How time-dependent
electric fields affect
the Schwinger mechanism?
Hidetoshi TAYA
Keio U. → RIKEN (from this April)What is the Schwinger mechanism?
[Sauter (1932)] [Heisenberg, Euler (1936)] [Schwinger (1951)]✔ Vacuum pair production occurs in the presence of strong fields
Intuitive picture for a slow electric field = quantum tunneling~ 2mpositive energy bandgap

Dirac sea

~ 2mslow & strong E~ 2m/eEtunnelingWhat is the Schwinger mechanism?
[Sauter (1932)] [Heisenberg, Euler (1936)] [Schwinger (1951)]✔ Vacuum pair production occurs in the presence of strong fields
Intuitive picture for a slow electric field = quantum tunneling~ 2mpositive energy band
gap

~ 2m

slow & strong E

Dirac sea

✔ For constant strong E-field 𝐸 𝑡, 𝒙

~ 2m/eEtunneling(at least,= 𝐸, it’s understood well theoretically)𝑒𝐸 2 𝑉𝑇𝑚2
𝑁𝑒 ± =
× exp −𝜋
∼ exp[−# × gap height × (gap length)]
2𝜋 3
𝑒𝐸・ Non-perturbative ⇒ Interesting, since it is the unexplored region of QED (or QFT)

・ Vacuum process ⇒ Fundamental, since all the physical processes occur on top of vacuum
・ However, exponentially suppressed i.t.o. mass ⇒ Not confirmed by experiments yet…
cf.) Guinness world record:
HERCULES laser 𝑒𝐸~ 1022 W/cm2 ∼ 0.01 MeV 2 ≪ 𝑚𝑒2 ∼[Yanovsky et al (2008)]

1029 WΤcm2 ∼ 0.5 MeV 2

Timeliness✔ Now is the best time to study the Schwinger mechanism !
Developments in experimental technologies ⇒ Novel strong-field sources

Timeliness✔ Now is the best time to study the Schwinger mechanism !
Developments in experimental technologies ⇒ Novel strong-field sourcesex.2) Heavy-ion collisionsex.1) Intense lasers
Schwinger’slimit・(ultra)peripheral collisions
𝑒𝐸, 𝑒𝐵 ∼𝛼𝑍𝑣𝛾∼ 𝛼𝑍𝛾 × 𝑚𝜋2
𝑟2[Skokov, Illarionov, Toneev (2009)][Deng, Huang (2012)]
Current
world
record・glasma (color flux tube)
𝑔𝐸color , 𝑔𝐵color ∼ 𝑄s2 ∼ 𝑂 1 GeV 2[Lappi, McLerran (2006)][Mourou, Tajima, Bulanov (2006)]

・Also in other geometries:- asymmetric coll. [Hirono, Hongo, Hirano (2014)] [Voronyuk, et al (2014)]- low-energy coll. [Review: Rafelski, et al (2014)] [Maltsev et al (2019)][Allor, Cohen, McGady (2008)][Solinas, Amoretti, Giazotto (2021)]ex.3) Cond-mat analogues: Graphene, Cold atom, Superconductor, Semiconductor, …
[Szpak, Shutzhold (2012)][Thesis by Linder; 1807.08050]

Timeliness✔ Now is the best time to study the Schwinger mechanism !
Developments in experimental technologies ⇒ Novel strong-field sourcesex.2) Heavy-ion collisionsex.1) Intense lasers
Schwinger’slimit・(ultra)peripheral collisions
𝑒𝐸, 𝑒𝐵 ∼𝛼𝑍𝑣𝛾∼ 𝛼𝑍𝛾 × 𝑚𝜋2
𝑟2[Skokov, Illarionov, Toneev (2009)][Deng, Huang (2012)]
Current
world
record・glasma (color flux tube)
𝑔𝐸color , 𝑔𝐵color ∼ 𝑄s2 ∼ 𝑂 1 GeV 2[Lappi, McLerran (2006)][Mourou, Tajima, Bulanov (2006)]

・Also in other geometries:- asymmetric coll. [Hirono, Hongo, Hirano (2014)] [Voronyuk, et al (2014)]- low-energy coll. [Review: Rafelski, et al (2014)] [Maltsev et al (2019)][Allor, Cohen, McGady (2008)][Solinas, Amoretti, Giazotto (2021)]ex.3) Cond-mat analogues: Graphene, Cold atom, Superconductor, Semiconductor, …
[Szpak, Shutzhold (2012)][Thesis by Linder; 1807.08050]✔ Schwinger’s result is insufficient for actual problems ⇒ Beyond Schwinger !
・ Inhomogeneous field (e.g., time- & space-dependence, more efficient field config., …)・ Realtime dynamics (e.g., backreaction, intermediate particle number, …)・ Higher loop effects (e.g., radiation, mass shift, …)・ Other observables (e.g., spin, chirality, high-harmonic generation…)・…

Today’s talkDiscuss the Schwinger mech. under time-depending E-field
Part I: Interplay b/w non-pert. & pert. production mechanisms[HT, Fujii, Itakura, PRD (2014)][HT et al., JHEP (2021)]

Part II: Dynamically assisted Schwinger mechanism[HT, PRD (2019)][Huang, HT, PRD (2019)][Huang, Matsuo, HT, PTEP (2019)]

Part III: Dynamical assistance to chirality production[HT, PRR (2020)]

Introduction⇓

Part I:

Interplay b/w non-pert. & pert. production mechanisms⇓

Part II:

Dynamically assisted Schwinger mechanism⇓

Part III:

Dynamical assistance to chirality production⇓

Summary

Background (1/2): Interplay b/w non-pert. & pert. mech.
✔ Consider time-dependent E-field, having strength 𝒆𝑬𝟎 and frequency 𝛀Background (1/2): Interplay b/w non-pert. & pert. mech.
✔ Consider time-dependent E-field, having strength 𝒆𝑬𝟎 and frequency 𝛀・ The tunneling picture should break down if the frequency Ω is large
Non-pert. tunneling 𝑁~ exp[#/𝑒𝐸0 ]~ 2m~ 2m/eEtunneling
Tunneling time Δ𝑡 ~2𝑚𝑒𝐸

⇒ E-field should be slower than Δ𝑡⇒ Ω−1 ≿ Δ𝑡
⇒1≿

Δ𝑡
Ω𝑚
=
≡ 𝛾 (Keldysh parameter)−1Ω𝑒𝐸

Background (1/2): Interplay b/w non-pert. & pert. mech.
✔ Consider time-dependent E-field, having strength 𝒆𝑬𝟎 and frequency 𝛀・ The tunneling picture should break down if the frequency Ω is large
Non-pert. tunneling 𝑁~ exp[#/𝑒𝐸0 ]Pert. multi-photon 𝑁~𝑒𝐸02𝑛

~ 2m

Ω
Ω
Ω

~ 2m/eEtunneling
Tunneling time Δ𝑡 ~2𝑚𝑒𝐸

⇒ E-field should be slower than Δ𝑡⇒ Ω−1 ≿ Δ𝑡
⇒1≿

~ 2m

𝑛>

2𝑚
is # of photons involved
Ω

Δ𝑡
Ω𝑚
=
≡ 𝛾 (Keldysh parameter)−1Ω𝑒𝐸・ For large Ω, E-field may behave like a photon and interact incoherently (perturbatively)

Background (2/2): Semi-classical analysis✔ The interplay has been “confirmed” by (but only by) semi-classical analysis
・ Steepest descent analysis [Brezin, Itzykson (1970)]
・ Imaginary-time method [Popov (1972)]
・ Divergent asymptotic series method [Berry (1989)]
・ Worldline instanton method [Dunne, Shubert (2005)]・ (exact) WKB [HT, Fujimori, Misumi, Nitta, Sakai (2020)]
・…

・ Trans-series expansion in terms of ℏ
𝑁 = ෍ 𝑁𝑛,𝑚 ℏ𝑛 e−𝑚𝑆/ℏ = (𝑁0,1 + 𝑂(ℏ)) 𝐞−𝑺/ℏ + 𝑂(e−2𝑆/ℏ )𝑛,𝑚・ Valid in the slow limit ℏ ≪ 1 ⇔ Ω ≪ 1 :∵ iℏ𝜕𝑡 𝜓 = 𝐻 Ω𝑡 𝜓𝜏 ≡ 𝑡/ℏi𝜕𝜏 𝜓 = 𝐻 ℏΩ𝜏 𝜓・ Production number 𝑁 is controlled solely by the Keldysh parameter 𝛾 ≡
𝑚2
𝑆=𝜋
𝑔 𝛾
𝑒𝐸0

2

𝜋
Example: 𝑒𝐸 𝑡 = 𝑒𝐸0 cos(Ω𝑡)See, e.g., [Dunne, Gies, Schubert, Wang (2006)]
for an explicit expression for 𝑔(𝛾)𝑚
+ 𝑂(𝛾 1 )𝑒𝐸02𝑚
ln 𝛾 2 + const. +𝑂(𝛾 −1 )Ω# of photons

pert. power-dep.
after exponentiation𝑚Ω
𝑒𝐸0[Keldysh (1965)](slow limit 𝛾 ≪ 1)
(fast limit 𝛾 ≫ 1)

Background (2/2): Semi-classical analysis✔ The interplay has been “confirmed” by (but only by) semi-classical analysis
・ Steepest descent analysis [Brezin, Itzykson (1970)]
・ Imaginary-time method [Popov (1972)]
・ Divergent asymptotic series method [Berry (1989)]
・ Worldline instanton method [Dunne, Shubert (2005)]・ (exact) WKB [HT, Fujimori, Misumi, Nitta, Sakai (2020)]
・…

・ Trans-series expansion in terms of ℏ
𝑁 = ෍ 𝑁𝑛,𝑚 ℏ𝑛 e−𝑚𝑆/ℏ = (𝑁0,1 + 𝑂(ℏ)) 𝐞−𝑺/ℏ + 𝑂(e−2𝑆/ℏ )𝑛,𝑚・ Valid in the slow limit ℏ ≪ 1 ⇔ Ω ≪ 1 :∵ iℏ𝜕𝑡 𝜓 = 𝐻 Ω𝑡 𝜓i𝜕𝜏 𝜓 = 𝐻 ℏΩ𝜏 𝜓𝜏 ≡ 𝑡/ℏ

・ Production number 𝑁 is controlled solely by the Keldysh parameter 𝛾 ≡
𝑚2
𝑆=𝜋
𝑔 𝛾
𝑒𝐸0

2

𝜋
Example: 𝑒𝐸 𝑡 = 𝑒𝐸0 cos(Ω𝑡)See, e.g., [Dunne, Gies, Schubert, Wang (2006)]
for an explicit expression for 𝑔(𝛾)𝑚
+ 𝑂(𝛾 1 )𝑒𝐸0

[Keldysh (1965)](slow limit 𝛾 ≪ 1)2𝑚
ln 𝛾 2 + const. +𝑂(𝛾 −1 )Ω# of photons𝑚Ω
𝑒𝐸0(fast limit 𝛾 ≫ 1)

pert. power-dep.
after exponentiation✔ However, you should not be convinced with the semi-classical argument
・ Fast limit γ ≫ 1 is dangerous𝑒𝐸ex). Sauter pulsed E-field 𝑒𝐸 𝑡 = cosh20(Ω𝑡)⇒ Q1: When are semi-classicalmethods really OK i.t.o. 𝛀 ?
・ Another dimensionless parameter should exist

𝛾<1

𝛾>1

∵ 3 dimensionful parameters 𝑚, 𝑒𝐸0 , Ω⇒ 2 dimensionless parameters ⇒ Q2: Why only 𝛄 ? The other has no role in the interplay ?

Idea: One-photon process✔ One-photon process is the key to answer Q1 & Q2 !

Idea: One-photon process✔ One-photon process is the key to answer Q1 & Q2 !

✔ Larger 𝛀, more perturbative ⇒ one-photon dominates eventually・because there is no time to interact many times, no matter how strong 𝑒𝐸0 is
𝑬

𝑬

𝑬

…… 𝑬

𝑬

𝑬

…… 𝑬

𝑬

……

𝑬

larger 𝛀 ≃ shorter E-field
𝑬

𝑬

……

𝑬

𝑬

time

multiple scatterings

𝑬

time

finite # of scatterings

・ should cover different region, as photon # should be huge

2𝑚
→ ∞ in the semi-classics
Ω・ analytical formula is available (for any field config. !)
𝑬𝒑, 𝑠

3

3

𝑁1−photon = ෍ නd 𝒑 d 𝒑′
𝑠,𝑠′

2
∞
𝑉
4𝑚2 1
4𝑚2
=
න d𝜔 1 − 2
2+ 2
4𝜋 2 2𝑚𝜔 3
𝜔𝑬𝒑′, 𝑠′𝑒𝐸෨ 𝜔

2[HT, Fujii, Itakura (2014)] [QFT textbook by Itzykson, Zuber]

Result: Interplay b/w tunneling, multi-, one-photon𝒆𝑬✔ An explicit demonstration for Sauter pulsed E-field 𝒆𝑬 𝒕 = 𝐜𝐨𝐬𝐡𝟐𝟎(𝛀𝒕)・ One-photon dominates for large 𝛀,
where semi-classical methods fail,・ One-photon is more efficientthan tunneling ; 𝑁 is the largest at Ω ∼ 2𝑚
𝛾>1

𝜈<1

𝒎𝛀

𝒆𝑬

✔ Two dimensionless parameters 𝜸 = 𝒆𝑬 , 𝝂 ≡ 𝛀𝟐𝟎 control the interplay
𝟎・ Sauter field is solvable, which can becompared w/ Schwinger & one-photon𝛾 ≫ 1, 𝜈𝜈 ≪ 1 ⇒ pert. one−photon𝛾 ≪ 1, 𝜈𝜈 ≫ 1 ⇒ non−pert. tunneling・𝜈 =(work done by E-field)
𝑒𝐸0 /Ω
=Ω(photon energy)= (# of photons involved)

Message of Part I(1) The interplay b/w non-pert. & pert. production mechanisms is controlledby 𝛾 =

𝑚Ω
𝑒𝐸(Keldysh parameter) and also by 𝜈 = 2 (~ # of photons involved)𝑒𝐸Ω(2) Semi-classical methods (e.g., worldline) are dangerous for 𝛾 ≫ 1;
It breaks down for 𝜈 ≿ 1, where one-photon process dominates(3) One-photon production is very efficient, compared to non-pert. tunneling[HT, Fujii, Itakura, PRD (2014)][HT, Fujimori, Misumi, Nitta, Sakai, JHEP (2021)]

Introduction⇓

Part I:

Interplay b/w non-pert. & pert. production mechanisms⇓

Part II:

Dynamically assisted Schwinger mechanism⇓

Part III:

Dynamical assistance to chirality production⇓

Summary

Background (1/2): Dynamically assisted Schwinger mechanism✔ In Part I, I discussed E-field having a single frequency mode 𝛀✔ What if E-field is bi-frequent; superposition of slow strong + weak fast E-fields ?

Background (1/2): Dynamically assisted Schwinger mechanism✔ In Part I, I discussed E-field having a single frequency mode 𝛀✔ What if E-field is bi-frequent; superposition of slow strong + weak fast E-fields ?
・ Combinatorial effects of non-pert. & pert. mechs. ⇒ dynamically assisted Schwinger mech.
[Dunne, Gies, Schutzhold (2008), (2009)][Piazza et al (2009)] [Monin, Voloshin (2010)]

perturbative scattering

quantum tunneling𝑁 ∼ exp[−# × gap height × (gap length)] ⇒ Enhancement in production
reduced by pert.

(even though the fast field is very weak)Background (1/2): Dynamically assisted Schwinger mechanism✔ In Part I, I discussed E-field having a single frequency mode 𝛀✔ What if E-field is bi-frequent; superposition of slow strong + weak fast E-fields ?
・ Combinatorial effects of non-pert. & pert. mechs. ⇒ dynamically assisted Schwinger mech.
[Dunne, Gies, Schutzhold (2008), (2009)][Piazza et al (2009)] [Monin, Voloshin (2010)]

perturbative scattering

quantum tunneling𝑁 ∼ exp[−# × gap height × (gap length)] ⇒ Enhancement in production
reduced by pert.

・ Typically, analyzed within semi-classical methods(even though the fast field is very weak)[Dunne, Gies, Schutzhold (2008), (2009)]

・ Phenomenological importance:ex1) laser: available E-field is still weak ⇒ needs enhancement to observe the Schwinger mechex2) heavy-ion collisions: (Mini-)jets on top of glasma, Event generators (e.g., PYTHIA)

Background (2/2): Franz-Keldysh effect in cond-mat✔ Apply strong slow E-field & a photon (~ weak fast E-field) onto a semi-conductor,
and measure photo-absorption rate
・ photo-absorption rate ～ Im[1-loop action] ～ particle production rate
TheoryExperiment (w/ Si)[Franz (1958)] [Keldysh (1958)]

difference(■ー■)

w/ strong slow E
w/o strong slow E

difference

photoabsorption rate

[Yacoby (1966)](photon energy) - (gap energy)

・ Enhancement below the mass gap⇒ Looks very similar to the dynamically assisted Schwinger mechanism (?)
・ Oscillation above the mass gap (Franz-Keldysh oscillation)

・ Enhancement is maximized around the mass gap

What I am going to do✔ Get a better understanding of the dynamical assistance by …
・ Clarifying the relation to the Franz-Keldysh effect in cond-mat
・ Establishing a novel analytical method, applicable for very fast E-field⇐ Why such method needed ?
(1) Conventional semi-classical approaches are invalid for very fast E-field (result of Part I)(2) Enhancement by pert. one-photon around Ω ∼ 2𝑚 may be important (result of Part I)(3) Such an enhancement is observed in Franz-Keldysh effect(4) The Franz-Keldysh oscillation occurs for very fast E-field
・ Revealing novel features, e.g., behavior at large Ω, spin generation, effective mass concept, …

What I am going to do✔ Get a better understanding of the dynamical assistance by …
・ Clarifying the relation to the Franz-Keldysh effect in cond-mat
・ Establishing a novel analytical method, applicable for very fast E-field⇐ Why such method needed ?
(1) Conventional semi-classical approaches are invalid for very fast E-field (result of Part I)(2) Enhancement by pert. one-photon around Ω ∼ 2𝑚 may be important (result of Part I)(3) Such an enhancement is observed in Franz-Keldysh effect(4) The Franz-Keldysh oscillation occurs for very fast E-field
・ Revealing novel features, e.g., behavior at large Ω, spin generation, effective mass concept, …

✔ Use “the perturbation theory in the Furry picture”
d3 𝑁𝑠†
Goal: Evaluate 3 = ⟨vac|𝑎ො𝒑,𝑠
𝑎ො𝒑,𝑠 |vac⟩ in the presence of strong slow 𝐸s & weak fast ℰfd𝒑

Idea: Perturbative expansion 𝑎ො𝑝,𝑠 i.t.o. ℰf , but no expansion i.t.o. 𝐸s[Furry (1951)] [Fradkin, Gitman, Shvartsman (1991)] [Torgrimsson, Schneider, Shutzhold (2017)]

Perturbation theory in Furry picture (1/3)
STEP 1

Separate the total 𝑬 into strong slow 𝑬𝐒 & weak fast 𝓔𝐟

𝑬 = 𝑬𝐬 + 𝓔𝐟

Perturbation theory in Furry picture (1/3)
STEP 1

Separate the total 𝑬 into strong slow 𝑬𝐒 & weak fast 𝓔𝐟

𝑬 = 𝑬𝐬 + 𝓔𝐟
STEP 2

Solve Dirac eq. non-pert. w.r.t. 𝑬𝐬 , but just pert. w.r.t. 𝓔𝐟𝑖𝜕/ − 𝒆𝑨/ 𝐬 − 𝑚 𝜓෠ = 𝒆𝓐
/ 𝐟 𝜓෠
(0) 𝑥 + ∞ d𝑦 4 𝑆 𝑥, 𝑦 𝒆𝓐 (𝑦)𝜓෠෠
෠ 0 𝑦 + 𝑂(|𝒆𝓐𝐟 |2 )
𝜓
𝑥
=
𝜓
‫׬‬−∞
⇒
/𝐟
R

Here, 𝜓෠ (0) and 𝑆R are non-perturbatively dressed by 𝑬𝐬 as𝑖𝜕/ − 𝒆𝑨/𝐬 − 𝑚 𝜓෠ (0) = 0𝑖𝜕/ − 𝒆𝑨/ 𝐬 − 𝑚 𝑆R (𝑥, 𝑦) = 𝛿 4 (𝑥 − 𝑦)Perturbation theory in Furry picture (2/3)
STEP 3in/outCompute in/out annihilation operators 𝑎ො 𝑝,𝑠in/out𝑎ො𝒑,𝑠in/out†𝑏෠−𝒑,𝑠≡lim𝑡→−∞Τ+∞

න d3 𝒙(𝑢𝒑,𝑠 e−𝑖𝜔𝒑 𝑡 e𝑖𝒑⋅𝒙 )†(𝑣𝒑,𝑠e+𝑖𝜔𝒑 𝑡 e𝑖𝒑⋅𝒙 )†in/out, 𝑏෠𝑝,𝑠෠𝜓(𝑥)from 𝜓෠

Perturbation theory in Furry picture (2/3)
STEP 3in/outCompute in/out annihilation operators 𝑎ො 𝑝,𝑠in/out𝑎ො𝒑,𝑠in/out†𝑏෠−𝒑,𝑠≡lim𝑡→−∞Τ+∞

න d3 𝒙(𝑢𝒑,𝑠 e−𝑖𝜔𝒑 𝑡 e𝑖𝒑⋅𝒙 )†(𝑣𝒑,𝑠e+𝑖𝜔𝒑 𝑡 e𝑖𝒑⋅𝒙 )†in/out, 𝑏෠𝑝,𝑠from 𝜓෠෠𝜓(𝑥)in , 𝑜out are inequivalent 𝑜in ≠ 𝑜out and related with each
ො𝒑,𝑠
ො𝒑,𝑠
ො𝒑,𝑠
⇒ 𝑜ො𝒑,𝑠other by a Bogoliubov transformation
out
𝑎ො𝒑,𝑠
𝛼𝒑,𝑠;𝒑′,𝑠′3out† = ෍ න d 𝒑′ −𝛽 ∗𝑏෠−𝒑,𝑠𝒑,𝑠;𝒑′,𝑠′𝑠′

𝛽𝒑,𝑠;𝒑′,𝑠′∗
𝛼𝒑,𝑠;𝒑′,𝑠′where, up to 1st order in 𝒆𝓐𝐟 ,(0)in4
ത 0 out 𝒆𝓐f + 𝜓 (0)in+ 𝑂(|𝒆𝓐𝐟 |2 )+ 𝜓𝒑′,𝑠′ − 𝑖 න 𝑑 𝑥 + 𝜓𝒑,𝑠
𝒑′ ,𝑠 ′

(0)out†(0)in4
ത 0 out 𝒆𝓐f + 𝜓 (0)in+ 𝑂(|𝒆𝓐𝐟 |2 )+ 𝜓𝒑′,𝑠′ − 𝑖 න 𝑑 𝑥 − 𝜓𝒑,𝑠
𝒑′ ,𝑠 ′

𝛽𝒑,𝑠;𝒑′,𝑠′ = න d3 𝒙 − 𝜓𝒑,𝑠
0 in/out𝑖𝜕/ − 𝒆𝑨/𝐬 − 𝑚/(0)out†𝛼𝒑,𝑠;𝒑′,𝑠′ = න d3 𝒙 + 𝜓𝒑,𝑠Here, ± 𝜓𝒑,𝑠in𝑎ො𝒑′,𝑠′in†𝑏෠−𝒑′,𝑠′/are sol. of Dirac eq. dressed by 𝒆𝑨𝐬 w/ different B.C.
0 in/out=0± 𝜓𝒑,𝑠w/

lim𝑡→−∞Τ+∞

0 in/out+ 𝜓𝒑,𝑠0 in/out− 𝜓𝒑,𝑠=𝑢𝒑,𝑠 e−𝑖𝜔𝒑 𝑡 e𝑖𝒑⋅𝒙𝑣𝒑,𝑠 e−𝑖𝜔𝒑 𝑡 e𝑖𝒑⋅𝒙Perturbation theory in Furry picture (3/3)
STEP 4

Evaluate the in-vacuum expectation value of # operator
d3 𝑁𝑒
2out† out
3 𝒑′ 𝛽
′
′
න
≡
vac;
in
𝑎
𝑎
vac;
in
=
෍
d
𝒑,𝑠
𝒑,𝑠
𝒑,𝑠;𝒑 ,𝑠d𝒑3
𝑠′Perturbation theory in Furry picture (3/3)
STEP 4

Evaluate the in-vacuum expectation value of # operator
d3 𝑁𝑒
2out† out
3 𝒑′ 𝛽
′
′
න
≡
vac;
in
𝑎
𝑎
vac;
in
=
෍
d
𝒑,𝑠
𝒑,𝑠
𝒑,𝑠;𝒑 ,𝑠d𝒑3𝑠′Assume 𝑬𝐬 is so slow that it can be approximated as a static E-field
0 in/out⇒ analytical sol. of Dirac eq. ± 𝜓𝒑,𝑠

⇒ one can evaluate 𝛽𝒑,𝑠;𝒑′ ,𝑠′ exactly !

is known

Perturbation theory in Furry picture (3/3)
STEP 4

Evaluate the in-vacuum expectation value of # operator
d3 𝑁𝑒
2out† out
3 𝒑′ 𝛽
′
′
න
≡
vac;
in
𝑎
𝑎
vac;
in
=
෍
d
𝒑,𝑠
𝒑,𝑠
𝒑,𝑠;𝒑 ,𝑠d𝒑3𝑠′Assume 𝑬𝐬 is so slow that it can be approximated as a static E-field
0 in/out⇒ analytical sol. of Dirac eq. ± 𝜓𝒑,𝑠

is known⇒ one can evaluate 𝛽𝒑,𝑠;𝒑′ ,𝑠′ exactly !

✔ Remarks
・ Directly computing VEV of # operator[Baltz, McLerran (2001)]⇒ inclusive quantity that includes all the processes up to 1st order in 𝓔𝐟
out† out
vac; in 𝑎𝒑,𝑠
𝑎𝒑,𝑠 vac; in = ෍ 𝑒𝒑,𝑠 𝑋; out|vac; in

2

=෍
𝑋

𝑋

X;out

=

2

2

+
Schwinger

2

2

2

+

𝓔𝐟

+

𝓔𝐟

+

2
𝓔𝐟

+…

multiple
Schwinger
Breit-Wheeler Schwinger
Schwinger+ absorption + annihilation

・ No approximation in evaluating 𝛽𝒑,𝑠;𝒑′ ,𝑠′within 0-th order WKB [Torgrimsson et al (2017)]

Result (1/6): Analytical formula for parallel 𝑬𝐬 || 𝓔𝐟✔ Analytical formula for 𝑬 = (0,0, 𝐸s + ℰf (𝑡)), with arbitrary time-dep.
・ is applicable even for very fast ℰf and reproduces numerics very well (show later)d3 𝑁𝑒𝑉𝜋(𝑚2 +𝒑2⊥ )
=
exp −
d𝒑3
2𝜋 3
𝑒𝐸s
1 𝑚2 + 𝒑2⊥ ∞ℰሚ𝐟 𝜔𝑖 𝜔2 + 4𝜔𝑝∥
× 1 +
න 𝑑𝜔
exp −
2 𝑒𝐸s
𝐸s
4
𝑒𝐸s
0

1 𝐹1

𝑖 𝑚2 + 𝒑2⊥
𝑖 𝜔2
1−
; 2;
2 𝑒𝐸s
2 𝑒𝐸s

2

Result (1/6): Analytical formula for parallel 𝑬𝐬 || 𝓔𝐟✔ Analytical formula for 𝑬 = (0,0, 𝐸s + ℰf (𝑡)), with arbitrary time-dep.
・ is applicable even for very fast ℰf and reproduces numerics very well (show later)d3 𝑁𝑒𝑉𝜋(𝑚2 +𝒑2⊥ )
=
exp −
d𝒑3
2𝜋 3
𝑒𝐸s
1 𝑚2 + 𝒑2⊥ ∞ℰሚ𝐟 𝜔𝑖 𝜔2 + 4𝜔𝑝∥
× 1 +
න 𝑑𝜔
exp −
2 𝑒𝐸s
𝐸s
4
𝑒𝐸s
0

Schwinger mech. by slow 𝑬𝐬

1 𝐹1

𝑖 𝑚2 + 𝒑2⊥
𝑖 𝜔2
1−
; 2;
2 𝑒𝐸s
2 𝑒𝐸s

2

Result (1/6): Analytical formula for parallel 𝑬𝐬 || 𝓔𝐟✔ Analytical formula for 𝑬 = (0,0, 𝐸s + ℰf (𝑡)), with arbitrary time-dep.
・ is applicable even for very fast ℰf and reproduces numerics very well (show later)d3 𝑁𝑒𝑉𝜋(𝑚2 +𝒑2⊥ )
=
exp −
d𝒑3
2𝜋 3
𝑒𝐸s
1 𝑚2 + 𝒑2⊥ ∞ℰሚ𝐟 𝜔𝑖 𝜔2 + 4𝜔𝑝∥
× 1 +
න 𝑑𝜔
exp −
2 𝑒𝐸s
𝐸s
4
𝑒𝐸s
0

Schwinger mech. by slow 𝑬𝐬

1 𝐹1

𝑖 𝑚2 + 𝒑2⊥
𝑖 𝜔2
1−
; 2;
2 𝑒𝐸s
2 𝑒𝐸s

2

Dynamical assistance by fast 𝓔𝐟

・ describes interplay b/w Schwinger & one-photon process smoothly:- Slow limit 𝜔Τ 𝑒𝐸s ≪ 1: ■ dominates ⇒ Schwinger formulad3 𝑁𝑒
𝑉𝜋(𝑚2 +𝒑2⊥ )𝜋 𝑚2 + 𝒑2⊥ ℰf
∼
exp −
1+
d𝒑3
2𝜋 3
𝑒𝐸s
2 𝑒𝐸s 𝐸s2

𝑉
𝜋(𝑚2 +𝒑2⊥ )
∼
exp −
2𝜋 3
𝑒(𝐸s + ℰf )- Fast limit 𝜔Τ 𝑒𝐸s ≫ 1: ■ dominates ⇒ one-photon process
d3 𝑁𝑒
𝑉 1 𝑚2 + 𝒑2⊥ |𝑒 ℰ෩f 2𝜔𝒑 |2∼d𝒑3
2𝜋 3 4 𝜔𝒑2
𝜔𝒑2

Result (2/6): Total N & Relation to Franz-Keldysh effect✔ For an oscillating perturbation 𝓔𝐟 𝒕 = 𝓔 𝐜𝐨𝐬(𝛀𝒕) as a demonstration𝑵𝒆𝑬𝒔 𝟐 𝑽𝑻For 𝑚Τ 𝑒𝐸s = 2.5; ℰ Τ𝐸s = 0.01

difference (■－■)
pert(𝑁𝑒 − 𝑁𝑒)/ 𝑒𝐸s 2 𝑉𝑇𝑚Τ 𝑒𝐸s = 3.0𝑚Τ 𝑒𝐸s = 4.0𝑚Τ 𝑒𝐸s = 5.0𝑚Τ 𝑒𝐸s = 10.0

perturbative(prod. w/o 𝐸s )𝑚Τ 𝑒𝐸s = 25.0

our formula(prod. w/ 𝐸s )Schwinger

frequency 𝛀/ 𝒆𝑬𝒔

frequency(𝛀 − 𝟐𝒎)/ 𝒆𝑬𝒔

Result (2/6): Total N & Relation to Franz-Keldysh effect✔ For an oscillating perturbation 𝓔𝐟 𝒕 = 𝓔 𝐜𝐨𝐬(𝛀𝒕) as a demonstration𝑵𝒆𝑬𝒔 𝟐 𝑽𝑻For 𝑚Τ 𝑒𝐸s = 2.5; ℰ Τ𝐸s = 0.01

difference (■－■)
pert(𝑁𝑒 − 𝑁𝑒)/ 𝑒𝐸s 2 𝑉𝑇𝑚Τ 𝑒𝐸s = 3.0𝑚Τ 𝑒𝐸s = 4.0𝑚Τ 𝑒𝐸s = 5.0𝑚Τ 𝑒𝐸s = 10.0

perturbative(prod. w/o 𝐸s )𝑚Τ 𝑒𝐸s = 25.0

our formula(prod. w/ 𝐸s )Schwinger

frequency 𝛀/ 𝒆𝑬𝒔

frequency(𝛀 − 𝟐𝒎)/ 𝒆𝑬𝒔

Result (2/6): Total N & Relation to Franz-Keldysh effect✔ For an oscillating perturbation 𝓔𝐟 𝒕 = 𝓔 𝐜𝐨𝐬(𝛀𝒕) as a demonstration
𝑵
𝒆𝑬𝒔

𝟐 𝑽𝑻

For 𝑚Τ 𝑒𝐸s = 2.5; ℰ Τ𝐸s = 0.01

difference (■－■)
pert(𝑁𝑒 − 𝑁𝑒)/ 𝑒𝐸s 2 𝑉𝑇𝑚Τ 𝑒𝐸s = 3.0𝑚Τ 𝑒𝐸s = 4.0𝑚Τ 𝑒𝐸s = 5.0𝑚Τ 𝑒𝐸s = 10.0

perturbative(prod. w/o 𝐸s )𝑚Τ 𝑒𝐸s = 25.0

our formula(prod. w/ 𝐸s )Schwinger

frequency 𝛀/ 𝒆𝑬𝒔✔ Completely the same as the Franz-Keldysh effect !
・ enhancement below the mass gap
・ oscillation above the mass gap

・ enhancement becomes the maximum at around the mass gap⇒ dynamically assisted Schwinger mechanism = Franz-Keldysh effectfrequency(𝛀 − 𝟐𝒎)/ 𝒆𝑬𝒔

Result (3/6): Momentum dist. & comp. w/ numerics✔ Similar enhancement/oscillating behavior for the momentum dist.
momentum dist.
𝟐𝝅 𝟑 𝐝𝟑 𝑵𝒆
𝑽 𝐝𝒑𝟑

For 𝑚Τ 𝑒𝐸s = 2.5; ℰ Τ𝐸s = 0.01

frequency 𝛀/ 𝒆𝑬𝒔

・ excellent agreement b/w our analytical formula and the numerics⇒ One-photon process it the essence of the dynamical assistance・ Another notable feature: the biggest peak (~ 1-photon peak) is above the gap Ω > 2𝜔𝑝cf) effective mass shift in the multi-photon regime[Kohlfurst, Gies, Alkofer (2014)]

Result (4/6): Physical interpretation

××

probability density

・ quantum tunneling ⇒ enhancement
・ quantum reflection ⇒ oscillation⇒ production occurs most efficiently at the maxima

Result (5/6): Analytical formula for general 𝑬𝐬 ∦ 𝓔𝐟
✔ Generalization of the analytical formula for 𝑬𝐬 || 𝓔𝐟 → 𝑬𝐬 ∦ 𝓔𝐟
d3 𝑁𝑒
𝑉𝜋(𝑚2 +𝒑2⊥ )
=
exp −
d𝒑3
2𝜋 3
𝑒𝐸s

×

∞

1 + න 𝑑𝜔 e

−𝑖

0
∞

+𝑖 න 𝑑𝜔 e

−𝑖

𝜔𝑝∥
2 ෩
2
1
𝑚
+
𝒑
⊥ 𝓔f 𝜔
𝑒𝐸s

2

𝜔2
4𝑒𝐸s

1 𝐹1

𝑖 𝑚2 + 𝒑2⊥
𝑖 𝜔2
1−
; 2;
2 𝑒𝐸s
2 𝑒𝐸s

2

𝜔𝑝 ෩
−𝑖 ∥ (𝓔f 𝜔𝑑𝜔 e 𝑒𝐸s0𝑥𝜔𝑝−𝑖 ∥ 𝑚 ℰሚf 𝜔𝑑𝜔 e 𝑒𝐸s

∞

0

e

−𝑖

2
2
𝜔
⋅ 𝒑⊥
𝑖
𝑚
+
𝒑
𝑖 𝜔2
−𝑖
⊥
4𝑒𝐸
s
Re e
; 1;
1 𝐹1 1 −
𝐸s 𝜔
2 𝑒𝐸s
2 𝑒𝐸s

∞

+ න

𝐸s2

𝑒𝐸s

𝜔𝑝∥ ෩
𝓔 𝜔
𝑒𝐸s f

0

+𝑠 × 𝑖 න

⋅ 𝑬s

𝜔

2

2

𝜔
× 𝒑⊥ ) ⋅ 𝑬s
𝑖 𝑚2 + 𝒑2⊥
𝑖 𝜔2
−𝑖
4𝑒𝐸
s 𝐹
Im e
; 1;
1 1 1−
2 𝑒𝐸s
2 𝑒𝐸s
𝐸s2 𝜔

𝑦
𝜔2
+ 𝑖𝑠ℰሚf 𝜔𝑖 𝑚2 + 𝒑2⊥
𝑖 𝜔2
−𝑖
4𝑒𝐸
s
Im e
; 1;
1 𝐹1 1 −
𝐸s
2 𝑒𝐸s
2 𝑒𝐸s

・ becomes complicated (green = new terms), but the basic structure is the same2

Result (5/6): Analytical formula for general 𝑬𝐬 ∦ 𝓔𝐟
✔ Generalization of the analytical formula for 𝑬𝐬 || 𝓔𝐟 → 𝑬𝐬 ∦ 𝓔𝐟
d3 𝑁𝑒
𝑉𝜋(𝑚2 +𝒑2⊥ )
=
exp −
d𝒑3
2𝜋 3
𝑒𝐸s

×

∞

1 + න 𝑑𝜔 e0∞+𝑖 න 𝑑𝜔 e
Schwingermechanismby slow 𝑬𝐬

−𝑖

−𝑖

𝜔𝑝∥
2 ෩
2
1
𝑚
+
𝒑
⊥ 𝓔f 𝜔
𝑒𝐸s

2

𝜔2
4𝑒𝐸s

1 𝐹1

𝑖 𝑚2 + 𝒑2⊥
𝑖 𝜔2
1−
; 2;
2 𝑒𝐸s
2 𝑒𝐸s

2

𝜔𝑝 ෩
−𝑖 ∥ (𝓔f 𝜔𝑑𝜔 e 𝑒𝐸s0𝑥𝜔𝑝−𝑖 ∥ 𝑚 ℰሚf 𝜔𝑑𝜔 e 𝑒𝐸s

∞

0

e

−𝑖

2
2
𝜔
⋅ 𝒑⊥
𝑖
𝑚
+
𝒑
𝑖 𝜔2
−𝑖
⊥
4𝑒𝐸
s
Re e
; 1;
1 𝐹1 1 −
𝐸s 𝜔
2 𝑒𝐸s
2 𝑒𝐸s

∞

+ න

𝐸s2

𝑒𝐸s

𝜔𝑝∥ ෩
𝓔 𝜔
𝑒𝐸s f

0

+𝑠 × 𝑖 න

⋅ 𝑬s

𝜔

2

2

𝜔
× 𝒑⊥ ) ⋅ 𝑬s
𝑖 𝑚2 + 𝒑2⊥
𝑖 𝜔2
−𝑖
4𝑒𝐸
s 𝐹
Im e
; 1;
1 1 1−
2 𝑒𝐸s
2 𝑒𝐸s
𝐸s2 𝜔

𝑦
𝜔2
+ 𝑖𝑠ℰሚf 𝜔𝑖 𝑚2 + 𝒑2⊥
𝑖 𝜔2
−𝑖
4𝑒𝐸
s
Im e
; 1;
1 𝐹1 1 −
𝐸s
2 𝑒𝐸s
2 𝑒𝐸s

・ becomes complicated (green = new terms), but the basic structure is the same2

Result (5/6): Analytical formula for general 𝑬𝐬 ∦ 𝓔𝐟
✔ Generalization of the analytical formula for 𝑬𝐬 || 𝓔𝐟 → 𝑬𝐬 ∦ 𝓔𝐟
d3 𝑁𝑒
𝑉𝜋(𝑚2 +𝒑2⊥ )
=
exp −
d𝒑3
2𝜋 3
𝑒𝐸s

×

∞

1 + න 𝑑𝜔 e0∞+𝑖 න 𝑑𝜔 e
Schwingermechanismby slow 𝑬𝐬−𝑖−𝑖

Dynamical assistance by fast 𝓔𝐟

𝜔𝑝∥
2 ෩
2
1
𝑚
+
𝒑
⊥ 𝓔f 𝜔
𝑒𝐸s

2

𝜔2
4𝑒𝐸s

1 𝐹1

𝑖 𝑚2 + 𝒑2⊥
𝑖 𝜔2
1−
; 2;
2 𝑒𝐸s
2 𝑒𝐸s

2

𝜔𝑝 ෩
−𝑖 ∥ (𝓔f 𝜔𝑑𝜔 e 𝑒𝐸s0𝑥𝜔𝑝−𝑖 ∥ 𝑚 ℰሚf 𝜔𝑑𝜔 e 𝑒𝐸s

∞

0

e

−𝑖

2
2
𝜔
⋅ 𝒑⊥
𝑖
𝑚
+
𝒑
𝑖 𝜔2
−𝑖
⊥
4𝑒𝐸
s
Re e
; 1;
1 𝐹1 1 −
𝐸s 𝜔
2 𝑒𝐸s
2 𝑒𝐸s

∞

+ න

𝐸s2

𝑒𝐸s

𝜔𝑝∥ ෩
𝓔 𝜔
𝑒𝐸s f

0

+𝑠 × 𝑖 න

⋅ 𝑬s

𝜔

2

2

𝜔
× 𝒑⊥ ) ⋅ 𝑬s
𝑖 𝑚2 + 𝒑2⊥
𝑖 𝜔2
−𝑖
4𝑒𝐸
s 𝐹
Im e
; 1;
1 1 1−
2 𝑒𝐸s
2 𝑒𝐸s
𝐸s2 𝜔

𝑦
𝜔2
+ 𝑖𝑠ℰሚf 𝜔𝑖 𝑚2 + 𝒑2⊥
𝑖 𝜔2
−𝑖
4𝑒𝐸
s
Im e
; 1;
1 𝐹1 1 −
𝐸s
2 𝑒𝐸s
2 𝑒𝐸s

・ becomes complicated (green = new terms), but the basic structure is the same2

Result (5/6): Analytical formula for general 𝑬𝐬 ∦ 𝓔𝐟
✔ Generalization of the analytical formula for 𝑬𝐬 || 𝓔𝐟 → 𝑬𝐬 ∦ 𝓔𝐟
d3 𝑁𝑒
𝑉𝜋(𝑚2 +𝒑2⊥ )
=
exp −
d𝒑3
2𝜋 3
𝑒𝐸s

×

∞

1 + න 𝑑𝜔 e0∞+𝑖 න 𝑑𝜔 e
Schwingermechanismby slow 𝑬𝐬−𝑖−𝑖

Dynamical assistance by fast 𝓔𝐟

𝜔𝑝∥
2 ෩
2
1
𝑚
+
𝒑
⊥ 𝓔f 𝜔
𝑒𝐸s

2

𝜔2
4𝑒𝐸s

1 𝐹1

𝑖 𝑚2 + 𝒑2⊥
𝑖 𝜔2
1−
; 2;
2 𝑒𝐸s
2 𝑒𝐸s

2

𝜔𝑝 ෩
−𝑖 ∥ (𝓔f 𝜔𝑑𝜔 e 𝑒𝐸s0𝑥𝜔𝑝−𝑖 ∥ 𝑚 ℰሚf 𝜔𝑑𝜔 e 𝑒𝐸s

∞

0

e

−𝑖

2
2
𝜔
⋅ 𝒑⊥
𝑖
𝑚
+
𝒑
𝑖 𝜔2
−𝑖
⊥
4𝑒𝐸
s
Re e
; 1;
1 𝐹1 1 −
𝐸s 𝜔
2 𝑒𝐸s
2 𝑒𝐸s

∞

+ න

𝐸s2

𝑒𝐸s

𝜔𝑝∥ ෩
𝓔 𝜔
𝑒𝐸s f

0

+𝑠 × 𝑖 න

⋅ 𝑬s

𝜔

𝑦
𝜔2
+ 𝑖𝑠ℰሚf 𝜔𝑖 𝑚2 + 𝒑2⊥
𝑖 𝜔2
−𝑖
4𝑒𝐸
s
Im e
; 1;
1 𝐹1 1 −
𝐸s
2 𝑒𝐸s
2 𝑒𝐸s

・ becomes complicated (green = new terms), but the basic structure is the same・ spin-dependence appears even without magnetic fields [Takayoshi, Wu, Oka (2020)]∵ Dirac particle has a spin-orbit coupling 𝒔 ⋅ 𝒑 × 𝑬[Foldy, Wouthuysen (1950)] [Tani (1951)]22

𝜔× 𝒑⊥ ) ⋅ 𝑬s
𝑖 𝑚2 + 𝒑2⊥
𝑖 𝜔2
−𝑖
4𝑒𝐸
s 𝐹
Im e
; 1;
1 1 1−
2 𝑒𝐸s
2 𝑒𝐸s
𝐸s2 𝜔

𝐸s

ℰf

・ can be applied to rotating E-fields: 𝐸 = 𝐸0 cos Ω𝑡 , sin Ω𝑡 , 0 ∼ (𝐸0 , 𝐸0 Ω𝑡, 0)

Numerical studies: [Blinne, Strobel (2015)] [Strobel, Xue (2015)] [Woller, Bauke, Keitel (2015)] [Kohlfurst (2019)]

2

Result (6/6): Spin-dependent productionℰ cos 𝜃ℰ⊥ × cos Ω𝑡✔ Non-parallel superposition : 𝑬 = ℰ sin 𝜃ℰ⊥ × cos Ω𝑡
𝐸s
1 d3 𝑁 ↑momentum dist. of spin up
𝑉 d𝒑𝟑

diff. b/w spin up and down

For 𝑒𝐸s /𝑚2 = 2.5; ℰ Τ𝐸s = 0.025

・ The dynamical assistance to N is basically the same as the parallel case・ Spin-dependence appears ⇒ O(10%) effect ⇒ non-negligible・ 𝜃𝒑⊥ -dependent because of the spin-orbit interaction 𝒔 ⋅ (𝒑 × 𝓔)

1 d 3 𝑁↑1 d3 𝑁 ↓
−
𝑉 d𝒑𝟑
𝑉 d𝒑𝟑

Message of Part II(1) The dynamically assisted Schwinger mechanism is an analogue ofthe Franz-Keldysh effect in cond-mat(2) Perturbation theory in the Furry picture provides a very powerful analytical formula(3) Not only the enhancement due to the quantum tunneling and one-photon assistance,
but also oscillation appears due to the quantum reflection(4) Spin-dependent production occurs when transverse E-field is time-dependent[HT, PRD (2019)][Huang, HT, PRD (2019)][Huang, Matsuo, HT, PTEP (2019)]

Introduction⇓

Part I:

Interplay b/w non-pert. & pert. production mechanisms⇓

Part II:

Dynamically assisted Schwinger mechanism⇓

Part III:

Dynamical assistance to chirality production⇓

Summary

Background: Chirality production✔ Chirality is produced through anomaly when 𝑬 ⋅ 𝑩 ≠ 0

Background: Chirality production✔ Chirality is produced through anomaly when 𝑬 ⋅ 𝑩 ≠ 0・ Microscopically, caused by the interplay b/w Schwinger mech. by E-field& Landau quantization by B-field

E-field
momentum

spin

-

+

momentumB-field⇒ chirality ~ helicity = +𝟐 × 𝑵𝐩𝐚𝐢𝐫 𝐢𝐧 𝐋𝐋𝐋

spin[Nielsen, Ninomiya (1983)][Tanji (2010)][Fukushima, Kharzeev, Warringa (2010)][Copinger, Fukushima, Pu (2018)]
etc …

・ 𝑬 ⋅ 𝑩 ≠ 0 may/can be realized at: heavy-ion collisions, early Universe, laser, …

Background: Chirality production✔ Chirality is produced through anomaly when 𝑬 ⋅ 𝑩 ≠ 0・ Microscopically, caused by the interplay b/w Schwinger mech. by E-field& Landau quantization by B-field

E-field
momentum

spin

-

+

momentumspin

B-field⇒ chirality ~ helicity = +𝟐 × 𝑵𝐩𝐚𝐢𝐫 𝐢𝐧 𝐋𝐋𝐋[Nielsen, Ninomiya (1983)][Tanji (2010)][Fukushima, Kharzeev, Warringa (2010)][Copinger, Fukushima, Pu (2018)]
etc …

・ 𝑬 ⋅ 𝑩 ≠ 0 may/can be realized at: heavy-ion collisions, early Universe, laser, …

✔ Chirality production is suppressed strongly by mass2

∵ 𝑁pair in LLL ∝ e−# 𝑚 /𝑒𝐸 ⇒ difficult to observe …

Background: Chirality production✔ Chirality is produced through anomaly when 𝑬 ⋅ 𝑩 ≠ 0・ Microscopically, caused by the interplay b/w Schwinger mech. by E-field& Landau quantization by B-field

E-field
momentum

spin

-

+

momentumspin

B-field⇒ chirality ~ helicity = +𝟐 × 𝑵𝐩𝐚𝐢𝐫 𝐢𝐧 𝐋𝐋𝐋[Nielsen, Ninomiya (1983)][Tanji (2010)][Fukushima, Kharzeev, Warringa (2010)][Copinger, Fukushima, Pu (2018)]
etc …

・ 𝑬 ⋅ 𝑩 ≠ 0 may/can be realized at: heavy-ion collisions, early Universe, laser, …

✔ Chirality production is suppressed strongly by mass2

∵ 𝑁pair in LLL ∝ e−# 𝑚 /𝑒𝐸 ⇒ difficult to observe …

Q: Any way to avoid the mass suppression ? ⇒ A: Dynamical assistance !

Theory: Perturbation theory in the Furry picture✔ Extend the E-field case (Part II) to 𝑬||𝑩 case・ Repeat essentially the same calculation, but with slight modifications:Theory: Perturbation theory in the Furry picture✔ Extend the E-field case (Part II) to 𝑬||𝑩 case・ Repeat essentially the same calculation, but with slight modifications:ഥ, 𝑩ഥ with perturbative weak fast E-field ℰ
SETUP: Parallel 𝑬
ഥ+ℰ 𝑡
𝑬 𝑡 =𝑬ഥ
𝑩 𝑡 =𝑩ഥ ഥ
STEP 1: Solve Dirac eq. under 𝑬, 𝑩 non-perturbatively, and include
effects of ℰ perturbatively
ഥ/ − 𝑚 𝜓෠ = 𝒆𝓐/ 𝜓෠𝑖𝜕/ − 𝒆𝑨(0) 𝑥 + ∞ d𝑦 4 𝑆 𝑥, 𝑦 𝒆𝓐(𝑦)𝜓
෠ 0 𝑦 + 𝑂(|𝒆𝓐|2 )‫׬‬−∞⇒ 𝜓෠ 𝑥 = 𝜓෠STEP 2: Compute VEV of chirality operatorത෠ 5 𝜓෠ vac; in = 𝑂 1 + 𝑂 𝒆𝓐 1 + 𝑂 𝒆𝓐 2 + ⋯𝑄5 ≡ lim ‫ ׬‬d𝒙3 vac; in 𝜓𝛾t→∞Result (1/2): Analytical formula22 ∞ሚ
ത 𝐵ത −𝜋𝑚2
𝑄5 𝑒𝐸𝑒
2𝜋 𝑚2
ℰ(𝜔)
i 𝑚2
i 𝜔2
=
e 𝑒𝐸ത × [1 +න d𝜔
.1 𝐹෨1 (1 −
; 2;
) ]
2
ത
ത
ത
ത
𝑉𝑇
2𝜋
𝑇 𝑒𝐸
2
2
𝐸
𝑒
𝐸
𝑒
𝐸
0

Dynamical assistance by fast E-field ℰ・ Dynamical assistance- is positive ⇒ chirality is always enhanced- goes away when 𝑚 → 0 ⇒ important only for massive case- is independent of B-field(∵ B-field does not supply energy, i.e., does not affect the prod. mech.)Result (2/2): Plot for chirality production✔ An oscillating pert. as demonstration:

𝑒𝐵 𝑡
𝑒 𝐵ത
𝑒𝐸 𝑡
𝑒𝐸ത
= 2
= 2 + 0.001 cos Ω𝑡 , .
𝑚2
𝑚
𝑚2
𝑚

✔ The same behavior as the dynamically assisted Schwinger mech.
・ Free from the exponential suppression due to the enhancement

・ Enhancement becomes largest around the mass gap
・ Oscillation above the mass gap

Message of Part III

Chirality production can be enhanced significantly via the dynamical assistance[HT, PRR (2020)]

Introduction⇓

Part I:

Interplay b/w non-pert. & pert. production mechanisms⇓

Part II:

Dynamically assisted Schwinger mechanism⇓

Part III:

Spin & chirality production⇓

SummarySummary
I discussed the Schwinger mechanism under time-depending E-field:
Part I: Interplay b/w non-pert. & pert. production mechanisms
・ The interplay is controlled by 𝛾 ≡𝑚Ω𝑒𝐸(Keldysh parameter) and also by 𝜈 ≡ 2𝑒𝐸Ω

・ Semi-classical methods are invalid when 𝜈 ≿ 1, where one-photon process dominates

・ One-photon production is very efficient, compared to non-pert. tunnelingPart II: Dynamically assisted Schwinger mechanism
・ Point out relation to Franz-Keldysh effect in cond-mat.

・ Analytical analysis based on perturbation theory in the Furry picture
・ Not only enhancement due to quantum tunneling & one-photon assist,
but also oscillation due to quantum reflection
・ Spin-dependent production occurs when transverse E-field is time-dependent

Part III: Dynamical assistance to chirality production
・ Chirality production can be enhanced significantly via the dynamical assistanceBACKUPStrong EM fields in HIC✔ 3 ways to produce (as far as I know)① (ultra)peripheral② asymmetric③ low-energy

B
×
・
・

𝑒𝐵 ∼

𝛼𝑍𝑣𝛾
∼ 𝛼𝑍𝛾 × 𝑚𝜋22𝑟[Deng, Huang (2012)]

𝑍1

E

×

𝑍2

𝛼 𝑍1 − 𝑍2 𝛾
𝑟2∼ 𝛼(𝑍1 − 𝑍2 )𝛾 × 𝑚𝜋2E𝑍tot = 𝑍1 + 𝑍2

𝑒𝐸 ∼[Voronyuk, Toneev, Voloshin, Cassing (2014)]

𝑒𝐸 ∼

𝛼𝑍tot
∼ 𝛼𝑍tot × 𝑚𝜋2
2𝑟[Review: Rafelski, Kirsch, Muller,
Reinhardt, Greiner (2014)]

Glasma✔ Gluon saturation of ultra-relativistic nuclei[McLerran, Venugopalan (1994)]⇒ something like a “color capacitor”
w/ huge color charge density = 𝑂 𝑄𝑠 = 𝑂(a few GeV)

[Lappi, McLerran (2006)]✔ High-energy heavy-ion collisions≃ formation of “color condenser”⇒ strong color flux tubes
Old ideas: [Low (1975)] [Nussinov (1975)] [Casher, Neuberger, Nussinov (1979)]