Symmetry relations between angular observables in B0 →k*μ +μ - And the LHCb P5′ anomaly

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 Pi and PiCP of the angular distribution of the four-body decay B→K*(→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 P1, P2, and P4,5′. 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 of performing a full angular fit of the observables with existing data sets. An important consequence of the found relations is that a priori two different measurements, namely the measured position of the zero (q02) of the forward-backward asymmetry AFB and the value of P5′ evaluated at this same point, are related by P42(q02)+P52(q02)=1. Under the hypotheses of real Wilson coefficients and P4′ being SM-like, we show that the higher the position of q02, the smaller should be the value of P5′ evaluated at the same point. A precise determination of the position of the zero of AFB together with a measurement of P4′ (and P1) at this position can be used as an independent experimental consistency test of the anomaly in P5′. We also point out the existence of upper and lower bounds for P1, namely P5′2-1≤P1≤1-P4′2, which constrains the physical region of the observables. © 2014 American Physical Society.