Implications for new physics from {bar B}⁰to π⁰π⁰ and {bar B}⁰to {bar K}⁰K⁰

We have analyzed the ${\bar B}^0\to \pi^0\pi^0$ puzzle in three kinds of models beyond the standard model (SM). It is shown that the minimal flavor violation (MFV) models, the minimal supersymmetric standard model (MSSM), and the two Higgs doublet models (2HDM) I and II can not give an explanation of the ${\bar B}^0\to \pi^0\pi^0$ puzzle within $1 \sigma$ experimental bounds and the model III 2HDM can explain the puzzle without a conflict with other experimental measurements. If the constraint on $C_{8g}$ from $b\to s g$ is not imposed, for all kinds of insertions considered there are regions of parameter space, where the scalar quark mass is larger (much larger) than the gluino mass in the case of LR or RL (LL or RR), in which the puzzle can be resolved within $1 \sigma$ experimental bounds.