JPS08autumnT.Umeda (Tsukuba)1 Thermodynamics at fixed lattice spacing Takashi Umeda (Univ. of Tsukuba) for WHOT-QCD Collaboration JPS meeting, Yamagata Univ., Yamagata, Sept /13 This talk is based on arXiv: [hep-lat]

JPS08autumnT.Umeda (Tsukuba)2 Introduction /13 Equation of State (EOS) is important for phenomenological study of QGP, etc. 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). e.g. Integral method J. Engels et al. (’90). Temperature T=1/(N τ a) is varied by a(β) at fixed N τ Temperature T=1/(N τ a) is varied by a(β) at fixed N τ The EOS calculation requires huge computational cost, in which T=0 calculations dominate despite T>0 study. in which T=0 calculations dominate despite T>0 study. Search for a Line of Constant Physics (LCP) Search for a Line of Constant Physics (LCP) beta functions at each temperature beta functions at each temperature T=0 subtraction at each temperature T=0 subtraction at each temperature

JPS08autumnT.Umeda (Tsukuba)3 T-integration method to calculate the EOS /13 We propose a new method (“T-integration method”) to calculate the EOS at fixed scales (*) Temperature T=1/(N τ a) is varied by N τ at fixed a(β) Temperature T=1/(N τ a) is varied by N τ 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. whose method is based on the derivative method.

JPS08autumnT.Umeda (Tsukuba)4 Notable points in T-integration method /13 Our method can reduce computational cost at T=0 drastically. Zero temperature subtraction is performed Zero temperature subtraction is performed using a common T=0 calculation. using a common T=0 calculation. Line of Constant Physics (LCP) is trivially exact (even in full QCD). Line of Constant Physics (LCP) is trivially exact (even in full QCD). Only the beta functions at the simulation point are required. Only the beta functions at the simulation point are required. However... Temperatures are restricted by integer N τ. Temperatures are restricted by integer N τ.  Sufficiently fine lattice is necessary. Example of Temp. resolution (a=0.07fm) Integer N τ provides - higher resolution at T~T c - higher resolution at T~T c - lower resolution at high T - lower resolution at high T T~T c is important for EOS

JPS08autumnT.Umeda (Tsukuba)5 Simulation parameters (isotropic lattices) /13 We present results from SU(3) gauge theory as a test of our method plaquette gauge action on N σ 3 x N τ lattices plaquette gauge action on N σ 3 x N τ lattices Jackknife analysis with appropriate bin-size Jackknife analysis with appropriate bin-size To study scale- & volume-dependence, we prepare 3-type of lattices. we prepare 3-type of lattices. (1) β=6.0, V=(16a) 3 a=0.094fm a=0.094fm (2) β=6.0, V=(24a) 3 a=0.094fm a=0.094fm (3) β=6.2, V=(22a) 3 a=0.078fm a=0.078fm

JPS08autumnT.Umeda (Tsukuba)6 Simulation parameters (anisotropic lattice) /13 Anisotropic lattice is useful to increase Temp. resolution, we also test our method on an anisotropic lattice a σ ≠ a τ we also test our method on an anisotropic lattice a σ ≠ a τ plaquette gauge action on N σ 3 x N τ lattices plaquette gauge action on N σ 3 x N τ lattices with anisotropy ξ=a σ /a τ =4 β=6.1, ξ=4 V=(20a σ ) 3 V=(20a σ ) 3 =(1.95fm) 3 =(1.95fm) 3 a=0.097fm a=0.097fm - EOS calculation - static quark free energy free energy V=(20a σ ) 3 =(1.95fm) 3 =(1.95fm) 3 V=(30a σ ) 3 =(2.92fm) 3 =(2.92fm) 3 V=(40a σ ) 3 =(3.89fm) 3 =(3.89fm) 3 - critical temp.

JPS08autumnT.Umeda (Tsukuba)7 Trace anomaly ( e - 3p )/T 4 on isotropic lattices /13 beta function : G.Boyd et al. (’96) lattice scale r 0 : R.Edwards et al. (’98) (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 dotted lines : cubic spline Excellent agreement Excellent agreement between (1) and (3) between (1) and (3)  scale violation is small  scale violation is small a=0.1fm is good a=0.1fm is good Finite volume effect Finite volume effect appears below & near T c appears below & near T c  volume size is important  volume size is important V=(2fm) 3 is necessary. V=(2fm) 3 is necessary.

JPS08autumnT.Umeda (Tsukuba)8 Trace anomaly ( e - 3p )/T 4 on aniso. lattice /13 (1) ξ=4, a=0.097fm, V=(2.0fm) 3 (2) ξ=1, a=0.094fm, V=(2.2fm) 3 dotted lines : cubic spline beta function : obtained by r 0 /a σ fit r 0 /a σ data H.Matsufuru et al. (’01) r 0 /a σ data H.Matsufuru et al. (’01) Anisotropic lattice is useful Anisotropic lattice is useful to increase Temp. resolution. to increase Temp. resolution. is required in SU(3) gauge theory. in SU(3) gauge theory. T.R.Klassen (’98)

JPS08autumnT.Umeda (Tsukuba)9 Pressure & Energy density /13 Integration Integration is performed with the cubic is performed with the cubic spline of (e-3p)/T 4 spline of (e-3p)/T 4 Cubic spline vs trapezoidal inte. Cubic spline vs trapezoidal inte. yields small difference ~ 1σ yields small difference ~ 1σ Our results are roughly Our results are roughly consistent with previous results. consistent with previous results. Unlike the fixed N τ approach, Unlike the fixed N τ approach, scale/temp. is not constant. scale/temp. is not constant.  Lattice artifacts increase  Lattice artifacts increase as temperature increases. as temperature increases.

JPS08autumnT.Umeda (Tsukuba)10 Transition temperature at fixed scale /13 T-dependence of the (rotated) Polyakov loop the (rotated) Polyakov loop and its susceptibility and its susceptibility No renormalization is No renormalization is required upto overall factor required upto overall factor due to the fixed scale. due to the fixed scale. Rough estimation of Rough estimation of critical temperature critical temperature is possible. is possible. T c = 280~300 MeV T c = 280~300 MeV at β=6.1, ξ=4 at β=6.1, ξ=4 (SU(3) gauge theory) (SU(3) gauge theory)

JPS08autumnT.Umeda (Tsukuba)11 Static quark free energy at fixed scale /13 Static quark free energies at fixed scale at fixed scale Due to the fixed scale, Due to the fixed scale, no renomalization constant no renomalization constant is required. is required.  small thermal effects in V(r)  small thermal effects in V(r) at short distance at short distance (without any matching) (without any matching) Easy to distinguish Easy to distinguish temperature effect of V(r) temperature effect of V(r) from scale & volume effects from scale & volume effects color singlet static quark free energy V(r)

JPS08autumnT.Umeda (Tsukuba)12 Conclusion /13 We studied thermodynamics of SU(3) gauge theory We studied thermodynamics of SU(3) gauge theory at fixed lattice scale at fixed lattice scale Our method ( T-integration method ) works well Our method ( T-integration method ) works well to calculate the EOS to calculate the EOS Fixed scale approach is also useful for Fixed scale approach is also useful for - critical temperature - critical temperature - static quark free energy - static quark free energy - etc. - etc. Our method is also available in full QCD !! Our method is also available in full QCD !! Therefore...

JPS08autumnT.Umeda (Tsukuba)13 Toward full QCD calculations /13 Our method is suited for Our method is suited for already performed high statistics full QCD results. already performed high statistics full QCD results. When beta functions are (able to be) known at a simulation point When beta functions are (able to be) known at a simulation point and T=0 configurations are open to the public, and T=0 configurations are open to the public, our method requires no additional T=0 simulation !! our method requires no additional T=0 simulation !! We are pushing forward in this direction We are pushing forward in this direction using CP-PACS/JLQCD results in ILDG using CP-PACS/JLQCD results in ILDG (N f =2+1 Clover+RG, a=0.07fm, m ps /m v =0.6) (N f =2+1 Clover+RG, a=0.07fm, m ps /m v =0.6) Our final goal is to study Our final goal is to study thermodynamics on the physical point thermodynamics on the physical point with 2+1 flavors of Wilson quarks  see PACS-CS talks with 2+1 flavors of Wilson quarks  see PACS-CS talks

JPS08autumnT.Umeda (Tsukuba)14 Pressure & Energy density /13

JPS08autumnT.Umeda (Tsukuba)15 Pressure & Energy density /13 G.Boyd et al. (’96)

JPS08autumnT.Umeda (Tsukuba)16 Simulation parameters (isotropic lattices) /13 We present results from SU(3) gauge theory as a test of our method plaquette gauge action on N σ 3 x N τ lattices plaquette gauge action on N σ 3 x N τ lattices Jackknife analysis with appropriate bin-size Jackknife analysis with appropriate bin-size To study scale- & volume-dependence, we prepare 3-type of lattices. we prepare 3-type of lattices.

JPS08autumnT.Umeda (Tsukuba)17 Pressure & Energy density /13 Integration Integration is performed with the cubic is performed with the cubic spline of (e-3p)/T 4 spline of (e-3p)/T 4 Our results are roughly Our results are roughly consistent with previous results. consistent with previous results. -- mild scale violation -- mild scale violation -- Large volume is important -- Large volume is important Unlike the fixed N τ approach, Unlike the fixed N τ approach, scale/temp. is not constant. scale/temp. is not constant.  Lattice artifacts increase  Lattice artifacts increase as temperature increases. as temperature increases.

JPS08autumnT.Umeda (Tsukuba)18 EOS on an anisotropic lattice /13 Anisotropic lattice is useful Anisotropic lattice is useful to increase Temp. resolution. to increase Temp. resolution. Results are roughly consistent Results are roughly consistent with previous & isotropic results with previous & isotropic results Additional coefficients are Additional coefficients are required to calculate (e-3p)/T 4 required to calculate (e-3p)/T 4 is required in SU(3) gauge theory. in SU(3) gauge theory. T.R.Klassen (’98) beta function : obtained by r 0 /a σ fit r 0 /a σ data H.Matsufuru et al. (’01) r 0 /a σ data H.Matsufuru et al. (’01)

JPS08autumnT.Umeda (Tsukuba)19 EOS on an anisotropic lattice /13 Anisotropic lattice is useful Anisotropic lattice is useful to increase Temp. resolution. to increase Temp. resolution. Results are roughly consistent Results are roughly consistent with previous & isotropic results with previous & isotropic results Additional coefficients are Additional coefficients are required to calculate (e-3p)/T 4 required to calculate (e-3p)/T 4 is required in SU(3) gauge theory. in SU(3) gauge theory. T.R.Klassen (’98) beta function : obtained by r 0 /a σ fit r 0 /a σ data H.Matsufuru et al. (’01) r 0 /a σ data H.Matsufuru et al. (’01) G.Boyd et al. (’96)