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We investigate non-thermal emission from the Fermi bubbles on a hadronicmodel. Cosmic-ray (CR) protons are accelerated at the forward shock of thebubbles. They interact with the background gas in the Galactic halo and create$\pi^0$-decay gamma-rays and secondary electrons through proton-protoninteraction. We follow the evolution of the CR protons and electrons bycalculating their distribution functions. We find that the spectrum and theintensity profile of $\pi^0$-decay gamma-rays are consistent with observations.We predict that the shock front is located far ahead of the gamma-ray boundaryof the Fermi bubbles. This naturally explains the fact that a clear temperaturejump of thermal gas was not discovered at the gamma-ray boundary in recentSuzaku observations. We also consider re-acceleration of the background CRs inthe Galactic halo at the shock front. We find that it can significantly affectthe gamma-rays from the Fermi bubbles, unless the density of the background CRsis $\lesssim 10$\% of that in the Galactic disk. We indicate that secondaryelectrons alone cannot produce the observed radio emission from the Fermibubbles. However, the radio emission from the outermost region of the bubblescan be explained, if electrons are directly accelerated at the shock front withan efficiency of $\sim 0.1$\% of that of protons.

A Hadronic-Leptonic Model for the Fermi Bubbles: Cosmic-Rays in the Galactic Halo and Radio Emission