Comment on the Prediction of Two-loop Standard Chiral Perturbation Theory for Low-energy

Four of the six parameters defining the two-loop ππ scattering amplitude have been determined using Roy dispersion relations. Combining this information with the Standard χPT expressions, we obtain the threshold parameters, low-energy phases and the O(p) constants l 1, l r 2. The result l r 2(Mρ) = (1.6 ± 0.4 ± 0.9) × 10 −3 (l̄2 = 4.17 ± 0.19 ± 0.43) reproduces the correct D-waves but it is incompatible with existing Standard χPT analyses of Kl4 form factors beyond one loop. Key-Words: chiral symmetries, sum rules, meson-meson interactions, chiral lagrangians Number of figures: 2 March 1997 CPT-97/P.3470 IPNO/TH 97-08 anonymous ftp or gopher: cpt.univ-mrs.fr ∗Unité Propre de Recherche 7061 Division de Physique Théorique, Institut de Physique Nucléaire, F-91406 Orsay Cedex, France. Division de Physique Théorique is Unité de Recherche des Universités Paris XI et Paris VI associée au CNRS 1. During the last few years there has been a noticeable revival of interest in the high precision analysis of low-energy ππ scattering [1]-[13]. There are at least two reasons for this. First, it has been shown [13, 1, 3] and repeatedly emphasized [14] that the ππ scattering amplitude in the threshold region is particularly sensitive to the strength of quark anti-quark pair condensation in the QCD vacuum: the smaller the condensate, the stronger the isoscalar S-wave ππ interaction. The accurate measurement of S-wave scattering lengths would, indeed, provide the first experimental evidence in favour of, or against, the standardly admitted hypothesis according to which the mechanism of spontaneous chiral symmetry breaking is dominated by the formation of a large < q̄q > condensate. Within QCD, this hypothesis is by no means a logical necessity and its experimental test might well become an important step towards a non-perturbative understanding of the quark-gluon dynamics. The second reason which makes detailed ππ studies topical, is that there are two new high precision experiments currently under preparation: i) The phase shift difference δ 0(E) − δ 1 1(E) at low energies (E < 400MeV) will be extracted from a new Kl4-decay experiment [15] performed with the KLOE detector at the Frascati φ-factory DAΦNE [16]. ii) At CERN, the project DIRAC [17] aims at the measurement of the lifetime of ππ atoms to 10%, implying the determination of the combination of scattering lengths | a0 − a 2 0 | with a 5% accuracy. On the theoretical side an even better precision can be reached by a systematic use of chiral perturbation theory [18, 19] (χPT). The low-energy expansion of the ππ scattering amplitude A(s|t, u) starts at order O(p) given by Weinberg more than 30 years ago [20]. Subsequently, the one-loop O(p) contribution to A(s|t, u) has been calculated by Gasser and Leutwyler [21, 19]. It is given by four low-energy constants l1, l2, l3, l4 besides the (charged) pion mass Mπ and the decay constant Fπ. The present state of the art involves the two-loop O(p) order and the present letter concerns this degree of accuracy. 2. The O(p) amplitude A(s|t, u) has been first given in Ref. [1] in the form A(s|t, u) = AKMSF (s|t, u;α, β;λ1, λ2, λ3, λ4) +O [ ( p ΛH )8 , ( Mπ ΛH )8 ]