# Schwinger effect

In the presence of a strong, constant electric field, electrons, e, and positrons, e+, will be spontaneously created.

The Schwinger effect is a predicted physical phenomenon whereby matter is created by a strong electric field. It is also referred to as the Sauter–Schwinger effect, Schwinger mechanism, or Schwinger pair production. It is a prediction of quantum electrodynamics (QED) in which electronpositron pairs are spontaneously created in the presence of an electric field, thereby causing the decay of the electric field. The effect was originally proposed by Fritz Sauter in 1931[1] and further important work was carried out by Werner Heisenberg and Hans Heinrich Euler in 1936,[2] though it was not until 1951 that Julian Schwinger gave a complete theoretical description.[3]

The Schwinger effect can be thought of as vacuum decay in the presence of an electric field. Although the notion of vacuum decay suggests that something is created out of nothing, physical conservation laws are nevertheless obeyed. To understand this, note that electrons and positrons are each other's antiparticles, with identical properties except opposite electric charge.

To conserve energy, the electric field loses energy when an electron–positron pair is created, by an amount equal to ${\displaystyle 2m_{\text{e}}c^{2}}$, where ${\displaystyle m_{\text{e}}}$ is the electron rest mass and ${\displaystyle c}$ is the speed of light. Electric charge is conserved because an electron–positron pair is charge neutral. Linear and angular momentum are conserved because, in each pair, the electron and positron are created with opposite velocities and spins. In fact, the electron and positron are expected to be created at (close to) rest, and then subsequently accelerated away from each other by the electric field.[4]

## Mathematical description

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 (one-loop) order 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, ${\displaystyle e}$ is the elementary charge, 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.

## Experimental prospects

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 1018 V/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,[5][6][7] and as such is being pursued by high-intensity laser experiments such as the Extreme Light Infrastructure.[8] Another possibility is to include a highly charged nucleus which itself produces a strong electric field.[9]

By electromagnetic duality, the same mechanism in a magnetic field should produce magnetic monopoles, if they exist. A search conducted using the Large Hadron Collider failed to detect monopoles, and analysis indicated a lower bound on monopole mass of 75 GeV/c2 at the 95% confidence level.[10]

In January 2022 researchers at the National Graphene Institute led by Andre Geim and a number of other collaborators reported the observation of an analog process between electron and holes at the Dirac point of a superlattice of graphene on hexagonal boron nitride (G/hBN) and another one of twisted bilayer graphene (TBG). An interpretation as Zener–Klein tunneling (a mix[11] between Zener tunneling and Klein tunneling) is also utilized.[12][13]

## References

1. ^ Sauter, Fritz (1931). "Über das Verhalten eines Elektrons im homogenen elektrischen Feld nach der relativistischen Theorie Diracs". Zeitschrift für Physik (in German). Springer Science and Business Media LLC. 69 (11–12): 742–764. doi:10.1007/bf01339461. ISSN 1434-6001.
2. ^ Heisenberg, W.; Euler, H. (1936). "Folgerungen aus der Diracschen Theorie des Positrons". Zeitschrift für Physik (in German). Springer Science and Business Media LLC. 98 (11–12): 714–732. arXiv:physics/0605038. doi:10.1007/bf01343663. ISSN 1434-6001.
3. ^ a b Schwinger, Julian (1951-06-01). "On Gauge Invariance and Vacuum Polarization". Physical Review. American Physical Society (APS). 82 (5): 664–679. doi:10.1103/physrev.82.664. ISSN 0031-899X.
4. ^ A.I. Nikishov (1970). "Pair Production by a Constant External Field". Journal of Experimental and Theoretical Physics. 30: 660.
5. ^ Brezin, E.; Itzykson, C. (1970-10-01). "Pair Production in Vacuum by an Alternating Field". Physical Review D. American Physical Society (APS). 2 (7): 1191–1199. doi:10.1103/physrevd.2.1191. ISSN 0556-2821.
6. ^ Ringwald, A. (2001). "Pair production from vacuum at the focus of an X-ray free electron laser". Physics Letters B. Elsevier BV. 510 (1–4): 107–116. doi:10.1016/s0370-2693(01)00496-8. ISSN 0370-2693.
7. ^ Popov, V. S. (2001). "Schwinger mechanism of electron–positron pair production by the field of optical and X-ray lasers in vacuum". Journal of Experimental and Theoretical Physics Letters. Pleiades Publishing Ltd. 74 (3): 133–138. doi:10.1134/1.1410216. ISSN 0021-3640.
8. ^ I. C. E. Turcu; et al. (2016). "High field physics and QED experiments at ELI-NP" (PDF). Romanian Reports in Physics. 68: S145-S231.
9. ^ Müller, C.; Voitkiv, A. B.; Grün, N. (2003-06-24). "Differential rates for multiphoton pair production by an ultrarelativistic nucleus colliding with an intense laser beam". Physical Review A. American Physical Society (APS). 67 (6): 063407. Bibcode:2003PhRvA..67f3407M. doi:10.1103/physreva.67.063407. ISSN 1050-2947.
10. ^ Acharya, B.; Alexandre, J.; Benes, P.; Bergmann, B.; Bertolucci, S.; et al. (2022-02-02). "Search for magnetic monopoles produced via the Schwinger mechanism". Nature. Springer Science and Business Media LLC. 602 (7895): 63–67. doi:10.1038/s41586-021-04298-1. ISSN 0028-0836.
11. ^ Vandecasteele, Niels; Barreiro, Amelia; Lazzeri, Michele; Bachtold, Adrian; Mauri, Francesco (2010-07-20). "Current-voltage characteristics of graphene devices: Interplay between Zener–Klein tunneling and defects". Physical Review B. 82 (4): 045416. arXiv:1003.2072. Bibcode:2010PhRvB..82d5416V. doi:10.1103/PhysRevB.82.045416. ISSN 1098-0121. S2CID 38911270.
12. ^ Berdyugin, Alexey I.; Xin, Na; Gao, Haoyang; Slizovskiy, Sergey; Dong, Zhiyu; Bhattacharjee, Shubhadeep; Kumaravadivel, P.; Xu, Shuigang; Ponomarenko, L. A.; Holwill, Matthew; Bandurin, D. A. (2022-01-28). "Out-of-equilibrium criticalities in graphene superlattices". Science. 375 (6579): 430–433. arXiv:2106.12609. Bibcode:2022Sci...375..430B. doi:10.1126/science.abi8627. ISSN 0036-8075. PMID 35084955. S2CID 235623859.
13. ^ "Schwinger effect seen in graphene". Physics World. 2022-03-25. Retrieved 2022-03-28.