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Bootstrapping SU(3) Lattice Yang-Mills Theory

Yuanhong Guo, Zeyu Li, Gang Yang, Guorui Zhu

TL;DR

This paper extends the positivity bootstrap program to SU(3) lattice Yang-Mills theory by incorporating multi-trace Wilson loops, introducing twist-reflection positivity, and employing dimensional-reduction truncation to manage computational complexity. Using Hermitian and reflection positivity together with Schwinger-Dyson loop equations, the authors formulate a large SDP framework to bound plaquette expectations across 2D, 3D, and 4D YM, with results converging and aligning with known analytic or numerical data. The study demonstrates that positivity-based methods can handle higher-rank gauge theories beyond single-trace sectors and provides a practical blueprint for future non-perturbative investigations, including potential extensions to QCD-like settings. While twist RP yields strong constraints in 2D, its applicability diminishes in higher dimensions, guiding the authors toward dimension-reduction strategies and careful path selection to achieve reliable bounds. Overall, the work offers a principled nonperturbative approach that complements traditional lattice simulations and deepens understanding of gauge dynamics through positivity and loop-reduction techniques.

Abstract

We apply the positivity bootstrap approach to SU(3) lattice Yang-Mills (YM) theory, extending previous studies of large N and SU(2) theories by incorporating multiple-trace Wilson loop operators. By utilizing Hermitian and reflection positivity conditions, alongside Schwinger-Dyson (SD) loop equations, we compute rigorous bounds for the expectation values of plaquette Wilson loops in 2D, 3D, and 4D YM theories. Our results exhibit clear convergence and are consistent with known analytic or numerical results. To enhance the approach, we introduce a novel twist-reflection positivity condition, which we prove to be exact in 2D YM theory. Additionally, we propose a dimensional-reduction truncation, where Wilson loop operators are effectively restricted to a lower-dimensional subplane, significantly simplifying computations. SD equations for double-trace Wilson loops are also derived in detail. Our findings suggest that the positivity bootstrap method is broadly applicable to higher-rank gauge theories beyond single-trace cases, providing a solid foundation for further non-perturbative investigations of gauge theories using positivity-based methods.

Bootstrapping SU(3) Lattice Yang-Mills Theory

TL;DR

This paper extends the positivity bootstrap program to SU(3) lattice Yang-Mills theory by incorporating multi-trace Wilson loops, introducing twist-reflection positivity, and employing dimensional-reduction truncation to manage computational complexity. Using Hermitian and reflection positivity together with Schwinger-Dyson loop equations, the authors formulate a large SDP framework to bound plaquette expectations across 2D, 3D, and 4D YM, with results converging and aligning with known analytic or numerical data. The study demonstrates that positivity-based methods can handle higher-rank gauge theories beyond single-trace sectors and provides a practical blueprint for future non-perturbative investigations, including potential extensions to QCD-like settings. While twist RP yields strong constraints in 2D, its applicability diminishes in higher dimensions, guiding the authors toward dimension-reduction strategies and careful path selection to achieve reliable bounds. Overall, the work offers a principled nonperturbative approach that complements traditional lattice simulations and deepens understanding of gauge dynamics through positivity and loop-reduction techniques.

Abstract

We apply the positivity bootstrap approach to SU(3) lattice Yang-Mills (YM) theory, extending previous studies of large N and SU(2) theories by incorporating multiple-trace Wilson loop operators. By utilizing Hermitian and reflection positivity conditions, alongside Schwinger-Dyson (SD) loop equations, we compute rigorous bounds for the expectation values of plaquette Wilson loops in 2D, 3D, and 4D YM theories. Our results exhibit clear convergence and are consistent with known analytic or numerical results. To enhance the approach, we introduce a novel twist-reflection positivity condition, which we prove to be exact in 2D YM theory. Additionally, we propose a dimensional-reduction truncation, where Wilson loop operators are effectively restricted to a lower-dimensional subplane, significantly simplifying computations. SD equations for double-trace Wilson loops are also derived in detail. Our findings suggest that the positivity bootstrap method is broadly applicable to higher-rank gauge theories beyond single-trace cases, providing a solid foundation for further non-perturbative investigations of gauge theories using positivity-based methods.

Paper Structure

This paper contains 58 sections, 185 equations, 17 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Setup and strategy
  3. Lattice YM theory and Wilson loops
  4. Positivity bootstrap
  5. Positivity conditions
  6. Hermitian positivity
  7. Reflection positivity
  8. Standard RP
  9. Twist RP
  10. Loop equations
  11. Normalization.
  12. Large $N$ theory
  13. SU(2) theory
  14. SU(3) theory
  15. Path selection
  16. ...and 43 more sections

Figures (17)

  • Figure 1: A Wilson loop on a 2D square lattice.
  • Figure 2: SDP bounds for plaquette in 2d YM.
  • Figure 3: Two paths sharing the same endpoints (red dots) can be sewn to form a Wilson loop. The figure on the right-hand side figure is a 3D lattice example in which both endpoints coincide.
  • Figure 4: Reflection positivity with mirror plane $\pi: x =0$.
  • Figure 5: Other two types of reflection.
  • ...and 12 more figures