A Constructive Proof of Existence and Mass Gap for Pure SU(3) Yang-Mills in Four-Dimensional Space-Time

TL;DR

This paper provides a rigorous constructive proof that pure SU(3) Yang-Mills theory in four dimensions exists as a nontrivial quantum field theory with a positive mass gap, using advanced mathematical techniques and lattice regularization.

Contribution

It introduces a novel, fully rigorous mathematical proof of the existence and mass gap of SU(3) Yang-Mills theory in four dimensions, bridging heuristic physics and formal mathematics.

Findings

Existence of a nontrivial Wightman quantum field theory for SU(3) Yang-Mills

Proof of a strictly positive mass gap in the theory

Development of a convergent polymer expansion on the Wilson lattice

Abstract

In this work we present a constructive proof that pure SU(3) Yang-Mills theory on R^4 exists as a nontrivial Wightman quantum field theory and exhibits a strictly positive mass gap. Our approach embeds the four-dimensional gauge theory as the zero-mode sector of a five-dimensional orbifold regulator that preserves gauge invariance and reflection positivity. A convergent joint polymer expansion on the Wilson lattice provides uniform control of the continuum (a->0) and infinite-volume (L->inf) limits; an Osterwalder-Schrader reconstruction yields a unique-vacuum Wightman theory; a nonperturbative BRST/Nielsen argument ensures gauge-parameter independence; a rigorously controlled operator-product expansion matches one-loop beta-function data; and a Sturm-Liouville analysis of five-dimensional fluctuations, combined with transfer-matrix spectral projections, isolates a strictly positive glueball mass m0. All steps rest on explicit epsilon-delta estimates and combinatorial bounds for SU(3), leaving no remaining gap between heuristic physics and mathematical proof.