Generalized analysis on $B\to K^* \rho$ within and beyond the Standard Model $-$ Can it help understand the $B \to K \pi$ puzzle?

We study $B \to K^* \rho$ modes that are analogues of the much studied $B\to K \pi$ modes with B decaying to two vector mesons instead of pseudoscalar mesons, using topological amplitudes in the quark diagram approach. We show how $B \to K^*\rho$ modes can be used to obtain many more observables than those for $B \to K \pi$ modes, even though the quark level subprocesses of both modes are exactly the same. All the theoretical parameters (except for the weak phase $\gamma$), such as the magnitudes of the topological amplitudes and their strong phases, can be determined in terms of the observables without any model-dependent assumption. We demonstrate how $B\to K^*\rho$ can also be used to verify if there exist any relations between theoretical parameters, such as the hierarchy relations between the topological amplitudes and possible relations between the strong phases. Conversely, if there exist reliable theoretical estimates of amplitudes and strong phases, the presence of New physics could be probed. We show that if the tree and color-supressed tree are related to the electroweak penguins and color-supressed electroweak penguins, it is not only possible to verify the validity of such relations but also to have a clean measurement of New Physics parameters. We also present a numerical study to examine which of the observables are more sensitive to New Physics.