Quark Pair Production fromExpanding Electromagnetic Flux Tube

Hidetoshi
TAYA(Univ. Tokyo)
h_taya@hep1.c.u-tokyo.ac.jp
Hirotsugu FUJII (Univ. Tokyo)

1. INTRODUCTION
1-1. Formation Process of QGP is a BIG QUESTION in HIC
Collision:
0 fm/c~1 fm/ctime 𝜏

1-2. Previous Studies on Quark Production via Schwinger Mech.
Some of the important properties of the flux tubes are
1. Longitudinal expansion (Bjorken expansion) of the flux tubes
2. Presence of magnetic fields in parallel to electric fields,?

QGPExpanding?

Color EM Flux Tubes(Glasma)

CGCHydrodynamics

gluon & quark PARTICLESclassical gluon FIELDS

Aim of this study

whose effects on the Schwinger mech. are NOT understood well in previous studies:

Try to reveal ? focusing on the quark production
based on the Schwinger mechanism

Magnetic Field?

Quark Mass

Mihaila et al.

Yes

NoTanjiNo

Yes (with arbitrary strength)

Gelis et al.

Yes, but only for

Yes

Limited:
𝒈𝑩
𝒈𝑬

∼𝟏

𝒎𝟐
𝒈𝑬

Backreaction?

𝟏=𝟒

Yes

Arbitrary

Yes

Limited: 𝒎 ≥ 60 MeV

No

ArbitraryYes

Our study gets over these limitations :Our Study

Yes

Yes (with arbitrary strength)

2. THEORY
2-1. Formulation✔

✔

𝝁𝟎 = 𝒊𝜸 (𝝏𝝁 +𝒊𝒈𝑨𝝁 ) − 𝒎 𝝍,Following simplifications are assumed hereafter:

1. U(1) gauge theory with 𝑵𝒇 = 𝟏 fermion
2. Homogeneity in space: Boost invariance & transverse symmetry
3. Single flux tube with infinite length in the transverse direction
4. Longitudinal EM-field at initial time 𝝉𝟎 : 𝑬(𝝉𝟎 ) = 𝟎, 𝟎, 𝑬𝟎 , 𝑩(𝝉𝟎 ) = (𝟎, 𝟎, 𝑩𝟎 )3. RESULT I – WithOUT Backreaction g=0 (Analytical)

𝝂𝝁𝑭 ;𝝁 = 𝒈𝝍𝜸𝝂 𝝍and canonically quantize 𝝍 → 𝝍 at each instant of (proper) time 𝜏:𝝍 𝒙 =𝜏𝜏𝒆𝒊𝝀𝜼

positive freq. mode

negative freq. mode𝟐𝝅𝒅𝝀𝒔𝒏

†
𝜓𝑥,
𝜏𝒂
𝜏
+
𝜓𝑥,
𝜏𝒃+ 𝑛,𝜆,𝑠 𝑇− 𝑛,𝜆,𝑠 𝑇𝑛,𝜆,𝑠𝑛,−𝜆,𝑠 𝜏REMARK 1: 3 labels = spin 𝒔, Landau level 𝒏, and (canonical) longitudinal momentum 𝝀.
𝜏REMARK 2: The instantaneous mode functions ±𝜓𝑛,𝜆,𝑠 𝑥𝑇 , 𝜏 are defined by theapproximate sol. of the Dirac eq. under an adiabatic assumption: 𝜕𝜏 𝐴𝜇 = 0Tanji;

3-1. Longitudinal Distribution 𝒅𝟐 𝑵 𝒅𝒚𝒅𝝀 /𝑺⊥ [𝐟𝐦−𝟐 ]Landau Level n=0

We solve the EoMs (non-perturbatively w.r.t the strong fields 𝑔𝐴𝜇 ):Schmidt, Blashke, Smolyansky …;

4. RESULT II – With Backreaction g≠0 (Numerical)

Landau Level n=24-1. Longitudinal Distribution 𝒅𝟐 𝑵 𝒅𝒚𝒅𝝀 /𝑺⊥ [𝐟𝐦−𝟐 ]Landau Level n=0

𝑔𝐸0 = 1 GeV 2 ,𝑔𝐵0 = 1 GeV 2 ,𝑚 = 0.001 GeV

𝑔𝐸0 = 1 GeV 2 ,𝑔𝐵0 = 1 GeV 2 ,𝑚 = 0.001 GeV✔ The distribution is saturated because of✔ 2-peak structure:Ιand .
Pauli’s principle.
✔ I : Particles which annihilate w/ their pair
✔ Particles are created constantly with 0
immediately after their creation.
longitudinal momentum.
This peak depends on the choice of the✔ After being created, particles are accelerated
instantaneous particle picture.
by electric fields in the long. direction✔ I has pwr-dep. on n: ∼ (𝑚2 + 2𝑛𝑔𝐵)/𝑔𝐸 −2obeying the classical EoM: 0 = 𝑑𝜆 𝑑𝜏3-2. Number Density 𝒅𝑵 𝒅𝒚 /𝑺⊥ [𝐟𝐦]𝟎. 𝟓 𝐆𝐞𝐕 𝟐
Dependence on Mag. Field 𝒈𝑩𝟎 = 𝟏 𝐆𝐞𝐕 𝟐𝟐 𝐆𝐞𝐕 𝟐

𝟎. 𝟎𝟎𝟏 𝐆𝐞𝐕
𝟎. 𝟏 𝐆𝐞𝐕Dependence on Mass 𝒎 =𝟎. 𝟓 𝐆𝐞𝐕
𝑔𝐸0 = 1 GeV 2 ,𝑔𝐵0 = 1 GeV 2−𝟐

𝒎𝟐 + 𝟐𝒏𝒈𝑩−𝟐n (𝒎𝑻 ), while it has a power-law tail ∝𝒈𝑬for large n (𝒎𝑻 ). cf. Mihaila et al. (2009)✔ Lowest Landau contribution (n=0) dominates the particle production.
✔ Larger particle production ∝ 𝑔𝐵 for increasing 𝑔𝐵.
✔ Very fast quark production: For example, 𝑑𝑁 𝑑𝑦/𝑆⊥ ∼ 1 fm−2 particles are produced atGelis et al.
𝜏 ∼ 0.2 fm, which roughly corresponds to 𝑑𝑁𝑞 𝑑𝑦 ∼ 𝑁𝑐 × 𝑁𝑓 × 1 × 𝑆⊥ ∼ 1000 in HIC. cf. (2006)✔ Combining with the experimental fact 𝑑𝑁ℎ 𝑑𝑦 ≲ 1000, classical gluon fields should

𝑔𝐸0 = 1 GeV 2 ,𝑔𝐵0 = 1 GeV 2 ,𝑚 = 0.001 GeV,
g=1𝑔2 𝑁𝑓 𝑁𝑐✔ Plasma oscillation with timescale 𝜏osc ∼ 1
𝑔𝐵 occurs as a result of the backreaction.
2𝜋2✔ In addition to the oscillating behavior, the longitudinal momentum 𝜋𝜂 decreases as ∼ 1 𝜏
reflecting the longitudinal expansion of the system.

−𝟐𝒅𝑵
𝒅𝒚/𝑺
[𝐟𝐦]
4-2. Number Density⊥𝟎. 𝟎𝟎𝟏 𝐆𝐞𝐕
Dependence on Mass 𝒎 =𝟎. 𝟏 𝐆𝐞𝐕
𝟎. 𝟓 𝐆𝐞𝐕

𝟎. 𝟓 𝐆𝐞𝐕 𝟐Dependence on Mag. Field 𝒈𝑩𝟎 = 𝟏 𝐆𝐞𝐕 𝟐𝟐 𝐆𝐞𝐕 𝟐
𝑔𝐸0 = 1 GeV 2 ,𝑚 = 0.001 GeV,
g=1𝑔𝐸0 = 1 GeV 2 ,𝑔𝐵0 = 1 GeV 2 ,g=1

𝑔𝐸0 = 1 GeV 2 ,𝑚 = 0.001 GeV✔ The density is consistent w/ Schwinger’s formula ∝ 𝐞𝐱𝐩[−𝝅|(𝒎𝟐 + 𝟐𝒏𝒈𝑩)/𝒈𝑬|] for small

Landau Level n=2

𝑔𝐸0 = 1 GeV 2 ,𝑔𝐵0 = 1 GeV 2 ,𝑚 = 0.001 GeV,
g=1Schwinger FormulaPower-law TailKluger, Mottola, Eisenberg …

✔ The oscillation is “flattened” with increasing the mass.
✔ Faster oscillation with increasing 𝑔𝐵.
✔ Effect of backreaction is small for small values of 𝜏 ≲ 0.2 fm, while it is not for large valuesof 𝜏 ≳ 0.2 fm, where 𝑑𝑁/𝑑𝑦/𝑆⊥ ∝ 𝜏 (𝜏 2 ) with(without) backreaction.

4-3. Decay of Electric Fields
𝟎. 𝟎𝟎𝟏 𝐆𝐞𝐕
Dependence on Mass 𝒎 = 𝟎. 𝟏 𝐆𝐞𝐕
𝟎. 𝟑 𝐆𝐞𝐕
𝟎. 𝟓 𝐆𝐞𝐕

𝟎. 𝟓 𝐆𝐞𝐕 𝟐
𝟐𝟏𝐆𝐞𝐕
Dependence on Mag. Field 𝒈𝑩𝟎 =𝟐 𝐆𝐞𝐕 𝟐𝟒 𝐆𝐞𝐕 𝟐

have decayed very fast 𝝉 ≲ 0.2 fm, otherwise we would have too huge # of particles.

5. SUMMARY
What we did
The quark production in the early stage dynamics of HIC based on the Schwinger
mechanism is extensively studied.
What we have learned
1. Effects of the longitudinal expansion and the longitudinal magnetic fields on theSchwinger mech., which are the missing pieces of previous studies, are clarified.
2. The Schwinger mech. can explain a fast quark production and, combining w/ theexperimental result on the multiplicity, it suggests that the classical gluon fields
should have decayed very fast (though we did not say anything about its physics).
3. The backreaction effects from the produced quarks are also investigated, and anapproximate formula is obtained in the LLL approximation and the massless limit.
There are several topics (energy density, pressure, anomalous current, etc) which I could
not cover in this poster. If you are interested in these topics, feel free to ask me!

𝑔𝐸0 = 1 GeV 2 ,𝑔𝐵0 = 1 GeV 2 ,g=1

𝑔𝐸0 = 1 GeV 2 ,𝑚 = 0.001 GeV,
g=1

✔ 𝒈𝑩 never decays: Rotational sym. always holds due to the simplification 2&3 (i.e., 𝑗 𝜃 = 0)
✔ Faster decay and slower oscillation with increasing mass.
✔ Faster oscillation for stronger 𝑔𝐵.
✔ Long. expansion makes the field decay faster ∝ 1/ 𝜏 compared to the non-expanding case.

Note: Approximate Solution for the Backreaction Problem
If we take LLL approx. and massless limit, we can analytically solve the backreaction eqs. to find𝐽1 𝑠𝜏0𝑌1 𝑠𝜏0
1 𝑑𝑁
𝑔𝐸0
𝐽1 𝑠𝜏0𝐸(𝜏) = 𝐸0𝑌 𝑠𝜏 −𝐽 (𝑠𝜏) ,(𝜏) =𝑠𝜏
𝑌 𝑠𝜏
𝐽1 𝑠𝜏0 𝑌0 𝑠𝜏0 0𝑌1 𝑠𝜏0 𝐽0 𝑠𝜏0 0𝑆⊥ 𝑑𝑦2𝐽1 𝑠𝜏0 𝑌0 𝑠𝜏0 1
𝑁𝑐 𝑁𝑓 𝑔2
𝑔222
where 𝑠 = 2𝜋2 𝑔𝐵 (which roughly corresponds to 𝑠 ∼ 2𝜋2 𝑔𝐵 in QCD).

𝑌1 𝑠𝜏0−𝐽 (𝑠𝜏) ,𝑌1 𝑠𝜏0 𝐽0 𝑠𝜏0 1