# nLab Schwinger effect

### Context

#### Fields and quanta

fields and particles in particle physics

and in the standard model of particle physics:

force field gauge bosons

scalar bosons

flavors of fundamental fermions in the
standard model of particle physics:
generation of fermions1st generation2nd generation3d generation
quarks ($q$)
up-typeup quark ($u$)charm quark ($c$)top quark ($t$)
down-typedown quark ($d$)strange quark ($s$)bottom quark ($b$)
leptons
chargedelectronmuontauon
neutralelectron neutrinomuon neutrinotau neutrino
bound states:
mesonslight mesons:
pion ($u d$)
ρ-meson ($u d$)
ω-meson ($u d$)
f1-meson
a1-meson
strange-mesons:
ϕ-meson ($s \bar s$),
kaon, K*-meson ($u s$, $d s$)
eta-meson ($u u + d d + s s$)

charmed heavy mesons:
D-meson ($u c$, $d c$, $s c$)
J/ψ-meson ($c \bar c$)
bottom heavy mesons:
B-meson ($q b$)
ϒ-meson ($b \bar b$)
baryonsnucleons:
proton $(u u d)$
neutron $(u d d)$

(also: antiparticles)

effective particles

hadrons (bound states of the above quarks)

solitons

in grand unified theory

minimally extended supersymmetric standard model

superpartners

bosinos:

dark matter candidates

Exotica

auxiliary fields

# Contents

## Idea

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 of a sequence $\mathcal{B} \to 0$, 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

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

Here

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:

Discussion via worldline formalism:

Review:

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)

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:

• 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)

• 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:

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:

• 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)