Hard-TeV BL Lacs are a new type of blazars characterized by a hard intrinsic TeV spectrum, locating the peak of their gamma-ray emission in the spectral energy distribution (SED) above 2-10 TeV. Such high energies are problematic for the Compton emission, using a standard one-zone leptonic model. We study six examples of this new type of BL Lacs in the hard X-ray band with the NuSTAR satellite. Together with simultaneous observations with the SWIFT satellite, we fully constrain the peak of the synchrotron emission in their SED, and test the leptonic synchrotron self-Compton (SSC) model. We confirm the extreme nature of 5 objects also in the synchrotron emission. We do not find evidence of additional emission components in the hard X-ray band. We find that a one-zone SSC model can in principle reproduce the extreme properties of both peaks in the SED, from X-ray up to TeV energies, but at the cost of i) extreme electron energies with very low radiative efficiency, ii) conditions heavily out of equipartition (by 3 to 5 orders of magnitude), and iii) not accounting for the simultaneous UV data, which then should belong to a different emission component, possibly the same as the far-IR (WISE) data. We find evidence of this separation of the UV and X-ray emission in at least two objects. In any case, the TeV electrons must not see the UV or lower-energy photons, even if coming from different zones/populations, or the increased radiative cooling would steepen the VHE spectrum.
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