Degeneracy Relations in QCD and the Equivalence of Two Systematic All-Orders Methods for Setting the Renormalization Scale

TL;DR

This paper discusses two systematic all-orders methods for setting the renormalization scale in QCD, demonstrating their practical equivalence and the resulting scheme-independent, rapidly convergent predictions for high-order calculations.

Contribution

It introduces and compares two all-orders PMC-scale setting methods, proving their equivalence and elucidating the degeneracy relations underlying conformal series in QCD.

Findings

PMC-I and PMC-II methods are practically equivalent.

Both methods produce identical conformal series up to all orders.

Degeneracy relations are fundamental properties of non-Abelian gauge theories.

Abstract

The Principle of Maximum Conformality (PMC) eliminates QCD renormalization scale-setting uncertainties using fundamental renormalization group methods. The resulting scale-fixed pQCD predictions are independent of the choice of renormalization scheme and show rapid convergence. The coefficients of the scale-fixed couplings are identical to the corresponding conformal series with zero $\beta$-function. Two all-orders methods for systematically implementing the PMC-scale setting procedure for existing high order calculations are discussed in this article. One implementation is based on the PMC-BLM correspondence \mbox{(PMC-I)}; the other, more recent, method \mbox{(PMC-II)} uses the ${\cal R}_\delta$-scheme, a systematic generalization of the minimal subtraction renormalization scheme. Both approaches satisfy all of the principles of the renormalization group and lead to scale-fixed and scheme-independent predictions at each finite order. In this work, we show that PMC-I and PMC-II scale-setting methods are in practice equivalent to each other. We illustrate this equivalence for the four-loop calculations of the annihilation ratio $R_{e^+ e^-}$ and the Higgs partial width $\Gamma(H\to b\bar{b})$. Both methods lead to the same resummed (`conformal') series up to all orders. The small scale differences between the two approaches are reduced as additional renormalization group $\{\beta_i\}$-terms in the pQCD expansion are taken into account. We also show that {\it special degeneracy relations}, which underly the equivalence of the two PMC approaches and the resulting conformal features of the pQCD series, are in fact general properties of non-Abelian gauge theory.