nLab Schwinger limit

Contents

Context

Fields and quanta

fields and particles in particle physics

and in the standard model of particle physics:

force field gauge bosons

photon - electromagnetic field (abelian Yang-Mills field)
W, Z, B-boson - electroweak field (Yang-Mills field)
gluon - strong nuclear force (Yang-Mills field)
graviton - field of gravity
infraparticle

scalar bosons

Higgs boson, inflaton (scalar field)

matter field fermions (spinors, Dirac fields)

flavors of fundamental fermions in the standard model of particle physics:
generation of fermions	1st generation	2nd generation	3d generation
quarks ( $q$ )
up-type	up quark ( $u$ )	charm quark ( $c$ )	top quark ( $t$ )
down-type	down quark ( $d$ )	strange quark ( $s$ )	bottom quark ( $b$ )
leptons
charged	electron	muon	tauon
neutral	electron neutrino	muon neutrino	tau neutrino

bound states:
mesons	light 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 ( $b \bar b$ )
baryons	nucleons: proton $(u u d)$ neutron $(u d d)$

(also: antiparticles)

effective particles

Goldstone bosons

hadrons (bound states of the above quarks)

meson
- scalar meson
  - σ-meson
  - pion
  - kaon
  - D-meson
  - B-meson
- vector meson
  - ω-meson
  - ρ-meson
  - f1-meson
  - a1-meson
  - b1-meson
  - h1-meson
  - K*-meson
- tensor meson
- quarkonium
  - charmonium
  - ϒ-meson
exotic meson
- XYZ meson
- tetraquark
baryon
- nucleon
  - proton
  - neutron
  - chemical element
    - carbon
    - nitrogen
- Lambda baryon
- pentaquark

solitons

in grand unified theory

minimally extended supersymmetric standard model

superpartners

bosinos:

gravitino - field of supergravity (Rarita-Schwinger field)
gaugino - super Yang-Mills field
gluino

sfermions:

squark

dark matter candidates

WIMP
axion

Exotica

auxiliary fields

Relation to DBI-limit

Related concepts
References

General
Experimental realization
Holographic Schwinger effect

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.

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)