We show that it is possible to accommodate the observed size of the phase in $B^0_s$--$bar B^0_s$, mixing in the framework of a model with violation of $3times 3$ unitarity. This violation is associated to the presence of a new $Q=2/3$ isosinglet qua
rk $T$, which mixes both with $t$ and $c$ and has a mass not exceeding 500 GeV. The crucial point is the fact that this framework allows for $chiequivarg(-V_{ts}V_{cb}V_{tb}^*V_{cs}^*)$ of order $lambda$, to be contrasted with the situation in the Standard Model, where $chi$ is constrained to be of order $lambda^2$. We point out that this scenario implies rare top decays $tto cZ$ at a rate observable at the LHC and $|V_{tb}|$ significantly different from unity. In this framework, one may also account for the observed size of $D^0$--$bar D^0$ mixing without having to invoke long distance contributions. It is also shown that in the present scenario, the observed size of $D^0$--$bar D^0$ mixing constrains $chi^primeequivarg(-V_{cd}V_{us}V_{cs}^*V_{ud}^*)$ to be of order $lambda^4$, which is significantly smaller than what is allowed in generic models with violations of $3times 3$ unitarity.
The extraction of the weak phase $alpha$ from $Btopipi$ decays has been controversial from a statistical point of view, as the frequentist vs. bayesian confrontation shows. We analyse several relevant questions which have not deserved full attention
and pervade the extraction of $alpha$. Reparametrization Invariance proves appropriate to understand those issues. We show that some Standard Model inspired parametrizations can be senseless or inadequate if they go beyond the minimal Gronau and London assumptions: the single weak phase $alpha$ just in the $Delta I=3/2$ amplitudes, the isospin relations and experimental data. Beside those analyses, we extract $alpha$ through the use of several adequate parametrizations, showing that there is no relevant discrepancy between frequentist and bayesian results. The most relevant information, in terms of $alpha$, is the exclusion of values around $alphasim pi/4$; this result is valid in the presence of arbitrary New Physics contributions to the $Delta I=1/2$ piece.
