(Non)-perturbative properties of high-T QCD from lattice ... - Physics

F. Karsch, xQCD, July 2008 – p. 1/32(Non)-perturbative properties of high-T QCDFrithjof Karsch, Brookhaven National Laboratory & Bielefeld UniversityIntroduction:thermal scales in QCD: T , gT , g 2 T ,...Heavy quark free energiesscreening and running couplingsBulk thermodynamicsthe equation of state: QCD and SU(3)Hadronic fluctuationsquark number and charge fluctuationsConclusions

F. Karsch, xQCD, July 2008 – p. 2/32Introduction: sQGPmatter created at RHIC is a dense, strongly interacting systemDoes the coupling at finite temperature become large?Are exotic bound states in the QGP responsible for largeinteraction cross sections?Is the QGP a liquid?Can the QGP be described in terms of a conformal field theory?overview on perturbative and non-perturbative features of the QGP

Thermal scales in QCDthe hard scale: p ∼ Tthermal modes, bulk thermodynamics, eg. pressurepT 4 = a SBf p (g(T))the soft scale: p ∼ gTstatic color-electric modes, eg. Debye screening√m D (T)g(T)T = N c3 + n f6 + n ( ) 2f µq· f2π 2 E (g(T))Tthe ultra-soft scale: p ∼ g 2 Tstatic color-magnetic modes, eg. spatial string tension√σsg 2 (T)T = c Mf M (g(T))F. Karsch, xQCD, July 2008 – p. 3/32

F. Karsch, xQCD, July 2008 – p. 5/32Hierachy of scale ?Perturbation theory provides a hierachy of length scalesT ≫ gT ≫ g 2 T ...⇒ guiding principle for effective theories,resummation, dimensional reduction...These scales are not well separated close to T c !!Early lattice results show that g 2 (T) > 1 even at T ∼ 5T cG. Boyd et al, NP B469 (1996) 419: SU(3) thermodynamics.....one has to conclude that the temperature dependent runningcoupling has to be large, g 2 (T) ≃ 2 even at T ≃ 5T cthe Debye screening mass is large close to T cthe spatial string tension does not vanish above T c√σs ≠ 0 ⇒ the QGP is ”non-perturbative” up to very high T

Screening of heavy quark free energies– remnant of confinement above T c –pure gauge: O.Kaczmarek, FK, P. Petreczky, F. Zantow, PRD70 (2005) 0745052-flavor QCD: O.Kaczmarek, F. Zantow, Phys. Rev. D71 (2005) 114510singlet free energy(in Coulomb gauge)1000T ≃ T c : screening forr> ∼0.5fmF 1 (r, T) ∼ α(T)r+const.500e −µ(T)r0-500F 1 [MeV]r [fm]0.76T c0.81T c0.90T c0.96T c1.00T c1.02T c1.07T c1.23T c1.50T c1.98T c4.01T c0 0.5 1 1.5 2 2.5 34α(r, T = 0)F 1 (r, T) follows linear rise of V¯qq (r, T = 0) = −3rfor T< ∼1.5T c , r< ∼0.3 fm+ σrF. Karsch, xQCD, July 2008 – p. 6/32

Non-perturbative Debye screeningleading order perturbation theory: m D = g D (T)T√1 + n f6T c < T< ∼10T c : non-perturbative effects are well representedby an ”A-factor”: m D ≡ Ag D (T)T, A ≃ 1.5perturbative limit is reached very slowly4.0(logarithms at work!!)SU(3)10 100 β G=144 32 310 10T/Λ MS1010.543m D /Tm D /T=Ag(T)A=1.42(2)N f =0N f =23.02.0m/gT210T/T c1 1.5 2 2.5 3 3.5 41.00.0xβ G=72 24 3β G=40 24 3β G=24 24 3β G=12 24 310 −3 10 −2 10 −1O.Kaczmarek,F.Zantow, PRD 71 (2005) 114510 K.Kajantie et al, PRL 79 (1997) 3130F. Karsch, xQCD, July 2008 – p. 7/32

Non-perturbative Debye screeningleading order perturbation theory: m D = g D (T)T√1 + n f6T c < T< ∼10T c : non-perturbative effects are well representedby an ”A-factor”: m D ≡ Ag D (T)T, A ≃ 1.5perturbative limit is reached very slowly4.0(logarithms at work!!)SU(3)10 100 β G=144 32 310 10T/Λ MS1010.543m D /Tm D /T=Ag(T)A=1.42(2)N f =0N f =23.02.0m/gT210g 2 D (T c) ≃ 4 , g 2 D (4T c) ≃ 2T/T c1 1.5 2 2.5 3 3.5 41.00.0xβ G=72 24 3β G=40 24 3β G=24 24 3β G=12 24 310 −3 10 −2 10 −1O.Kaczmarek,F.Zantow, PRD 71 (2005) 114510 K.Kajantie et al, PRL 79 (1997) 3130F. Karsch, xQCD, July 2008 – p. 7/32

Non-perturbative Debye screeningµ q -dependence43.532.521.510.50leading order perturbation√theory:m D = g(T)T1 + n f6 + n (f µq2π 2 TTaylor expansion, 2-flavor QCD:( ) 2 µqm D (T) = m 0 (T) + m 2 (T) + O(µ 4 qT)m 2 /m 0 = 3/8π 2 : agrees with perturbation theory for T> ∼1.5T cm 1 0 /T 21.510.50T/T c1 1.5 2 2.5 3 3.5 4M. Döring et al., Eur. Phys. J. C46 (2006) 179) 2m 1 2 /T1/2 ⋅ m av2 /TT/T c1 1.5 2 2.5 3 3.5 4F. Karsch, xQCD, July 2008 – p. 8/32

Non-perturbative Debye screeningµ q -dependence43.532.521.510.50leading order perturbation√theory:m D = g(T)T1 + n f6 + n (f µq2π 2 Tm 1 0 /T 21.510.50T/T c1 1.5 2 2.5 3 3.5 4M. Döring et al., Eur. Phys. J. C46 (2006) 179) 2non-perturbativeeffects are in the gluequark sectorTaylor expansion, 2-flavor QCD:( ) ”perturbative”2µqm D (T) = m 0 (T) + m 2 (T) + O(µ 4 qT) above T> ∼1.5T c ?m 2 /m 0 = 3/8π 2 : agrees with perturbation theory for T> ∼1.5T cm 1 2 /T1/2 ⋅ m av2 /TT/T c1 1.5 2 2.5 3 3.5 4F. Karsch, xQCD, July 2008 – p. 8/32

The spatial string tensionDoes dimensional reduction work with light quarks?Non-perturbative, vanishes in high-T perturbation theory:√σs = −lim ln W(R x, R y )R x ,R y →∞ R x R y1.2√σsg 2 σ (T)T = c σ , c σ ≡ c 3 , g 2 E ≡ g2 σ T_T c/ Λ MS= 1.10...1.35c 3 : 3-d SU(3), LGTg 2 σ : 2-loop dim. red. pert. th.1.02-loop4-d SU(3), LGT data:G. Boyd et al. Nucl. Phys. B469 (1996) 419T/σ s1/20.81-loop0.64d lattice, N τ1= 81.0 2.0 3.0 4.0 5.0T / T c3-d SU(3), dimensional reduction:M. Laine, Y. Schröder, JHEP 0503 (2005) 067F. Karsch, xQCD, July 2008 – p. 9/32

F. Karsch, xQCD, July 2008 – p. 9/32The spatial string tensionDoes dimensional reduction work with light quarks?Non-perturbative, vanishes in high-T perturbation theory:√σs = −lim ln W(R x, R y )R x ,R y →∞ R x R y√σsg 2 σ (T)T = c σ , c σ ≡ c 3 , g 2 E ≡ g2 σ T0.5 1 2c 3 : 3-d SU(3), LGTg 2 σ : 2-loop dim. red. pert. th.0.8r 0 Tg 2 σ(T c ) ≃ 3.7 , g 2 σ(5T c ) ≃ 20.70.60.50.41/2T/σ s (T)(2+1)-flavor QCDN τ =4N τ =6T/T 0 N τ =81 1.5 2 2.5 3 3.5 4 4.5 5dimensional reduction works for T> ∼2T c- c M (almost) flavor independent- g 2 σ(T) shows 2-loop runningc σ = 0.553(1) [SU(3)]c σ = 0.54(1) [QCD]M. Cheng et al (RBC-Bielefeld),arXiv:0806.3264

Screening of heavy quark free energies– remnant of confinement above T c –pure gauge: O.Kaczmarek, FK, P. Petreczky, F. Zantow, PRD70 (2005) 0745052-flavor QCD: O.Kaczmarek, F. Zantow, Phys. Rev. D71 (2005) 114510singlet free energy(in Coulomb gauge)1000T ≃ T c : screening forr> ∼0.5fmF 1 (r, T) ∼ α(T)r+const.500e −µ(T)r0-500F 1 [MeV]r [fm]0.76T c0.81T c0.90T c0.96T c1.00T c1.02T c1.07T c1.23T c1.50T c1.98T c4.01T c0 0.5 1 1.5 2 2.5 34α(r, T = 0)F 1 (r, T) follows linear rise of V¯qq (r, T = 0) = −3rfor T< ∼1.5T c , r< ∼0.3 fm+ σrF. Karsch, xQCD, July 2008 – p. 10/32

F. Karsch, xQCD, July 2008 – p. 11/32Singlet free energyand asymptotic freedompure gauge: O.Kaczmarek, FK, P. Petreczky, F. Zantow, PRD70 (2005) 0745052-flavor QCD: O.Kaczmarek, F. Zantow, Phys. Rev. D71 (2005) 114510singlet free energy defines a running coupling:α eff = 3r24dF 1 (r, T)dr(in Coulomb gauge)0.60.50.40.30.20.10α qq (r,T)r [fm]0.01 0.1T/T cT=01.051.101.201.301.501.603.006.009.0012.0

Singlet free energyand asymptotic freedompure gauge: O.Kaczmarek, FK, P. Petreczky, F. Zantow, PRD70 (2005) 0745052-flavor QCD: O.Kaczmarek, F. Zantow, Phys. Rev. D71 (2005) 114510singlet free energy defines a running coupling:α eff = 3r24dF 1 (r, T)dr(in Coulomb gauge)large distance: constantCoulomb term (string model)0.60.50.40.30.2short distance: running couplingα(r) from (T = 0), 3-loop0.1(S. Necco, R. Sommer,Nucl. Phys. B622 (2002) 328) 0111 000111 000111 000111 000111 000α qq (r,T)α ≡ π/16r [fm]0.01 0.1short distance physics ⇔ vacuum physics00 1100 1100 11000 111000 111000 111T/T cT=01.051.101.201.301.501.603.006.009.0012.0T-dependence starts in non-perturbativeregime for T< ∼ 3 T cF. Karsch, xQCD, July 2008 – p. 11/32

Singlet free energyand asymptotic freedompure gauge: O.Kaczmarek, FK, P. Petreczky, F. Zantow, PRD70 (2005) 0745052-flavor QCD: O.Kaczmarek, F. Zantow, Phys. Rev. D71 (2005) 114510singlet free energy defines a running coupling:α eff = 3r24dF 1 (r, T)dr(in Coulomb gauge)large distance: constantCoulomb term (string model)0.60.50.40.300 1100 1100 11111 000111 000111 000111 000111 000α qq (r,T)rise due toconfinementT/T cT=01.051.101.201.301.501.603.006.009.0012.0α eff ∼ σr 2α ≡ π/16regime for T< ∼ 3 T cF. Karsch, xQCD, July 2008 – p. 11/320.2short distance: running couplingα(r) from (T = 0), 3-loop0.1(S. Necco, R. Sommer,r [fm]Nucl. Phys. B622 (2002) 328) 00.01 0.1T-dependence starts in non-perturbativeshort distance physics ⇔ vacuum physics00 1100 1100 11000 111000 111000 111

F. Karsch, xQCD, July 2008 – p. 12/32QCD equation of statetwo prominent features of EoS that characterize thenon-perturbative structure of QCD at high temperaturestrong deviations from ideal gas behavior (ǫ = 3p) forT c ≤ T< ∼3T cdeviations from Stefan-Boltzmann limit persist even at high T16.014.012.010.08.06.04.02.00.0ε SB /T 4strong deviations from ε = 3p (10-15)%dev.ε/T 4 3p/T 43 flavor, N τ =4, p4 staggeredm π =770 MeVT/T c1.0 1.5 2.0 2.5 3.0 3.5 4.0

F. Karsch, xQCD, July 2008 – p. 12/32QCD equation of statetwo prominent features of EoS that characterize thenon-perturbative structure of QCD at high temperaturestrong deviations from ideal gas behavior (ǫ = 3p) forT c ≤ T< ∼3T cdeviations from Stefan-Boltzmann limit persist even at high Tstructure of EoS is ’universal’, i.e. shows little quark massdependence in ǫ/ǫ SB vs. T/T cquark content changes only ’details’

F. Karsch, xQCD, July 2008 – p. 13/32SU(3) Equation of Statepressure: LGT vs. HTLhigh T part of the pressure calculated on the lattice is in good agreementwith HTL-resummed perturbation theory for T > ∼3T c1.81.61.41.2p/T 4HTL1.00.80.60.40.20.0(1x1)(1x2)RG(1x2) tadT/T c1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5HTL:J.P. Blaizot,E. Iancu, A. Rebhan,PL B470 (99) 181no need for AdS/QCD to explain ’pressure gap’

The pressure revisitedT> ∼(2-3)T c : deviations from ideal gas understood in terms ofHTL-resummed perturbation theoryT< ∼2T c : strong deviations from ideal gasdeviations from p SB almost flavor independent54p/T 4 p SB /T 41.00.8p/p SB30.6213 flavour2+1 flavour2 flavourpure gauge0.40.23 flavour2 flavour2+1 flavourpure gauge0T [MeV]100 200 300 400 500 6000.0T/T c1.0 1.5 2.0 2.5 3.0 3.5 4.0FK, E. Laermann and A. Peikert, Phys. Lett. B478 (2000) 447F. Karsch, xQCD, July 2008 – p. 14/32

(2+1)-flavor QCD: ...towards the cont. limit (N τ = 8)with light quarks (m π ≃ 220 MeV)980.4 0.6 0.8 1.0 1.2Tr 04.54.00.8 1.0 1.2 1.4 1.6 1.8(ε-3p)/T 4Tr 07654(ε-3p)/T 4 asqtad: N τ =68p4: N τ =683.53.02.52.0p4: N τ =468asqtad: N τ =68321.51.0hotQCDpreliminary1T [MeV]0100 150 200 250 300 350 400 450 500 550trace anomaly0.50.0T [MeV]300 400 500 600 700 800high-T behaviornon-conformal:[(ǫ − 3p)/T 4 ] max ≃ (6 − 7)at T max ≃ 200 MeV(∼ softest point of EoS)some cut-off effects in the peak region;p4 and asqtad agreeT> ∼300 MeV: good agreement betweenN τ = 6 and 8 resultsnon-perturbative:(ǫ − 3p)/T 4 ∼ A/T 2 + B/T 4for 1.5T c < ∼T< ∼4T cF. Karsch, xQCD, July 2008 – p. 15/32

F. Karsch, xQCD, July 2008 – p. 16/32EoS: low T regime87654320.32 0.36 0.40 0.44 0.48(ε-3p)/T 4asqtad: N τ =68p4: N τ =68hotQCDpreliminaryTr 01T [MeV]0130 140 150 160 170 180 190 200LGT vs resonance gasapproach to physical quark masses:m q = 0.1m s → m q = 0.05m s(m π ≃ 220 MeV → 150 MeV)

F. Karsch, xQCD, July 2008 – p. 16/32EoS: low T regime870.32 0.36 0.40 0.44 0.48(ε-3p)/T 4Tr 0870.32 0.36 0.40 0.44 0.48(ε-3p)/T 4Tr 0654asqtad: N τ =68p4: N τ =686540.1m s : N τ =680.05m s : N τ =8HRG32hotQCDpreliminary321T [MeV]0130 140 150 160 170 180 190 200LGT vs resonance gasapproach to physical quark masses:m q = 0.1m s → m q = 0.05m s(m π ≃ 220 MeV → 150 MeV)→ O(5MeV) shift of T -scale1000 111000 111000 111000 111000 111000 111T [MeV]0130 140 150 160 170 180 190 200∼ physical quark massesgood agreement with HRG model inthe transition region;still need to control cut-off effects

F. Karsch, xQCD, July 2008 – p. 17/32Pressure, Energy and Entropyp/T 4 from integration over (ǫ − 3p)/T 5 ;p(T 0 ) = 0 at T 0 = 0 MeV(exponential extrapolation);systematic error on 3p/T 4 ≃ 0.33good scaling behavior; good agreement between different discretization schemes1614121086420.4 0.6 0.8 1 1.2ε/T 4 Tr 03p/T 4p4asqtadp4asqtadT [MeV]ε SB /T 4hotQCDpreliminary0100 150 200 250 300 350 400 450 500 5501614121086420.4 0.6 0.8 1 1.2 1.4 1.6Tr 0T [MeV]ε/T 4 : N τ =4683p/T 4 : N τ =468ε SB /T 40100 200 300 400 500 600 700

F. Karsch, xQCD, July 2008 – p. 17/32Pressure, Energy and Entropyp/T 4 from integration over (ǫ − 3p)/T 5 ;p(T 0 ) = 0 at T 0 = 0 MeV(exponential extrapolation);systematic error on 3p/T 4 ≃ 0.33good scaling behavior; good agreement between different discretization schemes1614121086420.4 0.6 0.8 1 1.2ε/T 4 Tr 03p/T 4p4asqtadp4asqtadT [MeV]ε SB /T 4hotQCDpreliminary0100 150 200 250 300 350 400 450 500 55020151050.4 0.6 0.8 1 1.2s/T 3 Tr 0s SB /T 3p4: N τ =68asqtad: N τ =68T [MeV]0100 150 200 250 300 350 400 450 500 550

F. Karsch, xQCD, July 2008 – p. 18/32Hadronic fluctuations at µ > 0 fromTaylor expansion coefficients at µ = 0n f = 2, m π ≃ 770 MeV: S. Ejiri, FK, K.Redlich, PLB633 (2006) 275n f = 2 + 1, m π ≃ 220 MeV: RBC-Bielefeld, preliminaryTaylor expansion of bulk thermodynamics in terms of µ u,d,spT 41≡V T ln Z(V, T, µ u, µ 3 d , µ s )= ∑ ( ) i ( ) j (µu µu µsc i,j,kT T Ti,j,k) kexpansion coefficients evaluated at µ u,d,s = 0 are related tofluctuations of B, S, Q at µ B,S,Q = 0:⇑ baryon number, strangeness, charge fluctuationsevent-by-event fluctuations at RHIC and LHC

F. Karsch, xQCD, July 2008 – p. 19/32Hadronic fluctuations at µ > 0 fromTaylor expansion coefficients at µ = 0n f = 2, m π ≃ 770 MeV: S. Ejiri, FK, K.Redlich, PLB633 (2006) 275n f = 2 + 1, m π ≃ 220 MeV: RBC-Bielefeld, preliminaryhigher derivatives ⇒ higher momentsmixed derivatives ⇒ correlations2c x 2 = ∂2 p/T 4∂(µ x /T) 2 = 1V T 3 〈(δN x) 2 〉 µ=0 = 1V T 3 〈N2 x 〉 µ=024c x 4 = ∂4 p/T 4∂(µ x /T) 4 = 1V T 3 `〈(δNx ) 4 〉 − 3〈(δN x ) 2 〉 2´µ=0 = 1V T 3 `〈N4x 〉 − 3〈N 2 x〉 2´µ=04c xy22 = ∂ 4 p/T 4∂(µ x /T) 2 ∂(µ y /T) 2 = 1V T 3 ˆ〈N2x N 2 y 〉 − 2〈N x N y 〉 2 − 〈N 2 x〉〈N 2 y 〉˜µ=0with x = q, s

Christian Schmidt (for RBC-Bielefeld), Lattice 2008F. Karsch, xQCD, July 2008 – p. 20/32• Results for expansion coefficients0.60.50.4c u 2Taylor expansion of the pressure 7SB0.120.10.08c u,d,si,j,kc u 4n f =2+1, m =220 MeVn f =2, m =770 MeVfilled: n t =4open: n t =60.30.20.10.0250.020.0150.010.0050-0.005-0.01-0.015-0.02n f =2+1, m =220 MeVn f =2, m =770 MeVfilled: n t =4T[MeV ]T[MeV ]open: n t =6Preliminary0150 200 250 300 350 400 450c u 6n f =2+1, m =220 MeVn f =2, m =770 MeVPreliminary-0.025150 200 250 300 350 400 450SB0.060.040.02T[MeV ]Preliminary0150 200 250 300 350 400 450Cut-off dependance:→ Small cut-off effects in the transitionregion (similar to p, e-3p, ...)Mass dependance:Tc decreases with decreasing mass→ Fluctuations increase with decreasingmassSBred: RBC-Bielefeld, preliminaryblue: PRD71:054508,2005.

Christian Schmidt (for RBC-Bielefeld), Lattice 2008F. Karsch, xQCD, July 2008 – p. 21/32Hadronic fluctuations and the QCD critical point 8•Baryon number fluctuations(µ B = 0)1.22c B 2 = 〈 B 2〉 24c B 4 = 〈 B 4〉 − 3 〈 B 2〉 2110SB120.88n f =2+1, m =220 MeV0.6n f =2+1, m =220 MeV6n f =2, m =770 MeVfilled: nt=40.4n f =2, m =770 MeV4open: nt=60.2T[MeV ]filled: nt=4open: nt=60150 200 250 300 350 400 4502T[MeV ]0150 200 250 300 350 400 450SB21.510.512 cB 4c B 2T[MeV ]n f =2+1, m =220 MeV=n f =2, m =770 MeVResonance gas〈B4 〉 − 3 〈 B 2〉 2〈B 2 〉filled: nt=4open: nt=60150 200 250 300 350SB→ fluctuations increase withdecreasing mass→ fluctuations increase overthe resonance gas valuered: RBC-Bielefeld, preliminaryblue: PRD71:054508,2005.

Quark number in Boltzmann approximationbaryonic sector of pressure in a hadron resonance gas;m B ≫ T ⇒ Boltzmann approximation: p B /T 4 =∑p m /T 4m≤m maxwith p m /T 4 = F(T, m, V ) cosh(Bµ B /T)χ B 2≡ ∂2 p m /T 4∂(µ B /T) 2 = B2 F(T, m, V ) cosh(Bµ B /T)χ B 4≡ ∂4 p m /T 4∂(µ B /T) 4 = B4 F(T, m, V ) cosh(Bµ B /T)ratio of fourth (χ B 4 ) and second (χB 2) cumulant of quark number fluctuationgives ”unit of charge” carried by the particle with mass ”m”:m ≫ T ⇒ R B 4,2 ≡ χB 4χ B 2= B 2 F. Karsch, xQCD, July 2008 – p. 22/32

Charge fluctuations inBoltzmann approximationhadronic resonance gas: contributions from isosinglet (G (1) : η, ...)and isotriplet (G (3) : π, ...) mesons as well as isodoublet(F (2) : p, n, ...) and isoquartet (F (4) : ∆, ...) baryonsp(T, µ q , µ I )T 4 ≃ G (1) (T) + G (3) (T) 1 3„ «+F (2) 3µq(T) cosh coshT+F (4) (T) 1 2 cosh „ 3µqT„ „ 2µI2 coshT“ µIT« »cosh”“ µIT«”«+ 1„ 3µI+ coshT«–charge fluctuations at µ q = µ I = 0;isospin quartet F (4) contains baryons carrying charge 2R Q 4,2 ≡ χQ 4χ Q 2= 4G(3) + 3F (2) + 27F (4)4G (3) + 3F (2) + 9F (4) → 1 for T → 0contribution of doubly charged baryons increases quartic relativeto quadratic fluctuationsF. Karsch, xQCD, July 2008 – p. 23/32

F. Karsch, xQCD, July 2008 – p. 24/32Ratios of quartic and quadratic fluctuationsof charges in (2+1)-flavor QCDRBC-Bielefeld, preliminarybaryon number fluctuationcharge fluctuation2n f =2+1, m π =220 MeVn f =2, m π =770 MeVResonance gas2n f =2+1, m π =220 MeVn f =2, m π =770 MeV1.51.5Resonance gas11filled: nt=40.5open: nt=6SB0150 200 250 300 3500.5SB050 100 150 200 250 300 350 400 450chiral limit: ratios ∼ |T − T c | −α + regular⇒ enhancement over resonance gas values⇒ may be observable in event-by-event fluctuations(valence) quark sector quickly (T > ∼1.5T c ) behaves perturbative

F. Karsch, xQCD, July 2008 – p. 25/32Hadronic fluctuations andchiral symmetry restorationexpect 2 nd order transition in 3-d, O(4) symmetry class;scaling field: t =˛T − T cT c„ « 2 µq˛˛˛˛ + A , µ crit = 0T csingular part: f s (T, µ u , µ d ) = b −1 f s (tb 1/(2−α) ) ∼ t 2−α∂ 2 ln Z∂µ 2 q∼ t 1−α ,∂ 4 ln Z∂µ 4 q∼ t −α (µ = 0)O(4)/O(2): α < 0, small ⇒〈(δN q ) 2 〉 dominated by T-dependence of regular part〈(δN q ) 4 〉 develops a cusp

F. Karsch, xQCD, July 2008 – p. 26/32Quadratic fluctuations of baryon numbercharge & strangeness in (2+1)-flavor QCDvanishing chemical potentials:RBC-Bielefeld, preliminary1.110.90.80.70.60.50.40.3Bχ 2Qχ 2Sχ 20.20.1T[MeV]0150 200 250 300 350 400 450SBfull: N t =4open: N t =6χ Q 2 = 1V T 3〈Q2 〉χ B 2 = 1V T 3〈N2 B 〉χ S 2 = 1V T 3 〈N2 S 〉rapid approach to SB limit⇒ smooth change of quadratic fluctuations across transition regionchiral limit: χ B 2 , χQ 2 ∼ |T − T c| 1−α + regular

F. Karsch, xQCD, July 2008 – p. 27/32Quartic fluctuations of baryon numbercharge & strangeness in (2+1)-flavor QCDvanishing chemical potentials:12108642T[MeV]0150 200 250 300 350 400 450χ 4Bχ 4Qχ 4Sfull: N t =4open: N t =6SBRBC-Bielefeld, preliminaryχ Q 4 = 1V T 3 (〈Q 4 〉 − 3〈Q 2 〉 2)χ B 4 = 1V T 3 (〈N4B 〉 − 3〈N2 B 〉2)χ S 4 = 1V T 3 (〈N4S 〉 − 3〈N2 S 〉2)rapid approach to SB limit⇒ large light quark number & charge fluctuations across transition regionchiral limit: χ B 4 , χQ 4 ∼ |T − T c| −α + regular

F. Karsch, xQCD, July 2008 – p. 28/32Deconfinement and χ-symmetryThe chiral phase transition (i.e. at m q = 0) is deconfiningtrue in QCD, i.e. SU(3) + fermions in the fundamentalrepresentationSU(3) + fermions in the adjoint representation: T deconf < T χThe transition in QCD with physical quark masses is a crossoverIn which sense is the transitiondeconfining and chiral symmetry restoring?deconfinement: heavy hadrons ⇒ light quarks and gluons;liberation of many new light degrees of freedom⇒ rapid change in ǫ/T 4 , s/T 3 , ....chiral symmetry restoration: vanishing mass splittings,no new degrees of freedom⇒ minor effect on bulk thermodynamics, butrapid change of chiral condensate

F. Karsch, xQCD, July 2008 – p. 28/32Deconfinement and χ-symmetryThe chiral phase transition (i.e. at m q = 0) is deconfiningtrue in QCD, i.e. SU(3) + fermions in the fundamentalrepresentationSU(3) + fermions in the adjoint representation: T deconf < T χThe transition in QCD with physical quark masses is a crossoverIn which sense is the transitiondeconfining and chiral symmetry restoring?deconfinement: heavy hadrons ⇒ light quarks and gluons;liberation of many new light degrees of freedom⇒ rapid change in ǫ/T 4 , s/T 3 , ....chiral symmetry restoration: vanishing mass splittings,no new degrees of freedom⇒ minor effect on bulk thermodynamics, butrapid change of chiral condensate

χ-symmetry restoration300250χ-symmetry restoration:drop in condensate;peak in susceptibilitiesχ singlet /T 2hotQCDpreliminary∆ l,s (T) = 〈 ¯ψψ〉 l,T − m lm s〈 ¯ψψ〉 s,T〈 ¯ψψ〉 l,0 − m lm s〈 ¯ψψ〉 s,01.00.80.35 0.40 0.45 0.50 0.55Tr 0∆ l,sN τ =82000.60.4asqtad: m l /m s =0.1p4: m l /m s =0.10.051501000.75*p4fat3asqtadT [MeV]140 160 180 200 220 2400.20.0140 160 180 200 220 240F. Karsch, xQCD, July 2008 – p. 29/32

Deconfinement and χ-symmetryand bulk thermodynamics1.801.601.401.201.000.800.60χ-symmetry restoration:drop in condensate;peak in susceptibilitiesχ l,discillustrative fit:A (T-T c ) -1.32∆ l,s (T) = 〈 ¯ψψ〉 l,T − m lm s〈 ¯ψψ〉 s,T〈 ¯ψψ〉 l,0 − m lm s〈 ¯ψψ〉 s,01.00.80.60.40.35 0.40 0.45 0.50 0.55Tr 0∆ l,sN τ =8asqtad: m l /m s =0.1p4: m l /m s =0.10.050.400.20T[MeV]0.00165 170 175 180 185 190 195 200 205 2100.20.0140 160 180 200 220 240F. Karsch, xQCD, July 2008 – p. 29/32

Deconfinement and χ-symmetryand bulk thermodynamicsmost prominent features of bulk thermodynamics are related todeconfinementχ-symmetry restoration:drop in condensate;peak in susceptibilities20150.4 0.6 0.8 1 1.2s/T 3 Tr 0s SB /T 3300250χ singlet /T 2hotQCDpreliminary105p4: N τ =68asqtad: N τ =68200T [MeV]0100 150 200 250 300 350 400 450 500 5501501000.75*p4fat3asqtadT [MeV]140 160 180 200 220 240do they stay closely relatedin the continuum limit?F. Karsch, xQCD, July 2008 – p. 29/32

F. Karsch, xQCD, July 2008 – p. 30/32Conclusionsglue sticksthe interesting non-perturbative physics in QCD happens in thegluon sectorquarks add flavorquarks add to the picture by ’modifying prefactors’ (in accord withdimensional reduction approach)no glue ⇒ no binding’glue-free’ observables show early onset of perturbative behaviour

F. Karsch, xQCD, July 2008 – p. 31/32Conclusionsnon-perturbative QCD-EoS ∼ pure gauge theory EoSthe interesting non-perturbative physics in QCD happens in thegluon sectornothing qualitatively new in QCD with light quarksquarks add to the picture by ’modifying prefactors’ (in accord withdimensional reduction approach) except close to T c !!quantum numbers are carried by ”quarks” already close to T c’glue-free’ observables show early onset of perturbative behaviour

F. Karsch, xQCD, July 2008 – p. 32/32Finally...the regime T c ≤ T< ∼(1.5 − 2.0)T c differs fromthe regime T> ∼(1.5 − 2.0)T cIt is more difficult (impossible?) to describe it quantitativelyin terms of conventional theoretical high-T concepts:perturbation theory, resummation, dimensional reduction

F. Karsch, xQCD, July 2008 – p. 32/32Finally...the regime T c ≤ T< ∼(1.5 − 2.0)T c differs fromthe regime T> ∼(1.5 − 2.0)T cIt is more difficult (impossible?) to describe it quantitativelyin terms of conventional theoretical high-T concepts:perturbation theory, resummation, dimensional reductionDo we see new physics? ⇒ Quark Gluon Liquidor, remnants of old physics? ⇒ confinement