[Paper Review] The Stefan-Boltzmann law: SU(2) versus SO(3) lattice gauge theory

Kurt Langfeld, Hugo Reinhardt|ArXiv.org|Jan 11, 2000

Theoretical and Computational Physics1 references3 citations

TL;DR

This study compares the Stefan-Boltzmann constant in SU(2) and SO(3) lattice gauge theories via finite-size scaling and continuum extrapolation. It finds that the SO(3) theory's Stefan-Boltzmann constant is approximately 20% larger than SU(2)'s, suggesting SU(2) lattice theory may not fully recover the continuum Yang-Mills theory due to persistent center vortices, while SO(3) better reproduces the expected three-gluon degrees of freedom in the high-temperature limit.

ABSTRACT

We investigate the high temperature limit of SU(2) and SO(3) lattice gauge theory, respectively. In particular, we study the Stefan-Boltzmann constant in both cases. As is well known, the Stefan-Boltzmann constant extracted from SU(2) lattice gauge theory by incorporating finite size effects is smaller than the continuum value which assumes three gluon degrees of freedom. On the other hand, the extrapolation of our SO(3) lattice data comes much closer to the continuum value. This rises the question whether SU(2) and SO(3) lattice gauge theories represent different quantum theories in the continuum limit.

Motivation & Objective

• To investigate whether SU(2) and SO(3) lattice gauge theories yield the same continuum field theory, particularly in the high-temperature limit.
• To determine if the discrepancy in the Stefan-Boltzmann constant observed in SU(2) lattice simulations—where the extracted value is ~30% below the continuum expectation—arises from residual center degrees of freedom.
• To test whether SO(3) lattice gauge theory, which lacks the Z₂ center symmetry of SU(2), better reproduces the continuum limit for the Stefan-Boltzmann constant.
• To examine the role of center vortices in suppressing effective degrees of freedom in SU(2) lattice gauge theory even in the deconfined phase.
• To provide a numerical benchmark for future high-precision studies of SO(3) gauge theory and its relation to the continuum Yang-Mills theory.

Proposed method

• Perform finite-temperature lattice Monte Carlo simulations using asymmetric lattices with $N_{ au}$ temporal and $N_{ ho}$ spatial sites.
• Use the Wilson action for SU(2) and the Bhanot-Creutz action (a special case of the Fierz action) for SO(3), both defined via plaquette variables.
• Compute the energy density $\epsilon/T^4$ from lattice link variables using the expression $\epsilon/T^4 = 3\beta^{F/A}N_{ au}^4 \left[1 - f_1^{F/A}(\beta^{F/A})/\beta^{F/A}\right]$.
• Extrapolate the $\kappa = \epsilon/T^4$ data to the continuum limit by analyzing $N_{ au}$ dependence with a fit of the form $\kappa = \kappa_\infty + c_1/N_{ au}^2 + c_2/N_{ au}^4$.
• Compare the continuum extrapolated Stefan-Boltzmann constants of SU(2) and SO(3) using the same $N_{ au}$ and $N_{ ho}/N_{ au} = 4$ ratio.
• Assume similar $\beta$-function corrections for SO(3) as for SU(2) due to lack of non-perturbative $\beta$-function data for SO(3), with a 10% error margin.

Experimental results

Research questions

• RQ1 Does the continuum limit of SU(2) lattice gauge theory reproduce the standard Yang-Mills theory with three gluon degrees of freedom?
• RQ2 Does SO(3) lattice gauge theory, which lacks the Z₂ center symmetry of SU(2), yield a Stefan-Boltzmann constant closer to the continuum expectation?
• RQ3 Can persistent center vortices in SU(2) lattice theory reduce the effective number of degrees of freedom in the high-temperature phase?
• RQ4 Is the discrepancy in the Stefan-Boltzmann constant in SU(2) simulations due to finite-volume or finite-lattice-spacing effects, or to a fundamental difference in the continuum limit?
• RQ5 Do SU(2) and SO(3) lattice gauge theories flow to the same fixed point in the continuum limit?

Key findings

• The continuum extrapolated Stefan-Boltzmann constant for SO(3) gauge theory is approximately 20% larger than that for SU(2) gauge theory, with $\kappa_{SO(3)}/\kappa_{SU(2)} = 1.26 \pm 0.03$ at $3 \times 12^3$.
• The ratio $\kappa_{SO(3)}/\kappa_{SU(2)}$ remains consistent across different lattice sizes, with values ranging from 1.21 to 1.27, indicating robustness of the result.
• The SU(2) lattice data yield a Stefan-Boltzmann constant about 30% below the continuum expectation for three gluon degrees of freedom, consistent with prior findings.
• The SO(3) data show a significantly better agreement with the continuum value, suggesting that the Z₂ center degrees of freedom in SU(2) may not decouple in the continuum limit.
• The observed discrepancy in SU(2) is attributed to persistent correlations from center vortices, which remain active even in the deconfined phase and reduce effective degrees of freedom.
• The results suggest that SU(2) and SO(3) lattice gauge theories may not flow to the same continuum field theory, challenging the assumption that they are equivalent in the continuum limit.