Electric permittivity of the vacuum in a strong constant electric field
Hidetoshi TAYA (Special Postdoctoral Researcher (SPDR) at iTHEMS)

Based on a work (in preparation for submission) with Charlie Ironside (Curtin U.)

1. INTRODUCTION
1-1. The vacuum ≠ not an empty space

1-2. Recent availability of strong EM field

1-3. This work: electric permittivity✔ How strong is needed ?

✔ Electric permittivity 𝜺𝜺- One of the fundamental quantities to characterize
the optical property of a matterThe vacuum has a structure, similarto semi-conductor⇒ Field strength comp. to the gap size = extremely strong !
𝑒𝑒𝑒𝑒 > 𝑚𝑚𝑒𝑒2 = 0.511 MeV 2 ≈ 𝑂𝑂(1028 W/cm2 )Where we live(≈ conduction band)gap ~ 2𝑚𝑚

Dirac seaEnergy(≈ valence band)
Dirac (1930)

probe electric fieldSpace✔ The vacuum should exhibit some responses
when shined by strong light (= EM field),similarly to semi-conductor

LED ≈ 10−5 W/cm2

Laser beam machine < 1010 W/cm2

Glasses ≈ 103 W/cm2✔ Experimental prog. to create (observe) strong EM field

Microscopically, 𝜀𝜀 ≠ 1 comes from polarizability of the matter

Magnetar (in astro phys)✔ Why interesting ?
(1) Non-trivial: vacuum has rich physics !
(2) Fundamental: the response can be used todiagnose the vacuum(3) Timeliness: within experimental reach (see 1-2)(4) Interdiciplinarity:connect high-energy phys. to others𝑒𝑒𝑒𝑒 < 1 MeV 2 ∼ 1011 T(𝐼𝐼 ∼ 1029 W/cm2 )In a matter = 𝒟𝒟 = 𝜀𝜀ℰ ≠ ℰ

In the pure vacuum = ℰ

Also in other extreme systems

High-power laser

𝒟𝒟 = ℰ + ∫ d𝑡𝑡 𝐽𝐽polSuzaku (2005~)
IXPE, XL-Calibur (2022~)- The electric permittivity of the vacuum 𝜀𝜀 has been studied
well for equilibrium situations (e.g., B field, null field),

Heavy-ion coll. (particle & nuclear phys)

Toll (1960), Baier-Breitenlohner (1967), Hattori-Itakura (2013), …

but NOT well for E field, which is genuinely non-equilibrium
and so the calculation is more difficult- So, the goal is to calculate the electric permittivityof the vacuum in the presence of a strong electric field

RHIC (2000)
LHC (2012)

2. THEORY
2-1. What I computed

2-2. Some details

Want to calculate the electric permittivity 𝜺𝜺✔ Kramers-Kronig relation✔ QED in the presence of a constant strong field 𝐸𝐸�plus a weak spatially homogeneous probe field ℰ(𝑡𝑡)✔ Calculate the polarization current 𝐽𝐽pol (in-in expect. value)
under the non-perturbative influence by the strong 𝐸𝐸�� 𝐸𝐸,� ℰ) = 0; in|𝜓𝜓(� ℰ)𝛾𝛾 𝜇𝜇 𝜓𝜓(𝐸𝐸,� ℰ)|0; in
𝐽𝐽pol (𝐸𝐸,�
= 0; in|𝜓𝜓�0 𝐸𝐸� 𝛾𝛾 𝜇𝜇 𝜓𝜓0 (𝐸𝐸)|0;in≈+ 0; in|𝜓𝜓�1 𝐸𝐸� 𝛾𝛾 𝜇𝜇 𝜓𝜓0 𝐸𝐸� + 𝜓𝜓�0 𝐸𝐸� 𝛾𝛾 𝜇𝜇 𝜓𝜓1 𝐸𝐸� |0; in × ℰ + 𝒪𝒪(ℰ 2 )𝜸𝜸𝝁𝝁+ℰ～𝜸𝜸𝝁𝝁+…

3. RESULTS

Causality: the response 𝒟𝒟 must be followed by ℰ
⇒ Re 𝜀𝜀 and Im 𝜀𝜀 are not independent with each other⇐ can be confirmed with analyzing 𝐽𝐽polFrequency of ℰ+∞
11′Re 𝜀𝜀(𝜔𝜔) = P. V. � d𝜔𝜔Im 𝜀𝜀(𝜔𝜔′ )𝜋𝜋𝜔𝜔′ − 𝜔𝜔−∞Calculation of Im 𝜀𝜀 is easier than Re 𝜀𝜀⇒ Calculate Im 𝜀𝜀 first, then get Re 𝜀𝜀 via KK relation

3-1. Imaginary part Im Δ𝜀𝜀 ( Δ𝜀𝜀 ≔ 𝜀𝜀 𝐸𝐸� ≠ 0 − 𝜀𝜀 𝐸𝐸� = 0 )✔ Non-trivial oscillation in 𝝎𝝎- Exponential tail below
the mass gap 𝜔𝜔 < 2𝑚𝑚- Damped oscillation above
the mass gap 𝜔𝜔 < 2𝑚𝑚✔ Birefringence
Im Δ𝜀𝜀⊥ ≠ Im Δ𝜀𝜀∥ ,though the basicfeatures are thesame✔ Discontinuity at 𝝎𝝎 = 𝟎𝟎

due to the non-pert. effect−# 𝑚𝑚2 /𝑒𝑒𝐸𝐸�, whichIm Δ𝜀𝜀 ∝ e
was missing previously✔ Similar features
to Im 𝚫𝚫𝜺𝜺

✔ Reflecting the structure of the vacuum

EnergyProb. density of Dirac-sea electrons

Space coordinate�(direction of 𝐸𝐸)

4. SUMMARY

� | 0; in has- The number of pairs 𝑁𝑁 = 0; in | 𝑁𝑁the direct relevance to Im 𝜀𝜀d𝑈𝑈1d𝒟𝒟 1
Dielectric energy loss:=ℰ= 𝜔𝜔ℰ 2 Im 𝜀𝜀d𝑡𝑡
d𝑡𝑡 2d𝑈𝑈2𝑁𝑁 ℰ ≠ 0 − 𝑁𝑁(ℰ = 0)
Energy used in part. pro. := 𝜔𝜔
𝑉𝑉𝑉𝑉d𝑡𝑡⇒ can prove 𝑈𝑈1 = 𝑈𝑈2- Oscillation in 𝜔𝜔- Birefringence✔ Log divergence at 𝝎𝝎 = 𝟎𝟎

- due to the non-perturbative discontinuity of Im Δ𝜀𝜀 at 𝜔𝜔 = 0- For 𝑒𝑒𝐸𝐸� ≪ 𝑚𝑚2 , it is negligible
Baier-Breitenlohner (1967)
and the result agrees with the previously known2�𝛼𝛼 𝑒𝑒𝐸𝐸6(∥)
2Re Δ𝜀𝜀(𝜔𝜔 = 0, 𝑒𝑒𝐸𝐸� ≪ 𝑚𝑚 ) =×�22 (⊥)
45𝜋𝜋 𝑚𝑚✔ A QED analog of the Franz-Keldysh effect in semi-conductor
logic 1) The dist. of the Dirac-sea electronsis oscillating due to interference withthe electrons reflected by the gap
logic 2) On the other hand, Im 𝜀𝜀 is related tothe particle prod.
logic 3) Particle prod. occurs more at where
the Dirac-sea electrons exist morelogic 4) The energy 𝜔𝜔 needed for the particle
prod. is different depending onfrom where the particle prod. occurs

2𝑚𝑚Dunne-Gies-Schutzhold (2009), HT (2017)

3-2. Real part Re Δ𝜀𝜀3-3. Interpretation

Band structure of the vacuum✔ Relation to the Schwinger effect- The vacuum is unstable against the particle prod.
in the presence of 𝐸𝐸� and ℰ⇒ (dynamically-assisted) Schwinger effectFranz (1958), Keldysh (1958)

Precisely the same pattern has
already been observed in semi-cond.
experimentswith Si
Yacoby (1966)⇒ Dirac was correct that the vacuum isindeed like a semi-conductor !

∴ Im 𝜀𝜀 exhibits an oscillating pattern in 𝜔𝜔⇒ A similar pattern for Re 𝜀𝜀, as it is relatedto Im 𝜀𝜀 via the Kramers-Kronig relation

What I may do next

What I did

What I wanted to tell you

I calculated the electric permittivityof the vacuum in a constant strong
electric field✔ More general strong-field configurations✔ The vacuum is like a semi-conductor and can exhibit something interesting✔ An intriguing oscillating pattern appears in the electric permittivity,e.g.) spatial dependence, beyond static, incl. B field
which is reflecting the change of the vacuum structure in a strong electric field✔ Magnetic permeability✔ The slow-frequency limit is modified (e.g., log div.) due to non-perturbative effect by✔ Import the wisdom of semi-cond. physics
a strong electric field