$e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$, $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$, and $K^{+}{K}^{-}{K}^{+}{K}^{-}$ cross sections at center-of-mass energies 0.5–4.5 GeV measured with initial-state radiation

We study the process $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}\gamma$, with a hard photon radiated from the initial state. About 60 000 fully reconstructed events have been selected from $89{\mathrm{f}\mathrm{b}}^{-1}$ of BABAR data. The invariant mass of the hadronic final state defines the effective $e^{+}{e}^{-}$ $e^{+}{e}^{-}$ center-of-mass energy, so that these data can be compared with the corresponding direct $e^{+}{e}^{-}$ measurements. From the $4\pi$ -mass spectrum, the cross section for the process $e^{+}{e}^{-}$ $\to {\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$ is measured for center-of-mass energies from 0.6 to 4.5 $\mathrm{G}\mathrm{e}\mathrm{V}$. The uncertainty in the cross section measurement is typically 5%. We also measure the cross sections for the final states $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ and $K^{+}{K}^{-}{K}^{+}{K}^{-}$. We observe the $J/\psi$ in all three final states and measure the corresponding branching fractions. We search for $X(3872)$ in $J/\psi (\to {\mu }^{+}{\mu }^{-}){\pi }^{+}{\pi }^{-}$ and obtain an upper limit on the product of the $e^{+}{e}^{-}$ width of the $X(3872)$ and the branching fraction for $X(3872)\to J/\psi {\pi }^{+}{\pi }^{-}$.