格子QCDによる高温でのクォーク間ポテンシャルの研究

Heavy quark potential at finite temperature

in full QCD lattice simulations

Tsukuba-Tokyo collaboration

前沢 祐 (東京大学)

筑波大学 青木慎也、金谷和至

東京大学 江尻信司、初田哲男、石井理修、浮田尚哉

MotivationsHeavy quark free energy in hot matter

1. Channel dependence of "potential" ( 1c, 8c, 3c, 6c)

2. Effective running coupling at

3. Debye screening mass at

4. Relation to p-QCD at high T

0≠T

0≠T

Full-QCD lattice simulation

We use improved Wilson fermion action.

1. not many works

2. Comparison with staggered fermion action

Free energy and Potential on a lattice

Free energy of the quark-antiquark pairMcLerran, Svetitsky, PRD 24 (1981) 450

)(4 nUtStatic

charged quark

Polyakov loop:

Quark- antiquark “potential” (normalized free energy)

Separation to each channel after Coulomb gauge fixing

Projection operator and 81 PP Nadkarni, PRD 34 (1986) 3904

Quark-quark potential

Numerical simulations

Lattice size:

Gauge action: RG Iwasaki improved action

Fermion action: Clover improved Wilson action (2-flavor)

Quark mass & Temperature

# of Configurations: 500 confs. (5000 traj.)

Gauge fixing: Coulomb gauge

aNTNN

tts

1,41633 =×=×

points) (6 5.20.1 :80.0/points) (9 2.30.1 :65.0/

cc

cc

TTTmmTTTmm

～

～

==

==

ρπ

ρπ

Parameters

cTr

cTTrV ),(

65.0=ρ

πm

m

Quark-antiquark potentialDebye

Screening

1c channel: attractive force8c channel: repulsive force

cTr

cTTrV ),(

65.0=ρ

πm

m

Quark-quark potentialDebye

Screening

3c channel: attractive force6c channel: repulsive force

c.f. Quenched actionNakamura and Saito (2004)

Screening effectPhenomenological potential

: screened Coulomb potential

31)6(,

32)3(,

61)8(,

34)1( =−==−= CCCC

M

aa

a ttMC 28

1 1)( ∑ = ⋅= : Casimir factor

mass screening Debye :)(mcoupling running effective"" :)(

D TTα

Fitting the potentials of each channel with and as free parameters.)( )( TmT Dα

)(Tα TTmD )(

cTT

cTT

Results of and . )( )( TmT Dα

For T > 2.5Tc, potentials of each channel can be written by the same parameters: and .)( )( TmT Dα

Relation between and . )( )( TmT Dα

)(4)(eff TTg πα≡Effective running coupling

TTgTm fND )(1)( eff6+=

?Screening mass

cTT

TTg

TmfND

)(1

)(

eff6+10%

for 1c

potential

is described by effective running coupling with 10% accuracy.

)(TmD )(eff Tg

on a lattice vs. perturbative screening mass )(TmD

2

0

12

02 lnlnln)(

+

=−

SMSMTg

ΛΛµ

ββµβ

MeV 2612 ≈=fNSMΛ

2-loop running coupling

TTT πππµ 3,2,=

Leading order perturbation

cTT

TTmD )(

)(1)(loop-26 Tg

TTm fND +=

Tπ Tπ2 Tπ3

Lattice screening mass

is not reproduced by the

LO-type screening mass.

on a lattice vs. perturbative screening mass )(TmD

Next-to-leading order perturbation Rebhan, PRD 48 (1993) 48

+

−

+++= )(

212ln

11

23)(1)(1)( 2

66 go

mmTgTg

TTm

mag

DN

NNLOD

f

f

π

TTgCm mmag )(2=Magnetic screening mass:

Fitting m )2.3( cD TT =T

TmD )(

cTT

Tπ Tπ2 Tπ3

37.0=mC

Lattice screening mass

is well reproduced by the

NLO-type screening mass

at T > 2Tc.

Comparison with staggered fermion

Kaczmarek and Zantow, PRD 71 (2005) 114510

70.0/,41633 ≈×=× ρπ mmNN tsImproved staggered fermion with

)(TαT

TmD )(

cTT

cTT

Systematic error due to the difference of actions

smaller when )0( →∞→ aT

SummaryHeavy quark "potential"

Lattice QCD simulation using 2-flavor Wilson fermion action

1c, 3c : attractive force 8c, 6c : repulsive force

rTmDerTMCTrV )()()(),( −=

αScreening effect potentials are fitted by

• and are independent on channel.)( )( TmT Dα

TTgTm fND )(1)( eff6+≈ ))(4)(( eff TTg πα≡

effective running coupling

[ ])(1)(1)( 6 TgCTTgTm NLON

Df ++≈

2-loop running coupling

Systematic error due to the difference of actions

smaller when )0( →∞→ aT

Back up Slides

Heavy quark potential at finite temperature

in full QCD lattice simulations

Tsukuba-Tokyo collaboration

Yuu Maezawa (The Univ. of Tokyo)

The Univ. of TokyoUniv. of TsukubaE. EjiriT. HatsudaN. IshiiN. Ukita

S. AokiK. KanayaY. Taniguchi

AbstractWe study the free energies and potentials between static quarks at finite temperature by performing numerical simulations of lattice QCD. The potentials can be separated to each channel: the color singlet QQ channel (1), octet QQ channel (8), anti-triplet QQ channel (3), and sextetQQ channel (6).

The results are fitted by the screened Coulomb form with an effective coupling constant and Debye screening mass above critical temperature. Then, we find that the perturbative form of the heavy quark free energy becomes valid at high temperature.

Furthermore, we discussed the relation between lattice QCD and perturbative QCD, and the Debye screening mass on a lattice is reproduced by the screening mass of next-to-leading order perturbation. The results are also compared with previous results of the staggered fermion action.

Formalism of lattice QCD

)](exp[)( nAiganU µµ = : link variable)3()( SUnU ∈µ

)(nUµ)(nψ)(n

Gluon field

Quark field ψ• Staggered fermion action

• Wilson fermion action

Staggered fermion < Wilson fermion×10

Simulation time

[ ] 4)(Det fNUDStaggered fermion trick

We use improved Wilson fermion action.

Compare with Staggered fermion action

Quark field on a lattice "doubling problem"

1 fermion + 15 degenerated fermionType of action

△ Chiral sym.Wilson

Non-local

Taste dep.

Staggered

(Kogut-Susskind)

fastNo dynamical

quarksQuench

Simulation time

FeatureFor doubling

problemAction

1)(Det =UD

fermions 16

spinor Dirac-4flavor-4×

∞→fermions 15m0at →a 0at ≠a

Extra term

)4( ≠fN ×10

×10

Fit range

Tr

ratio

Debye screening mass

rT

TV1

Fit range 5.16.0 ≤≤ RT