Corrections to Hawking radiation from asteroid-mass primordial black holes: analytic and numerical evaluation of the stochastic charge effect

Hawking radiation sets stringent constraints on Primordial Black Holes (PBHs) as a dark matter candidate in the $M\sim 10^{16}$ g regime based on the evaporation products produced by the black hole, motivating the need to rigorously model the photon, electron, and positron emission spectra. This manuscript is the second in a series of two papers (see Vasquez et al. [Phys. Rev. D, 112:063002 (2025)] for the first paper in the series) with the goal of proving the stochastic emission of electrons and positrons, known as the "stochastic charge effect" and first predicted by Page [Phys. Rev. D, 16:2402 (1977)] using semi-classical arguments, arises from quantum electrodynamics (QED) on a Schwarzschild spacetime. We derive the corrections to the $e^\pm$ from the relevant term in the Hamiltonian (the long-range monopole: $H_{\rm int,0L}$) to first order in the fine structure constant ($\alpha$), and show that the semi-classical terms are also present in the QED calculation. We also highlight which terms in our analysis only appear within our quantum mechanical approach and cannot be explained otherwise. We find that due to some cancellation of terms, over the range of $2.2\times 10^{16}$ - $1.7\times 10^{17}$ g, the net correction to the $e^\pm$ emission rate is less than 2%, versus the suppression of up to 5% previously found by semi-classical arguments.