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

Samuel E. Gralla and Morifumi Mizuno

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Phys. Rev. D 113, 085007 – Published 8 April, 2026

DOI: https://doi.org/10.1103/5jhz-cbf7

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 a 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.

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Phys. Rev. D 113, 085007– Published 8 April, 2026
  • Received 15 December 2025
  • Accepted 16 March 2026
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