1. JPS spring 2012T. Umeda (Hiroshima)1 固定格子間隔での有限温度格子 QCD の研究 Takashi Umeda (Hiroshima Univ.) for WHOT-QCD Collaboration JPS meeting, Kwansei-gakuin, Hyogo, 26 Mar. 2012 /13

2. JPS spring 2012 2 Quark Gluon Plasma in Lattice QCD Quark Gluon Plasma will be understood with theoretical models (e.g. hydrodynamic models) which require “physical inputs” Lattice QCD : first principle calculation 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,...) etc... etc... http://www.gsi.de/fair/experiments/ T. Umeda (Hiroshima) /13

3. JPS spring 2012T. Umeda (Hiroshima)3 Choice of quark actions on the lattice Most (T, μ≠0) studies done with staggerd-type quarks  N f =2+1, physical quark mass, (μ≠0)  N f =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 theoretically sound lattice quarks like Wilson-type quarks The objective of WHOT-QCD collaboration is finite T & μ calculations using Wilson-type quarks finite T & μ calculations using Wilson-type quarks /13 Equation of state in 2+1 flavor QCD with improved Wilson quarks by the fixed scale approach T. Umeda et al. (WHOT-QCD Collab.) [arXiv:1202.4719]

4. JPS spring 2012T. 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 ) - Lattice spacing at lower T - Lattice spacing at lower T - Finite volume effects - Finite volume effects Disadvantages Disadvantages - T resolution - T resolution - High T region - High T region a : lattice spacing N t : lattice size in temporal direction /13 LCP’s in fixed N t approach (N f =2 Wilson quarks at N t =4)

5. JPS spring 2012T. Umeda (Hiroshima)5 Fixed scale approach to study QCD thermodynamics Temperature T=1/(N t a) is varied by N t at fixed a a : lattice spacing N t : lattice size in temporal direction /13 lattice spacing at fixed N t 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 ) - Lattice spacing at lower T - Lattice spacing at lower T - Finite volume effects - Finite volume effects Disadvantages Disadvantages - T resolution - T resolution - High T region - High T region

6. JPS spring 2012T. Umeda (Hiroshima)6 Fixed scale approach to study QCD thermodynamics Temperature T=1/(N t a) is varied by N t at fixed a a : lattice spacing N t : lattice size in temporal direction /13 spatial volume at fixed N t 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 ) - Lattice spacing at lower T - Lattice spacing at lower T - Finite volume effects - Finite volume effects Disadvantages Disadvantages - T resolution - T resolution - High T region - High T region N s for V=(4fm) 3

7. JPS spring 2012T. Umeda (Hiroshima)7 Lattice setup 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 /13

8. JPS spring 2012T. Umeda (Hiroshima)8 Beta-functions from CP-PACS+JLQCD results fit β,κ ud,κ s as functions of /13 Direct fit method at a LCP LCPscale We estimate a systematic error using for scale dependence

9. JPS spring 2012T. Umeda (Hiroshima)9 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 /13

10. JPS spring 2012T. Umeda (Hiroshima)10 Equation of State in Nf=2+1 QCD T-integration T-integration is performed by Akima Spline is performed by Akima Spline interpolation. 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 A systematic error due to A systematic error due tobeta-functions SB limit /13

11. JPS spring 2012T. Umeda (Hiroshima)11 Polyakov loop and Susceptibility /13 Polyakov loop requires T dependent renormalization T dependent renormalization c(T) : additive normalization constant (self-energy of the (anti-)quark sources) : heavy quark free energy

12. JPS spring 2012T. Umeda (Hiroshima)12 Renormalized Polyakov loop and Susceptibility /13 Cheng et al.’s renormalization [PRD77(2008)014511] [PRD77(2008)014511] matching of V(r) to V string (r) at r=1.5r 0 N f =0 case ( c m is identical at a fixed scale )

13. JPS spring 2012T. Umeda (Hiroshima)13 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 beta-functions using reweighting method beta-functions using reweighting method 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 and renormalized Polyakov loop in N f =2+1 QCD using improve Wilson quarks in N f =2+1 QCD using improve Wilson quarks /13

14. JPS spring 2012T. Umeda (Hiroshima)14 Renormalized Polyakov loop and Susceptibility /13 Cheng et al.’s renormalization [PRD77(2008)014511] [PRD77(2008)014511] matching of V(r) to V string (r) at r=1.5r 0 V(r) = A –α/r + σr V string (r) = c diff -π/12r + σr L ren = exp(c diff N t /2) L ren = exp(c diff N t /2) N f =0 case