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

TL;DR

This paper extends the helicity formalism to effective field theories of rare B meson decays, deriving comprehensive angular distributions, and discusses how higher-spin operators and QED corrections impact the analysis of current anomalies.

Contribution

It generalizes the Jacob-Wick helicity formalism to include higher-spin operators and QED corrections in rare B decay angular distributions, providing a new framework for analysis.

Findings

Derived full angular distributions for B to K and B to K* decays.

Expressed amplitudes in terms of Wigner rotation matrices and moments.

Discussed potential impacts of higher-spin operators and QED corrections on angular analyses.

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 $B \to K^* \mu^+\mu^-$.