nLab Schwinger effect

Idea

from Hebenstreit 14

What came to be called the Schwinger effect (Sauter 31, Heisenberg-Euler 36, Schwinger 51, Affleck-Manton 82, Affleck-Alvarez-Manton 82) is a non-perturbative effect of vacuum polarization expected in quantum electrodynamics, where a strong electric field causes electron/positron-pairs to appear out of the vacuum. The analogous effect in quantum chromodynamics would lead to deconfinement of quarks in a strong electric field.

While the effect is clearly predicted by established theory, it has not been observed in experiment yet, since the required electric field strengths are so large. But recent experiments get close to the required intensities (Dunne 09).

The field strengths

We consider $(\vec E, \vec B)$ a constant electromagnetic field on 4d Minkowski spacetime in a given Lorentz frame.

Write

(1)$$\begin{aligned} E & \coloneqq \sqrt{ \vec E \cdot \vec E } \\ B & \coloneqq \sqrt{ \vec B \cdot \vec B } \end{aligned}$$

for the norm of these field vectors. These, of course, depend on the choice of Lorentz frame. For the Schwinger effect the relevant Lorentz invariants are

(2)$$\begin{aligned} \mathcal{E} \;\coloneqq\; \sqrt{ \sqrt{ \left( \tfrac{1}{2} \big( \vec E \cdot \vec E - \vec B \cdot \vec B \big) \right)^2 + \big( \vec E \cdot \vec B \big)^2 } + \tfrac{1}{2} \big( \vec E \cdot \vec E - \vec B \cdot \vec B \big) } \\ \mathcal{B} \;\coloneqq\; \sqrt{ \sqrt{ \left( \tfrac{1}{2} \big( \vec B \cdot \vec B - \vec E \cdot \vec E \big) \right)^2 + \big( \vec E \cdot \vec B \big)^2 } + \tfrac{1}{2} \big( \vec B \cdot \vec B - \vec E \cdot \vec E \big) } \end{aligned}$$

(reviewed in Dunne 04, (1.6))

Noticing that if $\vec E \cdot \vec B \neq 0$ then there is a Lorentz transformation to $(\vec E', \vec B')$ such that the electric field is strictly parallel to the magentic field $\vec E \parallel \vec B$, these invariants are more explicity given as follows:

$$\begin{aligned} \mathcal{E} \; = \; \left\{ \array{ \sqrt{ \vec E \cdot \vec E - \vec B \cdot \vec B } & \vert & \mathrlap{ \vec E \cdot \vec B = 0 } \\ E' &\vert& \mathrlap{ \vec E \cdot \vec B \neq 0 \;\; \Rightarrow \underset{ { Lorentz \atop transformation } \atop { (\vec E, \vec B) \mapsto (\vec E', \vec B') } }{\exists} \vec E' \parallel \vec B' } } \right. \\ \mathcal{B} \; = \; \left\{ \array{ \sqrt{ \vec B \cdot \vec B - \vec E \cdot \vec E } & \vert & \mathrlap{ \vec E \cdot \vec B = 0 } \\ B' &\vert& \mathrlap{ \vec E \cdot \vec B \neq 0 \;\; \Rightarrow \underset{ { Lorentz \atop transformation } \atop { (\vec E, \vec B) \mapsto (\vec E', \vec B') } }{\exists} \vec E' \parallel \vec B' } } \right. \end{aligned}$$

The Schwinger effect

For general field strength at small coupling

Assuming

• parallel field components: $\vec E \cdot \vec B \neq 0$;

and

• weak coupling: $e$ small;

the rate of pair creation out of the vacuum of spinor particles of electric charge $e$ and mass $m$ is

(3)$$\Gamma_{ { e\,small } } \;=\; \frac { e^2 \mathcal{E} \mathcal{B} }{ 8 \pi^2 } \underoverset {n = 1} {\infty} {\sum} \frac{1}{n} \coth \left( \frac{\mathcal{B}}{\mathcal{E}} n \pi \right) \exp \left( - \frac{ m^2 \pi n }{ e \mathcal{E} } \right)$$

For electron/positron pair-creation in electromagnetism this is due to Nikishov 69, Bunkin-Tugov 70, reviewed in Dunne 04, (1.28). For quark/ani-quark pair creation in quantum chromodynamics the analogous formula is due to Suganuma-Tatsumin 91, Suganuma-Tatsumi 93, reviewed in Hidaka-Iritani-Suganuma 11, (2).

In the limit $\mathcal{B} \to 0$, using that

$$\underset{ \underset{ x \to 0 }{ \longrightarrow } }{\lim} \, x \coth(x) \;=\; x \,,$$

this reduces to

(4)$$\Gamma_{ { { \mathcal{B} = 0 } } \atop { e\,small } } \;\coloneqq\; \underset{ \underset{ B \to 0 }{ \longrightarrow } }{\lim} \, \Gamma \;=\; \frac { e^2 \mathcal{E}^2 }{ 8 \pi^3 } \underoverset {n = 1} {\infty} {\sum} \frac{1}{n^2} \exp \left( - \frac{ m^2 \pi n }{ e \mathcal{E} } \right)$$

This is due to Schwinger 51 (review in Dunne 04, (1.25))

For small field strength at small coupling

Assuming in addition

• weak fields:: $\mathcal{E}$ small compared to $m$

the expression (4) simplifies to

(5)$$\Gamma_{ { { \mathcal{B} = 0 } \atop { \mathcal{E}\, small } } \atop { e\,small } } \;\coloneqq\; \frac { e^2 \mathcal{E}^2 }{ 8 \pi^3 } \exp \left( - \frac{ m^2 \pi }{ e \mathcal{E} } \right)$$

For electron/positron pair-creation in electromagnetism this is originally due to Heisenberg-Euler 36, reviewed in Dunne 04, (1.10). For quark/ani-quark pair creation in quantum chromodynamics the analogous formula is due to Suganuma-Tatsumin 91, Suganuma-Tatsumi 93, reviewed in Hidaka-Iritani-Suganuma 11, (3).

For small field strength at strong coupling

For strong coupling $e$ the expression (5) get corrected to

$$\Gamma_{ { { \mathcal{E}\, small } \atop { \mathcal{B} = 0 } } } \;=\; \frac { e^2 \mathcal{E}^2 }{ 8 \pi^3 } \exp \left( - \frac{ m^2 \pi }{ e \mathcal{E} } + \tfrac{1}{4}e^2 \right)$$

This was argued in Affleck-Alvarez-Manton 82.

Properties

Schwinger limit – Critical electric field strength

From (3) one deduces a critical electric field strength $\mathcal{E}_{crit}$ which sets the scale beyond which the vacuum polarization due to the Schwinger effect counteracts the ambient electric field and/or leads to vacuum decay.

As a Lorentz invariant (2) this Schwinger limit for the electric field strength is:

(6)$$\mathcal{E}_{crit} \;\coloneqq\; \frac{ m^2 c^3 }{ e \hbar }$$

(Dunne 04, (1.3), Martin 07, (40))

Here

• $e$ is the charge of the charged particles,

• $m$ is the mass of the charged particles,

• $c$ is the speed of light,

• $\hbar$ is Planck's constant.

This is such that the corresponding Lorentz force

$$F_{crit} \; \coloneqq \; e \, \mathcal{E}_{crit}$$

acting over the Compton wavelength $\lambda_{Comp} \;\coloneqq\; \frac{\hbar }{m c}$ equals the rest energy $m c^2$ of the given charged particle:

$$\begin{aligned} & F_{crit} \lambda_{Comp} \; = \; m c^2 \\ \Leftrightarrow \;\;\; & \mathcal{E}_{crit} \; = \; \frac{ m c^2 }{ e \lambda_{Comp} } \end{aligned}$$

Expressing (6) in terms of the corresponding critical value $E_{crit}$ of the actual electric field strength (1) in the given Lorentz frame yields (Hashimoto-Oka-Sonoda 14b, (2.17), check):

$$\mathcal{E}(E_{crit}, B) \;=\; \mathcal{E}_{crit} \phantom{AA} \Leftrightarrow \phantom{AA} E_{crit} \;=\; \mathcal{E}_{crit} \sqrt{ \frac{ \mathcal{E}_{crit}^2 + B^2 } { \mathcal{E}_{crit}^2 + B_{\parallel}^2 } }$$

This happens to coincide with the critical field strength of the DBI-action, see there.

In holographic QCD

It has been argued that in terms of intersecting D-brane models the Schwinger effect is what is reflected by the non-linearities in the DBI-action on probe branes in AdS/CFT (Semenoff-Zarembo 11) and on flavor branes in holographic QCD (Hashimoto-Oka 13, Hashimoto-Oka-Sonoda 14a, Hashimoto-Oka-Sonoda 14b). This is now referred to as the holographic Schwinger effect.

References

In quantum electrodynamics

Discussion of the Schwinger effect in quantum electrodynamics:

The original theoretical prediction:

• F. Sauter, Über das Verhalten eines Elektrons im homogenen elektrischen Feld nach der relativistischen Theorie Diracs, Z. Physik 69, 742–764 (1931) (doi:10.1007/BF01339461)

• Werner Heisenberg, Hans Euler, Folgerungen aus der Diracschen Theorie des Positrons, Z. Physik 98, 714–732 (1936) (doi:10.1007/BF01343663)

• Julian Schwinger, On Gauge Invariance and Vacuum Polarization, Phys. Rev. 82, 664 (1951) (doi:10.1103/PhysRev.82.664)

• A. I. Nikishov, Pair production by a constant external field, Zh. Eksp. Teor. Fiz. 57 (1969) 1210-1216 (spire:59436)

• F. V. Bunkin, I. I. Tugov, Possibility of creating electron-positron pairs in avacuum by the focusing of laser radiation, Sov. Phys. Dokl.14 (1970), 678

• Ian Affleck, Nicholas Manton, Monopole pair production in a magnetic field, Nuclear Physics B Volume 194, Issue 1, (1982) Pages 38-64 (doi:10.1016/0550-3213(82)90511-9

• Ian Affleck, Orlando Alvarez, Nicholas Manton, Pair production at strong coupling in weak external fields, Nuclear Physics B Volume 197, Issue 3 (1982) Pages 509-519 (doi:10.1016/0550-3213(82)90455-2)

Discussion via worldline formalism:

• Gerald Dunne, Christian Schubert, Worldline Instantons and Pair Production in Inhomogeneous Fields, Phys. Rev. D72 (2005) 105004 (arXiv:hep-th/0507174)

• Christian Schubert, The worldline formalism in strong-field QED [arXiv:2304.07404]

More theoretical background on QED in background fields:

• Uwe Hernandez Acosta, Burkhard Kämpfer, Strong-field QED in Furry-picture momentum-space formulation: Ward identities and Feynman diagrams [arXiv:2303.12941]

Review:

• Walter Dittrich, Holger Gies, Probing the Quantum Vacuum – Perturbative Effective Action Approach in Quantum Electrodynamics and its Application, Springer Tracts in Modern Physics, Vol. 166 (ISBN 978-3-540-45585-1)

• Gerald Dunne, Heisenberg-Euler Effective Lagrangians: Basics and Extensions, in: Misha Shifman, Arkady Vainshtein, John Wheater (eds.), From Fields to Strings – Circumnavigating Theoretical Physics, pp. 445-522, World Scientific 2005 (arXiv:hep-th/0406216, doi:10.1142/9789812775344_0014)

• Gerald Dunne, The Heisenberg-Euler Effective Action: 75 years on, Int. J. Mod. Phys. A27 (2012) 1260004 (arXiv:1202.1557)

• Francois Gelis, Naoto Tanji, Schwinger mechanism revisited, Progress in Particle and Nuclear Physics Volume 87, March 2016, Pages 1-49 (arXiv:1510.05451)

See also

• Wikipedia, Schwinger effect

• Wikipedia, Schwinger limit

Discussion of experiments that could/should see the Schwinger effect:

• Gerald Dunne, New Strong-Field QED Effects at ELI: Nonperturbative Vacuum Pair Production, Eur. Phys. J. D55:327-340, 2009 (arXiv:0812.3163)

• Hidetoshi Taya, Mutual assistance between the Schwinger mechanism and the dynamical Casimir effect (arXiv:2003.12061)

• Florian Hebenstreit, A space-time resolved view of the Schwinger effect, Frontiers of intense laser physics – KITP 2014 (pdf)

• A. Fedotov et al., Section 8 of: Advances in QED with intense background fields [arXiv:2203.00019]

• M. Pentia, C. R. Badita, D. Dumitriu, A. R. Ionescu, H. Petrascu, Nonperturbative QED Processes at ELI-NP [arXiv:2307.09315]

and in relation to magnetic monopoles:

• B. Acharya et al., First experimental search for production of magnetic monopoles via the Schwinger mechanism (arXiv:2106.11933)

Discussion in inflationary cosmology:

• Jerome Martin, Inflationary Perturbations: the Cosmological Schwinger Effect, Lect. Notes Phys. 738:193-241, 2008 (arXiv:0704.3540, doi:10.1007/978-3-540-74353-8_6)

See also:

• Shintaro Takayoshi, Jianda Wu, Takashi Oka, Twisted Schwinger Effect: Pair Creation in Rotating Fields (arXiv:2005.01755)

• Prasant Samantray, Suprit Singh, Schwinger Effect in Compact Space (arXiv:2010.13453)

In quantum chromodynamics

Discussion of the Schwinger effect in quantum chromodynamics:

• Asim Yildiz, Paul H. Cox, Vacuum Behavior in Quantum Chromodynamics, Phys. Rev. D21 (1980) 1095 (spire:7860)

• M. Claudson, Asim Yildiz, Paul H. Cox, Vacuum behavior in quantum chromodynamics. II, Phys. Rev. D 22, 2022 (1980) (doi:10.1103/PhysRevD.22.2022)

• Paul H. Cox, Asim Yildiz, $q \bar q$ pair creation: A field-theory approach, Phys. Rev. D 32, 819 (1985) (doi:10.1103/PhysRevD.32.819)

• Hideo Suganuma, Toshitaka Tatsumim, On the behavior of symmetry and phase transitions in a strong electromagnetic field, Annals of Physics Volume 208, Issue 2, June 1991, Pages 470-508 (doi:10.1016/0003-4916(91)90304-Q)

• Hideo Suganuma, Toshitaka Tatsumi, Chiral Symmetry and Quark-Antiquark Pair Creation in a Strong Color-Electromagnetic Field, Progress of Theoretical Physics, Volume 90, Issue 2, August 1993, Pages 379–404 (spire:318092, doi:10.1143/ptp/90.2.379)

• Naoto Tanji, Dynamical view of pair creation in uniform electric and magnetic fields, Annals Phys. 324:1691-1736, 2009 (arXiv:0810.4429)

• Yoshimasa Hidaka, Takumi Iritani, Hideo Suganuma, Fast Vacuum Decay into Quark Pairs in Strong Color Electric and Magnetic Fields, AIP Conference Proceedings 1388, 516 (2011) (arXiv:1103.3097, doi:10.1063/1.3647442)

• Sho Ozaki, Takashi Arai, Koichi Hattori, Kazunori Itakura, Euler-Heisenberg-Weiss action for QCD+QED, Phys. Rev. D 92, 016002 (2015) (arXiv:1504.07532)

• Koichi Hattori, Kazunori Itakura, Sho Ozaki, Note on all-order Landau-level structures of the Heisenberg-Euler effective actions for QED and QCD (arXiv:2001.06131)

• Patrick Copinger, Pablo Morales, Schwinger Pair Production in $SL(2,\mathbb{C})$ Topologically Non-Trivial Fields via Non-Abelian Worldline Instantons (arXiv:2011.12526)

In quantum hadrodynamics:

• William R. Tavares, Sidney S. Avancini, Schwinger mechanism in the $SU(3)$ Nambu–Jona-Lasinio model with an electric field, Phys. Rev. D 97, 094001 (2018) (arXiv:1801.10566)

Holographic Schwinger effect

Interpretation in holographic QCD of the Schwinger effect of vacuum polarization as exhibited by the DBI-action on flavor branes:

Precursor computation in open string theory:

• Constantin Bachas, Massimo Porrati, Pair Creation of Open Strings in an Electric Field, Phys. Lett. B296 (1992) 77-84 (arXiv:hep-th/9209032)

Relation to the DBI-action of a probe brane in AdS/CFT:

• Gordon Semenoff, Konstantin Zarembo, Holographic Schwinger Effect, Phys. Rev. Lett. 107, 171601 (2011) (arXiv:1109.2920, doi:10.1103/PhysRevLett.107.171601)

• S. Bolognesi, F. Kiefer, E. Rabinovici, Comments on Critical Electric and Magnetic Fields from Holography, J. High Energ. Phys. 2013, 174 (2013) (arXiv:1210.4170)

• Yoshiki Sato, Kentaroh Yoshida, Holographic description of the Schwinger effect in electric and magnetic fields, J. High Energ. Phys. 2013, 111 (2013) (arXiv:1303.0112)

• Yoshiki Sato, Kentaroh Yoshida, Holographic Schwinger effect in confining phase, JHEP 09 (2013) 134 (arXiv:1306.5512

• Yoshiki Sato, Kentaroh Yoshida, Universal aspects of holographic Schwinger effect in general backgrounds, JHEP 12 (2013) 051 (arXiv:1309.4629)

• Daisuke Kawai, Yoshiki Sato, Kentaroh Yoshida, The Schwinger pair production rate in confining theories via holography, Phys. Rev. D 89, 101901 (2014) (arXiv:1312.4341)

• Yue Ding, Zi-qiang Zhang, Holographic Schwinger effect in a soft wall AdS/QCD model (arXiv:2009.06179)

Relation to DBI-action of flavor branes in holographic QCD:

• Koji Hashimoto, Takashi Oka, Vacuum Instability in Electric Fields via AdS/CFT: Euler-Heisenberg Lagrangian and Planckian Thermalization, JHEP 10 (2013) 116 (arXiv:1307.7423)

• Koji Hashimoto, Takashi Oka, Akihiko Sonoda, Magnetic instability in AdS/CFT: Schwinger effect and Euler-Heisenberg Lagrangian of Supersymmetric QCD, J. High Energ. Phys. 2014, 85 (2014) (arXiv:1403.6336)

• Koji Hashimoto, Shunichiro Kinoshita, Keiju Murata, Takashi Oka, Electric Field Quench in AdS/CFT, J. High Energ. Phys. 2014 (arXiv:1407.0798)

• Koji Hashimoto, Takashi Oka, Akihiko Sonoda, Electromagnetic instability in holographic QCD, J. High Energ. Phys. 2015, 1 (2015) (arXiv:1412.4254)

See also:

• Xing Wu, Notes on holographic Schwinger effect, J. High Energ. Phys. 2015, 44 (2015) (arXiv:1507.03208, doi:10.1007/JHEP09(2015)044)

• Kazuo Ghoroku, Masafumi Ishihara, Holographic Schwinger Effect and Chiral condensate in SYM Theory, J. High Energ. Phys. 2016, 11 (2016) (doi:10.1007/JHEP09(2016)011)

• Le Zhang, De-Fu Hou, Jian Li, Holographic Schwinger effect with chemical potential at finite temperature, Eur. Phys. J. A54 (2018) no.6, 94 (spire:1677949, doi:10.1140/epja/i2018-12524-4)

• Wenhe Cai, Kang-le Li, Si-wen Li, Electromagnetic instability and Schwinger effect in the Witten-Sakai-Sugimoto model with D0-D4 background, Eur. Phys. J. C 79, 904 (2019) (doi:10.1140/epjc/s10052-019-7404-1)

• Zhou-Run Zhu, De-fu Hou, Xun Chen, Potential analysis of holographic Schwinger effect in the magnetized background (arXiv:1912.05806)

• Zi-qiang Zhang, Xiangrong Zhu, De-fu Hou, Effect of gluon condensate on holographic Schwinger effect, Phys. Rev. D 101, 026017 (2020) (arXiv:2001.02321)

Review:

• Daisuke Kawai, Yoshiki Sato, Kentaroh Yoshida, A holographic description of the Schwinger effect in a confining gauge theory, International Journal of Modern Physics A Vol. 30, No. 11, 1530026 (2015) (arXiv:1504.00459)

• Akihiko Sonoda, Electromagnetic instability in AdS/CFT, 2016 (spire:1633963, pdf)