Neutrino Oscillations

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 |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 two-flavour 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 CP-violating phase δ in the form

D = 1 0 0 ,
0 1 0
0 0 e^iδ
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 e^iδ c₁₂c₂₃-s₁₂s₁₃s₂₃ e^iδ s₂₃c₁₃
s₁₂s₂₃-s₁₃c₁₂c₂₃ e^iδ -s₂₃c₁₂-s₁₂s₁₃c₂₃ e^iδ c₁₃c₂₃
where c_jk = cos θ_jk and s_jk = sin θ_jk.

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 ∂ |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^-iE_jt |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^-iE_jt |ν_j> .

Simplified graphical representation of neutrino oscillations in the two-flavour 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 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 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 Δm²/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^-iE_jt 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^-iE_jt .

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_k² - m_j² ) / 2 p = Δm_kj² / 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_kj²L / 2E ) ,

which, in the two-flavour approximation, is written

P_α→β(E,L) = sin²( 2θ_jk ) sin²( Δm_jk² L / 4 E ) .

This is commonly expressed in the convenient form:

P_α→β(E,L) ≈ sin²( 2θ ) sin²( 1.27 Δm²_[eV²] L_[m] / E_[MeV] ) ,

with an oscillation length equal to L_o[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 ν_e → ν_μ 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

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 = sin²( 2θ_jk ) / 2 .

Oscillations probability P_e→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

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 two-neutrino approximation takes the form

H_v =	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 =	Δ_v		- cos 2θ	sin 2θ
	2		sin 2θ	cos 2θ

where

Δ_v = 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)> = 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 = 2^½GN 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 + 2^½GN )} |ν> .

The term of weak potential energy must be added to the vacuum Hamiltonian diagonal matrix elements: H_m,ee = U_W,ee = 2^½GN. For convenience and symmetry, a term ½U_W,ee is also subtracted from the diagonal elements so that the matter-only 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

sin²2θ_m =	sin²2θ	.
	( cos 2θ - 2^½ G N E / Δm² )² + sin²2θ

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 = 2^½πћc/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 = Δm² cos 2θ .
2 × 2^½ 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

i ∂ |ν_m,j(t)> = H |ν_m,j(t)> = ± Δ_m |ν_m,j(t)> .
∂t 2
In a slowly (adiabatic) density-varying 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 elastic-scattering cross sections are about 1/6^th of that of ν_e, 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 + 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 electron-neutrino flux of &Phi_{ν_e} = 1.76 (-0.5+0.5±0.9) × 10⁶ cm^-2s^-1 and a total neutrino flux of &Phi_ν = 5.14 (±0.45-0.45+0.48) × 10⁶cm^-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 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 3-σ errors on its position.

Solar neutrinos and &theta₁₂

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 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 ν_e 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⁺ + 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 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 ν_μ flux observed by Super-K, while the ν_e 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 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 Δm₁₃. 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 sin²2&theta₁₃. With the current knowledge of Δm₁₃, this limit is now about 0.1.

Comparison of the probabilities of oscillations P_e→μ and P_e→τ for Δm₁₃² = 3.13 × 10^-3 eV² and sin²2θ₁₃ arbitrarily set to 0.04.

More to come...

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	ν_α		=		cos θ_jk	sin θ_jk			ν_j		,
	ν_β				-sin θ_jk	cos θ_jk			ν_k

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 e^iδ	c₁₂c₂₃-s₁₂s₁₃s₂₃ e^iδ	s₂₃c₁₃
	s₁₂s₂₃-s₁₃c₁₂c₂₃ e^iδ	-s₂₃c₁₂-s₁₂s₁₃c₂₃ e^iδ	c₁₃c₂₃

Neutrino Oscillations

Oscillations Probabilities

Oscillations in matter

Measurements of the oscillation parameters

Flux measurement

Solar neutrinos and &theta12

Atmospheric neutrinos and &theta23

Reactor neutrinos and &theta13

Solar neutrinos and &theta₁₂

Atmospheric neutrinos and &theta₂₃

Reactor neutrinos and &theta₁₃