固定格子間隔での有限温度格子 qcd の研究

JPS spring 2012 T. Umeda (Hiroshima) 1

固定格子間隔での有限温度格子固定格子間隔での有限温度格子 QCDQCD の研の研究究

Takashi Umeda (Hiroshima Univ.)Takashi Umeda (Hiroshima Univ.)

for WHOT-QCD Collaborationfor WHOT-QCD Collaboration

JPS meeting, Kwansei-gakuin, Hyogo, 26 Mar. 2012JPS meeting, Kwansei-gakuin, Hyogo, 26 Mar. 2012

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JPS spring 2012 2

Quark Gluon Plasma in Lattice QCD

Quark Gluon Plasma will be Quark Gluon Plasma will be

understood with theoretical models understood with theoretical models

(e.g. hydrodynamic models)(e.g. hydrodynamic models)

which require “physical inputs”which require “physical inputs”

Lattice QCD : first principle calculationLattice QCD : first principle calculation

Phase diagram in (T, Phase diagram in (T, μμ, m, mudud, m, mss))

Transition temperatureTransition temperature

Equation of state ( ε/TEquation of state ( ε/T44, p/T, p/T44,...),...)

etc...etc...http://www.gsi.de/fair/experiments/

T. Umeda (Hiroshima) /13


Choice of quark actions on the lattice

Most (T, μ≠0) studies done with staggerd-type quarksMost (T, μ≠0) studies done with staggerd-type quarks

NNff=2+1, physical quark mass, (μ≠0)=2+1, physical quark mass, (μ≠0)

4th-root trick to remove unphysical “tastes”4th-root trick to remove unphysical “tastes”

non-locality “Validity is not guaranteed”non-locality “Validity is not guaranteed”

It is important to cross-check with It is important to cross-check with

theoretically sound lattice quarks like Wilson-type quarkstheoretically sound lattice quarks like Wilson-type quarks

The objective of WHOT-QCD collaboration is The objective of WHOT-QCD collaboration is

finite T & μ calculations using finite T & μ calculations using Wilson-type quarksWilson-type quarks

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Equation of state in 2+1 flavor QCD with improved Wilson quarks Equation of state in 2+1 flavor QCD with improved Wilson quarks by the fixed scale approachby the fixed scale approachT. UmedaT. Umeda et al. (WHOT-QCD Collab.) [arXiv:1202.4719]et al. (WHOT-QCD Collab.) [arXiv:1202.4719]


Fixed scale approach to study QCD thermodynamics

Temperature Temperature T=1/(NT=1/(Ntta)a) is varied by is varied by NNtt at fixed at fixed aa

AdvantagesAdvantages

- Line of Constant Physics- Line of Constant Physics

- T=0 subtraction for renorm.- T=0 subtraction for renorm. (spectrum study at T=0 )(spectrum study at T=0 )

- Lattice spacing at lower T- Lattice spacing at lower T

- Finite volume effects- Finite volume effects DisadvantagesDisadvantages

- T resolution- T resolution

- High T region - High T region

a : lattice spacinga : lattice spacingNNtt : lattice size in temporal direction : lattice size in temporal direction

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LCP’s in fixed NLCP’s in fixed Ntt approach approach (N(Nff=2 Wilson quarks at N=2 Wilson quarks at Ntt=4)=4)





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lattice spacing at fixed Nlattice spacing at fixed Ntt AdvantagesAdvantages











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spatial volume at fixed Nspatial volume at fixed Ntt AdvantagesAdvantages







NNss for V=(4fm) for V=(4fm)33


Lattice setup

T=0 simulation: on 28T=0 simulation: on 2833 x 56 x 56 by CP-PACS/JLQCDby CP-PACS/JLQCD Phys. Rev. D78 (2008) 011502Phys. Rev. D78 (2008) 011502

- - RG-improved Iwasaki glue + NP-improved Wilson quarksRG-improved Iwasaki glue + NP-improved Wilson quarks

- β=2.05, κ- β=2.05, κudud=0.1356, κ=0.1356, κss=0.1351=0.1351

- V~(2 fm)- V~(2 fm)33 , a~0.07 fm, , a~0.07 fm,

- configurations available on the - configurations available on the ILDG/JLDGILDG/JLDG

T>0 simulations: on 32T>0 simulations: on 3233 x N x Ntt (N (Ntt=4, 6, ..., 14, 16) lattices=4, 6, ..., 14, 16) lattices

RHMC algorithm, same parameters as T=0 simulationRHMC algorithm, same parameters as T=0 simulation

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Beta-functions from CP-PACS+JLQCD results

fit fit ββ,,κκudud,,κκss as functions of as functions of

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Direct fit method Direct fit method

at a LCPat a LCP

LCPLCPscalescale

We estimate a systematic error using We estimate a systematic error using

for scale dependencefor scale dependence


Beta-functions from CP-PACS/JLQCD results

χχ22/dof=5.3/dof=5.3χχ22/dof=1.6/dof=1.6 χχ22/dof=2.1/dof=2.1

Meson spectrum by CP-PACS/JLQCD Meson spectrum by CP-PACS/JLQCD Phys. Rev. D78 (2008) 011502Phys. Rev. D78 (2008) 011502. .

fit fit ββ,,κκudud,,κκss as functions of as functions of

3 (β) x 5 (κ3 (β) x 5 (κudud) x 2 (κ) x 2 (κss) = 30 data points) = 30 data points

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Equation of State in Nf=2+1 QCD

T-integrationT-integration

is performed by Akima Splineis performed by Akima Spline interpolation. interpolation.

ε/Tε/T44 is calculated from is calculated from

Large error in whole T regionLarge error in whole T region

A systematic error due to A systematic error due to beta-functionsbeta-functions

SB limitSB limit

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Polyakov loop and Susceptibility

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Polyakov loop requiresPolyakov loop requires T dependent renormalizationT dependent renormalization

c(T) : additive normalization constantc(T) : additive normalization constant(self-energy of the (anti-)quark sources)(self-energy of the (anti-)quark sources)

: heavy quark free energy: heavy quark free energy


Renormalized Polyakov loop and Susceptibility

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Cheng et al.’s renormalization Cheng et al.’s renormalization [PRD77(2008)014511][PRD77(2008)014511]

matching of V(r) to Vmatching of V(r) to Vstringstring(r) at r=1.5r(r) at r=1.5r00

NNff=0 case=0 case

( c( cmm is identical at a fixed scale ) is identical at a fixed scale )


Summary & outlook

Equation of stateEquation of state

More statistics More statistics are needed in the lower temperature region are needed in the lower temperature region

Results at different scales (β=1.90 by CP-PACS/JLQCD)Results at different scales (β=1.90 by CP-PACS/JLQCD)

NNff=2+1 QCD just at the physical point=2+1 QCD just at the physical point

the physical point (pion mass ~ 140MeV) by PACS-CSthe physical point (pion mass ~ 140MeV) by PACS-CS

beta-functions using reweighting methodbeta-functions using reweighting method

Finite densityFinite density

Taylor expansion method to explore EOS at Taylor expansion method to explore EOS at μμ≠≠00

We presented the EOS and renormalized Polyakov loopWe presented the EOS and renormalized Polyakov loop in Nin Nff=2+1 QCD using improve Wilson quarks=2+1 QCD using improve Wilson quarks

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Renormalized Polyakov loop and Susceptibility

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Cheng et al.’s renormalization Cheng et al.’s renormalization [PRD77(2008)014511][PRD77(2008)014511]

matching of V(r) to Vmatching of V(r) to Vstringstring(r) at r=1.5r(r) at r=1.5r00

V(r) = A –α/r + σrV(r) = A –α/r + σr

VVstringstring(r) = c(r) = cdiff diff -π/12r + σr-π/12r + σr

LLrenren = exp(c = exp(cdiffdiff N Ntt/2)<L>/2)<L>

NNff=0 case=0 case

固定格子間隔での有限温度格子 qcd の研究

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