Setting the Renormalization Scale in pQCD: Comparisons of the Principle of Maximum Conformality with the Sequential Extended Brodsky-Lepage-Mackenzie Approach

TL;DR

This paper compares the Principle of Maximum Conformality (PMC) and the sequential extended BLM (seBLM) methods for setting the renormalization scale in perturbative QCD, highlighting their theoretical foundations, advantages, and limitations.

Contribution

It provides a detailed comparison of PMC and seBLM approaches, demonstrating PMC's all-orders systematic scale setting and improved convergence over seBLM.

Findings

PMC eliminates scale and scheme ambiguities in pQCD.

PMC improves convergence by removing renormalon divergences.

seBLM requires auxiliary fields and is limited to low-order calculations.

Abstract

A key problem in making precise perturbative QCD (pQCD) predictions is how to set the renormalization scale of the running coupling unambiguously at each finite order. The elimination of the uncertainty in setting the renormalization scale in pQCD will greatly increase the precision of collider tests of the Standard Model and the sensitivity to new phenomena. Renormalization group invariance requires that predictions for observables must also be independent on the choice of the renormalization scheme. The well-known Brodsky-Lepage-Mackenzie (BLM) approach cannot be easily extended beyond next-to-next-to-leading order of pQCD. Several suggestions have been proposed to extend the BLM approach to all-orders. In this paper we discuss two distinct methods. One is based on the "Principle of Maximum Conformality" (PMC), which provides a systematic all-orders method to eliminate the scale- and scheme- ambiguities of pQCD. The PMC extends the BLM procedure to all orders using renormalization group methods; as an outcome, it significantly improves the pQCD convergence by eliminating renormalon divergences. An alternative method is the "sequential extended BLM" (seBLM) approach, which has been primarily designed to improve the convergence of pQCD series. The seBLM, as originally proposed, introduces auxiliary fields and follows the pattern of the $\beta_0$-expansion to fix the renormalization scale. However, the seBLM requires a recomputation of pQCD amplitudes including the auxiliary fields; due to the limited availability of calculations using these auxiliary fields, the seBLM has only been applied to a few processes at low-orders......