Precise Branching Ratio Measurements of the Decays $D^0 \to \pi^ - \pi^ + \pi^0$ and $D^0 \to K^ - K^ + \pi^0$

Using 232 fb-1 of e+e- collision data recorded by the BaBar experiment, we measure the ratios of three-body Cabibbo-suppressed decay rates of the D^0 meson relative to that of the Cabibbo-favored decay: B(D0 --> pi- pi+ pi0)/ B(D0 --> K- pi+ pi0) = (10.59 +/- 0.06 +/- 0.13).10^{-2} and B(D0 --> K- K+ pi0)/ B(D0 --> K- pi+ pi0) = (2.37 +/- 0.03 +/- 0.04). 10^{-2}, where the errors are statistical and systematic respectively. The precisions of these measurements are significantly better than those of the current world average values.We note that the second result differs significantly from the current world average value. Using the PDG-2006 value for D0 --> K- pi+ pi0 branching fraction, we obtain, B(D0 --> pi- pi+ pi0) = (1.493 +/- 0.008 +/- 0.018 +/- 0.053). 10^{-2}, B(D0 --> K- K+ pi0) = (0.334 +/- 0.004 +/- 0.006 +/- 0.012). 10^{-2}, where the errors are statistical, systematic, and due to the uncertainty of B(D0 --> K- pi+ pi0). The average squared matrix elements for both of the singly Cabibbo-suppressed decays are roughly a factor of sin^2 \theta_C smaller than that for the Cabibbo-favored decay and are therefore, in contrast to the corresponding two-body modes, consistent with the naive expectations.