# Constraints on Disconnected Contributions in $\pi\pi$ Scattering

## TL;DR

This paper combines Lüscher's formula and chiral perturbation theory to predict energy levels related to disconnected contractions in lattice QCD $$ $$ scattering, aiding the validation of lattice computations.

## Contribution

It provides the first precise theoretical predictions for disconnected contraction contributions in lattice QCD $$ $$ scattering using combined analytical methods.

## Key findings

- Predictions for rectangular and vacuum contraction energy levels.
- Expressions to extract low-energy constants from lattice data.
- Framework for testing lattice QCD calculations of $$ $$ scattering.

## Abstract

The accuracy of the lattice QCD computation of hadron-hadron scattering at low isospin depends critically on the ability to compute correlation functions with fermionic disconnected Wick contractions. This happens, for instance, in isospin $I=0$ $\pi\pi$ scattering, which receives contributions from rectangular and vacuum types of contractions among other easier calculable ones. Combining L\"{u}scher's formula and partially-quenched chiral perturbation theory, we provide precise theory predictions of the discrete energy levels extracted from specific linear combinations of lattice correlation functions corresponding to various types of contractions. Expressions are provided for extracting the unphysical low-energy constants in the partially-quenched chiral perturbation theory from the energy levels for these contractions. The predictions for the rectangular and vacuum contractions may serve as solid tests of the accuracy for existing and future lattice studies of $\pi\pi$ scattering.