Searching for Axion-Like Particles with the Blazar Observations of MAGIC and Fermi-LAT

We explore the axion-like particle (ALP)-photon oscillation effect in the $\gamma$-ray spectra of the blazars Markarian 421 (Mrk 421) and PG 1553+113, which are measured by the Major Atmospheric Gamma Imaging Cherenkov Telescopes (MAGIC) and Fermi Large Area Telescope (Fermi-LAT) with high precision. We investigate the constraints on the ALP parameter space using the Mrk 421 and PG 1553+113 observations of 15 and 5 phases, respectively. We find that the combined analysis with all the 15 phases improves the limits from the Mrk 421 observations. The combined limit set by the Mrk 421 observations has excluded the ALP parameter region with the ALP-photon coupling of $g_{a\gamma} \gtrsim 2 \times 10^{-11} \, \rm GeV^{-1}$ for the ALP mass of $\sim 8\times 10^{-9}\, {\rm eV} \lesssim m_a \lesssim 2\times 10^{-7}\rm \, eV$ at 95% $\rm C.L.$ We also find that the ALP hypothesis can slightly improve the fit to the PG 1553+113 results in some parameter regions, and do not set the limit in this case.