At kneemo's request, let us recall how triality forces the non standard CKM phase of $\textrm{sin} 2 \beta = 0.649$. In the CKM matrix $V_{ij}$ we are interested in phases of the form

$\beta = \textrm{arg} \frac{- V_{cd} V_{cb}^{*}}{V_{td} V_{tb}^{*}}$

Observe that multiplying the complex matrix $V_{ij}$ by any phase $\phi$ does not alter these characteristic phases. Now triality tells us that the $3 \times 3$ matrix should be built from three $2 \times 2$ factors. Since $V_{ij}$ is magic (along rows and columns) the $2 \times 2$ blocks must all be magic. That is, we start with general $2 \times 2$ complex circulants. Multiplication by a phase $\phi$ allows us to make the diagonal elements real. After division by a normalization constant, there is still potentially an arbitrary phase in the off diagonal slots. However, by scaling the diagonal one can always adjust this phase to some fixed value. We choose the fixed phase $i$, so that the $2 \times 2$ factors look like:

This phase is motivated by the $R_2$ MUB circulant, given by the eigenvectors of the Pauli matrix $\sigma_{Y}$. This is the case $a = 1$, and this factor appears in the very elementary tribimaximal neutrino mixing matrix. For three parameters $a$, $b$ and $c$ in a triple product of distinct $3 \times 3$ factors $R_{ij}$, we now fit the nine real CKM entries. The parameters required are $-0.231$, $24.00$ and $0.0035$.

All CKM phases are now fixed by the triple product. The value of $\sin 2 \beta$ is $0.649$, which any educated child may easily verify.

6 years ago

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