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 that contains information on the mismatch of quantum states of neutrinos when they propagate freely and when they take part in 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.
PMNS matrix The Standard Model of particle physics contains three generations or "flavors" of neutrinos, ν e {\displaystyle \nu _{\mathrm {e} }} , ν μ {\displaystyle \nu _{\mu }} , and ν τ {\displaystyle \nu _{\tau }} , each labeled with a subscript showing the charged lepton that it partners with 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 {\displaystyle \nu _{1}} , ν 2 {\displaystyle \nu _{2}} , and ν 3 {\displaystyle \nu _{3}} , which diagonalize the neutrino's free-particle Hamiltonian. Observations of neutrino oscillation established experimentally that for neutrinos, as for quarks, these two eigenbases are different – they are 'rotated' relative to each other. Consequently, each flavor eigenstate can be written as a combination of mass eigenstates, called a "superposition", and vice versa. The PMNS matrix, with components U α i {\displaystyle U_{\alpha \,i}} corresponding to the amplitude of mass eigenstate i = {\displaystyle i=} 1, 2, 3 in terms of flavor α = {\displaystyle \alpha =} "e", "μ", "τ"; parameterizes the unitary transformation between the two bases:
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