Worldline Instantons and Pair Production in Inhomogeneous Fields

TL;DR

This work develops and applies worldline instanton techniques to compute the leading nonperturbative contribution to vacuum pair production in QED for inhomogeneous electric fields. By solving the stationary-phase equations for both temporally and spatially inhomogeneous backgrounds, the authors obtain explicit worldline instanton trajectories and their actions, showing that temporal inhomogeneities shrink instantons and enhance local production while spatial inhomogeneities enlarge loops and suppress production. The formalism extends from scalar to spinor QED, with the spin factor yielding the standard spinor result and preserving the same instanton paths. The results connect smoothly with WKB analyses and provide a framework for including fluctuation determinants, higher-loop effects, and numerical Monte Carlo approaches in future work.

Abstract

We show how to do semiclassical nonperturbative computations within the worldline approach to quantum field theory using ``worldline instantons''. These worldline instantons are classical solutions to the Euclidean worldline loop equations of motion, and are closed spacetime loops parametrized by the proper-time. Specifically, we compute the imaginary part of the one loop effective action in scalar QED using ``worldline instantons'', for a wide class of inhomogeneous electric field backgrounds. We treat both time dependent and space dependent electric fields, and note that temporal inhomogeneities tend to shrink the instanton loops, while spatial inhomogeneities tend to expand them. This corresponds to temporal inhomogeneities tending to enhance local pair production, with spatial inhomogeneities tending to suppress local pair production. We also show how the worldline instanton technique extends to spinor QED.

Figures (11)

• Figure 1: Parametric plot of the stationary worldline instanton paths in the $(x_3, x_4)$ plane for the case of a constant electric field of strength $E$. The paths are circular and the radius has been expressed in units of $\frac{m}{eE}$.
• Figure 2: Parametric plot of the stationary worldline instanton paths (\ref{['psols-sech']}) in the $(x_3, x_4)$ plane for the case of a time dependent electric field $E(t)=E\, {\rm sech}^2(\omega t)$. The paths are shown for various values of the adiabaticity parameter $\gamma=\frac{m\omega}{eE}$ defined in (\ref{['ad']}), and $x_3$ and $x_4$ have been expressed in units of $\frac{m}{eE}$. Note that in the static limit, $\gamma\to 0$, the instanton paths reduce to the circular ones of the constant field case shown in Figure \ref{['fig1']}.
• Figure 3: Plot of the instanton action $S_0$, in units of $n\frac{m^2}{eE}$, in (\ref{['action-sech']}) for the time-dependent electric field $E(t)=E\, {\rm sech}^2(\omega t)$, plotted as a function of the adiabaticity parameter $\gamma$. Contrast this plot with the behavior in Figure \ref{['fig7']} for a spatial inhomogeneity of the same form.
• Figure 4: Parametric plot of the stationary worldline instanton paths (\ref{['psols-cos']}) in the $(x_3, x_4)$ plane for the case of a time dependent electric field $E(t)=E\, \cos(\omega t)$. The paths are shown for various values of the adiabaticity parameter $\gamma=\frac{m\omega}{eE}$ defined in (\ref{['ad']}), and $x_3$ and $x_4$ have been expressed in units of $\frac{m}{eE}$. Note that in the static limit, $\gamma\to 0$, the instanton paths reduce to the circular ones of the constant field case shown in Figure \ref{['fig1']}.
• Figure 5: Plot of the instanton action $S_0$, in units of $n\frac{m^2}{eE}$, for the time-dependent electric field $E(t)=E\, \cos(\omega t)$, plotted as a function of the adiabaticity parameter $\gamma$. Contrast this plot with the behavior in Figure \ref{['fig10']} for a spatial inhomogeneity of the same form.