[Paper Review] Thermodynamics and reference scale of SU(3) gauge theory from gradient flow on fine lattices

Masakiyo Kitazawa, Masayuki Asakawa+4 more|arXiv (Cornell University)|Nov 17, 2015

Quantum Chromodynamics and Particle Interactions12 references4 citations

TL;DR

This paper presents a high-precision determination of the lattice spacing and thermodynamic properties of SU(3) gauge theory using the Yang-Mills gradient flow on fine lattices. By introducing reference scales $w_{0.4}$ and $w_{0.2}$, the authors derive a new parametrization of the lattice spacing $a(\beta)$ over $6.3 \leq \beta \leq 7.5$, and apply the gradient flow to measure the energy-momentum tensor, enabling accurate extraction of thermodynamic observables like the trace anomaly and entropy density at $T = 1.66T_c$, with results consistent with continuum extrapolation.

ABSTRACT

We study the parametrization of lattice spacing and thermodynamics of SU(3) gauge theory on the basis of the Yang-Mills gradient flow on fine lattices. The lattice spacing of the Wilson gauge action is determined over a wide range $6.3\leβ\le7.5$ with high accuracy. The measurements of the flow time and lattice spacing dependences of the expectation values of the energy-momentum tensor are performed on fine lattices.

Motivation & Objective

• To establish a high-accuracy parametrization of the lattice spacing $a(\beta)$ for SU(3) Yang-Mills theory on fine lattices.
• To introduce and utilize $w_{0.4}$ and $w_{0.2}$ as reference scales based on the gradient flow to set physical units.
• To measure thermodynamic observables—such as the trace anomaly and entropy density—using the energy-momentum tensor defined via the gradient flow.
• To validate the gradient flow method for thermodynamics by comparing results with continuum-extrapolated values from the integral method.
• To assess finite volume and lattice discretization effects in the context of the gradient flow approach.

Proposed method

• The lattice spacing is determined using the gradient flow to define reference scales $w_{0.4}$ and $w_{0.2}$, where $t^2\langle E(t)\rangle = X$ and $t \frac{d}{dt} t^2\langle E(t)\rangle = X$ at $t = w_X^2$.
• The energy-momentum tensor $T_{\mu\nu}^R$ is reconstructed from the small flow-time expansion of the gauge-invariant operators $U_{\mu\nu}(t,x)$ and $E(t,x)$, using perturbative coefficients $\alpha_U(t)$ and $\alpha_E(t)$ up to next-to-leading order.
• Numerical simulations are performed on fine lattices with temporal extent $N_t = 12$ to $32$, using the Wilson gauge action and clover-type field strength tensors.
• Thermodynamic quantities—energy density $e$, pressure $p$, trace anomaly $\Delta/T^4$, and entropy density $s/T^3$—are extracted from the expectation values of $T_{\mu\nu}^R$.
• The flow-time dependence of observables is analyzed to identify the linear regime where $\sqrt{8t} \gg 2a$ and $\sqrt{8t} \ll 1/(2T)$, avoiding lattice artifacts and oversmearing.
• Continuum extrapolation is performed by taking the $y$-intercept of the linear fit in flow time, with results compared to the integral method.

Experimental results

Research questions

• RQ1 What is the most accurate parametrization of the lattice spacing $a(\beta)$ for SU(3) gauge theory in the range $6.3 \leq \beta \leq 7.5$ using gradient flow reference scales?
• RQ2 How do the reference scales $w_{0.4}$ and $w_{0.2}$ compare in suppressing lattice discretization effects?
• RQ3 Can the gradient flow method accurately reproduce thermodynamic observables like $\Delta/T^4$ and $s/T^3$ at $T = 1.66T_c$?
• RQ4 What is the optimal range of flow time $t$ to minimize lattice artifacts and oversmearing in thermodynamic measurements?
• RQ5 How well do the gradient flow results agree with the continuum-extrapolated values obtained via the integral method?

Key findings

• A new parametrization of the lattice spacing $a(\beta)$ is derived using a hybrid approach with $w_{0.4}$ and $w_{0.2}$, achieving high accuracy across $6.3 \leq \beta \leq 7.5$.
• The $y$-intercept of the linear fit in flow time for $\Delta/T^4$ and $s/T^3$ agrees well with the continuum-extrapolated result from the integral method, indicated by the blue arrow in Fig. 2.
• Lattice discretization effects are observed for $\sqrt{8t} \lesssim 2a$, while oversmearing effects appear for $\sqrt{8t} \gtrsim 1/(2T)$, defining the valid range for physical extraction.
• The method successfully extracts thermodynamic quantities with good statistical precision, demonstrating the viability of the gradient flow approach for SU(3) gauge theory thermodynamics.
• The small flow-time expansion of $U_{\mu\nu}(t,x)$ and $E(t,x)$ enables the reconstruction of the conserved energy-momentum tensor $T_{\mu\nu}^R$ with perturbative coefficients up to next-to-leading order.
• The results confirm that the gradient flow method provides a regularization-independent and robust framework for thermodynamic measurements in lattice gauge theory.