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TeV $γ$-ray fluxes from the long campaigns on Mrk421 as constraints on the emission of TeV-PeV Neutrinos and UHECRs

Long TeV $γ$-ray campaigns have been carried out to study the spectrum, variability and duty cycle of the BL Lac object Markarian 421. These campaigns have given some evidence of the presence of protons in the jet: i) Its spectral energy distribution which shows two main peaks; one at low energies ($\sim$ 1 keV) and the other at high energies (hundreds of GeV), has been described by using synchrotron proton blazar model. ii) The study of the variability at GeV $γ$-rays and X-rays has indicated no significant correlation. iii) TeV $γ$-ray detections without activity in X-rays, called "orphan flares" have been observed in this object.\\ Recently, The Telescope Array Collaboration reported the arrival of 72 ultra-high-energy cosmic rays with some of them possibly related to the direction of Markarian 421. The IceCube Collaboration reported the detection of 37 extraterrestrial neutrinos in the TeV - PeV energy range collected during three consecutive years. In particular, no neutrino track events were associated with this source. In this paper, we consider the proton-photon interactions to correlate the TeV $γ$-ray fluxes reported by long campaigns with the neutrino and ultra-high-energy cosmic ray observations around this blazar. Considering the results reported by The IceCube and Telescope Array Collaborations, we found that only from $\sim$ 25\% to 70\% of TeV fluxes described with a power law function with exponential cutoff can come from the proton-photon interactions.