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

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