B and Bs decays into three pseudoscalar mesons and the determination of the angle of the unitarity triangle

We reconsider two classical proposals for the determination of the angle $\ensuremath{\gamma}$ of the unitarity triangle: ${B}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}{\ensuremath{\chi}}_{c0}{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}$ and ${B}_{s}\ensuremath{\rightarrow}{\ensuremath{\rho}}^{0}{K}_{S}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{K}_{S}.$ We point out the relevance, in both cases, of nonresonant amplitudes, where the ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ pair is produced by weak decay of a ${B}^{*}{(J}^{P}{=1}^{\ensuremath{-}})$ or ${B}_{0}{(J}^{P}{=0}^{+})$ off-shell meson. In particular, for the B decay channel, the inclusion of the ${B}_{0}$ pole completes some previous analyses and confirms their conclusions, provided a suitable cut in the Dalitz plot is performed; for the ${B}_{s}$ decay the inclusion of the ${B}^{*},{B}_{0}$ amplitudes enhances the role of the tree diagrams as compared to penguin amplitudes, which makes the theoretical uncertainty related to the ${B}_{s}\ensuremath{\rightarrow}{\ensuremath{\rho}}^{0}{K}_{S}$ decay process less significant. While the first method is affected by theoretical uncertainties, the second one is cleaner, but its usefulness will depend on the available number of events to perform the analysis.