Neutrino oscillation arises from mixing between the flavor and mass
eigenstates of neutrinos. That is, the three neutrino states that interact with the charged leptons in weak interactions are each a different
superposition of the three (propagating) neutrino states of definite mass. Neutrinos are produced and detected in
weak interactions as flavour eigenstates but propagate as coherent superpositions of mass
eigenstates. As a neutrino superposition propagates through space, the quantum mechanical
phases of the three neutrino mass states advance at slightly different rates, due to the slight differences in their respective masses. This results in a changing superposition mixture of mass eigenstates as the neutrino travels; but a different mixture of mass eigenstates corresponds to a different mixture of flavor states. For example, a neutrino born as an electron neutrino will be some mixture of electron,
mu, and
tau neutrino after traveling some distance. Since the quantum mechanical phase advances in a periodic fashion, after some distance the state will nearly return to the original mixture, and the neutrino will be again mostly electron neutrino. The electron flavor content of the neutrino will then continue to oscillate – as long as the quantum mechanical state maintains
coherence. Since mass differences between neutrino flavors are small in comparison with long
coherence lengths for neutrino oscillations, this microscopic quantum effect becomes observable over macroscopic distances. In contrast, due to their larger masses, the charged leptons (electrons, muons, and tau leptons) have never been observed to oscillate. In nuclear beta decay, muon decay,
pion decay, and
kaon decay, when a neutrino and a charged lepton are emitted, the charged lepton is emitted in incoherent mass eigenstates such as , because of its large mass. Weak-force couplings compel the simultaneously emitted neutrino to be in a "charged-lepton-centric" superposition such as , which is an eigenstate for a "flavor" that is fixed by the electron's mass eigenstate, and not in one of the neutrino's own mass eigenstates. Because the neutrino is in a coherent superposition that is not a mass eigenstate, the mixture that makes up that superposition oscillates significantly as it travels. No analogous mechanism exists in the Standard Model that would make charged leptons detectably oscillate. In the four decays mentioned above, where the charged lepton is emitted in a unique mass eigenstate, the charged lepton will not oscillate, as single mass eigenstates propagate without oscillation. The case of (real)
W boson decay is more complicated: W boson decay is sufficiently energetic to generate a charged lepton that is not in a mass eigenstate; however, the charged lepton would lose coherence, if it had any, over interatomic distances () and would thus quickly cease any meaningful oscillation. More importantly, no mechanism in the Standard Model is capable of pinning down a charged lepton into a coherent state that is not a mass eigenstate, in the first place; instead, while the charged lepton from the W boson decay is not initially in a mass eigenstate, neither is it in any "neutrino-centric" eigenstate, nor in any other coherent state. It cannot meaningfully be said that such a featureless charged lepton oscillates or that it does not oscillate, as any "oscillation" transformation would just leave it the same generic state that it was before the oscillation. Therefore, detection of a charged lepton oscillation from W boson decay is infeasible on multiple levels.
Pontecorvo–Maki–Nakagawa–Sakata matrix The idea of neutrino oscillation was first put forward in 1957 by
Bruno Pontecorvo, who proposed that neutrino–antineutrino transitions may occur in analogy with
neutral kaon mixing. and further elaborated by Pontecorvo in 1967. and that was followed by the famous article by Gribov and Pontecorvo published in 1969 titled "Neutrino astronomy and lepton charge". The concept of neutrino mixing is a natural outcome of gauge theories with massive neutrinos, and its structure can be characterized in general. In its simplest form it is expressed as a
unitary transformation relating the flavor and mass
eigenbasis and can be written as : \left| \nu_i \right\rangle = \sum_{\alpha} U^*_{\alpha i} \left| \nu_\alpha \right\rangle : \left| \nu_\alpha \right\rangle = \sum_{i} U_{\alpha i} \left| \nu_i \right\rangle where : \ \left| \nu_\alpha \right\rangle is a neutrino with definite flavor \alpha = (electron), (muon) or (tauon) : \ \left| \nu_i \right\rangle is a neutrino with definite mass m_i with i = 1, 2 or 3 : the superscript asterisk (^*) represents a
complex conjugate; for
antineutrinos, the complex conjugate should be removed from the first equation and inserted into the second. The symbol U_{\alpha i} 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 . If only two neutrinos are considered, a matrix is used. If one or more
sterile neutrinos are added (see later), it is or larger. In the form, it is given by : \begin{align} U &= \begin{bmatrix} U_{e 1} & U_{e 2} & U_{e 3} \\ U_{\mu 1} & U_{\mu 2} & U_{\mu 3} \\ U_{\tau 1} & U_{\tau 2} & U_{\tau 3} \end{bmatrix} \\ &= \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} \\ 0 & 1 & 0 \\ -s_{13} e^{i\delta} & 0 & c_{13} \end{bmatrix} \begin{bmatrix} c_{12} & s_{12} & 0 \\ -s_{12} & c_{12} & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} e^{i\alpha_1 / 2} & 0 & 0 \\ 0 & e^{i\alpha_2 / 2} & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} \\ &= \begin{bmatrix} c_{12} c_{13} & s_{12} c_{13} & s_{13} e^{-i\delta} \\ - s_{12} c_{23} - c_{12} s_{23} s_{13} e^{i \delta} & c_{12} c_{23} - s_{12} s_{23} s_{13} e^{i \delta} & s_{23} c_{13}\\ s_{12} s_{23} - c_{12} c_{23} s_{13} e^{i \delta} & - c_{12} s_{23} - s_{12} c_{23} s_{13} e^{i \delta} & c_{23} c_{13} \end{bmatrix} \begin{bmatrix} e^{i\alpha_1 / 2} & 0 & 0 \\ 0 & e^{i\alpha_2 / 2} & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}, \end{align} 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 matrix to be not
unitary, a
sterile neutrino or some other new physics is required.
Propagation and interference Since \left|\, \nu_j \,\right\rangle are mass eigenstates, their propagation can be described by
plane wave solutions of the form : \left|\, \nu_j(t) \,\right\rangle = e^{-i\,\left(\, E_j t \,-\, \vec{p}_j \cdot \vec{x} \,\right) } \left|\, \nu_j(0) \,\right\rangle ~, where • quantities are expressed in
natural units (c = 1, \hbar = 1), and i^2 \equiv -1, • E_j is the
energy of the mass-eigenstate j, • t is the time from the start of the propagation, • \vec{p}_j is the three-dimensional
momentum, • \vec{x} is the current position of the particle relative to its starting position In the
ultrarelativistic limit, \left|\vec{p}_j\right| = p_j \gg m_j, we can approximate the energy as : E_j = \sqrt{\, p_j^2 + m_j^2 \;} \simeq p_j + \frac{ m_j^2 }{\, 2\,p_j \,} \approx E + \frac{ m_j^2 }{\, 2\,E \,} ~, 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 : \left| \,\nu_j(L) \,\right\rangle = e^{-im_j^2L/{(2E)}} \, \left| \, \nu_j(0) \, \right\rangle . 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. Constructive
interference causes it to be possible to observe a neutrino created with a given flavor to change its flavor during its propagation. The probability that a neutrino originally of flavor will later be observed as having flavor is : P_{\alpha\rightarrow\beta} \, = \, \Bigl|\, \left\langle\, \left. \nu_\beta \, \right| \, \nu_\alpha (L) \, \right\rangle \,\Bigr|^2 \, =\, \left|\, \sum_j\, U_{\alpha j}^*\, U_{\beta j}\,e^{-im_j^2L/(2E)} \, \right|^2 ~. This is more conveniently written as : \begin{align} P_{\alpha\rightarrow\beta} = \delta_{\alpha\beta} &{}- 4\,\sum_{j>k} \,\operatorname\mathcal{R_e}\left\{\, U_{\alpha j}^*\, U_{\beta j}\, U_{\alpha k}\, U_{\beta k}^* \,\right\}\, \sin^2 \left( \frac{\Delta_{jk} m^2\, L}{4E} \right) \\ &{}+ 2\,\sum_{j>k} \,\operatorname\mathcal{I_m}\left\{\, U_{\alpha j}^*\, U_{\beta j}\, U_{\alpha k}\, U_{\beta k}^* \,\right\}\, \sin \left( \frac{\Delta_{jk} m^2\, L}{2E} \right) ~, \end{align} where \Delta_{jk} m^2\ \equiv m_j^2 - m_k^2 ~. The phase that is responsible for oscillation is often written as (with and \hbar restored) : \frac{\Delta_{jk} (mc^2)^2 \, L}{4 \hbar c\,E} = \frac{{\rm GeV}\, {\rm fm}}{4 \hbar c} \times \frac{\Delta_{jk} m^2}{{\rm eV}^2} \frac{L}{\rm km} \frac{\rm GeV}{E} \approx 1.27 \times \frac{\Delta_{jk} m^2}{{\rm eV}^2} \frac{L}{\rm km} \frac{\rm GeV}{E} ~, where 1.27 is
unitless. In this form, it is convenient to plug in the oscillation parameters since: • The mass differences, , are known to be on the order of = () • Oscillation distances, , in modern experiments are on the order of
kilometres • Neutrino energies, , in modern experiments are typically on order of MeV or GeV. If there is no
CP-violation ( is zero), then the second sum is zero. Otherwise, the CP asymmetry can be given as : A^{(\alpha\beta)}_\mathsf{CP} = P(\nu_\alpha \rightarrow \nu_\beta) - P(\bar{\nu}_\alpha \rightarrow \bar{\nu}_\beta) = 4\,\sum_{j>k}\,\operatorname\mathcal{I_m} \left\{\, U_{\alpha j}^*\, U_{\beta j}\, U_{\alpha k}\, U_{\beta k}^* \,\right\} \,\sin \left( \frac{\Delta_{jk} m^2\,L}{2E} \right) In terms of
Jarlskog invariant : \operatorname\mathcal{I_m} \left\{\, U_{\alpha j}^*\, U_{\beta j}\, U_{\alpha k}\, U_{\beta k}^* \,\right\} = J \, \sum_{\gamma,\ell} \varepsilon_{\alpha\beta\gamma}\,\varepsilon_{jk\ell} ~, the CP asymmetry is expressed as : A^{(\alpha\beta)}_\mathsf{CP} = 16 \, \sin \left( \frac{\Delta_{21} m^2\,L}{4E} \right) \sin \left( \frac{\Delta_{32} m^2\,L}{4E} \right) \sin \left( \frac{\Delta_{31} m^2\,L}{4E} \right) \,J\, \sum_{\gamma} \,\varepsilon_{\alpha\beta\gamma}
Two-neutrino case The above formula is correct for any number of neutrino generations. Writing it explicitly in terms of mixing angles is extremely cumbersome if there are more than two neutrinos that participate in mixing. Fortunately, there are several meaningful cases in which only two neutrinos participate significantly. In this case, it is sufficient to consider the mixing matrix : U = \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix}. Then the probability of a neutrino changing its flavor is : P_{\alpha\rightarrow\beta, \alpha\neq\beta} = \sin^2(2\theta) \, \sin^2 \left(\frac{\Delta m^2 L}{4E}\right) \quad \text{ [natural units] .} Or, using
SI units and the convention introduced above : P_{\alpha\rightarrow\beta, \alpha\neq\beta} = \sin^2(2\theta) \, \sin^2 \left( 1.27\, \frac{\Delta m^2 L}{E}\, \frac{\rm [eV^{2}]\,[km]}{\rm [GeV]}\right) ~. This formula is often appropriate for discussing the transition in atmospheric mixing, since the electron neutrino plays almost no role in this case. It is also appropriate for the solar case of , where is a mix (superposition) of and . These approximations are possible because the mixing angle is very small and because two of the mass states are very close in mass compared to the third.
Classical analogue of neutrino oscillation The basic physics behind neutrino oscillation can be found in any system of coupled
harmonic oscillators. A simple example is a system of two
pendulums connected by a weak spring (a spring with a small
spring constant). The first pendulum is set in motion by the experimenter while the second begins at rest. Over time, the second pendulum begins to swing under the influence of the spring, while the first pendulum's amplitude decreases as it loses energy to the second. Eventually all of the system's energy is transferred to the second pendulum and the first is at rest. The process then reverses. The energy oscillates between the two pendulums repeatedly until it is lost to
friction. The behavior of this system can be understood by looking at its
normal modes of oscillation. If the two pendulums are identical then one normal mode consists of both pendulums swinging in the same direction with a constant distance between them, while the other consists of the pendulums swinging in opposite (mirror image) directions. These normal modes have (slightly) different frequencies because the second involves the (weak) spring while the first does not. The initial state of the two-pendulum system is a combination of both normal modes. Over time, these normal modes drift out of phase, and this is seen as a transfer of motion from the first pendulum to the second. The description of the system in terms of the two pendulums is analogous to the flavor basis of neutrinos. These are the parameters that are most easily produced and detected (in the case of neutrinos, by weak interactions involving the
W boson). The description in terms of normal modes is analogous to the mass basis of neutrinos. These modes do not interact with each other when the system is free of outside influence. When the pendulums are not identical the analysis is slightly more complicated. In the small-angle approximation, the
potential energy of a single pendulum system is \tfrac{1}{2}\tfrac{mg}{L} x^2, where is
standard gravity, is the length of the pendulum, is the mass of the pendulum, and is the horizontal displacement of the pendulum. As an isolated system the pendulum is a harmonic oscillator with a frequency of \textstyle \sqrt{g/L\ }. The potential energy of a spring is , where is the spring constant and is the displacement. With a mass attached it oscillates with a period of \textstyle \sqrt{k/m\ }. With two pendulums (labeled and ) of equal mass but possibly unequal lengths and connected by a spring, the total potential energy is : V = \frac{m}{2} \left( \frac{g}{L_\text{a}} x_\text{a}^2 + \frac{g}{L_\text{b}} x_\text{b}^2 + \frac{k}{m} (x_\text{b} - x_\text{a})^2 \right). This is a
quadratic form in and , which can also be written as a matrix product: : V = \frac{m}{2} \begin{pmatrix} x_\text{a} & x_\text{b} \end{pmatrix} \begin{pmatrix} \frac{g}{L_\text{a}} + \frac{k}{m} & -\frac{k}{m} \\ -\frac{k}{m} & \frac{g}{L_\text{b}} + \frac{k}{m} \end{pmatrix} \begin{pmatrix} x_\text{a} \\ x_\text{b} \end{pmatrix}. The matrix is real symmetric and so (by the
spectral theorem) it is
orthogonally diagonalizable. That is, there is an angle such that if we define : \begin{pmatrix} x_\text{a} \\ x_\text{b} \end{pmatrix} = \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} then : V = \frac{m}{2} \begin{pmatrix} x_1 \ x_2 \end{pmatrix} \begin{pmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} where and are the
eigenvalues of the matrix. The variables and describe normal modes which oscillate with frequencies of \sqrt{\lambda_1\,} and \sqrt{\lambda_2\,}. When the two pendulums are identical (), is 45°. The angle is analogous to the
Cabibbo angle (though that angle applies to quarks rather than neutrinos). When the number of oscillators (particles) is increased to three, the orthogonal matrix can no longer be described by a single angle; instead, three are required (
Euler angles). Furthermore, in the quantum case, the matrices may be
complex. This requires the introduction of complex phases in addition to the rotation angles, which are associated with
CP violation but do not influence the observable effects of neutrino oscillation. == Theory, graphically ==