Novel all-orders single-scale approach to QCD renormalization scale-setting

The principle of maximal conformality (PMC) provides a rigorous method for eliminating renormalization scheme-and-scale ambiguities for perturbative QCD (pQCD) predictions. The PMC uses the renormalization group equation to fix the $\beta$ pattern of each order in an arbitrary pQCD approximant, and it then determines the optimal renormalization scale by absorbing all $\{{\beta }_i\}$ terms into the running coupling at each order. The resulting coefficients of the pQCD series match the scheme-independent conformal series with $\beta =0$. As in QED, different renormalization scales appear at each order; we call this the multiscale approach. In this paper, we present a novel single-scale approach for the PMC, in which a single effective scale is constructed to eliminate all nonconformal $\beta$ terms up to a given order simultaneously. The PMC single-scale approach inherits the main features of the multiscale approach; for example, its predictions are scheme independent, and the pQCD convergence is greatly improved due to the elimination of divergent renormalon terms. As an application of the single-scale approach, we investigate the $e^{+}{e}^{-}$ annihilation cross-section ratio $R_{e^{+}{e}^{-}}$ and the Higgs decay width $\mathrm{\Gamma }(H\to b\overline{b})$, including four-loop QCD contributions. The resulting predictions are nearly identical to the multiscale predictions for both the total and differential contributions. Thus in many cases, the PMC single-scale approach PMC-s, which requires a simpler analysis, could be adopted as a reliable substitution for the PMC multiscale approach for setting the renormalization scale for high-energy processes, particularly when one does not need detailed information at each order. The elimination of the renormalization scale uncertainty increases the precision of tests of the Standard Model at the LHC.

Figure 1

The determined PMC scale $Q_{\star }$ for $R(Q)$ up to ${\mathrm{N}}^2\mathrm{LL}$ accuracy. $Q_{\star }^{(1)}$ is at the LL accuracy, $Q_{\star }^{(2)}$ is at the NLL accuracy, and $Q_{\star }^{(3)}$ is at the ${\mathrm{N}}^2\mathrm{LL}$ accuracy. $Q=31.6\mathrm{GeV}$.

Figure 2

The determined PMC scale ${\tilde{Q}}_{\star }$ for $H\to b\overline{b}$ up to ${\mathrm{N}}^2\mathrm{LL}$ accuracy. $Q_{\star }^{(1)}$ is at the LL accuracy, $Q_{\star }^{(2)}$ is at the NLL accuracy, and $Q_{\star }^{(3)}$ is at the ${\mathrm{N}}^2\mathrm{LL}$ accuracy. $\mu =M_H$.