$\eta \to 3\pi$: Study of the Dalitz Plot and Extraction of the Quark Mass Ratio $Q$

The $\eta \to 3\pi$ amplitude is sensitive to the quark mass difference $m_u-m_d$ and offers a unique way to determine the quark mass ratio $Q^2\equiv (m_s^2-m_{ud}^2)/(m_d^2-m_u^2)$ from experiment. We calculate the amplitude dispersively and fit the KLOE Collaboration data on the charged mode, varying the subtraction constants in the range allowed by chiral perturbation theory. The parameter-free predictions obtained for the neutral Dalitz plot and the neutral-to-charged branching ratio are in excellent agreement with experiment. Our representation of the transition amplitude implies $Q=22.0\pm 0.7$.

Figure 1

Prediction obtained from the KLOE measurements of $\eta \to {\pi }^{+}{\pi }^{-}{\pi }^0$ compared with the MAMI results for $\eta \to 3{\pi }^0$ ($Z\equiv X^2+Y^2$ represents the square of the distance from the center of the Dalitz plot).