  Radio observations represent a powerful probe of the physics occurring in the
intracluster medium (ICM) because they trace the relativistic cosmic rays in
the cluster magnetic fields, or within galaxies themselves. By probing the
low-energy cosmic rays, low-frequency radio observations are especially
interesting because they unveil emission powered by low-efficiency particle
acceleration processes, which are believed to play a crucial role in the origin
of diffuse radio emission. We investigate the origin of the radio mini-halo at
the center of the galaxy cluster 2A0335+096 and its connection to the central
galaxy and the sloshing cool core. We also study the properties of the
head-tail galaxy GB6 B0335+096 hosted in the cluster to explore the lifecycle
of the relativistic electrons in its radio tails. We use new LOw Frequency
ARray (LOFAR) observations from the LOFAR Two-meter Sky Survey at 144 MHz to
map the low-frequency emission with a high level of detail. The new data were
combined with archival Giant Metrewave Radio Telescope and Chandra observations
to carry out a multi-wavelength study. We have made the first measurement of
the spectral index of the mini-halo ($\alpha=-1.2\pm0.1$ between 144 MHz and
1.4 GHz) and the lobes of the central source ($\alpha\simeq-1.5\pm0.1$ between
144 and 610 MHz). Based on the low-frequency radio emission morphology with
respect to the thermal ICM, we propose that the origin of the diffuse radio
emission is linked to the sloshing of the cool core. The new data revealed the
presence of a Mpc-long radio tail associated with GB6 B0335+096. The observed
projected length is a factor 3 longer than the expected cooling length, with
evidence of flattening in the spectral index trend along the tail. Therefore,
we suggest that the electrons toward the end of the tail are kept alive by the
ICM gentle re-acceleration.
