Constraining relativistic protons and magnetic fields in galaxy clusters
  through radio and gamma-ray observations : the case of A2256
  Giant radio halos are the most relevant examples of diffuse synchrotron
emission from galaxy clusters. A number of these sources have very steep
spectrum, with spectral index $\alpha \geq 1.5-1.6$ ($F(\nu) \propto
\nu^{-\alpha}$), and are ideal targets to test current models for the origin of
the relativistic particles. A2256 hosts the nearest radio halo with very steep
spectrum, with $\alpha =1.61$, and a very large population of relativistic
protons in the cluster would be necessary to explain the halo as due to
synchrotron emission from secondary particles. In this case the 0.1-1 GeV
$\gamma$--ray luminosity is expected 10-20 times larger than that of clusters
hosting radio halos with similar radio power at GHz frequencies but with
spectra more typical of the presently observed halo population, $\alpha \sim
1.2$. Under these assumptions incoming FERMI/GLAST observations are expected to
detect A2256, provided that the magnetic field in the central cluster region is
$\leq$10-15 $\mu$G. We show that this will allow for a prompt test of hadronic
models for the origin of radio halos, and for complementary constraints on both
the cluster magnetic field and the physics of particle acceleration mechanisms.
