Penguin trapping with isospin analysis and $\mathrm{CP}$ asymmetries in $B$ decays

Isospin relations are used to eliminate hadronic uncertainties in various $\mathrm{CP}$ asymmetries in $B^0$ decays. In addition to the simple triangle relations for the $\pi \pi$ mode, we study quadrilateral relations for $K\pi$ and pentagon relations for $\rho \pi$. A combined angular and isospin analysis is required for $\rho \rho$. These methods are useful also for three-body decays such as $K\pi \pi$. The magnitude of the penguin amplitude can be extracted in various modes. The theoretical principles behind this analysis can be experimentally tested through sum rules for decay rates prior to the measurement of $\mathrm{CP}$ asymmetries.