Combined limit on the production of a light gauge boson decaying into $\mu^+\mu^-$ and $\pi^+\pi^-$

We searched for the $\mu^+\mu^-$ decay of a light vector gauge boson, also known as dark photon, in the $e^+ e^- \to \mu^+ \mu^- \gamma_{\rm ISR}$ process by means of the Initial State Radiation (ISR) method. We used 1.93~fb$^{-1}$ of data collected by the KLOE experiment at the DA$\Phi$NE $\phi$-factory. No structures have been observed over the irreducible $\mu^+ \mu^-$ background. A 90\% CL limit on the ratio $\varepsilon^2=\alpha^{\prime}/\alpha$ between the dark coupling constant and the fine structure constant of $ 3\times 10^{-6}-2\times 10^{-7}$ has been set in the dark photon mass region between 519 MeV and 973 MeV. This new limit has been combined with the published result obtained investigating the hypothesis of the dark photon decaying into hadrons in $e^+ e^- \to \pi^+ \pi^- \gamma_{\rm ISR}$ events. The combined 90\% CL limit increases the sensitivity especially in the $\rho-\omega$ interference region and excludes $\varepsilon^2$ greater than $(13-2)\times 10^{-7}$. For dark photon masses greater than 600 MeV the combined limit is lower than 8~$\times\, 10^{-7}$ resulting more stringent than present constraints from other experiments.