Strong-field physics
in heavy-ion collisions
Hidetoshi Taya
(Keio University)Today’s talk
Q: What happens if we make light (or “field” in general) stronger and stronger ?
Laser welding

LED

Cond.-mat(THz laser)

Guiness record???

?
HERCULES @ USA (2008)𝐼 ∼ 10−5 W/cm2

106 W/cm2

1010 W/cm2

1022 W/cm2

or 𝑒𝐸 ∼ 10−1 eVΤcm

104 eVΤcm

106 eVΤcm

1012 eVΤcm

10−1 eV 2

1 eV 2−3∼ 10 eV
Daily life2

Industry

1 keVScience2

Our limit
at the present

Today’s talk
Q: What happens if we make light (or “field” in general) stronger and stronger ?
Laser welding

LED

Cond.-mat(THz laser)???

Guiness record?
HERCULES @ USA (2008)𝐼 ∼ 10−5 W/cm2

106 W/cm2

1010 W/cm2

1022 W/cm2

or 𝑒𝐸 ∼ 10−1 eVΤcm

104 eVΤcm

106 eVΤcm

1012 eVΤcm

10−1 eV 2

1 eV 2−3∼ 10 eV
Daily life2

Industry

1 keV

2

Our limit
at the present

Science

Purpose: Review physics of such strong fieldTake-home messages:(1) Once 𝒆𝑬 > (typical energy scale), sthg extremely non-trivial occur(e.g., Schwinger effect ≈ “something” from “nothing” )(2) Such strong fields are now (or soon will be) within the exp. reach(3) Of relevance to hadron/QCD physics, in particular, heavy-ion collisions

Contents
1. Overview of strong-field physics
Review: [Fedotov, Ilderton, Karbstein, King, Seipt, HT, Torgrimsson, 2203.00019]

2. Recent development of the Schwinger effect・focus on the Schwinger effect with time-dependent E fields[HT, Itakura, Fujii, 1405.6182] [HT, 1812.03630] [HT, Fujimori, Misumi, Nitta, Sakai, 2010.16080][HT, Ironside, 2308.11248] [HT, Nishimura, Ohnishi, 2402.17136]

3. An application of the Schwinger effect to QCD:
the early-time dynamics of heavy-ion collisions
・Quark production is very fast !

[HT, 1609.06189] [HT, Ph. D thesis (2017)]

Contents
1. Overview of strong-field physics
Review: [Fedotov, Ilderton, Karbstein, King, Seipt, HT, Torgrimsson, 2203.00019]

2. Recent development of the Schwinger effect・focus on the Schwinger effect with time-dependent E fields[HT, Itakura, Fujii, 1405.6182] [HT, 1812.03630] [HT, Fujimori, Misumi, Nitta, Sakai, 2010.16080][HT, Ironside, 2308.11248] [HT, Nishimura, Ohnishi, 2402.17136]

3. An application of the Schwinger effect to QCD:
the early-time dynamics of heavy-ion collisions
・Quark production is very fast !

[HT, 1609.06189] [HT, Ph. D thesis (2017)]

What if field becomes strong ?
final

initialtime evol.

No fieldWhat if field becomes strong ?
final

initialtime evol.

No field

Weak field (𝒆𝑬, 𝒆𝑩Τ𝒎𝟐 ≪ 𝟏)

Strong field (𝒆𝑬, 𝒆𝑩Τ𝒎𝟐 ≫ 𝟏)

What if field becomes strong ?
final

initialtime evol.

𝑭

No field

Weak field (𝒆𝑬, 𝒆𝑩Τ𝒎𝟐 ≪ 𝟏)

Strong field (𝒆𝑬, 𝒆𝑩Τ𝒎𝟐 ≫ 𝟏)

Only minor changes⇒ Perturbative⇒ Very well understoodin both exp.& theor.
g−2ex.) Electron (anomalous) magnetic moment 𝑎 ≔2≈ Electron energy shift in a weak magnetic field𝑎 theor. = 1159652182.03 … × 10−12
𝑎 exp.
= 1159652180.73 … × 10−12[Aoyama, Kinoshita, Nio (2017)]

What if field becomes strong ?
final

initial𝑭
time evol.

𝑭

𝑭𝑭

No field

Weak field (𝒆𝑬, 𝒆𝑩Τ𝒎𝟐 ≪ 𝟏)

𝑭…

Strong field (𝒆𝑬, 𝒆𝑩Τ𝒎𝟐 ≫ 𝟏)

Only minor changes

Big change !

⇒ Perturbative⇒ Non-Perturbative⇒ Very well understoodin both exp.& theor.

⇒ Not understood wellg−2ex.) Electron (anomalous) magnetic moment 𝑎 ≔2≈ Electron energy shift in a weak magnetic field𝑎 theor. = 1159652182.03 … × 10−12
𝑎 exp.
= 1159652180.73 … × 10−12[Aoyama, Kinoshita, Nio (2017)]

If field becomes strong, physics becomes totally different & nontrivial

Novel processes with strong fields
In QED (for 𝒆𝑬, 𝒆𝑩 ∕ 𝒎𝟐𝑒 ≫ 𝟏)

Review: [Fedotov, Ilderton, Karbstein, King,
Seipt, HT, Torgrimsson, 2203.00019]

Ex1 Schwinger effect Ex2 photon splitting Ex3 vacuum birefringence Ex4 high-harmonic gen.

～

～

～

In hadron phys. / QCD (for 𝒆𝑬, 𝒆𝑩 ∕ 𝚲𝟐𝐐𝐂𝐃 ≫ 𝟏)
Ex 1 hadron properties⇒ mass, form factor,
nuclear force, ...

[Miura, Hongo, HT (in prep.)][HT , Hongo, Ikeda (2021)][HT (2015)]

Ex 2 QCD phase diagram⇒ (inverse-)catalysis of chiral condensate,order of phase trans., novel phase, ...

[D’Elia et al. (2022)]

Ex 3 Realtime dynamics⇒ Anomalous transports, spin polarization,
early-stage dynamics of HIC (QGP formation), ...

In other systems: Gravity, Curved spacetime, Inflaton, Cond-mat analog, ...

Novel processes with strong fields
In QED (for 𝒆𝑬, 𝒆𝑩 ∕ 𝒎𝟐𝑒 ≫ 𝟏)

Review: [Fedotov, Ilderton, Karbstein, King,
Seipt, HT, Torgrimsson, 2203.00019]

Ex1 Schwinger effect Ex2 photon splitting Ex3 vacuum birefringence Ex4 high-harmonic gen.

～

～

～

In hadron phys. / QCD (for 𝒆𝑬, 𝒆𝑩 ∕ 𝚲𝟐𝐐𝐂𝐃 ≫ 𝟏)
Ex 1 hadron properties⇒ mass, form factor,
nuclear force, ...

[Miura, Hongo, HT (in prep.)][HT , Hongo, Ikeda (2021)][HT (2015)]

Ex 2 QCD phase diagram⇒ (inverse-)catalysis of chiral condensate,order of phase trans., novel phase, ...

[D’Elia et al. (2022)]

Ex 3 Realtime dynamics⇒ Anomalous transports, spin polarization,
early-stage dynamics of HIC (QGP formation), ...

In other systems: Gravity, Curved spacetime, Inflaton, Cond-mat analog, ...

However, extremely strong fields needed⇒ Experimentally impossible in the 20th century

𝑒𝐸Guiness ≈ 1 keV 2≪ 𝑚𝑒2 , Λ2QCD

Recent availability of strong EM fields
The situation changing: becoming able to create/observe strong light∴ NOW is the BEST time to study strong-field physics

Recent availability of strong EM fields
The situation changing: becoming able to create/observe strong light

Laser intensity 𝑰 (𝐖/𝐜𝐦𝟐 )High-power laser
Strong regime (𝒆𝑬Τ𝒎𝟐𝒆 > 𝟏)

・ Heavy-ion collisions
RIC (2000~), LHC (2012~),FAIR/NICA/HIAF/J-Parc-HI/… (20XX~)
𝐼 ∼ 1035 W/cm2(𝑒𝐸, 𝑒𝐵 ∼ 100 MeV − 1 GeV 2 )●●Latest: ELI・ Electron collider + Laser●

Guiness record (2008年)
HERCULES @ USA●

Extreme physical systems

Start soon: LUXE @ DESY, FACET-II @ SLAC𝐼 ∼ 1029 W/cm2(𝑒𝐸, 𝑒𝐵 > 𝑚𝑒2 ∼ 1 MeV 2 )・ Magnetar

Invention of laser(Nobel prize in 1964)

Invention of CPA(Nobel prize in 2018)

Suzaku (2005~2015), NICER (2017~)
XL-Calibur (2018~), IXPE (2021~), …

𝐼 ∼ 1029 W/cm2(𝑒𝐸, 𝑒𝐵 > 𝑚𝑒2 ∼ 1 MeV 2 )∴ NOW is the BEST time to study strong-field physics

A bit more on heavy-ion collisions
Low-energy ( 𝒔𝐍𝐍 = 𝟐 − 𝟏𝟎) HIC is interesting among other strong-field systems

・ Sales points: (1) the only system that has supercritical 𝐹: = 𝑬𝟐 − 𝑩𝟐 > 0, 𝐺 ∶= 𝑬 ⋅ 𝑩 ≠ 0(2) sufficiently long lived ⇒ will discuss later
magnetar

High-energy HIC

Low-energy HIC(usually) 𝐹 = 𝐺 = 0

𝐹 < 0, 𝐺 = 0

𝐹 < 0, 𝐺 = 0

𝐹 > 0, 𝐺 ≠ 0

subcriticalsuper-longsupercriticalsuper-long

far-supercriticalsuper-shortsupercriticallonghigh-power laser
Field profile
strength
lifetime

・ Numerical estimation of the EM profile in low-energy HIC (with JAM)
𝐹 = 𝑬𝟐 − 𝑩𝟐

E・B<0

y
impact para. bx B

b = 0 fm

z

E・B>0

3 fm

6 fm

𝐺 = 𝑬⋅𝑩E@ 𝑠NN = 5.2 GeV[HT, Nishimura, Ohnishi (2024)] [HT (in prep)]

I’m interested in this and wanna study this further⇒ Chiral XXX ? Axion electrodynamics? Novel QCD phase? Let’s discuss if interested

Recent availability of strong EM fields
The situation changing: becoming able to create/observe strong light

Laser intensity 𝑰 (𝐖/𝐜𝐦𝟐 )High-power laser
Strong regime (𝒆𝑬Τ𝒎𝟐𝒆 > 𝟏)

・ Heavy-ion collisions
RIC (2000~), LHC (2012~),FAIR/NICA/HIAF/J-Parc-HI/… (20XX~)
𝐼 ∼ 1035 W/cm2(𝑒𝐸, 𝑒𝐵 ∼ 𝑚𝜋2 ∼ 140 MeV 2 )●●Latest: ELI・ Electron collider + Laser●

Guiness record (2008年)
HERCULES @ USA●

Extreme physical systems

Start soon: LUXE @ DESY, FACET-II @ SLAC𝐼 ∼ 1029 W/cm2(𝑒𝐸, 𝑒𝐵 > 𝑚𝑒2 ∼ 1 MeV 2 )・ Magnetar

Invention of laser(Nobel prize in 1964)

Invention of CPA(Nobel prize in 2018)

Suzaku (2005~2015), NICER (2017~)
XL-Calibur (2018~), IXPE (2021~), …

𝐼 ∼ 1029 W/cm2(𝑒𝐸, 𝑒𝐵 > 𝑚𝑒2 ∼ 1 MeV 2 )∴ NOW is the BEST time to study strong-field physics

Novel processes with strong fields
In QED (for 𝒆𝑬, 𝒆𝑩 ∕ 𝒎𝟐𝑒 ≫ 𝟏)

Review: [Fedotov, Ilderton, Karbstein, King,
Seipt, HT, Torgrimsson, 2203.00019]

Ex1 Schwinger effect Ex2 photon splitting Ex3 vacuum birefringence Ex4 high-harmonic gen.

～

～

～

In hadron phys. / QCD (for 𝒆𝑬, 𝒆𝑩 ∕ 𝚲𝟐𝐐𝐂𝐃 ≫ 𝟏)
Ex 1 hadron properties⇒ mass, form factor,
nuclear force, ...

[Miura, Hongo, HT (in prep.)][HT , Hongo, Ikeda (2021)][HT (2015)]

Ex 2 QCD phase diagram⇒ (inverse-)catalysis of chiral condensate,order of phase trans., novel phase, ...

[D’Elia et al. (2022)]

Ex 3 Realtime dynamics⇒ Anomalous transports, spin polarization,
early-stage dynamics of HIC (QGP formation), ...

In other systems: Gravity, Curved spacetime, Inflaton, Cond-mat analog, ...

However, extremely strong fields needed⇒ Experimentally impossible in the 20th century

𝑒𝐸Guiness ≈ 1 keV 2≪ 𝑚𝑒2 , Λ2QCD

Novel processes with strong fields
In QED (for 𝒆𝑬, 𝒆𝑩 ∕ 𝒎𝟐𝑒 ≫ 𝟏)

Review: [Fedotov, Ilderton, Karbstein, King,
Seipt, HT, Torgrimsson, 2203.00019]

Ex1 Schwinger effect Ex2 photon splitting Ex3 vacuum birefringence Ex4 high-harmonic gen.

～

～

～

In hadron phys. / QCD (for 𝒆𝑬, 𝒆𝑩 ∕ 𝚲𝟐𝐐𝐂𝐃 ≫ 𝟏)
Ex 1 hadron properties⇒ mass, form factor,
nuclear force, ...

[Miura, Hongo, HT (in prep.)][HT , Hongo, Ikeda (2021)][HT (2015)]

Ex 2 QCD phase diagram⇒ (inverse-)catalysis of chiral condensate,order of phase trans., novel phase, ...

[D’Elia et al. (2022)]

Ex 3 Realtime dynamics⇒ Anomalous transports, spin polarization,
early-stage dynamics of HIC (QGP formation), ...

In other systems: Gravity, Curved spacetime, Inflaton, Cond-mat analog, ...

However, extremely strong fields needed⇒ Experimentally impossible in the 20th century

𝑒𝐸Guiness ≈ 1 keV 2≪ 𝑚𝑒2 , Λ2QCD

Contents
1. Overview of strong-field physics[Fedotov, Ilderton, Karbstein, King, Seipt, HT, Torgrimsson, 2203.00019]

2. Recent development of the Schwinger effect・focus on the Schwinger effect with time-dependent E fields[HT, Itakura, Fujii, 1405.6182] [HT, 1812.03630] [HT, Fujimori, Misumi, Nitta, Sakai, 2010.16080][HT, Ironside, 2308.11248] [HT, Nishimura, Ohnishi, 2402.17136]

3. An application of the Schwinger effect to QCD:
the early-time dynamics of heavy-ion collisions
・Quark production is very fast !

[HT, 1609.06189] [HT, Ph. D thesis]

Basic of the Schwinger effect[Sauter (1932)][Heisenberg, Euler (1936)][Schwinger (1951)]

Vacuum pair production by quantum tunneling in electric fieldpositive energy states
gap

Dirac sea

Energyx~ 2m

V = -eE x~ 2mStrong slow E field~ 2m/eEtunnelingBasic of the Schwinger effect[Sauter (1932)][Heisenberg, Euler (1936)][Schwinger (1951)]

Vacuum pair production by quantum tunneling in electric fieldpositive energy states
gap

~ 2m

Strong slow E field

Dirac sea

EnergyV = -eE x~ 2m~ 2m/eEtunnelingx

Why important/interesting ?
・ Vacuum process⇒ Fundamental, since all the physical processes occur on top of vacuum
・ Non-perturbative⇒ Interesting, since it is the unexplored region of QFT
・ Interdisciplinarity⇒ Similar phenomena appear in many other areas of physics
e.g., Landau-Zener effect, Hawking radiation, broad resonance, ...

Basic of the Schwinger effect[Sauter (1932)][Heisenberg, Euler (1936)][Schwinger (1951)]

Vacuum pair production by quantum tunneling in electric fieldpositive energy states
gap

~ 2m

~ 2m/eE

Strong slow E field

Dirac sea

EnergyV = -eE x~ 2mtunnelingx

Well understood (only) for a constant E field (+ many assump.: weak coup., no backreac., ...)
Schwinger formula:𝑒𝐸 2 𝑉𝑇𝑚2
𝑁𝑒 ± =
× exp −𝜋
∼ exp[−# × gap height × (gap length)]
2𝜋 3𝑒𝐸[Schwinger (1951)] [Nikishov (1969)]

・ Simple theory: Calculate scattering amplitude for |0; inۧ → |𝑒 − 𝑒 + ; outۧ⇒ Evaluate

w/ a dressed wavefunc (or propagator) :[Furry (1951)]

=

𝑬

𝑬

…

𝑬

⇒ Sufficient to solve Dirac eq. w/ strong-field (classical c-number) potential: 0 = i𝜕 − 𝑒𝐴 − 𝑚 𝜓・ Notice the strong exp suppression ⇒ the reason why Schwinger has never been verified in experimentscf.) Guinness world record: 𝑒𝐸 ≈ 1 keV 2 ≪ 𝑚𝑒2 = 𝑂 1 MeV 2[Yanovsky et al (2008)]

Open problems
Strong-field guys are trying to go beyond the Schwinger formula
Relax the strong assumptions of the Schwinger formula・ to find a formula applicable to realistic situations

・ to find something new/interesting(not a splitting hair but rich physics appears beyond the Schwinger formula ! )Examples・ Beyond the constant-E-field approximation
・ Beyond the no-backreaction approximation
・ Beyond the weak-coupling limit

・ Observables other than N ?

Review: [Fedotov, Ilderton, Karbstein, King, Seipt, HT, Torgrimsson, 2203.00019]

Open problems
Strong-field guys are trying to go beyond the Schwinger formula
Relax the strong assumptions of the Schwinger formula・ to find a formula applicable to realistic situations

・ to find something new/interesting(not a splitting hair but rich physics appears beyond the Schwinger formula ! )Examples・ Beyond the constant-E-field approximation
・ Beyond the no-backreaction approximation
・ Beyond the weak-coupling limit

・ Observables other than N ?

Review: [Fedotov, Ilderton, Karbstein, King, Seipt, HT, Torgrimsson, 2203.00019]

What happens E field
is not constant in time ?

No Schwinger effect for “fast” E fields
Time-dependent E field with strength 𝒆𝑬𝟎 & frequency 𝛀 (or lifetime 𝝉 = 𝟏/𝛀)

No Schwinger effect for “fast” E fields
Time-dependent E field with strength 𝒆𝑬𝟎 & frequency 𝛀 (or lifetime 𝝉 = 𝟏/𝛀)
Slow = small Ω⇒ Non-pert. tunneling 𝑁~ exp[#/𝑒𝐸0 ]~ 2𝑚~ 2𝑚/𝑒𝐸0

Tunneling
Tunneling time Δ𝑡 ~2𝑚
𝑒𝐸0

⇒ E field must be slower than Δ𝑡⇒ Ω−1 ≿ Δ𝑡
[Keldysh (1965)]Δ𝑡Ω𝑚⇒ 1 ≿ −1 =≡ 𝛾 (Keldysh parameter)Ω𝑒𝐸0

No Schwinger effect for “fast” E fields
Time-dependent E field with strength 𝒆𝑬𝟎 & frequency 𝛀 (or lifetime 𝝉 = 𝟏/𝛀)
Slow = small Ω⇒ Non-pert. tunneling 𝑁~ exp[#/𝑒𝐸0 ]Fast = large Ω⇒ Pert. photon scattering 𝑁~𝑒𝐸02𝑛

~ 2𝑚

Ω
Ω
Ω

~ 2𝑚/𝑒𝐸0

Tunneling
Tunneling time Δ𝑡 ~2𝑚
𝑒𝐸0

⇒ Incoherent photons, rather thancoherent E field⇒ E field must be slower than Δ𝑡⇒ Ω−1 ≿ Δ𝑡
⇒1≿

~ 2m

[Keldysh (1965)]⇒ pair prod. when 𝑛Ω > 2𝑚Δ𝑡Ω𝑚=
≡ 𝛾 (Keldysh parameter)
−1
Ω
𝑒𝐸0

⇒ QED version of the photoelectric effect in material→

Development 1/5: A better understanding of
non-pert Schwinger vs pert pair prod
“Phase diagram” of the Schwinger effect
𝑆

𝑺

2𝑆

Theory: (1) Semi-classical approx.
𝑁 = ෍ 𝑁𝑛,𝑚 ℏ𝑛 e−𝑚ℏ = (𝑁0,1 + 𝑂(ℏ)) 𝐞−ℏ + 𝑂(e− ℏ )[Brezin, Itzykson (1970)] [Popov (1972)]= Trans-series expansion in ℏ𝑛,𝑚[Berry (1989)] [Dunne, Shubert (2005)](2) Compare with exactly solvable cases [HT, Fujiii, Itakura (2014)][HT, Fujimori, Misumi, Nitta, Sakai (2020)]

Development 1/5: A better understanding of
non-pert Schwinger vs pert pair prod
“Phase diagram” of the Schwinger effect
𝑆

𝑺

2𝑆

Theory: (1) Semi-classical approx.
𝑁 = ෍ 𝑁𝑛,𝑚 ℏ𝑛 e−𝑚ℏ = (𝑁0,1 + 𝑂(ℏ)) 𝐞−ℏ + 𝑂(e− ℏ )[Brezin, Itzykson (1970)] [Popov (1972)]= Trans-series expansion in ℏ𝑛,𝑚[Berry (1989)] [Dunne, Shubert (2005)](2) Compare with exactly solvable cases [HT, Fujiii, Itakura (2014)][HT, Fujimori, Misumi, Nitta, Sakai (2020)]
・Two dim-less params control the interplay⇐ 3 dimensionfull params in the system: 𝑒𝐸, 𝜏 ≔ 1/Ω, 𝑚𝑚Ω
: Keldysh parameter𝑒𝐸(Work done by field)
𝑒𝐸𝜏
𝜈=
=
Ω
(One photon energy)

Non-linear𝛾== (number of photons)

・Non-pert Schwinger if 𝜸 ≪ 𝟏 , 𝝂 ≫ 𝟏
Pert pair production if 𝜸 ≫ 𝟏 , 𝝂 ≪ 𝟏

Pert.

(LO is enough)Non-pert.

Development 1/5: A better understanding of
non-pert Schwinger vs pert pair prod
“Phase diagram” of the Schwinger effect
𝑆

𝑺

2𝑆

Theory: (1) Semi-classical approx.
𝑁 = ෍ 𝑁𝑛,𝑚 ℏ𝑛 e−𝑚ℏ = (𝑁0,1 + 𝑂(ℏ)) 𝐞−ℏ + 𝑂(e− ℏ )[Brezin, Itzykson (1970)] [Popov (1972)]= Trans-series expansion in ℏ𝑛,𝑚[Berry (1989)] [Dunne, Shubert (2005)](2) Compare with exactly solvable cases [HT, Fujiii, Itakura (2014)][HT, Fujimori, Misumi, Nitta, Sakai (2020)]
・Two dim-less params control the interplay⇐ 3 dimensionfull params in the system: 𝑒𝐸, 𝜏 ≔ 1/Ω, 𝑚𝑚Ω
: Keldysh parameter𝛾=𝑒𝐸(Work done by field)
𝑒𝐸𝜏
𝜈=
=
Ω
(One photon energy)

Non-linear

Non-pert.
Low-energy
HIC= (number of photons)

・Non-pert Schwinger if 𝜸 ≪ 𝟏 , 𝝂 ≫ 𝟏

Pert.

(LO is enough)Pert pair production if 𝜸 ≫ 𝟏 , 𝝂 ≪ 𝟏

・Implication: “Strong field ⇒ Non-pert strong-field physics” is necessarily correct⇒ Not only strength but also lifetime (& other dimful params, if any) is importante.g., High-energy HIC is not non-pert.: 𝑒𝐹 ∼ 1 GeV

2, 𝜏 ∼ 0.1 fmΤ𝑐 ⇒ 𝛾 ∼ ൝( 𝑠𝑁𝑁 > 𝑂(100 GeV − 1 TeV))

Low-energy HIC is non-pert.: 𝑒𝐹 ∼ 100 MeV( 𝑠𝑁𝑁 = 𝑂(1 − 10 GeV))2, 𝜏 ∼ 10 fmΤ𝑐 ⇒ 𝛾 ∼ ൝10−3 𝑚 = ΛQCD
10−5 𝑚 = 𝑚e, 𝜈 ∼ 0.1

10−1 𝑚 = ΛQCD
, 𝜈 ∼ 1010−4 𝑚 = 𝑚e[HT, Nishimura, Ohnishi, (2024)]

Development 2/5: Importance of pert pair production
Schwinger formula is inapplicable for fast E fields
Slow ⇒ Non-pert ⇒ Strong exp suppression 𝑁~ exp[− 𝑚2 /𝑒𝐸0 ]Fast ⇒ Pert

⇒ Weak power suppression 𝑁~(𝑒𝐸0 /𝑚2 )2𝑛⇒ (So long as 𝒆𝑬𝟎~< 𝒎𝟐 ) Fast E creates more particles than slow E does

Development 2/5: Importance of pert pair production
Schwinger formula is inapplicable for fast E fields
Slow ⇒ Non-pert ⇒ Strong exp suppression 𝑁~ exp[− 𝑚2 /𝑒𝐸0 ]Fast ⇒ Pert

⇒ Weak power suppression 𝑁~(𝑒𝐸0 /𝑚2 )2𝑛⇒ (So long as 𝒆𝑬𝟎~< 𝒎𝟐 ) Fast E creates more particles than slow E does
𝑒𝐸0 Τ𝑚2 = 0.1

Demonstration:
Pair prod from pulsed E field
w/lifetime 𝜏 = 1/Ω𝑒𝐸(Sauter field 𝑒𝐸 𝑡 = cosh20(Ω𝑡))[HT, Fujiii, Itakura (2014)][HT, Fujimori, Misumi, Nitta, Sakai (2020)]

Development 2/5: Importance of pert pair production
Schwinger formula is inapplicable for fast E fields
Slow ⇒ Non-pert ⇒ Strong exp suppression 𝑁~ exp[− 𝑚2 /𝑒𝐸0 ]Fast ⇒ Pert

⇒ Weak power suppression 𝑁~(𝑒𝐸0 /𝑚2 )2𝑛⇒ (So long as 𝒆𝑬𝟎~< 𝒎𝟐 ) Fast E creates more particles than slow E does
𝑒𝐸0 Τ𝑚2 = 0.1

Demonstration:
Pair prod from pulsed E field
w/lifetime 𝜏 = 1/Ω𝑒𝐸(Sauter field 𝑒𝐸 𝑡 = cosh20(Ω𝑡))[HT, Fujiii, Itakura (2014)][HT, Fujimori, Misumi, Nitta, Sakai (2020)]

Some application to actual physical problems
・ Enhancement of heavy quark prod in heavy-ion collisions [Levai, Skokov (2010)]
・ Use of fast fields to enhance the Schwinger effect in weak-field exp. (e.g., laser)⇒ Dynamically assisted Schwinger effect(next slide)Development 3/5: Dynamically assisted Schwinger effect (1/2)[Dunne, Gies, Schutzhold (2008), (2009)]

Significant enhancement of the Schwinger effectby superimposing fast (weak) E fields

Perturbative scattering

Tunneling

Tunneling

Fast E field𝑁 ∼ exp[−# × gap height × (gap length)] ⇒ Enhancement of pair prod

Reduced by the pert scattering

Development 3/5: Dynamically assisted Schwinger effect (1/2)(NEW theory)(old theory in 2010) Exact numerics
Furry-picture
pert theory
Semi-classical approx

Field config
𝐸 = 𝐸0 1 +1cos Ω𝑡100

Schwinger formula(for const E)

𝑒𝐸0 Τ𝑚2 = 0.25

Physics outcome
・ Expected: Huge enhancement, even for very weak fast field
・ Un-expected: Oscillating behavior above the mass gap⇐ Related to the Dirac-sea structure in strong E field (next slide)fast

2

Technical advancement[HT, (2019)] [Huang, HT, (2019)]

𝑁≃・ Dressed scattering theory w/ unstable vacuum≈ Expand w/ fast field, while keeping slow field exactly= slow+ slow+ …

Development 4/5: Modified Dirac-sea structure by E field (1/2)

Positive energy band

EnergyDirac seaGap

E field direction∴ The spectrum of the dynamically-assisted Schwinger effectreflects the modified Dirac-sea structure in strong E fieldDevelopment 4/5: Modified Dirac-sea structure by E field (1/2)

electrons in-coming wave

EnergyDirac sea

Positive energy bandGap

E field direction∴ The spectrum of the dynamically-assisted Schwinger effectreflects the modified Dirac-sea structure in strong E fieldDevelopment 4/5: Modified Dirac-sea structure by E field (1/2)

electrons in-coming wave

Positive energy band

Electron reflected wave
EnergyDirac seaGap

E field direction∴ The spectrum of the dynamically-assisted Schwinger effectreflects the modified Dirac-sea structure in strong E fieldDevelopment 4/5: Modified Dirac-sea structure by E field (1/2)

Positive energy band

Electrondistribution
EnergyDirac seaGap

E field direction∴ The spectrum of the dynamically-assisted Schwinger effectreflects the modified Dirac-sea structure in strong E fieldDevelopment 4/5: Modified Dirac-sea structure by E field (1/2)
cf. similar argument in Franz-Keldysh effect in semi-conductor

Positive energy band

Electrondistribution
EnergyDirac seaGap

E field direction

・ Tunneling ⇒ Enhancement∴ The spectrum of the dynamically-assisted Schwinger effectreflects the modified Dirac-sea structure in strong E fieldDevelopment 4: Modified Dirac-sea structure by E field (1/2)
cf. similar argument in Franz-Keldysh effect in semi-conductor

××
EnergyDirac sea

Positive energy band

ElectrondistributionGap

E field direction

・ Tunneling ⇒ Enhancement

・ Reflection ⇒ Oscillation∴ The spectrum of the dynamically-assisted Schwinger effectreflects the modified Dirac-sea structure in strong E fieldDevelopment 4/5: Modified Dirac-sea structure by E field (2/2)
Larger electron density ⇒ affects more

The modified Dirac-sea struc affects everything∵ Any process occurs on top of the vacuum

Development 4/5: Modified Dirac-sea structure by E field (2/2)
Larger electron density ⇒ affects more

The modified Dirac-sea struc affects everything∵ Any process occurs on top of the vacuum
Example: Photon birefringence (electric permittivity) in strong E field𝐸𝐸ℰ ∝ cos 𝜔𝑡𝐸[HT, Ironside, (2024)]
cf. Birefringence in B field [Hattori, Itakura (2013)]

𝒟 = 𝜀ℰ = 𝑛2 ℰ = 1 + 𝜒 ℰ

Real part

Kramers-Kronig rel.
𝑒𝐸 Τ𝑚2 = 0.25

Re 𝜀 𝐸 − 𝜀 0 /𝑒 2Im 𝜀 𝐸 − 𝜀 0 /𝑒 2

Imaginary part𝑒𝐸 Τ𝑚2 = 0.25

⇒ Characteristic oscillation, as expected from the modified Dirac-sea structure !

Contents
1. Overview of strong-field physics[Fedotov, Ilderton, Karbstein, King, Seipt, HT, Torgrimsson, 2203.00019]

2. Recent development of the Schwinger effect・focus on the Schwinger effect with time-dependent E fields[HT, Itakura, Fujii, 1405.6182] [HT, 1812.03630] [HT, Fujimori, Misumi, Nitta, Sakai, 2010.16080][HT, Ironside, 2308.11248] [HT, Nishimura, Ohnishi, 2402.17136]

3. An application of the Schwinger effect to QCD:
the early-time dynamics of heavy-ion collisions
・Quark production is very fast !

[HT, 1609.06189] [HT, Ph. D thesis]

Spacetime evolution of high-energy HIC (1/2)

QGP

before coll.

coll.

QGP

Hadron gas> 0 fm/c

0 fm/c~ 1 fm/c~ 10 fm/c

Spacetime evolution of high-energy HIC (1/2)

QGP

before coll.

coll.

QGP

Hadron gas> 0 fm/c

0 fm/c~ 1 fm/c~ 10 fm/c■ & ■ are well understoodSpacetime evolution of high-energy HIC (1/2)

QGP

before coll.

coll.

QGP

Hadron gas> 0 fm/c

0 fm/c~ 1 fm/c~ 10 fm/c■ & ■ are well understood
DIS @ HERASpacetime evolution of high-energy HIC (1/2)

QGP

before coll.

coll.

QGP

Hadron gas> 0 fm/c

0 fm/c~ 1 fm/c~ 10 fm/c■ & ■ are well understood
DIS @ HERA

Dense gluons⇒ pQCD (color glass condensate ?)Spacetime evolution of high-energy HIC (1/2)

QGP

before coll.

coll.

QGP

Hadron gas> 0 fm/c

0 fm/c~ 1 fm/c~ 10 fm/c■ & ■ are well understood
DIS @ HERA

Dense gluons⇒ pQCD (color glass condensate ?)Spacetime evolution of high-energy HIC (1/2)

QGP

before coll.

coll.

QGP

Hadron gas> 0 fm/c

0 fm/c~ 1 fm/c~ 10 fm/c■ & ■ are well understood
DIS @ HERA

Dense gluons⇒ pQCD (color glass condensate ?)Spacetime evolution of high-energy HIC (1/2)

QGP

before coll.

coll.

QGP

Hadron gas> 0 fm/c

0 fm/c~ 1 fm/c~ 10 fm/c■ & ■ are well understood
DIS @ HERA

Dense gluons⇒ pQCD (color glass condensate ?)Spacetime evolution of high-energy HIC (1/2)

QGP

before coll.

coll.

QGP

Hadron gas> 0 fm/c

0 fm/c~ 1 fm/c~ 10 fm/c■ & ■ are well understood
DIS @ HERA

Dense gluons⇒ pQCD (color glass condensate ?)(nearly) perfect liquid⇒ hydrodynamics( + hadron transport)Spacetime evolution of high-energy HIC (2/2)？

QGP

before coll.

coll.

QGP

Hadron gas> 0 fm/c

0 fm/c~ 1 fm/c~ 10 fm/c■ is less understoodcf. the weak-coupling kinetic picture from Heidelberg group⇒ Formation dynamics of QGP is still an open issue
・ How are the huge number of quarks & gluons produced dN/dy=O(1000) ?
・ How do they thermalize (hydrodynamize) to form the liquid-like QGP？
・ How to explain the “early thermalization” O(1fm/c), indicated by exp data？

Not only important for completing our spacetime picture of HIC
but also for deepening our understanding of QGP(e.g.: provide the initial cond. for hydro sim. ⇒ better determination of QGP properties)

Strong color EM field (glasma) (1/2)
Low-Nussinov model: Low, Nussinov, Casher, Neuberger (1970~80)
Glasma: McLerran, Lappi, Kovner, Weigert (~2005)

The key: Decay of the strong color EM field into particles⇒ Schwinger effect !

[Kerman, Matsui, Gatoff (1987)]

Strong color EM field (glasma) (1/2)
Low-Nussinov model: Low, Nussinov, Casher, Neuberger (1970~80)
Glasma: McLerran, Lappi, Kovner, Weigert (~2005)

The key: Decay of the strong color EM field into particles⇒ Schwinger effect !

[Kerman, Matsui, Gatoff (1987)]

can guessKnown
DIS @ HERA

high-energy heavy ion＝
dense gluon state≃∝ 1/ 𝑠NN“color plate”
with huge color density(𝜎 ∼ 𝑄𝑠2 ∼ 𝑂 1 GeV 2 )Strong color EM field (glasma) (2/2)
Low-Nussinov model: Low, Nussinov, Casher, Neuberger (1970~80)
Glasma: McLerran, Lappi, Kovner, Weigert (~2005)

The key: Decay of the strong color EM field into particles⇒ Schwinger effect !

[Kerman, Matsui, Gatoff (1987)]

can guess≃[Lappi (2006)]

an expanding“color capacitor”(𝜎 ∼ 𝑄𝑠2 ∼ 𝑂 1 GeV 2 )𝑄𝑠4 /𝑔2

Known

just after a collision⇒strong color EM field(𝑔𝐸, 𝑔𝐵 = 𝑂 1 GeV 2 )𝑄𝑠 𝜏What I am going to do
Purpose:Study the quark & gluon production in the early-time dynamics of HIC
by applying the Schwinger-effect realtime technique developed in QED to QCD

Setup:t

・ QCD with Nc=3 and Nf=6 (with actual quark masses)

𝝉 = 𝐜𝐨𝐧𝐬𝐭.

・ Boost-invariantly expanding color E field 𝐸(𝜏, 𝜂, 𝒙⊥ ) = 𝐸(𝜏)𝑬・ Neglect color magnetic field𝑬(⇐ problem due to gluon instability)
・ Solve QCD within mean-field approx. (next slides)𝜼 = 𝐜𝐨𝐧𝐬𝐭.

𝑬𝒕 𝟐 − 𝒛𝟐
𝟏𝒕+𝒛
𝜼 = 𝐥𝐧𝟐𝒕−𝒛𝝉=zMean-field theory (1/2)

QED: [Kluger, Eisenberg, Svetitsky,
Cooper, Mottola (~1990)][Tanji (2008)]
Same as:Bogoliubov-de Gennes (TD-BdG)
in cond-mat

Mean-field theory (1/2)
1
𝐿QCD = − tr 𝐹𝜇𝜈 𝐹𝜇𝜈 + 𝐿quark + 𝐿FP+GF
2

QED: [Kluger, Eisenberg, Svetitsky,
Cooper, Mottola (~1990)][Tanji (2008)]
Same as:Bogoliubov-de Gennes (TD-BdG)
in cond-mat

Mean-field theory (1/2)
1
𝐿QCD = − tr 𝐹𝜇𝜈 𝐹𝜇𝜈 + 𝐿quark + 𝐿FP+GF
2

QED: [Kluger, Eisenberg, Svetitsky,
Cooper, Mottola (~1990)][Tanji (2008)]
Same as:Bogoliubov-de Gennes (TD-BdG)
in cond-mat

STEP 1 Split the total gauge field 𝐴 into classical (strong) part 𝐴ҧ = 𝐴 and quantum
fluctuation on top of it 𝑎, i.e., 𝐴 = 𝐴ҧ + 𝑎
STEP 2 Expand 𝐿QCD i.t.o 𝑎

QED: [Kluger, Eisenberg, Svetitsky,
Cooper, Mottola (~1990)][Tanji (2008)]
Same as:Bogoliubov-de Gennes (TD-BdG)
in cond-mat

Mean-field theory (1/2)
1
𝐿QCD = − tr 𝐹𝜇𝜈 𝐹𝜇𝜈 + 𝐿quark + 𝐿FP+GF
2

STEP 1 Split the total gauge field 𝐴 into classical (strong) part 𝐴ҧ = 𝐴 and quantum
fluctuation on top of it 𝑎, i.e., 𝐴 = 𝐴ҧ + 𝑎
STEP 2 Expand 𝐿QCD i.t.o 𝑎
1
𝐿QCD = − tr 𝐹ത𝜇𝜈 𝐹ത𝜇𝜈
2
+ 𝑂 𝑎1

𝑎

𝑂 𝑎2

+

𝑎

+

𝑎

𝑔 × 𝑂 𝑎3

+

𝑎

𝑎

𝑎
𝑎 is couplted to 𝐴ҧ in non-perturbatively:=+

𝑔2 × 𝑂 𝑎4+ 𝐿quark +𝐿𝐹𝑃+𝐺𝐹

𝑎

𝑎

𝑎

𝑎

𝐴ҧ

𝐴ҧ

+

𝐴ҧ

+…

QED: [Kluger, Eisenberg, Svetitsky,
Cooper, Mottola (~1990)][Tanji (2008)]
Same as:Bogoliubov-de Gennes (TD-BdG)
in cond-mat

Mean-field theory (1/2)
1
𝐿QCD = − tr 𝐹𝜇𝜈 𝐹𝜇𝜈 + 𝐿quark + 𝐿FP+GF
2

STEP 1 Split the total gauge field 𝐴 into classical (strong) part 𝐴ҧ = 𝐴 and quantum
fluctuation on top of it 𝑎, i.e., 𝐴 = 𝐴ҧ + 𝑎
STEP 2 Expand 𝐿QCD i.t.o 𝑎
1
𝐿QCD = − tr 𝐹ത𝜇𝜈 𝐹ത𝜇𝜈
2
+ 𝑂 𝑎1

𝑎

𝑂 𝑎2

+

𝑎

+

𝑎

𝑔 × 𝑂 𝑎3

𝑔2 × 𝑂 𝑎4

+

𝑎

𝑎

𝑎
𝑎 is couplted to 𝐴ҧ in non-perturbatively:
STEP 3 Apply mean-field approx.
to the non-linear
terms 𝑂 𝑎3 , 𝑎4

=

𝑎

𝑎

𝑎

𝑎

𝐴ҧ

𝐴ҧ

+

𝐴ҧ

+…
𝑎

→

→
𝑎

𝑎

𝑎𝑎𝑎

𝑎 𝑎𝑎
color current 𝑗

𝐿QCD = (up to the second orer in 𝑎)

+
𝑎

𝑎

𝑎

+ 𝐿quark +𝐿𝐹𝑃+𝐺𝐹

𝑎

𝑎

𝑎𝑎𝑎𝑎

𝑎

𝑎

𝑎𝑎 𝑎𝑎

(local) self-energy 𝑀

Mean-field theory (2/2)
STEP 4 Get equation of motion

QED: [Kluger, Eisenberg, Svetitsky,
Cooper, Mottola (~1990)][Tanji (2008)]
Same as:Bogoliubov-de Gennes (TD-BdG)
in cond-mat

Coupled linear EoMs
EoM for quantum fluct. 𝑎0=

𝜕 + 𝑖𝑔𝐴ҧ 2 𝑔𝜇𝜈 + 𝑀𝜇𝜈 𝑎𝜈

EoM for classical field 𝐹ത
𝑗𝜇 = 𝜕𝜈 𝐹ത 𝜈𝜇 + 𝑓 𝜈𝜇

Mean-field theory (2/2)
STEP 4 Get equation of motion

QED: [Kluger, Eisenberg, Svetitsky,
Cooper, Mottola (~1990)][Tanji (2008)]
Same as:Bogoliubov-de Gennes (TD-BdG)
in cond-mat

Coupled linear EoMs
EoM for quantum fluct. 𝑎
0=

EoM for classical field 𝐹ത𝜕 + 𝑖𝑔𝐴ҧ 2 𝑔𝜇𝜈 + 𝑀𝜇𝜈 𝑎𝜈𝑗𝜇 = 𝜕𝜈 𝐹ത 𝜈𝜇 + 𝑓 𝜈𝜇Multiple scattering b/w 𝐴ҧ and 𝑎⇒ Particle production of 𝑎 from 𝐴ҧ
𝐴ҧ

=

+

𝐴ҧ

+

Current 𝑗𝜇 produced by 𝑎 screens out 𝐹ത⇒ Backreaction to 𝐹ത by 𝑎𝐴ҧ+…

Mean-field theory (2/2)
STEP 4 Get equation of motion

QED: [Kluger, Eisenberg, Svetitsky,
Cooper, Mottola (~1990)][Tanji (2008)]
Same as:Bogoliubov-de Gennes (TD-BdG)
in cond-mat

Coupled linear EoMs
EoM for quantum fluct. 𝑎
0=

EoM for classical field 𝐹ത𝜕 + 𝑖𝑔𝐴ҧ 2 𝑔𝜇𝜈 + 𝑀𝜇𝜈 𝑎𝜈𝑗𝜇 = 𝜕𝜈 𝐹ത 𝜈𝜇 + 𝑓 𝜈𝜇Multiple scattering b/w 𝐴ҧ and 𝑎⇒ Particle production of 𝑎 from 𝐴ҧ
𝐴ҧ

=

+

𝐴ҧ

+

Current 𝑗𝜇 produced by 𝑎 screens out 𝐹ത⇒ Backreaction to 𝐹ത by 𝑎𝐴ҧ+…

STEP 5 Solve EoM !
(1) assume 𝑂 𝑎 2 terms (𝑀𝜇𝜈 and 𝑗𝜇 ) are negligible (≈ no backreaction)⇒ analytically solvable ⇒ gives essentially the same to the Schwinger formula(2) don’t neglect 𝑂 𝑎2 terms (≈ w/ backreaction) ⇒ numerically doable (⇒ this talk)Results

Energy density 𝝐 𝐆𝐞𝐕 𝟒

Results (1/4): Energy balance

Total (■+■+■)

gluonsquarksclassical
E field (= 𝐸2 /2)Time 𝝉 𝐆𝐞𝐕 −𝟏

Due to the particle prod (+ the Bjorken exp), the initial classical fielddecays into quark & gluon particles rapidly 𝜏 ∼ 10 GeV −1 ∼ 2 fm/𝑐
cf. Non-expanding QED: [Kluger et al. (~1990)] [Tanji (2008)]

Decay with oscillation (plasma oscillation)∵ 𝐸ሶ = −𝐽 ∝ (particles′ velocity), but particles do not stop immediately at 𝐸 = 0

Results (2/4): (longitudinal) momentum dist. dN/dp
𝐝𝟔 𝑵(longitudinal) momentum dist 𝟐
at 𝒑𝑻 ∼ 𝟎𝐝𝒙𝑻 𝐝𝜼 𝐝𝒑𝟐𝑻 𝐝𝒑𝜼

Quark

Gluon

Plasma oscillation = particles are going back and forth
When comes back to p=0, where new particles are being produced,
quantum interference occurs (known as stuckelberg interference in cond-mat)・ gluon: Bose enhancement ⇒ increase of the production
・ quark: Pauli blocking ⇒ saturation behavior

𝐝𝑵Τ𝐝𝒚𝒑 /𝑺⊥ 𝐆𝐞𝐕 𝟐

Yields per unit rapidity

Results (3/4): yields N per unit rapidity

Total (■+■)
gluon
quark

time 𝝉 𝐆𝐞𝐕 −𝟏(cf. 𝑆⊥ ∼ 𝑂 4000 GeV −2 )A huge number of particles O(1000) can actually be produced at early time τ=O(1fm/c)⇒ Strong-field physics (= decay of strong field into particles) actually plays
an important role in the early-stage dynamics of HIC
Quark production is fast and abundant⇒ Quark DoG is non-negligible
・ should affect the early-stage dynamics ...
・ good news for CME search (∵ U(1) B field decays very rapidly )Results (4/4): anisotropy as a measure of thermalizationtrans pressure 𝑷𝑻 /𝝐long pressure 𝑷𝑳 /𝝐classical fieldtime 𝝉 𝐆𝐞𝐕 −𝟏particles

Even within mean-field approx. (i.e., no interaction),
anisotropy gets relaxed significantly (𝑃𝑇 Τ𝑃𝐿 ∼ 0.5)(∵ non-zero long pressure due to acceleration by E field ⇒ don’t simply go to the free streaming 𝑃𝐿 = 0)

Outlook: Need to go beyond mean-field approx. to really discuss
thermalization (hydrodynamization)(cf. go beyond MFA is new not only in QCD but also in QED, so should be interesting ...)

Contents
1. Overview of strong-field physics[Fedotov, Ilderton, Karbstein, King, Seipt, HT, Torgrimsson, 2203.00019]

2. Recent development of the Schwinger effect・focus on the Schwinger effect with time-dependent E fields[HT, Itakura, Fujii, 1405.6182] [HT, 1812.03630] [HT, Fujimori, Misumi, Nitta, Sakai, 2010.16080][HT, Ironside, 2308.11248] [HT, Nishimura, Ohnishi, 2402.17136]

3. An application of the Schwinger effect to QCD:
the early-time dynamics of heavy-ion collisions
・Quark production is very fast !

[HT, 1609.06189] [HT, Ph. D thesis]

Today’s talk
Q: What happens if we make light (or “field” in general) stronger and stronger ?
Laser welding

LED

Cond.-mat(THz laser)???

Guiness record?
HERCULES @ USA (2008)𝐼 ∼ 10−5 W/cm2

106 W/cm2

1010 W/cm2

1022 W/cm2

or 𝑒𝐸 ∼ 10−1 eVΤcm

104 eVΤcm

106 eVΤcm

1012 eVΤcm

10−1 eV 2

1 eV 2−3∼ 10 eV
Daily life2

Industry

1 keV

2

Our limit
at the present

Science

Purpose: Review physics of such strong fieldTake-home messages:(1) Once 𝒆𝑬 > (typical energy scale), sthg extremely non-trivial occur(e.g., Schwinger effect ≈ “something” from “nothing” )(2) Such strong fields are now (or soon will be) within the exp. reach(3) Of relevance to hadron/QCD physics, in particular, heavy-ion collisions