In-medium QCD forcesfor HQs at high T

Yukinao AkamatsuNagoya University, KMI

Y.Akamatsu, A.Rothkopf, PRD85(2012),105011 (arXiv:1110.1203[hep-ph] )Y.Akamatsu, PRD87(2013),045016 (arXiv:1209.5068[hep-ph]) arXiv:1303.2976[nuch-th]

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New Frontiers in QCD 2013

Contents

1. Introduction2. In-medium QCD forces3. Influence functional of QCD4. Perturbative analysis5. Summary & Outlook

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1. INTRODUCTION

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Confinement & Deconfinement

• HQ potential

4

R

V(R) Coulomb + LinearString tension K ~ 0.9GeVfm-1

2134)( MR

KRRV s

Singlet channel

T=0

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Debye ScreenedHigh T>>Tc

The Schrödinger equationExistence of bound states (cc, bb) J/Ψ suppression in heavy-ion collisions

_ _

Matsui & Satz (86)

21exp34)( MRR

RV Ds

Debye screened potential

Debye mass ωD ~ gT (HTL)

Quarkonium Suppression at LHC

• Sequential melting of bottomonia

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p+p A+A

)syst(02.0)stat(07.021.0)1()2(

)1()2(

pp

PbPb

SSSS

)CL%95(17.0)syst(06.0)stat(06.006.0)1()3(

)1()3(

pp

PbPb

SSSS

CMS

Time evolution of quarkonium states in medium is necessary

2. IN-MEDIUM QCD FORCES

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In-medium Potential

• Definition

7

T>0, M=∞r

t

RLong time dynamics

i=0 i=1 …

Lorentzian fit of lowest peak in SPF σ(ω;R,T)

σ(ω;R,T)

ωV(R,T)

Γ(R,T) (0<τ<β)

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Complex potential !Laine et al (07), Beraudo et al (08),Bramilla et al (10), Rothkopf et al (12).

In-medium Potential

• Stochastic potential

8

Noise correlation length ~ lcorr

Imaginary part= Local correlation only

Phase of a wave function gets uncorrelated at large distance > lcorr

Decoherence Melting (earlier for larger bound states)

Akamatsu & Rothkopf (‘12)

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

),(),(

),,(),(),(2

)(),(

22

RtRtRttiRtRt

RtRtRRiRVRtt

i

In-medium Forces

• Whether M<∞ or M=∞ matters

9

Debye screened force+

Fluctuating force

Drag force

Langevin dynamics

M=∞

M<∞

(Stochastic) Potential forceHamiltonian dynamics

Non-potential forceNot Hamiltonian dynamics

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3. INFLUENCE FUNCTIONAL OF QCD

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Open Quantum System

• Basics

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)(ˆ,ˆ)(ˆ

H

tottottot

envsystot

tHtdtdi

?)(ˆ

)(ˆTr)(ˆ

red

totenvred

tdtdi

tt

Hilbert space

von Neumann equation

Trace out the environment

Reduced density matrix

Master equation

(Markovian limit)

sys = heavy quarksenv = gluon, light quarks

Closed-time Path

• QCD on CTP

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],[],[],[

][][exp

][][exp

],[][~],[

ini2

ini*1sys

ini22

ini

1*1

eqenv

ini222

ini

1*1

*1tot

syseqenvtot

22112211

221121

ini222

ini

1*1

*12,12,12,121

AqAqAqAq

AjigAjigAqiSAqiS

iiiSiS

AqAqAqDZ

Factorized initial density matrix

Influence functional Feynman & Vernon (63)

Influence Functional

• Reduced density matrix

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1

2

)(),( 11 tt

)(),( 22 tt ],[ ini

2*ini1sys

sPath integrate until s, with boundary condition 2211 )(,)( ss

2red*12

*1red )(ˆ],,[ ss

Influence Functional

• Functional master equation

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],,[],[],,[ 2*1red2

*1

func212

*1red tHt

ti

Long-time behavior (Markovian limit)Analogy to the Schrödinger wave equation

Effective initial wave function

Effective action S1+2 Single time integral

Functional differential equation

Hamiltonian Formalism (skip)

• Order of operators = Time ordered

• Change of Variables (canonical transformation)

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),()~,~(~2

*2

*2

*222 cc QQQQ

),(),( ),,(),( 2*21

*1 xtxtxtxt

Instantaneous interaction

Kinetic term

or

Make 1 & 2 symmetric

Remember the original order

]~,[ *2

*1

func21 HDetermines without ambiguity

Technical issue

Hamiltonian Formalism (skip)

• Variables of reduced density matrix

• Renormalization

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*)(2red

*)(1

*)(2

*)(1red

*2red

*1

*2

*1red

~)(ˆ]~,,[

~)(ˆ]~,,[

cccc QtQQQt

tt

Convenient to move all the functional differential operators to the right in

]~,,[]~,[]~,,[ *)(2

*)(1red

*)(2

*)(1

func21

*)(2

*)(1red cccccc QQtQQHQQt

ti

Latter is better (explained later)

In this procedure, divergent contribution from e.g. Coulomb potential at the origin appears needs to be renormalized

Technical issue

Reduced Density Matrix

• Coherent state

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Source for HQs

Reduced Density Matrix

• A few HQs

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0~

*)(2

*)(1red*

2*1

red

*)(2

*)(1

~,,)(~)(

)(ˆ)(ˆ)(ˆ),,(

cc QQcc

Q

QQtyQxQ

yQtxQyxt

†One HQ

Similar for two HQs, …

),,,,,( 2121 yyxxt

cQQ

Master Equation

• From fields to particles

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]~,,[]~,[]~,,[ *)(2

*)(1red

*)(2

*)(1

func21

*)(2

*)(1red cccccc QQtQQHQQt

ti

Functional differentiation )(~)( *2

*1 yQxQ

Master equation

Functional master equation

For one HQ

Similar for two HQs, …

4. PERTURBATIVE ANALYSIS

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Approximation (I)

• Leading-order perturbation

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TATA

TATA

xAxAxxGxAxAxxG

xAxAxxGxAxAxxG

)(ˆ)(ˆT~)( ,)(ˆ)(ˆ)(

)(ˆ)(ˆ)( ,)(ˆ)(ˆT)(

2121F~

2121

12212121F

Leading-order result by HTL resummed perturbation theory

Expansion up to 4-Fermi interactions

Influence functional

Approximation (II)

• Heavy mass limit

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†

†

†

cc

c

cc

QMMiQ

QMMiQQQS

QQQQSS

]2[

]2[],[

),(~ ],[][

20

20

NRkin

NRkin

Non-relativistic kinetic term

Non-relativistic 4-current (density, current)

††

††

ca

ca

a

aca

ca

a

QtiM

QQtiM

Qj

QtQQtQj

22

0

��

(quenched)

GTgGQMTQ

)(~

~

MT~

Expansion up to

Approximation (III)

• Long-time behavior

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)()(

)(')()()(

)(' )()0,('~ )0,(~),(~

0000

00 yxyx

yxGiyxyxGyxG

yxGyxGyxGyxGyxG

t yxxy ytjxtjytjxtjyxGiytjxtjyxG

yjyxGxj

00 ),(),(),(),()('2

),(),()( )()()(

Low frequency expansion

Time-retardation in interaction

Scattering time ~ 1/q (q~gT, T)HQ time scale is slow: Color diffusion ~ 1/g2T Momentum diffusion ~ M/g4T2

Effective Action

• LO pQCD, NR limit, slow dynamics

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ababab

ababRab

yxVyxDyxGg

yxVyxGiyxGg

)(Im)()(

)()()(

,002

,00,002

Stochastic potential(finite in M∞)

Drag force(vanishes in M∞)

Physical Process

• Scatterings in t-channel

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Scatterings with hard particles contribute to drag, fluctuation, and screening

Q

Q

g q

q

g

g

. . .

Independent scatterings

. . .

. . . . . .

Single HQ

• Master equation

• Ehrenfest equations

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Moore et al (05,08,09)

Complex Potential

• Forward propagator

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0~

*)(2

*)(1red*

1*1

2

00†

*)(2

*)(1

,~,)()(

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

cc QQccc

TT

tQQyQxQ

yxJyxtJyxt

RiTR

eCTgRVCMaRV D

R

DF

F

D

4)( )()1(2)(

2

singlet

Time-evolution equation + Project on singlet state

Laine et al (07), Beraudo et al (08), Brambilla et al (10)

Stochastic Potential

• Stochastic representation

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M=∞ : Stochastic potential

D(x-y): Negative definite)()(),(),( yxDstysxt abba

Debye screened potential

Fluctuation

M<∞ : Drag forceTwo complex noises c1,c2

Non-hermitian evolution ),(),(~),(~),(),,(

*

,, 21

xtxt

ytxtyxtccQ

5. SUMMARY & OUTLOOK

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• Open quantum systems of HQs in medium– Stochastic potential, drag force, and fluctuation– Influence functional and closed-time path– Functional master equation, master equation, etc.

• Toward phenomenological application– Stochastic potential with color– Emission and absorption of real gluons

• More on theoretical aspects– Conform to Lindblad form– Non-perturbative definition

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Backup

In-medium Potential

• Definition

32

T=0, M=∞r

t

R

tRiV

mm

tRiE

tRiERJm

RJRtJRt

)(min

2†

†

e)(exp~

)(expvac);0(

vac);0();(vac);(

σ(ω;R)

ωV(R)

Long time dynamics

V(R) from large τ behavior

)(min

)(†

e)(exp~

e][~vac);0();(vac);(RV

AS

RE

ADRJRiJRG

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Closed-time Path

33

• Basics

...

),()(ˆ)(ˆ],[ln)()(

),()(ˆ)(ˆT],[ln)()(

)(ˆT],[ln)(

21conn120

212211

2

21F

conn210

212111

2

connC

021

2,1

2,1

2,1

xxGxxZxx

xxGxxZxx

xZx

T,j

T,j

T,iii

jiii

221121ini2

ini12,1

12

2121

][][exp],[~

ˆ);,(ˆ);,(ˆTr

);,(ˆˆ);,(ˆTr],[

iiiSiSD

UU

UUZ†

†

　

1

2

11,

22 ,],[ ini

2ini1

Partition function

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Functional Master Equation

• Renormalized effective Hamiltonian

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)()()( ),(lim2

1 )0()0()0(

0

F rVrVrVrVMCa TTT

r

Functional Master Equation

• Analogy to Schrödinger wave equation

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*)(2

)(22222

*)(1

)(11111

~~)()(~̂),(~̂)(~̂),(~̂

)()(ˆ),(ˆ)(ˆ),(ˆ

cccc

cccc

QQyxyQxQyQxQ

QQyxyQxQyQxQ

††

††

func2121

ˆ HH

]~,,[]~,[]~,,[ *)(2

*)(1red

*)(2

*)(1

func21

*)(2

*)(1red cccccc QQtQQHQQt

ti

Anti-commutator in functional space