  We consider two-loop renormalization of high-dimensional Lorentz scalar
operators in the gluonic sector of QCD. These operators appear also in the
Higgs effective theory obtained by integrating out the top quark loop in the
gluon fusion process. We first discuss the classification of operators and how
to construct a good set of basis using both off-shell field theory method and
on-shell form factor formalism. To study loop corrections, we apply efficient
unitarity-IBP strategy and compute the two-loop minimal form factors of
length-3 operators up to dimension sixteen. From the UV divergences of form
factor results, we extract the renormalization matrices and analyze the
operator mixing behavior in detail. The form factors we compute are also
equivalent to Higgs plus three-gluon amplitudes that capture high-order top
mass corrections in Higgs EFT. We obtain the analytic finite remainder
functions which exhibit several universal transcendentality structures.
