Why Padé approximants reduce the renormalization-scale dependence in QFT

We prove that in the limit where the $\beta$ function is dominated by the one-loop contribution (“large ${\beta }_0$ limit”) diagonal Padé approximants (PA’s) of perturbative series become exactly renormalization-scale (RS) independent. This symmetry suggests that diagonal PA’s are correctly resumming contributions from higher-order diagrams that are responsible for the renormalization of the coupling constant. Nondiagonal 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 Padé resummation with the Brodsky-Lepage-Mackenzie (BLM) method. We find that in the large-$N_f$ limit using the BLM scale is identical to resumming the series by an $x[0/n]$ nondiagonal PA.