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 $N_{\mathrm{Par}}=N_S\times N_I\times (p{r}_c{)}^2/2$ 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 $N_{\mathrm{Dat}}$ the number of degrees of freedom involved in a ${\chi }^2$-fit is given by $\nu =N_{\mathrm{Dat}}-N_{\mathrm{Par}}$ and confidence levels can be obtained accordingly by standard means. Namely a $1\sigma$ confidence level corresponds to ${\chi }_{\min }^2/\nu \in (1-\sqrt{2/\nu },1+\sqrt{2/\nu })$. 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.

Comments: 11 pages, 6 figures. Proceedings of the 9th Chiral Dynamics International Workshop, Durham, NC, USA, 17-21 September, 2018