Extremal dilaton metric photon capture ($B_c=2M,{\sigma }_{\mathrm{capture}}=4\pi M^2$)

For $f=1-2M/r$ and squared areal radius $C=r(r-b)$, an equatorial null geodesic obeys ${\dot{r}}^2=E^2[1-B^{2}f/C]$, with impact parameter $B=L/E$. At $b=2M$, $f/C=1/r^2$ in the exterior. Its supremum is $1/(4M^2)$ at the null singular boundary, rather than at a regular exterior photon sphere. Rays with $B>2M$ turn at $r=B$; rays with $B<2M$ reach the boundary. Assuming those rays are absorbed gives the displayed geometric-optics cross-section despite zero limiting horizon area.