Scale Factorized-Quantum Field Theory: Eliminating renormalization ambiguities in QCD and QED

Abstract

We introduce Scale Factorized-Quantum Field Theory (SF-QFT), a framework performing path-integral factorization of ultraviolet and infrared momentum modes at a physical scale $Q^*$ before perturbative expansion through Effective Dynamical Renormalization (EDR) with Principle of Observable Effective Matching (POEM) constraints. This yields completely scale and scheme invariant observables. Because the two-loop $\beta$-function is universal, $a_{\mathrm{eff}}(Q)$ evolves with scheme-independent equations, with higher-order $\beta$-coefficients absorbed into Wilson coefficients. For the inclusive ratio $R_{e^{+}e^{-}}$, SF-QFT gives $R^{\mathrm{SF-QFT}}(31.6\,\mathrm{GeV}) = 1.05262 \pm 0.0005$, in excellent agreement with experiment ($1.0527 \pm 0.005$) while requiring calculations orders of magnitude simpler than conventional four-loop $\overline{\mathrm{MS}}$ approaches. SF-QFT generates universal algebraic recursion relations producing all higher-order contributions without additional Feynman diagrams. For QED, the formalism yields scheme-independent predictions for the electron anomalous magnetic moment with $a_e^{\text{theory}} = 0.001\,159\,652\,180\,61(76)$, differing from experiment by only $0.15\sigma$. The framework enables self-consistent extraction of $\alpha_{\text{eff}}^{-1}(m_e) = 137.036005301$. SF-QFT represents a paradigm shift, replacing pursuit of ever-higher loop orders with a unified framework eliminating renormalization ambiguities through systematic EDR and POEM implementation.