[Paper Review] Finite ma Errors of the Overlap Fermion

S. J. Dongand, K. F. Liu|ArXiv.org|Oct 16, 2007

Particle physics theoretical and experimental studies3 citations

TL;DR

This study investigates finite $ma$ systematic errors in the overlap fermion formalism by analyzing the effective speed of light from meson dispersion relations and hyperfine splitting between vector and pseudoscalar mesons. It concludes that $ma < 0.5$ is necessary to keep $O(ma^2)$ and $O(m^2a^2)$ errors below 3–4%, ensuring reliable simulations for both light and heavy quarks.

ABSTRACT

In this talk, we shall assess the finite ma errors from the overlap fermion. We shall present results on the speed of light from the dispersion relation and hyperfine splitting between the vector and pseudoscalar mesons as a function to ma to reveal the mΛ_{QCD}a^2 and m^2a^2 errors. We conclude from this study that one should be limited to using ma less than 0.5 in order to keep the systematic ma errors below a few percent level.

Motivation & Objective

• To assess systematic $ma$ errors in the overlap fermion formalism, particularly $O(ma^2)$ and $O(m^2a^2)$ terms, which affect physical observables.
• To determine the maximum allowable $ma$ value that maintains $ma$-induced systematic errors below a few percent for reliable lattice QCD calculations.
• To evaluate the validity of the overlap fermion for heavy quarks by examining deviations in the speed of light and hyperfine splitting.
• To compare results across different lattice spacings and gauge actions to test the robustness of $ma$ error scaling.
• To validate the use of overlap fermions in large-scale simulations by quantifying how $ma$ affects key physical quantities like dispersion relations and spin splittings.

Proposed method

• Computed pseudoscalar and vector meson dispersion relations using overlap fermion propagators on quenched lattices with Iwasaki gauge action.
• Extracted the effective speed of light $c$ from fitting energies to the dispersion relation $ (E(p)a)^2 = c^2 (pa)^2 + (E(0)a)^2 $, where deviations from $c=1$ signal $ma$ errors.
• Fitted $c$ as a function of $ma$ using a quadratic form: $c = c_0 + b(ar{ ho}ma) + d m^2 a^2$, with $\bar{\rho} = \Lambda_{QCD}a$.
• Measured the hyperfine splitting between vector and pseudoscalar mesons as a function of $ma$ on two lattices with different spacings ($a = 0.0561$ fm and $a = 0.133$ fm).
• Fitted the hyperfine splitting to a form $h.f.s = \frac{a}{\sqrt{ma}}(1 + \frac{b}{ma})$ to isolate $ma$-dependent deviations from the expected $1/\sqrt{m}$ scaling.
• Used $\chi^2/\text{dof}$ to assess fit quality and identify regions where $ma$ errors dominate the deviation from continuum behavior.

Experimental results

Research questions

• RQ1 How do $O(ma^2)$ and $O(m^2a^2)$ errors affect the effective speed of light in the overlap fermion formalism?
• RQ2 At what value of $ma$ do systematic errors in the dispersion relation exceed 3–4%?
• RQ3 How do $ma$ errors influence the hyperfine splitting between vector and pseudoscalar mesons, particularly in the context of $1/\sqrt{m}$ scaling?
• RQ4 Does the $ma$ error behavior depend on lattice spacing or gauge action, and is the $ma < 0.5$ threshold robust across different simulations?
• RQ5 To what extent do $ma$ errors distort the expected $1/\sqrt{m}$ behavior of hyperfine splittings in quarkonium states?

Key findings

• The effective speed of light $c$ remains consistent with unity up to $ma \sim 0.4$, with deviations becoming significant only beyond $ma \sim 0.55$.
• For $ma < 0.5$, the $O(ma^2)$ and $O(m^2a^2)$ errors in the speed of light are less than 4%, as determined by fitting $c = c_0 + b(\Lambda_{QCD}a)ma + d m^2 a^2$.
• The hyperfine splitting deviates from the expected $1/\sqrt{m}$ scaling at large $ma$, with $ma = 0.6$ corresponding to a $m^2a^2$ error of about 7%.
• At $ma = 0.85$, the $m^2a^2$ error in the hyperfine splitting reaches approximately 50%, indicating strong systematic bias.
• The $ma$ error behavior is consistent across two lattices with different spacings ($a = 0.0561$ fm and $a = 0.133$ fm), suggesting robustness of the $ma < 0.5$ threshold.
• The study concludes that $ma < 0.5$ is a prudent upper bound to keep systematic $ma$ errors below 3–4% for reliable physical predictions using overlap fermions.