The Coma cluster at LOFAR frequencies II: the halo, relic, and a new
  accretion relic
  We present LOw Frequency ARray observations of the Coma cluster field at
144\,MHz. The cluster hosts one of the most famous radio halos, a relic, and a
low surface-brightness bridge. We detect new features that allow us to make a
step forward in the understanding of particle acceleration in clusters. The
radio halo extends for more than 2 Mpc, which is the largest extent ever
reported. To the North-East of the cluster, beyond the Coma virial radius, we
discover an arc-like radio source that could trace particles accelerated by an
accretion shock. To the West of the halo, coincident with a shock detected in
the X-rays, we confirm the presence of a radio front, with different spectral
properties with respect to the rest of the halo. We detect a radial steepening
of the radio halo spectral index between 144 MHz and 342 MHz, at $\sim
30^{\prime}$ from the cluster centre, that may indicate a non constant
re-acceleration time throughout the volume. We also detect a mild steepening of
the spectral index towards the cluster centre. For the first time, a radial
change in the slope of the radio-X-ray correlation is found, and we show that
such a change could indicate an increasing fraction of cosmic ray versus
thermal energy density in the cluster outskirts. Finally, we investigate the
origin of the emission between the relic and the source NGC 4789, and we argue
that NGC4789 could have crossed the shock originating the radio emission
visible between its tail and the relic.
