1. 格子QCDシミュレーションによる QGP媒質中のクォーク間ポテンシャルの研究
WHOT-QCD Collaboration
前沢祐 (理研)
in collaboration with
青木慎也, 金谷和至, 大野浩史 (筑波大物理)
浮田尚哉 (筑波大計算セ)
初田哲男,石井理修 (東京大学)
江尻信司 (BNL)
梅田貴士 (広島大学)
WHOT-QCD Coll. arXive:
WHOT-QCD Coll. PoS LAT2007 (2007) 207
WHOT-QCD Coll. Phys. Rev. D75 (2007)
KEK, 2009年8月11—13日

2. Introduction Study of Quark-Gluon Plasma (QGP)
Early universe after Big Bang
Relativistic heavy-ion collision
Theoretical study based on first principle (QCD)
Lattice QCD simulation at finite (T, mq)
Bulk properties of QGP (p, e, Tc,…) Internal properties of QGP
are well investigated are still uncertain.
Properties of quarks and gluons in QGP
Heavy-quark potential: free energy btw. static charged quarks in QGP
Relation of inter-quark interaction between zero and finite T
Temperature dependence of various color-channels
Heavy-quark potential at finite density (mq) in Taylor expansion method 2

3. Heavy-quark potential at finite T
interaction btw. static quark (Q) and antiquark (Q) in QCD matter
T = 0,
T > 0, string tension (s ) decreases,
string breaking at rc
At T > Tc, screening effect in QGP
Properties of heavy-quark potential in quark-gluon plasma
Heavy-quark bound state (J/y, Υ) in QGP
Screening effect in QGP
Inter-quark interaction btw. QQ and QQ
Lattice QCD simulations 3 3

4. Operator similar to the Wilson-loop
Heavy-quark free energy
To characterize an interaction b/w static quarks,
T = 0
Heavy-quark potential Brown and Weisberger (1979)
Wilson-loop operator
T > 0
Heavy-quark free energy Nadkarni (1986)
Polyakov-line : static quark at x
Correlation b/w Polyakov-lines projected to a color singlet channel in the Coulomb gauge
Operator similar to the Wilson-loop
at finite T
Heavy-quark free energy may behave in QGP as:
short r: F 1(r,T ) ~ V(r) (no effect from thermal medium)
mid r : screened by plasma
long r : two single-quark free energies w/o interactions 4

5. Approaches of lattice simulations at finite T
To vary temperature on the lattice,
a: lattice spacing
Nt: lattice size in E-time direction
Fixed Nt approach
Fixed scale approach WHOT-QCD, PRD 79 (2009)
Changing a (controlled by the gauge coupling (b = 6/g2)), fixing Nt
Advantages: T can be varied arbitrarily, because b is an input parameter.
Changing Nt , fixing a
Advantages: investigation of T dependence w/o changing spatial volume possible.
renormalization factor dose not change when T changes.

6. Results of numerical simulations
Heavy-quark potential at T = 0 and T > 0
Heavy-quark potential for various color-channels
Heavy-quark potential at finite density (mq) 6

7. Simulation details Nf = 2+1 full QCD simulations
Lattice size: Ns3 x Nt = 323 x 12, 10, 8, 6, 4
Temperature: T ~ MeV (5 points)
Lattice spacing: a = 0.07 fm
Light quark mass*: mp/mr = (38)
Strange quark mass*: mK/mK* = (28)
Scale setting: Sommer scale, r0 = 0.5 fm
*CP-PACS & JLQCD Coll., PRD78 (2008) 7

8. Heavy-quark potential at T = 0
Data calculated by
CP-PACS & JLQCD Coll.
on a 283 x 58 lattice
Phenomenological potential
Our fit results
= Coulomb term at short r
+ Linear term at long r
+ const. term 8

9. Heavy-quark free energy at T > 0
At short distance
F 1(r,T ) at any T
converges to
V(r) = F 1(r,T=0 ).
Short distance physics is
insensitive to T.
Note:
In the fixed Nt approach, this property is used to adjust the constant term of F 1(r,T ).
In our fixed scale approach, because the renormalization is common to all T.
no further adjustment of the constant term is necessary.
We can confirm the expected insensitivity!

10. Heavy-quark free energy at T > 0
At short distance
F 1(r,T ) at any T
converges to
V(r) = F 1(r,T=0 ).
Short distance physics is
insensitive to T.
T ~ 200 MeV: F 1 ~ V up to fm
T ~ 700 MeV: F 1 = V even at 0.1 fm
Range of thermal effect is T-dependent
Debye screening effect
WHOT-QCD Coll. arXive:
WHOT-QCD Coll. Phys. Rev. D75 (2007)

11. Heavy-quark free energy at T > 0
2FQ calculated from
Polyakov-line operator
Long distance
F 1(r,T ) becomes flat and has no increase linealy.
Confinement is destroyed due to a thermal medium effect.
At large r, correlation b/w Polyakov-lines will disappear,
F 1 converges to 2x(single-quark free energy) at long distance

12. Results of numerical simulations
Heavy-quark potential at T = 0 and T > 0
Heavy-quark potential for various color-channels
Heavy-quark potential at finite density mq 12 12

13. Heavy-quark potential for various color-channels
Interaction for various color-channels are induced in QGP
Separation to each color channel
Projection of Polyakov-line correlators Nadkarni (1986)
Normalized free energy
QQ correlator: 13

14. Heavy-quark potential for various color-channels
QQ potential QQ potential
Temperature increases becomes weak
1c, 3*c: attractive force, 8c, 6c: repulsive force
Single gluon exchange ansatz
Consistent with Casimir scaling
e.g.)
Casimir factor 14

15. Results of numerical simulations
Heavy-quark potential at T = 0 and T > 0
Heavy-quark potential for various color-channels
Heavy-quark potential at finite density (mq) 15 15

16. Heavy-quark potential at finite mq
Motivation
Taylor expansion method in terms of mq /T
Properties of QGP at 0 < mq /T << 1
Expectation values at finite mq
At mq /T << 1, Taylor expansion of quark determinant detD(mq)
where
Early universe after Big Bang
Relativistic heavy-ion collisions
Low density
e.g. mq /Tc ~ 0.1 at RHIC
c.f.) sign problem
detD becomes complex when mq≠0
As one of the reweighting method, we use the Taylor expansion method with respect to the chemical potential. Let me explain the Taylor expansion method.
The expectation value of observable O at finite \mu is written like this, where the “Delta” is the quark determinant. The Taylor expansion of the quark determinant can be written like this. The expansion coefficients can be expressed here, using the inverse of the quark determinant.
If “O” is independent of \mu, the expectation value can be expanded with respect to \mu, like this, and we can calculate the expansion coefficients from the simulation at zero chemical potential. 16

17. Heavy-quark potential at finite mq
Heavy-quark potential up to 2nd order
with expansion coefficients:
QQ potential (1c, 8c) QQ potential (3*c, 6c)
Odd term of QQ potential vanishes
because of symmetry under mq→ -mq

18. QQ potential 1c channel: attractive force 8c channel: repulsive force
becomes weak
at 0 < mq /T << 1
1c channel: attractive force
8c channel: repulsive force

19. QQ potential 3*c channel: attractive force 6c channel: repulsive force
becomes strong
at 0 < mq /T << 1
3*c channel: attractive force
6c channel: repulsive force

20. Heavy-quark potential at finite mq
QQ potential (1c, 8c)
QQ potential (3*c, 6c)
Attractive channel in heavy-quark potential
Heavy-meson correlation (1c) weak
at 0 < mq /T << 1
Diquark correlation (3*c) strong

21. Summary Heavy-quark potential at T = 0 and T > 0
Properties of quark-gluon plasma
Heavy-quark potential (F 1(r,T )) defined
by free energy btw. static charged quarks
Heavy-quark potential at T = 0 and T > 0
Heavy-quark potential for various color-channels
Heavy-quark potential at finite density (mq)
At short r: F 1 at any T converges to heavy-quark potential at T = 0
Short distance physics is insensitive to T.
At mid r : screening effect appears
At long r: F 1 becomes flat and has no linear behavior.
Correlation b/w Polyakov-lines disappears and F 1 converges to 2FQ
1c, 3*c: attractive force, 8c, 6c: repulsive force
Consistent with Casimir scaling based on single gluon exchange ansatz
Taylor expansion in terms of mq /T
Heavy-meson correlation (1c) weak
Diquark correlation (3*c) strong
at 0 < mq /T << 1