Implication of the running mass of the ${\rho }^0$ meson for the dilepton mass spectrum and the ${\mu }^{+}{\mu }^{-}{/e}^{+}{e}^{-}$ ratio in $K^{+}\to {\pi }^{+}{l}^{+}{l}^{-}$ decays

We make an attempt to resolve the discrepancy of the observed $e^{+}{e}^{-}$ mass spectrum in the $K^{+}\to {\pi }^{+}{e}^{+}{e}^{-}$ decay with that predicted by meson dominance. To this end we investigate the properties of the ${\rho }^0$ propagator. We use dispersion relations to evaluate the running mass $m_{\rho }^2\mathrm{}\left(t\right)\mathrm{}$ of the ${\rho }^0$ resonance without adjustable parameters. To improve the convergence of the dispersion integral, the momentum dependence of strong vertices is taken from the flux-tube-breaking model of Kokoski and Isgur. The obtained behavior of $m_{\rho }^2\mathrm{}\left(t\right)\mathrm{}$ at small momentum squared t makes the $K^{+}\to {\pi }^{+}{e}^{+}{e}^{-}$ form factor rise faster with increasing t than in the original meson dominance calculation and more in agreement with the published data. As a consequence, the meson dominance prediction of the ${\mu }^{+}{\mu }^{-}{/e}^{+}{e}^{-}$ ratio changes slightly, from 0.224 to 0.236. We do not see any possibility to accommodate into the meson dominance approach an even steeper $e^{+}{e}^{-}$ spectrum, indicated by the preliminary data of the E865 Collaboration at BNL AGS.