Chandra reveals a luminous Compton-thick QSO powering a $Ly\alpha$ blob in a $z=4$ starbursting protocluster

Galaxy clusters in the local universe descend from high-redshift overdense regions known as protoclusters. The large gas reservoirs and high rate of galaxy interaction in protoclusters are expected to trigger star-formation activity and luminous SMBH accretion in the host galaxies. We investigated the AGN content of a gas-rich and starbursting protocluster at $z=4$, known as the Distant Red Core (DRC). We observed with Chandra (139 ks) the 13 identified members of the structure, and searched for luminous and possibly obscured AGN among them. We also tested whether a hidden AGN can power the $Ly\alpha$ blob (LAB) detected with VLT/MUSE in the DRC. We detected obscured X-ray emission from the two most gas-rich members of the DRC, named DRC-1 and DRC-2. Both of them are resolved into multiple interacting clumps in high-resolution ALMA and HST observations. In particular, DRC-2 is found to host a luminous ($L_{2-10\,\mathrm{keV}}\approx3\times10^{45}\,\mathrm{erg\,s^{-1}}$) Compton-thick ($N_H\gtrsim10^{24}\,\mathrm{cm^{-2}}$) QSO, comparable to the most luminous QSOs known at all cosmic times. The AGN fraction among DRC members is consistent with results found for lower redshift protoclusters. However, X-ray stacking analysis reveals that SMBH accretion is likely also taking place in other DRC galaxies that are not detected individually by Chandra. Our results point toward the presence of a strong link between large gas reservoirs, galaxy interactions, and luminous and obscured nuclear activity in protocluster members. The powerful and obscured QSO detected in DRC-2 is likely powering the nearby LAB detected with VLT/MUSE, possibly through photoionization; however, we propose that the diffuse $Ly\alpha$ emission may be due to gas shocked by a massive outflow launched by DRC-2 over a $\approx10$ kpc scale.