Constructive Functional Renormalization Group and Infrared Mass Generation in Non-Abelian Gauge Theories

The constructive exact renormalization group provides a nonperturbative framework for studying the scale dependence of quantum and statistical field theories.Its central object is the effective average action, denoted by Γk, which interpolatesbetween a microscopic action at an ultraviolet scale and the full quantum effectiveaction in the infrared. Its evolution is governed by the Wetterich equation,∂tΓk[Φ] = 12STrΓ (2) k[Φ] + Rk−1∂tRk, t = lnkµ. (1)Although this equation has a one-loop form, it is nonperturbative because the fullfield-dependent inverse propagator occurs inside the functional trace.This paper develops a constructive formulation of the infrared sector of nonAbelian gauge theories. Particular attention is given to the possibility that thetransverse gauge-boson two-point function approaches a finite nonzero limit,limk→0m2(k) = m2g > 0. (2)Such behavior corresponds to a massive decoupling regime rather than a scaleinvariant infrared solution. The relationship between infrared fixed points, logarithmic determinants, dynamical vacuum polarization, the Schwinger mechanismand infrared decoupling is examined. The main conclusion is that the Wetterichequation alone does not constitute a mathematical proof of a nonzero mass gap.A reliable result requires a symmetry-consistent truncation, a regular regulator, acontrolled projection of the flow, compatibility with gauge identities and stabilityunder changes of regulator and truncation.