Bethe-Heitler emission in BL Lacs: filling the gap between X-rays and $\gamma$-rays

We present the spectral signatures of the Bethe-Heitler pair production ($pe$) process on the spectral energy distribution (SED) of blazars, in scenarios where the hard $\gamma$-ray emission is of photohadronic origin. If relativistic protons interact with the synchrotron blazar photons producing $\gamma$ rays through photopion processes, we show that, besides the $2-20$ PeV neutrino emission, the typical blazar SED should have an emission feature due to the synchrotron emission of $pe$ secondaries that bridges the gap betweeen the low-and high-energy humps of the SED, namely in the energy range 40 keV-40 MeV. We first present analytical expressions for the photopion and $pe$ loss rates in terms of observable quantities of blazar emission. For the $pe$ loss rate in particular, we derive a new approximate analytical expression for the case of a power-law photon distribution, which has an excellent accuracy with the numerically calculated exact one, especially at energies above the threshold for pair production. We show that for typical blazar parameters, the photopair synchrotron emission emerges in the hard X-ray/soft $\gamma$-ray energy range with a characteristic spectral shape and non negligible flux, which may be even comparable to the hard $\gamma$-ray flux produced through photopion processes. We argue that the expected "$pe$ bumps" are a natural consequence of leptohadronic models, and as such, they may indicate that blazars with a three-hump SED are possible emitters of high-energy neutrinos.