On lattice actions for static quarks

TL;DR

This work develops new lattice discretizations for static quarks that exponentially improve the signal-to-noise ratio in static-light correlation functions compared with the Eichten–Hill action, enabling more precise HQET analyses. It provides a detailed nonperturbative program—within the Schrödinger functional framework—to determine improvement coefficients and renormalization constants for the static-light axial current across multiple actions, including APE-like, one-link integral, and HYP-smeared constructions. Scaling tests show small O($a$) artifacts and near $O(a^2)$ behavior, with the HYP2 action offering the best noise reduction and favorable discretization properties; the study also furnishes step-scaling results and renormalization factors $Z_A^{stat}$ to connect finite-volume matrix elements to the continuum and to the RGI framework. Together, these results enable more reliable continuum extrapolations and facilitate the computation of $1/m_b$ corrections and the regularization-dependent part of the $b$-quark mass in static HQET, with potential applicability to dynamical-quark simulations in the future.

Abstract

We introduce new discretizations of the action for static quarks. They achieve an exponential improvement (compared to the Eichten-Hill regularization) on the signal to noise ratio in static-light correlation functions. This is explicitly checked in a quenched simulation and it is understood quantitatively in terms of the self energy of a static quark and the lattice heavy quark potential at zero distance. We perform a set of scaling tests in the Schroedinger functional and find scaling violations in the O(a) improved theory to be rather small -- for one observable significantly smaller than with the Eichten-Hill regularization. In addition we compute the improvement coefficients of the static light axial current up to O(g_0^4) corrections and the corresponding renormalization constants non-perturbatively. The regularization dependent part of the renormalization of the b-quark mass in static approximation is also determined.

Figures (9)

• Figure 1: Schematic representation of the correlation functions $f_{\rm A}^{\rm stat}(x_0)$ and $f_{1}^{\rm stat}$ in the SF. The double lines indicate the static quark propagators.
• Figure 2: The ratio $R_{\rm NS}$ for the correlation function $f_{\rm A}^{\rm stat}(x_0)$ for a statistics of 5000 measurements on a $16^3 \times 32$ lattice at $\beta=6$. Filled circles refer to $S_{\rm h}^{\rm EH}$, empty circles to $S_{\rm h}^{\rm A}$ ($S_{\rm h}^{\rm s}$ gives similar results), empty (filled) triangles to $S_{\rm h}^{\rm HYP1}$ ($S_{\rm h}^{\rm HYP2}$).
• Figure 3: Numerical results for the improvement coefficient $c_{\rm A}^{\rm stat}$. For $S_{\rm h}^{\rm EH}$ the result of castat is plotted, for $S_{\rm h}^{\rm s}$ the band represents the result eq. (\ref{['caA']}), while otherwise it refers to the numerical estimates eqs. (\ref{['caHYP1']}, \ref{['caHYP2']}).
• Figure 4: Numerical results for the improvement coefficient $b_{\rm A}^{\rm stat}$, presented as in figure \ref{['ca_fig']}.
• Figure 5: Scaling plots for $\xi_{\rm A}(0,0.5)$, $\xi_1(0,0.5)$ and $h(1/4)$. Symbols are as in figure \ref{['f_rns']}, some of them have been shifted on the horizontal axis for clarity. Again the results for $S_{\rm h}^{\rm s}$ are similar to the ones from $S_{\rm h}^{\rm A}$ and haven't been plotted.