  Measuring the Higgs couplings accurately at colliders is one of the best routes for finding physics Beyond the Standard Model (BSM). If the measured couplings deviate from the SM predictions, then this would give rise to energy-growing processes that violate tree-level unitarity at some energy scale, indicating new physics. In this paper, we extend previous work on unitarity bounds from the Higgs potential and the Higgs couplings to vector bosons and the top quark; to the Higgs couplings to $\gamma\gamma$ and $\gamma Z$. We find that while the HL-LHC might be able to find new physics in the $\gamma Z$ sector, the scale of new physics in both sectors is mostly beyond its reach. However, accurate measurements of the leading couplings of the two sectors in the HL-LHC can place stringent limits on both the scale of new physics and on other Higgs couplings that are difficult to measure. In addition, the scale of new physics is mostly within the reach of the $100$ TeV collider. 
The Scale of New Physics from the Higgs Couplings to $\gamma\gamma$ and   $\gamma Z$