in-medium qcd forces for hqs at high t
DESCRIPTION
New Frontiers in QCD 2013. In-medium QCD forces for HQs at high T. Yukinao Akamatsu Nagoya University, KMI. Y.Akamatsu , A.Rothkopf , PRD85(2012), 105011 (arXiv:1110.1203[ hep-ph ] ) Y.Akamatsu , PRD87(2013),045016 (arXiv:1209.5068[ hep-ph ]) - PowerPoint PPT PresentationTRANSCRIPT
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]
12013/12/11 NFQCD2013@YITP
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