Nonperturbative Renormalisation of Composite Operators in Lattice QCD

TL;DR

This work develops a nonperturbative renormalisation program for bilinear quark operators in lattice QCD using a MOM-type scheme and quenched Wilson fermions. It defines a broad operator basis, implements momentum-source techniques, and converts results to the MSbar scheme to compare with perturbation theory and Wilson coefficients. The results show mixed scaling behavior across operators: axial/implicit currents and the scalar density exhibit clearer scaling windows aided by tadpole improvement, while the local vector current and pseudoscalar density are more sensitive to lattice artifacts and chiral dynamics. The study highlights both the practicality of nonperturbative renormalisation for hadronic structure calculations and the need for further improvements, such as finer lattices, operator improvement, and nonperturbative Wilson coefficients, to fully control systematic effects.

Abstract

We investigate the nonperturbative renormalisation of composite operators in lattice QCD restricting ourselves to operators that are bilinear in the quark fields. These include operators which are relevant to the calculation of moments of hadronic structure functions. The computations are based on Monte Carlo simulations using quenched Wilson fermions.

Figures (24)

• Figure 1: Chiral extrapolation of $Z$ for ${\mathcal{O}}_{v_{2,a}}$ at $\beta = 6.2$. Some representative values of $\mu^2$ have been selected.
• Figure 2: $Z$ in the MOM scheme for ${\mathcal{O}}^V_\mu$, ${\mathcal{O}}^A_\mu$, ${\mathcal{O}}^P$, and ${\mathcal{O}}^S$ (from top to bottom) at $\beta = 6.2$. For the local vector current and the axial vector current, $Z$ has been determined from the transverse components (cf. Section \ref{['sec.vector']}).
• Figure 3: $Z$ in the MOM scheme for ${\mathcal{O}}_{r_{2,b}}$, ${\mathcal{O}}_{r_{2,a}}$, ${\mathcal{O}}_{v_{2,b}}$, and ${\mathcal{O}}_{v_{2,a}}$ (from top to bottom) at $\beta = 6.2$.
• Figure 4: $Z$ in the MOM scheme for ${\mathcal{O}}_{r_3}$, ${\mathcal{O}}_{a_2}$, ${\mathcal{O}}_{v_4}$, and ${\mathcal{O}}_{v_3}$ (from top to bottom) at $\beta = 6.2$.
• Figure 5: $Z$ for the operator ${\mathcal{O}}_{v_{2,a}}$ at $\beta = 6.2$. The filled circles represent the data corresponding to the MOM renormalisation condition (\ref{['defz']}), the open circles represent the values in the $\overline{\mathrm{MS}}$ scheme.