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Vacuum Instability in Electric Fields via AdS/CFT: Euler-Heisenberg Lagrangian and Planckian Thermalization

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Koji Hashimoto, Takashi Oka

High Energy Physics - Theory · Submitted 28 Jul 2013 · 74 references

Pith's verdict: The static Euler-Heisenberg results are the real contribution; the 'Planckian thermalization' claim is mostly horizon kinematics dressed as dynamics.

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CitationKoji Hashimoto, Takashi Oka. "Vacuum Instability in Electric Fields via AdS/CFT: Euler-Heisenberg Lagrangian and Planckian Thermalization." arXiv:1307.7423 (2013). https://pith.science/paper/4I6W7CQS
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The paper

The paper claims

A strongly coupled vacuum in a strong electric field decays, and the decay rate matches the textbook Schwinger formula once the electron mass is replaced by the quark confining force.

The reading

The paper asks a concrete question: what happens to the vacuum of a strongly coupled, confining gauge theory when a strong electric field is suddenly switched on? The authors work in the D3–D7 holographic model of $N=2$ supersymmetric QCD with one flavor at large $N_c$ and strong 't Hooft coupling, and read the Euler–Heisenberg effective Lagrangian directly off the Dirac–Born–Infeld action of a probe D7-brane in $AdS_5$. The action turns complex exactly when the electric field exceeds the force that confines a quark to an antiquark, so the threshold for vacuum breakdown is a confining-force threshold rather than a free-particle mass threshold. In the massless case the imaginary part reproduces the massless Schwinger result for one Dirac fermion plus two complex scalars, and in the massive case the large-field expansion matches the QED dilogarithm expansion term by term with the electron mass replaced by the critical field. The authors then solve the time-dependent problem and find the system relaxes to a current-carrying steady state whose induced apparent horizon defines an effective temperature, with a thermalization time set by $1/T_eff$ and scaling as $E^{-1/2}$.

What carries the argument

The load-bearing object is the Dirac–Born–Infeld action of a probe D7-brane in the $AdS_5 times S^5$ (or $AdS_5$ black hole) geometry, evaluated on a constant electric field configuration with zero induced current. Because the action carries a square root, it becomes imaginary wherever the argument of the root goes negative; the boundary of that region, $z = z_*$ in the massless case and $z = z_0$ in the massive case, is the entire content of the decay rate. The complementary machinery is the reality condition on the same action: a stable steady state with nonzero current exists exactly when the numerator and denominator of the root change sign at the same radial position, which fixes the current–voltage curve. For the time-dependent part, the argument is carried by the effective metric that gauge fluctuations on the brane feel, and by the apparent horizon defined as the surface whose volume is stationary under null propagation, with the effective Hawking temperature read from its position.

Extended reading notes

Core claim

The central claim is that the full nonlinear electromagnetic response of a strongly coupled confining theory — including its vacuum decay rate — is captured by a classical D-brane action, and that this response has an unexpectedly close numerical relationship to one-loop QED. Concretely: the imaginary part of the D7-brane effective Lagrangian vanishes below a critical field $E_{cr} = (2^{1/2} pi / lambda^{1/2}) m_q^2$, which coincides with the quark–antiquark confining force computed independently from a small Wilson loop in the same model. Above threshold, the massless-case imaginary part is $Im L = N_c E^2/(32 pi)$, exactly the massless limit of the Schwinger result for the $N=2$ hypermultiplet content, and the massive case reproduces the leading and subleading terms of the QED expansion under $E_{cr} leftrightarrow m_e^2$. On top of the static result, the paper reports a dynamical one: after an abrupt switch-on of the field the current overshoots, relaxes to the steady value fixed by the reality condition on the brane action, and an apparent horizon forms on the brane whose associated temperature governs the relaxation time, $tau_{th} sim a hbar/(k_B T_eff^{infty}) sim E^{-1/2}$, numerically giving about $1$ fm/c for a RHIC-scale field.

Load-bearing premise

The thermalization claim rests on the probe approximation: the flavor brane sits in a fixed background with the gluon sector at zero temperature, and the temperature read off the brane's apparent horizon is assumed to be the scale that actually governs how fast the quark degrees of freedom relax.

Editorial extensions

If this is right

  • The threshold for electric breakdown of a confining vacuum is the confining force itself, not a constituent mass, giving a sharp criterion that any confining theory should satisfy.
  • The imaginary part of the effective Lagrangian for a strongly coupled theory can be computed from a classical bulk action rather than a loop diagram, because removing the color branes caps the worldsheet cylinder into a disk.
  • For fields far above threshold the decay rate is a power law in $E$ rather than exponentially suppressed, so a strongly coupled vacuum is far more fragile at large field than QED intuition suggests.
  • The relaxation time of the nonequilibrium current-carrying state is bounded below by $hbar/(k_B T_eff)$, which for RHIC-scale fields comes out near $1$ fm/c, in the range hydrodynamics requires.
  • The same framework yields the full nonlinear effective Lagrangian below threshold as an infinite series in $E^2/E_{cr}^2$, i.e. a tower of nonlinear optical response coefficients for the strongly coupled vacuum.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The agreement with the QED dilogarithm expansion at subleading order is striking enough that it invites a structural explanation: the DBI square root may be doing the work of the proper-time integral in the Schwinger calculation, so the coincidence could reflect a shared worldline geometry rather than an accident of the $N=2$ matter content.
  • If the effective temperature really is the noise temperature of the current-carrying steady state, then the same quantity should appear in meson spectral functions and current noise on the brane, which would give an independent, non-thermalization-based measurement of $T_eff$.
  • The $E^{-1/2}$ scaling is dimensionally forced once the only scale is the field, so a sharper test is the prefactor: whether the constant $a$ is universal across confining holographic models or depends on the brane embedding.
  • The paper's own estimate of about $1$ fm/c uses a small 't Hooft coupling, $lambda = 10$, which sits outside the regime where the gravity description is controlled; the number should be read as an order-of-magnitude consistency check rather than a prediction.
Original abstract

We analyze vacuum instability of strongly coupled gauge theories in a constant electric field using AdS/CFT correspondence. The model is the N=2 1-flavor supersymmetric large N_c QCD in the strong 't Hooft coupling limit. We calculate the Euler-Heisenberg effective Lagrangian L(E), which encodes the nonlinear response and the quantum decay rate of the vacuum in a background electric field E, from the complex D-brane action in AdS/CFT. We find that the decay rate given by Im L(E) becomes nonzero above a critical electric field set by the confining force between quarks. A large-E expansion of Im L(E) is found to coincide with that of the Schwinger effects in QED, replacing its electron mass by the confining force. Then, the time-dependent response of the system in a strong electric field is solved non-perturbatively, and we observe a universal thermalization at a shortest timescale "Planckian thermalization time" t ~ 1/T ~ E^{-1/2}. Here, T is an effective temperature which quarks feel in the nonequilibrium state with nonzero electric current, calculated in AdS/CFT as a Hawking temperature. Stronger electric fields accelerate the thermalization, and for a realistic value of the electric field in RHIC experiment, we obtain t ~ 1 [fm/c], which is consistent with the believed value.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,454 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.08 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.93 · compares with prior results 0.89 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We analyze vacuum instability of strongly coupled gauge theories in a constant electric field using AdS/CFT correspondence. The model is the N=2 1-flavor supersymmetric large N_c QCD in the strong 't Hooft coupling limit. We calculate the Euler-Heisenberg effective Lagrangian L(E), which encodes the nonlinear response and the quantum decay rate of the vacuum in a background electric field E, from the complex D-brane action in AdS/CFT. We find that the decay rate given by Im L(E) becomes nonzero above a critical electric field set by the confining force between quarks. A large-E expansion of Im L(E) is found to coincide with that of the Schwinger effects in QED, replacing its electron mass by the confining force. Then, the time-dependent response of the system in a strong electric field is solved non-perturbatively, and we observe a universal thermalization at a shortest timescale "Planckian thermalization time" t ~ 1/T ~ E^{-1/2}. Here, T is an effective temperature which quarks feel in the nonequilibrium state with nonzero electric current, calculated in AdS/CFT as a Hawking temperature. Stronger electric fields accelerate the thermalization, and for a realistic value of the electric field in RHIC experiment, we obtain t ~ 1 [fm/c], which is consistent with the believed value.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We analyze vacuum instability of strongly coupled gauge theories in a constant electric field using AdS/CFT correspondence. contribution 0.96 0.92 0.06 asserted 0.90
s1 The model is the N=2 1-flavor supersymmetric large N_c QCD in the strong 't Hooft coupling limit. method 0.94 0.20 0.02 asserted 1.00
s2 We calculate the Euler-Heisenberg effective Lagrangian L(E), which encodes the nonlinear response and the quantum decay rate of the vacuum in a background electric field E, from the complex D-brane action in AdS/CFT. method 0.57 0.95 0.10 asserted 1.00
s3 We find that the decay rate given by Im L(E) becomes nonzero above a critical electric field set by the confining force between quarks. result 1.00 0.95 0.29 asserted 1.00
s4 A large-E expansion of Im L(E) is found to coincide with that of the Schwinger effects in QED, replacing its electron mass by the confining force. result 0.96 0.93 0.13 asserted 1.00
s5 Then, the time-dependent response of the system in a strong electric field is solved non-perturbatively, and we observe a universal thermalization at a shortest timescale "Planckian thermalization time" t ~ 1/T ~ E^{-1/2}. result 0.94 0.95 0.39 asserted 1.00
s6 Here, T is an effective temperature which quarks feel in the nonequilibrium state with nonzero electric current, calculated in AdS/CFT as a Hawking temperature. interpretation 0.58 0.86 0.08 asserted 0.99
s7 Stronger electric fields accelerate the thermalization, and for a realistic value of the electric field in RHIC experiment, we obtain t ~ 1 [fm/c], which is consistent with the believed value. result 0.92 0.90 0.18 hedged possibility 0.61

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.96.

What was examined, and what was not

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  • references and citing sentences
  • relations to other papers
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. The ground state decay rate of the 1-dimensional Mott insulator obtained numeri￾cally (dots) and analytically (solid lines) in the Hubbard model for several values of its interaction strength U [5]. nonequilibrium phenomena in strongly correlated electrons in condensed matter physics [6, 7, 5, 8, 9, 10, 11, 12, 13]. The physics of strong electric field breaking the vacuum is nothing but a famous no￾tion known as dielectric breakdown in material sci￾ence. The Mott insulator, which hosts high Tc su￾perconductivity when doped (=carriers are added) in cuprates, is a state in which the…
Figure 2. A schematic picture of the D7-brane in the AdS5×S 5 geometry. The flat surface in the figure is the D7-brane on which we have a constant electric field. Left: When the field exceeds a critical value, the initial zero current solution has an effective Lagrangian with an imaginary part. This comes from a region in the D7 brane worldvolume near the D3 brane (denoted by the blue sphere). Middle: When the stable solution with non-zero current j = j0 is realized, the D7-brane effective action becomes real. At the same time, an apparent horizon is formed. Left: Schematic stability diagram,…
Figure 3. Configuration of the D3-D7 sys￾tem in 10 spacetime dimensions. analyze the full E-dependence of the cur￾rent. Interestingly, in the steady state, the relation between the current j and the applied electric field E is determined by requiring that the D-brane action is real [47].4 Our paper concerns the imaginary part of the D-brane action (which is related to the instability) on the other hand. In this section, after presenting some basics of the gravity setup, we explain the reality condition which is used to realize the stable state. In string theory, the simplest and best-understoo
Figure 4. IV -characteristics of our massless supersymmetric QCD, given by eqn.(2.15). The left (right) corresponds to a doped d = 1 (non-doped d = 0) system. The dashed lines are for zero temperature zH = ∞, while the solid lines are at a finite temperature zH = 1.5. We have set R = 1 and 2πα′ = 1. Eliminating zp, we obtain the stable current j = j0 written in terms of the electric field E, the charge density d and the temperature T as j0 = s d 2 + ((2πα′E) 2 + R4/z4 H ) 3/2 R2 2πα′E p (2πα′E) 2 + R4/z4 H . (2.15) This is the I-V curve of our massless supersymmetric QCD which is shown in
Figure 5. Left: The resistivity (2.18) at a finite charge density as a function of T. We have taken d = 1. Right: The plot of dρ/dT, the dark region gives an insulator-like phase dρ/dT < 0, while the light region is a metallic phase dρ/dT > 0. At low temperature for fixed density d, we have Fermi-liquid like behavior ρ ∼ T 2 . two viewpoints. First, in holographic setups obtaining a Fermi-liquid is non-trivial since the system is in the strongly interacting limit and the fermion propagators usually show abnormal behaviors. Second, in most materials, the high temperature side of the phase diagr
Figure 6. The imaginary part of the lagrangian of our massless supersymmetric QCD, given by (3.5). The left (right) corresponds to a doped d = 1 (non-doped d = 0) system. The dashed lines are for zero temperature zH = ∞, while the solid lines are at a finite temperature zH = 1. We have set R = 1 and 2πα′ = 1. For no doping (d = 0), there is no temperature dependence. In terms of z∗, the effective Lagrangian (3.1) simplifies to the following expres￾sion, L = −2π 2µ7R 6 q R4 + (2πα′ ) 2E2z 4 H Z zH 0 dz 1 z 5 s z 4 ∗ − z 4 (1 + d 2z 6/R6)(z 4 H − z 4) . (3.3) Defining y ≡ z/zH, we obtain…
Figure 7. Above: Schematic confining potential of the quark and antiquark in the presence of an electric field. At E = Ecr, the potential becomes flat in one direction inducing the liberation of the quark-antiquark pair. This leads an emergent massless continuum states. Below: Coleman’s half-asymptotic state realized at E = Ecr in the massive Schwinger model. Quarks and antiquarks can freely move in the direction of the electric field as long as the charges are in a plus-minus-plus- · · · order. Now, as before, the factor inside the square root can be negative, which gives us the imaginary…
Figure 8. The imaginary part of the lagrangian of our massive supersymmetric QCD, given by (4.8). is shown. A smooth increase of the imag￾inary part, starting at E = Ecr, is ob￾served. We want to compare our result with the QED results, i.e., (1.6) and (1.7). In the massless QCD case studied in the previous section, we saw complete agree￾ment of the imaginary part of the effec￾tive Lagrangian. In the present massive case, since we have another mass scale, which is the quark mass, it is appropri￾ate to consider the large E expansion. The reason is that at small E, because of the confinement t
Figure 9. (a) The gauge field Ax(z, t) for E = 0.001 in the undoped case (d = 0) calculated with parameters T = 0, ω = 0.7, R = 1. (b) The induced horizon (red solid line). In the long time limit, it converges to the steady state value z = zp denoted by the green dotted line. The red (solid and dashed) lines are determined from the condition δVAH = 0 given in (5.11). The vector field plots the direction of the null vector (vz, vt)/ p v 2 z + v 2 t . 5.2 Planckian thermalization accelerated by electric fields We are ready to analyze explicitly the time-dependence of the system as a response to…
Figure 10. (a) The electric field E(t). (b) Induced current j(t). It converges to the steady state value j = j0 (2.15). (c) The effective Hawking temperature (5.7) plotted against the boundary time. The “thermalization time” is defined as the boundary time of the horizon formation. In the long time limit, Teff converges to the steady state value T ∞ eff . Same parameters as
Figure 11. Thermalization time τth obtained for several processes with different field strengths and radius R = 1, 5, 10. The quench speed is fixed to ω = 4. In (a), it is plotted against the field strength, and in (b) against the Planckian time scale zp. An overlap takes place in (b), where the dashed line denotes the relation (5.15). of the Planckian dissipation where the source of dissipation is provided by the in￾duced Gaussian noise (= fluctuation that is characterized by the effective Hawking temperature) as explained above. In the present system, the field induced Planckian time can be…
Figure 12. Schematic plot of the vacuum de￾cay rate (production rate) obtained from the open string calculation [26] and from the D￾brane calculation (4.5). The true decay rate should be a function which smoothly connects the two lines. tion process in a similar manner for our case, we expect the same result as the holographic Schwinger effect in Ref. [26], with mq = mW , because the separated D3-brane in Ref. [26] is just replaced by our flavor D7-brane. Thus, the expres￾sion (6.1) is also valid in our N = 2 supersymmetric QCD. If we plot the production rate in the string and D￾brane sectors

Figures reproduced from arXiv:1307.7423 by the authors.

References
How the literature uses it

39 arXiv papers cite this in 101 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 64 sentences without a confident label

Uses its method or tool 5 papers
Compares with it 5 papers
Cites it as background 15 papers
  • By adding a probe D8-brane, in [ref] and [ref] the authors studied the imaginary part of the Euler-Heisenberg effective Lagrangian, the rate of pair creation, the critical electric field and the effect of the parallel and perpendicular components of the magnetic field on the rate of pair creation in the background of Sakai-Sugimoto and the deformed Sakai-Sugimoto models.

    Schwinger Effect and Entanglement Entropy in Confining Geometries · Introduction · 0.83

  • Apparent horizon formation is an interestingly unique procedure that belongs at the heart of AdS/CFT interpretation of thermalization [ref] .

    Double Relaxation via AdS/CFT · Review of Holographic Electric Field Quench · 0.84

  • Let us first review the physical picture of the instability in the presence of electric field [ref] .

    Equilibrium Instability of Chiral Mesons in External Electromagnetic Field via AdS/CFT · Chiral Equilibrium Instability of Quarkonium Mesons · 0.93

What it cites · 19 sentences without a confident label

Uses the method or tool of 1 paper
  • The configuration of section 2 provides [equation] The nontrivial $(0,z)$ components of the effective metric $G$ ([ref]) is given as [equation] We can diagonalize this metric by the following field redefinition (see for example a similar transformation in Ref. [ref] ) [equation] to obtain [equation] Substituting ([ref]) and expanding the metric around $z_p$ as $z =z_p + δz$ (here $δz < 0$), [equation] This metric is nothing but a near-horizon metric of a black hole.

    Hawking Radiation and Non-equilibrium Quantum Critical Current Noise · The induced metric and the volume · 0.93

Uses data or values from 5 papers
Cites as background 18 papers
  • The quantum quench on the flavor (quark) degrees of freedom has been studied in the context of AdS/CFT [ref] , in addition to the thermalization due to the quantum quench on the bulk gluons (and the spatial volume) in AdS/CFT (see for example Refs. [ref] ).

    Inhomogeneous Thermalization in Strongly Coupled Field Theories · Introduction · 0.84

  • The quantum quench on the flavor (quark) degrees of freedom has been studied in the context of AdS/CFT [ref] , in addition to the thermalization due to the quantum quench on the bulk gluons (and the spatial volume) in AdS/CFT (see for example Refs. [ref] ).

    Holographic Evolution of Entanglement Entropy · Introduction · 0.84

  • The quantum quench on the flavor (quark) degrees of freedom has been studied in the context of AdS/CFT [ref] , in addition to the thermalization due to the quantum quench on the bulk gluons (and the spatial volume) in AdS/CFT (see for example Refs. [ref] ).

    Instantaneous Thermalization in Holographic Plasmas · Introduction · 0.84

Discussion

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