  We present a systematic study of the non-thermal electron-proton plasma and
its emission processes in starburst galaxies in order to explain the
correlation between the luminosity in the radio band and the recently observed
gamma luminosity. In doing so, a steady state description of the cosmic ray
electrons and protons within the spatially homogeneous starburst is considered
where continuous momentum losses are included as well as catastrophic losses
due to diffusion and advection. The primary source of the relativistic cosmic
rays, e.g. supernova remnants, provides a quasi-neutral plasma with a power law
spectrum in momentum where we account for rigidity dependent differences
between the electron and proton spectrum. We examine the resulting leptonic and
hadronic radiation processes by synchrotron radiation, inverse Compton
scattering, Bremsstrahlung and hadronic pion production. Finally, the
observations of NGC 253, M82, NGC 4945 and NGC 1068 in the radio and gamma-ray
band as well as the observed supernova rate are used to constrain a best-fit
model. In the case of NGC 253, M82, NGC 4945 our model is able to accurately
describe the data, showing that: (i) Supernovae are the dominant particle
accelerators for NGC 253, M82 and NGC 4945, but not in the case of NGC 1068.
(ii) All considered starburst galaxies are poor proton calorimeters in which
for NGC 253 the escape is predominantly driven by the galactic wind, whereas
the diffusive escape dominates in NGC 4945 and M82 (at energies > 1TeV). (iii)
Secondary electrons from hadronic pion production are important to model the
radio flux, but the associated neutrino flux is below the current observation
limit.
