Proper Hevay Quark Potential from Lattice QCD

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
Alexander Rothkopf
1. Oktober 2015
Motivation
Hadronic Thermometer for the QGP
Matsui & Satz (`86): Complete melting of J/ψ at T=TC
T
2TC pQGP
Convenient Separation of Scales
T=2TC
Large mass allows a non-relativistic description:
or
Temperatures are comparatively small (at least for
T TC sQGP
)
T=TC
Derive an effective Schroedinger equation for Heavy
Quarkonium at finite temperature
T<TC Hadronic phase
Alexander Rothkopf
At m=∞ separation distance of constituents is external parameter
1. Oktober 2015
2
Previous Studies
Hard Thermal Loop
Laine et al. JHEP 0703:054,2007
Real-time formulation based on the forward correlator
Resummed perturbation theory: applies at very high T
T
Quarkonia is transient:
2TC pQGP
T=2TC
Potential Models
eg. Petreczky et al. arXiv:0904.1748
Ad-hoc identification with Free Energies in Coulomb Gauge
T TC sQGP
Challenges: Gauge dependence, Entropy contributions
T=TC
T=0 NRQCD & pNRQCD
Brambilla et al. Rev.Mod.Phys. 77 (`05)
Expansion in 1/m (Minkowski time):
T<TC Hadronic phase
Alexander Rothkopf
Remember:
1. Oktober 2015
3
Previous Studies II
Spectral functions from LQCD
Asakawa, Hatsuda PRL 92 012001,(‘04)
Extract spectral functions directly from non-perturbative
Lattice QCD simulations at any temperature
T
Challenging but well understood: Maximum Entropy Method
2TC pQGP
Asakawa, Hatsuda: Prog.Part.Nucl.Phys.46:459,(‘01);
T=2TC
Unexpected: J/ψ seems to survive up to T > 1.6TC
Free Energies predict melting at 1.2 TC
This talk:
T TC sQGP
Combine the systematic expansion of the effective theory
with the non-perturbative power of Lattice QCD at T>0
T=TC
We propose a gauge invariant and non-perturbative definition
of the static proper in-medium heavy quark potential based
on the thermal Wilson loop.
Numerical results for the proper potential
T<TC Hadronic phase
Alexander Rothkopf
1. Oktober 2015
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Starting point
Forward quarkonia correlator D>(t,R) (gauge invariant)
time
M (R,t’)
x´
Gamma matrix
y´
spatial Wilson line
M†(R,t)
x
y
x,y,z
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
Heavy fermions do no appear in virtual loops: Heavy quark determinant can be neglected
Path integral measure is unchanged
2x2 sub matrix
Temperature
dependence
NRQCD heavy quark greens function (2x2)
Alexander Rothkopf
1. Oktober 2015
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QM Path Integral picture
Express Greens functions as quantum mechanical path integral
Barchielli et. al. PRB296 625, 1988
&
Combine the paths from both constituents:
time
x´
z1(s)
x
This is not just the Wilson loop: fluctuating paths
y´
z2(s)
y
x,y,z
To obtain a potential term in the time Hamiltonian operator for D> we need:
Alexander Rothkopf
1. Oktober 2015
6
The proper static potential
Expansion in the velocity:
Barchielli et. al. PRB296 625, 1988
For m ∞:
time
x´
z∞1(s)
Use the spectral function (Fourier transform) to obtain:
x
y´
z∞2(s)
Real-time thermal Wilson loop
R
y
x,y,z
The notion of a static potential is valid if the peak structure of ρ(ω,R) is well defined:
ρ(ω)
Γ0 = Im[V∞]
In the high T region e.g. a Breit Wigner shape
ω0 = Re[V∞]
Alexander Rothkopf
1. Oktober 2015
ω
7
Spetral function from LQCD
Definition of potential requires knowledge about the Wilson Loop spectral function
Spectral function connects imaginary and Minkowski time
Simulate in LQCD
Extraction via MEM
Exploring the proper potential
ρ
ρ
ω
T<TC
ω
T~TC
V∞(R)
R
R
V∞(R)
,
n‘=0
Alexander Rothkopf
1. Oktober 2015
,
n‘=1
, ...
n‘=2
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Numerical Results
Results: T=0.78TC
Confinement region:
Quenched
MEM
Reconstruction
QCD Simulations
(Bryan+SVD)
Anisotropic
N
Wilson Plaquette Action
ω=1500 Iω=[-10,20]
NX=20
N
NT=36 β=6.1 ξb=3.2108
τ=36 Iτ=(0,6.1]
Δx = 4Δτ
Prior:
m0=1/(ω-ω
0.1fm 0+1)
NC= 1500
Prior
dependence:
HB:OR = m
1:50={1e-5,..,1e-2}
Nsweep=200
Real Part coincides with the potential from Free Energies
in Coulomb Gauge
Imaginary part small: possibly artifact from the MEM
Alexander Rothkopf
1. Oktober 2015
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Numerical Results T=2.33TC
Deconfinement region: still s(Q)GP
Quenched QCD Simulations
Anisotropic
Wilson(Bryan+SVD)
Plaquette Action
MEM
Reconstruction
NX=20
β=6.1 ξb=3.2108
Nω=1500NT=12
Iω=[-10,20]
Δx
= 4Δτ
= 0.1fmand N =32 I =(0,7]
Nτ=12
Iτ=(0,6.1]
τ
τ
NC= 3500 HB:OR = 1:5 Nsweep=200
Prior: m0 1/(ω-ω0+1)
NX=20dependence:
Prior
NT=32 β=7.0
m0={1e-5,..,1e-2}
ξb=3.5
Δx = 4Δτ = 0.04fm
NC= 1100 HB:OR = 1:5 Nsweep=200
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
Alexander Rothkopf
1. Oktober 2015
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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
1. Oktober 2015
11
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
1. Oktober 2015
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The End
Thank you for your attention
ご清聴ありがとうございました
Alexander Rothkopf
1. Oktober 2015
13
Direct Spectral properties
How to extract spectral information from Euclidean Lattice data: Maximum Entropy Method
τ
D(τ)
J(x,τ)
J†(0,0)
x,y,z
O(10)
Ill posed problem: Cannot use usual
β
τ
O(1000)
χ2
fitting
Quenched QCD
Bayes Theorem can help:
Quenched QCD
Usual χ2 fitting term
Shannon Janes Entropy, makes prior knowledge explicit
Alexander Rothkopf
1. Oktober 2015
Asakawa, Hatsuda, Nakahara: Prog.Part.Nucl.Phys.46:459-508,2001
see also: Nickel, Annals Phys.322:1949-1960,2007
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Heavy Quark Potential Models
V(R)
Phenomenological expectations:
Confinement: Linear rising below TC
String breaking with dynamical fermions
R
Screening above TC due to free color charges
At T=0 rigorous result:
e.g. Brown, Weisberger PRD20:3239,1979.
Ad-hoc identification of potentials (No Schrödinger equation was derived)
Nf=2+1
Nf=2+1
Petreczky, arXiv:1001.5284v2
Alexander Rothkopf
1. Oktober 2015
Nf=2
Satz, J. Phys. G 36 (2009) 064011
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