One-loop renormalisation of quark bilinears for overlap fermions with improved gauge actions

TL;DR

This work delivers a comprehensive one-loop renormalisation analysis for local bilinear quark operators with overlap fermions in the presence of several improved gauge actions. By computing $Z_O$ in the MOM scheme and translating to $\overline{MS}$, the authors provide explicit expressions and finite parts across actions (Symanzik, TI-Lüscher-Weisz, Iwasaki, DBW2) and multiple $\rho$, revealing that gauge-dependent contributions are universal with respect to fermion representation. They also develop and test mean-field (tadpole) improvement for overlap fermions, deriving $Z_O^{TI}$ and $Z_O^{FTI}$ and discussing the perturbative convergence behavior for different actions; Symanzik (and possibly Lüscher-Weisz) show favorable convergence, while Iwasaki/DBW2 are less amenable. The results, supported by extensive numerical data and internal consistency tests, provide a robust perturbative foundation for matching lattice overlap computations to continuum schemes and guide action choices in simulations. Appendix C further derives a non-expanded mean-field overlap propagator, clarifying the relationship between $D_N^{\mathrm{MF}}$, $D_N^{\mathrm{tree}}$, and the tadpole-improved $\rho$.

Abstract

We compute lattice renormalisation constants of local bilinear quark operators for overlap fermions and improved gauge actions. Among the actions we consider are the Symanzik, Lüscher-Weisz, Iwasaki and DBW2 gauge actions. The results are given for a variety of $ρ$ parameters. We show how to apply mean field (tadpole) improvement to overlap fermions. The question, what is a good gauge action, is discussed from the perturbative point of view. Finally, we show analytically that the gauge dependent part of the self-energy and the amputated Green functions are independent of the lattice fermion representation, using either Wilson or overlap fermions.

Figures (5)

• Figure 1: The meaning of the four terms in the gauge field action (\ref{['gluonaction']}).
• Figure 2: One-loop diagrams contributing to the quark self-energy.
• Figure 3: One-loop vertex diagram contributing to the amputated Green functions.
• Figure 4: A test of the mean field approximation. We compare the perturbative $B_V$ (filled squares), as defined in (\ref{['BOdef']}), with the subtracted $B_V^{sub}$ (open circles) defined in (\ref{['Bsub']}). The subtracted coefficients are much smaller, showing that the mean field approximation is very good at the one-loop level.
• Figure 5: The subtracted points from Fig. \ref{['Btad']} shown in more detail. The open circles show results calculated with unimproved gauge actions, the open diamonds with gauge actions modified according to eqs. (\ref{['c0ti']}), (\ref{['cti']}) and (\ref{['rhoti']} ). All actions considered give very similar results.