Log-enhanced discretization errors in integrated correlation functions

TL;DR

This paper investigates discretization errors in integrated correlation functions used in lattice QCD, revealing log-enhanced errors at small lattice spacings and proposing modifications to improve continuum extrapolation.

Contribution

It derives the asymptotic behavior of discretization errors in integrated correlators and introduces a modified observable to achieve smoother continuum limits.

Findings

Log-enhanced discretization errors identified at small lattice spacings.

Modified observable improves the smoothness of the continuum limit.

Short distance behavior can be effectively handled by perturbation theory.

Abstract

Integrated time-slice correlation functions $G(t)$ with weights $K(t)$ appear, e.g., in the moments method to determine $\alpha_s$ from heavy quark correlators, in the muon g-2 determination or in the determination of smoothed spectral functions. For the (leading-order-)normalised moment $R_4$ of the pseudo-scalar correlator we have non-perturbative results down to $a=10^{-2}$ fm and for masses, $m$, of the order of the charm mass in the quenched approximation. A significant bending of $R_4$ as a function of $a^2$ is observed at small lattice spacings. Starting from the Symanzik expansion of the integrand we derive the asymptotic convergence of the integral at small lattice spacing in the free theory and prove that the short distance part of the integral leads to $\log(a)$-enhanced discretisation errors when $G(t)K(t) \sim\, t $ for small $t$. In the interacting theory an unknown, function $K(a\Lambda)$ appears. For the $R_4$-case, we modify the observable to improve the short distance behavior and demonstrate that it results in a very smooth continuum limit. The strong coupling and the $\Lambda$-parameter can then be extracted. In general, and in particular for $g-2$, the short distance part of the integral should be determined by perturbation theory. The (dominating) rest can then be obtained by the controlled continuum limit of the lattice computation.