Bottom and charm mass determinations with a convergence test

Abstract

We present new determinations of the $$ \overline{\mathrm{MS}} $$ charm quark mass using relativistic QCD sum rules at $$ \mathcal{O}\left({\alpha}_s^3\right) $$ from the moments of the vector and the pseudoscalar current correlators. We use available experimental measurements from e+ e− collisions and lattice simulation results, respectively. Our analysis of the theoretical uncertainties is based on different implementations of the perturbative series and on independent variations of the renormalization scales for the mass and the strong coupling. Taking into account the resulting set of series to estimate perturbative uncertainties is crucial, since some ways to treat the perturbative expansion can exhibit extraordinarily small scale dependence when the two scales are set equal. As an additional refinement, we address the issue that double scale variation could overestimate the perturbative uncertainties. We supplement the analysis with a test that quantifies the convergence rate of each perturbative series by a single number. We find that this convergence test allows to determine an overall and average convergence rate that is characteristic for the series expansions of each moment, and to discard those series for which the convergence rate is significantly worse. We obtain $$ \overline{m_c}\left(\overline{m_c}\right)=1.288\pm 0.020 $$ GeV from the vector correlator. The method is also applied to the extraction of the $$ \overline{\mathrm{MS}} $$ bottom quark mass from the vector correlator. We compute the experimental moments including a modeling uncertainty associated to the continuum region where no data is available. We obtain $$ \overline{m_b}\left(\overline{m_b}\right)=4.176\pm 0.023 $$ GeV.