[Paper Review] Scaling and low energy constants in lattice QCD with N_f=2 maximally twisted Wilson quarks

ETM Collaboration, P. Dimopoulos+4 more|ArXiv.org|Oct 12, 2007

Quantum Chromodynamics and Particle Interactions6 citations

TL;DR

This paper investigates scaling and low-energy constants in lattice QCD with $N_f=2$ maximally twisted Wilson fermions using simulations across multiple lattice spacings and pion masses down to 300 MeV. It demonstrates good scaling behavior with small cutoff effects, supporting the use of automatic $\mathcal{O}(a)$ improvement, and presents preliminary continuum-limit estimates for Gasser–Leutwyler low-energy constants and the chiral condensate.

ABSTRACT

We report on the scaling of basic hadronic observables in lattice QCD with N_f=2 maximally twisted Wilson dynamical quarks. We give preliminary results for some of the Gasser-Leutwyler low energy constants, the chiral condensate and the average mass of u and d quarks.

Motivation & Objective

• To assess the scaling behavior of hadronic observables in lattice QCD with $N_f=2$ maximally twisted Wilson fermions.
• To determine the continuum limit of low-energy constants (LECs) in the chiral perturbation theory framework.
• To estimate the chiral condensate and the average $u$- and $d$-quark mass using $\chi$PT-based analyses.
• To validate the reliability of the maximally twisted Wilson formulation for computing physical observables near the continuum limit.
• To provide preliminary results for charmed meson observables in a partially quenched setup with controlled cutoff effects.

Proposed method

• Simulations are performed using the tree-level improved gauge action and maximally twisted Wilson fermions with $\kappa$ tuned via the condition $m_{\text{PCAC}} = 0$ at $\mu_{\text{LOW}}$ to ensure automatic $\mathcal{O}(a)$ improvement.
• The critical mass is tuned non-perturbatively to suppress $\mathcal{O}(a)$ cutoff effects, with $\mu_{\text{LOW}} \gtrsim C a^2 \Lambda_{\text{QCD}}^3$ ensuring small residual artifacts.
• Physical observables are computed at multiple lattice spacings ($\beta = 3.8, 3.9, 4.05$) and spatial volumes ($L \sim 2-3$ fm), with $m_{\text{PS}}L \geq 3$.
• Chiral perturbation theory (χPT) is applied to extrapolate results to the physical pion mass and continuum limit, using $r_0$ as a scale-setting parameter.
• Partially quenched setups are employed for charmed meson observables, with valence quarks tuned to preserve $\mathcal{O}(a)$ improvement.
• Renormalization constants ($Z_P$, $Z_A$) and $r_0$ are determined non-perturbatively to ensure physical scale setting and accurate mass definitions.

Experimental results

Research questions

• RQ1 Do hadronic observables in $N_f=2$ maximally twisted Wilson lattice QCD exhibit good scaling behavior down to $m_{\text{PS}} \sim 300$ MeV?
• RQ2 What are the continuum-limit estimates for the Gasser–Leutwyler low-energy constants and the chiral condensate?
• RQ3 To what extent do cutoff effects remain small in the maximally twisted Wilson formulation, and is automatic $\mathcal{O}(a)$ improvement realized?
• RQ4 How reliable are the $\chi$PT-based extrapolations for the nucleon mass and $m_N/f_\pi$ ratio in the continuum limit?
• RQ5 What is the size of scaling violations in charmed meson observables such as $f_D$, $f_{D_s}$, and $m_{D_s}$?

Key findings

• Scaling violations in the light pseudoscalar meson sector are small, with deviations of order 1–2% at $m_{\text{PS}} \sim 300$ MeV, consistent with $\mathcal{O}(a^2)$ effects.
• For $f_D$ and $m_{D_s}$, scaling violations are at most 1–2% between $\beta = 4.05$ and $\beta = 3.9$, while $f_{D_s}$ shows 7–8% violations, suggesting larger systematic uncertainties.
• The nucleon mass data at $\beta = 4.05$ and $\beta = 3.9$ are consistent, supporting a smooth approach to the continuum limit.
• The $\chi$PT-based analysis suggests that the LECs and chiral condensate values obtained at a single lattice spacing are close to their continuum-limit values.
• The ratio $m_N/f_\pi$ after chiral extrapolation is in good agreement with the experimental value, validating the framework.
• The framework of maximally twisted Wilson fermions enables reliable, $\mathcal{O}(a)$-improved calculations of physical observables, including vector mesons and nucleons, with controlled cutoff effects.