Measurement of the phase difference between short- and long-distance amplitudes in the {{B} ^+} \!\rightarrow {{{K}} ^+} {\mu ^+\mu ^-} decay

A measurement of the phase difference between the short- and long-distance contributions to the \({{B} ^+} \!\rightarrow {{{K}} ^+} {\mu ^+\mu ^-} \) decay is performed by analysing the dimuon mass distribution. The analysis is based on pp collision data corresponding to an integrated luminosity of 3\(\mathrm{\,fb}^{-1}\) collected by the LHCb experiment in 2011 and 2012. The long-distance contribution to the \({{B} ^+} \!\rightarrow {{{K}} ^+} {\mu ^+\mu ^-} \) decay is modelled as a sum of relativistic Breit–Wigner 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 short-distance 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 short-distance component is measured.