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

The radio mini halos (MH) observed in relaxed clusters probe the presence of relativistic particles on scales of hundreds of kpc, beyond the scales directly influenced by the central AGN, but the nature of the mechanism that produces the relativistic electrons is still debated. In this work we explore the connection between thermal and non-thermal components of the ICM in a sample of MH and we study its implications for hadronic models for the origin of the relativistic electrons. 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. We extended the method generally applied to giant radio halos by considering the effects of a grid randomly generated through a Monte Carlo chain. Contrary to what is generally observed 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 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. Specifically, we focus on a model where cosmic rays are injected by the central AGN and they generate secondaries in the ICM, and we assume that the role of turbulent re-acceleration is negligible. This model allows us to constrain the AGN cosmic ray luminosity in the range $\sim10^{44-46}$ 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 ICM turbulence.