Corrections to Hawking radiation from asteroid-mass primordial black holes: description of the stochastic charge effect in quantum electrodynamics

TL;DR

This paper investigates the stochastic charge fluctuations of primordial black holes due to Hawking radiation, revealing quantum electrodynamics effects that influence their charge distribution and potential observational signatures.

Contribution

It provides a rigorous quantum electrodynamics derivation of the stochastic charge effect in black holes, connecting semi-classical and quantum operator approaches.

Findings

The variance of black hole charge approaches a finite value due to stochastic effects.

Quantum fluctuations in charge are mediated by Hawking-emitted particles, not the black hole itself.

The rms charge asymptotes to specific values depending on black hole mass.

Abstract

Hawking radiation sets stringent constraints on Primordial Black Holes (PBHs) as a dark matter candidate in the $M \sim 10^{16} \ \mathrm{g}$ regime based on the evaporation products such as photons, electrons, and positrons. This motivates the need for rigorous modeling of the Hawking emission spectrum. Using semi-classical arguments, Page [Phys. Rev. D 16, 2402 (1977)] showed that the emission of electrons and positrons is altered due to the black hole acquiring an equal and opposite charge to the emitted particle. The Poisson fluctuations of emitted particles cause the charge $Z|e|$ to random walk, but since acquisition of charge increases the probability of the black hole emitting another charged particle of the same sign, the walk is biased toward $Z=0$, and $P(Z)$ approaches an equilibrium probability distribution with finite variance $\langle Z^2\rangle$. This paper explores how this ``stochastic charge'' phenomenon arises from quantum electrodynamics (QED) on a Schwarzschild spacetime. We prove that (except for a small Fermi blocking term) the semi-classical variance $\langle Z^2 \rangle$ agrees with the variance of a quantum operator $\langle \hat{\cal Z}^2 \rangle$, where $\hat{\cal Z}$ may be thought of as an ``atomic number'' that includes the black hole as well as charge near it (weighted by a factor of $2M/r$). In QED, the fluctuations in $\hat{\cal Z}$ do not arise from the black hole itself (whose charge remains fixed), but rather as a collective effect in the Hawking-emitted particles mediated by the long-range electromagnetic interaction. We find the rms charge $\langle Z^2\rangle^{1/2}$ asymptotes to 3.44 at small PBH masses $M \lesssim 2\times 10^{16}\,$g, declining to 2.42 at $M=5.2\times 10^{17}\,$g.