クォーク・グルーオン・プラズマにおける
「力」の量子論的記述
赤松 幸尚
(名古屋大学素粒子宇宙起源研究機構)
Y.Akamatsu, A.Rothkopf, PRD85(2012),105011 (arXiv:1110.1203[hep-ph] )
Y.Akamatsu, arXiv:1209.5068[hep-ph]
2013/01/12
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Contents
1.
2.
3.
4.
5.
6.
Introduction
In-Medium QCD Forces
Influence Functional of QCD
Dynamical Equations (I)
Dynamical Equations (II)
Summary & Outlook
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1. INTRODUCTION
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Confinement & Deconfinement
• Vacuum Potential
Singlet channel
V(R)
T=0
4 s
1 M2
3 R
String tension K ~ 0.9GeVfm-1
V ( R)  KR 
R


Coulomb + Linear
The Schrödinger_ equation
_
Mass spectra (cc, bb)
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Confinement & Deconfinement
• In-Medium Potential
Debye screened potential
V(R)
T>TC
4 s
exp   D R    1 M 2
3 R
Debye mass ωD ~ gT (HTL)
V ( R)  
R


Debye Screened
Higher T
The Schrödinger equation _ _
Existence of bound states (cc, bb)
 J/Ψ suppression in heavy-ion collisions
What is the in-medium potential?
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Matsui & Satz (86)
5
Quarkonium Suppression at LHC
• Sequential melting of bottomonia
A+A
p+p
 (2S )  (1S ) PbPb
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CMS
 (2S )  (1S ) pp
 (3S )  (1S ) PbPb
 (3S )  (1S ) pp
 0.21 0.07(stat)  0.02(syst)
 0.06
 0.06(stat)  0.06(syst)  0.17(95%CL)
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What is the in-medium potential?
2. IN-MEDIUM QCD FORCES
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In-Medium Potential
• Definition
r
T=0, M=∞
 (t ; R)  vac J (t; R) J † (0; R) vac
  m J (0; R) vac exp iEm ( R)t 
†
R
2
m
~ exp iEmin ( R)t   e iV ( R )t
Long time dynamics
t
G( ; R)  vac J (i ; R) J † (0; R) vac ~  D[ A] e  S ( A)
σ(ω;R)
~ exp Emin ( R)   e V ( R )
V(R)
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ω
V(R) from large τ behavior
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In-Medium Potential
• Definition
T>0, M=∞
r
 (t ; R)
T
 J (t ; R) J † (0; R)
T
  m J (0; R) n e  En ( R ) expi{En ( R)  Em ( R)}t 
2
†
n ,m
R
W
~
 ( n ,m )
i 0
( ; T ) exp iE ( R)t 
lowest peak
~ exp i{V ( R, T )  i( R, T ) 2}t 
t
σ(ω;R,T)
i=0
G( ; R, T )  J (i ; R) J † (0; R)
i=1 …

Γ(R,T)
i
V(R,T)
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ω
T
W ( ; T ) exp E ( R) 


peak( i )
i
Long time dynamics
Lorentzian fit of
σ(ω;R,T)
~  D[ A] e  S ( A;T )
(0<τ<β)
Spectral decomposition
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In-Medium Potential
• Complex Potential
 (t ; R)
T
~
W
i 0
 lowest peak
Laine et al (07), Beraudo et al (08),
Bramilla et al (10), Rothkopf et al (12).
( ; T ) exp iE ( R)t 
~ exp i{V ( R, T )  i( R, T ) 2}t 
Long time dynamics
Lorentzian fit of σ(ω;R,T)
t
t
2



~  D( s) exp   ds ( s) ( R, T ) exp  i  dsV ( R, T )  ( s)
 0

 0

Suggests stochastic & unitary description
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In-Medium Potential
• Stochastic Potential
Akamatsu & Rothkopf (‘12)
 (t  t , R )  exp  itV ( R)  (t , R) (t , R),
(t , R )  0, (t , R)(t ' , R ' )  ( R, R' ) tt ' / t ,
(T omitted)
Introduce noise field Θ(t,R)
Density matrix: Non-local correlation relevant
 (t, R1, R2 )   * (t, R1 )(t, R2 )

i


 (t , R)  V ( R)  ( R, R)  (t , R) (t , R), Imaginary potential
t
2


= Local correlation
it
(t , R)  (t , R) 
(t , R) 2  (t , R) 2 , (t , R)  0
2
i

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
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In-Medium Forces
• M<∞
M=∞
Debye screened force
+
Fluctuating force
M<∞
Drag force
Langevin dynamics
(Stochastic) Potential force
Hamiltonian dynamics
Non-potential force
Not Hamiltonian dynamics
How to describe in-medium QCD forces?
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How to describe in-medium QCD forces?
3. INFLUENCE FUNCTIONAL OF QCD
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Open Quantum System
sys = heavy quarks
env = gluon, light quarks
• Basics
Hilbert space
H tot   sys   env
von Neumann equation
d
i ˆ tot (t )  Hˆ tot , ˆ tot (t )
dt


Trace out the environment
Reduced density matrix
ˆ red (t )  T renv ˆ tot (t )
Master equation
i
d
ˆ red (t )  ?
dt
(Markovian limit)
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Closed-Time Path
[1ini ,2ini ]
• Basics
Partition function


Z [1 , 2 ]  T r Uˆ (,;1 ) ˆUˆ (,; 2 )†
 T r Uˆ (,; )† Uˆ (,; ) ˆ

2
1
1
1 ,1
2
2 ,2


~  D1, 2  [1ini ,  2ini ] exp iS [1 ]  iS [ 2 ]  i  11  i   2 2

i

 i ( xi )
ln Z [1 , 2 ]
 TC  ˆi ( xi )
i
j1, 2  0
T, conn
2
ln Z [1 , 2 ]
1 ( x1 )1 ( x2 )
j
 Tˆ ( x1 )ˆ ( x2 )
2
ln Z [1 , 2 ]
1 ( x1 ) 2 ( x2 )
j
 ˆ ( x2 )ˆ ( x1 )
T, conn
 GF ( x1 , x2 )
1, 2  0
T, conn
 G ( x1 , x2 )
1, 2  0
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
Closed-Time Path
• Apply to QCD
Z [ , ] ~  D[
1
2
1, 2 q1, 2 A1, 2 ] [ q A1
* *
1 1
ini
, 2 q2 A2
ini
]


 expiS[q A ]  iS[q A ]  ig  j A  ig  j A 
 exp iS[ 1 ]  iS[ 2 ]  i  11  i  2 2
1
1
2
2
1
1
2
2
eq
 tot   env
  sys Factorized initial density matrix
  tot [ q A1
* *
1 1
ini
, 2 q2 A2
Influence functional

ini
] 
eq
env
*
1
[q A1
ini
, q2 A2
ini
]   sys [ 1*ini , 2ini ]
Feynman & Vernon (63)

 Z qA [ j1 , j2 ]  exp iS FV [ j1 , j2 ]


~
F
A 2
 exp  g 2  j1G j  j2G j  j1GA j2  j2GA j1   g 3GA(3) jjj  g 4GA( 4) jjjj  
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F
A 1
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Influence Functional
• Open Quantum System
Z [1 ,2 ] ~  D[ 1, 2 ]sys [ 1*ini , 2ini ]

 exp iS[ 1 ]  iS[ 2 ]  iS FV [ j1 , j2 ]  i  11  i  22
1
sys [ 1*ini , 2ini ]
s
2
 1 (t ),1 (t )
 2 (t ),2 (t )
Path integrate until s, with boundary condition
 1 (s)   1 , 2 (s)   2
  red [ s, 1* , 2 ]   1* ˆ red ( s )  2
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
Influence Functional
• Functional Master Equation
 red [t , 1* , 2 ] ~ 
t , 1* , 2


Effective initial wave function
D[ 1, 2 ] sys [ 1*ini , 2ini ]

 exp iS[ 1 ]  iS[ 2 ]  iS FV [ j1 , j2 ]
Effective action S1+2
Single time integral
Long-time behavior (Markovian limit)
Analogy to the Schrödinger wave equation
Functional differential equation

*
*
i  red [t , 1* , 2 ]  H1func
[

,

]

[
t
,

2
1
2
red
1 , 2 ]
t
How does this formalism work in perturbation theory?
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How does this formalism work in perturbation theory?
4. DYNAMICAL EQUATIONS (I)
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Approximations
• Leading-Order Perturbation
Influence Functional


exp iS FV [ j1 , j2 ]

 exp  g
2
~
F
A 2
2  j1G j  j2G j  j1G A j2  j2G A j1
F
A 1

Expansion up to 4-Fermi interactions
ˆ (x ) A
ˆ (x ) , G (x  x )  A
ˆ (x ) A
ˆ (x )
GAF ( x1  x2 )  TA
1
2
A
1
2
2
1
T
ˆ (x ) A
ˆ (x ) ,
G ( x1  x2 )  A
1
2

A
T
T
~ˆ
ˆ (x )
G ( x1  x2 )  TA( x1 ) A
2
~
F
A
T
Leading-order result by HTL resummed perturbation theory
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Approximations
Q ~ MT  Q
• Heavy Mass Limit
 G ~ ( g )T  G 
Non-relativistic kinetic term
NR
S[ ]  S kin
[Q, Qc ]  ~ (Q, Q†c )
NR
S kin
[Q, Qc ]  Q† [i 0  M   2 2M ]Q
 Qc [i 0  M   2M ]Q
2
†
c
Non-relativistic 4-current (density, current)
ja0  Q† t a Q  Qc t a Q†c   a



 
†    a
t Q  Qc 
ja  Q 

 2iM
2
iM



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a †
t Qc


Expansion up to
~
T
M
(quenched)
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Approximations
• Long-Time Behavior
Time-retardation in interaction
 
 
~  
~  
~  
G( x  y,  )  G( x  y,0)   G' ( x  y,0)  G ( x  y)   G ' ( x  y)
 
 

0
0
 G( x  y)  G ( x  y) ( x 0  y 0 )  iG ' ( x  y)

(
x

y
)
0
0
( x  y )
Low frequency expansion
 


G ( x  y ) j (t , x ) j (t , y )



j
(
x
)
G
(
x

y
)
j
(
y
)







i
xy
t xy  G ' ( x  y) j(t, x ) j(t, y)  j(t, x ) j(t, y)
0
0
 2

Using free equation of motion
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Effective Action
• LO pQCD, NR Limit, Slow Dynamics
NR
NR
LONR
S12  Skin
[Q1, Q1c ]  Skin
[Q2 , Q2c ]  SFV
[ j1, j2 ]
 


V
(
x

y
)

(
t
,
x
)

(
t
,
y
)  Stochastic potential

1a
1a
LONR
S FV [ j1 , j2 ]   1 2    
 

  (finite in M∞)
t ,x , y  V * ( x  y) 
2 a (t , x )  2 a (t , y ) 

 


iD
(
x

y
)

(
t
,
x
)

(
t
,
y
)


1a
2a





    j1a (t , x )  2 a (t , y ) 
  
t , x , y  1 4T D ( x  y )  





   (t , x ) j (t , y ) 
1a
2a





 
 
 

 g G ( x  y )  iG00,ab ( x  y )  V ( x  y ) ab
 
 
 
2 
 g G00,ab ( x  y )  D( x  y ) ab  ImV ( x  y ) ab
2
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00 , ab
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Drag force
(vanishes in M∞)
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Hamiltonian Formalism (technical)
• Order of Operators = Time Ordered




*
Kinetic term  (t , x)  1 (t , x ),  2 (t , x )  2 (t , x )
*
1
Instantaneous interaction
or
Remember the original order
• Change of Variables (canonical transformation)
Make 1 & 2 symmetric
~ ~*
~
 2  (Q2 , Q2c )  2*  (Q2* , Q2c )
* ~*
[

Determines H1func
2
1 , 2 ] without ambiguity
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Hamiltonian Formalism (technical)
• Variables of Reduced Density Matrix
 red [t , 1* ,~2* ]   1* ˆ red (t ) ~2*
  red [t , Q
*
1( c )
Latter is better (explained later)
~*
~*
*
, Q2( c ) ]  Q1( c ) ˆ red (t ) Q2 ( c )
• Renormalization
Convenient to move all the functional differential operators
to the right in

~
~*
~*
*
*
i  red [t , Q1*( c ) , Q2*( c ) ]  H1func
[
Q
,
Q
]

[
t
,
Q
,
Q
2
1( c )
2(c )
red
1( c )
2(c ) ]
t
In this procedure, divergent contribution from Coulomb
potential at the origin appears  needs to be renormalized
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Functional Master Equation
• Renormalized Effective Hamiltonian

 
~ˆ
~ˆ
2 M Q   (same for Q


Hˆ 1 2   aMQˆ†1 Qˆ1  Qˆ†1   2 2 M Qˆ1  (same for Qˆ1c )
x

~ˆ
~ˆ ~ˆ
  a * MQ†2 Q2  Q†2   2
2c )
2
x

  

a0  ˆ a0 
* 
a0  ˆ a0 
ˆ
ˆ
V ( x  y ) N j1 ( x ) j1 ( y )  V ( x  y ) N j2 ( x ) j2 ( y )
 
1 
a0  ˆ a0 
ˆ
   2iD( x  y ) N j1 ( x ) j2 ( y )
2 xy 
ˆ a 
ˆ a  a 0 

1   
0
a
D ( x  y )  N j1, NR ( x ) ˆj2 ( y )  ˆj1 ( x ) j2 ( y )

 2T

a  1
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






CF
limV (T 0 ) (r ), V (T 0 ) (r )  V (r )  V (T 0 ) (r )
2 M r 0
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






26
Functional Master Equation
• Schrödinger wave equation
Anti-commutator in functional space
Qˆ ( x),Qˆ ( y) Qˆ
1
†
1

 ˆ† 
 
(
x
),
Q
(
y
)


(
x
 y )  Q1( c ) 
1c
1c

Q1*( c )
 

~ˆ  ~ˆ†    ~ˆ  ~ˆ†  
~
Q
(
x
),
Q
(
y
)

Q
(
x
),
Q
(
y
)



(
x

y
)

Q


 2
  2c

~*
2
2c
2(c )

 

Q2( c )
Hˆ 12  H1func
2
i

~
~*
~*
*
*
 red [t , Q1*( c ) , Q2*( c ) ]  H1func
[
Q
,
Q
]

[
t
,
Q
,
Q
2
1( c )
2(c )
red
1( c )
2(c ) ]
t
So what?
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So what?
5. DYNAMICAL EQUATIONS (II)
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Density Matrix
• Coherent State


~
~
 red t , Q1*( c ) , Q2*( c )  Q1*( c ) ˆ red (t ) Q2*( c )
 
Q1*( c )   exp   Qˆ Q1*  Qˆ c Q1*( c )
x

~
~ * ~ˆ† ~ * ~ˆ† 
Q2*( c )  exp  
Q
 2 Q  Q2 c Qc  
x



Source for HQs


ˆ
  Q( x )
*
1( c )
 Q
Q1* ( x )
Q1*( c )  0

~*
Q
~ 
2(c )
Q2* ( x )
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
 Qˆ† ( x ) 
~
Q2*( c )  0
29
Density Matrix
• A few HQs
One HQ
 

† 
ˆ
ˆ
 Q (t , x , y )   Q( x ) ˆ red (t )Q ( y ) 




Q ( x ) Q2 ( y )
*
1

~
*
*

t
,
Q
,
Q
~ *  red
1( c )
2( c )

~
Q1*( c ) Q2*( c )  0
Similar for two HQs, …
   
QQc (t, x1, x2 , y1, y2 ),
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Master Equation
• Functional Master Equation

~*
~*
~*
*
func
*
*
i  red [t , Q1( c ) , Q2 ( c ) ]  H1 2 [Q1( c ) , Q2 ( c ) ] red [t , Q1( c ) , Q2 ( c ) ]
t


Functional differentiation

~ 
Q1* ( x ) Q2* ( y )
Color traced
Master equation
  2x   2y 
  
 


*


i t  Q (t , x , y )   a  a M  
 Q (t , x , y )


2 M 






  

   x D( x  y )  x   y 
 


 C F  iD( x  y ) 

  Q (t , x , y )
4T
iM 




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
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Master Equation
• HQ Number Conservation
 
T rˆ (t )    (t , x , x )

x
Q
i
Q
 
 
d
T rˆ Q (t )     ( x  y )i t  Q (t , x , y )   0
x, y
dt
• Ehrenfest Equation
d
dt
d
dt
d
dt

x 

p
M

p 
,


p ,
CF 2
g (T ) 2 CF 2 ~ 

 D( x)  
 G00,aa (  0, x)
x

0
x 0
3
9
2 MT
2
3
~
 
3T   g (T ) CF d k k 2G
E 
E

.


00 , aa (  0, k )
3

9
(
2

)
MT 
2 
Moore et al (05,08,09)
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Other Results
• Complex Potential
 
 (t ; x , y )
T
  †
 
 J (t ; x , y ) J (0; x0 , y0 )

2
~*
*


Q
,
Q
red
1( c )
2( c ) , t
* 
* 
Q1 ( x )Q1 ( y )

T
~
Q1*( c ) Q2*( c )  0
Time-evolution equation + Project on singlet state
g (T ) 2 CF
Vsinglet ( R)  2(a  1) M  CFV ( R)  
4


e  D R
 D 
 iT D R 
R


Laine et al (07), Beraudo et al (08), Brambilla et al (10)
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Other Results
• Stochastic Dynamics
M=∞ : Stochastic potential


Debye screened potential
LONR
exp iS FV
[ j1 , j2 ]
 



 

 exp  i 2    Re V ( x  y )1a (t , x ) 1a (t , y )   2 a (t , x )  2 a (t , y )


t,x, y



 exp i     a (t , x )1a (t , x )   2 a (t , x )
 t , x , y
  Fluctuation


 
a (t, x)b (s, y)   ab (t  s)D( x  y) D(x-y): Negative definite
 
M<∞ : Drag force
Two complex noises c1,c2
 Non-hermitian evolution
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 ~

Q (t , x, y)  (t , x )(t , y)

~ 
*
(t , x )   (t , x )
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 ,c1 ,c2
34
6. SUMMARY & OUTLOOK
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• Quantum Dynamics of HQs in Medium
– Stochastic potential, drag force
• Non-Equilibrium Quantum Field Theory
– Open quantum system, closed-time path, influence
functional
– Functional master equation, master equation, etc.
• Non-Perturbative Region
– Model the renormalized effective Hamiltonian
– Higher-order perturbative analyses (process involving
real gluons)
– Application to phenomenology
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ゲージ / ШЛ対応によS Wビークォーク対称性

Slides