Electric permittivity of the vacuum
in a strong constant electric field
Hidetoshi Taya
( RIKEN )with Charlie Ironside (Curtin U.)(paper in preparation)

This talkDiscuss the electric permittivity 𝝐𝝐 of thevacuum in a strong constant electric field
In the pure vacuum 𝒟𝒟 = ℰ～～

In an EM field 𝒟𝒟 = 𝜖𝜖ℰ ≠ ℰ～～

Motivations
Why interesting ?
(1) Is a well studied topic, but incomplete yet(2) Want to use as a tool to diagnose the structure of the QED vacuum(3) Semi-conductor experiments of electroreflectanceMotivations (1/3)
Why interesting ?
(1) Is a well studied topic, but incomplete yet(2) Want to use as a tool to diagnose the structure of the QED vacuum(3) Semi-conductor experiments of electroreflectance・ Previous studies: typically for equilibrium situations (e.g., B field, null field)
・ E-field case exists, but not satisfactory enough𝒟𝒟 = −

𝜕𝜕ℒEuler−Heisenberg𝜕𝜕𝜕

𝛼𝛼 𝑒𝑒𝐸𝐸�
2�
Re𝜖𝜖(𝜔𝜔≪𝑚𝑚,𝑒𝑒𝐸𝐸≪𝑚𝑚)=⇒
45𝜋𝜋 𝑚𝑚22×�6 (∥)
2 (⊥)[Baier-Breitenlohner (1967)][Toll (1960)] [Hattori-Itakura (2013)] …

😡😡 valid in the weak field limit
😡😡 valid for slow probes😡😡 imaginary part ?

Q: What happens if I go beyond the weak & slow limit ?

Motivations (2/3)
Why interesting ?
(1) Is a well studied topic, but incomplete yet(2) Want to use as a tool to diagnose the structure of the QED vacuum(3) Semi-conductor experiments of electroreflectanceIn an E field, the QED vacuum is tilted⇒ oscillating dist. due to interferenceelectric field direction ➡

𝑉𝑉 = −𝑒𝑒𝑒𝑒𝑒𝑒
electrons coming
electron distribution

reflected by gap

Q: How is the tilted QED-vacuum structure seen in the electric permittivity ?

Motivations (2/3)
Why interesting ?
(1) Is a well studied topic, but incomplete yet(2) Want to use as a tool to diagnose the structure of the QED vacuum(3) Semi-conductor experiments of electroreflectanceIn an E field, the QED vacuum is tilted

Can leave observable imprints⇒ oscillating dist. due to interference

・ ex.) dynamically-assisted Schwinger(pair prod. in const 𝐸𝐸� + fast ℰ)

electric field direction ➡

𝑉𝑉 = −𝑒𝑒𝑒𝑒𝑒𝑒
electrons coming
electron distribution

Oscillationreflected by gap[HT (2019)][Huang, HT (2019)]

Q: How is the tilted QED-vacuum structure seen in the electric permittivity ?

Motivations (3/3)
Why interesting ?
(1) Is a well studied topic, but incomplete yet(2) Want to use as a tool to diagnose the structure of the QED vacuum(3) Semi-conductor experiments of electroreflectanceSuch an oscillating behavior in 𝜖𝜖 has been observed in semi-conductor materials≈ Im Δ𝜖𝜖with Si𝜖𝜖 𝑒𝑒𝐸𝐸� ≠ 0 − 𝜖𝜖(𝑒𝑒𝐸𝐸� = 0)

[Yacoby (1966)]≈ Re Δ𝜖𝜖with Ge[Seraphin, Hess (1965)]

Q: Natural to expect semi-conductor-like behavior in QED. Is this true ?

Motivations
Why interesting ?
(1) Is a well studied topic, but incomplete yet

Q: What happens if I go beyond the weak & slow limit ?

(2) Want to use as a tool to diagnose the structure of the QED vacuum
Q: How is the tilted QED-vacuum structure seen in the electric permittivity ?
(3) Semi-conductor experiments of electroreflectance

Q: Natural to expect semi-conductor-like behavior in QED. Is this true ?

This talk: answer those questions

Theory (1/2)
Calculate the polarization via the linear response theory(linear in the probe ℰ but non-pert. in the strong field 𝐸𝐸� )Theory (1/2)
Calculate the polarization via the linear response theory(linear in the probe ℰ but non-pert. in the strong field 𝐸𝐸� )Setup: QED in the presence of a constant strong field 𝐸𝐸� + a weak spatially homo. probe ℰ(𝑡𝑡)Theory (1/2)
Calculate the polarization via the linear response theory(linear in the probe ℰ but non-pert. in the strong field 𝐸𝐸� )Setup: QED in the presence of a constant strong field 𝐸𝐸� + a weak spatially homo. probe ℰ(𝑡𝑡)Step 1: Definition of 𝒟𝒟 (or 𝜖𝜖)� ℰ) = 𝐸𝐸 + 𝑃𝑃0 (𝐸𝐸)� +𝑃𝑃1 (𝐸𝐸)ℰ� +⋯・ Total flux 𝐷𝐷 = 𝐸𝐸 + 𝑃𝑃(𝐸𝐸,� ℰ+⋯= 𝐸𝐸� + 𝑃𝑃0 𝐸𝐸� + (1 + 𝑃𝑃1 (𝐸𝐸))
𝐸𝐸� + ℰ�・ So, identify 𝜖𝜖 = 1 + 𝑃𝑃1 (𝐸𝐸)�𝐷𝐷𝒟𝒟

Theory (1/2)
Calculate the polarization via the linear response theory(linear in the probe ℰ but non-pert. in the strong field 𝐸𝐸� )Setup: QED in the presence of a constant strong field 𝐸𝐸� + a weak spatially homo. probe ℰ(𝑡𝑡)Step 1: Definition of 𝒟𝒟 (or 𝜖𝜖)� ℰ) = 𝐸𝐸 + 𝑃𝑃0 (𝐸𝐸)� +𝑃𝑃1 (𝐸𝐸)ℰ� +⋯・ Total flux 𝐷𝐷 = 𝐸𝐸 + 𝑃𝑃(𝐸𝐸,� ℰ+⋯= 𝐸𝐸� + 𝑃𝑃0 𝐸𝐸� + (1 + 𝑃𝑃1 (𝐸𝐸))
𝐸𝐸� + ℰ�・ So, identify 𝜖𝜖 = 1 + 𝑃𝑃1 (𝐸𝐸)

Step 2: Calculate the polarization 𝑃𝑃1

�
𝐷𝐷𝒟𝒟

� 𝐸𝐸,
� ℰ)𝛾𝛾 𝜇𝜇 𝜓𝜓(𝐸𝐸,� ℰ)|0; in
・ In QFT, 𝑃𝑃̇ = 0; in|𝜓𝜓(
�= 0; in|𝜓𝜓�0 𝐸𝐸� 𝛾𝛾 𝜇𝜇 𝜓𝜓0 (𝐸𝐸)|0;
ingives 𝑃𝑃1

�in × ℰ + 𝒪𝒪(ℰ 2 )+ 0; in|𝜓𝜓�1 𝐸𝐸� 𝛾𝛾 𝜇𝜇 𝜓𝜓0 𝐸𝐸� + 𝜓𝜓�0 𝐸𝐸� 𝛾𝛾 𝜇𝜇 𝜓𝜓1 (𝐸𝐸)|0;・ Diagrammatically, amounts to evaluateℰ～・ Some details

Theory (2/2)(1) Use of Kramers-Kronig relation+∞
11Im 𝜖𝜖(𝜔𝜔′ )Causality ⇒ Re 𝜖𝜖(𝜔𝜔) = P. V. � d𝜔𝜔′𝜋𝜋
𝜔𝜔′ − 𝜔𝜔−∞[Toll (1960)] [Heinzl, Schroeder (2006)] [Borysov et al. (2022)]⇒ Sufficient to calculate the imaginary part(Same approach has been adopted in semi-conductor) [Aspnes(1967)]

・ Some details

Theory (2/2)(1) Use of Kramers-Kronig relation+∞
11Im 𝜖𝜖(𝜔𝜔′ )Causality ⇒ Re 𝜖𝜖(𝜔𝜔) = P. V. � d𝜔𝜔′𝜋𝜋𝜔𝜔′ − 𝜔𝜔−∞[Toll (1960)] [Heinzl, Schroeder (2006)] [Borysov et al. (2022)]⇒ Sufficient to calculate the imaginary part(Same approach has been adopted in semi-conductor) [Aspnes(1967)](2) Im 𝜖𝜖 is directly related to the pair prod. via the dynamically-assisted Schwinger effect・ Im 𝜖𝜖 is related to the dielectric energy loss (= decay of probe)
See, e.g., textbook by Landau-Lefshitz

d𝑈𝑈1d𝒟𝒟 1=ℰ= 𝜔𝜔ℰ 2 Im 𝜖𝜖d𝑡𝑡
d𝑡𝑡
2d𝑈𝑈2
𝑁𝑁 ℰ ≠ 0 − 𝑁𝑁(ℰ = 0)=𝜔𝜔・ Energy of probe used in the pair productiond𝑡𝑡
𝑉𝑉𝑉𝑉・ Microscopically, the dielectric energy loss should be caused by the pair production⇒ 𝑈𝑈1 = 𝑈𝑈2 ⇒𝑁𝑁 ℰ≠0 −𝑁𝑁(ℰ=0)
1= ℰ 2 Im 𝜖𝜖𝑉𝑉𝑉𝑉2KK∴ Dynamically assisted Schwinger ↔ Im 𝜖𝜖 ↔ Re 𝜖𝜖

Results (1/2)
� ≠ 𝟎𝟎 − 𝝐𝝐(𝑬𝑬� = 𝟎𝟎)
Imaginary part of the change 𝚫𝚫𝝐𝝐 = 𝝐𝝐 𝑬𝑬Strong 𝐸𝐸�

～～～～Probe ℰProbe ℰStrong 𝐸𝐸�

・ Oscillation, as expected from the tilted vacuum
・ Essentially the same pattern as semi-conductor observation
・ Birefringent (Im Δ𝜖𝜖∥ ≠ Im Δ𝜖𝜖⊥ ) but the basically the same・ Non-vanishing even at 𝜔𝜔 → 0 due to the strong-field non-perturbative effectA simple explanation: In the slow limit, the Schwinger formula is valid⇒ Im 𝜖𝜖 ∝ (𝑁𝑁Schwinger 𝐸𝐸� + ℰ − 𝑁𝑁Schwinger 𝐸𝐸� ) ∝ (exp −𝜋𝜋𝑚𝑚2�𝑒𝑒(𝐸𝐸+ℰ)− exp[−𝜋𝜋𝑚𝑚2]) = (finite) × exp𝑒𝑒 𝐸𝐸�

−𝜋𝜋𝑚𝑚2𝑒𝑒𝐸𝐸�Results (2/2)
� ≠ 𝟎𝟎 − 𝝐𝝐(𝑬𝑬� = 𝟎𝟎)
Real part of the change 𝚫𝚫𝝐𝝐 = 𝝐𝝐 𝑬𝑬Strong 𝐸𝐸�

～～～～Probe ℰ

Probe ℰ

Strong 𝐸𝐸�

・ Again oscillation, which is again consistent with semi-conductor
・ Logarithmically divergent at 𝜔𝜔 → 0 due to the non-perturbative effect
1+∞

11+∞

1𝑚𝑚2

∵ Re 𝜖𝜖 0 = 𝜋𝜋 P. V. ∫−∞ d𝜔𝜔′ 𝜔𝜔′ Im 𝜖𝜖 𝜔𝜔′ ∼ 𝜋𝜋 ∫−∞ d𝜔𝜔′ 𝜔𝜔′ Im 𝜖𝜖 0 ∼ log div. × exp −𝜋𝜋 𝑒𝑒𝐸𝐸�

Note: agrees precisely with the known Euler-Heisenberg result at 𝜔𝜔 → 0 if the log div. was subtracted

This talk・ I’ve calculated the electric permittivity of the vacuum in a constant strong E field・ My answer to the 3 questions:
Q: What happens if I go beyond the weak & slow limit ?
A: Beyond slow ⇒ Oscillation appears
Beyond weak ⇒ Logarithmic divergence at 𝜔𝜔 → 0

Q: How is the tilted QED-vacuum structure seen in the electric permittivity ?
A: Oscillation in the high-frequency regime
Q: Natural to expect semi-conductor-like behavior in QED. Is this true ?
A: True.

(and similar plots for Δ𝜖𝜖⊥ )