Roy equation analysis of pi pi scattering

We analyse the Roy equations for the lowest partial waves of elastic pi pi scattering. In the first part of the paper, we review the mathematical prop erties of these equations as well as their phenomenological applications. I n particular, the experimental situation concerning the contributions from intermediate energies and the evaluation of the driving terms are discussed in detail. We then demonstrate that the two S-wave scattering lengths a(0) (0) and a(0)(2) are the essential parameters in the low energy region: Once these are known, the available experimental information determines the beh aviour near threshold to within remarkably small uncertainties. An explicit numerical representation for the energy dependence of the S- and P-waves i s given and it is shown that the threshold parameters of the D- and F-waves are also fixed very sharply in terms of a(0)(0) and a(0)(2). In agreement with earlier work, which is reviewed in some detail, we find that the Roy e quations admit physically acceptable solutions only within a band of the (a (0)(0), a(0)(2)) plane. We show that the data on the reactions e+e(-) --> p i pi and tau --> pi pi nu reduce the width of this band quite significantly . Furthermore, we discuss the relevance of the decay K --> pi pi ev in rest ricting 0 the allowed range of a(0)(0) preparing the grounds for an analysi s of the forthcoming precision data on this decay and on pionic atoms. We e xpect these to reduce the uncertainties in the two basic low energy paramet ers very substantially, so that a meaningful test of the chiral perturbatio n theory predictions will become possible. (C) 2001 Elsevier Science B.V. A ll rights reserved.