Lattice 2012T. Umeda (Hiroshima)1 Thermodynamics in 2+1 flavor QCD with improved Wilson quarks by the fixed scale approach Takashi Umeda (Hiroshima Univ.) for WHOT-QCD Collaboration Lattice2012, Cairns, Australia, 25 June 2012 /13

Lattice 2012T. Umeda (Hiroshima)2 QCD Thermodynamics with Wilson quarks Most (T, μ≠0) studies done with staggerd-type quarks 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 Aim of WHOT-QCD collaboration is finite T & μ calculations using Wilson-type quarks finite T & μ calculations using Wilson-type quarks /13 Phys. Rev. D (2012) [arXiv: ] Phys. Rev. D (2012) [arXiv: ] T. Umeda et al. (WHOT-QCD Collaboration) T. Umeda et al. (WHOT-QCD Collaboration) PTEP in press [ arXiv: (hep-lat) ] PTEP in press [ arXiv: (hep-lat) ] S. Ejiri, K. Kanaya, T. Umeda for WHOT-QCD Collaboration S. Ejiri, K. Kanaya, T. Umeda for WHOT-QCD Collaboration

Lattice 2012T. Umeda (Hiroshima)3 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 due to integer N t - T resolution due to integer N t - UV cutoff eff. at high T’s - UV cutoff eff. at high T’s  fixed scale scale

Lattice 2012T. Umeda (Hiroshima)4 Lattice setup T=0 simulation: on 28 3 x 56 by CP-PACS/JLQCD Phys. Rev. D78 (2008) T=0 simulation: on 28 3 x 56 by CP-PACS/JLQCD Phys. Rev. D78 (2008) RG-improved Iwasaki glue + NP-improved Wilson quarks - β=2.05, κ ud =0.1356, κ s = 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

Lattice 2012T. Umeda (Hiroshima)5 Trace anomaly for N f =2+1 improved Wilson quarks Noise method ( #noise = 1 for each color & spin indices ) Phys. Rev. D73, CP-PACS/JLQCD /13

Lattice 2012T. Umeda (Hiroshima)6 Beta-functions from CP-PACS/JLQCD results Meson spectrum by CP-PACS/JLQCD Phys. Rev. D78 (2008) κ ud x 2 κ s for each 3 β = 30 data points /13 fit β,κ ud,κ s as functions of LCPscale Direct fit method

Lattice 2012T. Umeda (Hiroshima)7 Beta-functions from CP-PACS/JLQCD results χ 2 /dof=1.6 χ 2 /dof=1.1 χ 2 /dof=1.7 fit β,κ ud,κ s as functions of /13 systematic error  for scale dependence with fixed

Lattice 2012T. Umeda (Hiroshima)8 Equation of State in N f =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 A systematic error A systematic error for beta-functions SB limit /13

Lattice 2012T. Umeda (Hiroshima)9 Polyakov loop in the fixed scale approach /13 Polyakov loop requires Polyakov loop requires T dependent renormalization T dependent renormalization c(T) : additive normalization factor of heavy quark free energy F qq of heavy quark free energy F qq matching of V(r) to V string (r) at r=1.5r 0 matching of V(r) to V string (r) at r=1.5r 0 Cheng et al. PRD77(2008) Cheng et al. PRD77(2008) ( c m is a constant at all temperatures )

Lattice 2012T. Umeda (Hiroshima)10 Renormalized Polyakov loop and Susceptibility /13 N f =0 case ( c m is a constant at all temperatures ) Roughly consistent with Roughly consistent with the Staggered result χ L increases around T~200MeV χ L increases around T~200MeV Similar behavior to N f =0 case Similar behavior to N f =0 case

Lattice 2012T. Umeda (Hiroshima)11 Chiral condensate in the fixed scale approach /13 Renormalization factors are constants at all temperatures preliminary Chiral condensate by the Wilson quarks requires additive & multiplicative renormalizations additive & multiplicative renormalizations

Lattice 2012T. Umeda (Hiroshima)12 Chiral susceptibility in the fixed scale approach /13 Peak positions in are identical to that in renormalized suscep. susceptibility peak around 200 MeV (?)  more statistics is needed  more statistics is needed preliminary Renormalization factors are constants at all temperatures

Lattice 2012T. Umeda (Hiroshima)13 Summary & outlook Equation of state Equation of state first result in N f =2+1 QCD with Wilson-type quarks first result in N f =2+1 QCD with Wilson-type quarks Thanks to the fixed scale approach, systematic errors Thanks to the fixed scale approach, systematic errors coming from lattice artifacts are well under control coming from lattice artifacts are well under control Renormalizations are common at all temperatures Renormalizations are common at all temperatures 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 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, Polyakov loop, chiral condensate in N f =2+1 QCD using improved Wilson quarks in N f =2+1 QCD using improved Wilson quarks /13

Lattice 2012T. Umeda (Hiroshima)14

Lattice 2012T. Umeda (Hiroshima)15 Renormalized Polyakov loop and Susceptibility 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 m -π/12r + σr L ren = exp(c m N t /2) L ren = exp(c m N t /2)

Lattice 2012T. Umeda (Hiroshima)16 Chiral condensate 3 Lattice2011 Borsanyi et al.

Lattice 2012T. Umeda (Hiroshima)17 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 /15 LCP’s in fixed N t approach (N f =2 Wilson quarks at N t =4)

Lattice 2012T. Umeda (Hiroshima)18 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 /15 spatial volume at fixed N t N s for V=(4fm) 3 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