The origin of the diffuse hard X-ray (2-10?keV) emission from starburst galaxies is a long-standing problem. We suggest that synchrotron emission of 10-100?TeV electrons and positrons (e ±) can contribute to this emission, because starbursts have strong magnetic fields. We consider three sources of e ± at these energies: (1) primary electrons directly accelerated by supernova remnants, (2) pionic secondary e ± created by inelastic collisions between cosmic ray (CR) protons and gas nuclei in the dense interstellar medium of starbursts, and (3) pair e ± produced between the interactions between 10 and 100?TeV γ-rays and the intense far-infrared (FIR) radiation fields of starbursts. We create one-zone steady-state models of the CR population in the Galactic center (R ≤ 112 pc), NGC?253, M82, and Arp?220's nuclei, assuming a power-law injection spectrum for electrons and protons. We consider different injection spectral slopes, magnetic field strengths, CR acceleration efficiencies, and diffusive escape times, and include advective escape, radiative cooling processes, and secondary and pair e ±. We compare these models to extant radio and GeV and?TeV γ-ray data for these starbursts, and calculate the diffuse synchrotron X-ray and inverse Compton (IC) luminosities of these starbursts in the models which satisfy multiwavelength constraints. If the primary electron spectrum extends to ~PeV energies and has a proton/electron injection ratio similar to the Galactic value, we find that synchrotron emission contributes 2%-20% of their unresolved, diffuse hard X-ray emission. However, there is great uncertainty in this conclusion because of the limited information on the CR electron spectrum at these high energies. IC emission is likewise a minority of the unresolved X-ray emission in these starbursts, from 0.1% in the Galactic center to 10% in Arp?220's nuclei, with the main uncertainty being the starbursts' magnetic field. We also model generic starbursts, including submillimeter galaxies, in the context of the FIR-X-ray relation, finding that anywhere between 0% and 16% of the total hard X-ray emission is synchrotron for different parameters, and up to 2% in the densest starbursts assuming an E –2.2 injection spectrum and a diffusive escape time of 10 Myr (E/3 GeV)–1/2 (h/100 pc). Neutrino observations by IceCube and?TeV γ-ray data from HESS, VERITAS, and CTA can further constrain the synchrotron X-ray emission of starbursts. Our models do not constrain the possibility of hard, second components of primary e ± from sources like pulsars in starbursts, which could enhance the synchrotron X-ray emission further.
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