Alexander Rothkopf
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 201
0年09月25日
Proper Heavy Quark Potentialfrom the Thermal Wilson Loop
21. April 2023
Alexander Rothkopf 21. April 2023 2
Motivation
Temperatures are comparatively small (at least for )
Convenient Separation of Scales
Large mass allows a non-relativistic description:
or
Hadronic Thermometer for the QGP
Matsui & Satz (`86): Complete melting of J/ψ at T=TC
T<TC Hadronic phase
TTC sQGP
T 2TC pQGP
T=TC
T=2TC
Derive an effective Schroedinger equation for Heavy Quarkonium at finite temperature
At m=∞ separation distance of constituents is external parameter
Alexander Rothkopf 21. April 2023 3
Previous Studies
T<TC Hadronic phase
TTC sQGP
T 2TC pQGP
T=TC
T=2TC
T=0 NRQCD & pNRQCD
Expansion in 1/m (Minkowski time):
Remember:
Brambilla et al. Rev.Mod.Phys. 77 (`05)
Potential Models
Ad-hoc identification with Free Energies in Coulomb Gauge
Challenges: Gauge dependence, Entropy contributions
eg. Petreczky et al. arXiv:0904.1748
Hard Thermal Loop
Resummed perturbation theory: applies at very high T
Quarkonia is transient:
Laine et al. JHEP 0703:054,2007
Real-time formulation based on the forward correlator
Alexander Rothkopf 21. April 2023 4
Previous Studies II
T<TC Hadronic phase
TTC sQGP
T 2TC pQGP
T=TC
T=2TC
Spectral functions from LQCD
Extract spectral functions directly from non-perturbativeLattice QCD simulations at any temperature
Asakawa, Hatsuda PRL 92 012001,(‘04)
Asakawa, Hatsuda: Prog.Part.Nucl.Phys.46:459,(‘01); Challenging but well understood: Maximum Entropy Method
Free Energies predict melting at 1.2 TC Unexpected: J/ψ seems to survive up to T > 1.6TC
This talk:
Combine the systematic expansion of the effective theorywith the non-perturbative power of Lattice QCD at T>0
We propose a gauge invariant and non-perturbative definitionof the static proper in-medium heavy quark potential based on the thermal Wilson loop.
Numerical results for the proper potential
Alexander Rothkopf
2x2 sub matrix
NRQCD heavy quark greens function (2x2)
Temperaturedependence
spatial Wilson line
Gamma matrix
21. April 2023 5
Starting point
Forward quarkonia correlator D>(t,R) (gauge invariant)
Foldy-Tani-Wouthuysen Transform: Expansion up to inverse rest mass 1/mc2
No coupling of upper and lower components of Dirac 4-spinor -> No creation/annihilation
Path integral measure is unchanged
Heavy fermions do no appear in virtual loops: Heavy quark determinant can be neglected
time
x,y,z
M†(R,t)
M (R,t’)
x
x´
y
y´
Alexander Rothkopf
This is not just the Wilson loop: fluctuating paths
21. April 2023 6
QM Path Integral picture
Express Greens functions as quantum mechanical path integral
&
Barchielli et. al. PRB296 625, 1988
time
x,y,z
x
x´
y
y´
z1(s)z2(s)
Combine the paths from both constituents:
To obtain a potential term in the time Hamiltonian operator for D> we need:
Alexander Rothkopf
ρ(ω)
ω
Real-time thermal Wilson loop
21. April 2023 7
The proper static potential
Expansion in the velocity:
Use the spectral function (Fourier transform) to obtain:
ω0
Γ0
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:
For m∞: time
x,y,zx
x´
y
y´
z∞1(s
) z∞
2 (s)
= Re[V∞]
= Im[V∞]
Barchielli et. al. PRB296 625, 1988
R
Alexander Rothkopf 21. April 2023 8
Spetral function from LQCD
Definition of potential requires knowledge about the Wilson Loop spectral functionSpectral function connects imaginary and Minkowski time
Extraction via MEM
Simulate in LQCD
Exploring the proper potential
ω
R
ρ
T<TCV∞(R)
ω
R
T~TC
ρ
V∞(R)
, , , ...n‘=1 n‘=2n‘=0
Alexander Rothkopf 21. April 2023 9
Numerical Results: T=0.78TCNumerical Results
Confinement region:
Real Part coincides with the potential from Free Energies in Coulomb Gauge
Imaginary part small: possibly artifact from the MEM
Quenched QCD Simulations
Anisotropic Wilson Plaquette Action
NX=20 NT=36 β=6.1 ξb=3.2108
Δx = 4Δτ = 0.1fm
NC= 1500 HB:OR = 1:5 Nsweep=200
MEM Reconstruction (Bryan+SVD)
Nω=1500 Iω=[-10,20]
Nτ=36 Iτ=(0,6.1]
Prior: m0 1/(ω-ω0+1)
Prior dependence: m0={1e-5,..,1e-2}
Alexander Rothkopf 21. April 2023 10
Numerical Results T=2.33TC
Deconfinement region: still s(Q)GP
Real Part much steeper than Free Energies in Coulomb Gauge
Imaginary part appears to be finite
Only solving the Schrödinger equation with both real and imaginary part can tell us about the survival of heavy Quarkonia
MEM Reconstruction (Bryan+SVD)
Nω=1500 Iω=[-10,20]
Nτ=12 Iτ=(0,6.1] and Nτ =32 Iτ=(0,7]
Prior: m0 1/(ω-ω0+1)
Prior dependence: m0={1e-5,..,1e-2}
Quenched QCD SimulationsAnisotropic Wilson Plaquette Action
NX=20 NT=12 β=6.1 ξb=3.2108Δx = 4Δτ = 0.1fm
NC= 1100 HB:OR = 1:5 Nsweep=200
NX=20 NT=32 β=7.0 ξb=3.5Δx = 4Δτ = 0.04fm
NC= 3500 HB:OR = 1:5 Nsweep=200
Alexander Rothkopf 21. April 2023 11
Schroedinger Equation
In the static case :
Note there is no kinetic term, since the paths collapsed to a straight line
Towards finite momentum corrections:
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.
Alexander Rothkopf 21. April 2023 12
Conclusion
Present approaches toward an in-medium potential
Either perturbative (HTL) or plagued by ambiguities (free energies, internal energies)
Proper heavy quark potential at finite T:
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)
Numerical results for purely gluonic medium:
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
Future work:Derivation of the 1/m correction to obtain a dynamic Schroedinger equation
Light fermions in the medium: Full QCD simulations
Alexander Rothkopf
The End
Thank you for your attention
ご清聴ありがとうございました
21. April 2023 13
Alexander Rothkopf
Asakawa, Hatsuda, Nakahara: Prog.Part.Nucl.Phys.46:459-508,2001see also: Nickel, Annals Phys.322:1949-1960,2007
Quenched QCD
Quenched QCD
Shannon Janes Entropy, makes prior knowledge explicit
Usual χ2 fitting term
O(10) O(1000)
Direct Spectral properties
How to extract spectral information from Euclidean Lattice data: Maximum Entropy Method
Ill posed problem: Cannot use usual χ2 fitting
Bayes Theorem can help:
D(τ)
τβ
τ J(x,τ)
J†(0,0)
x,y,z
21. April 2023 14
Alexander Rothkopf
Satz, J. Phys. G 36 (2009) 064011Petreczky, arXiv:1001.5284v2
Nf=2Nf=2+1Nf=2+1
Heavy Quark Potential Models
Phenomenological expectations: V(R)
R
Confinement: Linear rising below TC
String breaking with dynamical fermionsScreening above TC due to free color charges
Ad-hoc identification of potentials (No Schrödinger equation was derived)
21. April 2023 15
At T=0 rigorous result: e.g. Brown, Weisberger PRD20:3239,1979.