Stochastic Dynamics of Heavy Quarkonium in Quark-Gluon Plasma
Yukinao Akamatsu (KMI,Nagoya) In collaboration with
Alexander Rothkopf (Bielefeld)
2011/11/18QHEC111/17
Reference: arXiv:1110.1203[hep-ph]
Contents
2011/11/18QHEC112/17
Introduction Complex potential from lattice QCD Stochastic dynamics of heavy quarkonium Bound states in the medium Conclusion and discussion
Introduction
2011/11/18QHEC113/17
Matsui and Satz (‘86)
/
0 ),(2
12)(
)(exp
)()( :
2JrrQ
Q
D
eff
rErV
rMMrE
Trr
rT
rVTcT
No solution rJ/Ψ at T>1.2Tc
“Plasma formation thus prevents J/Ψ formation already just above Tc.”
• 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
2011/11/18QHEC114/17
New data from LHCALICE
CMS
Y(1S)
Y(2S,3S)
Introduction
2011/11/18QHEC115/17
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
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Rothkopf, et al. (‘11)
, ,)0,,(),,(),(
),,(],[),(),,(
00 yxryxMtyxMtrD
tyQyxUtxQtyxM
†
Meson operator (J/Ψ,ηc, …)
Forward correlator
In heavy quark limit, ω~2MQ describes 2-HQs physics ≈ described by Schroedinger equation
),(),(),(2
trDtrVM
trDt
i NRQ
rNR
□
In MQ=∞ 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
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Rothkopf, et al. (‘11) cont’dV□(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
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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
MQ
~T
fluctuation
Non-relativistic,Q and Q separately conserved
~(MQT)1/2
Due to this hierarchy, we expect unitary evolution of the reduced system
Stochastic dynamics of heavy quarkonium
2011/11/18QHEC119/17
Unitary evolution by stochastic Hamiltonian
tXXtXtXtX
XVM
XH
tXXHdtiTtU
xxXXtUtX
tt
Q
X
tX
QQX
/)',()','(),( ,0),(
hermite )(2
)(
)',()('exp)0|(
},{ ),0,()0|(),(
'
22
0
)(
21)(
Θ1Θ2
Θ3
Stochastic termlcorr ~ thermal wavelengthof medium particles
manifestly unitary
decays when |X-X’| > lcorr
stochastic
Stochastic dynamics of heavy quarkonium
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Stochastic differential equation
),(),(),(2
)(),(
0),( ,),(),(2
),(),(
)(),(),(2
)(1
)(),(21),()(1
),()(exp)0|(
22
2/3
2/3222
)(
tXtXXXiXHtXt
i
tXtXtXtitXtX
tOtXtiXXiXHti
tOtXttXXHti
tXXHtitU
QQQQ
X
Stochastic dynamics of heavy quarkonium
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Relation to complex potential
),(),( tXtXD QQNR
In MQ =∞ limit,
D>NR is ensemble average of wave function Ψ! Evolution of D>NR needs not be unitary.
),(2
)()(
),( ),(2
)(),(
),()(),(
XXiXVXV
tXXXiXVtXt
i
tXDXVtXDt
i
QQQQ
NRNR
□
□
complex potential = [real potential] + i[noise strength]
Stochastic dynamics of heavy quarknoium
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Remark2 : the observables of J/Ψ suppression Dilepton spectrum
① If charms are (chemically and kinetically) equilibrated, SPF of current correlator is enough to give dilepton spectrum.
② If not (and is not in heavy ion collisions), the stochastic dynamics is necessary.
J
① ②
initial
J/Ψ evolved
J/ΨJ
Remark1 : SPF of current correlatorCan be calculated only from the complex potential. no reference to lcorr
Bound states in the medium
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Fate of bound states
Θ
Low temperature
Bound state
Real potential : energy levels and sizes of bound statesNoise : excites modes with k~1/lcorr (spatial decoherence)
Θ1
High temperature
Θ2
Θ3 Θ4
Bound state
noise gives a nearly global phasedoes not change physics
noises excite the bound statebound state disappears
Argument here can be made more quantitative in terms of master equation.
Bound states in the medium
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1d simulation – set up
1||095.0)',(
)5.1|(| 0833.0
)5.1|(| ||1.0||
1.0)(
0.001dt 0.1,dxb.c. periodic 6],[-2.56,2.5x
'
Mxxx
x
xxxxv
xx
(Relative motion)
Initial condition
lcorr~dx=0.1 very(too) high temperature
Bound states in the medium
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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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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
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Conclusion Stochastic unitary evolution can explain complex potential
obtained by lattice simulation. Noise correlation length lcorr plays a crucial role in determining
the fate of bound states.
Discussion What is the first principle definition of lcorr? 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
2011/11/18QHEC1118/17
BACK UP
Master equation
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Master equation
),'(),(')(),',(
)(;)(;1)(
*
)(
tXtXXtXtXX
ttttN
t
QQQQQQQQ
tQQQQQQ
Reduced density matrix
2)','(),()',()',(
),',()',(),',()'()(),',(
XXXXXXXXF
tXXXXFtXXi
XHXHtXXt QQQQQQ
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
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Extracting relative motion
22
',0,0,0
22
,
',,3
),(),(2
),(),(
),(),(),(2
)(),(
),()',( ),()(
2)','(),()',()',( ),()(
),',(ˆ)',(),',(ˆ)'()(),',(ˆ
2,2
),',(),',(ˆ
trtrtitrtr
trtrrrirhtrt
i
XXrrXVrv
rrrrrrrrfrvM
rh
trrrrftrri
rhrhtrrt
rRrRX
tXXRdtrr
QQQQ
rrr
Q
r
QQQQQQ
rR
rRrRQQQQ