Renormalisation and off-shell improvement in lattice perturbation theory

TL;DR

This work develops a comprehensive one-loop perturbative framework for renormalisation and $O(a)$ off-shell improvement of flavour non-singlet quark operators in lattice QCD, using the SW (clover) action and comparing with Ginsparg–Wilson fermions. It provides explicit expressions for renormalisation factors and improvement coefficients for both point and one-link operators in ${ m ar{MS}}$ and MOM schemes, while systematically removing contact-term artefacts to achieve off-shell improvement. The authors demonstrate that GW fermions admit universal, operator-independent improvement coefficients and show that, for clover fermions, $c_{ m SW}=1+O(g^2)$ is required to realize full off-shell $O(a)$ improvement, with a substantial role for tadpole improvement. Comparisons with non-perturbative results indicate good agreement for $Z_V$ and related quantities, supporting the practical utility of the TI-augmented perturbative results for matching lattice calculations to continuum schemes and extracting moments of structure functions.

Abstract

We discuss the improvement of flavour non-singlet point and one-link lattice quark operators, which describe the quark currents and the first moment of the DIS structure functions respectively. Suitable bases of improved operators are given, and the corresponding renormalisation factors and improvement coefficients are calculated in one-loop lattice perturbation theory, using the Sheikholeslami-Wohlert (clover) action. To this order we achieve off-shell improvement by eliminating the effect of contact terms. We use massive fermions, and our calculations are done keeping all terms up to first order in the lattice spacing, for arbitrary m^2/p^2, in a general covariant gauge. We also compare clover fermions with fermions satisfying the Ginsparg-Wilson relation, and show how to remove O(a) effects off-shell in this case too, and how this is in many aspects simpler than for clover fermions. Finally, tadpole improvement is also considered.

Figures (4)

• Figure 1: The one-loop lattice Feynman diagrams needed for the contact Green's function $C_ {\mathcal{O}} (p,m)$, defined in eq. (\ref{['contactgreen']}).
• Figure 2: The one-loop lattice Feynman diagrams needed for the amputated Green's function (vertex function), $\Lambda^{ {\mathcal{O}} ^{\rm imp}}(p,m; c_0, c_1, \cdots ,c_n)$, as defined in eq. (\ref{['ampu']}).
• Figure 3: The renormalisation constant $Z_V$ and the improvement coefficient $c_0^V$ as a function of $c_1^V$. The solid lines are the results of tadpole improved perturbation theory. The dashed lines refer to the non-perturbative results of horslat97, and the symbols mark the non-perturbative results of LSSW (solid circle) and raklat97 (solid diamond).
• Figure 4: The renormalisation constant $Z_V$ as a function of $g^2$. The solid line is the result of tadpole improved perturbation theory, while the dotted line shows the one-loop perturbative result with no improvement. The dashed line is the non-perturbative result of LSSW.