Renormalization group scaling of Higgs operators and h → γγ decay

We compute the renormalization of dimension six Higgs-gauge boson operators that can modify the h -> gamma gamma rate at tree-level. Operator mixing is shown to lead to an important modification of new physics effects which has been neglected in past calculations. We also find that the usual formula for the S oblique parameter contribution of those Higgs-gauge boson operators needs additional terms to be consistent with renormalization group evolution. We study the implications of our results for Higgs phenomenology and for new physics models which attempt to explain a deviation in the h -> gamma gamma rate. We derive a new relation between the S parameter and the h -> gamma gamma and h -> Z gamma decay rates.