Observation of $e^+e^-$ Annihilations into the $\boldmath{C=+1}$ Hadronic Final States $\rho^0\rho^0$ and $\phi\rho^0$

We report the first observation of $e^+e^-$ annihilations into states of positive $C$-parity, namely $\rho^0 \rho^0$ and $\phi\rho^0$. The two states are observed in the $\pip\pim\pip\pim$ and $K^+K^-\pip\pim$ final states, respectively, in a data sample of 225~$fb^{-1}$ collected by the $BaBar$~experiment at the $PEP-II$ $e^+e^-$ storage rings at energies near $\sqrt{s} ~=$ 10.58~$GeV$. The distributions of $\cos\theta^*$, where $\theta^*$ is the center-of-mass polar angle of the $\phi$ meson or the forward $\rho^0$ meson, suggest production by two-virtual-photon annihilation. We measure cross sections within the range $|\cos\theta^*| ~<~ 0.8$ of $\sigma(\e^+e^- \to \rho^0 \rho^0) ~=~$$20.7 \pm 0.7 (\mathrm{stat})\pm 2.7(\mathrm{syst}) ~\mathrm{fb}$ and $\sigma(\e^+e^- \to \phi \rho^0) \~=~$$5.7\pm 0.5(\mathrm{stat}) \pm 0.8(\mathrm{syst}) ~\mathrm{fb}$.