Quantization of Beta Functions in Self-Dual Backgrounds and Emergent Non-Commutative EFT

We investigate the renormalization group flow and beta functions of Yang-Mills theory and adjoint QCD in a strong, stable, self-dual background field $F$. In deep UV, theory runs according to the standard beta function, $\beta_0$. Treating the background as a superselection sector, we find that the theory abelianizes below the scale $\sqrt{F}$ and remains strictly abelian in the deep infrared. In the intermediate weakly-coupled regime ($\Lambda_{\rm YM} \ll \mu \lesssim \sqrt{F}$), the gauge coupling remarkably continues to run despite the absence of propagating charged degrees of freedom. Because all non-zero Landau levels decouple, this running is driven exclusively by exact zero modes, resulting in an integer-quantized beta function coefficient, $\widetilde \beta_0$. Finally, we conjecture that this abelian dynamics is governed by an emergent non-commutative effective field theory that is free of pathological UV/IR mixing.