A novel demonstration of the renormalization group invariance of the fixed-order predictions using the principle of maximum conformality and the $C$-scheme coupling

TL;DR

This paper demonstrates that using the $C$-scheme coupling, the Principle of Maximum Conformality provides scheme-independent predictions at all orders, confirming renormalization group invariance for specific observables.

Contribution

It introduces the $C$-scheme coupling to prove the scheme-independence of PMC predictions at all orders, satisfying renormalization group invariance.

Findings

PMC predictions are scheme-independent to all orders using the $C$-scheme.

The approach is illustrated with the non-singlet Adler function.

Results are shown at the four-loop level for $ au$ decay.

Abstract

As a basic requirement of the renormalization group invariance, any physical observable must be independent of the choice of both the renormalization scheme and the initial renormalization scale. In this paper, we show that by using the newly suggested $C$-scheme coupling, one can obtain a demonstration that the {\it Principle of Maximum Conformality} prediction is scheme-independent to all-orders for any renormalization schemes, thus satisfying all of the conditions of the renormalization group invariance. We illustrate these features for the non-singlet Adler function and for $\tau $ decay to $\nu +$ hadrons at the four-loop level.