Schwinger effect

Schwinger pair production in a constant electric field takes place at a constant rate per unit volume, commonly referred to as ${\displaystyle \Gamma }$. The rate was first calculated by Schwinger[3] and at leading order in ${\displaystyle e^{2}}$, the square of the charge of an electron, is equal to

${\displaystyle \Gamma ={\frac {(eE)^{2}}{4\pi ^{3}c\hbar ^{2}}}\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}\mathrm {e} ^{-{\frac {\pi m^{2}c^{3}n}{eE\hbar }}}}$

where ${\displaystyle m}$ is the mass of an electron and ${\displaystyle E}$ is the electric field strength. This formula cannot be expanded in a Taylor series in ${\displaystyle e^{2}}$, showing the nonperturbative nature of this effect. In terms of Feynman diagrams, one can derive the rate of Schwinger pair production by summing the infinite set of diagrams shown below, containing one electron loop and any number of external photon legs, each with zero energy.

The infinite set of Feynman diagrams relevant for Schwinger pair production.

The Schwinger effect has never been observed due to the extremely strong electric-field strengths required. Pair production takes place exponentially slowly when the electric field strength is much below the Schwinger limit, corresponding to approximately ${\displaystyle 10^{18}\,\mathrm {V} /\mathrm {m} }$. With current and planned laser facilities, this is an unfeasibly strong electric-field strength, so various mechanisms have been proposed to speed up the process and thereby reduce the electric-field strength required for its observation.

The rate of pair production may be significantly increased in time-dependent electric fields,[4][5][6] and as such is being pursued by high-intensity laser experiments such as the Extreme Light Infrastructure.[7] Another possibility is to include a highly charged nucleus which itself produces a strong electric field.[8]

• Schwinger limit
• Vacuum polarization