1. JPS autumn 2010T. Umeda (Hiroshima)1 ウィルソンクォークを用いた N f =2+1 QCD の状態方程式の研究 Takashi Umeda (Hiroshima Univ.) for WHOT-QCD Collaboration JPS meeting, Kyushu-koudai, Fukuoka, 13 Sep. 2010 /12

2. JPS autumn 2010 2 Quark Gluon Plasma in Lattice QCD from the Phenix group web-site Observables in Lattice QCD Observables in Lattice QCD Phase diagram in (T, μ, m ud, m s ) Phase diagram in (T, μ, m ud, m s ) Transition temperature Transition temperature Equation of state ( ε/T 4, p/T 4,...) Equation of state ( ε/T 4, p/T 4,...) Hadronic excitations Hadronic excitations Transport coefficients Transport coefficients Finite chemical potential Finite chemical potential etc... etc... http://www.gsi.de/fair/experiments/ T. Umeda (Hiroshima) /12

3. JPS autumn 2010T. Umeda (Hiroshima)3 Choice of quark actions on the lattice Most (T, μ≠0) studies done with staggerd-type quarks less computational costs less computational costs a part of chiral sym. preserved... a part of chiral sym. preserved...  N f =2+1, almost physical quark mass, (μ≠0)  N f =2+1, almost 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 theoretically sound lattice quarks like Wilson-type quarks Our aim is to investigate QCD Thermodynamics with Wilson-type quarks QCD Thermodynamics with Wilson-type quarks WHOT-QCD Collaboration /12

4. JPS autumn 2010T. Umeda (Hiroshima)4 Fixed scale approach to study QCD thermodynamics Temperature T=1/(N t a) is varied by N t at fixed a Advantages Advantages - 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 ) - larger 1/a in whole T region - larger 1/a in whole T region Disadvantages Disadvantages - T resolution by integer N t - T resolution by integer N t - UV cutoff eff. at high T - UV cutoff eff. at high T a : lattice spacing N t : lattice size in temporal direction Temperatures in each approach /12 fine ~ coarse fixed N t approach fixed scale approach

5. JPS autumn 2010T. Umeda (Hiroshima)5 T=0 & T>0 configurations for N f =2+1 QCD T=0 simulation: on 28 3 x 56 by CP-PACS/JLQCD Phys. Rev. D78 (2008) 011502 T=0 simulation: on 28 3 x 56 by CP-PACS/JLQCD Phys. Rev. D78 (2008) 011502 - RG-improved Iwasaki glue + NP-improved Wilson quarks - β=2.05, κ ud =0.1356, κ s =0.1351 - V~(2 fm) 3, a=0.07 fm, - configurations available on the ILDG/JLDG T>0 simulations: on 32 3 x N t (N t =4, 6,..., 14, 16) lattices T>0 simulations: on 32 3 x N t (N t =4, 6,..., 14, 16) lattices RHMC algorithm, same parameters as T=0 simulation /12

6. JPS autumn 2010T. Umeda (Hiroshima)6 Formulation for N f =2+1 improved Wilson quarks Noise method ( #noise = 1 for each color & spin indices ) Phys. Rev. D73, 034501 CP-PACS/JLQCD /12

7. JPS autumn 2010T. Umeda (Hiroshima)7 Beta-functions from CP-PACS/JLQCD results Trace anomaly needs Beta-functions in N f =2+1 QCD Direct fit method Phys. Rev. D64 (2001) 074510 fit β,κ ud,κ s as functions of with fixed /12

8. JPS autumn 2010T. Umeda (Hiroshima)8 Beta-functions from CP-PACS/JLQCD results χ 2 /dof=5.3 χ 2 /dof=1.6 χ 2 /dof=2.1 Meson spectrum by CP-PACS/JLQCD Phys. Rev. D78 (2008) 011502. fit β,κ ud,κ s as functions of 3 (β) x 5 (κ ud ) x 2 (κ s ) = 30 data points only statistical error /12

9. JPS autumn 2010T. Umeda (Hiroshima)9 Trace anomaly in N f =2+1 QCD Nt config. (x 5MD traj.) Nt config. (x 5MD traj.) S g S q (**) S g S q (**) 56 1300 (*) 980 56 1300 (*) 980 16 1542 647 16 1542 647 14 1448 647 14 1448 647 12 1492 695 12 1492 695 10 863 487 10 863 487 8 628 520 8 628 520 6 657 360 6 657 360 4 802 295 4 802 295 (*) T=0 (Nt=56) by CP-PACS/JLQCD S g calculated with 6500traj. S g calculated with 6500traj. (**) thermal. = 1000 traj. Preliminary /12

10. JPS autumn 2010T. Umeda (Hiroshima)10 Trace anomaly in N f =2+1 QCD HotQCD,arXiv:1005.1131 HISQ stout peak height = 4~6 peak height = 4~6 in recent Staggered results in recent Staggered results ( m q ~ m q phys. ) ( m q ~ m q phys. ) Preliminary /12 S. Borsanyi et al., arXiv:1007.2580 WHOT-QCD Collaboration

11. JPS autumn 2010T. Umeda (Hiroshima)11 Equation of State in Nf=2+1 QCD Preliminary T-integration T-integration is performed by the trapezoidal is performed by the trapezoidal rule (straight line interpolation). rule (straight line interpolation). ε/T 4 is calculated from ε/T 4 is calculated from Large error in whole T region Large error in whole T region SB limit /12

12. JPS autumn 2010T. Umeda (Hiroshima)12 Summary & outlook Equation of state Equation of state More statistics are needed in the lower temperature region More statistics 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) N f =2+1 QCD just at the physical point N f =2+1 QCD just at the physical point the physical point (pion mass ~ 140MeV) by PACS-CS the physical point (pion mass ~ 140MeV) by PACS-CS β=1.90 is appropriate to control stat. error at lower T. β=1.90 is appropriate to control stat. error at lower T. Odd Nt config. generation is necessary. Odd Nt config. generation is necessary. Finite density Finite density Taylor expansion method to explore EOS at μ ≠0 Taylor expansion method to explore EOS at μ ≠0 We presented the EOS in N f =2+1 QCD using improve Wilson quarks /12

13. JPS autumn 2010T. Umeda (Hiroshima)13 T-integration method to calculate the EOS We propose a new method (“T-integration method”) to calculate the EOS at fixed scales Our method is based on the trace anomaly (interaction measure), and the thermodynamic relation. T.Umeda et al. (WHOT-QCD), Phys.Rev.D79 (2009) 051501(R)

14. JPS autumn 2010T. Umeda (Hiroshima)14 Test in quenched QCD Our results are roughly Our results are roughly consistent with previous results. consistent with previous results. Our results deviate from the Our results deviate from the fixed N t =8 results [*] fixed N t =8 results [*] at higher T ( aT~0.3 or higher ) at higher T ( aT~0.3 or higher ) Trace anomaly is sensitive to Trace anomaly is sensitive to spatial volume at lower T spatial volume at lower T (below T c ). (below T c ). V > (2fm) 3 is ncessarry. V > (2fm) 3 is ncessarry. ~ [*] G. Boyd et al., NPB469, 419 (1996)

15. JPS autumn 2010T. Umeda (Hiroshima)15 Quark mass dependence of Trace anomaly HotQCD arXiv1005.1131 0.05m s 0.2m s HotQCD PRD91,054504(2010) CP-PACS PRD64,074510 (2001) peak height of the Trace anomaly peak height of the Trace anomaly  small quark mass dependence  small quark mass dependence Our result seems to be reasonable ! Our result seems to be reasonable ! but small m q is necessary. but small m q is necessary.