Stochastic Dynamics of Heavy Quarkonium in Quark-Gluon Plasma

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Nov 14, 2013 (3 years and 7 months ago)

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Stochastic Dynamics of Heavy
Quarkonium

in Quark
-
Gluon Plasma

Yukinao Akamatsu (
KMI,Nagoya
)

In collaboration with


Alexander Rothkopf (Bielefeld)

2011/11/18

QHEC11

1
/17

Reference: arXiv:1110.1203[
hep
-
ph]

Contents

2011/11/18

QHEC11

2
/17


Introduction


Complex potential from lattice QCD


Stochastic dynamics of heavy
quarkonium


Bound states in the medium


Conclusion and discussion

Introduction

2011/11/18

QHEC11

3
/17


Matsui and
Satz

(‘86)



/
0

),
(
2
1
2
)
(
)
(
exp
)
(
)
(

:
2
J
r
r
Q
Q
D
eff
r
E
r
V
r
M
M
r
E
T
r
r
r
T
r
V
Tc
T



















No solution
r
J


at
T
>1.2
T
c

“Plasma formation
thus

prevents J/Ψ
formation already just above
T
c
.”



Underlying physics: Debye screening



Sensitive to color
deconfinement



All the discussion based on the potential
V
(
r
)



Propose J/Ψ suppression as a signal for
QGP formation

Introduction

2011/11/18

QHEC11

4
/17


New data from LHC

ALICE

CMS

Y(1S)

Y(2S,3S)

Introduction

2011/11/18

QHEC11

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/17


Potential Model Approaches


Provide clear physical picture!


Potential from QQ free energy, or internal energy, or
linear combination of both? Relation to first principle?



Spectral Function of Current
Correlator


Relation to first principle is clear!


How to discuss more than the shape of peak?

How to define the potential from first principle?

Complex potential from lattice QCD

2011/11/18

QHEC11

6
/17


Rothkopf, et al. (‘11)

,

,
)
0
,
,
(
)
,
,
(
)
,
(
),
,
(
]
,
[
)
,
(
)
,
,
(
0
0
y
x
r
y
x
M
t
y
x
M
t
r
D
t
y
Q
y
x
U
t
x
Q
t
y
x
M







Meson operator (J/Ψ,η
c
, …)

Forward
correlator

In heavy quark limit,

ω~2M
Q

describes 2
-
HQs physics



described by
Schroedinger

equation

)
,
(
)
,
(
)
,
(
2
t
r
D
t
r
V
M
t
r
D
t
i
NR
Q
r
NR

















In
M
Q
=

limit,

Fourier transformation (t

ω) of
D
>
NR
(
r,t
)

=Spectral decomposition of thermal Wilson loop

V

(
r
)

(
Lorentzian

fit)

Proper

potential from first principle

Complex potential from lattice QCD

2011/11/18

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7
/17


Rothkopf, et al. (‘11) cont’d

V

(
r
) =

Complex

potential !!

Complex potential also found by
perturbation theory [
Laine
, et al. (07’)]

What happened to unitarity?

In Coulomb gauge

Stochastic dynamics of heavy
quarkonium

2011/11/18

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8
/17

Stochastic unitary evolution of QQ?

Can stochastic unitary evolution explain
V

(
r
)?


Heavy quark(s) as an open quantum system

Integrated out

k

k

Heavy quarks

Gluons,

light quarks

M
Q

~
T

fluctuation

Non
-
relativistic,

Q and Q separately conserved

~(M
Q
T)
1/2

Due to this hierarchy,

we expect
unitary

evolution
of the reduced system

Stochastic dynamics of heavy
quarkonium

2011/11/18

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/17


Unitary evolution by stochastic Hamiltonian



t
X
X
t
X
t
X
t
X
X
V
M
X
H
t
X
X
H
dt
i
T
t
U
x
x
X
X
t
U
t
X
tt
Q
X
t
X
Q
Q
X
Q
Q





























/
)
'
,
(
)
'
,
'
(
)
,
(

,
0
)
,
(
hermite


)
(
2
)
(
)
'
,
(
)
(
'
exp
)
0
|
(
}
,
{

),
0
,
(
)
0
|
(
)
,
(
'
2
2
0
)
(
2
1
)
(






Θ
1

Θ
2

Θ
3

Stochastic term

l
corr

~ thermal wavelength

of medium particles



manifestly
unitary

decays when

|X
-
X’| >
l
corr



stochastic

Stochastic dynamics of heavy
quarkonium

2011/11/18

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/17


Stochastic differential equation







)
,
(
)
,
(
)
,
(
2
)
(
)
,
(

0
)
,
(

,
)
,
(
)
,
(
2
)
,
(
)
,
(
)
(
)
,
(
)
,
(
2
)
(
1


)
(
)
,
(
2
1
)
,
(
)
(
1


)
,
(
)
(
exp
)
0
|
(
2
2
2
/
3
2
/
3
2
2
2
)
(
t
X
t
X
X
X
i
X
H
t
X
t
i
t
X
t
X
t
X
t
i
t
X
t
X
t
O
t
X
t
i
X
X
i
X
H
t
i
t
O
t
X
t
t
X
X
H
t
i
t
X
X
H
t
i
t
U
Q
Q
Q
Q
X








































































Stochastic dynamics of heavy
quarkonium

2011/11/18

QHEC11

11
/17


Relation to complex potential





)
,
(
)
,
(
t
X
t
X
D
Q
Q
NR
In
M
Q


limit,

D
>
NR

is
ensemble average

of wave function

Ψ!



Evolution of
D
>
NR

needs
not

be unitary.

)
,
(
2
)
(
)
(
)
,
(

)
,
(
2
)
(
)
,
(
)
,
(
)
(
)
,
(
X
X
i
X
V
X
V
t
X
X
X
i
X
V
t
X
t
i
t
X
D
X
V
t
X
D
t
i
Q
Q
Q
Q
NR
NR



























complex potential = [real potential] +
i
[noise strength]

Stochastic dynamics of heavy
quarknoium

2011/11/18

QHEC11

12
/17


Remark2 : the observables of J/Ψ suppression

Dilepton

spectrum


If charms are (chemically and kinetically) equilibrated, SPF of
current
correlator

is enough to give
dilepton

spectrum.


If not (and is not in heavy ion collisions), the stochastic dynamics
is necessary.

J





initial


J/Ψ



evolved

J/Ψ

J


Remark1 : SPF of current
correlator

Can be calculated only from the complex potential.


no reference to
l
corr

Bound states in the medium

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/17


Fate of bound states






Θ

Low temperature

Bound
state

Real potential : energy levels and sizes of bound states

Noise : excites modes with
k
~1/
l
corr

(spatial
decoherence
)

Θ
1

High temperature

Θ
2

Θ
3

Θ
4

Bound
state

noise gives a nearly global phase


does not change physics

noises excite the bound state


bound state disappears

Argument here can be made more
quantitative in terms of master equation.

Bound states in the medium

2011/11/18

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/17


1d simulation


set up

1
|
|
095
.
0
)
'
,
(
)
5
.
1
|
(|


0833
.
0
)
5
.
1
|
(|

|
|
1
.
0
|
|
1
.
0
)
(
0.001
dt

0.1,
dx
b.c.

periodic

6],
[-2.56,2.5
x
'















M
x
x
x
x
x
x
x
x
v
xx


(Relative motion)

Initial condition

l
corr
~dx
=0.1


very(too) high temperature

Bound states in the medium

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/17


1d simulation


bound state probability P(t)

Probability of occupying bound states decays,
but saturates

at later time.

Bound states in the medium

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/17


1d simulation


norms, etc.

Norm of each trajectory = 1 (unitary)

Norm of average wave function decays. (noise


imaginary part
)

Energy average ~ 100! (due to high temperature)

Conclusion and discussion

2011/11/18

QHEC11

17
/17


Conclusion


Stochastic unitary evolution can explain complex potential
obtained by lattice simulation.


Noise correlation length
l
corr

plays a crucial role in determining
the fate of bound states.



Discussion


What is the first principle definition of
l
corr
?


Gauge dependence in introducing the color


Quantum Brownian motion of single heavy quark


Thermodynamic quantities (free energy, entropy, …)


Relation to heavy quark effective field theories

2011/11/18

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/17

BACK UP

Master equation

2011/11/18

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/17


Master equation


















)
,
'
(
)
,
(
'
)
(
)
,
'
,
(
)
(
;
)
(
;
1
)
(
*
)
(
t
X
t
X
X
t
X
t
X
X
t
t
t
t
N
t
Q
Q
Q
Q
Q
Q
Q
Q
t
Q
Q
Q
Q
Q
Q



Reduced density matrix

2
)
'
,
'
(
)
,
(
)
'
,
(
)
'
,
(
)
,
'
,
(
)
'
,
(
)
,
'
,
(
)
'
(
)
(
)
,
'
,
(
X
X
X
X
X
X
X
X
F
t
X
X
X
X
F
t
X
X
i
X
H
X
H
t
X
X
t
Q
Q
Q
Q
Q
Q
















Master equation

Equivalent master equation:

proposed as a modified quantum mechanics (
Ghirardi
, et al. ‘86)

derived in scattering model (
Gallis

& Fleming ‘90)


in random potential in Feynman
-
Vernon approach (
Gallis

‘92)

Master equation

2011/11/18

QHEC11

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/17


Extracting relative motion





2
2
'
,
0
,
0
,
0
2
2
,
'
,
,
3
)
,
(
)
,
(
2
)
,
(
)
,
(

)
,
(
)
,
(
)
,
(
2
)
(
)
,
(
)
,
(
)
'
,
(

),
(
)
(

2
)
'
,
'
(
)
,
(
)
'
,
(
)
'
,
(

),
(
)
(

)
,
'
,
(
ˆ
)
'
,
(
)
,
'
,
(
ˆ
)
'
(
)
(
)
,
'
,
(
ˆ
2
,
2
)
,
'
,
(
)
,
'
,
(
ˆ
t
r
t
r
t
i
t
r
t
r
t
r
t
r
r
r
i
r
h
t
r
t
i
X
X
r
r
X
V
r
v
r
r
r
r
r
r
r
r
f
r
v
M
r
h
t
r
r
r
r
f
t
r
r
i
r
h
r
h
t
r
r
t
r
R
r
R
X
t
X
X
R
d
t
r
r
Q
Q
Q
Q
r
r
r
Q
r
Q
Q
Q
Q
Q
Q
r
R
r
R
r
R
Q
Q
Q
Q