Opportunities of strong-field physics
In middle-energy heavy-ion collisions
Hidetoshi Taya
(RIKEN iTHEMS)

Plan
Middle-energy heavy-ion collisions 𝒔𝐍𝐍 = 𝑶 𝟏‐ 𝟏𝟎 𝐆𝐞𝐕 may be interesting
not only to QCD but also to strong-field physics
・ Estimation of EM field strength and spacetime volume with a hadron transport model (JAM)⇒ Strong O(30 MeV) and large spacetime volume O((20 fm) 4 )⇒ A nice setup to study strong-field QED. Non-negligible to hadronic/QCD processes as well.

1. Brief review of strong-field physics
2. Strong EM field at high-energy heavy-ion collisions
・ Clarify relationship between “non-perturbativeness” and strength and lifetime of EM field

3. Strong EM field at middle-energy heavy-ion collisions
・ Estimation of EM field profiles with JAM[Nishimura (Osaka), Ohnishi (Kyoto), HT, in progress]

4. An example of strong-field phenomenon:
Vacuum decay and modification to photon propagation[HT, Ironside (Curtin), in progress]

1. Brief review of strong-field physics

2. Strong EM field at high-energy heavy-ion collisions
3. Strong EM field at middle-energy heavy-ion collisions[Nishimura (Osaka), Ohnishi (Kyoto), HT, in progress]

4. An example of strong-field phenomenon:
Vacuum decay and modification to photon propagation[HT, Ironside (Curtin), in progress]

5. Summary

Strong-field physics
initialfinal

time evolution

Vacuum(=No EM field)

Images stolen from below
https://news.line.me/detail/oa-walkerplus/59d36b1b92da
https://ananweb.jp/anan/319760
https://twitter.com/1007hiro/status/1274596230136344576

Strong-field physics
initialfinal

time evolution

Vacuum(=No EM field)

Weak EM field(𝒆𝑭Τ𝒎𝟐 ≲ 𝟏)

Strong EM field(𝒆𝑭Τ𝒎𝟐 ≳ 𝟏)

Images stolen from below
https://news.line.me/detail/oa-walkerplus/59d36b1b92da
https://ananweb.jp/anan/319760
https://twitter.com/1007hiro/status/1274596230136344576

Strong-field physics
initialfinal

time evolution

Vacuum(=No EM field)

𝑭

Weak EM field(𝒆𝑭Τ𝒎𝟐 ≲ 𝟏)

Strong EM field(𝒆𝑭Τ𝒎𝟐 ≳ 𝟏)

Almost the same
⇒ Perturbative⇒ Understood wellex) Electron anomalous magnetic moment 𝑎 ≔𝑎 theor. = 1159652182.03 … × 10−12
𝑎 exp.
= 1159652180.73 … × 10−12

g−2
2[Aoyama, Kinoshita, Nio (2017)]

Images stolen from below
https://news.line.me/detail/oa-walkerplus/59d36b1b92da
https://ananweb.jp/anan/319760
https://twitter.com/1007hiro/status/1274596230136344576

Strong-field physics
initialfinal

𝑭time evolution

𝑭

𝑭
𝑭

Vacuum(=No EM field)

Weak EM field(𝒆𝑭Τ𝒎𝟐 ≲ 𝟏)

𝑭…

Strong EM field(𝒆𝑭Τ𝒎𝟐 ≳ 𝟏)

Almost the same

Completely different⇒ Perturbative⇒ Non-perturbative⇒ Understood well

⇒ Not understood wellex) Electron anomalous magnetic moment 𝑎 ≔𝑎 theor. = 1159652182.03 … × 10−12
𝑎 exp.
= 1159652180.73 … × 10−12

g−2
2[Aoyama, Kinoshita, Nio (2017)]

Images stolen from below
https://news.line.me/detail/oa-walkerplus/59d36b1b92da
https://ananweb.jp/anan/319760
https://twitter.com/1007hiro/status/1274596230136344576

Examples of strong-field phenomena
Novel QED processes (𝒆𝑭Τ𝒎𝟐𝒆 ≳ 𝟏)

ex)

Schwinger mechanism

Photon splitting

Review: [Fedotov, Ilderton, Karbstein, King, Seipt,
HT, Torgrimsson (2022)]

Vacuum birefringence(= Polarization dep. of reflective index)～

Vacuum

Impacts on QCD/hadron physics (𝒆𝑭Τ𝚲𝟐 ≳ 𝟏 (or deconfined phase ⇒ 𝒆𝑭Τ𝒎𝟐𝐜𝐮𝐫𝐫𝐞𝐧𝐭 𝐪 𝐦𝐚𝐬𝐬 ≳ 𝟏))[HT (2015)]

ex. 1) Hadron properties:e.g., mass, charge dist., decay mode, …
See also recent review [Iwasaki, Oka, Suzuki (2021)]

ex. 2) QCD phase diagram[D’Elia et al. (2022)]

e.g., (inverse) magnetic catalysis, new phase, …
ex. 3) Others: Anomalous transport,(for color EM field) Glasma, string breaking, …
Many reviews, e.g., [Kharzeev, Liao, Voloshin, Wang (2016)] [Hattori, Huang (2017)] …

Many theoretical predictions, but never observed experimentally yet

Need of extremely strong EM field
Minimum strength: 𝒆𝑬, 𝒆𝑩 𝐌𝐞𝐕 𝟐 > 𝒎𝟐𝒆 = 𝟎. 𝟓𝟏𝟏 𝐌𝐞𝐕 𝟐 ≈ 𝑶(𝟏𝟎𝟐𝟖 𝐖/𝐜𝐦𝟐 )LED ≈ 10−5 W/cm2

Glasses ≈ 103 W/cm2

Laser processing machine < 1010 W/cm2

Need of extremely strong EM field
Minimum strength: 𝒆𝑬, 𝒆𝑩 𝐌𝐞𝐕 𝟐 > 𝒎𝟐𝒆 = 𝟎. 𝟓𝟏𝟏 𝐌𝐞𝐕 𝟐 ≈ 𝑶(𝟏𝟎𝟐𝟖 𝐖/𝐜𝐦𝟐 )Laser processing machine < 1010 W/cm2

Glasses ≈ 103 W/cm2

LED ≈ 10−5 W/cm2

Intense laser is the most common source for strong EM field[Mourou, Tajima, Bulanov (2006)]

𝟏𝟎𝟐𝟒

Birth of laser

Laser intensity 𝐼 𝐞𝐕 𝟒

𝒆𝑬, 𝒆𝑩~ 𝟓𝟏𝟏 𝐤𝐞𝐕 𝟐

Near-future strongest
𝐼 > 1018 eV 4
e.g., ELI @ Europe

Strong EM field

Current strongest
𝐼 > 1017 eV 4
CoReLS @ Korea

𝟏𝟎𝟏𝟗[Yoon et al., (2021)]

𝟏𝟎𝟏𝟒

CPA technique𝟏𝟎𝟗

𝐼 ∼ 1016 eV 4
HERUCLES @ USA

𝟏𝟎𝟒
1960 ’70[Yanovsky et al., (2008)]‘80’90 2000 ’10‘20Need of extremely strong EM field
Minimum strength: 𝒆𝑬, 𝒆𝑩 𝐌𝐞𝐕 𝟐 > 𝒎𝟐𝒆 = 𝟎. 𝟓𝟏𝟏 𝐌𝐞𝐕 𝟐 ≈ 𝑶(𝟏𝟎𝟐𝟖 𝐖/𝐜𝐦𝟐 )Laser processing machine < 1010 W/cm2

Glasses ≈ 103 W/cm2

LED ≈ 10−5 W/cm2

Intense laser is the most common source for strong EM field[Mourou, Tajima, Bulanov (2006)]

𝟏𝟎𝟐𝟒

Birth of laser

Laser intensity 𝐼 𝐞𝐕 𝟒

𝒆𝑬, 𝒆𝑩~ 𝟓𝟏𝟏 𝐤𝐞𝐕 𝟐

Near-future strongest
𝐼 > 1018 eV 4
e.g., ELI @ Europe

Strong EM field

Current strongest
𝐼 > 1017 eV 4
CoReLS @ Korea

𝟏𝟎𝟏𝟗[Yoon et al., (2021)]

𝟏𝟎𝟏𝟒

CPA technique𝟏𝟎𝟗

𝐼 ∼ 1016 eV 4
HERUCLES @ USA

𝟏𝟎𝟒
1960 ’70[Yanovsky et al., (2008)]‘80’90 2000 ’10‘20

Weak. Any nice idea ? ⇒ Heavy-ion collisions !

1. Brief review of strong-field physics

2. Strong EM field at high-energy heavy-ion collisions
3. Strong EM field at middle-energy heavy-ion collisions[Nishimura (Osaka), Ohnishi (Kyoto), HT, in progress]

4. An example of strong-field phenomenon:
Vacuum decay and modification to photon propagation[HT, Ironside (Curtin), in progress]

5. Summary

Strong EM field at high-energy heavy-ion collisions
B∝J
J

b

J

Animation stolen from Internet

Strong magnetic field is createdStrong EM field at high-energy heavy-ion collisions[Deng, Huang (2012)]
See also [Bzdak, Skokov (2012)] [Hattori, Huang (2016)]

Strong magnetic field is created
Pro: Super strong 𝑒𝐵 ≫ Λ2QCD (Strongest in the Universe !)Cons: Extremely short-lived (𝜏 ≪ 1 fm/𝑐 )⇒ Affects “non-perturbativeness” of physics

Shorter lifetime ⇒ more perturbative
Intuition: No time for multiple interactions
𝑭

𝑭

𝑭

𝑭
𝑭

𝑭

…

𝑭

𝑭
𝑭

𝑭

Shorter lifetime ⇒ more perturbative
Intuition: No time for multiple interactions
𝑭

𝑭

𝑭

𝑭
𝑭

𝑭

𝑭

𝑭

…

𝑭

“Phase diagram” of strong-field physics
As example: Vacuum particle prod. by E field w/ finite lifetime[HT, Fujiii, Itakura (2014)][HT, Fujimori, Misumi, Nitta, Sakai (2021)]

Non-pert.

・ Three dimensionful parameters in the system: e𝐸, 𝜏, 𝑚

High-energy HIC(for 𝒎~𝚲𝐐𝐂𝐃 )⇒ Two dim.-less parameters determine the physics
𝛾=

𝑚
=
𝑒𝐸 𝜏

𝜈=

𝑒𝐸 𝜏
=
1/𝜏

(Typical energy)(Work by field)(Work by field)(Photon energy)

𝑭⇒ Characterize the magnitude of work

⇒ Characterize the number of photons

Pert.

(LO is enough)・ 𝛄 ≪ 𝟏 , 𝝂 ≫ 𝟏 ⇒ Non-perturbative・ High-energy HIC: 𝑒𝐹 ∼ 1 GeV

2, 𝜏 ∼ 0.1 fmΤ𝑐 ⇒ γ ∼ ൝10−3 𝑚 = Λ QCD
10−5 𝑚 = 𝑚e, 𝜈 ∼ 0.1

Shorter lifetime ⇒ more perturbative
Intuition: No time for multiple interactions
𝑭

𝑭

𝑭

𝑭
𝑭

𝑭

𝑭

𝑭

…

𝑭

“Phase diagram” of strong-field physics
As example: Vacuum particle prod. by E field w/ finite lifetime[HT, Fujiii, Itakura (2014)][HT, Fujimori, Misumi, Nitta, Sakai (2021)]

Non-pert.

・ Three dimensionful parameters in the system: e𝐸, 𝜏, 𝑚

High-energy HIC(for 𝒎~𝚲𝐐𝐂𝐃 )⇒ Two dim.-less parameters determine the physics
𝛾=

𝑚
=
𝑒𝐸 𝜏

𝜈=

𝑒𝐸 𝜏
=
1/𝜏

(Typical energy)(Work by field)(Work by field)(Photon energy)

𝑭⇒ Characterize the magnitude of work

⇒ Characterize the number of photons

Pert.

(LO is enough)・ 𝛄 ≪ 𝟏 , 𝝂 ≫ 𝟏 ⇒ Non-perturbative・ High-energy HIC: 𝑒𝐹 ∼ 1 GeV

2, 𝜏 ∼ 0.1 fmΤ𝑐 ⇒ γ ∼ ൝10−3 𝑚 = Λ QCD
10−5 𝑚 = 𝑚e, 𝜈 ∼ 0.1

⇒ High-energy heavy-ion collisions are meaningless for strong-field physics ?
⇒ Not necessarily. A good chance to study higher-order effects

Achievements

Experimental results(Prior to any other experiments; e.g., intense laser)・ very FIRST observation of high-order QED processes
ex. 1) Light-by-light scatteringex. 2) Breit-Wheeler process[ATLAS (2016)][STAR (2019)]

・ Unfortunately, nothing has been observed in QCDex) Negative result of CME search with isobar collisions[STAR (2021)]

OK, they are good. BUT, what we really want is something non-pert. !

...

⇒ Any good ideas?
⇒ Go to “middle energies” ！

1. Brief review of strong-field physics

2. Strong EM field at high-energy heavy-ion collisions
3. Strong EM field at middle-energy heavy-ion collisions[Nishimura (Osaka), Ohnishi (Kyoto), HT, in progress]

4. An example of strong-field phenomenon:
Vacuum decay and modification to photon propagation[HT, Ironside (Curtin), in progress]

5. Summary

Strong EM field at middle-energy heavy-ion collisions
Lower energy ⇒ Landau stopping pictureExp.: Proton rapidity dist. dN/dy[Esumi (2007)]AGS 5 GeV

Thr.: Numerics w/ hadron transport models
E = O(10 GeV/nucleon)

E = O(10 MeV/nucleon)

EnergySPS 20 GeV

RHIC 62 GeV[Nara (1999; proc)]
RHIC 200 GeV

LHC 5500 GeV[Maruyama, Bonasera, Papa, Chiba (2002)]⇒ Dense matter is formed for a long time 𝑶(𝟏𝟎 − 𝟏𝟎𝟎𝟎 𝐟𝐦/𝒄)

Strong EM field at middle-energy heavy-ion collisions
Lower energy ⇒ Landau stopping pictureExp.: Proton rapidity dist. dN/dy[Esumi (2007)]AGS 5 GeV

Thr.: Numerics w/ hadron transport models
E = O(10 GeV/nucleon)

E = O(10 MeV/nucleon)

EnergySPS 20 GeV

RHIC 62 GeV[Nara (1999; proc)]
RHIC 200 GeV

LHC 5500 GeV[Maruyama, Bonasera, Papa, Chiba (2002)]⇒ Dense matter is formed for a long time 𝑶(𝟏𝟎 − 𝟏𝟎𝟎𝟎 𝐟𝐦/𝒄)⇒ Charge density is also large (“High Z atom” s.t. 𝒁 ≳ 𝟏/𝜶) ⇒ Strong Coulomb field・ High energy ⇒ Magnetic vs Low/middle energy ⇒ Electric・ Rough order estimate: 𝑒𝐸 ∼⇒𝛾=𝑍𝛼∼ Λ2QCD ∼𝑟2

100 MeV 2

10−1 𝑚 = ΛQCD
< ൝ −4∼ 0.1, 𝜈 = 𝑒𝐸𝜏 2 ∼> 10 ⇒ Non.-pert both in QED & QCD ቊ𝜸 ≪ 𝟏𝑒𝐸𝜏 ∼ 10𝑚 = 𝑚e𝝂≫𝟏𝑚Strong EM field at middle-energy heavy-ion collisions
Lower energy ⇒ Landau stopping pictureExp.: Proton rapidity dist. dN/dy[Esumi (2007)]AGS 5 GeV

Thr.: Numerics w/ hadron transport models
E = O(10 GeV/nucleon)

E = O(10 MeV/nucleon)

EnergySPS 20 GeV

RHIC 62 GeV[Nara (1999; proc)]
RHIC 200 GeV

LHC 5500 GeV[Maruyama, Bonasera, Papa, Chiba (2002)]⇒ Dense matter is formed for a long time 𝑶(𝟏𝟎 − 𝟏𝟎𝟎𝟎 𝐟𝐦/𝒄)⇒ Charge density is also large (“High Z atom” s.t. 𝒁 ≳ 𝟏/𝜶) ⇒ Strong Coulomb field・ High energy ⇒ Magnetic vs Low/middle energy ⇒ Electric・ Rough order estimate: 𝑒𝐸 ∼⇒𝛾=𝑍𝛼∼ Λ2QCD ∼𝑟2

100 MeV 2

10−1 𝑚 = ΛQCD
< ൝ −4∼ 0.1, 𝜈 = 𝑒𝐸𝜏 2 ∼> 10 ⇒ Non.-pert both in QED & QCD ቊ𝜸 ≪ 𝟏𝑒𝐸𝜏 ∼ 10𝑚 = 𝑚e𝝂≫𝟏𝑚∴ Interesting to strong-field physics. May affect QCD/hadron processes as well
⇒ Need to think about this seriouslyEstimation with a hadron trans. model: JAM (1/5)[Nishimura (Osaka), Ohnishi (Kyoto), HT, in progress]

JAM (Jet AA Microscopic transport model)[Nara, Otsuka, Ohnishi, Nitta, Chiba (2000)]

・ A microscopic model to simulate heavy-ion collisions= Superposition of two body hadron scatterings + Inelastic scattering processes
Low energy: Resonance (𝐸CM ≲ 4 GeV)High energy: PYHITA → string breaking (𝐸CM ≳ 4 GeV)
mini jets(𝐸CM ≳ 20 GeV)・ Reliable at not-high energies, i.e., below RHIC energy O(100 GeV) scale
・ Anyway, phase space distribution of charged particles can be calculated

Phase space dist. ⇒ EM field
From JAM

Rem. 1: Don’t use Lienard-Wiechert potential for a point-like particle⇐ LW potential is not applicable, since particles are produced

Rem. 2: The following results are after N=100 event averaging: 𝐴 =1 𝑁σ𝐴𝑁 𝑛=1 𝑛

Estimation with a hadron trans. model: JAM (2/5)[Nishimura (Osaka), Ohnishi (Kyoto), HT, in progress]yTime evol. of charge density (central coll.) @ z=0zxPreliminary
results

ECM = 2.0 GeV

3.0 GeV

4.0 GeV

6.3 GeV

7.7 GeV・ Huge charge density with a long lifetime time

- O(10) times bigger than charge density of a single static ion 𝜌charge ~ 𝜌 nuclear /2 ~ 0.08 fm−3
- Energy⇒ρ(∵ Lorentz contraction)- Lifetime ~ O(20 fm/c)・ Switching Landau

Bjorken pictures at ~4 GeV

But, not completely transparent, so have finite ρ in the center⇒ Energy⇒ Spacetime volume

Saturation
densityEstimation with a hadron trans. model: JAM (3/5)[Nishimura (Osaka), Ohnishi (Kyoto), HT, in progress]yTime evolution of EM field (central coll. b=0) @ z=0
・ Lorentz inv. 𝐹 ≔ 𝐸 2 − 𝐵2 (F>0: Electric, F<0: Magnetic)

ECM = 2.0 GeV

3.0 GeVx

Preliminary
results

4.0 GeVz

6.3 GeV

7.7 GeV

50 MeV 4

50 MeV 4

50 MeV 4

50 MeV 4

50 MeV 4

0 MeV 40 MeV 40 MeV 40 MeV 40 MeV 4− 50 MeV 4− 50 MeV 4− 50 MeV 4− 50 MeV 4− 50 MeV 4・ Donut-shaped⇐ Gauss law 𝐸 ∝ ∫ d3 𝒙 𝜌・ Field strength can be strong as 𝑒 2 𝐹 ~ 𝑒𝐸 2 = 𝑂 50 MeV 4⇒ ・ Very strong for QED (𝒎𝒆 = 𝟎. 𝟓𝟏𝟏 𝐌𝐞𝐕)
・ May still weak for QCD (𝚲𝐐𝐂𝐃 = 𝟐𝟎𝟎 𝐌𝐞𝐕), but is non-negligible(If deconfined, very strong for current quark mass)Estimation with a hadron trans. model: JAM (4/5)[Nishimura (Osaka), Ohnishi (Kyoto), HT, in progress]

Spacetime volume of EM field (central coll. b=0)
Volume for 𝑒 2 𝐹 = |𝑒𝑬|2 − 𝑒𝑩 2 > 30 MeV 4

Before collision= Strength: so soVolume: so soCollision= Strength: strongest
Volume: small
After collision

Preliminary
results= Strength: weak
Volume: large

・ Strong E field with 𝒆𝟐 𝑭 ~ 𝒆𝑬 𝟐 ~ 𝟑𝟎 𝐌𝐞𝐕 𝟐 ,lifetime 𝝉 ≳ 𝟐𝟎 𝐟𝐦Τ𝒄 , volume 𝑽 ≳ 𝟏𝟓 𝐟𝐦 𝟑
⇒ Non-pert for QED. Non-linear for QCD
γ=

𝑚
𝑚
𝑂(1)
𝑚 = ΛQCDor=൝, 𝜈 = 𝑒𝐸𝜏 2 or 𝑒𝐸𝑙 2 = 𝑂(10)𝑒𝐸𝜏 𝑒𝐸𝑙
𝑂(0.1) 𝑚 = 𝑚e

・ Volume increases monotonically below 10 GeV⇒ A bit funny (?) Needs more investigation (future work)

Non-pert

Pert.

(LO is enough)QED

𝒎 = 𝒎𝒆QCD

Estimation with a hadron trans. model: JAM (5/5)[Nishimura (Osaka), Ohnishi (Kyoto), HT, in progress]

𝑒 2 𝐹 = |𝑒𝑬|2 − 𝑒𝑩 2

Non-central collision ⇒ B & E・B are produced

E・B<0

y
impact para. b

x B
E・B>0

50 MeV 4

50 MeV 4

0 MeV 40 MeV 40 MeV 4− 50 MeV 4− 50 MeV 4− 50 MeV 4

5 fm

b = 0 fmz
𝑒 2 𝐺 = 𝑒𝑬 ⋅ 𝑒𝑩

E

50 MeV 4

10 fm

50 MeV 4

50 MeV 4

50 MeV 4

0 MeV 40 MeV 40 MeV 4− 50 MeV 4− 50 MeV 4− 50 MeV 4

Preliminary
results@ ECM = 7.7 GeV・ 𝑒𝑬 ⋅ 𝑒𝑩 = 𝑂 50 MeV 4 ⇒ Non-negligible for QCD (Enough strong for electron and current quarks)
・ Perhaps, a nice place to study chiral-anomaly-related stuffs
𝑒𝑬 ⋅ 𝑒𝑩𝑚2ex) chirality production via the Schwinger mechanism 𝑁5 = 𝑉𝑇
exp −𝜋2𝜋 2𝑒𝐸

1. Brief review of strong-field physics

2. Strong EM field at high-energy heavy-ion collisions
3. Strong EM field at middle-energy heavy-ion collisions[Nishimura (Osaka), Ohnishi (Kyoto), HT, in progress]

4. An example of strong-field phenomenon:
Vacuum decay and modification to photon propagation[HT, Ironside (Curtin), in progress]

5. Summary

So far, what interesting?
Middle-energy HIC can be useful for studying “strong” “E-field” phys
・ point 1: Enables us to explorer the non-pert.
regime, where other experiments(e.g., high-energy HIC, intense laser)cannot coverNon-pert

QED

𝒎 = 𝒎𝒆QCD・ point 2: E- and B-field physics are different
High energy ⇒ B field ⇒ system is stable
Middle energy ⇒ E field ⇒ system is unstableO(10%) of QCD scale ⇒ May affect hadron/QCD processesex) B field at middle-energy heavy-ion collisions can affect flow and yield ratios[Sun, Wang, Li, Wang (2019)]

Ex. of E-field pheno.: Vacuum decay and photon propagation (1/4)
Vacuum decay = Vacuum becomes unstable against particle productionin the presence of strong E field・ Const E field ⇒ Schwinger mechanism [Sauter 1931] [Schwinger (1951)]
𝑉 = −𝑒𝐸𝑥

energyPositive energy states
gap

tunneling

Dirac sea
x=0

Const Ex

Ex. of E-field pheno.: Vacuum decay and photon propagation (1/4)
Vacuum decay = Vacuum becomes unstable against particle productionin the presence of strong E field・ Const E field ⇒ Schwinger mechanism [Sauter 1931] [Schwinger (1951)]
𝑉 = −𝑒𝐸𝑥

energyPositive energy states
gap

Const E
tunneling

Dirac seax=0energyxcontinuum

+m

𝑉=−

𝑍𝛼
𝑟

discrete-m・ Essentially the same for Coulomb field
Difference: Energy is discretized for 𝐸 < +𝑚

r=0

r=∞

⇒ Tunneling can occur only if there’re energy levels for 𝐸 < −𝑚
⇐ (under various simplifications) OK if 𝑍 is sufficiently large: 𝑍 > 𝛼 −1

[Pieper, Greiner (1969)][Gershtein, Zeldovich (1970)]

Ex. of E-field pheno.: Vacuum decay and photon propagation (1/4)
Vacuum decay = Vacuum becomes unstable against particle productionin the presence of strong E field・ Const E field ⇒ Schwinger mechanism [Sauter 1931] [Schwinger (1951)]
𝑉 = −𝑒𝐸𝑥

energyPositive energy states
gap

Const E
tunneling

Dirac seax=0energyxcontinuum

+m

𝑉=−

𝑍𝛼
𝑟

discrete-m・ Essentially the same for Coulomb field
Difference: Energy is discretized for 𝐸 < +𝑚

r=0

r=∞

⇒ Tunneling can occur only if there’re energy levels for 𝐸 < −𝑚
⇐ (under various simplifications) OK if 𝑍 is sufficiently large: 𝑍 > 𝛼 −1

[Pieper, Greiner (1969)][Gershtein, Zeldovich (1970)]∴ Extra positron production in middle-energy HIC
This is interesting, but not the whole story ⇒ Change of photon propagation

Ex. of E-field pheno.: Vacuum decay and photon propagation (2/4)“Tilting” of the vacuum affects photon propagation⇐ Interference in the Dirac sea due to the reflection by the gap[HT (2019)]

E field

𝑉 = −𝑒𝐸𝑥
in-coming

Positive energy states
Reflected
out-going
Electron distribution of the Dirac sea
energyDirac seax

gap

Ex. of E-field pheno.: Vacuum decay and photon propagation (2/4)“Tilting” of the vacuum affects photon propagation⇐ Interference in the Dirac sea due to the reflection by the gap[HT (2019)]

E field

𝑉 = −𝑒𝐸𝑥
in-coming

Positive energy states
Reflected
out-going
Electron distribution of the Dirac sea
energyDirac seagapx

⇒ Photon propagating in the vacuum shall interact with the Dirac sea
Expectation 1: Photon propagation (=real and imag. parts of reflective index)
should have signatures of the oscillating electron distribution
Expectation 2: Photon propagation should depend on the direction of the E field(vacuum birefringence)
For B field: [Hattori, Itakura (2013)]

Ex. of E-field pheno.: Vacuum decay and photon propagation (3/4)[HT, Ironside (Curtin), in progress]

E field

E fieldℰ ∝ e−i𝜔𝑡

E field

A simple setup: a probe E field in the vacuum under a strong const E field

𝒟 = 𝜀ℰ = 𝑛2 ℰ = 1 + 𝜒 ℰ

Observables: Real and imag. parts of electric permittivity
Im ⇒ Decay rate of probe=

Wavefunction dressed by the strong E-field non-perturbatively/ − 𝑚]Ψ
s.t. 0 = [i𝜕/ − 𝑒𝐴ഥ /Ψ~ ∫ 𝑑 3 𝒑 Ψ𝒜Re ⇒ Velocity and birefringence of photon
Optical theorem (Kramers-Kronig relation)=

Ex. of E-field pheno.: Vacuum decay and photon propagation (4/4)[HT, Ironside (Curtin), in progress]

ResultRe ⇒ Velocity and birefringence of photon

Im 𝜀 𝐸 − 𝜀 0 /𝑒 2

𝓔∥𝑬
𝓔⊥𝑬

𝑒𝐸 Τ𝑚2 = 0.25
Probe frequency

Re 𝜀 𝐸 − 𝜀 0 /𝑒 2Im ⇒ Decay rate of probe

𝓔∥𝑬
𝓔⊥𝑬
𝑒𝐸 Τ𝑚2 = 0.25

Probe frequency

・ Oscillating behavior⇐ result of the oscillating Dirac-sea structure
・ Probe-direction dependent response⇒ Birefringence

真空

Preliminary
results

Large density ⇒ Affects more

1. Brief review of strong-field physics

2. Strong EM field at high-energy heavy-ion collisions
3. Strong EM field at middle-energy heavy-ion collisions[Nishimura (Osaka), Ohnishi (Kyoto), HT, in progress]

4. An example of strong-field phenomenon:
Vacuum decay and modification to photon propagation[HT, Ironside (Curtin), in progress]

5. Summary

Summary
Middle-energy heavy-ion collisions 𝒔𝐍𝐍 = 𝑶 𝟏‐ 𝟏𝟎 𝐆𝐞𝐕 may be interesting
not only to QCD but also to strong-field physics
・ Estimation of EM field strength and spacetime volume with a hadron transport model (JAM)⇒ Strong O(30 MeV) and large spacetime volume O((20 fm) 4 )⇒ A nice setup to study strong-field QED. Non-negligible to hadronic/QCD processes as well.

Spacetime volume of EM field“Phase diagram” ofstrong-field physics

Photon propagation
on top of decaying vacuum

𝑒 2 𝐹 = |𝑒𝑬|2 − 𝑒𝑩 2 > 30 MeV 4

Non.-pert.

QED

𝒎 = 𝒎𝒆Pert.

QCD(LO is enough)[Nishimura (Osaka), Ohnishi (Kyoto), HT, in progress][HT, Ironside (Curtin), in progress]