ar X iv : h ep - p h / 03 08 13 6 v 1 1 2 A ug 2 00 3 S - wave π − π scattering lengths

An old calculation of the s-wave π − π scattering lengths is updated and supplemented with experimental data on the π−π s-wave phase shift. The results ma0 = .215 and ma2 = −.039 are in excellent agreement with those obtained from a recent analysis of the results published by the Brookhaven E865 collaboration. The subject of the s-wave π − π scattering lengths has witnessed two main peaks of both experimental and theoretical activity since they were first calculated by Weinberg [1] in 1966. On the experimental side the Geneva-Saclay experiment [2] carried in the mid 1970’s gave for the isoscalar scattering length ma0 = .26 ± .05. Recently a new measurement of Ke4 decay and of the π − π phase-shift difference δ00 − δ 1 1 has been published by the Brookhaven E865 collaboration [3] with statistics improved by more than a factor of 10. Analysis and interpretation of the new data has recently been carried out by Descotes et al. [4] who used solutions of the Roy equations [5] obtained by Ananthanarayan et al. [6]. Their results for the scattering lengths are ma0 = .228± .012 , ma2 = −.0382± .0038 (1) On the theoretical side considerable activity was devoted to the subject in the late 1960’s [7]. Subsequently the effective low energy theory of QCD, Chiral Perturbation Theory (ChPT) was applied to the problem. In ChPT the scattering amplitude is expanded in powers of the momenta and of the quark masses. ChPT is a non renormalizable theory and subtraction constants (low energy couplings) have to be introduced at each order of the calculation. The elaborate evaluation of the perturbation series to two loops was completed only recently [9], [10]. Four low energy couplings l1, l2, l3, l4 enter in the calculation the values of l1 and l2 were obtained in ref. [10] by solving the Roy equations [5]. l3 and l4 on the other hand are obtained only indirectly. l4 is expressible in terms of the scalar radius of the pion. r s = .61± .04 fm 2 (2)