Testing Explanations of the $B\to \varphi K^{\ast }$ Polarization Puzzle

$B\to \varphi K^{\ast }$ ($\text{btos}$) is three separate decays, one for each polarization of the final-state vector mesons (one longitudinal, two transverse). It is observed that the fraction of transverse decays, $\text{fT}$, and the fraction of longitudinal decays, $\text{fL}$, are roughly equal: $\text{fTfL}\simeq 1$, in opposition to the naive expectation that $\text{fT}\ll \text{fL}$. If one requires a single explanation of all polarization puzzles, two possibilities remain within the standard model: penguin annihilation and rescattering. In this paper we examine the predictions of these two explanations for $\text{fTfL}$ in $\text{btod}$ decays. In $B \to \rho\rho$decays,only$\bd \to \rho^0\rho^0$canpossiblyexhibitalarge$\fTfL$. In B decays related by U-spin, we find two promising possibilities: (i) $B^{+}\to K^{\ast 0}{\rho }^{+}$ ($\text{btos}$) and $B^{+}\to {\text{Kbar}}^{\ast 0}{K}^{\ast +}$ ($\text{btod}$) and (ii) $\text{bs}\to K^{\ast 0}{\text{Kbar}}^{\ast 0}$ ($\text{btos}$) and $\text{bd}\to {\text{Kbar}}^{\ast 0}{K}^{\ast 0}$ ($\text{btod}$). The measurement of $\text{fTfL}$ in these pairs of decays will allow us to test penguin annihilation and rescattering. Finally, it is possible to distinguish penguin annihilation from rescattering by performing a time-dependent angular analysis of $\text{bd}\to {\text{Kbar}}^{\ast 0}{K}^{\ast 0}$.