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QCD Thermodynamics at fixed lattice scale

QCD Thermodynamics at fixed lattice scale. Takashi Umeda (Univ. of Tsukuba) for WHOT-QCD Collaboration. This talk is based on arXiv:0809.2842 [hep-lat] T. Umeda, S. Ejiri, S. Aoki, T. Hatsuda, K. Kanaya,Y. Maezawa, H. Ohno (WHOT-QCD Collaboration).

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QCD Thermodynamics at fixed lattice scale

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  1. QCD Thermodynamics at fixed lattice scale Takashi Umeda (Univ. of Tsukuba) for WHOT-QCD Collaboration This talk is based on arXiv:0809.2842 [hep-lat] T. Umeda, S. Ejiri, S. Aoki, T. Hatsuda, K. Kanaya,Y. Maezawa, H. Ohno (WHOT-QCD Collaboration) ATHIC2008, Univ. of Tsukuba, Ibaraki, Japan, 13-15 Oct. 2008 T.Umeda (Tsukuba) /15

  2. Introduction • Equation of State (EOS) • is important for phenomenological study of QGP, etc. • Methods to calculate the EOS have been established, • e.g. Integral method J. Engels et al. (’90). • Temperature T=1/(Nta) is varied by a(β)at fixed Nt • The EOS calculation requires huge computational cost, • in which T=0 calculations dominate despite T>0 study. • Search for a Line of Constant Physics (LCP) • beta functions at each temperature • T=0 subtraction at each temperature T.Umeda (Tsukuba) /15

  3. Recent lattice calculations for Tc RBC-Bielefeld: Nt=4,6,8 Staggered (p4) quark pion mass ≥ 140MeV, Nf=2+1 MILC: Nt=4,6,8 Staggered (Asqtad) quark pion mass ≥ 220MeV, Nf=2+1 Wuppertal: Nt=4,6,8,10 Staggered (stout) quark pion mass ~ 140MeV, Nf=2+1 DIK: Nt=8,10,12 Wilson (NPI Clover) quark pion mass ≥ 500MeV, Nf=2 WHOT-QCD: Nt=4,6 Wilson (MFI Clover) quark pion mass ≥ 500MeV, Nf=2 T.Umeda (Tsukuba) /15

  4. Recent lattice calculations for EOS RBC-Bielefeld: Nt=4,6,8 Staggered (p4) quark pion mass ~ 220MeV, Nf=2+1 MILC: Nt=4,6,8 Staggered (Asqtad) quark pion mass ~ 220MeV, Nf=2+1 Wuppertal: Nt=4,6 Staggered (stout) quark pion mass ~ 140MeV, Nf=2+1 CP-PACS: Nt=4,6 Wilson (MFI Clover) quark pion mass ~ 500MeV, Nf=2 There are problems in Staggered quark formulations - Flavor symmetry violation - Rooted Dirac operator - etc. Wilson types quark results are important !!! T.Umeda (Tsukuba) /15

  5. T-integration method to calculate the EOS We propose a new method (“T-integration method”) to calculate the EOS at fixed scales (*) Temperature T=1/(Nta) is varied by Nt at fixed a(β) Our method is based on the trace anomaly (interaction measure), and the thermodynamic relation. (*) fixed scale approach has been adopted in L.Levkova et al. (’06) whose method is based on the derivative method. T.Umeda (Tsukuba) /15

  6. Notable points in T-integration method • Our method can reduce computational cost at T=0 drastically. • Zero temperature subtraction is performed • using a common T=0 calculation. • Line of Constant Physics (LCP) is trivially exact (even in full QCD). • Only the beta functions at the simulation point are required. • However ... • Temperatures are restricted by integer Nt. •  Sufficiently fine lattice is necessary. Example of Temp. resolution (a=0.07fm) Integer Nt provides - higher resolution at T~Tc - lower resolution at high T T~Tc is important for EOS T.Umeda (Tsukuba) /15

  7. Simulation parameters (isotropic lattices) We present results from SU(3) gauge theory as a test of our method • plaquette gauge action on Ns3 x Nt lattices • Jackknife analysis with appropriate bin-size To study scale- & volume-dependence, we prepare 3-type of lattices. (1)β=6.0, V=(16a)3 a=0.094fm (2)β=6.0, V=(24a)3 a=0.094fm (3)β=6.2, V=(22a)3 a=0.078fm T.Umeda (Tsukuba) /15

  8. Simulation parameters (anisotropic lattice) Anisotropic lattice is useful to increase Temp. resolution, we also test our method on an anisotropic lattice as≠ at • plaquette gauge action on Ns3 x Nt lattices • with anisotropy ξ=as/at=4 V=(20as)3 =(1.95fm)3 V=(30as)3 =(2.92fm)3 V=(40as)3 =(3.89fm)3 - critical temp. β=6.1, ξ=4 V=(20as)3 =(1.95fm)3 as=0.097fm - EOS calculation - static quark free energy T.Umeda (Tsukuba) /15

  9. Trace anomaly ( e - 3p )/T4 on isotropic lattices (1) β=6.0, a=0.094fm, V=(1.5fm)3 (2) β=6.0, a=0.094fm, V=(2.2fm)3 (3) β=6.2, a=0.068fm, V=(1.5fm)3 beta function : G.Boyd et al. (’96) lattice scale r0 : R.Edwards et al. (’98) • Excellent agreement • between (1) and (3) •  scale violation is small • a=0.1fm is good • Finite volume effect • appears below & near Tc •  volume size is important • V=(2fm)3 is necessary. dotted lines : cubic spline T.Umeda (Tsukuba) /15

  10. is required in SU(3) gauge theory. T.R.Klassen (’98) Trace anomaly ( e - 3p )/T4 on aniso. lattice (1) ξ=4, as=0.097fm, V=(2.0fm)3 (2) ξ=1, a=0.094fm, V=(2.2fm)3 beta function : obtained by r0/as fit r0/asdata H.Matsufuru et al. (’01) • Anisotropic lattice is useful • to increase Temp. resolution. dotted lines : cubic spline T.Umeda (Tsukuba) /15

  11. Pressure & Energy density • Integration • is performed with the cubic • spline of (e-3p)/T4 • Cubic spline vs trapezoidal inte. • yields small difference ~ 1σ • Our results are roughly • consistent with previous results. • Unlike the fixed Nt approach, scale/temp. is not constant. •  Lattice artifacts increase • as temperature increases. T.Umeda (Tsukuba) /15

  12. Transition temperature at fixed scale • T-dependence of • the (rotated) Polyakov loop • and its susceptibility • No renormalization is • required upto overall factor • due to the fixed scale. • Rough estimation of • critical temperature • is possible. • Tc = 280~300 MeV • at β=6.1, ξ=4 • (SU(3) gauge theory) T.Umeda (Tsukuba) /15

  13. Static quark free energy at fixed scale Static quark free energies at fixed scale color singlet static quark free energy V(r) • Due to the fixed scale, • no renomalization constant • is required. •  small thermal effects in V(r) • at short distance • (without any matching) • Easy to distinguish • temperature effect of V(r) • from scale & volume effects T.Umeda (Tsukuba) /15

  14. Conclusion • We studied thermodynamics of SU(3) gauge theory at fixed lattice scale • Our method ( T-integration method ) works well to calculate the EOS • Fixed scale approach is also useful for - critical temperature - static quark free energy - etc. • Our method is also available in full QCD !! Therefore ... T.Umeda (Tsukuba) /15

  15. Toward full QCD calculations • Our method is suited for • already performed high statistics full QCD results. • When beta functions are (able to be) known at a simulation point • and T=0 configurations are open to the public, • our method requires no additional T=0 simulation !! • We are pushing forward in this direction • using CP-PACS/JLQCD results in ILDG • (Nf=2+1 Clover+RG, a=0.07fm, pion mass ~ 500MeV) • Our final goal is to study • thermodynamics on the physical point (pion mass ~ 140MeV) • with 2+1 flavors of Wilson quarks T.Umeda (Tsukuba) /15

  16. Pressure & Energy density T.Umeda (Tsukuba) /15

  17. G.Boyd et al. (’96) Pressure & Energy density T.Umeda (Tsukuba) /15

  18. Simulation parameters (isotropic lattices) We present results from SU(3) gauge theory as a test of our method • plaquette gauge action on Nσ3 x Nτ lattices • Jackknife analysis with appropriate bin-size To study scale- & volume-dependence, we prepare 3-type of lattices. T.Umeda (Tsukuba) /15

  19. Pressure & Energy density • Integration • is performed with the cubic • spline of (e-3p)/T4 • Our results are roughly • consistent with previous results. -- mild scale violation -- Large volume is important • Unlike the fixed Nτ approach, scale/temp. is not constant. •  Lattice artifacts increase • as temperature increases. T.Umeda (Tsukuba) /15

  20. EOS on an anisotropic lattice beta function : obtained by r0/aσ fit r0/aσdata H.Matsufuru et al. (’01) • Anisotropic lattice is useful • to increase Temp. resolution. • Results are roughly consistent • with previous & isotropic results • Additional coefficients are • required to calculate (e-3p)/T4 is required in SU(3) gauge theory. T.R.Klassen (’98) T.Umeda (Tsukuba) /15

  21. EOS on an anisotropic lattice beta function : obtained by r0/aσ fit r0/aσdata H.Matsufuru et al. (’01) G.Boyd et al. (’96) • Anisotropic lattice is useful • to increase Temp. resolution. • Results are roughly consistent • with previous & isotropic results • Additional coefficients are • required to calculate (e-3p)/T4 is required in SU(3) gauge theory. T.R.Klassen (’98) T.Umeda (Tsukuba) /15

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