  The coupling of a pseudo-scalar Higgs boson to gluons is mediated through a
heavy quark loop. In the limit of large quark mass, it is described by an
effective Lagrangian that only admits light degrees of freedom. In this
effective theory, we compute the three-loop massless QCD corrections to the
form factor that describes the coupling of a pseudo-scalar Higgs boson to
gluons. Due to the axial anomaly, the pseudo-scalar operator for the gluonic
field strength mixes with the divergence of the axial vector current. Working
in dimensional regularization and using the 't~Hooft-Veltman prescription for
the axial vector current, we compute the three-loop pseudo-scalar form factors
for massless quarks and gluons. Using the universal infrared factorization
properties, we independently derive the three-loop operator mixing and finite
operator renormalisation from the renormalisation group equation for the form
factors, thereby confirming recent results in the operator product expansion.
The finite part of the three-loop form factor is an important ingredient to the
precise prediction of the pseudo-scalar Higgs boson production cross section at
hadron colliders. We discuss potential applications and derive the hard
matching coefficient in soft-collinear effective theory.
