  In some galaxy clusters powerful AGN have blown bubbles with cluster scale
extent into the ambient medium. The main pressure support of these bubbles is
not known to date, but cosmic rays are a viable possibility. For such a
scenario copious gamma-ray emission is expected as a tracer of cosmic rays from
these systems. Hydra A, the closest galaxy cluster hosting a cluster scale AGN
outburst, located at a redshift of 0.0538, is investigated for being a
gamma-ray emitter with the High Energy Stereoscopic System (H.E.S.S.) array and
the Fermi Large Area Telescope (Fermi-LAT). Data obtained in 20.2 hours of
dedicated H.E.S.S. observations and 38 months of Fermi-LAT data, gathered by
its usual all-sky scanning mode, have been analyzed to search for a gamma-ray
signal. No signal has been found in either data set. Upper limits on the
gamma-ray flux are derived and are compared to models. These are the first
limits on gamma-ray emission ever presented for galaxy clusters hosting cluster
scale AGN outbursts. The non-detection of Hydra A in gamma-rays has important
implications on the particle populations and physical conditions inside the
bubbles in this system. For the case of bubbles mainly supported by hadronic
cosmic rays, the most favorable scenario, that involves full mixing between
cosmic rays and embedding medium, can be excluded. However, hadronic cosmic
rays still remain a viable pressure support agent to sustain the bubbles
against the thermal pressure of the ambient medium. The largest population of
highly-energetic electrons which are relevant for inverse-Compton gamma-ray
production is found in the youngest inner lobes of Hydra A. The limit on the
inverse-Compton gamma-ray flux excludes a magnetic field below half of the
equipartition value of 16 muG in the inner lobes.
