Standard vs. Generalized Chiral Perturbation Theory : the Chell-Olsson test

A way of testing the $\pi \pi$ predictions of Chiral Perturbation Theory against experimental data is to use dispersion relations to continue experimental information into the subthreshold region where the theory should unambiguously apply. Chell and Olsson have proposed a test of the subthreshold behaviour of chiral expansions which highlights potential differences between the Standard and the Generalized forms of the theory. We illustrate how, with current experimental uncertainties, data cannot distinguish between these particular {\em discriminatory} coefficients despite their sensitivity. Nevertheless, the Chell--Olsson test does provide a consistency check of the chiral expansion, requiring that the ${\cal O}(p^6)$ corrections to the {\em discriminatory} coefficients in the Standard theory must be $\sim 100$. Indeed, some of these have been deduced from the new ${\cal O}(p^6)$ computations and found to give such large corrections. One can then check that the ${\cal O}(p^8)$ corrections must be much smaller. We conclude that this test, like others, cannot distinguish between the different forms of Chiral Symmetry Breaking embodied in the alternative versions of Chiral Perturbation Theory without much more precise experimental information near threshold.