Scale-invariant total decay width Γ(H → $$ b\overline{b} $$) using the novel method of characteristic operator

A bstract According to the renormalization group invariance, any physical observable should be invariant under different renormalization schemes and scales. Nevertheless, a fixed-order pQCD approximant cannot inherently satisfy this requirement, giving rise to the conventional scheme-and-scale ambiguities. The principle of maximum conformality (PMC) offers a process-independent approach to address these two types of ambiguities. In practice, it is necessary to handle the n f -terms in the pQCD series with caution to obtain accurate PMC predictions. In this paper, a novel method via using the characteristic operator (CO) $$ {\hat{D}}_{n_{\gamma },{n}_{\beta }} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> <mml:mrow> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>γ</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>β</mml:mi> </mml:msub> </mml:mrow> </mml:msub> </mml:math> is proposed to extend the applicability of PMC, which is a theoretical generalization of previous PMC single-scale setting approach. 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 α 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 Γ( H → $$ b\overline{b} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>b</mml:mi> <mml:mover> <mml:mi>b</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> ) under the $$ \overline{\textrm{MS}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>MS</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> -scheme. Using the CO framework, its effective coupling α s ( Q * ) and effective b -quark $$ \overline{\textrm{MS}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>MS</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> -mass $$ {\overline{m}}_b\left({Q}_{\ast}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mi>m</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mi>b</mml:mi> </mml:msub> <mml:mfenced> <mml:msub> <mml:mi>Q</mml:mi> <mml:mo>∗</mml:mo> </mml:msub> </mml:mfenced> </mml:math> are determined by absorbing all non-conformal { β i }-terms from the renormalization group equations for either α s or $$ {\overline{m}}_b $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mi>m</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mi>b</mml:mi> </mml:msub> </mml:math> 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, $$ \Gamma \left(H\to b\overline{b}\right)={2.3819}_{-0.0231}^{+0.0230} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> <mml:mfenced> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>→</mml:mo> <mml:mi>b</mml:mi> <mml:mover> <mml:mi>b</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> </mml:mfenced> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mn>2.3819</mml:mn> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>0.0231</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> <mml:mn>0.0230</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> MeV, whose errors are squared averages of the ones associated with ∆ α s ( M Z ) = ±0 . 0009, ∆ M H = 0 . 11 GeV, $$ \Delta {\overline{m}}_b\left({\overline{m}}_b\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Δ</mml:mi> <mml:msub> <mml:mover> <mml:mi>m</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mi>b</mml:mi> </mml:msub> <mml:mfenced> <mml:msub> <mml:mover> <mml:mi>m</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mi>b</mml:mi> </mml:msub> </mml:mfenced> </mml:math> = ±0 . 007 GeV, and the uncalculated N 5 LO contributions ∆Γ = ±0 . 0001 MeV predicted via Bayesian analysis with the degree-of-belief DoB = 95 . 5%.