A MeerKAT-meets-LOFAR study of Abell 1413: a moderately disturbed non-cool-core cluster hosting a similar to 500 kpc 'mini'-halo

Many relaxed cool-core clusters host diffuse radio emission on scales of hundreds of kiloparsecs: mini-haloes. However, the mechanism responsible for generating them, as well as their connection with central active galactic nuclei, is elusive and many questions related to their physical properties and origins remain unanswered. This paper presents new radio observations of the galaxy cluster Abell 1413 performed with MeerKAT (L-band; 872 to 1712 MHz) and LOFAR HBA (120 to 168 MHz) as part of a statistical and homogeneous census of mini-haloes. Abell 1413 is unique among mini-halo clusters as it is a moderately disturbed non-cool-core cluster. Our study reveals an asymmetric mini-halo up to 584 kpc in size at 1283 MHz, twice as large as first reported at similar frequencies. The spectral index is flatter than previously reported, with an integrated value of alpha = -1.01 +/- 0.06, shows significant spatial variation, and a tentative radial steepening. We studied the point-to-point X-ray/radio surface brightness correlation to investigate the thermal/non-thermal connection: our results show a strong connection between these components, with a superlinear slope of b = 1.63(-0.10)(+0.10) at 1283 MHz and b = 1.20(-0.11)(+0.13) at 145 MHz. We also explore the X-ray surface brightness/radio spectral index correlation, finding a slope of b = 0.59(-0.11)(+0.11). Both investigations support the evidence of spectral steepening. Finally, in the context of understanding the particle acceleration mechanism, we present a simple theoretical model which demonstrates that hybrid scenarios - secondary electrons (re-)accelerated by turbulence - reproduce a superlinear correlation slope.