In-medium QCD forces for HQs at high T

transcript

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

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

2134)( MR

KRRV s

Singlet channel

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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

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(

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

)1()3(

Time evolution of quarkonium states in medium is necessary

2. IN-MEDIUM QCD FORCES

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In-medium Potential

• Definition

T>0, M=∞r

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

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

RtRtRttiRtRt

RtRtRRiRVRtt

In-medium Forces

• Whether M<∞ or M=∞ matters

Debye screened force+

Fluctuating force

Drag force

Langevin dynamics

(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

tottottot

envsystot

tHtdtdi

)(ˆTr)(ˆ

totenvred

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

],[][~],[

ini*1sys

ini222

syseqenvtot

22112211

221121

ini222

*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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)(),( 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

func212

*1red tHt

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

*222 cc QQQQ

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

*1 xtxtxtxt

Instantaneous interaction

Kinetic term

Make 1 & 2 symmetric

Remember the original order

]~,[ *2

func21 HDetermines without ambiguity

Technical issue

Hamiltonian Formalism (skip)

• Variables of reduced density matrix

• Renormalization

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*)(2red

*)(1red

~)(ˆ]~,,[

cccc QtQQQt

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

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

*)(1red

func21

*)(1red cccccc QQtQQHQQt

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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*)(1red*

~,,)(~)(

)(ˆ)(ˆ)(ˆ),,(

cc QQcc

QQtyQxQ

yQtxQyxt

†One HQ

Similar for two HQs, …

),,,,,( 2121 yyxxt

Master Equation

• From fields to particles

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

*)(1red

func21

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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xAxAxxGxAxAxxG

)(ˆ)(ˆT~)( ,)(ˆ)(ˆ)(

)(ˆ)(ˆ)( ,)(ˆ)(ˆT)(

2121F~

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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QMMiQQQS

QQQQSS

]2[],[

),(~ ],[][

Non-relativistic kinetic term

Non-relativistic 4-current (density, current)

††

QtQQtQj

��

(quenched)

GTgGQMTQ

Expansion up to

Approximation (III)

• Long-time behavior

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)(')()()(

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

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)()(

)()()(

,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

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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*)(1red*

,~,)()(

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

cc QQccc

tQQyQxQ

yxJyxtJyxt

eCTgRVCMaRV D

4)( )()1(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 ),(),(~),(~),(),,(

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

T=0, M=∞r

tRiERJm

RJRtJRt

e)(exp~

)(expvac);0(

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

σ(ω;R)

ωV(R)

Long time dynamics

V(R) from large τ behavior

e)(exp~

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

ADRJRiJRG

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Closed-time Path

• Basics

),()(ˆ)(ˆ],[ln)()(

),()(ˆ)(ˆT],[ln)()(

)(ˆT],[ln)(

21conn120

212211

conn210

212111

xxGxxZxx

221121ini2

ini12,1

][][exp],[~

ˆ);,(ˆ);,(ˆTr

);,(ˆˆ);,(ˆTr],[

iiiSiSD

UUZ†

22 ,],[ ini

Partition function

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Functional Master Equation

• Renormalized effective Hamiltonian

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)()()( ),(lim2

1 )0()0()0(

F rVrVrVrVMCa TTT

Functional Master Equation

• Analogy to Schrödinger wave equation

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)(22222

)(11111

~~)()(~̂),(~̂)(~̂),(~̂

)()(ˆ),(ˆ)(ˆ),(ˆ

QQyxyQxQyQxQ

††

func2121

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

*)(1red

func21

Anti-commutator in functional space

In-medium QCD forces for HQs at high T

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