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We study the transverse momentum ($q_T$) spectrum of the Higgs boson producedvia the annihilation of heavy quarks ($s,c,b$) in proton-proton collisions.Using soft-collinear effective theory (SCET) and working in the five-flavourscheme, we provide predictions at three-loop order in resummed perturbationtheory (N$^3$LL$'$). We match the resummed calculation to full fixed-orderresults at next-to-next-to-leading order (NNLO), and introduce a decorrelationmethod to enable a consistent matching to an approximate N$^3$LO (aN$^3$LO)result. Since the $b$-quark initiated process exhibits large nonsingularcorrections, it requires special care in the matching procedure and estimationof associated theoretical uncertainties, which we discuss in detail. Ourresults constitute the most accurate predictions to date for these processes inthe small $q_T$ region and could be used to improve the determination of HiggsYukawa couplings from the shape of the measured Higgs $q_T$ spectrum.

The $q_T$ spectrum for Higgs production via heavy quark annihilation at N$^3$LL$'$+aN$^3$LO