Restoration of rotational symmetry in the continuum limit of lattice field theories

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 λφ⁴ theory and QCD. Contributions from these operators that violate rotational invariance occur at tree-level, with coefficients that are suppressed by O(a²) in the continuum limit. Quantum loops do not modify this behavior in λφ⁴, 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.