Achieving effective renormalization scale and scheme independence via the Principle of Observable Effective Matching

In this work, we explicate a new approach for eliminating renormalization scale and scheme (RSS) dependence in observables. We develop this approach by matching RSS dependent observables (such as cross-sections and decay rates) to a theory which is independent of both these forms of dependencies. We term the fundamental basis behind this approach as the principle of observable effective matching (POEM), which entails matching of a scale- and scheme-dependent observable with the fully physical scale (PS) and dynamical scale-dependent theory at loop orders at which RSS independence is guaranteed. This is aimed toward achieving so-called "effective" RSS-independent expressions as the resulting dynamical dependence is derived from a particular order in RSS-dependent perturbation theory. With this matching at a PS at which the coupling (and masses) is experimentally determined at this scale, we obtain an "effective theoretical observable (ETO)", a finite-order RSS-independent version of the RSS-dependent observable. We illustrate our approach with a study of the cross-section ratio $R_{e^{+}e^{-}}$ for $e^{+}e^{-}\rightarrow$ hadrons, which is demonstrated to achieve scale and scheme independence utilizing the three- and four-loop order MS scheme expression in QCD perturbation theory via matching at both one-loop and two-loop orders for obtaining the ETO. With two-loop matching, we obtain an ETO prediction of $\frac{3}{11}R_{e^{+}e^{-}}^{eff}=1.052431_{-0.0006}^{+0.0006}$ at $Q=31.6 GeV$, which is in excellent agreement with the experimental value of $\frac{3}{11}R_{e^{+}e^{-}}^{exp}=1.0527_{-0.005}^{+0.005}$.