or (tauon), is a neutrino with definite mass the asterisk () represents a complex conjugate; for antineutrinos, the complex conjugate should be removed from the first equation and inserted into the second. represents the Pontecorvo–Maki–Nakagawa–Sakata matrix (also called the PMNS matrix, lepton mixing matrix, or sometimes simply the MNS matrix). It is the analogue of the CKM matrix describing the analogous mixing of quarks. If this matrix were the identity matrix, then the flavor eigenstates would be the same as the mass eigenstates. However, experiment shows that it is not. When the standard three-neutrino theory is considered, the matrix is 3×3. If only two neutrinos are considered, a 2×2 matrix is used. If one or more sterile neutrinos are added (see later), it is 4×4 or larger. In the 3×3 form, it is given by where and The phase factors and are physically meaningful only if neutrinos are Majorana particles — i.e. if the neutrino is identical to its antineutrino (whether or not they are is unknown) — and do not enter into oscillation phenomena regardless. If neutrinoless double beta decay occurs, these factors influence its rate. The phase factor is non-zero only if neutrino oscillation violates CP symmetry; this has not yet been observed experimentally. If experiment shows this 3×3 matrix to be not unitary, a sterile neutrino or some other new physics is required. Propagation and interference Since are mass eigenstates, their propagation can be described by plane wave solutions of the form where quantities are expressed in natural units and is the energy of the mass-eigenstate , is the time from the start of the propagation, is the three-dimensional momentum, is the current position of the particle relative to its starting position In the ultrarelativistic limit, we can approximate the energy as where is the energy of the wavepacket (particle) to be detected. This limit applies to all practical (currently observed) neutrinos, since their masses are less than 1 eV and their energies are at least 1 MeV, so the Lorentz factor, , is greater than in all cases. Using also where is the distance traveled and also dropping the phase factors, the wavefunction becomes Eigenstates with different masses propagate with different frequencies. The heavier ones oscillate faster compared to the lighter ones. Since the mass eigenstates are combinations of flavor eigenstates, this difference in frequencies causes interference between the corresponding flavor components of each mass eigenstate.