Highly improved staggered quarks on anisotropic lattices

Technical Summary: Highly Improved Staggered Quarks on Anisotropic Lattices

Problem Statement
The primary motivation for this work is the development of a formalism to generate gauge field ensembles optimized for spectral reconstruction in lattice QCD, specifically for studying the melting patterns of heavy quarkonia and heavy open-flavor states in quark-gluon plasma. Spectral reconstruction from Euclidean correlation functions is an ill-posed problem requiring ensembles with very high statistics and large temporal extents ($N_\tau$). While isotropic ensembles with large $N_\tau$ are computationally prohibitive, anisotropic lattices (where the spatial lattice spacing $a_\sigma$ is significantly larger than the temporal spacing $a_\tau$, defined by the renormalized anisotropy $\xi = a_\sigma/a_\tau \sim 6-8$) offer a solution.

Existing large-scale anisotropic simulations typically employ Wilson fermions, which are computationally expensive and limit statistical precision. Conversely, while staggered fermions are efficient for finite-temperature QCD, previous attempts using anisotropic naive (unimproved) staggered fermions were abandoned due to large discretization effects, particularly taste-breaking. This paper addresses the gap by developing and tuning the Anisotropic Highly Improved Staggered Quark (aHISQ) action, which aims to combine the computational efficiency of staggered fermions with the reduced discretization errors of improved actions.

Methodology
The authors present a comprehensive study of tuning the parameters for both the gauge and fermion sectors of the aHISQ action on pure gauge (quenched) ensembles.

1. Gauge Anisotropy Tuning:

   • The study utilizes the tree-level Symanzik-improved gauge action with a bare gauge anisotropy parameter $\xi_0$.
   • Tuning is performed using the gradient flow method. The authors compare multiple flow schemes (Wilson, Symanzik, Zeuthen) and observable discretizations (Wilson, Symanzik, Clover, Improved Clover).
   • The renormalized anisotropy $\xi$ is determined by imposing the condition $w_{0,\sigma}/w_{0,\tau} = 1$, where $w_0$ is the gradient flow scale.
   • The authors identify the "Wilson flow with Clover observable" (WC) scheme as optimal due to its monotonic behavior and stability in reaching the continuum limit.

2. Fermion Anisotropy Tuning:

   • The aHISQ Dirac operator is constructed by introducing the bare fermion anisotropy $\xi^f_0$ at the outermost level of the smearing (Fat7 and Naik terms) to preserve the suppression of coupling to gluons at the Brillouin zone corners.
   • Tuning involves mapping the bare parameters $(a_\sigma m_s, \xi^f_0)$ to the renormalized anisotropy $\xi^f$ and the physical strange quark mass.
   • Two methods for determining $\xi^f$ are discussed: the ratio of ground state energies in temporal vs. spatial directions and the dispersion relation method. The authors prefer the dispersion relation method, fitting the energy-momentum relation $a_\tau^2 E^2(p) = a_\tau^2 E^2(0) + a_\sigma^2 p^2 / (\xi^f)^2$.
   • A full algorithm for the aHISQ fermion force is derived and implemented for the Rational Hybrid Monte Carlo (RHMC) algorithm, utilizing a recursive chain rule to handle the anisotropy factors.

3. Spectrum Analysis:

   • The study measures pion taste mass splittings for both naive and aHISQ actions across a range of anisotropies ($\xi = 1$ to $8$).
   • The authors analyze the dependence of taste splittings on $\xi$ and compare the behavior of the two actions.

Key Contributions and Results

• Gauge Tuning: The authors successfully mapped the Lines of Constant Renormalized Anisotropy (LCRA) for $\xi=2$ and $\xi=4$ in the $(\beta, \xi_0)$ plane. They reproduced existing results for $\xi=2$ and extended the tuning to higher anisotropies ($\xi=8$) using the WC scheme.
• Fermion Tuning Efficiency: A significant finding is the behavior of the aHISQ action compared to the naive action. For naive staggered fermions, the bare fermion anisotropy $\xi^f_0$ depends strongly on the quark mass, requiring a two-dimensional scan in the $(a_\sigma m_s, \xi^f_0)$ parameter space. In contrast, for aHISQ, the dependence of $\xi^f_0$ on the quark mass is negligible (flat) across a wide mass range. This allows the aHISQ tuning to be reduced to two one-dimensional scans, drastically simplifying the computational cost for future dynamical simulations.
• Taste Spectrum Behavior:
  • Naive Action: As anisotropy increases, heavier pion tastes move down in mass, approaching a new degeneracy pattern expected for infinite anisotropy.
  • aHISQ Action: The behavior is qualitatively different. While some tastes (e.g., $\xi_4\xi_5$) move down, others (e.g., $\xi_i\xi_5$, $\xi_i\xi_4$, $\xi_4$) move up or remain relatively constant at low anisotropy before eventually following the new pattern.
• Empirical Modeling: The authors developed two models to describe the aHISQ taste splittings:
  1. Chiral Perturbation Theory (ChPT) Inspired: An extension of the Lee-Sharpe analysis where low-energy constants ($C_i, D_i$) are made dependent on $\xi$. They found that for aHISQ, splittings are dominated by $C_4(\xi)$ and $D_4(\xi)$, with other constants near zero.
  2. Symmetry-Breaking Perturbation Model: An empirical model treating anisotropy as a perturbation that redistributes energy based on the number of unbalanced spatial ($l_\sigma$) and temporal ($l_\tau$) links in the interpolating operators. This model successfully captures the observed equidistant pattern of splittings and the specific dependence of $C_4$ and $D_4$ on $\xi$.

Significance
The paper claims that the developed aHISQ formalism and tuning procedures provide a robust foundation for generating fully dynamical anisotropic ensembles. The primary significance lies in the demonstrated efficiency: the ability to tune the aHISQ action with a simplified one-dimensional search for the fermion anisotropy makes the generation of high-statistics, large-$N_\tau$ ensembles feasible. This capability is crucial for addressing the spectral reconstruction problem in the quark-gluon plasma regime, where current Wilson fermion approaches are limited by computational cost and temporal extent. The paper validates the code by reproducing literature results for pure gauge, quenched naive, and quenched dynamical naive spectra, and presents the first detailed tuning and spectral analysis for the aHISQ action on anisotropic lattices.