A Rigorous Holonomy Mechanism for a Non-Perturbative Mass Gap inFour-Dimensional SU(N) Yang-Mills Theory

This paper presents a rigorous non-local formulation of four-dimensional Euclidean SU(N) Yang-Mills theory, termed "Agawa'sHolonomy Mechanism," providing a mathematically well-definedframework for demonstrating the existence of a mass gap. Central tothis approach is the introduction of a non-local gauge field operatormeticulously constructed from the holonomy around a closed loop ofcharacteristic size L. This non-locality, parameterized by L, modifies theconventional Yang-Mills action while strictly preserving gaugeinvariance. We demonstrate that the resulting non-local action isrigorously bounded from below by zero and possesses a unique vacuumstate. The continuum limit is rigorously defined using a novelholonomy-based regularization scheme. We establish the theory'srenormalizability by analyzing the beta function up to one-loop order,confirming its asymptotic freedom. A detailed examination of theOsterwalder-Schrader axioms, including a complete and self-containedproof of reflection positivity, further solidifies the theory's mathematicalfoundation. We propose a novel gauge fixing condition based on fixingthe eigenvalues of the holonomy, which demonstrably eliminates Gribovambiguities. A non-perturbative variational method, employing acarefully constructed gauge-invariant trial wave functional based on theholonomy, is developed to analyze the strong coupling regime. Ouranalytical results strongly indicate the existence of a mass gap, with aleading-order behavior of m ~ g² N / L in the strong coupling regime.We further discuss the implications of these findings for the continuumlimit, demonstrating that the mass gap, when properly scaled, remainsnon-zero. This work establishes a mathematically rigorous frameworkfor a non-local SU(N) Yang-Mills theory, offering a new perspective onthe mass gap problem and outlining clear paths for future research.