  We study the decay of the SM Higgs boson to massive charm quark pair at the
next-to-next-to-leading order QCD and next-to-leading order electroweak. At the
second order of QCD coupling, we consider the exact calculation of
flavour-singlet contributions where the Higgs boson couples to the internal top
and bottom quark. Some helpful information on the running mass effects related
to Yukawa coupling may be obtained by analyzing this process. High precision
production for $h\to c\bar{c}$ within SM makes it possible to search for new
physics that may induce relatively large interaction related to the charm
quark. As an example, we investigate the axion-like particle associate
production with charm quark pair in Higgs decay and obtain some constraints for
the corresponding parameters under some assumptions.
