重イオン衝突における散逸と揺らぎ

RHIC-LHC 高エネルギー原子核反応の物理研究会2013年 6月 22-23日

平野哲文（上智大理工）共同研究者 : 村瀬功一 , 橘保貴

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Outline1. はじめに2. Initial fluctuation3. Relativistic fluctuating

hydrodynamics4. Jet propagation in expanding

medium5. Summary and outlook

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はじめに発見のステージ 精密科学に向けて　　　　　　　 新奇な現象の発見

重イオン衝突における QGP の物理　

より詳細な QGP ダイナミクスの記述に向けた試み3

INITIAL FLUCTUATION

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Eccentricity Fluctuation in Small System

𝜀 part>𝜀stdPHOBOS, Phys. Rev. Lett. 98, 242302 (2007) 5

Fluctuation in Initial Conditions

B.Alver et al., Phys. Rev. C 77, 014906 (2008)

𝜀𝑛=⟨𝑟2𝑒𝑖𝑛 𝜙 ⟩

⟨𝑟 2 ⟩

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Higher Harmonics is Finite!

Figures adapted from talkby J.Jia (ATLAS) at QM2011

Two particle correlation function is composed solely of higher harmonics

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Impact of Finite Higher Harmonics• Most of the people

did not believe hydro description of the QGP (~ 1995)

• Hydro at work to describe elliptic flow (~ 2001)

• Hydro at work (?) to describe higher harmonics (~ 2010)

𝑑≲5 fm

coarsegraining

sizeinitialprofile

𝑑≲1 fm8

Open Questions• Can system respond

hydrodynamically to such a fine structure?

• Why is local thermalization achieved at such a short length/time scale (~1 fm)?

• Concept of hydrodynamics? Is ensemble average taken?

• Thermal fluctuation on an event-by-event basis ? 9

RELATIVISTIC FLUCTUATING HYDRODYNAMICS

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Fluctuation Appears Everywhere

0collision axis

time Finite number of

hadrons

Propagation of jets

Hydrodynamic fluctuation

Initial fluctuation andparticle production

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Green-Kubo Formula

Slow dynamics How slow?Macroscopic time scale ~ Microscopic time scale ~

cf.) Long tail problem (liquid in 2D, glassy system, super-cooling, etc. )

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Relaxation and CausalityConstitutive equations at Navier-Stokes level

…

𝑡

thermodynamics force

𝑡 0

𝐹∨𝑅

/𝜅

Realistic response

Instantaneous response violates causality Critical issue in

relativistic theory Relaxation plays an

essential role

𝜏 13

Causal Hydrodynamics

Π (𝑡 )=∫𝑑𝑡′𝐺𝑅 (𝑡 , 𝑡′ ) 𝐹 (𝑡 ′ )Linear response to thermodynamic force

Retarded Green function𝐺𝑅 (𝑡 ,𝑡 ′ )= 𝜅

𝜏 exp(− 𝑡− 𝑡′𝜏 )𝜃 (𝑡−𝑡 ′)

Differential formΠ̇ (𝑡 )=− Π (𝑡 )−𝜅 𝐹 (𝑡 )

𝜏 ,

Maxwell-Cattaneo Eq. (simplified Israel-Stewart Eq.)

𝑣 signal=√ 𝜅𝜏 <𝑐

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Relativistic Fluctuating Hydrodynamics (RFH)

)

⟨ 𝛿Π (𝑥 )𝛿Π (𝑥′ )⟩=𝑇𝐺∗(𝑥 , 𝑥′ )

Generalized Langevin Eq.dissipative current

hydrodynamicfluctuationthermodynamic force

For non-relativistic case, see Landau-Lifshitz, Fluid Mechanics

Fluctuation-Dissipation Relation (FDR)

K.Murase and TH, arXiv:1304.3243[nucl-th]

𝐺∗: Symmetrized correlation function15

Coarse-Graining in Time

𝑡

𝐺𝑅

𝑡 ′

𝐺𝑅

𝐺𝑅=𝜅𝜏 𝑒

− 𝑡𝜏 𝐺𝑅≈ 𝜅𝛿 (𝑡 ′ )coarse

graining???

Existence of upper bound in coarse-graining time (or lower bound of frequency) in relativistic theory???

Navier Stokes causality

Non-Markovian Markovian

Maxwell-Cattaneo

𝑡→ 𝑡 ′

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Colored Noise in Relativistic System

⟨𝛿Π 𝜔 ,𝒌∗ 𝛿Π 𝜔 ′ ,𝒌 ′ ⟩=2𝜅 (2𝜋 )4 𝛿(𝜔−𝜔 ′)𝛿(3 )(𝒌−𝒌 ′)

1+𝜔2𝜏2

𝐺𝑅 (𝑡 ,𝑡 ′ )= 𝜅𝜏 exp(− 𝑡− 𝑡′

𝜏 )𝜃 (𝑡−𝑡 ′)

: Extention to t<t’Correlation in Fourier space

Colored noise! (Indirect) consequence of causalityNote: white noise in differential form

K.Murase and TH, arXiv:1304.3243[nucl-th]

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A Comment on Heat Conductivity

Phenomenological constitutive eqs.,

for Gluonic system () ???𝜏 𝐷𝑞𝜇

𝐷𝑡 +𝑞𝜇=𝜅𝑇𝑋 𝜇

1st order:

2nd order:

𝜏 𝐷𝑞𝜇

𝐷𝑡 +𝑞𝜇=𝜅𝑇𝑋 𝜇+𝜉𝜇

Getting meaningful solely in RFH Fluctuation of ? 18

RFH: ,

Need Fluctuation?

Rayleigh-Taylor instabilityNon-linearity, instability, dynamic critical phenomena,…

Kelvin-Helmhortz instability

Figures from J.B.Bell et al., ESAIM 44-5 (2010)1085

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Seeds for instabilities

Outlook for RFH

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• Numerical implementation and its consequences in observables• Langevin simulation in constrained

system?• Dynamic critical

phenomenaSeparation of systematic motion from noise?• RFH from quantum Langevin eq.?• Heat conductivity in gluonic system

from Hosoya-Kajantie lattice QCD? AdS/CFT?

JET PROPAGATION INEXPANDING MEDIUM

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Fluctuation Appears Everywhere

0collision axis

time Finite number of

hadrons

Propagation of jets

Hydrodynamic fluctuation

Initial fluctuation andparticle production

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Jet Quenching as Missing pT

Figures adapted from talk

by C.Roland (CMS) at QM2011

Jet momentum balanced by low pT hadrons out of jet cone!

in-coneout-of-cone

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Momentum Balance Restored

Lost energy goes to low pT particles at large angle

Figure adapted from talk by C.Roland (CMS) at QM2011

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Horizontal: Jet asymmetryVertical: Transverse momenta (anti-)parallel to jet axis

𝜕𝜇𝑇 𝜇𝜈= 𝐽𝜈

Energy-Momentum Flow from Jets• Jet quenching could affect QGP

expansion at LHC (and even at RHIC)

• Many jets could disturb fluid evolution (or even heat them up?)

• Beyond linearized hydro

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𝐽 0 (𝑥 )= 𝐽 1 (𝑥 )=−𝑑𝑝0𝑑𝑡 𝛿(3 )(𝒙− 𝒙 (𝑡 ))

𝐽 2 (𝑥 )= 𝐽 3 (𝑥 )=0in the case that jet propagates parallel to x-axis

QGP Wake by Jet

A 50 GeV jet traverses a background with T=0.5 GeV Mach-cone like structure

Vortex ring behind a jet

Y.Tachibana and TH, Nucl.Phys.A904-905(2013)1023c

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April 13, 2013 at Mt. Etna

http://photoblog.nbcnews.com/_news/2013/04/12/17720274-mount-etna-blows-smoke-ring-during-volcanic-eruptions

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QGP Mach Cone Induced by a Jet

First fully non-linear and fully 3D hydro simulation of Mach cone

Y.Tachibana and TH, Nucl.Phys.A904-905(2013)1023c

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Large Angle Emission from Expanding Medium

Jet pair created atoff central position Enhancement of low

momentum particles at large angle from jet axis

Δ ( 𝑑𝑝∥

𝑑cos𝜃  )= 𝑑𝑝∥w . jet

𝑑 cos𝜃 −𝑑𝑝∥

w .o . jet

𝑑cos𝜃

out-of-coneqcos𝜃=−1cos𝜃=1

Y.Tachibana and TH, Nucl.Phys.A904-905 (2013)1023c

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Momentum Balance

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preliminaryin cone out of cone

Pb+Pb central collisions at sqrt(s_NN) = 2.76 TeV

Outlook for Interplay between Soft and Hard• Alternative constraint for shear

viscositythrough propagation of jets?

• Heat-up from mini-jets propagation similar to neutrino reheating in SNE?

Jet propagationmomentumtransfer

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Summary and Outlook• Fluctuation appears everywhere• Hydrodynamic fluctuation• Fluctuation induced by jet

propagation• Development of a more

sophisticated dynamical model towards precision/new heavy-ion physics

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BACKUPS

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HYDRO-BASED EVENT GENERATOR

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Current Status: E-by-E H2C

Hadronic cascade

3D ideal hydro

Monte Carlo I.C.(MC-KLN or MC-Glauber)0

collision axis

time

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Initial Density Fluctuation

1𝑁 ∑ initial condition single initial condition

conventionalhydro

event-by-eventhydro 36

Eccentricity, Triangularity, …𝜀𝑛 , part=

|⟨𝑟 2𝑒𝑖𝑛 𝜙 ⟩|⟨𝑟2 ⟩ 𝑛𝜓𝑛=arg ⟨𝑟2𝑒𝑖𝑛 𝜙 ⟩

40-50%

Almost boost invariant37

v2{***}v2{EP}, v2{2}, v2{4}, v2{6}, v2{LYZ}, …

Hydro-based event generator Analysis of the outputs almost in the sameway as experimental people do.

Demonstration of vn analysis according toevent plane method by ATLAS setup.E.g.) Centrality cut using ET in FCal region

ATLAS, arXiv:1108.601838

Resolution of Event Plane

𝑅𝑛=√ ⟨cos [𝑛 (𝛹 𝑛❑1−𝛹 𝑛

2 ) ]⟩

Reaction (Event) plane is not known experimentally nor in outputs from E-by-E H2C. Event plane method

Event plane resolution using two subevents

𝑛𝛹𝑛=tan−1(∑ 𝐸𝑇 ,𝑖 sin𝑛𝜙𝑖

∑ 𝐸𝑇 ,𝑖cos𝑛𝜙𝑖)

c.f.) A.M.Poskanzer and S.Voloshin, Phys. Rev. C 58, 1671 (1998)39

Resolution of Event Plane from E-by-E H2C

ATLAS, arXiv:1108.6018

Even Harmonics Odd Harmonics

# of events: 80000 (Remember full 3D hydro+cascade!)Subevent “N”: charged, -4.8< h < -3.2 (FCal in ATLAS)Subevent “P”: charged, 3.2< h < 4.8 (FCal in ATLAS)

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Vn{EP}(h)

Even Harmonics Odd Harmonics

Not boost inv. almost boost inv. for epsilon

40-50%40-50%

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vn{EP} vs. vn{RP}

Even Harmonics Odd Harmonics40-50% 40-50%

vodd{RP}~0veven{EP}>veven{RP}due to fluctuation

vn{RP}: vn w.r.t. reaction plane known in theory

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