Neutrino Oscillations
Fabrice Fleurot
Oscillations Probabilities
Neutrinos have this very special property that their flavour eigenstates
do not coincide with their mass eigenstates.
Flavour states can be expressed in the masseigenstate system and vice versa.
Consequently, for a given energy the mass states propagate at different velocities and the flavour states
change with time. This effect is known as oscillations.
The following text is a somewhat simpified description of neutrino oscillations.
The neutrino flavour states ν_{α}> (α = e, μ, τ)
are related to the mass states ν_{j}> (j = 1, 2, 3) by the linear combinations
ν_{α}> =
Σ_{j=1,2,3} U_{αj} ν_{j}> ,
where U is the PMNS unitary mixing matrix.
Thus U^{†} = U^{1}, and one can also write
the mass states with respect to the flavour states:
ν_{j}> =
Σ_{α=e,μ,τ} U^{*}_{αj} ν_{α}> .
The transformation is analogous to a rotation between the bases about the third axis, thus
the mixing matrix can be written in the form of a rotation matrix.
In the twoflavour approximation, this is
 ν_{α}

 =

 cos θ_{jk}
 sin θ_{jk}


 ν_{j}

 ,

ν_{β}
 sin θ_{jk}
 cos θ_{jk}
 ν_{k}

reducing the four unknowns to one: θ_{jk} is the mixing angle of the two mass states involved.
Introducing a Dirac CPviolating phase δ in the form
D =

 1
 0
 0

 ,

0
 1
 0

0
 0
 e^{iδ}

the threeflavour mixing matrix is written
U =

 1
 0
 0

 D

 c_{13}
 0
 s_{13}

 D^{†}

 c_{12}
 s_{12}
 0


0
 c_{23}
 s_{23}
 0
 1
 0
 s_{12}
 c_{12}
 0

0
 s_{23}
 c_{23}
 s_{13}
 0
 c_{13}
 0
 0
 1

=

 c_{12}c_{13}
 s_{12}c_{13}
 s_{13}e^{iδ}

 ,

s_{12}c_{23}s_{13}s_{23}c_{12}
e^{iδ}
 c_{12}c_{23}s_{12}s_{13}s_{23}
e^{iδ}
 s_{23}c_{13}

s_{12}s_{23}s_{13}c_{12}c_{23}
e^{iδ}
 s_{23}c_{12}s_{12}s_{13}c_{23}
e^{iδ}
 c_{13}c_{23}

where c_{jk} = cos θ_{jk} and s_{jk} = sin θ_{jk}.

Graphical representation of the neutrino mixing angles. Here,
θ_{12} = 32.5°,
θ_{23} = 25°,
and, arbitrarily, θ_{13} = 5°.

Each mass state has a definite mass and energy, thus its propagation can be described by the timedependent Schrödinger equation.
In vacuum, with ћ = 1:
i
 ∂
 ν_{j}(t)>
= H_{v} ν_{j}(t)>
= E_{j} ν_{j}(t)> ,

∂t

where H_{V} is the Hamiltonian operator in vacuum, the solutions of which are
ν_{j}(t)> = e^{iEjt} ν_{j}> ,
where ν_{j}> is the state at t = 0. Consequently, the flavour states propagate as
ν_{α}(t)>
= Σ_{j=1,2,3} U_{αj} e^{iEjt}
ν_{j}> .

Simplified graphical representation of neutrino oscillations in the twoflavour approximation.
A ν_{e} is created in the left of the figure and propagates along the black vector,
changing flavour with respect to the phases of mass eigenstates ν_{1} (in blue)
and ν_{2} (in green). The neutrino state
vector (in red) is the superposition of these two mass states. Two points in the red curve
correspond to pure ν_{e}, in the rest, the flavour state is not defined.
ν_{μ} is shown, but no points correpond to 100% of this flavour because the mixing
is not maximal, so the probability of transition can never reach 1.
In this figure, θ = 32.5°, and Δm^{2}/E is arbitrarily
set to a very large value.

Inserting the previous equations, one gets to
ν_{α}(t)>
= Σ_{β=e,μ,τ}
Σ_{j=1,2,3} U_{αj} e^{iEjt}
U^{*}_{βj} ν_{β}> .
The transition amplitude from ν_{α}(t)> to
ν_{β}> is then given by
<ν_{β}ν_{α}(t)> =
Σ_{j=1,2,3}
U_{αj} U^{*}_{βj}
e^{iEjt} .
Finally, the transition probability is
P_{α→β} =
<ν_{β}ν_{α}(t)>^{2}.
With c = 1,
t ≈ L (distance from source),
E_{k}  E_{j} ≈ ( m_{k}^{2}  m_{j}^{2} ) / 2 p
= Δm_{kj}^{2} / 2 p (with p >> m_{j,k}),
and p ≈ E (the neutrino energy, this is assuming that all the mass states have the same momentum), one finds
P_{α→β}(E,L) =
Σ_{j=1,2,3}
Σ_{k=1,2,3}
U_{αj} U^{*}_{βj}
U^{*}_{αk} U_{βk}
exp( i Δm_{kj}^{2}L / 2E ) ,
which, in the twoflavour approximation, is written
P_{α→β}(E,L) =
sin^{2}( 2θ_{jk} ) sin^{2}( Δm_{jk}^{2} L / 4 E ) .
This is commonly expressed in the convenient form:
P_{α→β}(E,L) ≈
sin^{2}( 2θ )
sin^{2}( 1.27 Δm^{2}_{[eV2]} L_{[m]} / E_{[MeV]} ) ,
with an oscillation length equal to
L_{o[m]} ≈ 2.48 × E_{[MeV]} / Δm^{2}_{[eV2]}
(half the period of the sine). The sin^{2}2θ factor is the oscillation amplitude.

This figure shows the twoflavour oscillation probabilities for the ν_{e} → ν_{μ}
transition at energies 0.3 and 8 MeV, using parameters θ_{12} ≈ 32.5°
and Δm_{12}^{2} ≈ 8.0×10^{5} eV^{2}.

However, neutrino signals are rarely monoenergetic so the probability of transition should be averaged over the whole spectrum.
For a density of states ρ(E),
The average oscillation probability is given by
P_{α→β}(L) =
 1
 ∫
 E_{max}
 P_{α→β}(E) ρ(E) dE ,

E_{max}
 0

which, far away from the source, simplifies to
P_{α→β} = sin^{2}( 2θ_{jk} ) / 2 .

Oscillations probability P_{e→μ} with respect to distance from the source
after averaging over the Solar ^{8}B neutrino spectrum (for the demonstration, the MSW effect is not
taken into account.)

Oscillations in matter
For any given neutrino state, the vacuum Hamiltonian can be written
H_{v} ν_{α}(t)> =
 Σ_{j=1,2,3}
U_{αj} E_{j} ν_{j} (t)>
so the Hamiltonian matrix elements are given by
H_{v,αβ} =
<ν_{β} H_{v} ν_{α}(t)> =
Σ_{j=1,2,3} U_{αj} U^{*}_{βj} E_{j} .
After some trivial trigonometric juggling, and leaving out the j, k indices for clarity,
the Hamiltonian in the twoneutrino approximation takes the form
H_{v} =
 Δm^{2}

  cos 2θ
 sin 2θ

 +

 E
 0

 .

4E
 sin 2θ
 cos 2θ
 0
 E

The second member only adds a same phase to all the flavour states and does not play any
role in oscillations, it can thus be discarded.
Only the first member is left and written as
H'_{v} =
 Δ_{v}

  cos 2θ
 sin 2θ


2
 sin 2θ
 cos 2θ

where
Δ_{v} = Δm^{2} / 2 E .
In matter, neutrinos propagate with forward elastic scattering like photons do.
All the neutrino flavours scatter on protons, neutrons and electrons via Z^{0} exchange,
but only electron neutrinos scatter on electrons via W^{} or W^{+} exchange.
In vacuum, the time development of a mass state is simply written
ν(t)> = e^{i(px  Et)} ν>
≈ e^{i m2t / 2p} ν> .
In matter, by analogy with optics, a refraction index can be defined as
n = 1 + U / p, where p is the neutrino momentum and
U is a weak interaction potential.
The weak potential U_{W,ee} = 2^{½}GN
acts on electron neutrinos only, where N is the electron density and
G ≈ 1.17×10^{11} MeV^{2} is Fermi's constant.
A U_{Z,αα} term should also be added
but this term is identical for all the flavours and does not add a phase between the states
so it can be discarded too. so the propagation equation in matter can be written
ν(t)> = e^{i(npx  Et)} ν>
≈ e^{i( m2 / 2p + 2½GN )} ν> .
The term of weak potential energy must be added to the vacuum Hamiltonian diagonal matrix elements:
H_{m,ee} = U_{W,ee} = 2^{½}GN.
For convenience and symmetry, a term ½U_{W,ee} is also subtracted from the diagonal elements
so that the matteronly Hamiltonian can finally be written
H'_{m} =
 G N

 1
 0

 .

2^{½}
 0
 1

The resulting Hamiltonian in matter is thus
H = H'_{v} + H'_{m}.
This matrix can be diagonalized by the transformations:
ν_{m,j} =
ν_{α} cos θ_{m}  ν_{β} sin θ_{m} =
ν_{j} cos(θ_{m}θ)  ν_{k} sin(θ_{m}θ) ,
ν_{m,k} =
ν_{α} sin θ_{m} + ν_{β} cos θ_{m} =
ν_{j} sin(θ_{m}θ) + ν_{k} cos(θ_{m}θ) ,
where
sin^{2}2θ_{m} =
 sin^{2}2θ
 .

( cos 2θ  2^{½} G N E / Δm^{2} )^{2}
+ sin^{2}2θ

ν_{m,jk} are the matter mass eigenstates propagating in matter as plane waves.
The matter Hamiltonian can also be written
 ν_{m,j}

 =

 cos θ_{m}
 sin θ_{m}


 ν_{α}

 ,

ν_{m,k}
 sin θ_{m}
 cos θ_{m}
 ν_{β}

so θ_{m} is a new matter mixing angle.
The eigenvalues of this Hamiltonian are ±Δ_{m}/2, where
Δ_{m} =
[ ( Δ_{v} cos 2θ  2^{½} G N )^{2}
+ ( Δ_{v} sin 2θ )^{2} ]^{½}
Finally, the total matter Hamiltonian can be written
H =
 Δ_{m}

 cos 2θ_{m}
 sin 2θ_{m}

 ,

2
 sin 2θ_{m}
 cos 2θ_{m}

The θ_{m} angle is thus the effective mixing angle in matter for electron density N.
The matter mixing angle relationship can be rewritten in the form
tan 2θ_{m} =
 tan 2θ_{ }
 ,

1  L_{o} / ( L_{e} cos 2θ )

where L_{e} = 2^{½}πћc/GN
is the ν_{e}e^{} interaction length.
This formula shows a "resonance" at L_{o}/L_{e} = cos 2θ
where the matter mixing angle is maximum. Therefore, the
"MSW resonant density" is given by
N_{r} =
 Δm^{2} cos 2θ
 .

2 × 2^{½} G E

For &theta = 32.5°, Δm^{2} = 8×10^{5} eV^{2},
and a 10MeV neutrino, the resonant density is about 1.3×10^{25} cm^{3},
i.e. a density in the Sun of about 26 g·cm^{3}
(density at the centre: 150 g·cm^{3}).
The Schrödinger equation for each matter mass eigenstate can be written
i
 ∂
 ν_{m,j}(t)>
= H ν_{m,j}(t)> = ±
 Δ_{m}
 ν_{m,j}(t)> .

∂t
 2

In a slowly (adiabatic) densityvarying medium, the solutions are written
ν_{m,j}(t)> = exp( ± i ½ ∫_{t} Δ_{m} dt )
ν_{m,j}> .
These are "adiabatic" states independently evolving in time and thus chosen as the basis.
To be continued...
Measurements of the oscillation parameters
That was theory. Now, how do we measure the oscillation parameters?
Flux measurement
Early chemical experiments studying Solar neutrinos
were only sensitive to ν_{e}
and observed only about 34% of the total flux (±experimental errors).
Due to the fact that the ν_{μ} and ν_{τ}
elasticscattering cross sections are about 1/6^{th} of that of ν_{e},
lightwater based experiment such as the Kamiokande series could see
34+66/6 = 45% of the total flux (±error bars).
These low fluxes with respect to the expected value from the standard Solar model
gave birth to the socalled "Solar neutrino problem".
To date, only SNO has been able to see and partially separate the complete flux from the ^{8}B reaction in the Sun,
thanks to its D_{2}O target sensitive to all flavours via the 'Neutral Current' reaction (NC):
d + ν_{eμτ} → p + n + ν_{eμτ} ,
equally sensitive to all flavours. SNO is also able to independently measure the electron neutrino flux via
the 'Charged Current' reaction (CC):
d + ν_{e} → p + p + e^{} .
It can also detect electrons from the elastic scattering reaction (ES).
The SNO measurements found an electronneutrino flux of
&Phi_{νe} = 1.76 (0.5+0.5±0.9) × 10^{6} cm^{2}s^{1}
and a total neutrino flux of
&Phi_{ν} = 5.14 (±0.450.45+0.48) × 10^{6 }cm^{2}s^{1},
in good agreement with the standard Solar model prediction.
To be continued...

^{8}B neutrino flux measurements by SNO for the three types of reactions.
SNO measured a reconstructed rate of about 1.76×10^{6} cm^{2}s^{1} with the
CC reaction, which is only sensitive to ν_{e}.
For the two other reactions, the reconstructed flux depends on the relative ratio of flavours, as
ES is 6 times less sensitive to
ν_{μ} and ν_{τ} than to ν_{e},
and NC is equaly sensitive to all flavours. The figure
shows the possible fluxes, including the error bars. The intersection of the
three bands allows us to determine each flux. The + sign is the most probable value,
while the ellipses are the 1, 2 and 3σ errors on its position.

Solar neutrinos and &theta_{12}
The Sun emits a large amount of ν_{e} from various fusion reactions
(see Stellar Evolution). Due to the
MSW effect, neutrinos leaving the Sun
are mostly in state ν_{2},
thus, when they are measured via the weak interaction,
the neutrino wave functions collapse to one of the flavour states.
The probability for a ν_{2} to collapse to a ν_{e}
is approximately sin^{2}θ_{12}, making the flux
measurement a direct measurement of this value.
In 2002, SNO published the first measurement
of nonelectron Solar neutrinos and could estimate the mixing parameters.
The most accurate estimate of these socalled "Solar parameters" today have been provided by combining these
measurements with those of
SuperK
and the latest results from KamLAND:
Δm_{12}^{2} = 7.9 (0.5+0.6) × 10^{5} eV^{2}
and sin^{2}2θ_{12} = 0.82 (0.07+0.07).
To be continued...
Atmospheric neutrinos and &theta_{23}
Atmospheric neutrinos are created by the reaction between highenergy cosmic rays and upperatmosphere nuclei, in particular:
p + ^{14}N → π^{+} + A ,
π^{+} → μ^{+} + ν_{μ} ,
μ^{+} → e^{+} + ν_{e} + ν_{μ} ,
and respectively for their antiparticles.
Without oscillations, it is thus expected that the ratio of muon (anti)neutrinos over electron (anti)neutrinos
be equal to 2. However due to oscillations, this ratio depends on the trajectory angle with respect to zenith:
neutrinos coming upward have travelled 13,000 km, while those coming downward have only travelled about 15 km.
The electron neutrino flux does not show any updown asymetry because small Δm_{12} makes the
e↔μ oscillation length much larger than the radius of the Earth at the high energy of atmospheric neutrinos.
On the contrary, Δm_{23} is greater and allows μ↔τ oscillations while traversing
the Earth, resulting in an updown asymetry in the muon neutrino flux. This particularity allows one to
apply the twoneutrino approximation.
In 1998, the angular dependence of the ν_{μ} flux observed by
SuperK,
while the ν_{e} flux was found constant, was accepted as clear evidence of neutrino flavour conversion.
The latest measurements of the socalled "atmospheric parameters" by the
MINOS collaboration found
Δm_{23}^{2} = 3.05 (0.55+0.60±0.12) × 10^{3} eV^{2}
and sin^{2}2θ_{23} = 0.88 (0.15+0.12±0.06).

After traversing the Earth, a large fraction of highenergy muon neutrinos have oscillated into tau neutrinos.

Reactor neutrinos and &theta_{13}
Nuclear reactors emit about 5 × 10^{20} ν_{e} per second and per GW
of thermal power. Although e→μ oscillation amplitudes are greater than that of e→τ, the oscillation
length of the latter is shorter, due to the smaller value of Δm_{13}. Therefore, close enough
to the detector, the probability of oscillations P_{e→μ} is still relatively small while
P_{e→τ} can already be close to the maximum.
In 1999, detector Chooz could set an upper limit on
sin^{2}2&theta_{13}. With the current knowledge of Δm_{13},
this limit is now about 0.1.

Comparison of the probabilities of oscillations
P_{e→μ} and
P_{e→τ} for
Δm_{13}^{2} = 3.13 × 10^{3} eV^{2}
and sin^{2}2θ_{13} arbitrarily set to 0.04.

More to come...
Send questions and comments to