# Pontecorvo–Maki–Nakagawa–Sakata matrix

 Flavour in particle physics Flavour quantum numbers: Isospin: I or I3 Charm: C Strangeness: S Topness: T Bottomness: B′ Related quantum numbers: Baryon number: B Lepton number: L Weak isospin: T or T3 Electric charge: Q X-charge: X Combinations: Hypercharge: Y Y = (B + S + C + B′ + T) Y = 2 (Q − I3) Weak hypercharge: YW YW = 2 (Q − T3) X + 2YW = 5 (B − L) Flavour mixing CKM matrix PMNS matrix Flavour complementarity This box:

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 matrix[note 1] which contains information on the mismatch of quantum states of leptons when they propagate freely and when they take part in the weak interactions. It is important in the understanding of neutrino oscillations. This matrix was introduced in 1962 by Ziro Maki, Masami Nakagawa and Shoichi Sakata,[1] to explain the neutrino oscillations predicted by Bruno Pontecorvo.[2][3]

## The matrix

For three generations of leptons, the matrix can be written as:

$\begin{bmatrix} {\nu_e} \\ {\nu_\mu} \\ {\nu_\tau} \end{bmatrix} = \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} \nu_1 \\ \nu_2 \\ \nu_3 \end{bmatrix}. \$

On the left are the neutrino fields participating in the weak interaction, and on the right is the PMNS matrix along with a vector of the neutrino fields diagonalizing the neutrino mass matrix. The PMNS matrix describes the probability of a neutrino of given flavor α to be found in mass eigenstate i. These probabilities are proportional to |Uαi|2.

Various parametrizations of this matrix exist,[4] however 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). The PMNS matrix is most commonly parameterized by three mixing angles(Θ12, Θ23 and Θ13) and a single phase called δCP 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).

Experimentally, the mixing angles were established to be approximately Θ12=34 degrees, Θ23=45 degrees, and Θ13=9.1 +/- 0.6 degrees (as of April 3, 2013).[5] The charge parity violating phase of the PMNS matrix and the mass hierarchy of the neutrino masses have not been determined experimentally and remain unsolved questions in physics that are the subject of multiple major ongoing experimental efforts to determine.[6] These mixing angles are much larger than the corresponding value of the CKM matrix for quarks, which means that while quark flavors mix with each other nearly minimally, neutrino flavors mix nearly maximally.

Based on less current data (28 June 2012) mixing angles are:[7]

$s_{12}^{2}=0.307\,,\; s_{23}^{2}=\begin{cases} 0.386 & (\mathrm{NH})\\ 0.392 & (\mathrm{IH}) \end{cases}\,,\ s_{13}^{2}=\begin{cases} 0.0241 & (\mathrm{NH})\\ 0.0244 & (\mathrm{IH}) \end{cases}\,,\ \delta=\begin{cases} 1.08\pi & (\mathrm{NH})\\ 1.09\pi & (\mathrm{IH}) \end{cases}$

where NH indicates $\Delta m^2>0$ normal hierarchy and IH $\Delta m^2<0$ inverted hierarchy in the mass spectrum with $\delta m^{2}=m_{2}^{2}-m_{1}^{2}>0$ and $\Delta m^{2}=m_{3}^{2}-(m_{1}^{2}+m_{2}^{2})/2$.

These values lead to following PMNS matrices:

$\mathbf{U}_{\mathrm{NH}}= \begin{bmatrix}0.822 & 0.547 & -0.150+0.0381\mathrm{i}\\ -0.356+0.0198\mathrm{i} & 0.704+0.0131\mathrm{i} & 0.614\\ 0.442+0.0248\mathrm{i} & -0.452+0.0166\mathrm{i} & 0.774 \end{bmatrix}$
$\mathbf{U}_{\mathrm{IH}}= \begin{bmatrix}0.822 & 0.547 & -0.150+0.0429\mathrm{i}\\ -0.354+0.0224\mathrm{i} & 0.701+0.0149\mathrm{i} & 0.618\\ 0.444+0.0278\mathrm{i} & -0.456+0.0186\mathrm{i} & 0.770 \end{bmatrix}.$

## Notes

1. ^ The PMNS matrix is not unitary in the seesaw model

## References

1. ^ Z. Maki, M. Nakagawa, and S. Sakata (1962). "Remarks on the Unified Model of Elementary Particles". Progress of Theoretical Physics 28: 870. Bibcode:1962PThPh..28..870M. doi:10.1143/PTP.28.870.
2. ^ B. Pontecorvo (1957). "Mesonium and anti-mesonium". Zh. Eksp. Teor. Fiz. 33: 549–551. reproduced and translated in Sov. Phys. JETP 6: 429. 1957.
3. ^ B. Pontecorvo (1967). "Neutrino Experiments and the Problem of Conservation of Leptonic Charge". Zh. Eksp. Teor. Fiz. 53: 1717. reproduced and translated in Sov. Phys. JETP 26: 984. 1968. Bibcode:1968JETP...26..984P.
4. ^ J.W.F. Valle (2006). "Neutrino physics overview". Journal of Physics: Conference Series 53: 473. arXiv:hep-ph/0608101. Bibcode:2006JPhCS..53..473V. doi:10.1088/1742-6596/53/1/031.
5. ^ The T2K Collaboration (3 April 2013). "Evidence of Electron Neutrino Appearance in a Muon Neutrino Beam". arXiv:1304.0841.
6. ^ R. Das and Jo˜ao Pulido (4 February 2013). "Long baseline neutrino experiments, mass hierarchy and δCP". arXiv:1302.0779.
7. ^ Fogli et al: Global analysis of neutrino masses, mixings and phases. 2012 http://arxiv.org/abs/1205.5254v3