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The $e^+ e^-\to 2(π^+π^-)π^0$, 2(π^+π^-)η$, $K^+ K^-π^+π^-π^0$ and $K^+ K^-π^+π^-η$ Cross Sections Measured with Initial-State Radiation

We study the processes $e^+ e^-\to 2(π^+π^-)π^0γ$, $2(π^+π^-)ηγ$, $K^+ K^-π^+π^-π^0γ$ and $K^+ K^-π^+π^-ηγ$ with the hard photon radiated from the initial state. About 20000, 4300, 5500 and 375 fully reconstructed events, respectively, are selected from 232 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 obtained cross sections from the threshold to about 5 GeV can be compared with corresponding direct \epem measurements, currently available only for the $ηπ^+π^-$ and $ωπ^+π^-$ submodes of the $e^+ e^-\to 2(π^+π^-)π^0$ channel. 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 $e^+ e^-\toω(782)π^+π^-$ and study the $ω(1420)$ and $ω(1650)$ resonances. In the charmonium region, we observe the $J/ψ$ in all these final states and several intermediate states, as well as the $ψ(2S)$ in some modes, and we measure the corresponding branching fractions.