θ Dependence of the Confinement-Deconfinement Transition Temperature in 4D SU(2) Yang-Mills Theory Determined Using Lattice Simulations

Core Concepts

The confinement-deconfinement transition temperature in 4D SU(2) Yang-Mills theory exhibits a quadratic dependence on the θ angle, determined to be Tc(θ)/Tc(0) = 1−0.016(3) θ2+O(θ4) using lattice simulations and the sub-volume method to mitigate the sign problem.

Abstract

• Bibliographic Information: Yamada, N., Yamazaki, M., & Kitanoa, R. (2024). θ dependence of Tc in SU(2) Yang-Mills theory. arXiv preprint arXiv:2411.00375v1.

• Research Objective: This study investigates the θ dependence of the confinement-deconfinement transition temperature (Tc) in 4D SU(2) pure Yang-Mills theory using lattice numerical simulations. The research aims to determine whether Tc(π) > 0, which would indicate spontaneous parity symmetry breaking in the vacuum at θ = π.

• Methodology: The study employs lattice numerical simulations with a tree-level Symanzik improved action on three spatial lattice sizes (NS = 24, 32, 48) and a fixed temporal size (NT = 8). A non-zero θ-angle is introduced using the re-weighting method combined with the sub-volume method to address the sign problem. The fourth-order Binder cumulant (B4) of the Polyakov loop is calculated to identify the critical point at various βg and θ values. The universality of the second-order phase transition in 4D SU(2) pure YM theory and the 3d Ising model is exploited, utilizing the critical exponents of the 3d Ising model to analyze the data.

• Key Findings: The study confirms the universality of the critical point between 4D SU(2) gauge theory and the 3d Ising model. The θ dependence of Tc is determined to be Tc(θ)/Tc(0) = 1−0.016(3) θ2+O(θ4), indicating a quadratic decrease in the critical temperature with increasing θ. This finding suggests that the θ dependence in SU(2) theory is weaker than the naive extrapolation from SU(Nc ≥ 3) theories.

• Main Conclusions: The research successfully determines the θ dependence of Tc in 4D SU(2) pure Yang-Mills theory, providing valuable insights into the θ-T phase diagram. The results suggest a weaker θ dependence compared to SU(Nc ≥ 3) theories and lay the groundwork for further investigations into the nature of the phase transition and potential emergence of a gapless theory at θ = π.

• Significance: This study contributes significantly to understanding the non-perturbative dynamics of pure Yang-Mills theory, a fundamental component of the Standard Model of particle physics. The findings have implications for exploring the θ-T phase diagram and the behavior of the theory at finite θ.

• Limitations and Future Research: The study acknowledges limitations in estimating systematic uncertainties associated with the sub-volume extrapolation. Future research could focus on a more detailed analysis of systematic uncertainties, exploring larger lattice volumes, and investigating the behavior of the theory at larger θ values, particularly around θ = π. Further investigation into the temperature dependence of the topological susceptibility and its singularity at the critical point is also warranted.

Stats

Tc(θ)/Tc(0) = 1−0.016(3) θ2+O(θ4)
βcrit
g
(θ) = d0 + ˜d2 ( θ / π )2
d0 = 1.919(2)
˜d2 = −0.052(9)
Tc(θ) / Tc(0) = c0 + ˜c2 ( θ / π )2
c0 = 0.996(7)
˜c2 = −0.16(3)

Quotes

"The θ dependence of the critical temperature has been studied in the vicinity of θ = 0 for SU(Nc) theories with Nc ≥3 [4–8], where the system undergoes the first order phase transition. The outcome is that the dependence can be written as Tc(θ)/Tc(0) = 1 −R θ2 + O(θ4) with R ∼0.17/N 2
c for (Nc ≥3) [4, 5, 8]."
"Since R increases with 1/N 2
c , Tc(θ) for Nc = 2 may vanish at θ ∼O(1) and a gapless theory may emerge at θ = π, as for the 2d CP1 model [9–12] which shares many features with the 4d SU(2) YM theory."
"In the analysis of SU(Nc ≥3) theories R is computed as the ratio of the jump of the topological susceptibility at Tc(0) to the latent heat via the Clausius-Clapeyron relation, which is available only for the first order phase transitions, and there is no such jump for the second-order transition for the SU(2) YM theory."

Key Insights Distilled From

$\theta$ dependence of $T_c$ in SU(2) Yang-Mills theory

by Norikazu Yam... at arxiv.org 11-04-2024

https://arxiv.org/pdf/2411.00375.pdf

Deeper Inquiries

How does the θ dependence of Tc in SU(2) Yang-Mills theory relate to the larger context of quantum field theories and the Standard Model?

The study of the θ dependence of Tc in SU(2) Yang-Mills theory holds significant implications for our understanding of quantum field theories (QFTs) and the Standard Model of particle physics. Here's how:

Understanding the QCD Vacuum: Yang-Mills theories, particularly SU(3), form the foundation of Quantum Chromodynamics (QCD), the theory describing the strong force. The θ term in these theories is a source of CP violation, a fundamental symmetry violation observed in nature. Studying how Tc changes with θ provides insights into the non-perturbative structure of the QCD vacuum and its response to CP-violating effects. This is crucial because the QCD vacuum is thought to be a complex entity with potential connections to phenomena like the strong CP problem and axion physics.

Phase Transitions in the Early Universe: In the early universe, when temperatures were extremely high, the strong force is believed to have been in a deconfined phase. As the universe cooled, it underwent a phase transition to the confined phase we observe today. Understanding the θ dependence of Tc helps us model this cosmological phase transition more accurately. It could potentially impact our understanding of baryogenesis, the process that led to the matter-antimatter asymmetry in the universe.

Beyond the Standard Model Physics: The Standard Model, while remarkably successful, has limitations. It fails to explain certain observed phenomena like the existence of dark matter and neutrino masses. Many extensions of the Standard Model, such as Peccei-Quinn theory, introduce new particles and interactions that can modify the θ dependence of Tc. Therefore, precise measurements of Tc(θ) can provide indirect hints for these beyond-the-Standard Model scenarios.

Universality Class and Critical Phenomena: The study of phase transitions in QFTs like Yang-Mills theories allows us to explore the concept of universality classes. Systems belonging to the same universality class exhibit similar critical behavior near their phase transition points, regardless of their microscopic details. The observed second-order nature of the confinement-deconfinement transition in SU(2) and its connection to the 3D Ising model highlights these universal aspects of QFTs.

Could the observed weaker θ dependence in SU(2) compared to SU(Nc ≥ 3) be an artifact of the sub-volume method or finite lattice spacing, and how can these systematic uncertainties be further investigated?

Yes, the observed weaker θ dependence of Tc in SU(2) compared to SU(Nc ≥ 3) could potentially be influenced by systematic uncertainties arising from the sub-volume method and finite lattice spacing used in lattice QCD calculations. Here's a breakdown of these uncertainties and how to investigate them:

Sub-volume Method:

Issue: The sub-volume method, while crucial for mitigating the sign problem at large θ, introduces a finite volume effect. The extrapolation of results from a smaller sub-volume to the full volume might not perfectly capture the true θ dependence, especially near the critical region where long-range correlations are important.
Investigation: To address this, one can perform simulations with multiple sub-volume sizes and study the systematic effects of the extrapolation. By comparing results from different sub-volume extrapolations, one can estimate the magnitude of this uncertainty.

Finite Lattice Spacing:

Issue: Lattice QCD calculations are performed on a discrete spacetime lattice, introducing a finite lattice spacing (a). This discretization can distort the continuum physics, potentially affecting the θ dependence of Tc.
Investigation: To investigate this, simulations at different lattice spacings are necessary. By performing calculations at progressively finer lattice spacings and extrapolating the results to the continuum limit (a → 0), one can assess the impact of finite lattice spacing effects.

Combined Approach: Ideally, a comprehensive study would involve varying both the sub-volume size and the lattice spacing. This allows for a systematic investigation of both uncertainties and a more reliable extrapolation to the continuum limit at each θ.

Other Systematics: It's also essential to consider other potential sources of systematic uncertainty, such as:

Statistical Errors: Increasing the statistics of the simulations can reduce statistical uncertainties and improve the reliability of the results.
Choice of Topological Charge Operator: Different discretization schemes for the topological charge operator on the lattice can lead to slightly different results. Exploring different operators can help quantify this uncertainty.

How does the concept of confinement in QCD, as explored in this study, connect to the emergence of complex structures and information in the universe?

The concept of confinement in QCD, as explored through the study of the θ dependence of Tc, has profound implications for our understanding of the emergence of complex structures and information in the universe. Here's the connection:

Confinement and Hadron Formation: Confinement is the fundamental property of QCD that dictates quarks and gluons, the fundamental particles of the strong force, cannot exist in isolation. They are always confined within composite particles called hadrons, such as protons and neutrons. This confinement arises from the non-linear nature of the strong force, leading to a strong increase in the interaction strength at large distances.

Emergence of Structure: The formation of hadrons due to confinement is the first step towards the emergence of complex structures in the universe. Without confinement, the universe would be a sea of free quarks and gluons, lacking the diversity and complexity we observe. Hadrons, in turn, interact through residual strong forces to form atomic nuclei, paving the way for the formation of atoms, molecules, and eventually, stars, galaxies, and planets.

Information Encoding: The diversity of hadrons and their interactions allows for the encoding of information. The specific arrangement of quarks within hadrons, their quantum numbers, and their interactions determine the properties of matter. This information content, arising from the confined nature of QCD, is essential for the formation of complex systems and the emergence of life itself.

Phase Transitions and Cosmic Evolution: The confinement-deconfinement transition in the early universe, as studied through the θ dependence of Tc, played a crucial role in shaping the evolution of the cosmos. The transition from a deconfined quark-gluon plasma to a confined phase led to the formation of hadrons, influencing the subsequent formation of light nuclei during Big Bang nucleosynthesis. This, in turn, determined the abundance of elements in the universe, a crucial factor in the formation of stars and galaxies.

Fundamental Questions: The study of confinement in QCD and its connection to the emergence of complexity raises fundamental questions:

How does the information content of the universe arise from the fundamental laws of physics?
What is the role of phase transitions in shaping the complexity of the cosmos?
Can we develop a deeper understanding of the emergence of information and structure from the underlying principles of quantum field theory?
By exploring these questions through studies like the one presented, we gain a deeper appreciation for the profound connection between the fundamental laws of physics and the emergence of the rich and complex universe we inhabit.