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

physical review. d/physical review. d.

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