Rigorous pion - pion scattering lengths from threshold πN → ππN data

A new evaluation of the universal ππ scattering length relation is used to extract the ππ s-wave scattering lengths from threshold pion production data. Previous work has shown that the chiral perturbation series relating threshold pion production to ππ scattering lengths appears to converge well only for the isospin-2 case, giving a2 = −0.031±0.007m−1 π . A model-independent and data-insensitive universal curve then implies a0 = 0.235 ± 0.03m−1 π for the isospin-0 scattering length. Pion-pion scattering is the simplest non-trivial hadron scattering process. It provides an ideal laboratory for the concepts of chiral symmetry. The ππ scattering lengths, first predicted by Weinberg [1], were improved by the development of chiral perturbation theory (ChPT) where higher order corrections could be computed [2], recently to the two loop accuracy [3,4]. In the case of the s-wave ππ scattering lengths aI , where isospin I = 0, 2, these higher-order corrections are reasonably small and the predictions are robust. It is the purpose of this paper to discuss how one can test these predictions using experimental data already at hand. The principal present source of ππ scattering information is from the reaction πN → ππN . For high-energy incident pions one can extrapolate to the kinematic region where single pion exchange is dominant [6]. The most accurate experiments of this sort were done in the 70’s by the CERN-Munich Collaboration [7]. These phase shifts do not extend close enough to threshold to determine a0 and a2 directly, but they do provide a powerful constraint with the help of a dispersion relation. The close relation between the πN → ππN amplitude at pion production threshold and the ππ scattering lengths was point out long ago [8]. In modern ChPT language this result is the lowest (tree level) order with no excited nucleon contributions. Recently the possibility of a more ambitious calculation of πN → ππN and its relation to the ππ scattering lengths has been discussed [9] and subsequently established [10]. This result takes into account loop corrections as well as πN resonant effects. In summary, it was found that the isospin-2 scattering length was related to the threshold πN → ππN amplitude in almost exactly the same way as the lowest order result [8]. The ChPT series in this case 1The relatively rare decay K → ππlν can be used to extract the isospin δ0 − δ1 phase shift difference below the K mass [5]. Future experiments using this process, e.g. at DAPHNE, will accurately determine a0. 2The ChPT formalism requires ξ = 0 in [8]. This would yield the Weinberg ππ scattering lengths.