On the Challenges of Cosmic-Ray Proton Shock Acceleration in the
  Intracluster Medium
  Galaxy clusters host the largest particle accelerators in the Universe: Shock
waves in the intracluster medium (ICM), a hot and ionised plasma, that
accelerate particles to high energies. Radio observations pick up synchrotron
emission in the ICM, proving the existence of accelerated cosmic-ray electrons.
However, a sign of cosmic-ray protons, in form of $\gamma$-rays. remains
undetected. This is know as the missing $\gamma$-ray problem and it directly
challenges the shock acceleration mechanism at work in the ICM. Over the last
decade, theoretical and numerical studies focused on improving our knowledge on
the microphysics that govern the shock acceleration process in the ICM. These
new models are able to predict a $\gamma$-ray signal, produced by shock
accelerated cosmic-ray protons, below the detection limits set modern
$\gamma$-ray observatories. In this review, we summarise the latest advances in
solving the missing $\gamma$-ray problem.
