An analytic resolution to the competition between Lyman-Werner radiation and metal winds in direct collapse black hole hosts

A near pristine atomic cooling halo close to a star forming galaxy offers a natural pathway for forming massive direct collapse black hole (DCBH) seeds which could be the progenitors of the $z>6$ redshift quasars. The close proximity of the haloes enables a sufficient Lyman-Werner flux to effectively dissociate H$_2$ in the core of the atomic cooling halo. A mild background may also be required to delay star formation in the atomic cooling halo, often attributed to distant background galaxies. In this letter we investigate the impact of metal enrichment from both the background galaxies and the close star forming galaxy under extremely unfavourable conditions such as instantaneous metal mixing. We find that within the time window of DCBH formation, the level of enrichment never exceeds the critical threshold (Z$_{cr} \sim 1 \times 10^{-5} \ \rm Z_{\odot})$, and attains a maximum metallicity of Z $\sim 2 \times 10^{-6} \ \rm Z_{\odot}$. As the system evolves, the metallicity eventually exceeds the critical threshold, long after the DCBH has formed.