$\eta \to 3 \pi$ at Two Loops In Chiral Perturbation Theory

We calculate the decay $\eta\to3\pi$ at next-to-next-to-leading order or order $p^6$ in Chiral Perturbation Theory. The corrections are somewhat larger than was indicated by dispersive estimates. We present numerical results for the Dalitz plot parameters, the ratio $r$ of the neutral to charged decay and the total decay rate. In addition we derive an inequality between the slope parameters of the charged and neutral decay. The experimental charged decay rate leads to central values for the isospin breaking quantities $R=42.2$ and $Q=23.2$.