Study of the pion-pion scatterings with a combination of all-to-all propagators and the HAL QCD method

TL;DR

This paper advances the HAL QCD method by integrating all-to-all propagators with stochastic estimators to study pion-pion scatterings, effectively managing statistical fluctuations in complex quark annihilation processes.

Contribution

It introduces a hybrid all-to-all propagator approach within the HAL QCD framework, enabling precise potential calculations for systems with quark annihilation.

Findings

Successfully evaluated HAL QCD potentials for $q=1,2$ $$ scatterings.

Demonstrated reduction of statistical fluctuations through stochastic noise dilution.

Confirmed the method's effectiveness despite increased noise in annihilation channels.

Abstract

In this paper, we report recent developments of the HAL QCD method for two hadron systems which contain quark annihilation processes using all-to-all quark propagators. We employ the hybrid method for all-to-all propagators, which combines a low-mode spectral decomposition of the quark propagator and stochastic estimators for remaining high modes, to evaluate the HAL QCD potentials for the first time. Using this method, we investigate the $I= 1,2$ $\pi \pi$ scatterings at $m_{\pi} \approx 870$ MeV. In the $I=2$ study, we study how statistical fluctuations of the HAL QCD potentials are increased due to stochastic estimators in the hybrid method, compared with the conventional one without them. We find that we can reduce statistical fluctuations by dilutions of stochastic noises in order to obtain sufficiently precise results, which turn out to be consistent with conventional results without all-to-all propagators. In the $I=1$ $\pi \pi$ case, which contains quark annihilation processes, we find that statistical fluctuations are further enhanced due to noise contaminations in annihilation processes. We, however, confirm that we can also reduce such statistical fluctuations to obtain the potential with a reasonable precision as long as we further increase a degree of dilutions at a price of large numerical costs and take an appropriate scheme for the potential.