Constraints on Axion-like Particles from Observations of Mrk 421 using the ${\rm CL_s}$ Method

Axion-like particles (ALPs) may undergo mixing with photons in the presence of astrophysical magnetic fields, leading to alterations in the observed high energy $\gamma$-ray spectra. In this study, we investigate the ALP-photon oscillation effect using the spectra of the blazar Mrk 421 over 15 observation periods measured by Major Atmospheric Gamma Imaging Cherenkov Telescopes (MAGIC) and Fermi Large Area Telescope (Fermi-LAT). Compared with previous studies, we generate mock data under the ALP hypothesis and employ the ${\rm CL_s}$ method to set constraints on the ALP parameters. This method is widely utilized in high energy experiments and avoids the exclusion of specific parameter regions where distinguishing between the null and ALP hypotheses is challenging. We find that the ALP-photon coupling $g_{a\gamma}$ is constrained to be smaller than $\sim 2\times10^{-11}$ GeV$^{-1}$ for ALP masses ranging from $10^{-9}$ eV to $10^{-7}$ eV at the 95\% confidence level. We also present the constraints derived from the TS distribution under the null hypothesis, which is commonly utilized in previous astrophysical ALP studies. Our results reveal that the combined constraints of all the periods obtained from both methods are consistent. However, the ${\rm CL_s}$ method remains effective in cases where the latter method fails to provide constraints for specific observation periods.