# Connected and Disconnected Contractions in Pion-Pion Scattering

## TL;DR

This paper combines chiral effective field theory and lattice QCD to analyze disconnected diagrams in pion-pion scattering, aiding in improving lattice calculations of QCD matrix elements.

## Contribution

It introduces a method using partially-quenched chiral perturbation theory to distinguish and analyze contraction diagram effects in pion-pion scattering.

## Key findings

- Disentangles effects of different contraction diagrams on scattering amplitudes.
- Provides criteria for assessing and improving lattice calculations of disconnected diagrams.
- Offers insights into the energy dependence of the imaginary part of the scattering amplitude.

## Abstract

We show that the interplay of chiral effective field theory and lattice QCD can be used in the evaluation of so-called disconnected diagrams, which appear in the study of the isoscalar and isovector channels of pion-pion scattering and have long been a major challenge for the lattice community. By means of partially-quenched chiral perturbation theory, we distinguish and analyze the effects from different types of contraction diagrams to the pion-pion scattering amplitude, including its scattering lengths and the energy-dependence of its imaginary part. Our results may be used to test the current degree of accuracy of lattice calculation in the handling of disconnected diagrams, as well as to set criteria for the future improvement of relevant lattice computational techniques that may play a critical role in the study of other interesting QCD matrix elements.