
,
ℏ
=
1
)
{\displaystyle (c=1,\hbar =1)}
E
i
{\displaystyle E_{i}}
is the energy of the mass-eigenstate
i
{\displaystyle i}
,
t
{\displaystyle t}
is the time from the start of the propagation,
p
→
i
{\displaystyle {\vec {p}}_{i}}
is the three-dimensional momentum,
x
→
{\displaystyle {\vec {x}}}
is the current position of the particle relative
to its starting positionIn the ultrarelativistic
limit,
|
p
→
i
|
=
p
i
≫
m
i
{\displaystyle |{\vec {p}}_{i}|=p_{i}\gg m_{i}}
, we can approximate the energy as
E
i
=
p
i
2
+
m
i
2
≃
p
i
+
m
i
2
2
p
i
≈
E
+
m
i
2
2
E
,
{\displaystyle E_{i}={\sqrt {p_{i}^{2}+m_{i}^{2}}}\simeq
p_{i}+{\frac {m_{i}^{2}}{2p_{i}}}\approx E+{\frac
{m_{i}^{2}}{2E}},}
where E is the total energy of the particle.
This limit applies to all practical (currently
observed) neutrinos, since their masses are
less than 1 eV and their energies are at least
1 MeV, so the Lorentz factor γ is greater
than 106 in all cases. Using also t ≈ L,
where L is the distance traveled and also
dropping the phase factors, the wavefunction
becomes:
|
ν
i
(
L
)
⟩
=
e
−
i
m
i
2
L
/
2
E
|
ν
i
(
0
)
⟩
.
{\displaystyle |\nu _{i}(L)\rangle =e^{-im_{i}^{2}L/2E}|\nu
_{i}(0)\rangle .}
Eigenstates with different masses propagate
with different frequencies. The heavier ones
oscillate faster compared to the lighter ones.
Since the mass eigenstates are combinations
of flavor eigenstates, this difference in
frequencies causes interference between the
corresponding flavor components of each mass
eigenstate. Constructive interference causes
it to be possible to observe a neutrino created
with a given flavor to change its flavor during
its propagation. The probability that a neutrino
originally of flavor α will later be observed
as having flavor β is
P
α
→
β
=
|
⟨
ν
β
(
L
)
|
ν
α
⟩
|
2
=
|
∑
i
U
α
i
∗
U
β
i
e
−
i
m
i
2
L
/
2
E
|
2
.
{\displaystyle P_{\alpha \rightarrow \beta
}=\left|\left\langle \nu _{\beta }(L)|\nu
_{\alpha }\right\rangle \right|^{2}=\left|\sum
_{i}U_{\alpha i}^{*}U_{\beta i}e^{-im_{i}^{2}L/2E}\right|^{2}.}
This is more conveniently written as
P
α
→
β
=
δ
α
β
−
4
∑
i
>
j
R
e
(
U
α
i
∗
U
β
i
U
α
j
U
β
j
∗
)
sin
2
⁡
(
Δ
m
i
j
2
L
4
E
)
+
2
∑
i
>
j
I
m
(
U
α
i
∗
U
β
i
U
α
j
U
β
j
∗
)
sin
⁡
(
Δ
m
i
j
2
L
2
E
)
,
{\displaystyle {\begin{aligned}P_{\alpha \rightarrow
\beta }=\delta _{\alpha \beta }&{}-4\sum _{i>j}{\rm
{Re}}(U_{\alpha i}^{*}U_{\beta i}U_{\alpha
j}U_{\beta j}^{*})\sin ^{2}\left({\frac {\Delta
m_{ij}^{2}L}{4E}}\right)\\&{}+2\sum _{i>j}{\rm
{Im}}(U_{\alpha i}^{*}U_{\beta i}U_{\alpha
j}U_{\beta j}^{*})\sin \left({\frac {\Delta
m_{ij}^{2}L}{2E}}\right),\end{aligned}}}
where
Δ
m
i
j
2
≡
m
i
2
−
m
j
2
{\displaystyle \Delta m_{ij}^{2}\ \equiv m_{i}^{2}-m_{j}^{2}}
. The phase that is responsible for oscillation
is often written as (with c and
ℏ
{\displaystyle \hbar }
restored)
Δ
m
2
c
3
L
4
ℏ
E
=
G
e
V
f
m
4
ℏ
c
×
Δ
m
2
e
V
2
L
k
m
G
e
V
E
≈
1.27
×
Δ
m
2
e
V
2
L
k
m
G
e
V
E
,
{\displaystyle {\frac {\Delta m^{2}\,c^{3}\,L}{4\hbar
E}}={\frac {{\rm {GeV}}\,{\rm {fm}}}{4\hbar
c}}\times {\frac {\Delta m^{2}}{{\rm {eV}}^{2}}}{\frac
{L}{\rm {km}}}{\frac {\rm {GeV}}{E}}\approx
1.27\times {\frac {\Delta m^{2}}{{\rm {eV}}^{2}}}{\frac
{L}{\rm {km}}}{\frac {\rm {GeV}}{E}},}
where 1.27 is unitless. In this form, it is
convenient to plug in the oscillation parameters
since:
The mass differences, Δm2, are known to be
on the order of 1×10−4 eV2
Oscillation distances, L, in modern experiments
are on the order of kilometers
Neutrino energies, E, in modern experiments
are typically on order of MeV or GeV.If there
is no CP-violation (δ is zero), then the
second sum is zero. Otherwise, the CP asymmetry
can be given as
A
CP
(
α
β
)
=
P
(
ν
α
→
ν
β
)
−
P
(
ν
¯
α
→
ν
¯
β
)
=
4
∑
i
>
j
I
m
(
U
α
i
∗
U
β
i
U
α
j
U
β
j
∗
)
sin
⁡
(
Δ
m
i
j
2
L
2
E
)
{\displaystyle A_{\text{CP}}^{(\alpha \beta
)}=P(\nu _{\alpha }\rightarrow \nu _{\beta
})-P({\bar {\nu }}_{\alpha }\rightarrow {\bar
{\nu }}_{\beta })=4\sum _{i>j}\mathrm {Im}
{\big (}U_{\alpha i}^{*}U_{\beta i}U_{\alpha
j}U_{\beta j}^{*}{\big )}\sin {\Big (}{\tfrac
{\Delta m_{ij}^{2}L}{2E}}{\Big )}}
In terms of Jarlskog invariant
I
m
(
U
α
i
U
β
i
∗
U
α
j
∗
U
β
j
)
=
J
∑
γ
,
k
ε
α
β
γ
ε
i
j
k
{\displaystyle \mathrm {Im} {\big (}U_{\alpha
i}U_{\beta i}^{*}U_{\alpha j}^{*}U_{\beta
j}{\big )}=J\sum _{\gamma ,k}\varepsilon _{\alpha
\beta \gamma }\varepsilon _{ijk}}
,the CP asymmetry is expressed as
A
CP
(
α
β
)
=
16
J
∑
γ
ε
α
β
γ
sin
⁡
(
Δ
m
21
2
L
4
E
)
sin
⁡
(
Δ
m
32
2
L
4
E
)
sin
⁡
(
Δ
m
31
2
L
4
E
)
{\displaystyle A_{\text{CP}}^{(\alpha \beta
)}=16J\sum _{\gamma }\varepsilon _{\alpha
\beta \gamma }\sin {\Big (}{\tfrac {\Delta
m_{21}^{2}L}{4E}}{\Big )}\sin {\Big (}{\tfrac
{\Delta m_{32}^{2}L}{4E}}{\Big )}\sin {\Big
(}{\tfrac {\Delta m_{31}^{2}L}{4E}}{\Big )}}
=== Two neutrino case ===
The above formula is correct for any number
of neutrino generations. Writing it explicitly
in terms of mixing angles is extremely cumbersome
if there are more than two neutrinos that
participate in mixing. Fortunately, there
are several cases in which only two neutrinos
participate significantly. In this case, it
is sufficient to consider the mixing matrix
U
=
(
cos
⁡
θ
sin
⁡
θ
−
sin
⁡
θ
cos
⁡
θ
)
.
{\displaystyle U={\begin{pmatrix}\cos \theta
&\sin \theta \\-\sin \theta &\cos \theta \end{pmatrix}}.}
Then the probability of a neutrino changing
its flavor is
P
α
→
β
,
α
≠
β
=
sin
2
⁡
(
2
θ
)
sin
2
⁡
(
Δ
m
2
L
4
E
)
(natural units)
.
{\displaystyle P_{\alpha \rightarrow \beta
,\alpha \neq \beta }=\sin ^{2}(2\theta )\sin
^{2}\left({\frac {\Delta m^{2}L}{4E}}\right)\,{\text{(natural
units)}}.}
Or, using SI units and the convention introduced
above
P
α
→
β
,
α
≠
β
=
sin
2
⁡
(
2
θ
)
sin
2
⁡
(
1.27
Δ
m
2
L
E
[
e
V
2
]
[
k
m
]
[
G
e
V
]
)
.
{\displaystyle P_{\alpha \rightarrow \beta
,\alpha \neq \beta }=\sin ^{2}(2\theta )\sin
^{2}\left(1.27{\frac {\Delta m^{2}L}{E}}{\frac
{\rm {[eV^{2}]\,[km]}}{\rm {[GeV]}}}\right).}
This formula is often appropriate for discussing
the transition νμ ↔ ντ in atmospheric
mixing, since the electron neutrino plays
almost no role in this case. It is also appropriate
for the solar case of νe ↔ νx, where νx
is a superposition of νμ and ντ. These
approximations are possible because the mixing
angle θ13 is very small and because two of
the mass states are very close in mass compared
to the third.
=== Classical analogue of neutrino oscillation
===
The basic physics behind neutrino oscillation
can be found in any system of coupled harmonic
oscillators. A simple example is a system
of two pendulums connected by a weak spring
(a spring with a small spring constant). The
first pendulum is set in motion by the experimenter
while the second begins at rest. Over time,
the second pendulum begins to swing under
the influence of the spring, while the first
pendulum's amplitude decreases as it loses
energy to the second. Eventually all of the
system's energy is transferred to the second
pendulum and the first is at rest. The process
then reverses. The energy oscillates between
the two pendulums repeatedly until it is lost
to friction.
The behavior of this system can be understood
by looking at its normal modes of oscillation.
If the two pendulums are identical then one
normal mode consists of both pendulums swinging
in the same direction with a constant distance
between them, while the other consists of
the pendulums swinging in opposite (mirror
image) directions. These normal modes have
(slightly) different frequencies because the
second involves the (weak) spring while the
first does not. The initial state of the two-pendulum
system is a combination of both normal modes.
Over time, these normal modes drift out of
phase, and this is seen as a transfer of motion
from the first pendulum to the second.
The description of the system in terms of
the two pendulums is analogous to the flavor
basis of neutrinos. These are the parameters
that are most easily produced and detected
(in the case of neutrinos, by weak interactions
involving the W boson). The description in
terms of normal modes is analogous to the
mass basis of neutrinos. These modes do not
interact with each other when the system is
free of outside influence.
When the pendulums are not identical the analysis
is slightly more complicated. In the small-angle
approximation, the potential energy of a single
pendulum system is
1
2
m
g
L
x
2
{\displaystyle {\tfrac {1}{2}}{\tfrac {mg}{L}}x^{2}}
, where g is the standard gravity, L is the
length of the pendulum, m is the mass of the
pendulum, and x is the horizontal displacement
of the pendulum. As an isolated system the
pendulum is a harmonic oscillator with a frequency
of
g
/
L
{\displaystyle {\sqrt {g/L}}}
. The potential energy of a spring is
1
2
k
x
2
{\displaystyle {\tfrac {1}{2}}kx^{2}}
where k is the spring constant and x is the
displacement. With a mass attached it oscillates
with a period of
k
/
m
{\displaystyle {\sqrt {k/m}}}
. With two pendulums (labeled a and b) of
equal mass but possibly unequal lengths and
connected by a spring, the total potential
energy is
V
=
m
2
(
g
L
a
x
a
2
+
g
L
b
x
b
2
+
k
m
(
x
b
−
x
a
)
2
)
.
{\displaystyle V={\frac {m}{2}}\left({\frac
{g}{L_{a}}}x_{a}^{2}+{\frac {g}{L_{b}}}x_{b}^{2}+{\frac
{k}{m}}(x_{b}-x_{a})^{2}\right).}
This is a quadratic form in xa and xb, which
can also be written as a matrix product:
V
=
m
2
(
x
a
x
b
)
(
g
L
a
+
k
m
−
k
m
−
k
m
g
L
b
+
k
m
)
(
x
a
x
b
)
.
{\displaystyle V={\frac {m}{2}}{\begin{pmatrix}x_{a}&x_{b}\end{pmatrix}}{\begin{pmatrix}{\tfrac
{g}{L_{a}}}+{\tfrac {k}{m}}&-{\tfrac {k}{m}}\\[8pt]-{\tfrac
{k}{m}}&{\tfrac {g}{L_{b}}}+{\tfrac {k}{m}}\end{pmatrix}}{\begin{pmatrix}x_{a}\\x_{b}\end{pmatrix}}.}
The 2×2 matrix is real symmetric and so (by
the spectral theorem) it is "orthogonally
diagonalizable". That is, there is an angle
θ such that if we define
(
x
a
x
b
)
=
(
cos
⁡
θ
sin
⁡
θ
−
sin
⁡
θ
cos
⁡
θ
)
(
x
1
x
2
)
{\displaystyle {\begin{pmatrix}x_{a}\\x_{b}\end{pmatrix}}={\begin{pmatrix}\cos
\theta &\sin \theta \\-\sin \theta &\cos \theta
\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}}}
then
V
=
m
2
(
x
1
x
2
)
(
λ
1
0
0
λ
2
)
(
x
1
x
2
)
{\displaystyle V={\frac {m}{2}}{\begin{pmatrix}x_{1}\
x_{2}\end{pmatrix}}{\begin{pmatrix}\lambda
_{1}&0\\0&\lambda _{2}\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}}}
where λ1 and λ2 are the eigenvalues of the
matrix. The variables x1 and x2 describe normal
modes which oscillate with frequencies of
λ
1
{\displaystyle {\sqrt {\lambda _{1}}}}
and
λ
2
{\displaystyle {\sqrt {\lambda _{2}}}}
. When the two pendulums are identical (La
= Lb), θ is 45°.
The angle θ is analogous to the Cabibbo angle
(though that angle applies to quarks rather
than neutrinos).
When the number of oscillators (particles)
is increased to three, the orthogonal matrix
can no longer be described by a single angle;
instead, three are required (Euler angles).
Furthermore, in the quantum case, the matrices
may be complex. This requires the introduction
of complex phases in addition to the rotation
angles, which are associated with CP violation
but do not influence the observable effects
of neutrino oscillation.
== Theory, graphically ==
=== 
Two neutrino probabilities in vacuum ===
In the approximation where only two neutrinos
participate in the oscillation, the probability
of oscillation follows a simple pattern:
The blue curve shows the probability of the
original neutrino retaining its identity.
The red curve shows the probability of conversion
to the other neutrino. The maximum probability
of conversion is equal to sin22θ. The frequency
of the oscillation is controlled by Δm2.
=== Three neutrino probabilities ===
If three neutrinos are considered, the probability
for each neutrino to appear is somewhat complex.
The graphs below show the probabilities for
each flavor, with the plots in the left column
showing a long range to display the slow "solar"
oscillation, and the plots in the right column
zoomed in, to display the fast "atmospheric"
oscillation. The parameters used to create
these graphs (see below) are consistent with
current measurements, but since some parameters
are still quite uncertain, some aspects of
these plots are only qualitatively correct.
The illustrations were created using the following
parameter values:
sin2(2 θ13) = 0.10 (Determines the size
of the small wiggles.)
sin2(2 θ23) = 0.97
sin2(2 θ12) = 0.861
δ = 0 (If the actual value of this phase
is large, the probabilities will be somewhat
distorted, and will be different for neutrinos
and antineutrinos.)
Normal mass hierarchy: m1 ≤ m2 ≤ m3
Δm212 = 7.59×10−5 eV2
Δm232 ≈ Δm213 = 2.32×10−3 eV2
== Observed values of oscillation parameters
==
sin2(2 θ13) = 0.093±0.008. PDG combination
of Daya Bay, RENO, and Chooz results.
sin2(2 θ12) = 0.846±0.021. This corresponds
to θsol (solar), obtained from KamLand, solar,
reactor and accelator data.
sin2(2 θ23) > 0.92 at 90% confidence level,
corresponding 
to θ23 ≡ θatm = 45±7.1° (atmospheric)
Δm221 ≡ Δm2sol = (7.53±0.18)×10−5
eV2
|Δm231| ≈ |Δm232| ≡ Δm2atm = (2.44±0.06)×10−3
eV2 (normal mass hierarchy)
δ, α1, α2, and the sign of Δm232 are currently
unknown.Solar neutrino experiments combined
with KamLAND have measured the so-called solar
parameters Δm2sol and sin2θsol. Atmospheric
neutrino experiments such as Super-Kamiokande
together with the K2K and MINOS long baseline
accelerator neutrino experiment have determined
the so-called atmospheric parameters Δm2atm
and sin2θatm. The last mixing angle, θ13,
has been measured by the experiments Daya
Bay, Double Chooz and RENO as sin2(2 θ13).
For atmospheric neutrinos the relevant difference
of masses is about Δm2 = 2.4×10−3 eV2
and the typical energies are ~1 GeV; for these
values the oscillations become visible for
neutrinos traveling several hundred kilometres,
which would be those neutrinos that reach
the detector traveling through the earth,
from below the horizon.
The mixing parameter θ13 is measured using
electron anti-neutrinos from nuclear reactors.
The rate of anti-neutrino interactions is
measured in detectors sited near the reactors
to determine the flux prior to any significant
oscillations and then it is measured in far
detectors (placed kilometres from the reactors).
The oscillation is observed as an apparent
disappearance of electron anti-neutrinos in
the far detectors (i.e. the interaction rate
at the far site is lower than predicted from
the observed rate at the near site).
From atmospheric and solar neutrino oscillation
experiments, it is known that two mixing angles
of the MNS matrix are large and the third
is smaller. This is in sharp contrast to the
CKM matrix in which all three angles are small
and hierarchically decreasing. Nothing is
currently known about the CP-violating phase
of the MNS matrix.
If the neutrino mass proves to be of Majorana
type (making the neutrino its own antiparticle),
it is possible that the MNS matrix has more
than one phase.
Since experiments observing neutrino oscillation
measure the squared mass difference and not
absolute mass, one might claim that the lightest
neutrino mass is exactly zero, without contradicting
observations. This is however regarded as
unlikely by theorists.
== Origins of neutrino mass ==
The question of how neutrino masses arise
has not been answered conclusively. In the
Standard Model of particle physics, fermions
only have mass because of interactions with
the Higgs field (see Higgs boson). These interactions
involve both left- and right-handed versions
of the fermion (see chirality). However, only
left-handed neutrinos have been observed so
far.
Neutrinos may have another source of mass
through the Majorana mass term. This type
of mass applies for electrically neutral particles
since otherwise it would allow particles to
turn into anti-particles, which would violate
conservation of electric charge.
The smallest modification to the Standard
Model, which only has left-handed neutrinos,
is to allow these left-handed neutrinos to
have Majorana masses. The problem with this
is that the neutrino masses are surprisingly
smaller than the rest of the known particles
(at least 500,000 times smaller than the mass
of an electron), which, while it does not
invalidate the theory, is widely regarded
as unsatisfactory as this construction offers
no insight into the origin of the neutrino
mass scale.
The next simplest addition would be to add
into the Standard Model right-handed neutrinos
that interact with the left-handed neutrinos
and the Higgs field in an analogous way to
the rest of the fermions. These new neutrinos
would interact with the other fermions solely
in this way, so are not phenomenologically
excluded. The problem of the disparity of
the mass scales remains.
=== Seesaw mechanism ===
The most popular conjectured solution currently
is the seesaw mechanism, where right-handed
neutrinos with very large Majorana masses
are added. If the right-handed neutrinos are
very heavy, they induce a very small mass
for the left-handed neutrinos, which is proportional
to the inverse of the heavy mass.
If it is assumed that the neutrinos interact
with the Higgs field with approximately the
same strengths as the charged fermions do,
the heavy mass should be close to the GUT
scale. Because the Standard Model has only
one fundamental mass scale, all particle masses
must arise in relation to this scale.
There are other varieties of seesaw and there
is currently great interest in the so-called
low-scale seesaw schemes, such as the inverse
seesaw mechanism.The addition of right-handed
neutrinos has the effect of adding new mass
scales, unrelated to the mass scale of the
Standard Model, hence the observation of heavy
right-handed neutrinos would reveal physics
beyond the Standard Model. Right-handed neutrinos
would help to explain the origin of matter
through a mechanism known as leptogenesis.
=== Other sources ===
There are alternative ways to modify the standard
model that are similar to the addition of
heavy right-handed neutrinos (e.g., the addition
of new scalars or fermions in triplet states)
and other modifications that are less similar
(e.g., neutrino masses from loop effects and/or
from suppressed couplings). One example of
the last type of models is provided by certain
versions supersymmetric extensions of the
standard model of fundamental interactions,
where R parity is not a symmetry. There, the
exchange of supersymmetric particles such
as squarks and sleptons can break the lepton
number and lead to neutrino masses. These
interactions are normally excluded from theories
as they come from a class of interactions
that lead to unacceptably rapid proton decay
if they are all included. These models have
little predictive power and are not able to
provide a cold dark matter candidate.
== Oscillations in the early universe ==
During the early universe when particle concentrations
and temperatures were high, neutrino oscillations
can behave differently. Depending on neutrino
mixing-angle parameters and masses, a broad
spectrum of behavior may arise including vacuum-like
neutrino oscillations, smooth evolution, or
self-maintained coherence. The physics for
this system is non-trivial and involves neutrino
oscillations in a dense neutrino gas.
== See also ==
MSW effect
Majoron
Neutral kaon mixing
Neutral particle oscillation
Neutrino astronomy
== Notes
