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.
