Heavy	  quark	  master	  equa.ons	  in	  the	  Lindblad	  form	
Yukinao	  Akamatsu	  (KMI,	  Nagoya)	  	  Refs:	  Y.A.,	  arXiv:1403.5783	  [hep-­‐ph],	  	  Y.A.,	  PRD87(2013),045016	  [arXiv:1209.5068	  [hep-­‐ph]]	  Y.A.	  and	  A.Rothkopf,	  PRD85(2012),105011	  [arXiv:1110.1203	  [hep-­‐ph]]		

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Contents	
1. Introduc.on	  2. Inﬂuence	  func.onal	  for	  the	  Lindblad	  form	  3. Stochas.c	  poten.al	  with	  color	  4. Summary	  &	  outlook	

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熱場の量子論とその応用＠理化学研究所	

2	Events / ( 0.1 GeV/c2 )800
he tracker, the muon pT measurement based
CMS PbPb sNN = 2.76 TeVfrom the tracker alone has a resolution
700
2% for a typical muon in this analysis.
Cent. 0-100%, |y| < 2.4µapparatus also has extensive forwardp > 4 GeV/c
600
Tcluding two steel–quartz-fiber Čerenkov
Lint = 150 µb-1
calorimeters (HF), which cover the range
500. These detectors are used for event selecdata
lity determination in PbPb collisions. The400
total fitobservable corresponds to the fraction ofbackground300
ic cross section, starting at 0% for the mostns and evaluated as percentiles of the dis200
e energy deposited in the HF [7,8]. The
es used in this analysis are 50–100%, 40–100
20–30%, 10–20%, 5–10%, and 0–5%,
e lowest to the highest HF energy deposit.
0r-model calculation as described in Ref. [7],7891011121314week ending
S I C 2A] L R E V I E W L E T T E R S
30 NOVEMBER
PRL 109, 222301 (2012)mber of nucleons participating in the colliMass(µP+µH-)Y[GeV/cMatsui,	  S2012atz	  (86)	
d the average nuclear overlap function (TAA )800
granularity of the50tracker, the muon pT measurement based
ated for each centrality class. The TAA factor
CMSpps=2.76
TeV
CMS PbPb sNN = 2.76 TeV
on information from the tracker alone has a resolution
umber of elementary nucleon-nucleon (NN)700
between 1 and 2% for a typical muon in |y|thisanalysis.
< 2.4
Cent. 0-100%, |y| < 2.4
ns divided by the elementary NN crossµµThe CMS apparatus
also has extensiveforward
40p>
4GeV/cp > 4 GeV/cbe interpreted as the NN-equivalent inte600
TTcalorimetry, including two steel–quartz-fiberČerenkov-1
Lint = 230 nb
ty per heavy-ion collision, at a given event
Lint = 150 µb-1
hadron forward calorimeters (HF), which cover the range
500
2:9 < j!j < 5:2.30
These detectors are used for event selecare identified through their dimuon decay.
data
data
tion and centrality determination in PbPb collisions.
The400e selected online with a hardware-based
total fit
total fit
event centrality observable corresponds to the fraction
ofbackgroundg two muon candidates in the muon detecbackground20300
the total inelastic cross section, starting at 0% for the mostringent muon quality requirements arecentral collisions and evaluated as percentiles of the disPbPb case relative to the pp online selec200
tribution of the energy deposited in the HF [7,8]. The
cit momentum or rapidity thresholds are
10centrality classes used in this analysis are 50–100%, 40–ger level. For the PbPb data, events are
10050%, 30–40%, 20–30%, 10–20%, 5–10%, and 0–5%,
ine if they contain a reconstructed primaryordered from the lowest to the highest HF energy deposit.
00ng at least two tracks, and the presence ofUsing a Glauber-model
calculation
as10described78
91011121314
7
8
911 in12Ref. [7],1314larger than 3 GeV in at least three towers in
2+ 2the colli+ the average number of nucleonsparticipatingin
Mass(µµ)[GeV/c]Mass(µ µ ) [GeV/c ]o HF calorimeters. These criteria reducesions (Npart ) and the average nuclear overlap function (TAA )om single-beam interactions, ultraperiph50
have been
estimatedeach centrality
The TAA factor
FIG.
1 (colorforonline).
Dimuon class.
invariant-mass
distributions inCMS pp s = 2.76 TeV
netic interactions, and cosmic-ray muons.
pﬃﬃﬃﬃﬃﬃﬃﬃ
TeV. The
PbPb(top) andofpp(bottom) nucleon-nucleon
data at sNN ¼ 2:76
isequal
tothenumber
elementary(NN)
constructed by matching tracks in the muon|y| < 2.4
same reconstruction
and analysis NN
selection
are appliedbinary collisions
divided algorithm
by the elementarycrosslicon tracker. The same offline reconstrucµ40
p > 4 GeV/cto bothincludingtransverse
momentumrequirement
section andcandata
be sets,
interpreted
as atheNN-equivalentinteT
nd selection criteria are applied to the PbPb> 4 GeV=c.
Thesolid(signal þon single permuonsof pTcollision,
Lint = 230 nb-1
gratedluminosity
heavy-ion
atagivenevent
mples. The muon candidates are required toand dashed (background-only) curves show thecentralitybackground)[9].
erse (longitudinal) distance of closest
30
results of the simultaneous fit to the two data sets.
The ! states are identified through their dimuon decay.
datae event vertex
smaller than 3 (15) cm.
14/09/04	
熱場の量子論とその応用＠理化学研究所	
3	The events are selected online with a hardware-based
total fit
y kept if the part of their trajectory in thebackgroundtrigger requiring two muon candidates in the muon detec21.	  Introduc.on	• 重イオン衝突でQGPが生成されたら？	  

– カラー荷が遮蔽されJ/ΨやΥの収量が減少	  PbPb	

Events / ( 0.1 GeV/c2 )CMS@LHC	Events / ( 0.1 GeV/c2 )pp	1S	

2S+3S	s / ( 0.1 GeV/c2 )	  – Open	  HQが多くなると逆に増加も(regenera.on)	  1.	  Introduc.on	• 環境（QGP）と相互作用するクォーコニウム	  – 量子開放系	  H

ρQ (t) ≡ TrqA [ ρ tot (t)]= qA ⊗ Q

ρ Q (t) = L !"ρQ (t)#$Markov極限	

• Lindblad形式：密度行列の正値性を保つ形	  &)11
ρ Q (t) = −i "# H, ρQ $% + ∑γ i ( Li ρQ L†i − L†i Li ρQ − ρQ L†i Li +'*22i– 時間スケールの階層性	  Lindblad	  (76)	

(γ i > 0)

システムの緩和時間の間の束縛状態の古典的軌道	

• 量子光学系：束縛状態|n>で記述	  • 量子Brown運動：波動関数Ψ(x)で記述	
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1.	  Introduc.on	• 古典的描像	  – 遮蔽ポテンシャルで相互作用する2つのBrown粒子	  à対応する量子論的記述	
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2.	  Inﬂuence	  func.onal	  for	  the	  Lindblad	  form	• 影響汎関数 F[x,y]	  – 経路積分形式の量子開放系の記述	 Feynman	  and	  Vernon	  (63)	
S[x, R] = ∫ d τ L(x, R) =S [x]+ S [x, R]+ S [R]A：システム、B：環境	
密度行列	 ρ (t, x, y) = ∫ dx 'dy' J(t, x, y;0, x ', y')ρ (0, x ', y') システムの初期条件	
tAIB0redsysx,y"i%
プロパゲータ	 J(t, x, y;0, x ', y') = ∫ DxDy exp $ ( SA [ x ]− SA [ y ])' F[ x, y ]#&x ',y'x,yのそれぞれの軌道について	

R’	
Q’	

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R	F[x, y] = ∫ dR'dQ'dRρ B (0, R',Q')R,Rt	

環境の初期条件	$i ' Q]exp
 S [y]− S [y, Q]× ∫ D[ R,S[x]+S[x,R]−
BIBI&%)(
R',Q'()

影響汎関数	
熱場の量子論とその応用＠理化学研究所	

6	2.	  Inﬂuence	  func.onal	  for	  the	  Lindblad	  form	• 量子Brown運動：Caldeira-­‐Leggel模型	  τ# i t&%− ∫ d τ ∫ ds [ x(τ ) − y(τ )] α I (τ − s) [ x(s) + y(s)] ( 00(F[x, y] = exp %τ% 1 t(%− ∫ d τ ∫ ds [ x(τ ) − y(τ )] α R (τ − s) [ x(s) − y(s)]($  0'0

Caldeira	  and	  Leggel	  (83)	

α：環境系の2点関数	
時間の粗視化	# iη t&ηkBT t2≅ exp %−d τ (xx − yy + xy − yx ) − 2 ∫ d τ (x − y) (∫2 0$'0• Lindblad形式にするには？	

時間微分展開の2次まで	# iη t
ηkBT t 2 ( x − y )2 &2
F[x, y] = exp %−d τ (xx − yy + xy − yx ) − 2 ∫ d τ (x − y) +(∫ 012(kBT )2 '$ 2 014/09/04	

熱場の量子論とその応用＠理化学研究所	

Diosi	  (93)	

7	

2.	  Inﬂuence	  func.onal	  for	  the	  Lindblad	  form	• QGP中の重クォーク系の場合	  時間スケールの階層性	  1/gT	  <<	  1/Mα2	– CL模型のx⇔カラー密度ρa	# V + iD
1IFaa
S [ ρ1, ρ 2 ] ≅ − ∫ ! ! ( ρ 1, ρ 2 ) ! %(t, x )
2 t, x, y$ −iD) ρ 1a ,−iD &++ a ..
(−V + iD '( x−! ! *ρ - !
2 (t, y )y)遮蔽ポテンシャルと熱揺らぎ（デコヒーレンス）	  近似	  • 摂動展開	  ) ρ" 1a ,# −D −D &1aa
• Velocity展開	   −++ a ..
(! ! ( ρ 1, ρ 2 ) ! %∫t, x, y(t,x)
4T
D '( x−! 2 -(t, y! )! ! *ρ
$ D• 時間粗視化	y)
運動量の拡散	# −D
iaa"!
+∫ ! ! ( ρ1, ρ 2 )(t, x! ) %$ D
24T 2 t, x, y) ρ" 1a ,D &++ a ..
(−D '( x−! 2 -(t, y! )! ! *ρy)

運動量の拡散の補正	
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熱場の量子論とその応用＠理化学研究所	

Lindblad	  form	

Lindblad	  form	
8	

3.	  Stochas.c	  poten.al	  with	  color	• 遮蔽ポテンシャルと熱揺らぎ（Recoilless	  limit）	  # V + iD
1IFaa
S [ ρ1, ρ 2 ] ≅ − ∫ ! ! ( ρ 1, ρ 2 ) ! %(t, x )
2 t, x, y$ −iDa,)&ρ−iD++ 1a ..
(−V + iD '( x−! ! *ρ - !
2 (t, y )y)

確率的変数（熱揺らぎ）を使って書き換える	# i! ! a ! a !&e = exp %− ∫ ! ! V ( x − y )ρ1 (t, x)ρ 2 (t, y)(
確率ポテンシャル	
t,x,y
$ 2'!
!
! &aaa#× exp $−i ∫ ! ξ (t, x) ( ρ 1 (t, x) − ρ 2 (t, x))'
t, x
ξ
! b !
! ! abaξ (t, x)ξ (s, y) = −D( x − y)δ δ (t − s)iSIF

（D:負定値）	

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3.	  Stochas.c	  poten.al	  with	  color	• 確率ポテンシャル中の量子力学	  – 確率的Schrödinger方程式	  & ∇ r2(aa (&+ iCF D(0) + (V (r) + i D(r) 2 ) ' t ⊗ (−t * ))+! *−!
∂i Ψ(t, r ) = * M
Ψ(t,r)+∂t*+ξ a (t, r! 2)&' t a ⊗1() + ξ a (t, − r! 2)&'1 ⊗ (−t *a )() +')! ! !
(Ψ : 3 ⊗ 3*, r = x − y)

• 熱揺らぎによるカラー回転：singlet	  ⇄	  octet	  • 典型的な熱揺らぎの空間スケール：lﬂuct~1/gT	  • 束縛状態のサイズ：lcoh	  (コヒーレンス長)	  14/09/04	

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3.	  Stochas.c	  poten.al	  with	  color	• カラー射影密度行列	  – デコヒーレンスの時間スケール	  • lcoh	  <<	  lﬂuctのとき、tdec	  ~	  1/g4T3lcoh2	  • lcoh	  >>	  lﬂuctのとき、tdec	  ~	  1/g2T	  （Sop	  scaleringの時間スケール）	

安定	

不安定	

QGPの寿命	#! !
! * ! &ρ1,8 (t, r, s ) ≡ Trcolor % Psinglet, Ψ(t, r )Ψ (t, s ) (ξ'$ octet! !
∂ρ1,8 (t, r, s ) = "tdec	
∂tlcoh	lﬂuct~1/gT	
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4.	  Summary	  &	  outlook	• 量子開放系としてのQGP中のクォーコニウム：影響汎関数、Lindblad形式	  • 有限温度のポテンシャル描像には、遮蔽以外に熱揺らぎも必要（確率ポテンシャル）	  • デコヒーレンスの時間スケールからクォーコニウムの安定性を議論	  • 現象論的応用：流体発展＋クォーコニウム	

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熱場の量子論とその応用＠理化学研究所	

12	Backup	  slides	

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熱場の量子論とその応用＠理化学研究所	

13	Subtle	  issues	  (personal	  views)	
• Time	  ordering	  in	  coarse	  graining	  Later	  	  Earlier	

τ
i t
iη t
− ∫ d τ ∫ ds [ x(τ ) − y(τ )] α I (τ − s) [ x(s) + y(s)] ≅ −∫ dτ (xx − yy + xy − yx ) 02
00Later	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  Earlier	τ
$1 t
ηkBT t 2 ( x − y )2 '2− ∫ d τ ∫ ds [ x(τ ) − y(τ )] α R (τ − s) [ x(s) − y(s)] ≅ − 2 ∫ d τ &(x − y) +2 )
 012(kT)%(B00CL	  model	  	  	  	  Diosi’s	  term	• Time	  diﬀeren.a.on	  in	  coarse	  graining	  – Should	  not	  be	  treated	  as	  one	  of	  the	  kine.c	  terms	  – If	  treated	  so,	  master	  equa.on	  does	  not	  change	  in	  CL	  model	  while	  it	  becomes	  diﬀerent	  with	  Diosi’s	  term.	  
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14	Subtle	  issues	  (personal	  views)	
• Integra.on	  by	  parts?	  
# iη t&ηkBT t2F[x, y] = exp %−d τ (xx − yy + xy − yx ) − 2 ∫ d τ (x − y) (∫ 0$ 2 0't# iη t&η
kT2B
 y] = exp %−or F[x,d τ (− xx + yy − xy + yx) − 2 ∫ d τ (x − y) (∫ 0$ 2 0'⇒ ρ red = exp #$iη x̂ 2 2&' ρred exp #$−iη x̂ 2 2&'– Inﬂuence	  func.onal	  should	  be	  determined	  without	  ambiguity	  with	  total	  deriva.ve	  terms.	

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(par.al)	  gauge	  invariance	• Gauge	  transforma.on	  (local	  only	  in	  .me)	  ∵Sint = g ∫ d 4 xρ a (x)Aa0 (x)– Density	  matrix	  ! !
! !
! !
ρQ (t, x, y) → ρ! Q (t, x, y) = U(t)ρQ (t, x, y)U −1 (t)! −1 (t)ρ! + ρ! UU! −1 )ρ! Q = L #$ρQ %& → ρ!" Q = U L #$ρQ %&U −1 + (UUQ
Q– Physical	  observable	  (singlet)	  	  ! ! !
!$"Osinglet ρ (t) = ∫ d xd yTrcolor #ρQ (t, x, y) y Osinglet x %33d"−1$ = 0 ∵U(t)U"! −1 (t) = 0O
(t)
−O(t)
(t)
+U(t)
Usinglet ρsinglet ρ!
%dt #U	  and	  singlet	  observable	  O	  commute	

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16