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Proper Heavy Quark Potential from the Thermal Wilson Loop

Proper Heavy Quark Potential from the Thermal Wilson Loop. Alexander Rothkopf. In collaboration with T. Hatsuda and S. Sasaki see also arXiv:0910.2321. 次世代格子ゲージシミュレーション研究会　・　理化学研究所 仁科加速器研究センター　２０１０年０９月２５日. Motivation. T  2 T C pQGP. T=2T C. or. T  T C sQGP. T=T C.

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Proper Heavy Quark Potential from the Thermal Wilson Loop

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  1. Proper Heavy Quark Potentialfrom the Thermal Wilson Loop Alexander Rothkopf In collaboration with T. Hatsuda and S. Sasaki see also arXiv:0910.2321 次世代格子ゲージシミュレーション研究会　・　理化学研究所 仁科加速器研究センター　２０１０年０９月２５日

  2. Motivation T  2TC pQGP T=2TC or TTC sQGP T=TC Large mass allows a non-relativistic description: Temperatures are comparatively small (at least for ) Matsui & Satz (`86): Complete melting of J/ψ at T=TC At m=∞ separation distance of constituents is external parameter Convenient Separation of Scales Hadronic Thermometer for the QGP Derive an effective Schroedinger equation for Heavy Quarkonium at finite temperature T<TC Hadronic phase

  3. Previous Studies Laine et al. JHEP 0703:054,2007 T=0 NRQCD & pNRQCD Hard Thermal Loop Potential Models T  2TC pQGP T=2TC eg. Petreczky et al. arXiv:0904.1748 TTC sQGP T=TC Brambilla et al. Rev.Mod.Phys. 77 (`05) Ad-hoc identification with Free Energies in Coulomb Gauge Expansion in 1/m (Minkowski time): Remember: Quarkonia is transient: Resummed perturbation theory: applies at very high T Real-time formulation based on the forward correlator Challenges: Gauge dependence, Entropy contributions T<TC Hadronic phase

  4. Previous Studies II Asakawa, Hatsuda PRL 92 012001,(‘04) Spectral functions from LQCD This talk: T  2TC pQGP Asakawa, Hatsuda: Prog.Part.Nucl.Phys.46:459,(‘01); T=2TC Free Energies predict melting at 1.2 TC TTC sQGP T=TC Combine the systematic expansion of the effective theory with the non-perturbative power of Lattice QCD at T>0 Extract spectral functions directly from non-perturbative Lattice QCD simulations at any temperature Challenging but well understood: Maximum Entropy Method Unexpected: J/ψ seems to survive up to T > 1.6TC Numerical results for the proper potential We propose a gauge invariant and non-perturbative definition of the static proper in-medium heavy quark potential based on the thermal Wilson loop. T<TC Hadronic phase

  5. Starting point time M(R,t’) y´ x´ Gamma matrix spatial Wilson line M†(R,t) y x x,y,z 2x2 sub matrix Temperature dependence No coupling of upper and lower components of Dirac 4-spinor -> No creation/annihilation Heavy fermions do no appear in virtual loops: Heavy quark determinant can be neglected Path integral measure is unchanged Foldy-Tani-Wouthuysen Transform: Expansion up to inverse rest mass 1/mc2 Forward quarkonia correlator D>(t,R) (gauge invariant) NRQCD heavy quark greens function (2x2)

  6. QM Path Integral picture Barchielli et. al. PRB296 625, 1988 time & x,y,z y´ x´ z1(s) z2(s) y x This is not just the Wilson loop: fluctuating paths Combine the paths from both constituents: To obtain a potential term in the time Hamiltonian operator for D> we need: Express Greens functions as quantum mechanical path integral

  7. The proper static potential time Barchielli et. al. PRB296 625, 1988 R y´ x´ Real-time thermal Wilson loop x,y,z z∞1(s) z∞2(s) Γ0 y x ω0 ρ(ω) = Im[V∞] In the high T region e.g. a Breit Wigner shape The notion of a static potential is valid if the peak structure of ρ(ω,R) is well defined: Use the spectral function (Fourier transform) to obtain: For m∞: Expansion in the velocity: ω = Re[V∞]

  8. Spetral function from LQCD Simulate in LQCD Extraction via MEM , , , ... n‘=0 n‘=2 n‘=1 ρ ρ ω ω Spectral function connects imaginary and Minkowski time Exploring the proper potential Definition of potential requires knowledge about the Wilson Loop spectral function T~TC V∞(R) T<TC V∞(R) R R

  9. Numerical Results: T=0.78TC Numerical Results Quenched QCD Simulations MEM Reconstruction (Bryan+SVD) Confinement region: NX=20 NT=36 β=6.1 ξb=3.2108 Δx = 4Δτ = 0.1fm NC= 1500 HB:OR = 1:5 Nsweep=200 Nτ=36 Iτ=(0,6.1] Prior: m0 1/(ω-ω0+1) Prior dependence: m0={1e-5,..,1e-2} Real Part coincides with the potential from Free Energies in Coulomb Gauge Imaginary part small: possibly artifact from the MEM Anisotropic Wilson Plaquette Action Nω=1500 Iω=[-10,20]

  10. Numerical Results T=2.33TC MEM Reconstruction (Bryan+SVD) Quenched QCD Simulations Deconfinement region: still s(Q)GP NC= 3500 HB:OR = 1:5 Nsweep=200 Δx = 4Δτ = 0.04fm Nτ=12 Iτ=(0,6.1] and Nτ =32 Iτ=(0,7] Prior: m0 1/(ω-ω0+1) NX=20 NT=12 β=6.1 ξb=3.2108 Δx = 4Δτ = 0.1fm NX=20 NT=32 β=7.0 ξb=3.5 Prior dependence: m0={1e-5,..,1e-2} NC= 1100 HB:OR = 1:5 Nsweep=200 Nω=1500 Iω=[-10,20] Anisotropic Wilson Plaquette Action Only solving the Schrödinger equation with both real and imaginary part can tell us about the survival of heavy Quarkonia Imaginary part appears to be finite Real Part much steeper than Free Energies in Coulomb Gauge

  11. Schroedinger Equation Note there is no kinetic term, since the paths collapsed to a straight line We need to systematically expand in the velocity, which is controlled by Work in progress: How to obtain the first correction term v/c from the Wilson loop with electrical field strength insertions. In the static case : Towards finite momentum corrections:

  12. Conclusion Derivation of the 1/m correction to obtain a dynamic Schroedinger equation Light fermions in the medium: Full QCD simulations Either perturbative (HTL) or plagued by ambiguities (free energies, internal energies) Separation of scales used to derive effective theory for Heavy Quarks: NRQCD In the m∞ limit: Spectral function of the thermal Wilson Loop determines the applicability of a potential picture, i.e. existence of V∞(R) In the most naïve case: peak position corresponds to real part, peak width to imaginary part of V(R) Below TC: Proper potential coincides with potential from free energies in Coulomb Gauge Above TC: Real part of the potential much steeper than free energies potential and imaginary part increases significantly Present approaches toward an in-medium potential Proper heavy quark potential at finite T: Numerical results for purely gluonic medium: Future work:

  13. The End Thank you for your attention ご清聴ありがとうございました

  14. Direct Spectral properties D(τ) τ J(x,τ) J†(0,0) τ x,y,z β O(10) O(1000) Quenched QCD Quenched QCD Usual χ2 fitting term How to extract spectral information from Euclidean Lattice data: Maximum Entropy Method Ill posed problem: Cannot use usual χ2 fitting Bayes Theorem can help: Asakawa, Hatsuda, Nakahara: Prog.Part.Nucl.Phys.46:459-508,2001 see also: Nickel, Annals Phys.322:1949-1960,2007 Shannon Janes Entropy, makes prior knowledge explicit

  15. Heavy Quark Potential Models V(R) Confinement: Linear rising below TC String breaking with dynamical fermions Screening above TC due to free color charges e.g. Brown, Weisberger PRD20:3239,1979. R Nf=2+1 Nf=2+1 Nf=2 Phenomenological expectations: Ad-hoc identification of potentials (No Schrödinger equation was derived) At T=0 rigorous result: Petreczky, arXiv:1001.5284v2 Satz, J. Phys. G 36 (2009) 064011

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