Finite-Size Effects in Lattice QCD with Dynamical Wilson Fermions

TL;DR

This study investigates finite-size effects in unquenched lattice QCD with two dynamical Wilson flavors at two lattice spacings, examining pion and nucleon masses across multiple volumes up to about $L=2.1$ fm. By comparing volume dependence to Lüscher-type mass-shift formulas and ChPT (including NNLO inputs for pions and $O(p^4)$ for nucleons), the authors find that an exponential ansatz $m_H(L)=m_H+c L^{-3/2} e^{-m_{PS}L}$ generally describes the data better than a simple power law and yields reliable infinite-volume estimates when plausible. For $L\gtrsim 1.5$–2 fm, finite-size corrections are typically a few percent or smaller, with pion masses most sensitive to volume at smaller volumes while nucleon shifts align well with relativistic baryon ChPT predictions. The results provide practical guidance for choosing volumes and applying finite-volume corrections in future unquenched Wilson-QCD simulations, potentially enabling volume extrapolations and reducing computational costs.

Abstract

As computing resources are limited, choosing the parameters for a full Lattice QCD simulation always amounts to a compromise between the competing objectives of a lattice spacing as small, quarks as light, and a volume as large as possible. Aiming to push unquenched simulations with the Wilson action towards the computationally expensive regime of small quark masses we address the question whether one can possibly save computing time by extrapolating results from small lattices to the infinite volume, prior to the usual chiral and continuum extrapolations. In the present work the systematic volume dependence of simulated pion and nucleon masses is investigated and compared with a long-standing analytic formula by Luescher and with results from Chiral Perturbation Theory. We analyze data from Hybrid Monte Carlo simulations with the standard (unimproved) two-flavor Wilson action at two different lattice spacings of a=0.08fm and 0.13fm. The quark masses considered correspond to approximately 85 and 50% (at the smaller a) and 36% (at the larger a) of the strange quark mass. At each quark mass we study at least three different lattices with L/a=10 to 24 sites in the spatial directions (L=0.85-2.08fm).

Figures (20)

• Figure 1: The effective potential, $V_\mathrm{eff}(R,\tau)$, at $(\beta,\kappa)=(5.32144,0.1665)$ on the $16^3$ lattice, for selected values of $R$ in the range $\tau=1,\dots,4$. Larger values of $\tau$ are dominated by sta-ti-sti-cal noise.
• Figure 2: The static quark potential obtained from Wilson loops at $(\beta,\kappa)=(5.32144,0.1665)$ on the $16^3$ lattice.
• Figure 3: The parameters $N$, $\alpha$ for the Wuppertal smearing scheme were chosen such as to yield approximately the same wave function shapes for both the smaller lattices and the SESAM lattice ($L=16$).
• Figure 4: Box-size dependence of the pseudoscalar and vector meson masses and of the nucleon mass at $(\beta,\kappa)=(5.6,0.1575)$. The solid lines result from fits to an exponential, Eq. \ref{['eqn:exp']}, while the dashed lines represent fits to a power law, Eq. \ref{['eqn:pow']}. The curves are dotted outside the fit interval.
• Figure 5: Fits as in Fig. \ref{['fig:ldep_56_1575']} for $(\beta,\kappa)=(5.6,0.158)$.