nLab Schwinger limit

Idea

In quantum electrodynamics and quantum chromodynamics the Schwinger limit is a maximal scale

$$\;\approx\; \frac{ m^2 c^3 }{ e \hbar }$$

of the field strength of an electric background field beyond which the vacuum polarization caused by pair creation electrons/positrons or quarks/anti-quarks (of charge $e$ and mass $m$) out of the vacuum, via the Schwinger effect, becomes sizeable, leading to non-linearities and/or vacuum decay.

From the perspective of geometric engineering of QCD in intersecting D-brane models (holographic QCD) the Schwinger limit corresponds to the limiting field strength in the DBI-action of the Chan-Paton gauge field on D-branes (“holographic Schwinger effect”).

The Schwinger effect and its resulting Schwinger limit are securely prediuted by established quantum field theory, but have not been observed in experiment yet. However, recent experiments are getting very close and upcoming experiments might see the effect.

Details

From the rate

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

(here) of particle pair creation via the Schwinger effect 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 (here) this Schwinger limit for the electric field strength is:

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

(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:

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

Properties

Relation to DBI-limit

Expressing (2) in terms of the corresponding critical value $E_{crit}$ of the actual electric field strength (here) 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.

Related concepts

fundamental scales (fundamental/natural physical units)

• speed of light$\,$ $c$

• Planck's constant$\,$ $\hbar$

• gravitational constant$\,$ $G_N = \kappa^2/8\pi$

• Planck scale

  • Planck length$\,$ $\ell_p = \sqrt{ \hbar G / c^3 }$

  • Planck mass$\,$ $m_p = \sqrt{\hbar c / G}$

• depending on a given mass $m$

  • Compton wavelength$\,$ $\lambda_m = \hbar / m c$

  • Schwarzschild radius$\,$ $2 m G / c^2$

  • depending also on a given charge $e$

    • Schwinger limit$\,$ $E_{crit} = m^2 c^3 / e \hbar$

• GUT scale

• string scale

  • string tension$\,$ $T = 1/(2\pi \alpha^\prime)$

  • string length scale$\,$ $\ell_s = \sqrt{\alpha'}$

  • string coupling constant$\,$ $g_s = e^\lambda$

References

General

Review:

• 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)

• Jerome Martin, around (40) in: 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)

For more see the references at Schwinger effect.

See also

• Wikipedia, Schwinger limit

Experimental realization

Discussion of experiments that could/should see physics at the Schwinger limit:

• 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)

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)