$e^{+}{e}^{-}\to K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$, $K^{+}{K}^{-}{\pi }^0{\pi }^0$ and $K^{+}{K}^{-}{K}^{+}{K}^{-}$ cross sections measured with initial-state radiation

We study the processes $e^{+}{e}^{-}\to K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}\gamma$, $K^{+}{K}^{-}{\pi }^0{\pi }^0\gamma$ and $K^{+}{K}^{-}{K}^{+}{K}^{-}\gamma$, where the photon is radiated from the initial state. About 34 600, 4400 and 2300 fully reconstructed events, respectively, are selected from $232{\mathrm{fb}}^{-1}$ of BABAR data. The invariant mass of the hadronic final state defines the effective $e^{+}{e}^{-}$ center-of-mass energy, so that the $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}\gamma$ data can be compared with direct measurements of the $e^{+}{e}^{-}\to K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ reaction; no direct measurements exist for the $e^{+}{e}^{-}\to K^{+}{K}^{-}{\pi }^0{\pi }^0$ or $e^{+}{e}^{-}\to K^{+}{K}^{-}{K}^{+}{K}^{-}$ reactions. Studying the structure of these events, we find contributions from a number of intermediate states, and we extract their cross sections where possible. In particular, we isolate the contribution from $e^{+}{e}^{-}\to \varphi (1020)f_0(980)$ and study its structure near threshold. In the charmonium region, we observe the $J/\psi$ in all three final states and several intermediate states, as well as the $\psi (2S)$ in some modes, and measure the corresponding branching fractions. We see no signal for the $Y(4260)$ and obtain an upper limit of ${\mathcal{B}}_{Y(4260)\to \varphi {\pi }^{+}{\pi }^{-}}·{\Gamma }_{ee}^Y<0.4\mathrm{eV}$ at 90% C.L.