Radio and X-ray connection in radio mini-halos: Implications for hadronic models

Context. A large fraction of cool-core clusters are known to host diffuse, steep-spectrum radio sources, called radio mini-halos, in their cores. Mini-halos reveal the presence of relativistic particles on scales of hundreds of kiloparsecs, beyond the scales directly influenced by the central active galactic nucleus (AGN), but the nature of the mechanism that produces such a population of radio-emitting, relativistic electrons is still debated. It is also unclear to what extent the AGN plays a role in the formation of mini-halos by providing the seeds of the relativistic population.Aims. In this work we explore the connection between thermal and non-thermal components of the intra-cluster medium in a sample of radio mini-halos and we study the implications within the framework of a hadronic model for the origin of the emitting electrons.Methods. For the first time, we studied the thermal and non-thermal connection by carrying out a point-to-point comparison of the radio and the X-ray surface brightness in a sample of radio mini-halos. We extended the method generally applied to giant radio halos by considering the effects of a grid randomly generated through a Monte Carlo chain. Then we used the radio and X-ray correlation to constrain the physical parameters of a hadronic model and we compared the model predictions with current observations.Results. Contrary to what is generally reported in the literature for giant radio halos, we find that the mini-halos in our sample have super-linear scaling between radio and X-rays, which suggests a peaked distribution of relativistic electrons and magnetic field. We explore the consequences of our findings on models of mini-halos. We use the four mini-halos in the sample that have a roundish brightness distribution to constrain model parameters in the case of a hadronic origin of the mini-halos. Specifically, we focus on a model where cosmic rays are injected by the central AGN and they generate secondaries in the intra-cluster medium, and we assume that the role of turbulent re-acceleration is negligible. This simple model allows us to constrain the AGN cosmic ray luminosity in the range similar to 10(44-46) erg s(-1) and the central magnetic field in the range 10-40 mu G. The resulting gamma -ray fluxes calculated assuming these model parameters do not violate the upper limits on gamma -ray diffuse emission set by the Fermi-LAT telescope. Further studies are now required to explore the consistency of these large magnetic fields with Faraday rotation studies and to study the interplay between the secondary electrons and the intra-cluster medium turbulence.