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 matrix 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, to explain the neutrino oscillations predicted by Bruno Pontecorvo.

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_{e1}&U_{e2}&U_{e3}\\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 |U_αi|².

Various parametrizations of this matrix exist, 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 matrix is most commonly parameterized by three mixing angles (Θ₁₂, Θ₂₃ and Θ₁₃) and a single phase. Experimentally, the mixing angles were established to be approximately Θ₁₂=45 degrees, Θ₂₃=34 degrees, and Θ₁₃<4 degrees.

As a starting point, one recent particle physics course provided the following estimated values for the matrix (assuming the mixing angle Θ₁₃=0, which before 2011 was in good agreement with experiments, but thanks to T2K, Double Chooz and Daya Bay it is known to be around 4.4 deg):

${\begin{bmatrix}U_{e1}&U_{e2}&U_{e3}\\U_{\mu 1}&U_{\mu 2}&U_{\mu 3}\\U_{\tau 1}&U_{\tau 2}&U_{\tau 3}\end{bmatrix}}={\begin{bmatrix}0.85&0.53&0\\-0.37&0.60&0.71\\0.37&-0.60&0.71\end{bmatrix}}\$.

See also

• Neutrino oscillations
• CKM matrix