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 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

α> = Σj=1,2,3 Uαjj> ,
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:
j> = Σα=e,μ,τ U*αjα> .

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
  να   =   cos θjk sin θjk     νj   ,
νβ -sin θjk cos θjk νk
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
D =   1 0 0   ,
0 1 0
0 0 eiδ
the three-flavour mixing matrix is written
U =   1 0 0   D   c13 0 s13   D   c12 s12 0  
0 c23 s23 0 1 0 -s12 c12 0
0 -s23 c23 -s13 0 c13 0 0 1
  =   c12c13 s12c13 s13e-iδ   ,
-s12c23-s13s23c12 eiδ c12c23-s12s13s23 eiδ s23c13
s12s23-s13c12c23 eiδ  -s23c12-s12s13c23 eiδ   c13c23 
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:
i j(t)> = Hvj(t)> = Ejj(t)> ,
where HV is the Hamiltonian operator in vacuum, the solutions of which are

j(t)> = e-iEjtj> ,
where |νj> is the state at t = 0. Consequently, the flavour states propagate as
α(t)> = Σj=1,2,3 Uαj e-iEjtj> .

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

α(t)> = Σβ=e,μ,τ Σj=1,2,3 Uαj e-iEjt U*βjβ> .
The transition amplitude from |να(t)> to |νβ> is then given by
βα(t)> = Σj=1,2,3 Uαj U*βj e-iEjt .

Finally, the transition probability is Pα→β = |<νβα(t)>|2. 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

Pα→β(E,L) = Σj=1,2,3 Σk=1,2,3 Uαj U*βj U*αk Uβk exp( i Δmkj2L / 2E ) ,
which, in the two-flavour approximation, is written
Pα→β(E,L) = sin2( 2θjk ) sin2( Δmjk2 L / 4 E ) .
This is commonly expressed in the convenient form:
Pα→β(E,L) ≈ 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
Pαβ(L) = 1 Emax Pα→β(E) ρ(E) dE ,
Emax 0
which, far away from the source, simplifies to

Pαβ = 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
Hvα(t)> = - Σj=1,2,3 Uαj Ejj (t)>
so the Hamiltonian matrix elements are given by
Hv,αβ = <νβ| Hvα(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
Hv = Δm2   - 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 = Δv   - cos 2θ sin 2θ  
2 sin 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
H'm = G N   1 0   .
2½ 0 -1
The resulting Hamiltonian in matter is thus H = H'v + H'm. This matrix can be diagonalized by the transformations:

νm,j = να cos θm - νβ sin θm = νj cos(θm-θ) - νk sin(θm-θ) ,
νm,k = να sin θm + νβ cos θm = νj sin(θm-θ) + νk cos(θm-θ) ,
sin22θmsin22θ .
( cos 2θ - 2½ G N E / Δm2 )2 + sin22θ
νm,jk are the matter mass eigenstates propagating in matter as plane waves. The matter Hamiltonian can also be written
  νm,j   =   cos θm -sin θm     να   ,
νm,k sin θm cos θm νβ
so θm is a new matter mixing angle. The eigenvalues of this Hamiltonian are ±Δm/2, where
Δm = [ ( Δv cos 2θ - 2½ G N )2 + ( Δv sin 2θ )2 ]½
Finally, the total matter Hamiltonian can be written
H = Δm   cos 2θm -sin 2θm   ,
2 sin 2θm cos 2θm
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
tan 2θmtan 2θ  ,
1 - Lo / ( Le cos 2θ )
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
Nr Δm2 cos 2θ .
2 × 2½ G E
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).

The Schrödinger equation for each matter mass eigenstate can be written
 |νm,j(t)> = Hm,j(t)> = ±  Δm  |νm,j(t)> .
t 2
In a slowly (adiabatic) density-varying medium, the solutions are written

m,j(t)> = exp( ± i ½ t Δm dt )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 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 Super-K 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 ,
π+ → μ+ + νμ ,
μ+ → e+ + νe + νμ ,
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.

In 1998, the angular dependence of the νμ flux observed by Super-K, 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 Pe→μ and Pe→τ for Δm132 = 3.13 × 10-3 eV2 and sin22θ13 arbitrarily set to 0.04.

More to come...

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