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Synchrotron pair halo and echo emission from blazars in the cosmic web: application to extreme TeV blazars

High frequency peaked high redshift blazars, are extreme in the sense that their spectrum is particularly hard and peaks at TeV energies. Standard leptonic scenarios require peculiar source parameters and/or a special setup in order to account for these observations. Electromagnetic cascades seeded by ultra-high energy cosmic rays (UHECR) in the intergalactic medium have also been invoked, assuming a very low intergalactic magnetic field (IGMF). Here we study the synchrotron emission of UHECR secondaries produced in blazars located in magnetised environments, and show that it can provide an alternative explanation to these challenged channels, for sources embedded in structured regions with magnetic field strengths of the order of $10^{-7}$ G. To demonstrate this, we focus on three extreme blazars: 1ES 0229+200, RGB J0710+591, and 1ES 1218+304. We model the expected gamma-ray signal from these sources through a combination of numerical Monte Carlo simulations and solving the kinetic equations of the particles in our simulations, and explore the UHECR source and intergalactic medium parameter space to test the robustness of the emission. We show that the generated synchrotron pair halo/echo flux at the peak energy is not sensitive to variations in the overall IGMF strength. This signal is unavoidable in contrast to the inverse Compton pair halo/echo intensity, which is appealing in view of the large uncertainties on the IGMF in voids of large scale structure. It is also shown that the variability of blazar gamma-ray emission can be accommodated by the synchrotron emission of secondary products of UHE neutral beams if these are emitted by UHECR accelerators inside magnetised regions.