Linearly polarized gluons at next-to-next-to leading order and the Higgs transverse momentum distribution

We calculate the small-b (or large-qT) matching of transverse momentum dependent (TMD) distribution for linearly polarized gluons to the integrated gluon distributions at the next -to-next -to-leading order (NNLO). This is the last missing part for the complete NNLO prediction of the Higgs spectrum within TMD factorization. We discuss the numerical impact of the correction so derived to the qT-differential cross-section for Higgs boson production and to the positivity bound for linearly polarized gluon transverse momentum distribution.