Quark and Gluon Production froman Expanding Strong Color Electric Flux Tube

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

1. INTRODUCTION
1-1. Pre-equilibrium dynamics of HIC is a BIG MISSING PIECE

1-2. Aim of this study✔ Many studies on this topic have been done:Collision
0 fm/c~1 fm/ctime 𝜏

Physical picture: Schwinger mech. ; NO instability; Weibel instability; bottom up thermalization …

Theory: classical statistical approx.; 2PI; effective kinetic theory; classical YM; flux tube model …

?

QGP❓ How the transition from the classical field to quantum particles occur❓ How (not only gluons but also) quarks are produced

Color EM Flux Tubes(Glasma)

CGC✔ But, there remain many unanswered questions such as❓ How the system isotropizes (or not)Hydrodynamics✔ We discuss these points by studying quark & gluon particle production from anexpanding classical gauge field based on QCD within mean field approximation

gluon & quark PARTICLESclassical gluon FIELDS

2. THEORY＊ Only the gluon part is explained here. The quark and ghost parts can be formulated in the same way.

1𝐿QCD = − tr 𝐹𝜇𝜈 𝐹𝜇𝜈 + 𝐿quark + 𝐿FP+GF
2
STEP 1: Separate the total gauge field (𝐴) into a classical field (𝐴 ≡ 𝐴 ) and a quantum fluctuation (𝑎) as 𝐴 = 𝐴 + 𝑎
STEP 2: Expand Lagrangian in terms of the quantum fluctuation 𝑎1
𝜇𝜈𝐬t𝐧𝐝𝐫𝐝2𝐭𝐡𝐿QCD = − tr 𝐹 𝐹𝜇𝜈 + 𝟏 order in 𝑎 + 𝟐 order in 𝑎 + 𝑔 × 𝟑 order in 𝑎 + 𝑔 × 𝟒 order in 𝑎 + 𝐿quark + 𝐿FP+GF
2𝑎𝑎𝑎𝐴＊ 𝑎’s propagator () non-perturbatively includes multiple-interactions b/w the classical field 𝐴 as

STEP 3: Adopt mean field approximation for the non-=→linear (higher than 3rd order in 𝑎) terms＋𝐴𝐴＋＋…

→𝑎𝑎𝑎𝑎𝑎𝑎 → 𝑎𝑎 𝑎𝑎𝑎𝑎𝑎 → 𝑎 𝑎𝑎

＝ current from produced particles 𝒋＝ (partial) effects of scatterings 𝑴

𝐿QCD = (up to 2nd order in the quantum fluctuation 𝑎)
STEP 4: Solve the Euler-Lagrange equations for 𝑎 and 𝐴

𝟎=𝟐 𝝁𝝂

𝛁 + 𝒊𝒈𝑨 𝒈𝝁𝝂+ 𝑴

Advantages: The equations𝒂𝝂for the quantum fluctuation 𝑎,and𝒋𝝁= 𝛁𝝂 𝑭𝝂𝝁+ 𝒇𝝂𝝁for the classical field 𝐴✔ are simple and easy to understand their physical meanings✔ include higher order quantum effects (within the mean field treatment)✔ describe a smooth transition from the classical field to quantum particles✔ are numerically feasible

3. NUMERICAL RESULTS
3-1. Numerical Setup

3-4. Number Density 𝒅𝑵 𝒅𝒚𝒑 /𝑺𝑻

3-2. Energy Density

3-3. Longitudinal Distributionenergy density [𝐆𝐞𝐕 𝟐 ]quark

𝟐𝝅 𝟑𝒅 𝑵

𝒅𝒑𝟐𝑻 𝒅𝒑𝜼 𝒅𝒙𝟐𝑻 𝒅𝜼 at 𝒑𝑻 ∼ 𝟎✔ Fast particle production

𝒅𝑵 𝒅𝒚𝒑 /𝑺𝑻 [𝐆𝐞𝐕 𝟐 ]✔ 𝑺𝑼(𝑵𝒄 = 𝟑) QCD with 𝑵𝒇 = 𝟑 massless quarks✔ Homogeneity in space: Boost invariance & transverse symmetry✔ Longitudinal color electric field at an initial time 𝝉𝟎 : 𝑬(𝝉𝟎 ) = 𝟎, 𝟎, 𝑬𝟎 , 𝑩(𝝉𝟎 ) = (𝟎, 𝟎, 𝟎)✔ Neglect the higher order term 𝑴 = 𝟎 (i.e., collisionless limit)✔ Parameter setting: 𝑬𝟎 = 𝟏 𝐆𝐞𝐕 𝟐 , 𝒈 = 𝟏, 𝝉𝟎 = 𝟎. 𝟏 𝐆𝐞𝐕 −𝟏
𝟔- In particular, quark production
is very fast 𝝉 <∼ 𝟏 𝐟𝐦/𝒄

✔ Huge number of particles- Gluon dominatestime [𝐟𝐦/𝒄]

gluon- 1,000 quarks + 2,000 gluons( for 𝑺𝑻 ∼ 𝝅 𝟕 𝒇𝒎 𝟐 )3-5. Isotropization✔ System becomes lessanisotropic as the classicalfield decays

time [𝐟𝐦/𝒄]- 𝑷𝑳 𝑷𝑻 ∼ 𝟎. 𝟓classical fieldtransient particles✔ Plasma oscillation✔ The classical field decays into quark ✔ Pauli blocking for quarks & Bose enhancement for gluons& gluon particles within a few 𝐟𝐦 𝒄 ✔ Red shift due to the longitudinal expansion(even if collisionless)time [𝐟𝐦/𝒄]

4. SUMMARY
What we did

What we have learned

What to do

We studied quark & gluon production
from an expanding classical gauge
field to discuss the pre-equilibrium
dynamics of HIC

Adopting mean field approx. to QCD, we have learned that✔ higher order effects (e.g. collisions)✔ inhomogeneity in transverse plane✔ longitudinal magnetic field✔ photon/di-lepton production✔ classical field decays into quarks & gluons within a few 𝐟𝐦 𝒄
✔ how the transition from the classical field to particles occurs✔ huge number of quarks & gluons are produced quickly✔ the anisotropy becomes moderate as 𝑷𝑳 𝑷𝑻 ∼ 𝟎. 𝟓