Recent Progress in Lattice QCD at Finite Density

Transcription

1 Recent Progress in Lattice QCD at Finite Density Shinji Ejiri (Brookhaven ational Laboratory) Lattice 008, July 14-19, 008

2 QCD thermodynamics at 0 Heavy-ion experiments (HIC) (low energy RHIC, FAIR) Properties of QCD at finite density Important roles of Lattice QCD study Critical temperature (c) and Equation of State (EoS) in low density region Important for Hydrodynamic calculations in HIC. Study by Lattice simulations: available Propose interesting observations measureable properties of QCD in HIC Critical point at finite density Large fluctuation in quark number? Large bulk viscosity? SPS RHIC LHC hadron phase AGS quark-gluon plasma phase low-e RHIC FAIR color super nuclear matter conductor? q

3 Lattice QCD at 0 Many interesting results in QCD thermodynamics at =0 Study at 0: in the stage of development. Problem of Complex Determinant at 0 M ( ) 5 M ( ) 5 Boltzmann weight: complex at 0 Monte-Carlo method is not applicable. Configuration cannot be generated. hree approaches aylor expansion in (5-conjugate) * detm ( ) detm ( ) detm ( ) Reweighting method: Simulations at =0, Modify the Boltzmann weight Analytic continuation from imaginary chemical potential simulations

4 Interesting studies in finite density QCD (16 parallel talks, 5 posters, and a lot of s) Equation of State - MILC, RBC-Bielefeld, Hot-QCD, WHO-QCD QCD Critical point P. de Forcrand (Fri), A. Li (ue), X. Meng (ue) Stochastic quantization for QCD at finite G. Aarts (ue); G. Aarts, I.-O. Stamatescu, arxiv: wo Color QCD: Di-quark condensation K. Fukushima (hu), S. Hands, J. Skullerud and S. Kim P. Cea, L. Cosmai, M. D'Elia, A. Papa, Pjys. Rev. D77, (008) M.P. Lombardo, M.L. Paciello, S. Petrarca, B. aglienti, arxiv: Isospin chemical potential - Y. Sasai (ue) W. Detmold, M. Savage, A. orok, S. Beane,. Luu, K. Orginos, A. Parreno, arxiv: High temperature effective theory A. Hietanen and K. Rummukainen, arxiv: Strong coupling limit M. Fromm (Wed), A. Ohnishi (Wed), K. Miura (Pos) Chiral perturbation theory - J. Verbaarschot (ue) Chiral fermions (Domain-Wall, overlap) R. Gavai (ue); arxiv: , P. Hegde (Pos)

5 Plan of talk Introduction Equation of State at finite density aylor expansion method QCD critical point at finite density Quark mass dependence of the critical point Plaquette effective potential Canonical approach Summary and Outlook

6 Equation of State at finite density aylor expansion method (Bielefeld-Swansea Collab., 0-06, Gavai-Gupta, 03-05) Systematic studies using p4-imploved staggered action with rather heavy quark masses. (Bielefeld-Swansea Collab., 0-06) 1. Useful for EoS study for Heavy-ion collisions Low density region is important for HIC. Large fluctuations in the quark number at high density Existence of a critical point: suggested Recent Progress Simulations near physical mass point MILC Collab., RBC-Bielefeld Collab., Hot QCD Collab. Isentropic equation of state, Fluctuations Simulations with a Wilson-type quark action WHO-QCD Collab., Quark number fluctuations (Bielefeld-Swansea Collab., 06)

7 aylor expansion method for EoS Heavy-ion collisions: low density In the heavy-ion collision at RHIC, the interesting regime of q is around q/c 0.1. aylor expansion in at =0. Simulations at 0: Free from the complex determinant problem. Calculation of the derivatives: a basic technique for QCD thermodynamics. e.g. Energy density, Quark number, Quark number susceptibility aylor expansion method Useful for EoS study 6 q 6 4 q 4 q c c c p p : odd for 0 ) ( ln q n n n, ) ( ln 4!, ) ( ln 4 q q 3 3 c c s t s t V p ln 1 3 4, ) ( ln q 3 3 q n s t q 3 3 B q ) ( ln 9 s t, ln ln a p s t

8 Isentropic Equation of State EoS along lines of constant entropy per baryon number (S/B) ero-viscosity hydro calculations explain experimental results. o entropy production in a heavy-ion collisions (in equibilium) S/B 300(RHIC), S/B 45(SPS), S/B 30 (AGS) MILC Collab. S.Gottlieb s talk (Monday) f=+1 Asqtad action, t=4,6, m 0MeV Lattice discretization error: small. (Open: t=4, Filled: t=6)

9 Isentropic Equation of State RBC-Bielefeld Collab. C.Schmidt s talk (Monday) f=+1 p4-imploved staggered, t=4,6, m 0MeV Consistent with Asqtad results. p/ vs is important for hydrodynamic calculations in HIC. Density dependence: small Velocity of sound cs dp d( p ) c s d d p preliminary (Filled: t=4, Open: t=6)

10 Hadronic fluctuations and the QCD critical point RBC-Bielefeld Collab. C.Schmidt s talk (Monday) Hadronic fluctuations at 0 increase with decreasing mass. ( ( B B ) ) B Low : hadron resonance gas High : quark-gluon gas RBC-Bielefeld Collab.: f=+1, m=0mev Bielefeld-Swansea, ( 06).: f=, m=770mev hadron resonance gas quark-gluon gas Fluctuations for m=0mev increase over the hadron resonance gas value at c. preliminary

11 Equation of state by Wilson quark action WHO-QCD Collab. K.Kanaya s poster RG gauge + -flavor Clover quark actions, lattice, m m 0.65 Hybrid method of Reweighting and aylor expansion up to 4 O q Large enhancement in the quark number fluctuations at high density. Critical point at finite

12 QCD critical point in the (,) plane Quark mass dependence of the critical line Reweighting method and Sign problem hadron QGP CSC Plaquette effective potential Canonical approach

13 Quark mass dependence of the critical point ms nd order Physical point 1 st order f= Quenched f=3 Crossover 0 0 mud 1 st order 0 Quark mass dependence near =0 Fodor, Katz, 01-04; Reweighting method Bielefeld-Swansea Collab., 0, 03; Reweighting method de Forcrand, Philipsen, 03-07; Imaginary chemical potential Kogut, Sinclair, 05-07; Phase-quenched approximation Philipsen s plenary talk in Lattice 005 Schmidt s plenary talk in Lattice 006

14 Curvature of the critical surface mc m 0 C 0 Usual expectation Critical point: exists de Forcrand - Philipsen, JHEP01(007)077; PoS(LA007)178 Curvature: slightly negative. (3-flavor, 8 3 x4 lattice) ew result de Forcrand s talk (Friday) ew result by 1 3 x4 lattice is consistent with 8 3 x4 result. Curvature: egative.

15 Imaginary chemical potential approach (de Forcrand, Philipsen, 03-08) Binder cumulant: Critical point ( universality): B4=1.604 [Crossover (m>mc): B4=3, Strong first order (m<mc): B4=1] Simulations: possible for imaginary = ii, detm(ii): real Assumption: Analytic continuation: Fit the simulation results m C b01 b 10 B m m b B b10 c 01 b 0 Curvature: egative b 01 0 t=4 B 4 i Other assumption for analytic continuation, (D Elia, Di Renzo, Lombardo, PRD76,114509(007)) 0 4 s 8 s b 10 0 de Forcrand s talk 1 1 i s

16 Reweighting method for 0 and Sign problem (Ferrenberg-Swendsen Glasgow group, Fodor-Katz) Reweighting method Boltzmann weight: Complex for >0 Monte-Carlo method is not applicable directly. partition function: DU f S g det M e Perform Simulation at =0. O 1 DUO detm ( ), f e S g Oe e i i det det f f M M det M det M det det f f M M 0 0,0,0 e i Sign problem i i If e changes its sign frequently, Oe become smaller than their statistical errors. hen O cannot be computed., and e i,0,0

17 Sign problem and phase fluctuations Complex phase of detm aylor expansion: odd terms of ln det M (Bielefeld-Swansea, PRD66, (00)) Good definition (staggered quarks: 4 th root trick, /4?) f Im ln det M ( ) Im dlndet M 1 d f 5 3 d 3 lndet M 1 d lndet M 3 3! d 5! d 5 5 : O in the range of [-, ] > /: Sign problem happens. i e changes its sign. Gaussian distribution Results for p4-improved staggered aylor expansion up to O( 5 ) Dashed line: fit by a Gaussian function Well approximated W e C histogram of

18 Complex phase distribution he Gaussian distribution is also suggested by chiral perturbation theory. (K. Splittorff and J. Verbaarschot, Phys.Rev.D77, (007)) Assume: Gaussian distribution (S.E., Phys.Rev.D77, (008)) Sign problem: J.Verbaarschot s talk (uesday) Sign problem is avoided. det M f i e F (statistical error) C histogram Gaussian integral: W F, e W ( F ) e i F df d e i FW 1 ( 4 ) F, df e FW ( F ) width F e i F e F F real and positive (o sign problem)

19 Effective potential of plaquette V(P) Plaquette histogram First order phase transition wo phases coexists at c e.g. SU(3) Pure gauge theory SU(3) Pure gauge theory QCDPAX, PRD46, 4657 (199) histogram Gauge action Partition function S g 6 site P, dpw P,, histogram Effective potential W f S P', DU detm e g P P' V ( P) ln W P

20 Distribution function and Effective potential at 0 (S.E., Phys.Rev.D77, (008)) Distributions of plaquette P (1x1 Wilson loop for the standard action) W R dp RP, W P, P DUP-P detm 0 P, DU DU Sg, f e P-P detm P-P detm 0 Effective potential: f detm f detm 0 P-P S g P-P (, 0) f 6 (, 0) site (Weight factor at R(P,): independent of, R(P,) can be measured at any. P, W V ( P) ln R P, =0 crossover non-singular ln W P, ln R P 1 st order phase transition? +? =? P, (Reweight factor

21 -dependence of the effective potential Crossover lnw P, Critical point lnw P, ln R P, QGP + = =0 reweighting Curvature: ero 1 st order phase transition hadron CSC ln W P, ln R P, + = =0 reweighting Curvature: egative

22 Effective potential at (S.E., Phys.Rev.D77, (008)) Results of f= p4-staggared, m/m0.7 [data in PRD71,054508(005)] lnw at =0 V P,, lnwp, lnrp, detm: aylor expansion up to O( 6 ) lnr he peak position of W(P) moves left as increases at =0. Solid lines: reweighting factor at finite /, R(P,) Dashed lines: reweighting factor without complex phase factor.

23 Curvature of the effective potential f= p4-staggared, m/m0.7 d lnw dp at q=0 + =? Critical point:, d V P,, d ln W P, d ln R P dp dp dp First order transition for q/.5 Existence of the critical point: suggested Quark mass dependence: large Study near the physical point is important. 0

24 Slope of lnr(p,) at low density P, ln RP lnw, + = Minimum point moves, P large Same effect as eff 1 6 site (ln R) P -dependence of c Bielefeld-Swansea Collab., PRD66, ( 0) Low phase High phase he phase transition point becomes lower as increases.

25 Canonical approach Canonical partition function (Laplace transformation ) Effective potential as a function of the quark number. At the minimum, First order phase transition: wo phases coexist. C GC W exp,, W V C ), ( ln ) ( ln ) ( 0 ), ( ln ) ( ln ) ( W V C C ) ( ln ( ) V

26 First order phase transition line V ( In the thermodynamic limit, 0, ) cp * ln C (, ) * 3 s cp Mixed state First order transition Inverse Laplace transformation by Glasgow method Kratochvila, de Forcrand, PoS (LA005) 167 (005) f=4 staggered fermions, f=4: First order for all lattice ew results: f= Direct simulations with fixed Inverse Laplace transformation in a Saddle point approximation

27 Simulations with Canonical partition function QCD collab. (Kentucky group), A. Li and X. Meng s talk (uesday) Canonical partition function with fixed (Alexandru, Faber, Horvath and Liu, Phys. Rev. D7, (005)) with Fourier coefficients, C S, det g DU e M f f ii det M d e detmi 1 I I f I 0 0 detm R Integral 0.90c 0.9c 0.83c 0.94c 0.86c f =4: first order transition f =: crossover Wilson quark 6 3 x4 lattice

28 Inverse Laplace transformation with a saddle point approximation (S.E., arxiv: ) Approximations: aylor expansion: ln det M up to O( 6 ) Gaussian distribution: Saddle point approximation Much easier calculations wo states at the same q/ First order transition at /c < 0.83 Study near the physical point important f= p4-staggered, m/m0.7, lattice Solid line: multi- reweighting Dashed line: spline interpolation Dot-dashed line: the free gas limit umber density

29 Summary and outlook Equation of State at finite density Isentropic EoS for heavy-ion collisions Simulations near physical quark mass point: studied Large hadronic fluctuation near c: observed 1. Staggered quark with Small quark mass,. Wilson-type quark QCD critical point at finite density echnical developments Quark mass dependence of the critical line Avoidance of the Sign problem Plaquette effective potential Canonical approach Existence of the QCD critical point: suggested Future studies ew echnique for high density: required ew phenomena at high density