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![Page 1: 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,](https://reader035.fdocuments.in/reader035/viewer/2022062721/56649f225503460f94c3b495/html5/thumbnails/1.jpg)
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
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Contents
1. Introduction2. In-medium QCD forces3. Influence functional of QCD4. Perturbative analysis5. Summary & Outlook
22013/12/11 NFQCD2013@YITP
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1. INTRODUCTION
32013/12/11 NFQCD2013@YITP
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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)
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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
![Page 6: 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,](https://reader035.fdocuments.in/reader035/viewer/2022062721/56649f225503460f94c3b495/html5/thumbnails/6.jpg)
2. IN-MEDIUM QCD FORCES
62013/12/11 NFQCD2013@YITP
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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)
2013/12/11 NFQCD2013@YITP
Complex potential !Laine et al (07), Beraudo et al (08),Bramilla et al (10), Rothkopf et al (12).
![Page 8: 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,](https://reader035.fdocuments.in/reader035/viewer/2022062721/56649f225503460f94c3b495/html5/thumbnails/8.jpg)
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
![Page 9: 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,](https://reader035.fdocuments.in/reader035/viewer/2022062721/56649f225503460f94c3b495/html5/thumbnails/9.jpg)
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
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3. INFLUENCE FUNCTIONAL OF QCD
102013/12/11 NFQCD2013@YITP
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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
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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)
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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
![Page 14: 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,](https://reader035.fdocuments.in/reader035/viewer/2022062721/56649f225503460f94c3b495/html5/thumbnails/14.jpg)
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
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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
![Page 16: 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,](https://reader035.fdocuments.in/reader035/viewer/2022062721/56649f225503460f94c3b495/html5/thumbnails/16.jpg)
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
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Reduced Density Matrix
• Coherent state
2013/12/11 NFQCD2013@YITP 17
Source for HQs
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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
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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, …
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4. PERTURBATIVE ANALYSIS
202013/12/11 NFQCD2013@YITP
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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
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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
![Page 23: 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,](https://reader035.fdocuments.in/reader035/viewer/2022062721/56649f225503460f94c3b495/html5/thumbnails/23.jpg)
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
![Page 24: 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,](https://reader035.fdocuments.in/reader035/viewer/2022062721/56649f225503460f94c3b495/html5/thumbnails/24.jpg)
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∞)
![Page 25: 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,](https://reader035.fdocuments.in/reader035/viewer/2022062721/56649f225503460f94c3b495/html5/thumbnails/25.jpg)
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
. . .
. . . . . .
![Page 26: 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,](https://reader035.fdocuments.in/reader035/viewer/2022062721/56649f225503460f94c3b495/html5/thumbnails/26.jpg)
Single HQ
• Master equation
• Ehrenfest equations
2013/12/11 NFQCD2013@YITP 26
Moore et al (05,08,09)
![Page 27: 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,](https://reader035.fdocuments.in/reader035/viewer/2022062721/56649f225503460f94c3b495/html5/thumbnails/27.jpg)
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)
![Page 28: 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,](https://reader035.fdocuments.in/reader035/viewer/2022062721/56649f225503460f94c3b495/html5/thumbnails/28.jpg)
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
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5. SUMMARY & OUTLOOK
292013/12/11 NFQCD2013@YITP
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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
2013/12/11 NFQCD2013@YITP 30
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2013/12/11 NFQCD2013@YITP 31
Backup
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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
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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
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Functional Master Equation
• Renormalized effective Hamiltonian
2013/12/11 NFQCD2013@YITP 34
)()()( ),(lim2
1 )0()0()0(
0
F rVrVrVrVM
Ca TTT
r
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Functional Master Equation
• Analogy to Schrödinger wave equation
2013/12/11 NFQCD2013@YITP 35
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Anti-commutator in functional space