[Paper Review] Theta dependence of 4D SU(N) gauge theories at finite temperature

Claudio Bonati, Massimo D’Elia+2 more|arXiv (Cornell University)|Sep 24, 2013

Quantum Chromodynamics and Particle Interactions28 references3 citations

TL;DR

This paper investigates the θ dependence of 4D SU(N) gauge theories at finite temperature using lattice Monte Carlo simulations and large-N scaling. It shows that the θ dependence undergoes a sharp transition across the deconfinement temperature: at low T, it follows large-N scaling with θ/N as the relevant variable, while at high T, it is well described by the dilute instanton gas approximation, yielding a periodic F(θ,T)−F(0,T)∼1−cosθ dependence. The transition becomes sharper with increasing N, indicating a crossover to the instanton gas regime just above Tc.

ABSTRACT

We report a study of the dependence of 4D SU(N) gauge theories on the topological theta term at finite temperature, and in particular in the large-N limit. We show that the theta dependence drastically changes across the deconfinement transition. The low-temperature phase is characterized by a large-N scaling with theta/N as relevant variable, while in the high-temperature phase the free energy is essentially determined by the dilute instanton-gas approximation, with a simple theta dependence of the free-energy density proportional to 1-Cos(theta).

Motivation & Objective

• To understand the θ dependence of 4D SU(N) gauge theories at finite temperature, particularly across the deconfinement transition.
• To clarify the role of large-N scaling and the dilute instanton gas (DIG) approximation in determining the free energy's θ dependence.
• To determine whether the topological susceptibility χ(T) and higher-order coefficients b2j(T) exhibit distinct behaviors in the low- and high-temperature phases.
• To test the validity of the DIG approximation at high T and its connection to the effective restoration of U(1)A symmetry.

Proposed method

• Lattice Monte Carlo simulations of SU(N) gauge theories at finite temperature, using the θ term added to the Euclidean action as a source for topological charge density.
• Computation of the free energy difference F(θ,T)−F(0,T) via correlation functions of the topological charge density at θ=0.
• Use of the parametrization F(θ,T)−F(0,T) = ½χ(T)θ²s(θ,T), where s(θ,T) is an even, dimensionless function with s(0,T)=1.
• Large-N scaling analysis to test whether χ(T)/σ² and b2j(T) scale as C∞ + O(N⁻²) and b̄2j/N²j + O(N⁻²j⁻²), respectively.
• Comparison of numerical results with the dilute instanton gas prediction F(θ,T)−F(0,T) ≈ χ(T)(1−cosθ), with χ(T) ∼ T⁴ exp(−8π²/g²(T)).
• Incorporation of higher-order corrections via a virial-like expansion: F(θ,T) ≈ χ(1−cosθ) + χ²κ(θ) + O(χ³), with κ(θ) parametrized as ∑k c2k sin(θ/2)^{2k}.

Experimental results

Research questions

• RQ1 How does the θ dependence of the free energy in 4D SU(N) gauge theories change across the deconfinement transition at finite temperature?
• RQ2 Is the large-N scaling behavior with θ/N as the relevant variable valid in the low-temperature phase, and does it break down in the high-temperature phase?
• RQ3 To what extent does the dilute instanton gas approximation accurately describe the θ dependence in the high-temperature phase?
• RQ4 How do the coefficients b2(T) and b4(T) of the θ² and θ⁴ expansions evolve with temperature and N, and do they approach the DIG prediction b2 = −1/12 and b4 = 1/360?
• RQ5 Does the topological susceptibility χ(T) show exponential suppression at high T, consistent with the DIG model and large-N scaling?

Key findings

• The θ dependence undergoes a qualitative change across the deconfinement transition: low-temperature phase exhibits large-N scaling with θ/N as the relevant variable, while high-temperature phase follows the dilute instanton gas (DIG) approximation.
• The high-temperature free energy difference is well described by F(θ,T)−F(0,T) ≈ χ(T)(1−cosθ), with χ(T) ∼ T⁴ exp(−8π²/g²(T)) and g²(T) ∼ (11/3)N ln(T/Λ), confirming exponential suppression of instanton density.
• The coefficient b2(T) approaches the DIG prediction b2 = −1/12 from below, with a negative correction from hard-core instanton interactions, consistent with c4 < 0 in the virial expansion.
• The coefficient b4(T) is consistent with the DIG value b4 = 1/360, supporting the validity of the instanton gas model at high T.
• The transition from low- to high-temperature θ dependence becomes sharper with increasing N, indicating that the DIG regime sets in just above Tc in the large-N limit.
• Numerical results confirm that χ(T) is exponentially suppressed at high T, supporting the DIG approximation and suggesting that U(1)A symmetry breaking is largely suppressed but not fully restored at high temperatures.