- #1
Doofy
- 74
- 0
I have been trying to find out about the MNS / PMNS matrix that governs neutrino oscillation:
[tex] \left( \begin{array}{c} \nu_{e} \\ \nu_{\mu} \\ \nu_{\tau} \end{array} \right) = \left( \begin{array}{ccc} 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{array} \right) \left( \begin{array}{c} \nu_{1} \\ \nu_{2} \\ \nu_{3} \end{array} \right) [/tex]
I am aware that there is more than one parameterization exists, but the most common one seems to be the one stated on wikipedia:
[tex]\left( \begin{array}{ccc} 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{array} \right) \left( \begin{array}{ccc} e^{i\alpha_{1}/2} & 0 & 0 \\ 0 & e^{i\alpha_{2}/2} & 0 \\ 0 & 0 & 1 \end{array} \right) [/tex]
My problem is that every paper, every website, every textbook I have consulted all just seem to state that this is what the matrix looks like, but offer no derivation / explanation of where it comes from. I feel uncomfortable just blindly accepting this.
The [itex]s_{ij}, c_{ij}[/itex] parts I am fine with, they are just abbreviations for [itex]sin\theta_{ij}, cos\theta_{ij}[/itex] and come from having multiplied together rotation matrices for rotations around 3 axes. Nothing unusual there.
My issue is with the [itex]e^{i\delta}[/itex] and [itex]e^{i\alpha}[/itex] parts. I am aware that the [itex]e^{i\delta}[/itex] part has something to do with allowing for possible CP violation, but I need to understand how it got there as it means very little to me at this moment.
I know even less about the [itex]e^{i\alpha}[/itex] parts - something to do with the possible Majorana-ness of neutrinos?
I'm so surprised by how little information there is out there on this little topic, especially given the amount of money being spent on experiments to measure these matrix elements. What's going on here?
[tex] \left( \begin{array}{c} \nu_{e} \\ \nu_{\mu} \\ \nu_{\tau} \end{array} \right) = \left( \begin{array}{ccc} 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{array} \right) \left( \begin{array}{c} \nu_{1} \\ \nu_{2} \\ \nu_{3} \end{array} \right) [/tex]
I am aware that there is more than one parameterization exists, but the most common one seems to be the one stated on wikipedia:
[tex]\left( \begin{array}{ccc} 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{array} \right) \left( \begin{array}{ccc} e^{i\alpha_{1}/2} & 0 & 0 \\ 0 & e^{i\alpha_{2}/2} & 0 \\ 0 & 0 & 1 \end{array} \right) [/tex]
My problem is that every paper, every website, every textbook I have consulted all just seem to state that this is what the matrix looks like, but offer no derivation / explanation of where it comes from. I feel uncomfortable just blindly accepting this.
The [itex]s_{ij}, c_{ij}[/itex] parts I am fine with, they are just abbreviations for [itex]sin\theta_{ij}, cos\theta_{ij}[/itex] and come from having multiplied together rotation matrices for rotations around 3 axes. Nothing unusual there.
My issue is with the [itex]e^{i\delta}[/itex] and [itex]e^{i\alpha}[/itex] parts. I am aware that the [itex]e^{i\delta}[/itex] part has something to do with allowing for possible CP violation, but I need to understand how it got there as it means very little to me at this moment.
I know even less about the [itex]e^{i\alpha}[/itex] parts - something to do with the possible Majorana-ness of neutrinos?
I'm so surprised by how little information there is out there on this little topic, especially given the amount of money being spent on experiments to measure these matrix elements. What's going on here?