# Galilean invariance restoration on the lattice

## TL;DR

This paper investigates the breaking of Galilean invariance in nuclear lattice simulations and demonstrates that it can be effectively restored by introducing specific operators, improving the accuracy of scattering calculations.

## Contribution

The study identifies the sources of Galilean invariance breaking in lattice calculations and proposes a method to restore it using Galilean invariance restoration operators.

## Key findings

- Galilean invariance breaking effects partially cancel due to lattice artifacts and smearing.
- Restoration operators effectively reduce Galilean invariance breaking.
- Small residual breaking effects are manageable with the proposed method.

## Abstract

We consider the breaking of Galilean invariance due to different lattice cutoff effects in moving frames and a nonlocal smearing parameter which is used in the construction of the nuclear lattice interaction. The dispersion relation and neutron-proton scattering phase shifts are used to investigate the Galilean invariance breaking effects and ways to restore it. For $S$-wave channels, ${}^1S_0$ and ${}^3S_1$, we present the neutron-proton scattering phase shifts in moving frames calculated using both L\"uscher's formula and the spherical wall method, as well as the dispersion relation. For the $P$ and $D$ waves, we present the neutron-proton scattering phase shifts in moving frames calculated using the spherical wall method. We find that the Galilean invariance breaking effects stemming from the lattice artifacts partially cancel those caused by the nonlocal smearing parameter. Due to this cancellation, the Galilean invariance breaking effect is small, and the Galilean invariance can be restored by introducing Galilean invariance restoration operators.