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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
22013/12/11 NFQCD2013@YITP
1. INTRODUCTION
32013/12/11 NFQCD2013@YITP
Confinement & Deconfinement
• HQ potential
4
R
V(R) Coulomb + Linear
String tension K ~ 0.9GeVfm-1
213
4)( M
RKRRV s
Singlet channel
T=0
2013/12/11 NFQCD2013@YITP
Debye ScreenedHigh T>>Tc
The Schrödinger equationExistence of bound states (cc, bb) J/Ψ suppression in heavy-ion collisions
_ _
Matsui & Satz (86)
21exp3
4)( MR
RRV D
s Debye screened potential
Debye mass ωD ~ gT (HTL)
Quarkonium Suppression at LHC
• Sequential melting of bottomonia
2013/12/11 NFQCD2013@YITP 5
p+p A+A
)syst(02.0)stat(07.021.0)1()2(
)1()2(
pp
PbPb
SS
SS
)CL%95(17.0)syst(06.0)stat(06.006.0)1()3(
)1()3(
pp
PbPb
SS
SS
CMS
Time evolution of quarkonium states in medium is necessary
2. IN-MEDIUM QCD FORCES
62013/12/11 NFQCD2013@YITP
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)
2013/12/11 NFQCD2013@YITP
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)
2013/12/11 NFQCD2013@YITP
0),( ,),(),(2
),(),(
),,(),(),(2
)(),(
22
RtRtRtti
RtRt
RtRtRRi
RVRtt
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
2013/12/11 NFQCD2013@YITP
3. INFLUENCE FUNCTIONAL OF QCD
102013/12/11 NFQCD2013@YITP
Open Quantum System
• Basics
112013/12/11 NFQCD2013@YITP
)(ˆ,ˆ)(ˆ
H
tottottot
envsystot
tHtdt
di
?)(ˆ
)(ˆTr)(ˆ
red
totenvred
tdt
di
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
2013/12/11 NFQCD2013@YITP 12
],[],[],[
][][exp
][][exp
],[][~],[
ini2
ini*1sys
ini
22
ini
1*1
eqenv
ini
222
ini
1*1
*1tot
syseqenvtot
22112211
221121
ini
222
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
2013/12/11 NFQCD2013@YITP 13
1
2
)(),( 11 tt
)(),( 22 tt ],[ ini
2*ini1sys
s
Path integrate until s, with boundary condition 2211 )(,)( ss
2red*12
*1red )(ˆ],,[ ss
Influence Functional
• Functional master equation
2013/12/11 NFQCD2013@YITP 14
],,[],[],,[ 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)
2013/12/11 NFQCD2013@YITP 15
),()~
,~
(~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
2013/12/11 NFQCD2013@YITP 16
*)(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
2013/12/11 NFQCD2013@YITP 17
Source for HQs
Reduced Density Matrix
• A few HQs
2013/12/11 NFQCD2013@YITP 18
0
~*
)(2*
)(1red*2
*1
red
*)(2
*)(1
~,,
)(~
)(
)(ˆ)(ˆ)(ˆ),,(
cc QQcc
Q
QQtyQxQ
yQtxQyxt
†
One HQ
Similar for two HQs, …
),,,,,( 2121 yyxxtcQQ
Master Equation
• From fields to particles
2013/12/11 NFQCD2013@YITP 19
]~
,,[]~
,[]~
,,[ *)(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
202013/12/11 NFQCD2013@YITP
Approximation (I)
• Leading-order perturbation
2013/12/11 NFQCD2013@YITP 21
TA
TA
TA
TA
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
2013/12/11 NFQCD2013@YITP 22
†
†
†
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)
GTgG
QMTQ
)(~
~
M
T~
Expansion up to
Approximation (III)
• Long-time behavior
2013/12/11 NFQCD2013@YITP 23
)()(
)(')()()(
)(' )()0,('~
)0,(~
),(~
0000
00 yxyx
yxGiyxyxGyxG
yxGyxGyxGyxGyxG
t yxxy ytjxtjytjxtjyxGi
ytjxtjyxGyjyxGxj
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
2013/12/11 NFQCD2013@YITP 24
ababab
ababRab
yxVyxDyxGg
yxVyxGiyxGg
)(Im)()(
)()()(
,002
,00,002
Stochastic potential(finite in M∞)
Drag force(vanishes in M∞)
Physical Process
• Scatterings in t-channel
2013/12/11 NFQCD2013@YITP 25
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
2013/12/11 NFQCD2013@YITP 26
Moore et al (05,08,09)
Complex Potential
• Forward propagator
2013/12/11 NFQCD2013@YITP 27
0
~*
)(2*
)(1red*1
*1
2
00†
*)(2
*)(1
,~
,)()(
),;0(),;(),;(
cc QQcc
c
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
2013/12/11 NFQCD2013@YITP 28
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
292013/12/11 NFQCD2013@YITP
• 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
2013/12/11 NFQCD2013@YITP 30
2013/12/11 NFQCD2013@YITP 31
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
2013/12/11 NFQCD2013@YITP
Closed-time Path
33
• Basics
...
),()(ˆ)(ˆ],[ln)()(
),()(ˆ)(ˆT],[ln)()(
)(ˆT],[ln)(
21conn12
0
212211
2
21F
conn21
0
212111
2
connC
0
21
2,1
2,1
2,1
xxGxxZxx
xxGxxZxx
xZx
T,j
T,j
T,ii
ijii
i
221121ini2
ini12,1
12
2121
][][exp],[~
ˆ);,(ˆ);,(ˆTr
);,(ˆˆ);,(ˆTr],[
iiiSiSD
UU
UUZ†
†
1
2
11,
22 ,],[ ini
2ini1
Partition function
2013/12/11 NFQCD2013@YITP
Functional Master Equation
• Renormalized effective Hamiltonian
2013/12/11 NFQCD2013@YITP 34
)()()( ),(lim2
1 )0()0()0(
0
F rVrVrVrVM
Ca TTT
r
Functional Master Equation
• Analogy to Schrödinger wave equation
2013/12/11 NFQCD2013@YITP 35
*)(2
)(22222
*)(1
)(11111
~~
)()(~̂
),(~̂
)(~̂
),(~̂
)()(ˆ),(ˆ)(ˆ),(ˆ
c
ccc
cccc
QQyxyQxQyQxQ
QQyxyQxQyQxQ
††
††
func2121
ˆ HH
]~
,,[]~
,[]~
,,[ *)(2
*)(1red
*)(2
*)(1
func21
*)(2
*)(1red cccccc QQtQQHQQt
ti
Anti-commutator in functional space