Evaluating the quark neutrino mixing matrix

[James1238765]

TL;DR

How to resolve the complex trigonometric exponential $\exp{i\sigma{cp}}$ in the CKM and PMNS matrix parameters?

The mixing of the 3 generations of fermions are tabulated into the CKM matrix for quarks:

$$ \begin{bmatrix}
c_{12}c_{13} & s_{12}c_{13} & s_{13}e^{-i\sigma_{13}} \\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\sigma_{12}} & c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\sigma_{13}} & s_{23}c_{13} \\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\sigma_{13}} & -c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\sigma_{13}} & c_{23}c_{13}
\end{bmatrix} $$

where c12 is shorthand for the $cos(\sigma_{12})$ function, and s is shorthand for the $sin(\sigma_{12})$ function, and with experimentally fitted values as follows:

$$ \begin{bmatrix}
0.97370 & 0.2245 & 0.00382 \\
0.221 & 0.987 & 0.041 \\
0.008 & 0.0388 & 1.013
\end{bmatrix} $$

Similarly the PMNS matrix tabulates the mixing statistics for neutrinos:

$$ \begin{bmatrix}
c_{12}c_{13} & s_{12}c_{13} & s_{13}e^{-i\sigma_{cp}} \\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\sigma_{cp}} & c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\sigma_{cp}} & s_{23}c_{13} \\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\sigma_{cp}} & -c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\sigma_{cp}} & c_{23}c_{13}
\end{bmatrix} $$

with experimentally fitted values as follows:

$$ \begin{bmatrix}
0.801 & 0.513 & 0.143 \\
0.232 & 0.459 & 0.629 \\
0.260 & 0.470 & 0.609
\end{bmatrix} $$

Could anyone explain how the complex trigonometric $e^{i\sigma_{13}}$ and $e^{i\sigma_{cp}}$ having the form

$$e^{i\sigma_{13}} = \cos \sigma_{13} + i \sin \sigma_{13} $$

can morph into real values in the final numerical matrices, please?

 

[vanhees71]

Where did you get these values from? It's most probably giving the moduli of the CKM/PMNS matrices as in the particle data booklet:

https://pdg.lbl.gov/2022/web/viewer.html?file=../reviews/rpp2022-rev-ckm-matrix.pdf

 

[James1238765]

@vanhees71 thank you. Very oddly the $e^{-i\sigma_{13}}$ has no constant fixed value throughout.

$\sigma_{13}$ is a fixed numerical angle at 68.8 degrees, but even if we set $e^{-i\sigma_{13}}$ to a particular value to correctly match a particular matrix element, the other matrix elements having $e^{-i\sigma_{13}}$ term will still output wrong answers.

So $e^{-i\sigma_{13}}$ changes for every matrix element calculation. It seems never defined what $e^{-i\sigma_{13}}$ means in the above paper, and other papers like [this] , so I guess I will just leave it there for now.

 

[James1238765]

Neutrino flavor oscillations are clock-like precise as a function of distance. For two-neutrinos oscillation:

$$Prob_{switch} = \sin^2{(2\theta})\sin^2{(\frac{\triangle m^2L}{4E})}$$

which is sinusoidal with respect to distance traveled L.

(source: wiki)