t case, the emission probability for charges $\pm q$ is proportional to a Boltzmann factor of the form $$\begin{aligned}
e^{-\frac{1}{T}\left(m\pm\frac{qQ}{r_+}\right)}= e^{-\frac{m}{T}\left(1\pm\frac{q}{\sqrt{2}m}(1-e^{2\Delta\phi})\right)}.\end{aligned}$$ If $|\Delta\phi|\gg 1$, the discharge rate is fast if the WGC is satisfied and the black hole is hot.

Now we consider the case $m/T\gg 1$. Here the discharge rate is governed by the Schwinger process, and the rate exponent can be determined by barrier penetration arguments for a mode of frequency equal to the electrostatic potential energy at the horizon, $\omega_+=-qQ/r_+$ [@Gibbons:1975kk]. For $\omega_+$ to be a scattering state, we must have $q>\sqrt{2}m$ near extremality.

The Klein-Gordon equation for the $s$-wave mode of frequency $\omega_+$ is $$\begin{aligned}
\Phi''(r)+W\Phi(r)=0\;,\;\;\;\;\;W=\frac{q^2}{2}-\frac{m^2r}{r-2M}+\frac{(Me^{2\Delta\phi})^2}{(r-2M)^2(r-2M+2Me^{2\Delta\phi})^2}\;.\end{aligned}$$ Here we have put the equation in normal form and taken $e^{2\Delta\phi}\ll 1$. The barrier $W<0$ extends approximately from $r\sim 2M+e^{2\Delta\phi}M \equiv \alpha$ to $r\sim 2M\left(1+ \frac{2m^2}{q^2-2m^2}\right)\equiv \beta$. In the WKB approximation, the barrier penetration factor is $$\begin{aligned}
e^{-2\int_\alpha^\beta \sqrt{W}dr}.\end{aligned}$$ We can approximately evaluate the WKB integral by splitting it into regions where the last two terms in $W$ dominate ($\equiv W_{23}$, valid near $\alpha$) and where the first two terms in $W$ dominate ($\equiv W_{12}$, valid near $\beta$). The two regions overlap where the first and third terms are of similar order, near $r\sim 2M+1/\sqrt{2}q$. Putting the pieces together and keeping only the dominant terms, we find the production rate is of order $$\begin{aligned}
\Gamma_{Einstein} \sim e^{-2\int_\alpha^\gamma \sqrt{W_{23}}dr-2\int_\gamma^\beta \sqrt{W_{12}}dr}\approx e^{-\frac{2\sqrt{2}\pi  M m^2}{\sqrt{q^2-2m^2}}}\;.
\label{eq:einsteinrate}\end{aligned}$$ This is similar to the Schwinger exp