Coarse graining hadronic scattering

Abstract

We show that it makes sense to coarse grain hadronic interactions such as $\pi\pi$ and $\pi N$ reactions following previous work on NN scattering. Moreover, if the interaction is taken to be given by chiral dynamics at long distances above a given value $r > r_c$ larger than the elementary radii of the interaction hadrons the unknown short distance region $r< r_c$ is characterized by a {\it finite} number of fitting parameters. This number of independent parameters needed for a presumably complete description of scattering data for a CM energy below $\sqrt{s}$ has been found to be given by $N_{\rm Par} = N_S \times N_I \times (p r_c )^2 /2 $ with $N_S$ and $N_I$ the number of spin and isospin channels, and $p$ the CM momentum respectively. Therefore, for an experiment (or sets of experiments) with a total number of data $N_{\rm Dat}$ the number of degrees of freedom involved in a $\chi^2$-fit is given by $\nu = N_{\rm Dat}-N_{\rm Par}$ and confidence levels can be obtained accordingly by standard means. Namely a $1 \sigma$ confidence level corresponds to $\chi_{\rm min}^2/\nu \in (1- \sqrt{2/\nu},1+\sqrt{2/\nu})$. We discuss the approach for $\pi\pi$ and $\pi N$ with an eye put on a data selection program and the eventual validation of chiral symmetry.