Momentum correlation in pair production by spacetime dependent fields from scattered wave functions

Abstract

We consider Sauter-Schwinger pair production by electric fields that depend on both time and space, $E(t,z)$ and $E(t,x,y)$. For space-independent fields, $E(t)$, momentum conservation, $\hat{I}´(\mathbf{p}+{\mathbf{p}}^{\prime })$, fixes the positron momentum, ${\mathbf{p}}^{\prime }$, in terms of the electron momentum, $\mathbf{p}$. For $E(t,z)$, on the other hand, $p_z$ and $p_z^{\prime }$ are independent. However, previous exact-numerical studies have considered only the probability as a function of a single momentum variable, $P(p_z)$, $P(p_z^{\prime })$ or $P(p_z^{\prime }-p_z)$, but not the correlation $P(p_z,p_z^{\prime })$. In this paper, we show how to obtain $P(p_z,p_z^{\prime })$ by solving the Dirac equation numerically. To do so, we split the wave function into a background and a scattered wave, $\ddot{I}ˆ(t,\mathbf{x})=\ddot{I}{ˆ}_{back.}(t,\mathbf{x})+\ddot{I}{ˆ}_{scat.}(t,\mathbf{x})$, where $\ddot{I}{ˆ}_{back.}\propto \exp (\pm ipx+\text{gauge term})$. $\ddot{I}{ˆ}_{scat.}$ vanishes outside a past light cone and is obtained by solving $(iγ^μD_μ-m)ψ_{\rm scat.}=-(iγ^μD_μ-m)ψ_{\rm back.}$ backwards in time starting with $\ddot{I}{ˆ}_{scat.}(t\to +\infty ,\mathbf{x})=0$.

11 pages, 6 figures. v2: added link to code for solving the Dirac equation on a GPU