A complete description of P- and S-wave contributions to the $B^0\to K^{+}\pi^{-}\ell^{+}\ell^{-}$ decay

TL;DR

This paper provides a comprehensive analysis of P- and S-wave contributions in the decay $B^0 o K^{+}\pi^{-}\ell^{+}\ell^{-}$, developing new observables and symmetry relations to improve the understanding of potential new physics effects.

Contribution

It introduces a complete set of experimentally accessible observables, including new S-wave observables, and establishes symmetry relations to reduce dependence on uncertain form factors.

Findings

Defined the full decay distribution with P- and S-wave contributions.

Constructed two theoretically clean S-wave observables.

Derived bounds on S-wave observables and identified zero-crossing points.

Abstract

In this paper we present a detailed study of the four-body decay $B^0\to K^{+}\pi^{-}\ell^{+}\ell^{-}$, where tensions with the Standard Model predictions have been observed. Our analysis of the decay with P- and S-wave contributions to the $K^{+}\pi^{-}$ system develops a complete understanding of the symmetries of the distribution, in the case of massless and massive leptons. In both cases, the symmetries determine relations between the observables in the $B^0\to K^{+}\pi^{-}\ell^{+}\ell^{-}$ decay distribution. This enables us to define the complete set of observables accessible to experiments, including several that have not previously been identified. The new observables arise when the decay rate is written differentially with respect to $m_{K\pi}$. We demonstrate that experiments will be able to fit this full decay distribution with currently available data sets and investigate the sensitivity to new physics scenarios given the experimental precision that is expected in the future. The symmetry relations provide a unique handle to explore the behaviour of S-wave observables by expressing them in terms of P-wave observables, therefore minimising the dependence on poorly-known S-wave form factors. Using this approach, we construct two theoretically clean S-wave observables and explore their sensitivity to new physics. By further exploiting the symmetry relations, we obtain the first bounds on the S-wave observables using two different methods and highlight how these relations may be used as cross-checks of the experimental methodology. We identify a zero-crossing point that would be at a common dilepton invariant mass for a subset of P- and S-wave observables, and explore the information on new physics and hadronic effects that this zero point can provide.