# Improved action and Hamiltonian in finite volumes

## TL;DR

This paper introduces a new Symanzik improved lattice action with a 2x2 plaquette, simplifying Feynman rules and reducing scaling violations, with comparisons to continuum and Wilson actions in finite volumes.

## Contribution

The paper presents the square Symanzik action, a novel improved lattice action that simplifies calculations and reduces discretization errors in finite volume simulations.

## Key findings

- Simplified Feynman rules in covariant gauge.
- Comparison of Lambda parameters with continuum and Wilson actions.
- Partial results for one-loop improvement coefficients.

## Abstract

We introduce a new Symanzik improved action by adding a 2x2 plaquette in such a way that the Feynman rules in the covariant gauge simplify. We call this the square Symanzik action. Some comparisons with the continuum and the standard Wilson action are made in intermediate volumes, where mass ratios are accurately known and the precise amount of improvement can be determined. Ratios of the Lambda parameters will be presented, as well as partial results for the one-loop improvement coefficients. We discuss some of the intricacies that arise because of violations of unitarity at the scale of the cutoff. In particular we show how a field redefinition in the zero-momentum effective action allows one to remove scaling violations linear in the lattice spacing.