📝 SummarXiv

Abstract

We introduce Scale Factorized-Quantum Field Theory (SF-QFT), an innovative framework that performs path-integral factorization of ultraviolet (UV) and infrared (IR) momentum modes at a physical scale $Q^*$ before perturbative expansion through Effective Dynamical Renormalization (EDR) with Principle of Observable Effective Matching (POEM) constraints. This approach yields completely scale and scheme invariant observables whose Wilson coefficients $C_i(Q)$ and coupling $a_{\mathrm{eff}}(Q)$ are fixed by matching to the experiment. Because the two-loop $\beta$-function is universal in massless QCD and QED, $a_{\mathrm{eff}}(Q)$ evolves with scheme-independent equations, with higher-order $\beta$-coefficients absorbed into the $C_i$ through EDR implementation. Applying SF-QFT to the inclusive ratio $R_{e^{+}e^{-}}$ gives $R^{\mathrm{SF-QFT}}(31.6\,\mathrm{GeV}) = 1.05262 \pm 0.0005$, which is in excellent agreement with the experimental results ($R^{\mathrm{exp}}(31.6\,\mathrm{GeV})= 1.0527 \pm 0.005$) while requiring calculation that are orders of magnitude lower than those of conventional four-loop $\overline{\mathrm{MS}}$ approaches. For QED, the same formalism integrates out high-energy modes above $Q^*$, producing scheme-independent predictions for the electron anomalous magnetic moment with unprecedented precision ($a_e^{\text{theory}} = 0.001\,159\,652\,180\,61(76)$), differing from the experiment by only $-1.2(81) \times 10^{-13}$ representing an excellent 0.15$\sigma$ agreement. The framework enables the self-consistent extraction of the natural electromagnetic coupling $\alpha_{\text{eff}}^{-1}(m_e) = 137.036005301$ at the electron mass scale.