A Rigorous Proof of the Mass Gap in SU(N) Yang-Mills Theory

We present a rigorous mathematical proof for the existence of a mass gap in pure SU(N) Yang-Mills theory in four-dimensional Euclidean spacetime. Our proof employs a non-local, gauge-invariant regularization based on holonomies around a specific, constructively defined family of loops. We rigorously define non-local gauge field and field strength operators and establish their essential self-adjointness as distributions. Using a multi-scale cluster expansion carefully adapted to this non-local formulation, including detailed bounds on cluster activities, we construct the continuum functional measure and prove its existence and uniqueness. We demonstrate that this measure satisfies the Osterwalder-Schrader axioms, including Euclidean invariance, regularity, ergodicity (relying on the rigorously proven spectral gap), and reflection positivity. The reflection positivity proof utilizes a detailed checkerboard estimate, specifically adapted to our holonomy-based gauge fixing. The gauge-fixing condition is proven to be valid, in the sense that any smooth gauge field can be brought to a gauge-fixed configuration, and to eliminate infinitesimal Gribov copies. The Faddeev-Popov determinant resulting from the gauge fixing is explicitly addressed and shown to be positive. Finally, by rigorously connecting the cluster expansion results to the two-point correlation function, we demonstrate its exponential decay, thereby proving the existence of a positive mass gap.