The PARADIGM Project II: Characterising Nuclear and Diffuse Radio Components in Local U/LIRGs

Disentangling SF and AGN emission is essential for understanding galaxy evolution, yet remains challenging in merging systems where both processes are enhanced and spatially intertwined. Galaxy mergers drive gas inflows that simultaneously fuel nuclear SBs and BH accretion, shaping morphology from nuclear ($\lesssim 250$~pc) to large-scale ($\gtrsim 500$~pc) regions. Radio interferometry provides an unobscured view, but separating compact nuclear SBs, AGN, and diffuse SF requires multiscale, multi-frequency observations. We present a systematic method to characterise multiscale radio properties in 15 local ($z\lesssim 0.1$) U/LIRGs ($L_{\mathrm{IR}} > 10^{11}\mathrm{L}_{\odot}$). Using \emph{e}-MERLIN and VLA at 1.4, 6.0 and 33.0~GHz, we probe physical scales from $\sim 10$--$250$~pc to $\sim 0.5$--$3.0$~kpc. We decompose radio emission into nuclear (compact cores and nuclear extended) and large-scale (total and diffuse) components, comparing morphological properties (emission fractions, sizes, luminosities, surface densities) and investigating correlations with source classes, merger stages, and $L_{\mathrm{IR}}$. We find: i) nuclear emission contributes $\sim$50\% of total radio emission on average; ii) total multiscale diffuse emission (SF-related) contributes $\sim$80\% to total power; iii) nuclear emission components act together to correlate with total radio and infrared luminosities, which increase with merger stage, whilst diffuse emission at larger scales shows no clear dependence on nuclear processes; iv) sources with radio excess (lower $q_{\mathrm{IR}}$) show lower nuclear luminosity ratios $L_{\mathrm{R,33}}^{\mathrm{N}}/L_{\mathrm{R,6}}^{\mathrm{N}}$, indicating a deficit of high-$\nu$ radio emission; since 33.0~GHz traces recent SF, this suggests the radio excess is dominated by non-thermal emission at lower $\nu$, likely AGN-related, rather than enhanced SF.