Gluon helicity $\mathrm{\Delta }G$ from a universality class of operators on a lattice

We show that the total gluon helicity $\mathrm{\Delta }G$ in a polarized nucleon can be calculated on a Euclidean lattice through a universality class of QCD operators that describe the helicity or polarization of the on-shell gluon radiation. We in particular find some operators whose matrix elements in a nucleon of momentum $P^z$ are directly related to $\mathrm{\Delta }G$ with only power-law $(1/P^z{)}^n(n\ge 2)$ corrections.