Schwinger effect with backreaction in 1+1D massive QED with a strong external field

TL;DR

The paper investigates backreaction in the Schwinger effect for 1+1D massive QED under a strong external field by bosonizing the theory and treating the fermion mass perturbatively. It shows that the vacuum expectation value of the boson field satisfies a sine-Gordon type PDE with external driving, yielding dissipation-free plasma oscillations and an analytically tractable plasma frequency, with a key $O(m)$ correction to the frequency. The work demonstrates a precise quantum treatment that reveals quantitative deficiencies in the semiclassical approximation, especially in capturing the $O(m)$ frequency shift. These results illuminate the dynamics of field-matter energy exchange in a reduced-dimensional setting and offer a framework for comparing quantum, semiclassical, and lattice approaches to backreaction in strong-field QED.

Abstract

In the presence of a strong electric field, the vacuum is unstable to the production of pairs of charged particles -- the Schwinger effect. The created pairs extract energy from the electric field, resulting in nontrivial backreaction. In this paper, we study 1+1D massive QED subject to strong external electric fields in a self-consistent and fully quantum manner. We use the bosonized version of the theory, which attains a cosine interaction term in the presence of nonzero fermion mass $m$. However, the assumption of strong electric field justifies a perturbative treatment of the cosine interaction, i.e., an expansion in $m$. We calculate the vacuum expectation value of the electric field to first order in $m$ and show that -- surprisingly -- it satisfies a classical nonlinear partial differential equation (related to the sine-Gordon equation). We show that the electric field exhibits dissipation-free oscillations (analogous to ordinary plasma oscillations) and calculate the plasma frequency analytically. We also compare to the semiclassical approximation commonly used to study backreaction, showing that it fails to capture the $O(m)$ shift in the plasma frequency.

Figures (3)

• Figure 1: The expectation value of the electric field $\braket{E}=E_{C}+q\braket{\phi}/\sqrt{\pi}$. Two charged plates start at $x=0$ and move in opposite directions with velocity $v=0.8$. The plates are stopped when they reach $x=\pm20$. The electric field eventually reaches a static state as all the dynamical part spreads out to infinity. The static solution (black line) shows exponential screening of the plates from the both sides. We set $q=1.0,m=0.1,E_{0}=1.0$.
• Figure 2: Same as Fig. \ref{['fig: non reflective plates']}, except for impenetrable places ($\braket{\phi}$ is forced to vanish outside the plates). In this case, the electric field inside the capacitor never reaches a static state; the dynamical part of the field bounces back and forth inside the capacitor.
• Figure 3: The potential $V(\phi)$ given in Eq. (\ref{['eq;potential']}) in the case of constant $E_{C}$. The massless and massive cases are shown by the red and black lines. Under this potential, $\phi$ oscillates back and forth around its minimum. For the massless case, the frequency is given by $\omega=M$. For the massive case, the potential is modified due to the third term in Eq. (\ref{['eq;potential']}), and the frequency is corrected to be Eq. (\ref{['eq: plasma frequency']}). We set $m=0.5,q=1$, and $E_{C}=1$ for this plot.