Hunting for $B^+\to K^+ \tau^+\tau^-$ imprints on the $B^+ \to K^+ \mu^+\mu^-$ dimuon spectrum.

We investigate the possibility of indirectly constraining the $B^{+}\to K^{+}\tau^+\tau^-$ decay rate using precise data on the $B^{+}\to K^{+}\mu^+\mu^-$ dimuon spectrum. To this end, we estimate the distortion of the spectrum induced by the $B^{+}\to K^{+}\tau^+\tau^-\to K^{+} \mu^+\mu^-$ re-scattering process, and propose a method to simultaneously constrain this (non-standard) contribution and the long-distance effects associated to hadronic intermediate states. The latter are constrained using the analytic properties of the amplitude combined with data and perturbative calculations. Finally, we estimate the sensitivity expected at the LHCb experiment with present and future datasets. We find that constraints on the branching fraction of $O(10^{-3})$, competitive with current direct bounds, can be achieved with the current dataset, while bounds of $O(10^{-4})$ could be obtained with the LHCb upgrade-II luminosity.