$h\to gg$ and $h\to \gamma \gamma$ with Anomalous Couplings at Next-to-Leading Order in QCD

We generalize the next-to-leading order QCD calculations for the decay rates of $h\to gg$ and $h\to \gamma \gamma$ to the case of anomalous couplings of the Higgs boson. We demonstrate how this computation can be done in a consistent way within the framework of an electroweak chiral Lagrangian, based on a systematic power counting. It turns out that no additional coupling parameters arise at NLO in QCD beyond those already present at leading order. The impact of QCD is large for $h\to gg$ and the uncertainties from QCD are significantly reduced at NLO. $h\to \gamma \gamma$ is only mildly affected by QCD; here the NLO treatment practically eliminates the uncertainties. Consequently, our results will allow for an improved determination of anomalous Higgs couplings from these processes. The relation of our framework to a treatment in Standard Model effective field theory is also discussed.