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 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. 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 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 Standard Model of particle physics contains three generations or 'flavors' of neutrinos, ν e { extstyle u _{e}} , ν μ { extstyle u _{mu }} , and ν τ { extstyle u _{ au }} 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, ν 1 { extstyle u _{1}} , ν 2 { extstyle u _{2}} , and ν 3 { extstyle u _{3}} , 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 a i {displaystyle U_{ai}} corresponding to the amplitude of mass eigenstate i {displaystyle i} in flavor a { extstyle a} , parameterizes the unitary transformation between the two bases: 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 { extstyle alpha } 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 { extstyle m_{i}} with probability | U a i | 2 { extstyle |U_{ai}|^{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). As noted above, PMNS matrix is unitary. That is, 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%,