[Paper Review] Effective lattice action for the configurations smeared by the Wilson flow

Aya Kagimura, Akio Tomiya+1 more|arXiv (Cornell University)|Aug 20, 2015

Fluid Dynamics and Turbulent Flows17 references3 citations

TL;DR

This paper determines the effective lattice action for gauge configurations smeared via the Wilson flow using the demon method, a non-perturbative inverse Monte Carlo technique. It finds that the plaquette coefficient increases while the rectangular term coefficient turns negative with increasing flow time, forming a straight-line trajectory in the two-coupling theory space—indicating a flow-induced improvement similar to known improved actions without perturbative approximations.

ABSTRACT

We investigate a trajectory for the Wilson flow in the theory space. For this purpose, we determine the coefficient of the plaquette and rectangular terms in the action for the configurations defined by the solution of the Wilson flow. The demon method regarded as one of the inverse Monte Carlo methods is used for the determination of them. Starting from the conventional Wilson plaquette action of quenched QCD, we find that the coefficient of the plaquette grows while that of the rectangular tends to negative with the development of the flow as the known improved actions. We also find that the trajectory forms a straight line in the two-coupling theory space.

Motivation & Objective

• To non-perturbatively determine the effective lattice action for gauge configurations evolved via the Wilson flow, avoiding perturbative approximations.
• To investigate how the couplings of the plaquette and rectangular Wilson loops evolve with flow time in the theory space.
• To assess whether the Wilson flow induces an improvement in lattice artifacts, analogous to known improved actions.
• To examine the universality and structure of the trajectory in the two-coupling parameter space.
• To evaluate the feasibility of using the Wilson flow as a continuous blocking procedure for effective action determination.

Proposed method

• The demon method is employed to extract effective couplings from a given set of Wilson flow-smeared gauge configurations, treating the system as a microcanonical ensemble with a demon degree of freedom.
• The effective action is defined via a delta function constraint that maps the original configurations to the smeared ones, enabling the derivation of $ S_{\text{eff}}[V_t] $ from the original $ S_{\text{W}}[U] $.
• The method involves microcanonical updates that conserve the total energy $ S_{\text{W}}[U] + E_d $, where $ E_d $ is the demon energy, to estimate the effective couplings $ \beta_{\text{plaq}} $ and $ \beta_{\text{rect}} $.
• Configurations are generated using the standard Wilson plaquette action at $ \beta = 6.0 $, then evolved via the Wilson flow up to $ \sqrt{8\hat{t}} \lesssim 1.3 $, with lattice sizes $ 4^4 $, $ 8^4 $, and $ 16^4 $ to assess finite-size effects.
• The effective couplings are extracted by measuring the probability distribution of the demon energy and fitting to the Boltzmann weight of the effective action.
• The trajectory in the two-coupling space is analyzed to determine its geometric structure, particularly whether it forms a straight line.

Experimental results

Research questions

• RQ1 How do the effective couplings of the plaquette and rectangular Wilson loops evolve with increasing Wilson flow time in the absence of perturbative approximations?
• RQ2 Does the trajectory traced by the effective action in the two-coupling theory space form a straight line, and what does this imply about the nature of the flow-induced improvement?
• RQ3 To what extent does the Wilson flow-smeared effective action resemble known improved lattice actions in its coupling evolution?
• RQ4 Can the demon method reliably extract effective couplings from configurations that have undergone continuous smearing, especially when the original action is lowered by the flow?
• RQ5 Is the effective action obtained via the Wilson flow and demon method suitable for reducing lattice artifacts in cutoff-sensitive observables like the topological charge or energy-momentum tensor?

Key findings

• The coefficient of the plaquette term in the effective action increases with increasing flow time, indicating a growing dominance of the standard Wilson term.
• The coefficient of the rectangular Wilson loop term evolves to negative values with increasing flow time, a hallmark of improved actions that suppress lattice artifacts.
• The trajectory of the effective couplings in the two-coupling theory space forms a straight line, suggesting a universal, linear renormalization group-like flow for the Wilson flow.
• The observed evolution of couplings matches the behavior seen in known improved actions such as Iwasaki and Weisz-type actions, indicating that the Wilson flow induces a similar improvement mechanism.
• The effective action is non-perturbatively defined and free from truncation ambiguities in blocking or projection, offering a well-defined alternative to standard improvement schemes.
• Finite-size effects are small across $ 4^4 $, $ 8^4 $, and $ 16^4 $ lattices, supporting the robustness of the extracted couplings.