Upper limit on the axion-photon coupling from Markarian 421

Markarian 421 is a well-known nearby BL Lac blazar at the redshift $z=0.031$. Many previous works were investigated to constrain the axion-photon coupling from its TeV gamma-ray observations, showing the upper limit on the coupling constant $g_{a\gamma} \lesssim 2.0\times 10^{-11} \rm \, GeV^{-1}$ for the axion mass $[5.0\times10^{-10} \, {\rm eV} \lesssim m_a \lesssim 5.0\times10^{-7} \, {\rm eV}]$. While in this work, we obtain a more stringent upper limit on the axion-photon coupling from the 1038 days gamma-ray observations of the blazar Markarian 421. The long-term gamma-ray spectra are measured by the collaborations Large Area Telescope on board NASA's Fermi Gamma-ray Space Telescope (Fermi-LAT) and High Altitude Water Cherenkov (HAWC) Gamma-Ray Observatory from 2015 June to 2018 July. We show the best-fit spectral energy distributions (SEDs) of Markarian 421 under the null and axion hypotheses. Then we set the axion-photon limit in the $\{m_a, \, g_{a\gamma}\}$ plane. The 99% $\rm C.L.$ upper limit set by Markarian 421 is $g_{a\gamma} \lesssim 4.0\times 10^{-12} \rm \, GeV^{-1}$ for the axion mass $[1.0\times10^{-9} \, {\rm eV} \lesssim m_a \lesssim 1.0\times10^{-8} \, {\rm eV}]$. It is the most stringent upper limit in this axion mass region.