Stochastic Dynamics of Heavy Quarkonium in Quark-Gluon Plasma

Yukinao Akamatsu (KMI,Nagoya) In collaboration with

Alexander Rothkopf (Bielefeld)

2011/11/18QHEC111/17

Reference: arXiv:1110.1203[hep-ph]

Contents

2011/11/18QHEC112/17

Introduction Complex potential from lattice QCD Stochastic dynamics of heavy quarkonium Bound states in the medium Conclusion and discussion

Introduction

2011/11/18QHEC113/17

Matsui and Satz (‘86)

/

0 ),(2

12)(

)(exp

)()( :

2JrrQ

Q

D

eff

rErV

rMMrE

Trr

rT

rVTcT

No solution rJ/Ψ at T>1.2Tc

“Plasma formation thus prevents J/Ψ formation already just above Tc.”

• Underlying physics: Debye screening• Sensitive to color deconfinement• All the discussion based on the potential V(r)

Propose J/Ψ suppression as a signal for QGP formation

Introduction

2011/11/18QHEC114/17

New data from LHCALICE

CMS

Y(1S)

Y(2S,3S)

Introduction

2011/11/18QHEC115/17

Potential Model Approaches Provide clear physical picture! Potential from QQ free energy, or internal energy, or

linear combination of both? Relation to first principle?

Spectral Function of Current Correlator Relation to first principle is clear! How to discuss more than the shape of peak?

How to define the potential from first principle?

Complex potential from lattice QCD

2011/11/18QHEC116/17

Rothkopf, et al. (‘11)

, ,)0,,(),,(),(

),,(],[),(),,(

00 yxryxMtyxMtrD

tyQyxUtxQtyxM

†

Meson operator (J/Ψ,ηc, …)

Forward correlator

In heavy quark limit, ω~2MQ describes 2-HQs physics ≈ described by Schroedinger equation

),(),(),(2

trDtrVM

trDt

i NRQ

rNR

□

In MQ=∞ limit, Fourier transformation (t⇔ω) of D>NR(r,t)=Spectral decomposition of thermal Wilson loop

V□(r)(Lorentzian fit)

Proper potential from first principle

Complex potential from lattice QCD

2011/11/18QHEC117/17

Rothkopf, et al. (‘11) cont’dV□(r) = Complex potential !!

Complex potential also found by perturbation theory [Laine, et al. (07’)]

What happened to unitarity?In Coulomb gauge

Stochastic dynamics of heavy quarkonium

2011/11/18QHEC118/17

Stochastic unitary evolution of QQ?Can stochastic unitary evolution explain V□(r)?

Heavy quark(s) as an open quantum system

Integrated out

k

k

Heavy quarks

Gluons,light quarks

MQ

~T

fluctuation

Non-relativistic,Q and Q separately conserved

~(MQT)1/2

Due to this hierarchy, we expect unitary evolution of the reduced system

Stochastic dynamics of heavy quarkonium

2011/11/18QHEC119/17

Unitary evolution by stochastic Hamiltonian

tXXtXtXtX

XVM

XH

tXXHdtiTtU

xxXXtUtX

tt

Q

X

tX

QQX

QQ

/)',()','(),( ,0),(

hermite )(2

)(

)',()('exp)0|(

},{ ),0,()0|(),(

'

22

0

)(

21)(

Θ1Θ2

Θ3

Stochastic termlcorr ~ thermal wavelengthof medium particles

manifestly unitary

decays when |X-X’| > lcorr

stochastic

Stochastic dynamics of heavy quarkonium

2011/11/18QHEC1110/17

Stochastic differential equation

),(),(),(2

)(),(

0),( ,),(),(2

),(),(

)(),(),(2

)(1

)(),(21),()(1

),()(exp)0|(

22

2/3

2/3222

)(

tXtXXXiXHtXt

i

tXtXtXtitXtX

tOtXtiXXiXHti

tOtXttXXHti

tXXHtitU

QQQQ

X

Stochastic dynamics of heavy quarkonium

2011/11/18QHEC1111/17

Relation to complex potential

),(),( tXtXD QQNR

In MQ ＝∞ limit,

D>NR is ensemble average of wave function Ψ! Evolution of D>NR needs not be unitary.

),(2

)()(

),( ),(2

)(),(

),()(),(

XXiXVXV

tXXXiXVtXt

i

tXDXVtXDt

i

QQQQ

NRNR

□

□

complex potential = [real potential] + i[noise strength]

Stochastic dynamics of heavy quarknoium

2011/11/18QHEC1112/17

Remark2 : the observables of J/Ψ suppression Dilepton spectrum

① If charms are (chemically and kinetically) equilibrated, SPF of current correlator is enough to give dilepton spectrum.

② If not (and is not in heavy ion collisions), the stochastic dynamics is necessary.

J

① ②

initial

J/Ψ evolved

J/ΨJ

Remark1 : SPF of current correlatorCan be calculated only from the complex potential. no reference to lcorr

Bound states in the medium

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Fate of bound states

Θ

Low temperature

Bound state

Real potential : energy levels and sizes of bound statesNoise : excites modes with k~1/lcorr (spatial decoherence)

Θ1

High temperature

Θ2

Θ3 Θ4

Bound state

noise gives a nearly global phasedoes not change physics

noises excite the bound statebound state disappears

Argument here can be made more quantitative in terms of master equation.

Bound states in the medium

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1d simulation – set up

1||095.0)',(

)5.1|(| 0833.0

)5.1|(| ||1.0||

1.0)(

0.001dt 0.1,dxb.c. periodic 6],[-2.56,2.5x

'

Mxxx

x

xxxxv

xx

(Relative motion)

Initial condition

lcorr~dx=0.1 very(too) high temperature

Bound states in the medium

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1d simulation – bound state probability P(t)

Probability of occupying bound states decays, but saturates at later time.

Bound states in the medium

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1d simulation – norms, etc.

Norm of each trajectory = 1 (unitary)Norm of average wave function decays. (noise imaginary part)Energy average ~ 100! (due to high temperature)

Conclusion and discussion

2011/11/18QHEC1117/17

Conclusion Stochastic unitary evolution can explain complex potential

obtained by lattice simulation. Noise correlation length lcorr plays a crucial role in determining

the fate of bound states.

Discussion What is the first principle definition of lcorr? Gauge dependence in introducing the color Quantum Brownian motion of single heavy quark Thermodynamic quantities (free energy, entropy, …) Relation to heavy quark effective field theories

2011/11/18QHEC1118/17

BACK UP

Master equation

2011/11/18QHEC1119/17

Master equation

),'(),(')(),',(

)(;)(;1)(

*

)(

tXtXXtXtXX

ttttN

t

QQQQQQQQ

tQQQQQQ

Reduced density matrix

2)','(),()',()',(

),',()',(),',()'()(),',(

XXXXXXXXF

tXXXXFtXXi

XHXHtXXt QQQQQQ

Master equation

Equivalent master equation:proposed as a modified quantum mechanics (Ghirardi, et al. ‘86)derived in scattering model (Gallis & Fleming ‘90) in random potential in Feynman-Vernon approach (Gallis ‘92)

Master equation

2011/11/18QHEC1120/17

Extracting relative motion

22

',0,0,0

22

,

',,3

),(),(2

),(),(

),(),(),(2

)(),(

),()',( ),()(

2)','(),()',()',( ),()(

),',(ˆ)',(),',(ˆ)'()(),',(ˆ

2,2

),',(),',(ˆ

trtrtitrtr

trtrrrirhtrt

i

XXrrXVrv

rrrrrrrrfrvM

rh

trrrrftrri

rhrhtrrt

rRrRX

tXXRdtrr

QQQQ

rrr

Q

r

QQQQQQ

rR

rRrRQQQQ