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

Resummation and Matching of $b$-quark Mass Effects in $b\bar{b}H$ Production