Probing $HZ\gamma$ and $H\gamma\gamma$ anomalous couplings in the process of $e^+e^- \to H\gamma$

Rare decay of the Higgs boson is a powerful tool to probe new physics beyond the standard model. The rare decays occur through quantum loops in which the standard model and new physics contributions might cancel each other out. We consider a faked-no-new-physics scenario that the new physics contributions are about minus two times the standard model contribution such that partial widths of Higgs boson rare decays are the same as the standard model predictions. We propose to measure the $HZ\gamma$ and $H\gamma\gamma$ anomalous couplings in the process of $e^+ e^- \to H\gamma$ that is strongly correlated to Higgs boson rare decays. We show that the faked-no-new-physics scenario can be fully probed at a high energy electron-positron collider.