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A self-consistent interpretation of the GeV-TeV emission from a distant blazar PKS 1424+240

We propose a scenario for self-consistent interpretation of GeV - TeV spectrum of a distant blazar PKS 1424+240. In this scenario, ultra-high energy (UHE) protons are assumed to exist in the blazar jet and produce gamma rays through synchrotron emission emitted by relativistic protons and pair cascades (resulted from $pγ$ interaction); meanwhile, some of these UHE protons may escape from the jet and are injected into intergalactic space. Therefore, we assume that UHE cosmic rays (CR) originate from relativistic protons in the jet and use energy-independent escape timescale to obtain UHECR injection spectrum. Both contributions of gamma rays injected by the source and secondary gamma rays produced in interactions of UHECRs emitted by the blazar with photon background during their propagation through intergalactic space are calculated. Our results show that this scenario is able to reproduce the GeV-TeV spectrum of PKS 1424+240 self-consistently in a broad range of redshifts $0.6<z<1.3$. The required relativistic jet power $L_p\simeq3\times10^{46}\ \rm erg\ s^{-1}$ only moderately depends on the assumed source redshift, while the proton escape timescale (and the injected UHECR luminosity) strongly depends on $z$. We compare the integral TeV fluxes predicted in our scenario with the sensitivity of the planned Cherenkov Telescope Array (CTA), and discuss the implications to future observations.