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.