If we plot the nine entries for the two factor matrix $R(a)R(b)$, we obtain a nice grid in the complex plane where it is easy to see that the four interesting phases are given by

$\textrm{tan}^{-1} (a)$, $\textrm{tan}^{-1} (b)$, $\textrm{tan}^{-1} (1/a)$, $\textrm{tan}^{-1} (1/b)$.

Observe that the angle between the $(-1 + bi)$ line and the $(ab + ai)$ line is $\pi / 2$. Similarly for the other two interesting angles. That is, the eight non zero matrix entries form four sets of right angles. Thus the parameters $a$ and $b$ account for essentially only two independent phases in the mixing matrix $R(a)R(b)$.

Note also that for the CKM values, the three phases $(tb)$, $(bc)$ and $(cs)$ cancel out, leaving the $(ts)$ term to account for $2 \beta_s = 0.04$. Moreover, the $(cs)$ phase is $\pi / 2$, and $(tb)$ and $(bc)$ sum to $\pi /2$.

8 years ago

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