Analytical & numerical study of strong-field QED
and its application to high-energy physics

Hidetoshi TAYA

(Theoretical physics; High-energy particle and nuclear physics)

Introduction to strong-field QED
3. About this project and presentation

1. Non-trivial response of the vacuum by strong field 2. Availability of strong EM field

Purpose of this project

How strong is needed ?

The vacuum is NOT an empty pace but hasa structure similar to semi-conductor⇒ Field strength comp. to the gap size = extremely strong !

(1) Develop a better formulation to discuss thevacuum physics by strong EM field𝑒𝑒𝑒𝑒 > 𝑚𝑚𝑒𝑒2 = 0.511 MeV 2 ≈ 𝑂𝑂(1028 W/cm2 )Where we live(≈ conduction band)gap ~ 2𝑚𝑚(2) Use (1) to predict something new(3) Propose possible experimental setups
and/or signals of (2)Dirac seaEnergy(≈ valence band)
Dirac (1930)

LED ≈ 10−5 W/cm2

SpaceAlso in other extreme systems

Explain what I have accomplished regarding(1) - (3) during SPDR (2021 - 2023)

Magnetar

References:Realization in nature

The vacuum should exhibit some responses
when shined by strong light (= electromagnetic field),similarly to semi-conductor

High-power laser experiments

About this presentation

Laser beam machine < 1010 W/cm2Glasses ≈ 103 W/cm2[1] Taya, Ironside, Phys. Rev. D 108, 096005 (2023) [arXiv:2308.11248][2] Fukushima, Shimazaki, Hidaka, Taya,

Review: [Fedotov, Ilderton, Karbstein, King, Seipt, HT, Torgrimsson (2023)]
Why interesting ?
(1) Non-trivial: Our vacuum has rich physics !
(2) Fundamental: The response can be used to diagnose the vacuum𝑒𝑒𝑒𝑒 < 1 MeV 2 ∼ 1011 T(𝐼𝐼 ∼ 1029 W/cm2 )Ann. Phys. 458, 169494 (2023) [arXiv:2305.11432]

Suzaku (2005~)
IXPE, XL-Calibur (2022~)[3] Taya, Yamada, JHEP 02, 048 (2023) [arXiv:2207.03831][4] Cao, Hattori, Hongo, Huang, Taya, PTEP 7, 071D01 (2022) [arXiv:2205.08051]

Early Universe& any physical processes occur on top of the vacuum[5] Fedotov, Ilderton, Karbstein, King, Seipt, Taya,
Phys. Rep. 1010, 1 (2023) [arXiv:2203.00019](3) Timeliness: within the experimental reach in the future(4) Interdiciplinarity:connect various disciplines of physics[6] Taya, Hongo, Ikeda, Phys. Rev. B 104, L140305 (2021) [arXiv:2105.12446][7] Du, Huang, Taya Phys. Rev. D 104, 056022 (2021) [arXiv:2104.12534][8] Taya, Fujimori, Misumi, Nitta, Sakai, JHEP 03, 082 (2021) [arXiv:2010.16080][9] Hattori, Taya, Yoshida, JHEP 01, 093 (2021) [arXiv:2010.13492]

Achievement on (1): Theoretical formulation of strong-field QED
1. Background

2. Typical calc. of strong-field QED
Observable = Integral of wavefunctions

Strong field supplies large
energy to the Dirac sea

Typically, of the form of・Non-perturbative treatment
is indispensable†�𝑂𝑂 = < 𝜓𝜓 𝑂𝑂𝜓𝜓 > =⇒ inapplicability of standard
calculation-method= perturbative(free-field wavefunction)

cf. [Voros (1983)] [Pham, Dillinger, Delabaere, Aoki, Koike, Takei, …]Exact WKB = non-pert. method to solve ODE = standard WKB + Borel resummation
・ Borel resum. = make a div. series well-defined・ Standard WKB = expansion in ℏ�

𝑖𝑖 ∈quantum number𝜓𝜓𝑖𝑖† 𝑂𝑂𝑖𝑖 𝜓𝜓𝑖𝑖

𝜓𝜓𝑖𝑖,± 𝑡𝑡; ℏ ≔ expi 𝑡𝑡∓ ∫𝑡𝑡 d𝑡𝑡 ′ 𝐸𝐸𝑖𝑖 (𝑡𝑡𝑡)ℏ 0∞(𝑛𝑛)
𝑛𝑛× ∑∞𝜓𝜓
𝑛𝑛=0 𝑖𝑖,± (𝑡𝑡)ℏ

𝐵𝐵 𝜓𝜓𝑖𝑖,± 𝑡𝑡; 𝜂𝜂 ≔ �𝑛𝑛

where 𝐸𝐸𝑖𝑖 is the (instantaneous)
eigenvalue of 𝐻𝐻(𝑛𝑛)・ However, 𝜓𝜓𝑖𝑖,± is in general divergent(𝑛𝑛)𝜓𝜓𝑖𝑖,± ∝ 𝑛𝑛!

iℏ𝜕𝜕𝑡𝑡 𝜓𝜓𝑖𝑖 (𝑡𝑡) = 𝐻𝐻(𝑡𝑡)𝜓𝜓𝑖𝑖 (𝑡𝑡)Strategy: Use a math. Tech.
of the exact WKB method[HT Fujimori, Misumi, Nitta, Sakai (2021)](i) Construct “Borel transformation”:

where 𝜓𝜓𝑖𝑖 is the so-called wavefunction
which is determined by solving the socalled Schrodinger equation = ODE・Need to use
dressed wavefunction

3. Exact WKB method𝑛𝑛!

𝜂𝜂 𝑛𝑛(ii) Laplace transformation gives “Borel sum”:∞d𝜂𝜂 −𝜂𝜂/ℏeΨ𝑖𝑖,± 𝑡𝑡; ℏ ≔ �𝐵𝐵[𝜓𝜓𝑖𝑖,± ](𝑡𝑡; 𝜂𝜂)0 ℏ⇒ Ψ𝑖𝑖,± is well-def. & is a natural analytic cont. of 𝜓𝜓𝑖𝑖,±⇐ Indeed, reduces to the original result if ∑∫ = ∫ ∑∞∞∞𝑛𝑛𝑛𝑛d𝜂𝜂 −𝜂𝜂/ℏ𝜓𝜓±,𝑛𝑛 (𝑡𝑡) 𝑛𝑛e𝜂𝜂 ∼ �
Ψ𝑖𝑖,± = ��ℏ𝑛𝑛!
0⇒ zero radius of convergenceThe essential problem is to solve ODE(𝑛𝑛)𝜓𝜓𝑖𝑖,± (𝑡𝑡)(𝑛𝑛)𝜓𝜓𝑖𝑖,± (𝑡𝑡)𝑛𝑛!

∞∞d𝜂𝜂 −𝜂𝜂/ℏ 𝑛𝑛(𝑛𝑛)�e𝜂𝜂 = � 𝜓𝜓𝑖𝑖,± (𝑡𝑡)ℏ𝑛𝑛0 ℏ𝚿𝚿𝒊𝒊,± gives a well-defined solution of ODE !

Need to make the series well-defined

𝑛𝑛=0

Achievement on (2): Predictions of novel vacuum processes
2. Predictions

1. Background
Analogies among various
physical systemsex 1) QED vacuum
Semi-conductor(materials w/ band)・ Ω → smth → 𝜔𝜔 = 𝑛𝑛 × Ω
Input light freq.

Output・Of increasing interestin sold-state physics
and material sciencee.g., atto-second light (Nobel prize 2023)

・Electroreflectancein materials
𝜖𝜖 = 𝜖𝜖 𝐸𝐸 ≠ 𝜖𝜖(0)oldest exp.
by Yacoby
w/ Si in 1966

・Vacuum permittivity
is not a const., either,under strong E field⇐ reflecting a tilted
vac. strc. by E

- plateau strc.
- cutoff law
2𝑒𝑒𝑒𝑒
𝑁𝑁cut ∼・Inflation is driven bystrong inflaton field

Inflaton field・Decay of inflaton= origin of matterin our Universe

・Predicted HHG spectrum for QED vacuumex 2) Strong EM field
Other types of strongfields/forces

Ex 3: New decay channel in preheating
Matter particles

If smth happens in an analog
system, it should happen in QED(or the vacuum) and vice versa

Ex 2: Electric permittivity of the vacuum

Ex 1: Vacuum high-harmonic generation

・(instant) preheating:one of possible scenarios[Felder, Kofman, Linde (1998)]⇐ strong-field technique applicable !
NEW !

at k=0ℏΩ[HT, Ironside (2023)][HT, Hongo, Ikeda (2021)][HT, Yamada (2022)]

Achievement on (3): Proposal of a new laboratory system to test strong-field QED1. Background

2. Numerical confirmation based on JAM (Jet AA Microscopic transport model)

・ Based on the hadron cascade model= superposition of elastic

Idea: Use heavy-ion collisions !
Baryon stopping at middle-energies (planned at FAIR, NICA, HIAF, J-Parc-HI, …)⇒ Formation of a highly-charged matter for a long time
𝑍𝑍2⇒
𝑍𝑍tot = 𝑍𝑍1 + 𝑍𝑍2

Rough order estimate:(𝑍𝑍1 +𝑍𝑍𝑧𝑧 )𝛼𝛼2∼Λ∼𝑒𝑒𝑒𝑒 ∼QCD2𝑟𝑟[Nara, Otsuka, Ohnishi,Nitta, Chiba (2000)]・ An open source code to simulate heavy-ion collisions

・ Laser: still too weak …
・ Extreme systems: not under control …

𝑍𝑍1

Hadron transport model: JAM

100 MeV 2⇒ Strongest E field on Earth !
・supercritical to QED・may also affect QCD/hadron& inelastic collisions ofResults“phase diagram” of strong-field QED & sensitivity region of HIC

𝑠𝑠NN

time

High energy

hadrons, which moveclassically in space[HT, Ohnishi, Nishimura (in preparation)]cf. [HT, Fujii, Itakura (2014)] [HT Fujimori, Misumi, Nitta, Sakai (2021)]

200 − 2500 GeV

From webpage of Hokkaido U., nuclear theory group

・ Output: Phase-space distributions of hadrons (𝑥𝑥, 𝑝𝑝)⇒ EM field is obtained by calc. the retarded pot.
∞𝜇𝜇 (𝑥𝑥 0 − 𝒙𝒙 − 𝒙𝒙′ , 𝒙𝒙𝒙)
1𝐽𝐽� d3 𝒙𝒙′
𝐴𝐴𝜇𝜇 𝑥𝑥 0 , 𝒙𝒙 =4𝜋𝜋 −∞|𝒙𝒙 − 𝒙𝒙𝒙|

obtained by JAM

Field strength

No good system to test strong-field QED at the presentMiddle energy𝑚𝑚𝛾𝛾 ==1𝑒𝑒𝑒𝑒 𝜏𝜏lifetime𝜈𝜈 = 𝑒𝑒𝑒𝑒𝜏𝜏 2 = 1