Time-dependent analysis of $B^0 \to \rho^0 \rho^0$ decays

We study the decay $B^0 \to \rho^0 \rho^0$ in a sample of about 454 million $\Upsilon (4S)\rightarrow B \bar{B}$ decays collected with the Babar detector at the PEP-II asymmetric-energy $e^+e^-$ collider at SLAC. We measure the branching fraction and longitudinal polarization fraction. We investigate the proper-time dependence of the longitudinal component in the decay and measure the $C\!P$-violating coefficients $S^{00}_L$ and $C^{00}_L$, corresponding to the sine and cosine terms in the time evolution of the CP asymmetry. We study the implication of these results for the CKM unitarity angle $\alpha$.