Scale-invariant total decay width $\Gamma(H\to b\bar{b})$ using the novel method of characteristic operator

In this paper, a novel method via using the characteristic operator~(CO) ${\cal \hat{D}}_{n_{\gamma}, n_{\beta}}$ is proposed to extend the applicability of PMC, which is a theoretical generalization of previous PMC single-scale setting approach. Using the CO formulism, we are able to facilitate the derivation of complex scenarios within a structured theoretical framework, leading to simpler procedures and more compact expressions. The CO framework not only streamlines derivations for complex scenarios, yielding simplified procedures and more compact expressions, but also achieves a scheme-and-scale invariant pQCD series by fixing the correct effective magnitude of $\alpha_s$ and the running mass simultaneously. Both are well matched with the expansion coefficients of the series, leading to the wanted scheme-and-scale invariant conformal series. As an example, we show the achievement of scale-invariant N$^{4}$LO total decay width $\Gamma(H\to b\bar{b})$ under the $\overline{\rm MS}$-scheme. Using the CO framework, its effective coupling $\alpha_{s}(Q_{*})$ and effective $b$-quark $\overline{\rm MS}$-mass $\overline{m}_{b}(Q_{*})$ are determined by absorbing all non-conformal $\{\beta_{i}\}$-terms from the renormalization group equations for either $\alpha_s$ or $\overline{m}_{b}$ simultaneously. The PMC scale is fixed up to N$^3$LL-accuracy, $Q_{*} = 55.2916$~GeV and a scale-invariant total decay width is obtained, $\Gamma(H \to b\bar{b}) = 2.3819 _{-0.0231}^{+0.0230}$~MeV, whose errors are squared averages of the ones associated with $\Delta \alpha_{s}(M_{Z}) = \pm 0.0009$, $\Delta M_{H} = 0.11$~GeV, $\Delta \overline{m}_{b}(\overline{m}_{b}) = \pm 0.007$~GeV, and the uncalculated N$^{5}$LO contributions $\Delta\Gamma= \pm0.0001$~MeV predicted via Bayesian analysis with the degree-of-belief ${\rm DoB}=95.5\%$.