Restoration of Rotational Symmetry in the Continuum Limit of Lattice Field Theories

TL;DR

This paper investigates how rotational symmetry is restored in the continuum limit of lattice field theories using smeared operators, with perturbative analysis in phi^4 theory and QCD showing suppression of symmetry violations as lattice spacing decreases.

Contribution

It demonstrates that smeared lattice operators can systematically recover rotational invariance in the continuum limit, with suppressed violations at tree-level and unaffected by quantum loops.

Findings

Violations occur at tree-level with O(a^2) suppression

Quantum loops do not alter the suppression behavior

Smeared operators enable higher moments calculation in Lattice QCD

Abstract

We explore how rotational invariance is systematically recovered from calculations on hyper-cubic lattices through the use of smeared lattice operators that smoothly evolve into continuum operators with definite angular momentum as the lattice-spacing is reduced. Perturbative calculations of the angular momentum violation associated with such operators at tree-level and at one-loop are presented in phi^4 theory and QCD. Contributions from these operators that violate rotational invariance occur at tree-level, with coefficients that are suppressed by O(a^2) in the continuum limit. Quantum loops do not modify this behavior in phi^4, nor in QCD if the gauge-fields are smeared over a comparable spatial region. Consequently, the use of this type of operator should, in principle, allow for Lattice QCD calculations of the higher moments of the hadron structure functions.