Bulk Thermodynamics: What do we (want to) know?

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Thomasine Goodman
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1 Bulk Thermodynamics: What do we (want to) know? µ = : properties of transition in, ( + 1)-flavor QCD: crossover or phase transition, deconfinement vs. chiral symmetry restoration, universality,... T c, ɛ c, EoS, v s, η,...: confront resonance gas, quasi-particle gas, high-t pert. theory, HTL-resummation with lattice calculations µ > : T c (µ) T freeze (µ) : location of the chiral critical point, direct evidence for 1 st order regime; density/charge fluctuations; T c (µ) T freeze? F. Karsch p.1/19

2 Critical behavior in hot and dense matter: phase diagram crossover vs. T ~17 MeV quark-gluon plasma deconfined, χ-symmetric phase transition hadron gas confined, χ-sb color superconductor µ o few times nuclear matter density µ F. Karsch p./19

3 Critical behavior in hot and dense matter: phase diagram T ~17 MeV continuous/rapid (crossover) transition confined, χ-sb quark-gluon plasma hadron gas µ o deconfined, χ-symmetric chiral critical point color superconductor few times nuclear matter density µ continuous transition for small chemical potential and small quark masses at T c 17 MeV ɛ c.7 GeV/fm 3 nd order phase transition; Ising universality class T c (µ) under investigation 1st order phase transition??? expected - however, so far no direct evidence from lattice QCD attractive 1 gluon exchange => qq condensates F. Karsch p./19

4 Critical temperature, equation of state 8 6 T c [MeV] n f =, p4 n f =3, p4 n f =, std m PS [MeV] ɛ c (6 ± ) T 4 c (.3 1.3) GeV /fm 3 energy density for, and 3-flavor QCD T c = (173 ± 8 ± sys) MeV FK, E. Laermann, A. Peikert, Nucl. Phys. B65 (1) 579 T c = 167(13) [177(11)] MeV MILC, hep-lat/ ε/t 4 RHIC ε SB /T 4 SPS 3 flavor flavor +1-flavor flavor LHC T [MeV] F. Karsch p.3/19

5 Critical temperature, equation of state 8 6 T c [MeV] 4 18 n f =, p4 n f =3, p4 n f =, std ɛ c (6 ± ) T 4 c (.3 1.3) GeV /fm 3 16 m PS [MeV] T c = (173 ± 8 ± sys) MeV FK, E. Laermann, A. Peikert, Nucl. Phys. B65 (1) 579 T c = 167(13) [177(11)] MeV MILC, hep-lat/459 Can we be satisfied energy density for, and 3-flavor QCD with these results? ε/t 4 RHIC ε SB /T 4 SPS 3 flavor flavor +1-flavor flavor LHC T [MeV] F. Karsch p.3/19

6 Critical temperature, equation of state T c [MeV] ɛ c m P S 77 MeV (!!!) n f =, p4 n f =3, p4 n f =, std m PS [MeV] V (4 fm) 3 (thermodynamic limit) energy density for, and 3-flavor QCD T c m P S > 3 MeV (chiral limit??) a. fm (continuum limit??) improved staggered fermions, flavor symmetry breaking (need even better fermion actions) ε/t 4 RHIC ε SB /T 4 SPS 3 flavor flavor +1-flavor flavor LHC T [MeV] F. Karsch p.3/19

7 The pressure revisited 5 4 p/t 4 p SB /T p/p SB flavour +1 flavour flavour pure gauge.4. 3 flavour flavour +1 flavour pure gauge T [MeV] T/T c status: FK, E. Laermann and A. Peikert, Phys. Lett. B478 () 447. F. Karsch p.4/19

8 The pressure revisited p/t 4 p SB /T T > (-3)T c : deviations from ideal gas understood in terms of non-perturbative (ordinary) QCD contributions p/p SB 1 3 flavour +1 flavour flavour pure gauge.4. 3 flavour flavour +1 flavour pure gauge T [MeV] T/T c status: FK, E. Laermann and A. Peikert, Phys. Lett. B478 () 447. T < T c : strong deviations from ideal gas large screening masses, remnants of confinement F. Karsch p.4/19

9 Singlet free energy and asymptotic freedom pure gauge: O.Kaczmarek, FK, P. Petreczky, F. Zantow, hep-lat/4636 -flavor QCD: O.Kaczmarek, F. Zantow, hep-lat/5317 singlet free energy defines a running coupling: α eff = 3r 4 df 1 (r, T ) dr α qq (r,t) r [fm].1.1 T/T c T= F. Karsch p.5/19

10 Singlet free energy and asymptotic freedom pure gauge: O.Kaczmarek, FK, P. Petreczky, F. Zantow, hep-lat/4636 -flavor QCD: O.Kaczmarek, F. Zantow, hep-lat/5317 singlet free energy defines a running coupling: α eff = 3r 4 df 1 (r, T ) dr large distance: constant Coulomb term (string model). short distance: running coupling α(r) from (T = ), 3-loop.1 (S. Necco, R. Sommer, Nucl. Phys. B6 () 38) α qq (r,t) α π/1 r [fm].1.1 short distance physics vacuum physics T/T c T= T-dependence starts in non-perturbative regime for T < 3 T c F. Karsch p.5/19

11 )) = ) - ed ts llo M { o} v o Q Š Š ƒ Š š Š Š Ž Š Œ Ž Œ Š ŒŽ Š Žš Š Š F. Karsch p.6/19!('! & # % "$#! 6 L K+ )FG H E)FG 3DC /1 *A +B- 5 4 JJ)FG 3DC /1 *A +I- /T 4 3 3/4 s/t 3 < 897 :?@5 > / *+ 3p/T 4 1 n kml ij h j fg < 89;: /1 *+,.- ) x vwu u p.u prq T/T c ~ {l yz c `ba _S ^ \ ] [ ZV SUT VWYX RSUT Q NPO Ž Œ ˆb Š Ž Œ 1Œ ˆ bš ƒ ƒ ƒ Ž Ž œ ƒ ž š š ƒ ˆƒ ŽŸ Œ ƒ Š ƒ Œ NPO ^ N Ž Œ Œ status:

12 Deviation from ideal gas (pure gauge) (ε - 3p)/T 4 (x) (1x) (1x1) (1x) tad T/T c evidence for significant deviations from ideal gas behaviour at least up to T T c ; i.e. the regime accessible to RHIC F. Karsch p.7/19

13 Velocity of sound (pure gauge) G. Boyd et al., NPB 1996 F. Karsch p.8/19

14 Viscosity (pure gauge, exploratory) η s Perturbative Theory KSS bound T Tc A. Nakamura and S. Sakai, PRL 94 (5) 735 F. Karsch p.9/19

15 Extending the phase diagram to non-vanishing chemical potential non-zero baryon number density: µ > Z(V, T, µ) = DADψD ψ e S E(V,T,µ) = DAD det M(µ) e S E(V,T ) three (partial) solutions for large T, small µ complex fermion determinant; long standing problem F. Karsch p.1/19

16 Extending the phase diagram to non-vanishing chemical potential non-zero baryon number density: µ > Z(V, T, µ) = = DADψD ψ e S E(V,T,µ) DAD det M(µ) e S E(V,T ) three (partial) solutions for large T, small µ complex fermion determinant; long standing problem exact evaluation of detm: works well on small lattices; requires reweighting Z. Fodor, S.D. Katz, JHEP 3 () 14 Taylor expansion around µ = : works well for small µ; requires reweighting C. R. Allton et al. (Bielefeld-Swansea), Phys. Rev. D66 () 7457 imaginary chemical potential: works well for small µ; requires analytic continuation Ph. deforcrand, O. Philipsen, Nucl. Phys. B64 () 9 F. Karsch p.1/19

17 Extending the phase diagram to non-vanishing chemical potential non-zero baryon number density: µ > Z(V, T, µ) = DADψD ψ e S E(V,T,µ) = DAD det M(µ) e S E(V,T ) T [MeV] 15 1 RHIC, LHC T c (µ) T c () : 1.56(4)(µ B/T ) deforcrand, Philipsen (imag. µ) GSI future 1.78(38)(µ B /T ) Bielefeld-Swansea (O(µ ) reweighting) 5 T c, Bielefeld-Swansea T c, Forcrand, Philipsen T f, J.Cleymans et. al. µ B [GeV] F. Karsch p.1/19

18 Extending the phase diagram to non-vanishing chemical potential non-zero baryon number density: µ > Z(V, T, µ) = = DADψD ψ e S E(V,T,µ) DAD det M(µ) e S E(V,T ) T [MeV] F,K 4 / Bi-Sw 3 chiral critical point chiral critical point F,K T c, Bielefeld-Swansea T c, Forcrand, Philipsen T f, J.Cleymans et. al. µ B [GeV] F. Karsch p.11/19

19 Extending the phase diagram to non-vanishing chemical potential non-zero baryon number density: µ > Z(V, T, µ) = = DADψD ψ e S E(V,T,µ) DAD det M(µ) e S E(V,T ) T [MeV] F,K 4 / Bi-Sw 3 chiral critical point chiral critical point F,K 15 µ c shifted from 7 MeV to 4 MeV Fodor, Katz (exact determinants small lattices) 1 5 T c, Bielefeld-Swansea T c, Forcrand, Philipsen T f, J.Cleymans et. al. µ B [GeV] F. Karsch p.11/19

20 Extending the phase diagram to non-vanishing chemical potential non-zero baryon number density: µ > Z(V, T, µ) = = DADψD ψ e S E(V,T,µ) DAD det M(µ) e S E(V,T ) T [MeV] F,K 4 / Bi-Sw 3 chiral critical point chiral critical point F,K 15 µ c shifted from 7 MeV to 4 MeV Fodor, Katz (exact determinants small lattices) 1 µ c (m q ): discrepancies between calculations with real and imaginary µ 5 T c, Bielefeld-Swansea T c, Forcrand, Philipsen T f, J.Cleymans et. al. µ B [GeV] F. Karsch p.11/19

21 The pressure for µ q /T > C.R. Allton et al. (Bielefeld-Swansea), PRD68 (3) 1457 µ q =, lattice improved staggered fermions; n f =, m π 77 MeV contribution from µ q /T > Taylor expansion, O((µ/T ) 4 ) 5 4 p/t 4 p SB /T 4.8 p/t 4 µ q /T= flavour +1 flavour flavour pure gauge T [MeV] µ q /T=.8 µ q /T=.6 µ q /T=.4 high-t, ideal gas limit p 7π T 4 = n f µq T 1 µq + 4π T 4 « T/T µ q /T=. F. Karsch p.1/19

22 The pressure for µ q /T > C.R. Allton et al. (Bielefeld-Swansea), PRD68 (3) 1457 µ q =, lattice improved staggered fermions; n f =, m π 77 MeV PRD71 (5) 5458 contribution from µ q /T > NEW: Taylor expansion, O((µ/T ) 6 ) 5 4 p/t 4 p SB /T 4.8 p/t 4 µ q /T= flavour +1 flavour flavour pure gauge T [MeV] pattern for µ q = and µ q > similar; quite large contribution in hadronic phase; O((µ/T ) 6 ) correction small for µ q /T < µ q /T=.8 µ q /T=.6 µ q /T=.4 µ q /T= T/T RHIC: µ q /T <.1 F. Karsch p.1/19

23 3) Hadronic fluctuations at µ q = : Taylor expansion coefficients for µ q > S. Ejiri, FK, K.Redlich, in preparation quark number and isospin chemical potentials: µ q = 1 (µ u + µ d ), µ I = 1 (µ u µ d ) expansion coefficients evaluated at µ q,i = are related to hadronic fluctuations at µ = : baryon number, isospin, charge event-by-event fluctuations at RHIC and LHC ln Z (µ q,i /T ) = (δn q,i ) µ= = N q,i µ= 4 ln Z (µ q,i /T ) 4 = (δn q,i ) 4 µ= = N 4 q,i µ= 3 N q,i µ= k Q ln Z = (δq)k µ=, Q 3 µ u /T 1 3 µ d /T F. Karsch p.13/19

24 generic QCD phase diagram T m> T m= crossover critical point Z(), nd order T c O(4), nd order tri critical point Z(), nd order 1st order 1st order color superconductor color superconductor µ o few times nuclear matter density µ µ o few times nuclear matter density µ F. Karsch p.14/19

25 Hadronic fluctuations and chiral symmetry restoration expect nd order transition in 3-d, O(4) symmetry class; scaling field: t = T T c T c «+ A µu + T c µd T c «! + B µ u T c µ d T c singular part: f s (T, µ u, µ d ) = b 1 f s (tb 1/( α) ) t α ln Z µ q,i t 1 α, 4 ln Z µ 4 q,i t α (µ = ) O(4)/O(): α <, small (δn q,i ) dominated by T-dependence of regular part (δn q,i ) 4 develops a cusp F. Karsch p.15/19

26 Hadronic fluctuations and chiral symmetry restoration expect nd order transition in 3-d, O(4) symmetry class; scaling field: t = T T c T c «+ A µq T c µcrit T c «! singular part: f s (T, µ u, µ d ) = b 1 f s (tb 1/( α) ) t α ln Z µ q,i t α, 4 ln Z µ 4 q,i t (+α) (µ > ) O(4)/O(): α <, small (δn q,i ) develops a cusp (δn q,i ) 4 diverges on the O(4) critical line; strength µcrit T c «4 ( 1 4 at RHIC) F. Karsch p.16/19

27 Quark number and isospin fluctuations at µ B = in -flavor QCD (m π 77 MeV ) C. Allton et al. (Bielefeld-Swansea), PRD71 (5) d (p/t 4 )/d(µ q /T) 1 d (p/t 4 )/d(µ I /T) T/T T/T d 4 (p/t 4 )/d(µ q /T) d 4 (p/t 4 )/d(µ I /T) T/T T/T F. Karsch p.17/19

28 4 3 1 Fluctuations of the quark number density (µ q > ) quark number density fluctuations: up to O `(µ q /T ) χ q /T µ q /T=1. µ q /T=.5 µ q /T=. T/T χ q T = ( (µ q /T ) p T 4 ) T fixed high-t, massless limit: polynomial in (µ q /T ) χ q,sb T = n f + 3n f π ( ) µq large density fluctuations closer to the chiral critical point χ q T = 1 ( N V T 3 q N q ) ( ) p = n q n q χ q T χ q will diverge on chiral critical point T F. Karsch p.18/19

29 4 3 1 Fluctuations of the quark number density (µ q > ) quark number density fluctuations: up to O `(µ q /T ) 4 χ q /T µ q /T=1. µ q /T=.5 µ q /T=. T/T χ q T = ( (µ q /T ) p T 4 ) T fixed high-t, massless limit: polynomial in (µ q /T ) χ q,sb T = n f + 3n f π ( ) µq large density fluctuations closer to the chiral critical point χ q T = 1 ( N V T 3 q N q ) ( ) p = n q n q χ q T χ q will diverge on chiral critical point T F. Karsch p.18/19

30 Fluctuations of the quark number density (µ q > ) quark number density fluctuations: χ q T = ( (µ q /T ) p T 4 ) T fixed 4 3 χ q /T µ q /T=1. µ q /T=.5 µ q /T=. T [MeV] RHIC, 15 LHC F,K 4 / Bi-Sw 3 chiral critical point F,K SPS GSI future 1 1 T/T T c, Bielefeld-Swansea T c, Forcrand, Philipsen T f, J.Cleymans et. al. µ B [GeV] F. Karsch p.18/19

31 Conclusions - lattice calculations at µ = produced a lot of evidence for substantial deviations from ideal gas behavior for T < T c, i.e. in the temperature regime accessible to RHIC - the next generation of lattice calculations on QCDOC and apenext will produce results for QCD thermodynamics with almost realistic quark masses (m π MeV) - QCD thermodynamics for µ and large T can be analyzed reliably - bulk thermodynamics at RHIC corresponds to µ q - higher moments of baryon number and charge fluctuations show large variations at T c F. Karsch p.19/19

F. Karsch for USQCD, LQCD II p. 1/27. Lattice QCD at High Temperature and Density. Frithjof Karsch for USQCD Brookhaven National Laboratory

F. Karsch for USQCD, LQCD II p. 1/27 Lattice QCD at High Temperature and Density Frithjof Karsch for USQCD Brookhaven National Laboratory F. Karsch for USQCD, LQCD II p. 2/27 Towards A New State of Matter