Towards the QED beta function and renormalons at $1\mathrm{/}{N}_f^2$ and $1\mathrm{/}{N}_f^3$

We determine the $1\mathrm{/}{N}_f^2$ and $1\mathrm{/}{N}_f^3$ contributions to the QED beta function stemming from the closed set of nested diagrams. At order $1\mathrm{/}{N}_f^2$ we discover a new logarithmic branch-cut closer to the origin when compared to the $1\mathrm{/}{N}_f$ results. The same singularity location appears at $1\mathrm{/}{N}_f^3$, and these correspond to a UV renormalon singularity in the finite part of the photon two-point function.