Let us recall why the mixing matrices should be expressed as a magic matrix. As Carl Brannen points out, and as was proved by Philip Gibbs in the $3 \times 3$ case, a unitary matrix can be written in the form

$D_1 M D_2$

for a magic matrix $M$, with $D_{1}$ and $D_{2}$ phased diagonals. That is, $D_{1}$ multiplies each row of $M$ by a phase, and $D_{2}$ multiplies each column by a phase. But the CKM parameters of most interest ($\alpha$, $\beta$ and $\gamma$) are invariant under these row and column rephasings. For instance, in the definition

$\beta = \textrm{arg} \frac{- V_{cd} \overline{V}_{cb}}{V_{td} \overline{V}_{tb}}$

we see that the charm quark row contributes two canceling phases: one for $V_{cd}$ and the complex conjugate for $\overline{V}_{cb}$. Similarly for all the other phases in $D_1$ and $D_2$. Thus it is proved that these CKM parameters can be encoded in a magic matrix, as often discussed. As previously noted, the resulting value for $\beta$ is non standard, but in good agreement with experiment.

6 years ago

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