Final State Interactions and Khuri-Treiman Equations in $\eta \to 3\pi$ decays

Abstract

Using extended Khuri-Treiman equations, we evaluate the final state interactions due to two-pion rescatterings to the decays $\eta \to {\pi }^0{\pi }^{+}{\pi }^{-}$ and $\eta \to {\pi }^0{\pi }^0{\pi }^0$. As subtraction to the dispersion relation we take the one-loop chiral perturbation theory result of Gasser and Leutwyler. The calculated corrections are moderate and amount to about $14\%$ in the amplitude at the center of the decay region. A careful analysis of the errors inherent to our approach is given. As a consequence, the experimental rate of the decay can only be reproduced if the double quark mass ratio $Q^{-2}\equiv \frac{m_d-m_u}{m_s-\hat{m}}\cdot \frac{m_d+m_u}{m_s+\hat{m}}$ is increased from the usual value of $1/(24.1{)}^2$ to $1/(22.4\pm 0.9{)}^2$. We have also calculated the ratio of the rates of the two decays and various Dalitz Plot parameters. In particular, the linear slope $a$ in the charged decay is different from the one-loop value and agrees better with experiment.

55 pages, 4 Postscript figures

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• On the Foundations of Chiral Perturbation Theory
• Virtual Photons in Chiral Perturbation Theory
• Violations of Dashen's Theorem
• Electromagnetic Corrections to the decays $\eta \to 3\pi$

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