Roy equation analysis of pi pi scattering

TL;DR

This work uses Roy equations to constrain low-energy ππ scattering, showing that the two S-wave scattering lengths $a_0^0$ and $a_0^2$ control near-threshold behavior when combined with experimental input for higher energies. The authors decompose the amplitude into a leading SP piece and a small background, solve a coupled set of integral equations for the three lowest partial waves up to a matching point $s_0$, and define a universal band in the $(a_0^0,a_0^2)$ plane that survives data and consistency checks. They further relate these parameters to sum rules, notably the Olsson sum rule, and compute threshold parameters for the S-, P-, D-, and F-waves, as well as phase shifts at the kaon mass; their framework provides a path to high-precision tests of chiral perturbation theory as new data become available. The analysis emphasizes crossing symmetry, analyticity, and unitarity as central constraints, and highlights the potential of upcoming $K_{\ell4}$ and pionium measurements to substantially tighten the allowed region for the basic low-energy parameters.

Abstract

We analyze the Roy equations for the lowest partial waves of elastic pi pi scattering and 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 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 D- and 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_0^2) plane. We show that the data on the reactions e+e- -> pi 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 e nu 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.

Figures (16)

• Figure 1: Driving terms versus energy in GeV. The full lines show the result of the calculation described in appendix B. The shaded regions indicate the uncertainties associated with the input of that calculation. The dashed curves represent the contributions from the $D$- and $F$-waves below 2 GeV.
• Figure 3: Comparison of the different input we used for the imaginary parts of the $I=0$ and $I=1$ lowest partial waves above the matching point at 0.8 GeV.
• Figure 4: Comparison of the results obtained for the dispersion integral $f^0_0$ with the various imaginary parts shown in fig. \ref{['fig:ImAHSZ']}.
• Figure 5: Different data sets for the $S$-wave in the $I=2$ channel and curves that we have used as input in the Roy equation analysis.
• Figure 6: Numerical solution of the Roy equations for $a_0^0=0.225$, $a_0^2=-0.0371$ (the value of $a_0^0$ corresponds to the center of the range considered while the one of $a_0^2$ results if the input used for $\hbox{Im}\,t^2_0$ is taken from the central curve in fig. \ref{['fig:I2']}). The arrow indicates the limit of validity of the Roy equations.