Generalised helicity formalism, higher moments and the $B \to K_{J_K}(\to K \pi) \bar{\ell}_1 \ell_2$ angular distributions

Abstract

We generalise the Jacob-Wick helicity formalism, which applies to sequential decays, to effective field theories of rare decays of the type $B \to K_{J_K}(\to K \pi) \bar{\ell}_1 \ell_2$. This is achieved by reinterpreting local interaction vertices $\bar b \Gamma'_{\mu_1 ..\mu_n} s \bar \ell \Gamma^{\mu_1 ..\mu_n} \ell$ as a coherent sum of $1 \to 2$ processes mediated by particles whose spin ranges between zero and $n$. We illustrate the framework by deriving the full angular distributions for $B \to K\bar{\ell}_1 \ell_2$ and $B \to K^*(\to K \pi) \bar{\ell}_1 \ell_2$ for the complete dimension-six effective Hamiltonian for non-equal lepton masses. Amplitudes and decay rates are expressed in terms of Wigner rotation matrices, leading naturally to the method of moments in various forms. We discuss how higher-spin operators and QED corrections alter the standard angular distribution used throughout the literature, potentially leading to differences between the method of moments and the likelihood fits. We propose to diagnose these effects by assessing higher angular moments. These could be relevant in investigating the nature of the current LHCb anomalies in $R_K = {\cal B}( B \to K \mu^+\mu^-) /{\cal B}( B \to K e^+e^-)$ as well as angular observables in $B \to K^* \mu^+\mu^-$.