[Paper Review] SO(4), SO(3) and SU(2) gauge theories in 2+1 dimensions: comparing glueball spectra and string tensions

M. Teper|arXiv (Cornell University)|Jan 17, 2018

Quantum Chromodynamics and Particle Interactions4 citations

TL;DR

This paper improves lattice calculations of glueball spectra and string tensions in 2+1D SO(4), SO(3), and SU(2) gauge theories using optimized, blocked, and smeared operators. It finds that the low-lying spectra and corrected string tensions of SO(4) and SO(3) are in excellent agreement with those of SU(2), indicating that global group differences (e.g., topology) do not significantly affect low-energy physics in these theories.

ABSTRACT

We improve upon recent calculations of the low-lying `glueball' spectra of SO(3) and SO(4) lattice gauge theories in 2+1 dimensions, and compare the resulting continuum extrapolations with SU(2). We find that these are reasonably consistent, as are the SU(2) and SO(4) string tensions when these are corrected for the differing representations of the flux. All this indicates that the different global properties of these groups do not play a significant role in the low-lying physics.

Motivation & Objective

• To resolve discrepancies in low-lying glueball spectra between SO(4), SO(3), and SU(2) gauge theories reported in prior work, which were attributed to poor operator overlap.
• To improve the accuracy of lattice calculations for SO(4) and SO(3) by optimizing smearing parameters for gauge link operators, enhancing projection onto glueball states.
• To compare continuum-extrapolated glueball masses and string tensions across SO(4), SO(3), and SU(2), focusing on whether global topological differences affect low-energy physics.
• To test whether the observed agreement in spectra implies that the global structure of gauge groups (e.g., SO(3) vs. SU(2)) is irrelevant for low-lying hadron-like states in 2+1D.

Proposed method

• Used improved blocked and smeared gauge link operators with optimized smearing parameters (2.5× and 1.7× larger than prior values for SO(3) and SO(4), respectively) to enhance overlap with glueball states.
• Applied mean-field improved coupling βI = β⟨up⟩ to reduce lattice artifacts and improve continuum extrapolation accuracy.
• Performed continuum extrapolations of glueball masses and string tensions using O(a²m²) corrections, with ratios of masses used to minimize systematic errors.
• Calculated effective mass plots from time-correlation functions to extract ground and excited glueball states, using variational methods for excited states.
• Used the mass gap (0⁺) as a reference scale to form dimensionless mass ratios, avoiding reliance on the less precise string tension for SO(3).
• Compared results for SO(4) and SO(3) with SU(2) using continuum-extrapolated values, assessing agreement within statistical uncertainties (2σ).

Experimental results

Research questions

• RQ1 Do the low-lying glueball spectra of SO(4) and SO(3) gauge theories agree with those of SU(2) when improved lattice operators are used?
• RQ2 To what extent do global topological differences between SO(N) and SU(N') groups with the same Lie algebra affect the low-energy spectrum?
• RQ3 Can improved operator smearing reduce systematic errors in glueball mass extraction, especially for heavier states?
• RQ4 Is the string tension in SO(3) reliably extractable, and how does it compare to SU(2) after correcting for representation differences?
• RQ5 Do the observed spectra of SO(4) and SO(3) agree with SU(2) within 2σ, indicating insensitivity to global group structure?

Key findings

• The improved operator construction reduced systematic errors in glueball mass extraction, particularly for heavier states like the 0⁻ and 1±, with clearer effective mass plateaus.
• The continuum extrapolated 0⁺ glueball mass in SO(3) is m₀⁺/g² = 0.1287(16) with a small O(a²) correction, consistent with SU(2) when scaled by a coupling factor of 4.108(51), close to the expected 4.
• Mass ratios mG/m₀⁺ for SO(3) and SU(2) agree within ~2σ for the 0⁺⋆, 0⁺⋆⋆⋆, 0⁺⋆⋆⋆⋆, 2⁺, 2⁻, 2⁺⋆, and 2⁻⋆ states, indicating strong spectral consistency.
• The only significant deviation is for the 0⁺⋆⋆ state, which lies more than 3σ above the SU(2) value, likely due to residual systematic errors in SO(3) mass estimation.
• For SO(4), the improved operators led to excellent agreement with SU(2) and with the SU(2)×SU(2) theory, confirming that the Lie algebra equivalence implies spectral equivalence.
• The results support the conclusion that global group structure (e.g., SO(3) vs. SU(2)) plays no significant role in the low-lying spectrum of 2+1D gauge theories.