Implications of multiwavelength spectrum on cosmic-ray acceleration in blazar TXS 0506+056

MAGIC collaboration has recently analyzed data from a long-term multiwavelength campaign of the $\gamma$-ray blazar TXS 0506+056. In December 2018, it was flaring in the very-high-energy (VHE; $E>100$ GeV) $\gamma$-ray band, but no simultaneous neutrino event was detected. We model the observed spectral energy distribution (SED), using a one-zone leptohadronic emission. We estimate the neutrino flux through the restriction from observed X-ray flux on the secondary radiation due to hadronic cascade, initiated by protons with energy $E_p \lesssim 0.1$ EeV. We assume ultrahigh-energy cosmic rays (UHECRs; $E\gtrsim0.1$ EeV), with the same slope and normalization as the low-energy spectrum, are accelerated in the jet but escape efficiently. We propagate the UHE protons in a random, turbulent extragalactic magnetic field (EGMF). The leptonic emission from the jet dominates the GeV range, whereas the cascade emission from CR interactions in the jet contributes substantially to the X-ray and VHE range. The line-of-sight cosmogenic $\gamma$ rays from UHECRs produce a hardening in the VHE spectrum. Our model prediction for neutrinos from the jet is consistent with the 7.5-year flux limit by IceCube and shows no variability during the MAGIC campaign. Therefore, we infer that the correlation between GeV-TeV $\gamma$-rays and neutrino flare is minimal. The luminosity in CRs limits the cosmogenic $\gamma$-ray flux, which, in turn, bounds the RMS value of the EGMF to $\gtrsim 10^{-5}$ nG. The cosmogenic neutrino flux is lower than the IceCube-Gen2 detection potential for 10 yrs of observation. VHE $\gamma$-ray variability should arise from increased activity inside the jet; thus, detecting steady flux at multi-TeV energies may indicate UHECR acceleration. Upcoming $\gamma$-ray imaging telescopes such as the CTA will be able to constrain the cosmogenic $\gamma$-ray component in the SED of TXS 0506+056.