Asymptotic Padé approximants validated on five-loop QCD data now yield six-loop QCD beta-function and quark-mass anomalous-dimension predictions plus eight-loop results in scalar theory.
Why Pad\'e Approximants reduce the Renormalization-Scale dependence in QFT?
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abstract
We prove that in the limit where the beta function is dominated by the 1-loop contribution (``large beta_0 limit'') diagonal Pad\'e Approximants (PA's) of perturbative series become exactly renormalization scale (RS) independent. This symmetry suggest that diagonal PA's are resumming correctly contributions from higher order diagrams that are responsible for the renormalization of the coupling-constant. Non-diagonal PA's are not exactly invariant, but generally reduce the RS dependence as compared to partial-sums. In physical cases, higher-order corrections in the beta function break the symmetry softly, introducing a small scale and scheme dependence. We also compare the Pad\'e resummation with the BLM method. We find that in the large-N_f limit using the BLM scale is identical to resumming the series by a $x[0/n]$ non-diagonal PA.
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Asymptotic Pad\'e Predictions up to Six Loops in QCD and Eight Loops in $\lambda\phi^4$
Asymptotic Padé approximants validated on five-loop QCD data now yield six-loop QCD beta-function and quark-mass anomalous-dimension predictions plus eight-loop results in scalar theory.