Dalitz plot analyses of $J/ψ\to π^+ π^- π^0$, $J/ψ\to K^+ K^- π^0$, and $J/ψ\to K^0_S K^{\pm} π^{\mp}$ produced via $e^+ e^-$ annihilation with initial-state radiation

Abstract: We study the processes $$e⁺ e^- \to \ \gamma_{\rm ISR} J/\psi$$ where $$J/\psi \to \pi⁺ \pi^- \pi<sup>0$$, $J/\psi \to K⁺ K^- \pi<sup>0$, and $$J/\psi \to K^0_S K^{\pm} \pi<sup>{\mp}$$ using a data sample of 519 $fb<sup>{-1}$ recorded with the BaBar detector operating at the SLAC PEP-II asymmetric-energy $e⁺ e<sup>-$ collider at center-of-mass energies at and near the $\Upsilon(nS)$ ($n = 2,3,4$) resonances. We measure the ratio of branching fractions $$R_1 = \frac{B(J/\psi \ \to \ K⁺ K^- \pi^0)}{B(J/\psi \ \to \ \pi⁺ \pi^- \pi<sup>0)}$$ and $$R_2 = \frac{B(J/\psi \ \to \ K^0_S K^{\pm} \pi^{{\mp})}{B(J/\psi} \ \to \ \pi⁺ \pi^- \pi<sup>0)}$$. We perform Dalitz-plot analyses of the three $J/\psi$ decay modes and measure fractions for resonances contributing to the decays. We also analyze the $$J/\psi \to \pi⁺ \pi^- \pi<sup>0$$ decay using the Veneziano model. We observe structures compatible with the presence of $\rho(1450)$ in all the three $J/\psi$ decay modes and measure the relative branching fraction: $$R(\rho(1450)) = \frac{B(\rho(1450) \ \to \ K⁺ K^{-)}{B(\rho(1450)} \ \to \ \pi⁺ \pi^-)} = 0.307 \pm 0.084 ({\rm stat}) \pm 0.082 ({\rm sys})$$.