Testing the fixed-point QCD action and the construction of chiral currents

TL;DR

This paper demonstrates the viability of a parametrized fixed-point Dirac operator $D^{\rm FP}$ in quenched QCD and provides a practical framework for covariant densities and conserved currents under the Ginsparg-Wilson relation. Through spectroscopy, finite-volume chiral condensate extractions, and topological susceptibility measurements, it shows that chiral symmetry can be preserved and controlled on relatively coarse lattices, especially when combined with an overlap improvement $D_{\rm ov}^{\rm FP}$. The work also furnishes explicit, covariant current constructions and Ward identities for chiral lattice actions, including general GW cases with $2R\neq1$, enabling robust chiral and topological analyses. Collectively, these results suggest fixed-point actions can reduce lattice artifacts and broaden the practical use of chiral lattice fermions, while highlighting ongoing questions about full QCD and chiral fermion costs.

Abstract

We present the first set of quenched QCD measurements using the recently parametrized fixed-point Dirac operator D^FP. We also give a general and practical construction of covariant densities and conserved currents for chiral lattice actions. The measurements include (a) hadron spectroscopy, (b) corrections of small chiral deviations, (c) the renormalized quark condensate from finite-size scaling and, independently, spectroscopy, (d) the topological susceptibility, (e) small eigenvalue distributions and random matrix theory, and (f) local chirality of near-zero modes and instanton-dominance.

Figures (19)

• Figure 1: The full eigenvalue spectrum of $D^{\rm FP}$ for a $S^{\rm FP}_g$ gauge configuration of volume $4^4$ (crosses) and the smallest 100 eigenvalues on a $8^4$ volume (circles) at a lattice spacing $0.16\,{\rm fm}$.
• Figure 2: Ratio of the 50 smallest $A^{\dagger} A$ eigenvalues to the largest eigenvalue using $D^{\rm FP}$ (filled squares) and $D^{\rm Wilson}$ (open squares) for 5 different $S^{\rm FP}_g$ gauge configurations of volume $12^4$ at a lattice spacing $0.16\,{\rm fm}$.
• Figure 3: Breaking of the Ginsparg-Wilson relation as measured by $\Delta_{\rm GW}(N)$ on $10^4$ lattices at different lattice spacings and different number of exactly projected $A^{\dagger} A$ eigenvalues. Keeping the number of exactly projected $A^{\dagger} A$ eigenvalues constant, the exponential fall-off is steeper for smaller lattice spacings. If the number of exactly projected $A^{\dagger} A$ eigenvalues is adjusted such that the range of the not exactly treated $A^{\dagger} A$ eigenvalues is approximately the same for the different lattice spacings, the fall-off is roughly equal in all cases, which illustrates that the approximation is governed by the ratio $\lambda_{\rm min}(A^{\dagger} A)/\lambda_{\rm max}(A^{\dagger} A)$.
• Figure 4: The dependence of the deviation of the $D_{\rm ov}^{\rm FP}$ eigenvalues from the Ginsparg-Wilson circle on the order of the polynomial approximation, as measured by $|{\rm Re}(\Lambda)|$.
• Figure 5: The locality of $D_{\rm ov}$, as measured by the expectation value $f(r)/f(0)$ defined in Eq. \ref{['eq:locality']}, using $D^{\rm FP}$ and $D^{\rm Wilson}$ as input on $12^4$ lattices at $a=0.16\,{\rm fm}$.