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We present a study of the branching fraction of the decay B^0->rho0rho0 andthe fraction of longitudinally polarized rho0 mesons in this decay. The resultsare obtained from the final data sample containing 772 million BBbar pairscollected at the Y(4S) resonance with the Belle detector at the KEKBasymmetric-energy e+e- collider. We find 166 +- 59 B^0 -> rho0 rho0 events(including systematic uncertainties), corresponding to a branching fraction ofB(B^0->rho0rho0) = (1.02 +- 0.30 (stat) +- 0.15 (syst)) x 10^{-6} with asignificance of 3.4 standard deviations and a longitudinal polarizationfraction fL = 0.21^{+0.18}_{-0.22} (stat) +- 0.15 (syst). We use thelongitudinal polarization fraction to determine the Cabibbo-Kobayashi-Maskawamatrix angle phi_2 = (84.9 +- 13.5) degrees through an isospin analysis in theB->rhorho system. We furthermore find 149 +- 49 B^0->f0rho0 events,corresponding to B(B^0->f0rho0) x B(f0->pi+pi-) = (0.78 +- 0.22 (stat) +- 0.11(syst)) x 10^{-6}, with a significance of 3.1 standard deviations. We find noother significant contribution with the same final state, and set upper limitsat 90% confidence level on the (product) branching fractions,B(B^0->pi+pi-pi+pi-)<11.2 x 10^{-6}, B(B^0->rho0pi+pi-)<12.0 x 10^{-6},B(B^0->f0pi+pi-) x B(f0->pi+pi-) <3.0 x 10^{-6} and B(B^0->f0f0) xB(f0->pi+pi-)^{2} < 0.2 x 10^{-6}.

Study of B^0 -> rho^0 rho^0 decays, implications for the CKM angle phi_2 and search for other B^0 decay modes with a four-pion final state