Neutrinos have this very special property that their flavour eigenstates
do not coincide with their mass eigenstates.
Flavour states can be expressed in the mass-eigenstate 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 |να> (α = e, μ, τ)
are related to the mass states |νj> (j = 1, 2, 3) by the linear combinations
Σj=1,2,3 Uαj |νj> ,
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:
Σα=e,μ,τ U*αj |να> .
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 two-flavour approximation, this is
reducing the four unknowns to one: θjk is the mixing angle of the two mass states involved.
Introducing a Dirac CP-violating phase δ in the form
the three-flavour mixing matrix is written
where cjk = cos θjk and sjk = sin θjk.
|Graphical representation of the neutrino mixing angles. Here,
θ12 = 32.5°,
θ23 = 25°,
and, arbitrarily, θ13 = 5°.
Each mass state has a definite mass and energy, thus its propagation can be described by the time-dependent Schrödinger equation.
In vacuum, with ћ = 1:
where HV is the Hamiltonian operator in vacuum, the solutions of which are
= Hv |νj(t)>
= Ej |νj(t)> ,
|νj(t)> = e-iEjt |νj> ,
where |νj> is the state at t = 0. Consequently, the flavour states propagate as
= Σj=1,2,3 Uαj e-iEjt
|Simplified graphical representation of neutrino oscillations in the two-flavour approximation.
A νe is created in the left of the figure and propagates along the black vector,
changing flavour with respect to the phases of mass eigenstates ν1 (in blue)
and ν2 (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 νe, in the rest, the flavour state is not defined.
νμ 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 Δm2/E is arbitrarily
set to a very large value.
Inserting the previous equations, one gets to
Σj=1,2,3 Uαj e-iEjt
U*βj |νβ> .
The transition amplitude from |να(t)> to
|νβ> is then given by
Finally, the transition probability is
With c = 1,
t ≈ L (distance from source),
Ek - Ej ≈ ( mk2 - mj2 ) / 2 p
= Δmkj2 / 2 p (with p >> mj,k),
and p ≈ E (the neutrino energy, this is assuming that all the mass states have the same momentum), one finds
exp( i Δmkj2L / 2E ) ,
which, in the two-flavour approximation, is written
sin2( 2θjk ) sin2( Δmjk2 L / 4 E ) .
This is commonly expressed in the convenient form:
sin2( 2θ )
sin2( 1.27 Δm2[eV2] L[m] / E[MeV] ) ,
with an oscillation length equal to
Lo[m] ≈ 2.48 × E[MeV] / Δm2[eV2]
(half the period of the sine). The sin22θ factor is the oscillation amplitude.
|This figure shows the two-flavour oscillation probabilities for the νe → νμ
transition at energies 0.3 and 8 MeV, using parameters θ12 ≈ 32.5°
and Δm122 ≈ 8.0×10-5 eV2.
However, neutrino signals are rarely mono-energetic 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
which, far away from the source, simplifies to
||Pα→β(E) ρ(E) dE ,
Pα→β = sin2( 2θjk ) / 2 .
|Oscillations probability Pe→μ with respect to distance from the source
after averaging over the Solar 8B 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
Hv |να(t)> =
Uαj Ej |νj (t)>
so the Hamiltonian matrix elements are given by
<νβ| Hv |να(t)> =
Σj=1,2,3 Uαj U*βj Ej .
After some trivial trigonometric juggling, and leaving out the j, k indices for clarity,
the Hamiltonian in the two-neutrino approximation takes the form
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
||- cos 2θ
||- cos 2θ
Δv = Δm2 / 2 E .
In matter, neutrinos propagate with forward elastic scattering like photons do.
All the neutrino flavours scatter on protons, neutrons and electrons via Z0 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)> = ei(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 UW,ee = 2½GN
acts on electron neutrinos only, where N is the electron density and
G ≈ 1.17×10-11 MeV-2 is Fermi's constant.
A UZ,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)> = ei(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:
Hm,ee = UW,ee = 2½GN.
For convenience and symmetry, a term ½UW,ee is also subtracted from the diagonal elements
so that the matter-only Hamiltonian can finally be written
The resulting Hamiltonian in matter is thus
H = H'v + H'm.
This matrix can be diagonalized by the transformations:
να cos θm - νβ sin θm =
νj cos(θm-θ) - νk sin(θm-θ) ,
να sin θm + νβ cos θm =
νj sin(θm-θ) + νk cos(θm-θ) ,
νm,jk are the matter mass eigenstates propagating in matter as plane waves.
The matter Hamiltonian can also be written
|( cos 2θ - 2½ G N E / Δm2 )2
so θm is a new matter mixing angle.
The eigenvalues of this Hamiltonian are ±Δm/2, where
[ ( Δv cos 2θ - 2½ G N )2
+ ( Δv sin 2θ )2 ]½
Finally, the total matter Hamiltonian can be written
The θm angle is thus the effective mixing angle in matter for electron density N.
The matter mixing angle relationship can be rewritten in the form
where Le = 2½πћc/GN
is the νe-e- interaction length.
This formula shows a "resonance" at Lo/Le = cos 2θ
where the matter mixing angle is maximum. Therefore, the
"MSW resonant density" is given by
|tan 2θm =
|1 - Lo / ( Le cos 2θ )
For &theta = 32.5°, Δm2 = 8×10-5 eV2,
and a 10-MeV neutrino, the resonant density is about 1.3×1025 cm-3,
i.e. a density in the Sun of about 26 g·cm-3
(density at the centre: 150 g·cm-3).
|| Δm2 cos 2θ
|2 × 2½ G E
The Schrödinger equation for each matter mass eigenstate can be written
In a slowly (adiabatic) density-varying medium, the solutions are written
= H |νm,j(t)> = ±
|| |νm,j(t)> .
|νm,j(t)> = exp( ± i ½ ∫t Δm dt )
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?
Early chemical experiments studying Solar neutrinos
were only sensitive to νe
and observed only about 34% of the total flux (±experimental errors).
Due to the fact that the νμ and ντ
elastic-scattering cross sections are about 1/6th of that of νe,
light-water 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 so-called "Solar neutrino problem".
To date, only SNO has been able to see and partially separate the complete flux from the 8B reaction in the Sun,
thanks to its D2O target sensitive to all flavours via the 'Neutral Current' reaction (NC):
d + νeμτ → p + n + νeμτ ,
equally sensitive to all flavours. SNO is also able to independently measure the electron neutrino flux via
the 'Charged Current' reaction (CC):
d + νe → p + p + e- .
It can also detect electrons from the elastic scattering reaction (ES).
The SNO measurements found an electron-neutrino flux of
&Phiνe = 1.76 (-0.5+0.5±0.9) × 106 cm-2s-1
and a total neutrino flux of
&Phiν = 5.14 (±0.45-0.45+0.48) × 106 cm-2s-1,
in good agreement with the standard Solar model prediction.
To be continued...
|8B neutrino flux measurements by SNO for the three types of reactions.
SNO measured a reconstructed rate of about 1.76×106 cm-2s-1 with the
CC reaction, which is only sensitive to νe.
For the two other reactions, the reconstructed flux depends on the relative ratio of flavours, as
ES is 6 times less sensitive to
νμ and ντ than to νe,
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 3-σ errors on its position.
Solar neutrinos and &theta12
The Sun emits a large amount of νe from various fusion reactions
(see Stellar Evolution). Due to the
MSW effect, neutrinos leaving the Sun
are mostly in state ν2,
thus, when they are measured via the weak interaction,
the neutrino wave functions collapse to one of the flavour states.
The probability for a ν2 to collapse to a νe
is approximately sin2θ12, making the flux
measurement a direct measurement of this value.
In 2002, SNO published the first measurement
of non-electron Solar neutrinos and could estimate the mixing parameters.
The most accurate estimate of these so-called "Solar parameters" today have been provided by combining these
measurements with those of
and the latest results from KamLAND:
Δm122 = 7.9 (-0.5+0.6) × 10-5 eV2
and sin22θ12 = 0.82 (-0.07+0.07).
To be continued...
Atmospheric neutrinos and &theta23
Atmospheric neutrinos are created by the reaction between high-energy cosmic rays and upper-atmosphere nuclei, in particular:
p + 14N → π+ + A ,
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 up-down asymetry because small Δm12 makes the
e↔μ oscillation length much larger than the radius of the Earth at the high energy of atmospheric neutrinos.
On the contrary, Δm23 is greater and allows μ↔τ oscillations while traversing
the Earth, resulting in an up-down asymetry in the muon neutrino flux. This particularity allows one to
apply the two-neutrino approximation.
π+ → μ+ + νμ ,
μ+ → e+ + νe + νμ ,
In 1998, the angular dependence of the νμ flux observed by
while the νe flux was found constant, was accepted as clear evidence of neutrino flavour conversion.
The latest measurements of the so-called "atmospheric parameters" by the
MINOS collaboration found
Δm232 = 3.05 (-0.55+0.60±0.12) × 10-3 eV2
and sin22θ23 = 0.88 (-0.15+0.12±0.06).
|After traversing the Earth, a large fraction of high-energy muon neutrinos have oscillated into tau neutrinos.
Reactor neutrinos and &theta13
Nuclear reactors emit about 5 × 1020 νe 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 Δm13. Therefore, close enough
to the detector, the probability of oscillations Pe→μ is still relatively small while
Pe→τ can already be close to the maximum.
In 1999, detector Chooz could set an upper limit on
sin22&theta13. With the current knowledge of Δm13,
this limit is now about 0.1.
|Comparison of the probabilities of oscillations
Δm132 = 3.13 × 10-3 eV2
and sin22θ13 arbitrarily set to 0.04.
More to come...
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