Determination of Two-Loop $\pi \pi$ Scattering Amplitude Parameters

Abstract

The chiral expansion of the $\pi \pi$ amplitude to the order of two loops was expressed in terms of six independent parameters in a previous paper: four of these are shown here to satisfy sum rules. Their derivation, where crossing symmetry plays a key role, is explained. Their convergence properties are studied and their practical evaluation, in terms of the available data on $\pi \pi$ phase shifts above 0.5 GeV, is discussed. Below 0.5 GeV, the chiral amplitude itself is employed, such that the parameters are determined in a self-consistent way. Some care is devoted to the estimate of the errors.

A mistake in formulas (48) is corrected, as well as the corresponding numerical values (49)

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