  The discovery of the Higgs boson has opened a new window to test the SM
through the measurements of its couplings. Of particular interest is the
measured Higgs coupling to photons which arises in the SM at the one-loop
level, and can then be significantly affected by new physics. We calculate the
one-loop renormalization of the dimension-six operators relevant for
$h\rightarrow \gamma\gamma, \gamma Z$, which can be potentially important since
it could, in principle, give log-enhanced contributions from operator mixing.
We find however that there is no mixing from any current-current operator that
could lead to this log-enhanced effect. We show how the right choice of
operator basis can make this calculation simple. We then conclude that
$h\rightarrow \gamma\gamma, \gamma Z$ can only be affected by RG mixing from
operators whose Wilson coefficients are expected to be of one-loop size, among
them fermion dipole-moment operators which we have also included.
