QCD sum rules and the properties of $\rho$- and $\phi$-meson excited states

The recently proposed method of extracting more detailed information on $\rho$- and $\phi$-meson properties from the Laplace QCD sum rules is extended to include also the lowest-lying radial excitations in both channels. The approach is based on a simultaneous numerical matching of the corresponding sum rule and its derivative in a suitable range of the sum-rule parameter $M$ as well as on the use of smooth parametrizations for the hadronic cross sections $\sigma (e^{+}{e}^{-}\to 2\pi , 4\pi )$ and $\sigma (e^{+}{e}^{-}\to K\overline{K})$. The predicted masses, widths, and some other decay characteristics are close to the experimental values.