Galaxy clusters are considered to be gigantic reservoirs of cosmic rays (CRs). Some of the clusters are found with extended radio emission, which provides evidence for the existence of magnetic fields and CR electrons in the intra-cluster medium (ICM). The mechanism of radio halo (RH) emission is still under debate, and it has been believed that turbulent reacceleration plays an important role. In this paper, we study the reacceleration of CR protons and electrons in detail by numerically solving the Fokker-Planck equation, and show how radio and gamma-ray observations can be used to constrain CR distributions and resulting high-energy emission for the Coma cluster. We take into account the radial diffusion of CRs and follow the time evolution of their one-dimensional distribution, by which we investigate the radial profile of the CR injection that is consistent with the observed RH surface brightness. We find that the required injection profile is non-trivial, depending on whether CR electrons have the primary or secondary origin. Although the secondary CR electron scenario predicts larger gamma-ray and neutrino fluxes, it is in tension with the observed RH spectrum. In either scenario, we find that galaxy clusters can make a sizable contribution to the all-sky neutrino intensity if the CR energy spectrum is nearly flat.
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