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Considering the constraints from the flavor physics, precision electroweakmeasurements, Higgs data and dark matter detections, we scan over the parameterspace of the MSSM and calculate the cross section of $e^+e^- \to h \gamma$ inthe allowed parameter space. Since the loop-induced gauge couplings$h\gamma\gamma$ and $hZ\gamma$ can simultaneously contribute to the process$e^+e^- \to h \gamma$, we find the cross section can be sizably enhanced by alight stau, maximally 1.47(1.38) times larger than the SM prediction at$\sqrt{s}=240(350)$ GeV. So with the high luminosity, the measurement of$e^+e^- \to h \gamma$ may be used to test the anomalous gauge couplings$h\gamma\gamma$ and $hZ\gamma$ in the MSSM at a Higgs Factory.

Revisiting Associated Production of 125 GeV Higgs Boson with a Photon at a Higgs Factory