Electric permittivity of the vacuum
in a strong constant electric field
Hidetoshi Taya( Keio U. )HT, C. Ironside, PRD 108, 096005 (2023) [2308.11248]

This talkDiscuss the electric permittivity 𝝐 of the vacuum
in a strong constant electric field
In the pure vacuum 𝒟 = ℰ

In an EM field 𝒟 = 𝜖ℰ ≠ ℰ
ഥ𝑬～～～～

ത due to the vacuum polarization
・ 𝜖 is no longer a const. 𝜖 = 𝜖(𝐸)・many studies since the early days,[Heisenberg-Euler (1936)] [Toll 1952] [Klein-Nigam (1964)] [Baier-Breitenlohner (1967)] …
Review: [King-Heinzl (2015)] [Ejlli et al. (PVLAS) (2020)] [Fedotov et al. (2023)] …

but is still worthwhile to be investigated ⇒ 3 motivations

Motivations (1/3)(1) The current understanding is limited to weak/slow regime
The most famous formula = based on Euler-Heisenberg Lagrangian
𝜕ℒEH
𝒟=−
𝜕ℰ

⇒

𝛼 𝑒𝐸ത
𝜖=45𝜋 𝑚22×ቊ6 (∥)
2 (⊥)[Baier-Breitenlohner (1967)]

Motivations (1/3)(1) The current understanding is limited to weak/slow regime
The most famous formula = based on Euler-Heisenberg Lagrangian
𝜕ℒEH
𝒟=−𝜕ℰ⇒𝛼 𝑒𝐸ത𝜖=45𝜋 𝑚22×ቊ

6 (∥)
2 (⊥)

[Baier-Breitenlohner (1967)]

ഥ ≪ 𝒎𝟐
Problem 1: Valid only in the weak limit 𝒆𝑬
power corrections could be included 𝑒𝐸ത 𝑛 , but
・is factorially divergent ⇒ does not necessarily improve the formula・non-pert. factor like 𝑒 1/𝑒𝐸ത can never be included

cf. [Heinzl-Schroder (2006)]

Problem 2: Valid only in the slow limit 𝝎 ≪ 𝒎・not possible to discuss 𝜔 dependence

cf. [King-Heinzl-Blackburn (2023)]

・the physics must be different above the pair-production threshold 𝜔 > 2𝑚Problem 3: Neglecting the imaginary part 𝐈𝐦 𝓛𝐄𝐇
𝜖 (in the coordinate space) must be real, so one must set Im ℒ Euler−Heisenberg⇒ pair production and “non-equilibrium-ness” of E field are completely dismissed

Q: What happens if I go beyond those limitations ?

Motivations (2/3)(2) As a signature of non-trivial QED vacuum structure in E field
The QED vacuum (= the Dirac sea) has a non-trivial electron dist. in an E field,
which can leave observable imprints
ഥ=𝟎QED vacuum at 𝑬

← positive energy states← mass gap ~ 2m

← Dirac sea

Motivations (2/3)(2) As a signature of non-trivial QED vacuum structure in E field
The QED vacuum (= the Dirac sea) has a non-trivial electron dist. in an E field,
which can leave observable imprints
ഥ≠𝟎
QED vacuum at 𝑬
E field direction

Motivations (2/3)(2) As a signature of non-trivial QED vacuum structure in E field
The QED vacuum (= the Dirac sea) has a non-trivial electron dist. in an E field,
which can leave observable imprints
ഥ≠𝟎
QED vacuum at 𝑬
E field direction

electrons coming𝜓 ∝ 𝑒 −i∫ d𝑧 𝑃Motivations (2/3)(2) As a signature of non-trivial QED vacuum structure in E field
The QED vacuum (= the Dirac sea) has a non-trivial electron dist. in an E field,
which can leave observable imprints
ഥ≠𝟎
QED vacuum at 𝑬
E field direction

electrons coming𝜓 ∝ 𝑒 −i∫ d𝑧 𝑃
∝ 𝑒 +i∫ d𝑧 𝑃
reflected by the gap

Motivations (2/3)(2) As a signature of non-trivial QED vacuum structure in E field
The QED vacuum (= the Dirac sea) has a non-trivial electron dist. in an E field,
which can leave observable imprints
ഥ≠𝟎QED vacuum at 𝑬
E field direction

electrons coming𝜓 ∝ 𝑒 −i∫ d𝑧 𝑃
∝ 𝑒 +i∫ d𝑧 𝑃
reflected by the gap
2∝ |# 𝑒 +i∫ d𝑧 𝑃 +# 𝑒 −i∫ d𝑧 𝑃 ቚ ∝ cos ∫ d𝑧 𝑃interfered electron distribution

Motivations (2/3)(2) As a signature of non-trivial QED vacuum structure in E field
The QED vacuum (= the Dirac sea) has a non-trivial electron dist. in an E field,
which can leave observable imprints
ഥ≠𝟎
QED vacuum at 𝑬E field direction

Observable imprintEx.) spectrum of the dynamically-assisted Schwinger(= pair prod. in const 𝐸ത + fast probe ℰ)[HT (2019)] [Huang-HT (2020)]

Exact numericalresults

electrons comingAnalytic(Furry picture)𝜓 ∝ 𝑒 −i∫ d𝑧 𝑃
∝ 𝑒 +i∫ d𝑧 𝑃
reflected by the gap

Oscillation2∝ |# 𝑒 +i∫ d𝑧 𝑃 +# 𝑒 −i∫ d𝑧 𝑃 ቚ ∝ cos ∫ d𝑧 𝑃interfered electron distribution

probe frequency 𝜔/𝑚Motivations (2/3)(2) As a signature of non-trivial QED vacuum structure in E field
The QED vacuum (= the Dirac sea) has a non-trivial electron dist. in an E field,
which can leave observable imprints
ഥ≠𝟎QED vacuum at 𝑬
E field direction

Observable imprint
Ex.) spectrum of the dynamically-assisted Schwinger(= pair prod. in const 𝐸ത + fast probe ℰ)[HT (2019)] [Huang-HT (2020)]

Exact numericalresults

electrons comingAnalytic(Furry picture)𝜓 ∝ 𝑒 −i∫ d𝑧 𝑃
∝ 𝑒 +i∫ d𝑧 𝑃
reflected by the gap

Oscillation2∝ |# 𝑒 +i∫ d𝑧 𝑃 +# 𝑒 −i∫ d𝑧 𝑃 ቚ ∝ cos ∫ d𝑧 𝑃interfered electron distribution

probe frequency 𝜔/𝑚

Q: What happens to the electric permittivity ?
Note: The motivation (1) (in particular, going beyond 𝜔 ≪ 𝑚) is important to achieve thisMotivations (3/3)(3) Pursue analogy between strong-field QED and semicond. phys・ Ground-state structure of semicond. = The QED vacuum⇒ the QED vacuum should response against external field in a similar way
to a semiconductor and vice versa

Motivations (3/3)(3) Pursue analogy between strong-field QED and semicond. phys・ Ground-state structure of semicond. = The QED vacuum⇒ the QED vacuum should response against external field in a similar way
to a semiconductor and vice versa[Seraphin-Hess (1965)] with GeRe Δ𝜖[Yacoby (1966)]with SiIm Δ𝜖Δ𝜖 ≔ 𝜖 𝑒𝐸ത ≠ 0 − 𝜖(𝑒𝐸ത = 0)・ Nontrivial oscillating change in 𝜖 (i.e., motivation (2)) has already been observed
more than 50 yrs ago in semicond. physics !
(⇒ the Franz-Keldysh effect and electroreflectance) [Franz (1958)] [Keldysh (1958)]

probe frequency 𝜔probe frequency 𝜔

Q: Natural to expect this change in QED. Is this analogy true ?

This talkDiscuss the electric permittivity 𝝐 of the vacuum
in a strong constant electric field

I. Introduction(1) What happens if I go beyond weak/slow-field limit ?
(2) As a signature of non-trivial QED vacuum structure in E field(3) Pursue analogy between strong-field QED and semicond. phys

II. Theory Linear response theory based on in-in formalism of QFT

III. Results will give positive statements to the motivations (1) - (3)IV. Summary

Theory (1/2)
Linear response theory based on in-in formalism of QFT

Theory (1/2)
Linear response theory based on in-in formalism of QFT
Setup: QED under a constant strong field 𝐸ത + a weak spatially homo. probe ℰ(𝑡)Theory (1/2)
Linear response theory based on in-in formalism of QFT
Setup: QED under a constant strong field 𝐸ത + a weak spatially homo. probe ℰ(𝑡)Step 1: Definitions of 𝒟 and 𝜖ത ℰ) = 𝐸 + 𝑃0 (𝐸)ത +𝑃1 (𝐸)ℰത +⋯・ Total flux 𝐷 = 𝐸 + 𝑃(𝐸,
ത + (1 +𝑃1 𝐸ത )ℰ + ⋯= 𝐸ത + 𝑃0 (𝐸)𝐸ത + ℰ𝒟ത⇒ 𝜖 = 1 +𝑃1 (𝐸)Theory (1/2)
Linear response theory based on in-in formalism of QFT
Setup: QED under a constant strong field 𝐸ത + a weak spatially homo. probe ℰ(𝑡)Step 1: Definitions of 𝒟 and 𝜖ത ℰ) = 𝐸 + 𝑃0 (𝐸)ത +𝑃1 (𝐸)ℰത +⋯・ Total flux 𝐷 = 𝐸 + 𝑃(𝐸,
ത + (1 +𝑃1 𝐸ത )ℰ + ⋯= 𝐸ത + 𝑃0 (𝐸)𝐸ത + ℰ𝒟ത⇒ 𝜖 = 1 +𝑃1 (𝐸)Step 2: Calculate 𝑃1
ത 𝐸,ത ℰ)𝛾 𝜇 𝜓(𝐸,ത ℰ)|0; in・Ampere law: −𝑃ሶ = 𝐽 = 0; in|𝜓(ത= 0; in|𝜓ത0 𝐸ത 𝛾 𝜇 𝜓0 (𝐸)|0;in𝑃1ത+ 0; in|𝜓ത1 𝐸ത 𝛾 𝜇 𝜓0 𝐸ത + 𝜓ത0 𝐸ത 𝛾 𝜇 𝜓1 (𝐸)|0;in × ℰ + 𝒪(ℰ 2 )・Not in-out amplitude, but in-in !
⇒ crucial when pair creating(or in non-equil.):・Diagrammatically, evaluate

cf. [Copinger-Fukushima-Shi (2018)]|0; outۧ = |0; inۧ + (pair states like |𝑒 + 𝑒 − ; inۧ)ℰ～

Theory (2/2)
Detail 1: Loop diagram ⇒ UV divergent ⇒ use Kramers-Kronig relation[Toll (1960)] [Heinzl, Schroeder (2006)] [Borysov et al. (2022)]+∞11′Im 𝜖(𝜔′ )Causality ⇒ Re 𝜖(𝜔) = P. V. න d𝜔𝜋𝜔′ − 𝜔−∞⇒ sufficient to calculate the imaginary part(same approach has been adopted in semicond.)[Aspnes(1967)]

2Im=

Theory (2/2)
Detail 1: Loop diagram ⇒ UV divergent ⇒ use Kramers-Kronig relation[Toll (1960)] [Heinzl, Schroeder (2006)] [Borysov et al. (2022)]+∞

11′Im 𝜖(𝜔′ )Causality ⇒ Re 𝜖(𝜔) = P. V. න d𝜔𝜋𝜔′ − 𝜔−∞⇒ sufficient to calculate the imaginary part(same approach has been adopted in semicond.)

2Im=[Aspnes(1967)]

Detail 2: Im 𝜖 is directly related to the (dynamically-assisted) Schwinger effect・Electromagnetism tells us:Im 𝜖 is related to the dielectric energy loss

d𝑈1d𝒟 1=ℰ= 𝜔ℰ 2 Im 𝜖d𝑡d𝑡2

・ Energy loss of probe due to the pair prod.

d𝑈2
𝑁 ℰ ≠ 0 − 𝑁(ℰ = 0)=𝜔d𝑡
𝑉𝑇・ Microscopically, the dielectric energy loss
should be caused by the pair prod.

𝑈1 = 𝑈2 ⇒See, e.g., textbook by Landau-Lefshitz∴ SchwingerKK𝐈𝐦 𝝐𝑁 ℰ≠0 −𝑁(ℰ=0)
1= ℰ 2 Im 𝜖𝑉𝑇2𝐑𝐞 𝝐 ⇒ 𝝐 as an indirect sign. of Schwinger & vice versa

This talkDiscuss the electric permittivity 𝝐 of the vacuum
in a strong constant electric field

I. Introduction(1) What happens if I go beyond weak/slow-field limit ?
(2) As a signature of non-trivial QED vacuum structure in E field(3) Pursue analogy between strong-field QED and semicond. phys

II. Theory Linear response theory based on in-in formalism of QFT

III. Results will give positive statements to the motivations (1) - (3)IV. Summary

Results (1/3): Imaginary part of 𝚫𝝐ഥ ≠ 𝟎 − 𝝐(𝑬ഥ = 𝟎)
Imaginary part of the change 𝚫𝝐 = 𝝐 𝑬～～～～

Probe ℰ

Probe ℰ

Strong 𝐸ത

Strong 𝐸ത

・ Oscillation, as expected from the tilted vacuum

・ Essentially the same pattern as semi-conductor observation
・ Birefringent (Im Δ𝜖∥ ≠ Im Δ𝜖⊥ ) but the basically the same・ Non-vanishing even at 𝜔 → 0 due to the strong-field non-perturbative effectA simple explanation: In the slow limit, the standard Schwinger formula is valid𝑚2𝑚2𝑚2⇒ Im 𝜖 ∝ (𝑁Schwinger 𝐸ത + ℰ − 𝑁Schwinger 𝐸ത ) ∝ (exp −𝜋 𝑒(𝐸+ℰ)− exp[−𝜋 𝑒𝐸ത ]) = (finite) × exp −𝜋 𝑒 𝐸തതResults (2/3): Real part of 𝚫𝝐ഥ ≠ 𝟎 − 𝝐(𝑬ഥ = 𝟎)
Real part of the change 𝚫𝝐 = 𝝐 𝑬
Strong 𝐸ത

・ Oscillation, consistent with semi-conductor
・ Logarithmically divergent at 𝜔 → 0due to the non-perturbative effect∵+∞
11Re 𝜖 0 = P. V. න d𝜔′ ′ Im 𝜖 𝜔′𝜋𝜔−∞

1 +∞ ′ 1𝑚2∼ න d𝜔 ′ Im 𝜖 0 ∼ log div. × exp −𝜋𝜋 −∞𝜔𝑒 𝐸ത

Probe ℰ～～～～Probe ℰ

Strong 𝐸ത

Results (3/3): Comparison w/ EH
Comparison w/ Euler-Heisenberg around 𝝎 ≈ 𝟎Strong 𝐸ത

Probe ℰ～～～～

Probe ℰ

Strong 𝐸ത

・ After the log subtraction, the result is consistent with the EH result・ Significant deviation for finite 𝜔 and/or stronger 𝐸ത

𝛼 𝑒 𝐸ത
Δ𝜖EH =45𝜋 𝑚22×ቊ

6 (∥)
2 (⊥)

This talkDiscuss the electric permittivity 𝝐 of the vacuum
in a strong constant electric field

I. Introduction(1) What happens if I go beyond weak/slow-field limit ?
(2) As a signature of non-trivial QED vacuum structure in E field(3) Pursue analogy between strong-field QED and semicond. phys

II. Theory Linear response theory based on in-in formalism of QFT

III. Results will give positive statements to the motivations (1) - (3)IV. Summary

Summary
Discussed the electric permittivity 𝝐 of the vacuum
in a strong constant electric field
Three motivations and my answers to them:
Motivations: (1) The current understanding is limited to weak/slow regime(2) As a signature of non-trivial QED vacuum structure in E field(3) Pursue analogy between strong-field QED and semicond. physAnswers: (1): Succeeded with a linear response theory based on in-in formalism(2)&(3): Yes, a characteristic oscillating structure in 𝜖, similarly to semicond.

Other lessons (and further comments)
・Higher frequency gives a bigger signal of 𝜖 (or vacuum birefringence, in general)
e.g. peak at 𝜔 ∼ 2𝑚:(PW laser)≫ Δ𝜖EH = 𝑂(10−7 ) for 𝐼 = 𝑂(1 × 1023 W/cm2 )・ Need to go beyond EH for large 𝜔 or strong 𝐸ത ⇒ implications to heavy-ion coll., magnetar, … ?
・ Schwinger

Im 𝜖KKRe 𝜖 ⇒ 𝜖 as an indirect signature of Schwinger & vice versa

Peak value
Behaviors of the peaks

Numerical fit says (similar numbers for Im 𝜖⊥ and Re 𝜖∥,⊥ )where 𝐼 = 𝐸ത 2 /2 is the focused intensity

・much larger than EH e.g., Δ𝜖EH = 𝑂(10−7 ) for 𝐼 = 𝑂(1 × 1023 W/cm2) (PW laser)⇒ High-frequency probe is useful to study 𝜖 (or vacuum birefringence, in general)・weak E dependence Δ𝜖 ∝ 𝐼

1 1~5 6⇒ the peaks of Δ𝜖 can still be large for subcritical fields
e.g., only one-order smaller Δ𝜖 = 𝑂 1 × 10−5 ≫ Δ𝜖EH at GW scale