I Evaluating the quark neutrino mixing matrix

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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?
 
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@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.
 
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320px-Oscillations_muon_short.svg.png

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