  We study the decay of the SM Higgs boson to a 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. 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 the SM makes it possible to search for
new physics that may induce relatively large interactions related to the charm
quark. As an example, we evaluate the axion-like particle associate production
with a charm quark pair in the Higgs decay and obtain some constraints for the
corresponding parameters under some assumptions.
