Multi-scale VLBI observations of the candidate host galaxy of GRB 200716C

We present the discovery and the subsequent follow up of radio emission from SDSS J130402.36$+$293840.6 (J1304+2938), the candidate host galaxy of the gamma-ray burst GRB~200716C. The galaxy is detected in the RACS (0.89 GHz), the NVSS, the Apertif imaging survey, and the FIRST (1.4 GHz), the VLASS (3 GHz), and in public LOFAR (130-170 MHz), WISE (3.4-22 $\mu$m), and SDSS (z, i, r, g, u filters) data. The luminosity inferred at 1.4 GHz is (5.1$\pm0.2)\times10^{30}$ erg s$^{-1}$ Hz$^{-1}$. To characterise the emission and distinguish between different components within the galaxy, we performed dedicated, high-sensitivity and high-resolution observations with the European VLBI Network + e-MERLIN at 1.6 and 5 GHz. We did not detect any emission from a compact core, suggesting that the presence of a radio-loud AGN is unlikely, and therefore we ascribe the emission observed in the public surveys to star-forming regions within the galaxy. We confirm and refine the redshift estimate, $z=0.341\pm0.004$, with a dedicated TNG spectroscopic observation. Finally, we compiled a list of all the known hosts of GRB afterglows detected in radio and computed the corresponding radio luminosity: if GRB~200716C belongs to J1304+2938, this is the third most radio-luminous host of a GRB, implying one of the highest star-formation rates (SFRs) currently known, namely SFR$\sim$324$\pm$61 M$_{\odot}$ yr$^{-1}$. On the other hand, through the analysis of the prompt emission light curve, recent works suggest that GRB~200716C might be a short-duration GRB located beyond J1304$+$2938 and gravitationally lensed by an intermediate-mass black hole (IMBH) hosted by the galaxy. Neither the public data nor our VLBI observations can confirm or rule out the presence of an IMBH acting as a (milli-)lens hosted by the galaxy, a scenario still compatible with the set of radio observations presented in this work.