  We extract MSbar quark masses mqbar (q=b,c) from the evaluation of the masses
of quarkonia Upsilon(1S) and J/\psi(1S), performed in two analytic QCD models,
and in perturbative QCD in two renormalization schemes. In analytic QCD the
running coupling has no unphysical singularities in the low-momentum regime. We
apply the analytic model of Shirkov et al.[Analytic Perturbation Theory (APT)],
extended by Bakulev et al.[Fractional Analytic Perturbation Theory (FAPT)], and
the two-delta analytic model (2delanQCD). The latter, in contrast to (F)APT, at
higher energies basically coincides with the perturbative QCD (in the same
scheme). We use the renormalon-free mass mqbar as input. The separation of the
soft and ultrasoft parts of the binding energy E is performed by the
requirement of the cancellation of the leading infrared renormalon. The
analysis in the 2delanQCD model indicates that the low-momentum ultrasoft
regime is important for the extraction of the masses mqbar, especially mbbar.
The 2delanQCD model gives us clues on how to estimate the influence of the
ultrasoft sector on mqbar in general. These effects lead to relatively large
values mbbar(mbbar) =approx (4.35 +- 0.08) GeV in the 2delanQCD model, which,
however, are compatible with recent lattice calculations. In perturbative QCD
in MSbar scheme these effects are even stronger and give larger uncertainties
in mbbar. The (F)APT model gives small ultrasoft effects and the extracted
values of mbbar agree with those in most of the literature: mbbar(mbbar)
=approx 4.2 GeV. The extracted values of mcbar(mcbar) in all four models are
about 1.26-1.27 GeV and agree well with those in the literature.
