A new relation between the zero of $A_{FB}$ in $B^0 \to K^* μ^+μ^-$ and the anomaly in $P_5^\prime$
We present two exact relations, valid for any dilepton invariant mass region (large and low-recoil) and independent of any effective Hamiltonian computation, between the observables $P_i$ and $P_i^{CP}$ of the angular distribution of the 4-body decay $B \to K^*(\to Kπ) l^+l^-$. These relations emerge out of the symmetries of the angular distribution. We discuss the implications of these relations under the (testable) hypotheses of no scalar or tensor contributions and no New Physics weak phases in the Wilson coefficients. Under these hypotheses there is a direct relation among the observables $P_{1}$,$P_2$ and $P_{4,5}^\prime$. This can be used as an independent consistency test of the measurements of the angular observables. Alternatively, these relations can be applied directly in the fit to data, reducing the number of free parameters in the fit. This opens up the possibility to perform a full angular fit of the observables with existing datasets. An important consequence of the found relations is that a priori two different measurements, namely the measured position of the zero ($q_0^2$) of the forward-backward asymmetry $A_{FB}$ and the value of $P_5^\prime$ evaluated at this same point, are related by $P_4^2(q_0^{2})+P_5^2(q_0^{2})=1$. Under the hypotheses of real Wilson coefficients and $P_4^\prime$ being SM-like, we show that the higher the position of $q_0^{2}$ the smaller should be the value of $P_5^\prime$ evaluated at the same point. A precise determination of the position of the zero of $A_{FB}$ together with a measurement of $P_4^\prime$ (and $P_1$) at this position can be used as an independent experimental test of the anomaly in $P_5^\prime$. We also point out the existence of upper and lower bounds for $P_1$, namely $P_5^{\prime 2}-1 \leq P_1 \leq 1-P_4^{\prime 2}$, which constraints the physical region of the observables.