How isospin violation mocks “new” physics: ${\pi }^0-\eta ,{\eta }^{\prime }$ mixing in $\overrightarrow{B}\pi \pi$ decays

An isospin analysis of $\overrightarrow{B}\pi \pi$ decays yields $\sin 2\alpha ,$ where $\alpha \equiv \arg \left[-V_{\mathrm{td}}{V}_{\mathrm{tb}}^{*}{/(V}_{\mathrm{ud}}{V}_{\mathrm{ub}}^{*}\right)]$ and $V_{\mathrm{ij}}$ is a CKM matrix element, without hadronic uncertainty if isospin is a perfect symmetry. Isospin, however, is broken not only by electroweak effects but also by the u and d quark mass difference. The latter drives ${\pi }^0-\eta ,{\eta }^{\prime }$ mixing and converts the isospin-perfect triangle relation between the $\overrightarrow{B}\pi \pi$ amplitudes to a quadrilateral. The combined isospin-violating effects impact the extracted value of $\sin 2\alpha$ in a significant manner, particularly if the latter is small.