Roy equation analysis of ππ scattering

Abstract

We analyze the Roy equations for the lowest partial waves of elastic ππ scattering. In the first part of the paper, we review the mathematical properties of these equations as well as their phenomenological applications. In 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 2 0 are the essential parameters in the low energy region: Once these are known, the available experimental information determines the behaviour near threshold to within remarkably small uncertainties. An explicit numerical representation for the energy dependence of the S-and P -waves is given and it is shown that the threshold parameters of the Dand F -waves are also fixed very sharply in terms of a 0 0 and a 2 0 . In agreement with earlier work, which is reviewed in some detail, we find that the Roy equations admit physically acceptable solutions only within a band of the (a 0 0 ,a 2 0 ) plane. We show that the data on the reactions e + e − → π π and τ → π π ν reduce the width of this band quite significantly. Furthermore, we discuss the relevance of the decay K → π π e ν in restricting the allowed range of a 0 0 , preparing the grounds for an analysis of the forthcoming precision data on this decay and on pionic atoms. We expect these to reduce the uncertainties in the two basic low energy parameters very substantially, so that a meaningful test of the chiral perturbation theory predictions will become possible.

On the precision of the theoretical predictions forππscattering

“…We have demonstrated that the low energy properties of the ππ scattering amplitude can be predicted to a remarkable degree of accuracy [1,2] (in the following these papers are referred to as ACGL and CGL, respectively). In our opinion, this work represents a breakthrough in a field that hitherto was subject to considerable uncertainties.…”

Low-energy ππ and πK scatterings revisited in three-flavour resummed chiral perturbation theory

“…Recent data on ππ scattering [2] together with older data and numerical solutions of the Roy equations [3] allowed us to determine the N f = 2 order parameters expressed in suitable physical units [4]:…”

“…In sec. 4, we explain how we determine the same amplitudes dispersively in subthreshold regions, building upon the solutions of Roy and Roy-Steiner equations [3,10]. In sec.…”

Towards a dispersive determination of the η and η′ transition form factors

“…by matching to chiral perturbation theory near s = 0. The Omnès function is entirely given in terms of the appropriate pion-pion phase shift, which is particularly useful as we today have excellent information on pion-pion scattering at our disposal [21][22][23][24]. The pion vector form factor F V π (s) as extracted from τ − → π − π 0 ν τ decays, e.g., can be described very accurately by a representation (5) up to √ s = 1 GeV, employing a linear polynomial R(s) ≡ P 1 1 (s) = 1 + α V s-at higher energies, the nonlinear effects of higher, inelastic (ρ ′ , ρ ′′ ) resonances become important [25].…”