  The Fermi-LAT collaboration recently published deep upper limits to the
gamma-ray emission of the Coma cluster, a cluster that hosts the prototype of
giant radio halos. In this paper we extend previous studies and use a formalism
that combines particle reacceleration by turbulence and the generation of
secondary particles in the intracluster medium to constrain relativistic
protons and their role for the origin of the radio halo. We conclude that a
pure hadronic origin of the radio halo is clearly disfavoured as it would
require magnetic fields that are too strong. For instance $B_0 > 21 \mu$G is
found in the cluster center assuming that the magnetic energy density scales
with thermal density, to be compared with $B_0 \sim 4-5 \mu$G as inferred from
Rotation Measures (RM) under the same assumption. However secondary particles
can still generate the observed radio emission if they are reaccelerated. For
the first time the deep gamma-ray limits allow us to derive meaningful
constraints if the halo is generated during phases of reacceleration of
relativistic protons and their secondaries by cluster-scale turbulence. In this
paper we explore a relevant range of parameter-space of reacceleration models.
Within this parameter space a fraction of model configurations is already ruled
out by current gamma-ray limits, including the cases that assume weak magnetic
fields in the cluster core, $B \leq 2-3 \mu$G. Interestingly, we also find that
the flux predicted by a large fraction of model configurations that assume a
magnetic field consistent with RM is not far from the limits. This suggests
that a detection of gamma rays from the cluster might be possible in the near
future, provided that the electrons generating the radio halo are secondaries
reaccelerated and the magnetic field in the cluster is consistent with that
inferred from RM.
