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We consider Adler $D$-function for the vector and scalar
correlators with recent QCD analysis at $N^4LO$ approximation. The
portion of perturbative coefficients of these observable
containing the leading power of $b$, the first beta-function
coefficient, is resummed to all-orders. The factorial behavior of
coefficients allows us to use the Borel transformation to reveal
the existent singularities as renormalons. CORGI approach is
employed to avoid the renormalisation dependence of the
observable.In addition to the CORGI approach, the calculations of
the standard perturbative QCD approach which uses the $\overline MS$
scheme with a physical choice of renormalization scale, are also presented.
A comparison between the results of the standard and
CORGI approaches for $R_{\tau}$ and Higgs decay width is done.
The comparison anticipates a better result for the CORGI approach.
As an adjunct to these studies, the difference between the all
order and fixed order result at $N^4LO$ approximation is used to
estimate the uncertainty in ${\alpha}_{s}({M_Z^2})$ extracted
from ${R}_{\tau}$ measurements. We find at $N^4LO$ approximation a
smaller uncertainty with respect to the previous result which has
been done at $N^2LO$ approximation.
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