Heavy QQbar potentialfrom a spectral decomposition of the thermal Wilson loop
T. Hatsuda, A. Rothkopf, S. Sasaki

ATHIC 2008 at Tsukuba University 2008年10月14日

Alexander Rothkopf

14. Oktober 2008

Motivation: Probing the QGP
Heavy quarkonia: static case

Heavy quarks: dynamical case

RextR(t)Quarks and Gluons

hadronic phase
Alexander Rothkopf

TCsQGP

14. Oktober 2008

2TCpQGP2

Outline

1. Motivation: Heavy Quark potential

2. Potential from the Spectral function
3. Lattice Reconstruction
4. Conclusion and Outlook

Alexander Rothkopf

14. Oktober 20083Heavy Quarks
Apriori: At the phase transition tightly bound states do not have to vanish
See e.g. Hatsuda, Kunihiro (PRL 55, 1985)

However: Matsui & Satz predicted J/Psi melting at crtitcal T
comparison Vcoulomb+lin.(r,T) vs. VDebye(r ,T)‘‘J/Psi Suppression by Quark-Gluon plasma formation“ (Phys.Lett. 178, 1986)

Heavy Quarks allow separation of scales: Non-relativistic (1/mQ)

see e.g. Laine et al. JHEP03, 2007

How do we find the potential to use in that Schrödinger Equation?

Alexander Rothkopf

14. Oktober 20084Potential Searches
Use Lattice QCD since non-perturbative:

First approach: Free Energy as static potentialτFree Energy from polyakov correlator
McLerran, Svetitsky

Digal, Kaczmarek, Karsch & Satz,
Hep-ph/0505193‘‘Quark liberation at high temperature“ (Phys. Rev. D 24, 1981)

R

0.5fm 1fm

x,y,z

2fm

Rs1/2

Different approaches: Internal Energy, Linear combination Internal/Free energy(Shuryak, Petreczky, Wong)

However: No direct relation to potential available!

Alexander Rothkopf

14. Oktober 2008

5J/Psi from first principles:
2004: J/Psi survives in the deconfined phase up to  2TC

Asakawa, Hatsuda‘‘J/Psi and EtaC in the deconfined plasma from Lattice QCD“ (PRL 92, 2004)

Fact: Spectral function contains all information of the system.

Fact: Vector channel SPF directly accessible by experiment:

Dilepton production rate

Feinberg (1976)

J/Psi(3.1 GeV)

Lattice simulations in Euclidean time τ, no direct access todynamical quantities. Reconstruction is ill-posed problem.

However possible with MEM (Maximum Entropy Method)
Asakawa, Nakahara, Hatsuda PRD 60 (1999) & Prog. Part. Nucl. Phys. 46 (2001)
Alexander Rothkopf

14. Oktober 2008

6

What is the true potential?
Challenge 1: Free energy potential has no direct theroetical basis (also J/Psi melts too quickly)
Challenge 2: Lattice potential does not include realtime dynamics, are we missing something?
Probably: Heavy Quark potential can be complex, i.e. J/Psi is transient(using the Hard Thermal Loop approximation, i.e perturbative)
Laine et al.
"Realtime static potential in hot QCD" JHEP 0703 (2007)
Beraudo, Balizot, Ratti"Real and imaginary-time QQbar correlators in a thermal medium" NPA 806 (2008)
HTL Potential:
Debye screening: (can be derived also from free energy)

Additional effect: Landau damping (scattering with light medium particles)

Alexander Rothkopf

14. Oktober 20087

A potential from Lattice QCD
Through MEM we can access dynamical information (SPF) directly from the lattice
What we need: Definition of the Heavy Quark Potential in terms of the spectral function

From NRQCD:

Heavy mass limit:

Lattice system is finite,
therefore SPF discrete:

What we expect:position:
Re[V]Alexander Rothkopf

14. Oktober 2008

ωwidth:Im[V]ω8

The heavy QQbar correlator
Can we simplify the QQbar correlator?

In the heavy mass limit: YES!

1. Medium is vaccum state for Q and Qbar fields: No thermal excitation possible
2. QQbar correlator G(x,x',y,y') in termsof heavy quark correlators S = <Qbar Q>

<y<

0x0y<

<<xτ'<

τ'<τ'x0y<<τ<<τ<y<<τ<<β<0β<βHandlesβτStaple<Loop<3. Use m-> to simplify the form of S (periodic boundary condition in τ direction!)τ'xconstant: ρ  ω δ(ω) Umeda PRD75 2007
Alexander Rothkopf

14. Oktober 20089

Exploring the QQbar potential
Combining both spectral function and the correlator gives:
β<0τ<<Tr<yτ'x

R

Reconstruct spectral function for different R: map the shape of the potential++ ...
l=2

l=1

ρρ
T<TCω

TTCω

V0(R)
R
Alexander Rothkopf

R
14. Oktober 2008

V0(R)
10

Conclusion & Outlook
1. Free energy as heavy quark potential not consistent
with Lattice QCD and HTL
2. Nonpertrubative definition of the potential possible:
Wilson contour (loop) Lattice Spectral Function MEM Potential
3. Obtaining Wilson Loop from Lattice Simulations (quenched)
203x46 β=7.0 ξ=3.5
and203x40 β=7.0 ξ=3.5
4. Reconstructing the spectral function from the lattice datausing the Maximum Entropy Method (in progress)

5. Extracting the QQbar potential both Re[V] and Im[V]Alexander Rothkopf

14. Oktober 200811End

Thank you for your attention
ありがとうございました감사합니다
谢谢
Alexander Rothkopf

14. Oktober 2008

12

Backup Slides

Additional Material

Alexander Rothkopf

14. Oktober 200813

Extracting the SPF
Maximum Entropy Method:

Deconvolution prescription using Inference:
What is the most probable image for given data?

τD(R,τi)

ill - posedρ (R,ωj)cannot use χ2 fittingω

x,y,zRNτ  O(10)Nω  O(1000)

1. No need for low ω region parametrization (for ω-> : ρ(ω) ω2 )2. If a solution for ρ exists it is unique
3. Satistical errors for ρ can be estimates
Alexander Rothkopf

14. Oktober 2008

14

MEM: Bayes Law
Probability of the Data given thesolution SPF (fit ρ onto the data set)

Probability of the SPF given the priorknowledge. (fit ρ onto the prior model m)Given by Shannon-Janes Entropy
For many lattice measurements, central
limit theorem gives gaussian distribution

Probability of SPF given
the Data D and prior knowledge mAlexander Rothkopf

Normalization,irrelevant

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