The low-energy ππ amplitude is computed explicitly to two-loop accuracy in the chiral expansion. It depends only on six independent (combinations of) low-energy constants which are not fixed by chiral symmetry. Four of these constants are determined via sum rules which are evaluated using ππ scattering data at higher energies. Dependence of the low-energy phase shifts and of the threshold parameters on the remaining two constants (called α and β) are discussed and compared to the existing data from K l4 experiments. Using generalised χPT, the constants α and β are related to fundamental QCD parameters such as the quark condensate 0|qq|0 and the quark mass ratio m s / m. It is shown that forthcoming accurate low-energy ππ data can be used to provide, for the first time, experimental evidence in favour of or against the existence of a large quarkantiquark condensate in the QCD vacuum.

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 (xPT) are relegated to higher orders. The new expansion represents a systematic and unambiguous generalization of the standard xPT, 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 m(qq) and the quark mass ratio m,/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 xPT values. The importance of new, more accurate, experimental information on low-energy T-T scattering is stressed.

Abstract. The recently published E865 data on charged Ke4 decays and ππ phases are reanalyzed to extract values of the two S-wave scattering lengths, of the subthreshold parameters α and β, of the lowenergy constantsl3 andl4 as well as of the main two-flavour order parameters: ūu and Fπ in the limit mu = m d = 0 taken at the physical value of the strange quark mass. Our analysis is exclusively based on direct experimental information on ππ phases below 800 MeV and on the new solutions of the Roy equations by Ananthanarayan et al. The result is compared with the theoretical prediction relating 2a 0 0 − 5a 2 0 and the scalar radius of the pion, which was obtained in two-loop Chiral Perturbation Theory. A discrepancy at the 1-σ level is found and commented upon.

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Due to its light mass of order Λ QCD , the strange quark can play a special role in Chiral Symmetry Breaking (χSB): differences in the pattern of χSB in the limits N f = 2 (m u , m d → 0, m s physical) and N f = 3 (m u , m d , m s → 0) may arise due to vacuum fluctuations of ss pairs, related to the violation of the Zweig rule in the scalar sector and encoded in particular in the O(p 4 ) low-energy constants L 4 and L 6 . In case of large fluctuations, we show that the customary treatment of SU(3) × SU(3) chiral expansions generate instabilities upsetting their convergence. We develop a systematic program to cure these instabilities by resumming nonperturbatively vacuum fluctuations of ss pairs, in order to extract information about χSB from experimental observations even in the presence of large fluctuations. We advocate a Bayesian framework for treating the uncertainties due to the higher orders. As an application, we present a three-flavour analysis of the low-energy ππ scattering and show that the recent experimental data imply a lower bound on the quark mass ratio 2m s /(m u + m d ) ≥ 14 at 95% confidence level. We outline how additional information may be incorporated to further constrain the pattern of χSB in the N f = 3 chiral limit.

The chiral expansion of the ππ 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 ππ 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.

Kπ scattering and K µ4 decays are studied at leading order of improved chiral perturbation theory. It is shown that high precision K µ4 experiments at, e.g., DAΦNE should allow for a direct measurement of the quark mass ratio m s /m.

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