Magnetic Black Hole Pair-Production: One-Loop Tunneling Rate in the Weak Field Limit

Pair-production of magnetic Reissner-Nordstr"{o}m black holes (of charges $\pm q$) is studied in the next-to-leading WKB approximation. We consider generic quantum fluctuations in the corresponding instanton geometry, a detailed study of which suggests that, for sufficiently weak field $B$, the problem can be reduced to that of quantum fluctuations around a single truncated near-extremal Euclidean black hole in thermal equilibrium. Typical one-loop contributions are such that the leading WKB exponent is corrected by a small fraction $\sim \mathrm{\hslash }\mathrm{/}{q}^2$. We show that this effect is merely due to a semiclassical shift of the black hole mass-to-charge ratio that persists even in the extremal limit. We close with a few final comments.