$\hat{I}\cdot \to \ddot{I}{€}^0{\hat{I}}^3{\hat{I}}^3$ and ${\hat{I}}^3{\hat{I}}^3\to \ddot{I}{€}^0\ddot{I}{€}^0$ in $O(p^6)$ chiral perturbation theory

Abstract

$\hat{I}\cdot \to \ddot{I}{€}^0{\hat{I}}^3{\hat{I}}^3$ and $\ggpipi$ are considered in $O(p^6)$ chiral perturbation theory. In addition to the usual $\ddot{I},\ddot{I}‰$ contributions, there are two $O(p^6)$ operators (${\mathcal{L}}_{6,m}$) arising from explicit chiral symmetry breaking. Since only one of the two operators contributes to $\etapigg$, the coefficient of this operator ($\equiv d_3$) can be determined in two ways : (i) from the measured decay rate of $\hat{I}\cdot \to \ddot{I}{€}^0{\hat{I}}^3{\hat{I}}^3$, and (ii) by assuming the resonance saturations of the low energy coefficients in the $O(p^6)$ chiral lagrangian. We find that two methods lead to vastly different values of $d_3$, which would indicate that either the measured decay rate for $\hat{I}\cdot \to \ddot{I}{€}^0{\hat{I}}^3{\hat{I}}^3$ is too large by a factor of $2\sim 3$, or the resonance saturation assumptions do not work for $d_3$.

13 pages in LATEX, one figure can be faxed upon request (To appear in Phys. Lett. B)