  [Abdriged] 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 CR protons and gas nuclei in the dense ISMs of starbursts;
(3) pair e+/- produced between the interactions between 10 - 100 TeV gamma-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 compare these models to extant radio and
GeV and TeV gamma-ray data for these starbursts, and calculate the diffuse
synchrotron X-ray and Inverse Compton (IC) luminosities of these starbursts. 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 contributes 2 - 20% of their unresolved, diffuse hard X-ray
emission. Inverse Compton 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. We also model generic starbursts, including submillimeter
galaxies, in the context of the FIR--X-ray relation, finding that up to 2% in
the densest starbursts with our fiducial assumptions. Neutrino and TeV
gamma-ray data can further constrain the synchrotron X-ray emission of
starbursts. Our models do not constrain hard synchrotron X-ray emission from
any additional hard components of primary e+/- from sources like pulsars in
starbursts.
