Bootstrapping SU(3) Lattice Yang-Mills Theory

TL;DR

This paper applies the positivity bootstrap approach to SU(3) lattice Yang-Mills theory, deriving rigorous bounds on Wilson loop expectations and introducing new positivity conditions and dimensional reduction techniques for non-perturbative analysis.

Contribution

It extends the positivity bootstrap method to SU(3) gauge theories, incorporating multiple-trace operators and novel positivity conditions, advancing non-perturbative gauge theory studies.

Findings

Derived rigorous bounds for Wilson loop expectation values in 2D, 3D, and 4D.

Introduced a new twist-reflection positivity condition proven in 2D.

Proposed a dimensional-reduction truncation simplifying computations.

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.