Nearby blue compact dwarf (BCD) galaxies are arguably our best local analogues of galaxies in the earlier Universe that may host relics of black hole (BH) seeds. Here we present high-resolution Chandra X-ray Observatory and Karl G. Jansky Very Large Array (VLA) observations of five nearby BCDs with stellar masses of less than the Small Magellanic Cloud ($M_star sim 10^{7} - 10^{8.4}$ $M_odot$). We search for signatures of accreting massive BHs at X-ray and radio wavelengths, which are more sensitive to lower BH accretion rates than optical searches. We detect a total of 10 hard X-ray sources and 10 compact radio sources at luminosities consistent with star-formation-related emission. We find one case of a spatially-coincident X-ray and radio source within the astrometric uncertainties. If the X-ray and radio emission are indeed coming from the same source, the origin of the radiation is plausibly from an active massive BH with log $(M_{rm BH}/M_{odot}) sim 4.8 pm 1.1$. However, given that the X-ray and radio emission are also coincident with a young star cluster complex, we consider the combination of an X-ray binary and a supernova remnant (or HII region) a viable alternative explanation. Overall, we do not find compelling evidence for active massive BHs in our target BCDs, which on average have stellar masses more than an order of magnitude lower than previous samples of dwarf galaxies found to host massive BHs. Our results suggest that moderately accreting massive BHs in BCDs are not so common as to permit unambiguous detection in a small sample.
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