Do radio mini-halos and gas heating in cool-core clusters have a common
  origin?
  In this letter we present a study of the central regions of cool-core
clusters hosting radio mini-halos, which are diffuse synchrotron sources
extended on cluster-scales surrounding the radio-loud brightest cluster galaxy.
We aim to investigate the interplay between the thermal and non-thermal
components in the intra-cluster medium in order to get more insights into these
radio sources, whose nature is still unclear. It has recently been proposed
that turbulence plays a role for heating the gas in cool cores. By assuming
that mini-halos are powered by the same turbulence, we expect that the
integrated radio luminosity of mini-halos, $\nu P_{\nu}$, depends on the
cooling flow power, $P_{\rm CF}$, which in turn constrains the energy available
for the non-thermal components and emission in the cool-core region. We carried
out a homogeneous re-analysis of X-ray Chandra data of the largest sample of
cool-core clusters hosting radio mini-halos currently available ($\sim$ 20
objects), finding a quasi-linear correlation, $\nu P_{\nu} \propto P_{\rm
CF}^{0.8}$. We show that the scenario of a common origin of radio mini-halos
and gas heating in cool-core clusters is energetically viable, provided that
mini-halos trace regions where the magnetic field strength is $B \gg 0.5\,
\mu$G .
