Whatπ−πscattering tells us about chiral perturbation theory

We describe a rearrangement of the standard expansion of the symmetry-breaking part of the QCD effective Lagrangian that includes into each order additional terms which in the standard chiral perturbation theory ($\chi \mathrm{PT}$) are relegated to higher orders. The new expansion represents a systematic and unambiguous generalization of the standard $\chi \mathrm{PT}$, and is more likely to converge rapidly. It provides a consistent framework for a measurement of the importance of additional "higher order" terms whose smallness is usually assumed but has never been checked. A method of measuring, among other quantities, the QCD parameters $\hat{m}〈\overline{q}q〉$ and the quark mass ratio $\frac{m_s}{\hat{m}}$ is elaborated in detail. The method is illustrated using various sets of available data. Both of these parameters might be considerably smaller than their respective leading-order standard $\chi \mathrm{PT}$ values. The importance of new, more accurate, experimental information on low-energy $\pi -\pi$ scattering is stressed.