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 454 fb⁻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)f⁰(980) and K⁺ K⁻ f⁰ (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.