Cross sections for the reactions e ⁺e ^-→K ⁺K ^-π ⁺π ^-, K ⁺K ^-π0π0, and K ⁺K ^-K ⁺K ^- measured using initial-state radiation events

We study the processes e ⁺e ^-→K ⁺K ^-π ⁺π ^-γ, K ⁺K ^-π0π0γ, and K ⁺K ^-K ⁺K ^-γ, where the photon is radiated from the initial state. About 84000, 8000, and 4200 fully reconstructed events, respectively, are selected from 454fb ^-1 of BABAR data. The invariant mass of the hadronic final state defines the e ⁺e ^- center-of-mass energy, so that the K ⁺K ^-π ⁺π ^-γ data can be compared with direct measurements of the e ⁺e ^-→K ⁺K ^-π ⁺π ^- reaction. No direct measurements exist for the e ⁺e ^-→K ⁺K ^-π0π0 or e ⁺e ^-→K ⁺K ^-K ⁺K ^- reactions, and we present an update of our previous result based on a data sample that is twice as large. Studying the structure of these events, we find contributions from a number of intermediate states and extract their cross sections. In particular, we perform a more detailed study of the e ⁺e ^-→φ(1020)ππγ reaction and confirm the presence of the Y(2175) resonance in the φ(1020)f ₀(980) and K ⁺K ^-f ₀(980) modes. In the charmonium region, we observe the J/ψ in all three final states and in several intermediate states, as well as the ψ(2S) in some modes, and measure the corresponding products of branching fraction and electron width.