Thermodynamics at fixed lattice spacing Takashi Umeda Univ

Thermodynamics at fixed lattice spacing Takashi Umeda (Univ. of Tsukuba) for WHOT-QCD Collaboration This talk is based on ar. Xiv: 0809. 2842 [hep-lat] JPS meeting, Yamagata Univ. , Yamagata , 20 -23 Sept. 2008 JPS 08 autumn T. Umeda (Tsukuba) 1 /13

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/(Nτa) is varied by a(β) at fixed Nτ The EOS calculation requires huge computational cost, in which T=0 calculations dominate despite T>0 study. JPS 08 autumn n Search for a Line of Constant Physics (LCP) n beta functions at each temperature n T=0 subtraction at each temperature T. Umeda (Tsukuba) 2 /13

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/(Nτa) is varied by Nτ at fixed a(β) Our method is based on the trace anomaly (interaction measure), and thermodynamic relation. (*) fixed scale approach has been adopted in L. Levkova et al. (’ 06) whose method is based on the derivative method. JPS 08 autumn T. Umeda (Tsukuba) 3 /13

Notable points in T-integration method Our method can reduce computational cost at T=0 drastically. n Zero temperature subtraction is performed using a common T=0 calculation. n Line of Constant Physics (LCP) is trivially exact (even in full QCD). n Only the beta functions at the simulation point are required. However. . . n Temperatures are restricted by integer Nτ. Sufficiently fine lattice is necessary. Example of Temp. resolution (a=0. 07 fm) Integer Nτ provides - higher resolution at T~Tc - lower resolution at high T T~Tc is important for EOS JPS 08 autumn T. Umeda (Tsukuba) 4 /13

Simulation parameters (isotropic lattices) We present results from SU(3) gauge theory as a test of our method n plaquette gauge action on Nσ3 x Nτ lattices n Jackknife analysis with appropriate bin-size To study scale- & volume-dependence, we prepare 3 -type of lattices. (1) β=6. 0, V=(16 a)3 a=0. 094 fm JPS 08 autumn (2) β=6. 0, V=(24 a)3 a=0. 094 fm T. Umeda (Tsukuba) (3) β=6. 2, V=(22 a)3 a=0. 078 fm 5 /13

Simulation parameters (anisotropic lattice) Anisotropic lattice is useful to increase Temp. resolution, we also test our method on an anisotropic lattice a σ≠ aτ n plaquette gauge action on Nσ3 x Nτ lattices with anisotropy ξ=aσ/aτ=4 β=6. 1, ξ=4 V=(20 aσ)3 =(1. 95 fm)3 a=0. 097 fm - EOS calculation - static quark free energy JPS 08 autumn T. Umeda (Tsukuba) V=(20 aσ)3 =(1. 95 fm)3 V=(30 aσ)3 =(2. 92 fm)3 V=(40 aσ)3 =(3. 89 fm)3 - critical temp. 6 /13

Trace anomaly ( e - 3 p )/T 4 on isotropic lattices (1) (2) (3) β=6. 0, β=6. 2, a=0. 094 fm, a=0. 068 fm, V=(1. 5 fm)3 V=(2. 2 fm)3 V=(1. 5 fm)3 beta function : G. Boyd et al. (’ 96) lattice scale r 0 : R. Edwards et al. (’ 98) n Excellent agreement between (1) and (3) scale violation is small a=0. 1 fm is good n Finite volume effect appears below & near Tc volume size is important V=(2 fm)3 is necessary. dotted lines : cubic spline JPS 08 autumn T. Umeda (Tsukuba) 7 /13

Trace anomaly ( e - 3 p )/T 4 on aniso. lattice (1) ξ=4, a=0. 097 fm, V=(2. 0 fm)3 (2) ξ=1, a=0. 094 fm, V=(2. 2 fm)3 beta function : obtained by r 0/aσ fit r 0/aσdata H. Matsufuru et al. (’ 01) is required in SU(3) gauge theory. T. R. Klassen (’ 98) n Anisotropic lattice is useful to increase Temp. resolution. dotted lines : cubic spline JPS 08 autumn T. Umeda (Tsukuba) 8 /13

Pressure & Energy density n Integration is performed with the cubic spline of (e-3 p)/T 4 n Cubic spline vs trapezoidal inte. yields small difference ~ 1σ n Our results are roughly consistent with previous results. n Unlike the fixed Nτ approach, scale/temp. is not constant. Lattice artifacts increase as temperature increases. JPS 08 autumn T. Umeda (Tsukuba) 9 /13

Transition temperature at fixed scale T-dependence of the (rotated) Polyakov loop and its susceptibility n No renormalization is required upto overall factor due to the fixed scale. n Rough estimation of critical temperature is possible. Tc = 280~300 Me. V at β=6. 1, ξ=4 (SU(3) gauge theory) JPS 08 autumn T. Umeda (Tsukuba) 10 /13

Static quark free energy at fixed scale color singlet static quark free energy V(r) Static quark free energies at fixed scale n Due to the fixed scale, no renomalization constant is required. small thermal effects in V(r) at short distance (without any matching) n Easy to distinguish temperature effect of V(r) from scale & volume effects JPS 08 autumn T. Umeda (Tsukuba) 11 /13

Conclusion n We studied thermodynamics of SU(3) gauge theory at fixed lattice scale n Our method ( T-integration method ) works well to calculate the EOS n Fixed scale approach is also useful for - critical temperature - static quark free energy - etc. n Our method is also available in full QCD !! Therefore. . . JPS 08 autumn T. Umeda (Tsukuba) 12 /13

Toward full QCD calculations n Our method is suited for already performed high statistics full QCD results. n 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 !! n We are pushing forward in this direction using CP-PACS/JLQCD results in ILDG (Nf=2+1 Clover+RG, a=0. 07 fm, mps/mv=0. 6) n Our final goal is to study thermodynamics on the physical point with 2+1 flavors of Wilson quarks see PACS-CS talks JPS 08 autumn T. Umeda (Tsukuba) 13 /13

Pressure & Energy density JPS 08 autumn T. Umeda (Tsukuba) 14 /13

Pressure & Energy density G. Boyd et al. (’ 96) JPS 08 autumn T. Umeda (Tsukuba) 15 /13

Simulation parameters (isotropic lattices) We present results from SU(3) gauge theory as a test of our method n plaquette gauge action on Nσ3 x Nτ lattices n Jackknife analysis with appropriate bin-size To study scale- & volume-dependence, we prepare 3 -type of lattices. JPS 08 autumn T. Umeda (Tsukuba) 16 /13

Pressure & Energy density n Integration is performed with the cubic spline of (e-3 p)/T 4 n Our results are roughly consistent with previous results. -- mild scale violation -- Large volume is important n Unlike the fixed Nτ approach, scale/temp. is not constant. Lattice artifacts increase as temperature increases. JPS 08 autumn T. Umeda (Tsukuba) 17 /13

EOS on an anisotropic lattice beta function : obtained by r 0/aσ fit r 0/aσdata H. Matsufuru et al. (’ 01) n Anisotropic lattice is useful to increase Temp. resolution. n Results are roughly consistent with previous & isotropic results n Additional coefficients are required to calculate (e-3 p)/T 4 is required in SU(3) gauge theory. T. R. Klassen (’ 98) JPS 08 autumn T. Umeda (Tsukuba) 18 /13

EOS on an anisotropic lattice beta function : obtained by r 0/aσ fit r 0/aσdata H. Matsufuru et al. (’ 01) G. Boyd et al. (’ 96) n Anisotropic lattice is useful to increase Temp. resolution. n Results are roughly consistent with previous & isotropic results n Additional coefficients are required to calculate (e-3 p)/T 4 is required in SU(3) gauge theory. T. R. Klassen (’ 98) JPS 08 autumn T. Umeda (Tsukuba) 19 /13