Polarisation fractions in $B\to V_1 V_2$: U-Spin constraints and new physics signatures
We investigate the decays of $B$ mesons, {\em i.e.}, $B_d$, $B_s$, $B^+$, and their antiparticles, to two light vector mesons ($B \to V_1V_2$). We use the SU(2) U-spin symmetry, which relates $ΔS = 0$ and $ΔS = 1$ decay amplitudes through the interchange $d \leftrightarrow s$ and is an approximate symmetry of the Standard Model (SM), to relate the helicity amplitudes of these decays. Treating all the helicity amplitudes for these decays, and hence the reduced matrix elements, as free parameters, we find an acceptable solution within the SM, although this is driven by the fact that the number of observables is smaller than what is needed for a meaningful fit. To reduce the number of free parameters, we then use some apparently reasonable and theoretically motivated approximations, like the dominance of factorisable contributions over the non-factorisable ones, and hence a distinct hierarchy between the helicity amplitudes. We find that once the assumption of hierarchy is imposed, there is no acceptable solution. This is due to the longitudinal polarisation fractions in almost all $ΔS = 1$ decays. This is particularly true for $B_s \to K^{*0} \overline{K^{*0}}$, for which the individual disagreement with U-spin based expectation is more than $7σ$. Within SM, the only effective resolution would be to allow for large nonfactorisable contributions to all these decay amplitudes. We also explore whether some new physics (NP) in the $b\to s$ sector that does not respect the hierarchy among the helicity amplitudes can reduce the tension for all the $ΔS=1$ modes. While such an option helps, we find that for simplistic new physics scenarios, the tension still exists and the fit remains poor enough, if the hierarchy exists among the SM amplitudes. Some possible scenarios for a complete solution of the puzzle are also suggested.