  We use a systematic effective field theory setup to derive the $b\bar{b}H$
production cross section. Our result combines the merits of both fixed 4-flavor
and 5-flavor schemes. It contains the full 4-flavor result, including the exact
dependence on the $b$-quark mass, and improves it with a resummation of
collinear logarithms of $m_b/m_H$. In the massless limit, it corresponds to a
reorganized 5-flavor result. While we focus on $b\bar{b}H$ production, our
method applies to generic heavy-quark initiated processes at hadron colliders.
Our setup resembles the variable flavor number schemes known from heavy-flavor
production in deep-inelastic scattering, but also differs in some key aspects.
Most importantly, the effective $b$-quark PDF appears as part of the
perturbative expansion of the final result where it effectively counts as an
$O(\alpha_s)$ object. The transition between the fixed-order (4-flavor) and
resummation (5-flavor) regimes is governed by the low matching scale at which
the $b$-quark is integrated out. Varying this scale provides a systematic way
to assess the perturbative uncertainties associated with the resummation and
matching procedure and reduces by going to higher orders. We discuss the
practical implementation and present numerical results for the $b\bar{b}H$
production cross section at NLO+NLL. We also provide a comparison to the
corresponding predictions in the fixed 4-flavor and 5-flavor results and the
Santander matching prescription. Compared to the latter, we find a slightly
reduced uncertainty and a larger central value, with its central value lying at
the lower edge of our uncertainty band.
