Convergence Of The Phase Space Cluster Expansion For Lattice Yang-Mills Theory

We provide a proof for the absolute convergence of the phase space cluster expansion for a lattice-regularized Yang-Mills theory with a compact, simple Lie group. The theory
is mapped to a statistical mechanical system of interacting polymers, and we prove convergence
for sufficiently small bare couplings using the Kotecký-Preiss criterion. The core of the proof is
a combinatorial bound on the sum over all Feynman diagram topologies, which circumvents the
factorial growth of naive perturbation theory. As a result, and under a standard assumption
of vacuum stability, we establish uniform bounds on the connected Schwinger functions, controlled by a weighted norm on the space of continuum action functionals. This result provides
a key analytic tool for the non-perturbative construction of the quantum theory by establishing
rigorous control over the theory’s perturbative sector. It serves as the foundational step in
a program to construct the full massive theory, from which properties like the mass gap can be
analyzed in the zero-mass limit.