Pontecorvo–Maki–Nakagawa–Sakata matrix

In particle physics, the Pontecorvo–Maki–Nakagawa–Sakata matrix (PMNS matrix), Maki–Nakagawa–Sakata matrix (MNS matrix), lepton mixing matrix, or neutrino mixing matrix, is a unitary matrix which contains information on the mismatch of quantum states of neutrinos when they propagate freely and when they take part in the weak interactions. It is important in the understanding of neutrino oscillation. This matrix was introduced in 1962 by Ziro Maki, Masami Nakagawa and Shoichi Sakata, to explain the neutrino oscillations predicted by Bruno Pontecorvo.

The PMNS matrix

The Standard Model of particle physics contains three generations or "flavors" of neutrinos, ν_e, ν_μ, and ν_τ labeled according to the charged leptons with which they partner in the charged-current weak interaction. These three eigenstates of the weak interaction form a complete, orthonormal basis for the Standard Model neutrino. Similarly, one can construct an eigenbasis out of three neutrino states of definite mass, ν₁, ν₂, and ν₃, which diagonalize the neutrino's free-particle Hamiltonian. Observations of neutrino oscillation have experimentally determined that for neutrinos, like the quarks, these two eigenbases are not the same - they are "rotated" relative to each other. Each flavor state can thus be written as a superposition of mass eigenstates, and vice-versa. The PMNS matrix, with components U_ai corresponding to the amplitude of mass eigenstate i in flavor a, parameterizes the unitary transformation between the two bases:

${\begin{bmatrix}{\nu _{e}}\\{\nu _{\mu }}\\{\nu _{\tau }}\end{bmatrix}}={\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}\nu _{1}\\\nu _{2}\\\nu _{3}\end{bmatrix}}.\$

The vector on the left represents a generic neutrino state expressed in the flavor basis, and on the right is the PMNS matrix multiplied by a vector representing the same neutrino state in the mass basis. A neutrino of a given flavor α is thus a "mixed" state of neutrinos with different mass: if one could measure directly that neutrino's mass, it would be found to have mass m_i with probability |U_αi|².

The PMNS matrix for antineutrinos is identical to the matrix for neutrinos under CPT symmetry.

Due to the difficulties of detecting neutrinos, it is much more difficult to determine the individual coefficients than in the equivalent matrix for the quarks (the CKM matrix).

Assumptions

As noted above, PMNS matrix is unitary (i.e. the sum of the square of the values in each row and in each column, which represent the probabilities of different possible events given the same starting point, add up to 100%) in the simplest Standard Model case in which there are three generations of neutrinos with Dirac mass that oscillate between three neutrino mass eigenvalues, an assumption that is made when best fit values for its parameters are calculated.

The PMNS matrix is not necessarily unitary and additional parameters are necessary to describe all possible neutrino mixing parameters, in other models of neutrino oscillation and mass generation, such as the see-saw model, and in general, in the case of neutrinos that have Majorana mass rather than Dirac mass.

There are also additional mass parameters and mixing angles in a simple extension of the PMNS matrix in which there are more than three flavors of neutrinos, regardless of the character of neutrino mass. As of July 2014, scientists studying neutrino oscillation are actively considering fits of the experimental neutrino oscillation data to an extended PMNS matrix with a fourth, light "sterile" neutrino and four mass eigenvalues, although the current experimental data tends to disfavor that possibility.

Parameterization

In general, there are nine degrees of freedom in any three by three matrix, and in the PMNS matrix, because it is a matrix whose directly physically observable values (the square of the respective entries) are real numbers between zero and 1 form a unitary matrix, the matrix can thus be fully described by four free parameters from which all physically observable properties of the matrix can be discerned. The PMNS matrix is most commonly parameterized by three mixing angles (θ₁₂, θ₂₃ and θ₁₃) and a single phase called δ_CP related to charge-parity violations (i.e. differences in the rates of oscillation between two states with opposite starting points which makes the order in time in which events take place necessary to predict their oscillation rates), in which case the matrix can be written as:

${\begin{aligned}&{\begin{bmatrix}1&0&0\\0&c_{23}&s_{23}\\0&-s_{23}&c_{23}\end{bmatrix}}{\begin{bmatrix}c_{13}&0&s_{13}e^{-i\delta _{CP}}\\0&1&0\\-s_{13}e^{i\delta _{CP}}&0&c_{13}\end{bmatrix}}{\begin{bmatrix}c_{12}&s_{12}&0\\-s_{12}&c_{12}&0\\0&0&1\end{bmatrix}}\\&={\begin{bmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta _{CP}}\\-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta _{CP}}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta _{CP}}&s_{23}c_{13}\\s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta _{CP}}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta _{CP}}&c_{23}c_{13}\end{bmatrix}}.\end{aligned}}$

where s_ij and c_ij are used to denote sinθ_ij and cosθ_ij respectively. In the case of Majorana neutrinos, two extra complex phases are needed, as the phase of Majorana fields cannot be freely redefined due to the condition $\nu =\nu ^{c}$. An infinite number of possible parameterizations exist; one other common example being the Wolfenstein parameterization.

The mixing angles have been measured by a variety of experiments (see neutrino mixing for a description). The CP-violating phase δ_CP has not been measured directly, but estimates can be obtained by fits using the other measurements.

Experimentally measured parameter values

As of July 2014, the current best directly measured values are:

${\begin{aligned}\sin ^{2}2\theta _{12}&=0.857\pm 0.024\\\sin ^{2}2\theta _{23}&>0.95\\\sin ^{2}2\theta _{13}&=0.095\pm 0.010\\\end{aligned}}$

while the current best-fit values, using direct and indirect measurements, from NuFit are:

${\begin{aligned}\theta _{12}[^{\circ }]&=33.36_{-0.78}^{+0.81}\\\theta _{23}[^{\circ }]&=40.0_{-1.5}^{+2.1}~{\textrm {or}}~50.4_{-1.3}^{+1.3}\\\theta _{13}[^{\circ }]&=8.66_{-0.46}^{+0.44}\\\delta _{\textrm {CP}}[^{\circ }]&=300_{-138}^{+66}\\\end{aligned}}$

So the current matrix will be:

$U={\begin{bmatrix}U_{e1}&U_{e2}&U_{e3}\\U_{\mu 1}&U_{\mu 2}&U_{\mu 3}\\U_{\tau 1}&U_{\tau 2}&U_{\tau 3}\end{bmatrix}}={\begin{bmatrix}0.82\pm 0.01&0.54\pm 0.02&0.15\pm 0.03\\0.35\pm 0.06&0.70\pm 0.06&0.62\pm 0.06\\0.44\pm 0.06&0.45\pm 0.06&0.77\pm 0.06\end{bmatrix}}$

Notes regarding the best fit parameter values

• These best fit values imply that there is much more neutrino mixing than there is mixing between the quark flavors in the CKM matrix (in the CKM matrix, the corresponding mixing angles are θ₁₂ = 13.04°±0.05°, θ₂₃ = 2.38°±0.06°, θ₁₃ = 0.201°±0.011°).
• These values are inconsistent with tribimaximal neutrino mixing (i.e. θ₁₂ = θ₂₃ = 45°, θ₁₃ = 0°) at a statistical significance of more than five standard deviations. Tribimaximal neutrino mixing was a common assumption in theoretical physics papers analyzing neutrino oscillation before more precise measurements were available.
• A value of θ₂₃ equal to exactly 45 degrees, which would imply maximal mixing between the second and third neutrino mass eigenstates, is ruled out with a statistical significance in excess of 2 standard deviations.
• The alternative choices for θ₂₃ are referred to as "first quadrant" and "second quadrant" values. The data favor the first quadrant value over the second quadrant value with a statistical significance of 1.5 standard deviations in a "normal mass hierarchy" context (i.e. where the second neutrino mass eigenstate is lighter than the third neutrino mass eigenstate), but there is not a statistically significant preference between the two values in the case of an "inverted mass hierarchy" (i.e. where the second neutrino mass eigenstate is heavier than the third neutrino mass eigenstate). This is the only PMNS matrix parameter which is strongly sensitive to the mass hierarchy of the neutrino masses given the currently available experimental data.
• The extent to which the best fit value for δ_CP is meaningful should not be overstated. The best fit value for δ_CP is consistent with zero at the 0.9 standard deviation level, since in circular coordinates 0 degrees and 360 degrees are equivalent. Generally speaking, in particle physics, experimental results that are within 2 standard deviations of each other are called "consistent" with each other. Currently, all possible values for δ_CP are with 1.8 standard deviations of the best fit values, so all possible values of δ_CP are "consistent" with the experimental data, even though those values closer to the best fit value are somewhat more likely to be correct.

See also

• Neutrino oscillations
• Koide formula
• Cabibbo–Kobayashi–Maskawa matrix