A measurement of the phase difference between the short and longdistance contributions to the $B^{+}\to K^{+}\mu^{+}\mu^{}$ decay is performed by analysing the dimuon mass distribution. The analysis is based on $pp$ collision data corresponding to an integrated luminosity of 3 $\rm fb^{1}$ collected by the LHCb experiment in 2011 and 2012. The longdistance contribution to the $B^{+}\to K^{+}\mu^{+}\mu^{}$ decay is modelled as a sum of relativistic BreitWigner amplitudes representing different vector meson resonances decaying to muon pairs, each with their own magnitude and phase. The measured phases of the $J/\psi$ and $\psi(2S)$ resonances are such that the interference with the shortdistance component in dimuon mass regions far from their pole masses is small. In addition, constraints are placed on the Wilson coefficients, $\mathcal{C}_{9}$ and $\mathcal{C}_{10}$, and the branching fraction of the shortdistance component is measured.
Reconstructed $ K ^+ \mu ^+\mu ^ $ mass of the selected $ B ^+ \rightarrow K ^+ \mu ^+\mu ^ $ candidates. The fit to the data is described in the text. 
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Efficiency to reconstruct, trigger and select simulated $ B ^+ \rightarrow K ^+ \mu ^+ \mu ^ $ decays as a function of the true dimuon mass. The efficiency is normalised to the efficiency at the $ { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu}$ meson mass. The band indicates the efficiency parameterisation used in this analysis and its statistical uncertainty. 
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Fits to the dimuon mass distribution for the four different phase combinations that describe the data equally well. The plots show cases where the $ { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu}$ and $\psi {(2S)}$ phases are both negative (top left); the $ { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu}$ phase is positive and the $\psi {(2S)}$ phase is negative (top right); the $ { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu}$ phase is negative and the $\psi {(2S)}$ phase is positive (bottom left); and both phases are positive (bottom right). The component labelled interference refers to the interference between the short and longdistance contributions to the decay. The $\chi^2$ value of the four solutions is almost identical, with a value of 110 for 78 degrees of freedom. 
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Twodimensional likelihood profile for the Wilson coefficients $\mathcal{C}_{9}$ and $\mathcal{C}_{10}$. The SM point is indicated by the blue marker. The intervals correspond to $\chi^2$ probabilities with two degrees of freedom. 
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Animated gif made out of all figures. 
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Resolution parameters of the different convolution regions in units of $ {\mathrm{ Me V /}c^2}$ . The $\alpha_{\rm l}$ and $\alpha_{\rm u}$ parameters are shared between the $ { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu}$ and $\psi {(2S)}$ regions. The parameters without uncertainties are fixed from fits to the simulated events. 
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Parameters describing the efficiency to trigger, reconstruct and select simulated $ B ^+ \rightarrow K ^+ \mu ^+\mu ^ $ decays as a function of $m_{\mu\mu}$. 
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Branching fractions and phases for each resonance in the fit for the four solutions of the $ { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu}$ and $\psi {(2S)}$ phases. Both statistical and systematic contributions are included in the uncertainties. There is a common systematic uncertainty of 4.5%, dominated by the uncertainty on the $ B ^+ \rightarrow { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} K ^+ $ branching fraction, which provides the normalisation for all measurements. 
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Coefficients of the form factor $f_{+}( q^2 )$ as introduced in Eq. 8 with both prior (from Ref. [42]) and posterior values shown. 
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Summary of systematic uncertainties. The branching fraction refers to the shortdistance SM contribution. A dash indicates that the uncertainty is negligible. 
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Created on 20 February 2021.