The MIni-Halos in MIni-Clusters (MIMIC) project: New insights into the origin of mini-halos from a LOFAR, uGMRT, and MeerKAT study of diffuse radio emission in Abell 496 and ZwCl 235

We aim at investigating the presence of diffuse radio emission in low-mass (M$_{500}\leq3\times10^{14}$ M$_{\odot}$), relaxed galaxy clusters. We present a pilot study for the MIMIC (MIni-halos in MIni-Clusters) project, focusing on Abell 496 and ZwCl 235, using LOFAR (54, 144 MHz), uGMRT (400, 650 MHz), and MeerKAT (1.28 GHz) data, as well as Chandra X-ray data. We detect diffuse radio emission in the central 100 kpc of both Abell 496 and ZwCl 235, spatially confined within sloshing-induced cold fronts seen in X-ray images. The diffuse sources satisfy the classification criteria for radio mini-halos and represent the least powerful known to date, with 1.4 GHz radio powers of $(2.2\pm0.2)\times10^{22}$ W/Hz in Abell 496 and $(1.3\pm0.3)\times10^{22}$ W/Hz in ZwCl 235. The mini-halos have steep spectra ($-1.9\leqα\leq-1.4$), and in Abell 496 we find outward steepening of the spectral index. In both clusters, the radio and X-ray surface brightness are correlated over the diffuse source extent, with sub-linear or nearly linear slopes depending on frequency. Separately, we present the properties of two extended (>200 kpc) tailed radio galaxies in both clusters. The steep spectra, sub-linear radio-X-ray correlations, and confinement within cold fronts suggest that contribution from fossil plasma advected from the central radio galaxy and plausible turbulent re-acceleration are at play in Abell 496 and ZwCl 235. These two systems allow us to extend mini-halo scaling relations toward the low-power regime. We confirm a strong correlation between the mini-halo radio power and the cool core X-ray luminosity, with $P_{1.4} \propto L_{X,70}^{1.5\pm0.2}$. We find, for the first time, tentative indications that mini-halos with steeper radio spectra may preferentially be associated with lower core X-ray luminosities and lower radio power. (abridged)