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[a] mixing 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 a model 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 matrixEdit
The Standard Model of particle physics contains three generations or "flavors" of neutrinos, , , and 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, , , and , which diagonalize the neutrino's free-particle Hamiltonian. Observations of neutrino oscillation have experimentally determined that for neutrinos, as for quarks, these two eigenbases are not the same - they are "rotated" relative to each other. Each flavor eigenstate can thus be written as a superposition of mass eigenstates, and vice versa. The PMNS matrix, with components corresponding to the amplitude of mass eigenstate in flavor , parameterizes the unitary transformation between the two bases:
The vector on the left represents a generic neutrino expressed in the flavor-eigenstate basis, and on the right is the PMNS matrix multiplied by a vector representing that same neutrino in the mass-eigenstate basis. A neutrino of a given flavor is thus a "mixed" state of neutrinos with distinct mass: If one could measure directly that neutrino's mass, it would be found to have mass with probability .
As noted above, PMNS matrix is unitary. This implies that the sum of the squares 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 case, the Standard Model posits 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.
In general, there are nine degrees of freedom in any unitary three by three matrix. However, in the case of the PMNS matrix five of those real parameters can be absorbed as phases of the lepton fields and thus the PMNS matrix can be fully described by four free parameters. The PMNS matrix is most commonly parameterized by three mixing angles ( , , and ) and a single phase angle called 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:
where and are used to denote and 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 . 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 has not been measured directly, but estimates can be obtained by fits using the other measurements.
Experimentally measured parameter valuesEdit
The 3σ ranges (99.7% confidence) for the magnitudes of the elements of the current matrix are:
Notes regarding the best fit parameter valuesEdit
- 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 13.04°±0.05°, 2.38°±0.06°, 0.201°±0.011°).
- These values are inconsistent with tribimaximal neutrino mixing (i.e. , ) 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.
- The value of is somewhat poorly constrained; a value equal to exactly 45° is currently consistent with the data.
- The PMNS matrix is not unitary in the seesaw model.
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