High-precision scale setting in lattice QCD (1203.4469v2)

Abstract: Scale setting is of central importance in lattice QCD. It is required to predict dimensional quantities in physical units. Moreover, it determines the relative lattice spacings of computations performed at different values of the bare coupling, and this is needed for extrapolating results into the continuum. Thus, we calculate a new quantity, $w_0$, for setting the scale in lattice QCD, which is based on the Wilson flow like the scale $t_0$ (M. Luscher, JHEP 1008 (2010) 071). It is cheap and straightforward to implement and compute. In particular, it does not involve the delicate fitting of correlation functions at asymptotic times. It typically can be determined on the few per-mil level. We compute its continuum extrapolated value in 2+1-flavor QCD for physical and non-physical pion and kaon masses, to allow for mass-independent scale setting even away from the physical mass point. We demonstrate its robustness by computing it with two very different actions (one of them with staggered, the other with Wilson fermions) and by showing that the results agree for physical quark masses in the continuum limit.

• The paper introduces a high-precision scale setting method using the Wilson flow-derived w₀ parameter for lattice QCD simulations.
• It demonstrates the reliability of the new method through continuum extrapolation at physical mass points, achieving precision within 1%.
• The results position w₀ as a cost-effective and versatile tool, simplifying cross-comparisons across various QCD simulation frameworks.

High-Precision Scale Setting in Lattice QCD

The paper "High-precision scale setting in lattice QCD," authored by the Budapest-Marseille-Wuppertal collaboration, addresses the critical task of scale setting in lattice Quantum Chromodynamics (QCD). The work explores the challenges of and solutions for accurate scale determination, which is imperative for predicting physical quantities in lattice QCD with high precision.

Overview of the Wilson Flow for Scale Setting

The authors introduce a new scale setting methodology utilizing the quantity denoted as $w_0$, derived from the Wilson flow. The Wilson flow, proposed by M. Lüscher, offers a way to smooth gauge fields on the lattice, facilitating a scale that avoids the complexities typically encountered in QCD calculations. The Wilson flow calculates the gauge fields as a function of a fictitious "flow time" and is particularly advantageous due to its relative simplicity and cost-effectiveness, not requiring the intricate fitting of correlation functions.

Methodological Distinctions

The paper elaborates on the motivation and advantages of the $w_0$ scale over existing methods such as the Sommer scale $r_0$. The $w_0$ scale offers precise and robust calculations, unaffected by the lattice gauge routine or quark propagator-associated computational costs. Unlike the Sommer scale, it avoids complications related to Wilson loop asymptotic behavior. The implementation by Wilson flow offers consistently high precision, reaching errors as low as a few per-mil, showcasing its viability for high-stakes QCD computations.

Numerical Findings and Implications

The results presented in the paper suggest that the $w_0$ scale can be reliably determined with minimal computational expense. Continuum extrapolation at the physical mass point—calculated using two different fermionic actions (staggered and Wilson)—demonstrates excellent agreement between methodologies, reinforcing the method's validity.

The $w_0$ scale's robustness is illustrated with computational data using both Wilson and staggered fermions. The authors report a continuum extrapolated value of $w_0 = 0.1755(18)(04)$ fm, with overall uncertainties of approximately 1%. This precision confirms $w_0$'s utility in setting relative scales between simulations and supports the comparison of computations across different groups, fostering collaborative advancements in the field.

Implications and Future Directions

This work pragmatically contributes a credible and efficient scale setting method for lattice QCD, potentially transforming how QCD computations handle errors and uncertainties in dimensional predictions. The method's mass independence up to a reasonable deviation makes it versatile and applicable across a range of quark mass scenarios, negating the need to adjust scale settings with each change in physical parameters.

The paper suggests further investigation into the implications of $w_0$ for simulations with different fermion counts (other than the evaluated $N_f=2+1$), anticipating broader utility. Additionally, the potential applications of $w_0$ in anisotropic actions and its integration into universal scale settings marks a forward-thinking contribution to QCD computations, rich with future research opportunities.

Conclusion

In summary, this paper delivers an essential advancement in the precision scale setting of lattice QCD. By leveraging the Wilson flow to define $w_0$, it offers a computationally efficient, reliable, and precise method that can resolve the current scale setting challenges in high-precision lattice QCD studies. The findings reinforce the necessity of such scale definitions in progressing high precision QCD-derived predictions and enhancing the theoretical and practical applications of lattice QCD.