New physics effects in tree-level decays

We critically review the assumption that no new physics is acting in tree-level $B$-meson decays and study the consequences for the ultimate precision in the direct determination of the CKM angle $\gamma$. In our exploratory study we find that sizable universal new physics contributions, $\mathrm{\Delta }{C}_{1,2}$, to the tree-level Wilson coefficients $C_{1,2}$ of the effective Hamiltonian describing weak decays of the $b$ quark are currently not excluded by experimental data. In particular we find that Im $\mathrm{\Delta }{C}_1$ and Im $\Delta C_2 $caneasilybeoforder$\pm 10%$ without violating any constraints from data. Such a size of new physics effects in $C_1$ and $C_2$ corresponds to an intrinsic uncertainty in the CKM angle $\gamma$ of the order of $|\delta \gamma| \approx 4^\circ$, which is slightly below the current experimental precision. The accuracy in the determination of $\gamma$ can be improved by putting stronger constraints on the tree-level Wilson coefficients, in particular $C_1$. To this end we suggest a more refined theoretical study as well as a more precise measurements of the observables that currently provide the strongest bounds on hypothetical new weak phases in $C_1$ and $C_2$. We note that the semi-leptonic CP asymmetries seem to have the best prospect for improving the bound on the weak phase in $C_1$.