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1 The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density From Lattices to Stars INT, University of Washington, Seattle, 28. April 2004 Christian Schmidt Universität Wuppertal Publications: Karsch, Laermann, CS, The chiral critical point in 3-flavor QCD, Phys. Lett. B520 (2001) 41 [hep-lat/ ]. Allton et. al., The QCD thermal phase transition in the presence of a small chemical potential, Phys. Rev. D66 (2002) [hep-lat/ ]. Allton et. al., The equation of state for two flavor QCD at non-zero chemical potential, Phys. Rev. D68 (2003) [hep-lat/ ]. Allton et. al., Where is the chiral critical point in 3-flavor QCD?, hep-lat/ The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 1/28

2 The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density From Lattices to Stars INT, University of Washington, Seattle, 28. April 2004 Thanks to: Bielefeld Swansea Collaboration: C. R. Allton, S. Ejiri, S. J. Hands, F. Karsch, O. Kaczmarek, E. Laermann, CS The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 1/28

3 The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density From Lattices to Stars INT, University of Washington, Seattle, 28. April 2004 Introduction / Motivation 1) Lattice QCD at non-zero density via Taylor expansion reweighting and Taylor-expanded reweighting in chemical potential, quark mass,... 2) Investigation of the 3-flavor Phase Diagram the universality-class, a first determination of the chiral critical point,... 3) Conclusions for 2+1 flavor QCD the critical surface, the physical point,... Summary The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 1/28

4 Motivation The phase diagram of hot and dens matter K.Rajagopal, F. Wilczek, The condensed matter physics of QCD, hep-ph/ T quark qluon plasma Why lattice calculations? crossover quark matter vacuum hadronic fluid n = 0 n B> 0 B µ o nuclear matter 922 MeV crossover superfluid/superconducting phases? 2SC CFL µ The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 2/28

5 Motivation The phase diagram of hot and dens matter K.Rajagopal, F. Wilczek, The condensed matter physics of QCD, hep-ph/ T Why lattice calculations? quantitative approach to QCD at T T c quark qluon plasma crossover quark matter vacuum hadronic fluid n = 0 n B> 0 B µ o nuclear matter 922 MeV crossover superfluid/superconducting phases? 2SC CFL µ The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 2/28

6 Motivation The phase diagram of hot and dens matter K.Rajagopal, F. Wilczek, The condensed matter physics of QCD, hep-ph/ T LHC RHIC crossover vacuum quark qluon plasma hadronic fluid n = 0 n B> 0 B SPS µ o nuclear matter 922 MeV future GSI experiments? quark matter crossover superfluid/superconducting phases? 2SC Why lattice calculations? quantitative approach to QCD at T T c CFL µ important for Heavy Ion Collisions fire ball s thermal equilibrium only for the strong processes µ s = 0 The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 2/28

7 Motivation The phase diagram of hot and dens matter K.Rajagopal, F. Wilczek, The condensed matter physics of QCD, hep-ph/ T early universe LHC RHIC crossover vacuum quark qluon plasma hadronic fluid n = 0 n B> 0 B SPS µ o nuclear matter 922 MeV future GSI experiments? quark matter crossover superfluid/superconducting phases? 2SC Why lattice calculations? quantitative approach to QCD at T T c CFL µ important for Heavy Ion Collisions Cosmology QCD phasetransition in the early universe 10 4 s after the big bang inhomogeneities of the universe? The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 2/28

8 Motivation The phase diagram of hot and dens matter K.Rajagopal, F. Wilczek, The condensed matter physics of QCD, hep-ph/ T early universe LHC RHIC crossover quark qluon plasma SPS future GSI experiments? quark matter Why lattice calculations? quantitative approach to QCD at T T c important for Heavy Ion Collisions Cosmology Astrophysics vacuum hadronic fluid n = 0 n B> 0 B µ o nuclear matter 922 MeV crossover superfluid/superconducting phases? 2SC neutron / quarkstars? CFL µ The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 2/28

9 Motivation The phase diagram of hot and dens matter K.Rajagopal, F. Wilczek, The condensed matter physics of QCD, hep-ph/ T early universe LHC RHIC crossover quark qluon plasma SPS future GSI experiments? quark matter Why lattice calculations? quantitative approach to QCD at T T c important for Heavy Ion Collisions Cosmology Astrophysics vacuum hadronic fluid n = 0 n B> 0 B µ o nuclear matter 922 MeV crossover superfluid/superconducting phases? 2SC neutron / quarkstars? CFL µ Lattice results: T c 170 MeV ɛ c 0.7 GeV/fm 3 (Karsch, Laermann, Peikert) The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 2/28

10 Introduction I 3-flavor QCD: m u = m d = m s µ 1st order crossover m crit m The goal of this work: Investigation of the (m, µ)-dependence of the chiral critical end-point in 3-flavor QCD on relative large volumes (12 4 4, ) with an improved action (p4) The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 3/28

11 Introduction I 3-flavor QCD: m u = m d = m s µ 1st order crossover m crit m The goal of this work: Investigation of the (m, µ)-dependence of the chiral critical end-point in 3-flavor QCD on relative large volumes (12 4 4, ) T chiral symmetric with an improved action (p4) chiral broken µ The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 3/28

12 Introduction I 3-flavor QCD: m u = m d = m s µ 1st order crossover m crit m The goal of this work: Investigation of the (m, µ)-dependence of the chiral critical end-point in 3-flavor QCD on relative large volumes (12 4 4, ) T chiral symmetric with an improved action (p4) chiral broken µ The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 3/28

13 Introduction I 3-flavor QCD: m u = m d = m s µ 1st order crossover m crit m The goal of this work: Investigation of the (m, µ)-dependence of the chiral critical end-point in 3-flavor QCD on relative large volumes (12 4 4, ) T chiral symmetric with an improved action (p4) chiral broken µ The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 3/28

14 Introduction I 3-flavor QCD: m u = m d = m s µ 1st order crossover m crit m The goal of this work: Investigation of the (m, µ)-dependence of the chiral critical end-point in 3-flavor QCD on relative large volumes (12 4 4, ) T chiral symmetric with an improved action (p4) chiral broken µ The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 3/28

15 Introduction II Adding irrelevant operators ( a 0 0 ): Quarks: S F (N τ, N σ ) = n,n µ η(n µ ) χ n ω + ω ν µ ν µ χ n + n m q χ n χ n The coefficients are determined that way to make the free quark propagator rotational invariant up to O(p 4 ). p4-action [Karsch, Heller, Sturm (1999)] The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 4/28

16 Introduction II Adding irrelevant operators ( a 0 0 ): Quarks: Properties of the p4-action: 3 S F (N τ, N σ ) = η(n µ ) χ n n,n 8 improved rotational µ symmetry of the free quark propagator: O(p 4 ) improved flavor symmetry due to fat-links ν µ (tree-level improvement: O(g 0 )) ~ E(p) ~ 1 1+6ω π π/2 + ω Kontinuum Standard p4 ν µ χ n + n m q χ n χ n dispersions-relation: The coefficients are determined that way to make the free quark propagator rotational invariant up to O(p 4 ). p4-action p~ [Karsch, Heller, Sturm (1999)] 0 π/2 π The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 4/28

17 Introduction II Adding irrelevant operators ( a 0 0 ): Quarks: Properties of the p4-action: 3 S F (N τ, N σ ) = η(n µ ) χ n n,n 8 improved rotational µ symmetry of the free quark propagator: O(p 4 ) improved flavor symmetry due to fat-links ν µ (tree-level improvement: O(g 0 )) ω P / P SB Standard p4 + ω ν µ χ n + n m q χ n χ n 1.0 cut-off effects of the pressure: The coefficients are determined that way to make the free quark 0.5 propagator rotational invariant up to O(p 4 ). N τ p4-action [Karsch, Heller, Sturm (1999)] The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 4/28

18 Introduction II Adding irrelevant operators ( a 0 0 ): Quarks: Properties of the p4-action: S F (N τ, N σ ) = Gluons: S G (N τ, N σ ) = 1 6 n,n µ n µ,ν>µ ν µ η(n µ ) χ n Re Tr ω Re Tr ω ν µ Symanzik improved eliminates χ n + the m q cut-off χ n χ n n effects of order O(a 2 ) The coefficients are determined that way to make the free quark [Weisz, Wohlert (1984)] propagator rotational invariant up to O(p 4 ). p4-action [Karsch, Heller, Sturm (1999)] (tree-level improvement O(g 0 )) The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 4/28

19 The mass scale: m π /σ 1/2 fit: a m 1/2 + b m 1 m The critical temperature: T c / σ n f =2, p4 n f =3, p4 n f =2, std Introduction III m PS / σ Karsch, Laermann, Peikert (2001) assume: m π / σ = 5.71(9) m, with σ = 425 MeV. simulation point: am = m π 170 MeV mass dependence of T c : T c / σ = 0.40(1)+0.039(4)m π / σ, T c in the chiral limit of n f = 3: T c = 154(8) MeV T c at m π 170 MeV: T c 160 MeV The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 5/28

20 1) Lattice QCD at non-zero Density the Sign-Problem Monte Carlo simulations at µ > 0: Z(T, V, µ) = D ψdψda exp{ S E (T, V, µ)} = DA detm(µ) exp{ S G (T, V )} fermion determinant complex for µ > 0 detm = detm e iθ Phase θ is strongly fluctuating Oe iθ..., e iθ... 0 Sign-Problem! The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 6/28

21 1) Lattice QCD at non-zero Density the Sign-Problem Monte Carlo simulations at µ > 0: Z(T, V, µ) = D ψdψda exp{ S E (T, V, µ)} = DA detm(µ) exp{ S G (T, V )} T µ/t < 1 fermion determinant complex for µ > 0 detm = detm e iθ Phase θ is strongly fluctuating Oe iθ..., e iθ... 0 µ Sign-Problem! The problem may not be serve for small µ/t! The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 6/28

22 1) Lattice QCD at non-zero Density solutions for small µ/t exact reweighting: the calculation of the fermion determinant is numerical demanding the reweighting technique works for small volumes Fodor, Katz (2002) Taylor-expansion around µ = 0: The Taylor expanded version of reweighting reduces the numerical effort Bielefeld-Swansea (2002) direct calculations of Taylor coefficients of observables avoids reweighting QCD TARO (2002) Bielefeld-Swansea (2003) Gavai, Gupta (2003) imaginary chemical potential: analytical continuation required deforcrand, Philipsen (2002) D Elia, Lombardo (2003) The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 7/28

23 1) Lattice QCD at non-zero Density reweighting O β,m,µ = 1 Z(T, V ) DU O (det M(m, µ)) N f /4 e βs G = O exp exp N f 4 (ln det M(µ, m) ln det M(µ 0, m 0 )) S G N f 4 (ln det M(µ, m) ln det M(µ 0, m 0 )) S G β 0,m 0,µ 0 β 0,m 0,µ 0. β = OR / R quark gluon plasma Ferrenberg, Swendsen (1988) Fodor, Katz (2002) hadronic phase best weight lines transition line Fodor and Katz (2002), see for instance Katz, proceedings of Lattice 2003, hep-lat/ µ high numerical effort problems with V large The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 8/28

24 1) Lattice QCD at non-zero Density Taylor expansion Taylor-expanded reweighting up to order O(ω 2 ), ω = (m, µ): O ω,β = (O 0 + O 1 ω + O 2 ( ω) 2 ) exp{r 1 ω + R 2 ( ω) 2 } exp{ S G } exp{r 1 ω + R 2 ( ω) 2 } exp{ S G } ω0,β 0 ω 0,β 0 +O(( ω) 3 ) Bielefeld-Swansea, Phys. Rev. D66 (2002) R 1 ω = n f 4 ln det M m m=0 m + n f 4 ln det M µ µ=0 µ R 10 + R 01 R 2 ( ω) 2 = n f 8 2 ln det M m 2 m=0 ( m) 2 + n f 8 2 ln det M µ 2 µ 2 µ=0 n f 4 2 ln det M m µ m=µ=0 m µ R 20 + R 02 + R 11 The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 9/28

25 1) Lattice QCD at non-zero Density Taylor expansion reweighting operators in detail: R 10 = Tr M 1 R 01 = Tr 1 M M µ R 20 = Tr M 1 M 1 R 02 = Tr R 11 = Tr M 1 2 M µ 2 Tr 1 M M µ M 1 1 M 1 M M M µ µ reduces the numerical effort (R ij can be calculated by random noise method) still a reweighting technique The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 10/28

26 1) Lattice QCD at non-zero Density Taylor expansion Remark: reweighting operators R i1, R i3, R i5,... are pure imaginary, R i2, R i4, R i6,... are pure real. Phase fluctuations of the fermion determinant: detm = detm e iθ ; phase is given by R H I C θ = N f 4 Im [µr 01 + µ 3 R ] < cos( θ) > β=3.250, , am=0.005 β=3.260, , am=0.005 β=3.265, , am=0.005 β=3.270, , am=0.005 β=3.275, , am=0.005 β=3.250, , am=0.005 β=3.260, , am=0.005 β=3.270, , am=0.005 β=3.280, , am=0.005 β=3.460, , am=0.100 β=3.470, , am=0.100 β=3.480, , am=0.100 β=3.490, , am=0.100 aµ u = a µ d The sign-problem becomes crucial when the phase fluctuations σ θ O(π/2) The sign-problem is not crucial for RHIC physics µ max B MeV The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 11/28

27 2) Investigation of the Phase Diagram Determination of the 3-flavor chiral critical point at µ = Chiral Condensate O( m): < ψψ> / (N σ 3 Nτ ) m=0.000 m=0.001 m=0.002 m=0.003 m=0.004 m=0.005 m=0.006 m= <L> / N σ 3 Polyakov Loop: rapid change in the critical region β Simulations at quark mass m = (m q 172 MeV) = mass-reweighting, R : O(( m) 2 ) β m=0.000 m=0.001 m=0.002 m=0.003 m=0.004 m=0.005 m=0.006 m=0.007 transition becomes stronger for m 0 finite volume effect truncation error The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 12/28

28 2) Investigation of the Phase Diagram Determination of the 3-flavor chiral critical point at µ = 0 Susceptibilities: χ ψψ m=0.000 m=0.001 m=0.002 m=0.003 m=0.004 m=0.005 m=0.006 m= χ L m=0.000 m=0.001 m=0.002 m=0.003 m=0.004 m=0.005 m=0.006 m=0.007 pronounced peak in χ ψψ, χ L m-dependence of the peak position increasing peak height with m β Simulations at quark mass m = (m q 172 MeV) = mass-reweighting, R : O(( m) 2 ) β finite volume effect truncation error The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 12/28

29 2) Investigation of the Phase Diagram Determination of the 3-flavor chiral critical point at µ = 0 Indications for critical behavior: scaling behavior of the susceptibilities: χ ψψ V γ/(3ν), with γ/ν 1.96 Binder-Cumulants: B 4 (δ ψψ) 4 (δ ψψ) 2 2, with δ ψψ ψψ ψψ universality classes: 3d-Ising 3d-O(2) 3d-O(4) γ/ν 1.963(3) 1.962(5) 1.975(4) B (1) 1.242(2) 1.092(3) The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 13/28

30 2) Investigation of the Phase Diagram Determination of the 3-flavor chiral critical point at µ = 0 Indications for critical behavior: scaling behavior of the susceptibilities: Binder-Cumulants: B 4 χ ψψ V γ/(3ν), with γ/ν 1.96 Not useful for the determination of the universality class! (δ ψψ) 4 (δ ψψ) 2 2, with δ ψψ ψψ ψψ universality classes: 3d-Ising 3d-O(2) 3d-O(4) γ/ν 1.963(3) 1.962(5) 1.975(4) B (1) 1.242(2) 1.092(3) The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 13/28

31 2) Investigation of the Phase Diagram Determination of the 3-flavor chiral critical point at µ = 0 Results from standard-action: χ ψψ chiral susceptiblilies 16 3 x x4 8 3 x B x x x4 Binder cumulants m m u = m d = 0.03 Ising O(2) m = standard-action enables high statistics The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 14/28

32 2) Investigation of the Phase Diagram Determination of the 3-flavor chiral critical point at µ = 0 Results from standard-action: χ ψψ chiral susceptiblilies 16 3 x x4 8 3 x B x x x4 Binder cumulants m ISING m u = m d = 0.03 Ising O(2) m = standard-action enables high statistics The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 14/28

33 2) Investigation of the Phase Diagram Determination of the 3-flavor chiral critical point at µ = 0 Results from standard-action: χ ψψ chiral susceptiblilies 16 3 x x4 8 3 x B x x x4 Binder cumulants m ISING m u = m d = 0.03 Ising m crit O(2) m s = (34) 290 MeV m crit π = standard-action enables high statistics The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 14/28

34 2) Investigation of the Phase Diagram Determination of the 3-flavor chiral critical point at µ = Results from the p4-action: χ ψψ 12 3 x x4 3.0 B 4 (0) 12 3 x x4 very low statistics Ising 10.0 m m finite volume effect Simulations at quark mass m = (m q 172 MeV) = mass-reweighting, R : O(( m) 2 ) truncation error The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 15/28

35 2) Investigation of the Phase Diagram Determination of the 3-flavor chiral critical point at µ = Results from the p4-action: χ ψψ 12 3 x x4 3.0 B 4 (0) 12 3 x x4 very low statistics m crit = (38) = 67(17) MeV m crit π Ising huge cut-off effects Simulations at quark mass m = (m q 172 MeV) = mass-reweighting, R : O(( m) 2 ) m m finite volume effect truncation error The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 15/28

36 2) Investigation of the Phase Diagram Determination of the 3-flavor chiral critical point at µ = 0 large cut-off effect in the critical pion mass m crit 290(20) MeV standard-action π = 67(18) MeV p4-action unimproved results of Binder cumulants have been confirmed by other groops: Christ, Liao (2003); de Frocrand, Philipsen (2003) improved results are not in contradiction to other calculations with improved actions: Hasenfratz, Knechtli (2001); MILC (2003) a critical pion mass of 67 MeV is consistent with results from effective models, e.g. the SU(3) SU(3) linera sigma model see for instance Lenaghan, Phys. Rev. D63 (2001) The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 16/28

37 2) Investigation of the Phase Diagram Determination of the 3-flavor chiral critical point at m π 172 MeV, µ χ ψψ Results from the p4-action: 12 3 x x B 4 (0) 12 3 x x4 very low statistics sign-problem for V = µ u,d µ u,d finite volume effects Simulations at quark mass m = (m q 172 MeV) = µ-reweighting, R : O(µ 2 ) truncation errors The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 17/28

38 2) Investigation of the Phase Diagram Determination of the 3-flavor chiral critical point at m π 172 MeV, µ χ ψψ Results from the p4-action: 12 3 x x B 4 (0) 12 3 x x4 very low statistics sign-problem for V = µ crit = 0.074(13) = 52(10) MeV µ crit q 15 1 µ u,d µ u,d Simulations at quark mass m = (m q 172 MeV) = µ-reweighting, R : O(µ 2 ) finite volume effects truncation errors The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 17/28

39 2) Investigation of the Phase Diagram The line of second order phase transitions hallo µ crit u,d [MeV] n f =3 p4 action phys. line hallo quadratic interpolation: dm crit /d(µ 2 u,d ) = 0.87(23) 100 at a physical pion mass: 80 first order µ crit u,d = 40(9) MeV crossover m π [MeV] NOT CONSISTENT WITH: m crit (µ) m crit (µ = 0) = (36) µ πt de Forcrand, Philipsen (2003) 2 The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 18/28

40 2) Investigation of the Phase Diagram Dependence of the critical temperature from the chemical potential Determined by the change of the peak position of χ ψψ (β) as a function of µ: β c (µ). reweighting O(µ 2 ) yields the leading order dβ c /dµ 2 exact. for the conversion to the temperature scale, the lattice β-function is required, resp. the derivative: dβ/da perturbative, string tension,... in leading order we have T [MeV] dt c dµ 2 = 1 N τ T c (0) dβ c dµ 2 a dβ c da T c (µ q )/T c (0) = (35)[µ q /T c (0)] 2 Results are consistent with deforcrand and Philipsen. Region of a strong interacting hadron gas in the phase diagram?, 50 T c, Bielefeld Swansea T c, Forcrand, Philipsen T f, J.Cleymans et. al. µ B [GeV] N f = 2, m = 0.1 (m π 760 MeV) high statistics µ-reweighting: O(µ 2 ) The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 19/28

41 2) Investigation of the Phase Diagram Dependence of the critical temperature from the chemical potential Determined by the change of the peak position of χ ψψ (β) as a function of µ: β c (µ). reweighting O(µ 2 ) yields the leading order dβ c /dµ 2 exact. for the conversion to the temperature scale, the lattice β-function is required, resp. the derivative: dβ/da perturbative, string tension,... in leading order we have T [MeV] dt c dµ 2 = 1 N τ T c (0) dβ c dµ 2 a dβ c da T c (µ q )/T c (0) = (35)[µ q /T c (0)] 2 Results are consistent with deforcrand and Philipsen. Region of a strong interacting hadron gas in the phase diagram?, 50 T c, Bielefeld Swansea T c, Forcrand, Philipsen T f, J.Cleymans et. al. µ B [GeV] N f = 2, m = 0.1 (m π 760 MeV) high statistics µ-reweighting: O(µ 2 ) The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 19/28

42 2) Investigation of the Phase Diagram Dependence of the critical temperature from the chemical potential N f = 3, m = 0.005, T c / T 0 mass dependence of T c (µ q ) : T c (µ q ) T c (0) = (6) (46) µ q T c (0) µ q T c (0) 2, m = 0.1 2, m = steeper for m n f =3, m=0.005 n f =3, m=0.100 µ u,d / T (scale perturbative) The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 20/28

43 2) Investigation of the Phase Diagram Dependence of the critical temperature from the chemical potential N f = 3, m = 0.005, 0.1 in physical units: T c / T T [MeV] Pure Gauge mass dependence of T c (0) [Karsch, Laermann, Peikert (2001)] no flavor dependence n n f =3, m=0.005 f =3, m= n f =3, m=0.100 n f =3, m=0.100 µ u,d / T 0 0 µ B [GeV] behavior of the end-point analytical [deforcrand (2002)] The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 20/28

44 2) Investigation of the Phase Diagram Baryon chemical potential versus Isospin chemical potential µ µ u = µ d, µ s = 0 from peak in χ L from peak in χ ψψ fit-range m N s dβ pc /dµ 2 β pc (0) dβ pc /dµ 2 β pc (0) [µ 2 min, µ2 max] (648) (12) (620) (11) [0,0.0006] (538) (11) (689) (8) [0,0.002] (64) (26) (55) (23) [0,0.0006] µ µ I from peak in χ L from peak in χ ψψ fit-range m N s dβ pc /dµ 2 β pc (0) dβ pc /dµ 2 β pc (0) [µ 2 min, µ2 max ] (68) (12) (60) (11) [0,0.0006] (99) (11) (70) (8) [0,0.002] (93) (26) (55) (24) [0,0.0006] The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 21/28

45 2) Investigation of the Phase Diagram Baryon chemical potential versus Isospin chemical potential µ µ u = µ d, µ s = 0 from peak in χ L from peak in χ ψψ fit-range m N s dβ pc /dµ 2 β pc (0) dβ pc /dµ 2 β pc (0) [µ 2 min, µ2 max] (648) (12) (620) (11) [0,0.0006] (538) (11) (689) (8) [0,0.002] (64) (26) (55) (23) [0,0.0006] µ µ I from peak in χ L from peak in χ ψψ fit-range m N s dβ pc /dµ 2 β pc (0) dβ pc /dµ 2 β pc (0) [µ 2 min, µ2 max ] (68) (12) (60) (11) [0,0.0006] (99) (11) (70) (8) [0,0.002] (93) (26) (55) (24) [0,0.0006] = T c (µ q ) < T c (µ I )!! (for µ q small) The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 21/28

46 3) Conclusions for 2+1 flavor QCD The critical surface The 3-flavor order of phase the phase diagram, transition: µ B = 0 T n f =2 χ T n f =3 χ 175 MeV m s m s tric? 0 0? 2nd order O(4)? phys. point 1st order 155 MeV N = 2 f crossover 2nd order Z(2) m, m u d N = 3 f Pure Gauge 1st order N = 1 f m crit PS mcrit no phase transition in a huge quark mass range (crossover region) the physical point most probability in the crossover region T d 270 MeV PS 2.5 GeV 70 MeV The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 22/28

47 3) Conclusions for 2+1 flavor QCD The critical surface The 3-flavor order of phase the phase diagram, transition: µ B = 0 T n f =2 χ T n f =3 χ 175 MeV m s m s tric? 0 0? 2nd order O(4)? phys. point 1st order 155 MeV N = 2 f crossover 2nd order Z(2) m, m u d N = 3 f Pure Gauge 1st order N = 1 f m crit PS mcrit no phase transition in a huge quark mass range (crossover region) the physical point most probability in the crossover region T d 270 MeV µ PS 2.5 GeV FIRST ORDER 70 MeV Extension to µ 0: N = 2 f phys. line CROSSOVER N = 3 f 0 u,d o m 0 m s o The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 22/28

48 3) Conclusions for 2+1 flavor QCD Reweighting in (m u,d, m s ) The reweighting factor: R = 2 [ ln det M(m u,d ) ln det M(m u,d0 ) ] = + ln det M(m s ) ln det M(m s 0)) R n (u,d) m n u,d + R n (s) m n s, n=1 n=1 with m = m m 0 and R (u,d,s) n = 1 n! n ln det M m n m=mu,d,s0 The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 23/28

49 3) Conclusions for 2+1 flavor QCD The quark mass plane m s two-state, Columbia [80] no two-state, Columbia [80] crossover like, JLQCD [81] 1st order like, JLQCD [81] two-state, JLQCD [81] B 4 =1.604 Rew(0.03) m crit linear σ model m u,d The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 24/28

50 3) Conclusions for 2+1 flavor QCD The quark mass plane m s two-state, Columbia [80] no two-state, Columbia [80] crossover like, JLQCD [81] 1st order like, JLQCD [81] two-state, JLQCD [81] B 4 =1.604 Rew(0.03) m crit linear σ model 2 critical points have been determined, not in contradiction to earlier calculation [Brown et. al. (1990), Aoki et. al. (1999)] m u,d The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 24/28

51 3) Conclusions for 2+1 flavor QCD The quark mass plane m s two-state, Columbia [80] no two-state, Columbia [80] crossover like, JLQCD [81] 1st order like, JLQCD [81] two-state, JLQCD [81] B 4 =1.604 Rew(0.03) m crit linear σ model 2 critical points have been determined, not in contradiction to earlier calculation [Brown et. al. (1990), Aoki et. al. (1999)] slope at the 3-flavor critical point 2, from mass-reweighting O( m), expected results m u,d The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 24/28

52 3) Conclusions for 2+1 flavor QCD The quark mass plane m s two-state, Columbia [80] no two-state, Columbia [80] crossover like, JLQCD [81] 1st order like, JLQCD [81] two-state, JLQCD [81] B 4 =1.604 Rew(0.03) m crit linear σ model 2 critical points have been determined, not in contradiction to earlier calculation [Brown et. al. (1990), Aoki et. al. (1999)] slope at the 3-flavor critical point 2, from mass-reweighting O( m), expected results linear σ-model suggests a const. slope in a huge mass region (σ-model results have been rescaled!) m u,d The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 24/28

53 3) Conclusions for 2+1 flavor QCD The quark mass plane m s two-state, Columbia [80] no two-state, Columbia [80] crossover like, JLQCD [81] 1st order like, JLQCD [81] two-state, JLQCD [81] B 4 =1.604 Rew(0.03) m crit linear σ model 2 critical points have been determined, not in contradiction to earlier calculation [Brown et. al. (1990), Aoki et. al. (1999)] slope at the 3-flavor critical point 2, from mass-reweighting O( m), expected results linear σ-model suggests a const. slope in a huge mass region (σ-model results have been rescaled!) m u,d mass-reweighting O(( m) 2 ) location of the tri-critical point possible? The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 24/28

54 3) Conclusions for 2+1 flavor QCD Line of minimum fluctuations Line of constant physics is roughly in proportion to the line of minimum fluctuation of the reweighting operator (R). Ejiri (2004) Determination of the Line of minimum fluctuations of R in the quark mass plane: using the condition φ (δr) 2 = 0 assuming with φ = m s / m u,d and δx = X X. φ min m s / m u,d = a 0 + a 1 m + a 2 m 2 + The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 25/28

55 3) Conclusions for 2+1 flavor QCD Line of minimum fluctuations we obtain a 0 = 2 and a 1 = 6 δr 1δR 2 (δr 1 ) a m m s n f = O(( m) 0 ) O(( m) 1 ) m u,d the effect of a 1 is less than 1% for m u,d < The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 26/28

56 3) Investigation of the Phase Diagram Approximation of the critical surface in the vicinity of the 3-flavor critical point µ crit u,d [MeV] first order n f =3 n f =2+1 p4-action phys. line The critical surface: in the (m u,d, m s )-plane: m crit s = 3m crit u,d 2m u,d chiral perturbation theory: m 2 K /m2 π = (m u,d + m s )/2m u,d crossover m π [MeV] in the (m 2 π, m2 K )-plane: (m crit K )2 = 3 2 (mcrit π ) m2 π quadratic interpolation of the 3-flavor critical points: (m crit π ) 2 [µ q ] = (m crit π ) 2 [0] + Aµ 2 q = surface defined by (1) and (2) (1) (2) The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 27/28

57 3) Investigation of the Phase Diagram Approximation of the critical surface in the vicinity of the 3-flavor critical point µ crit u,d [MeV] first order n f =3 n f =2+1 p4-action phys. line interpolation ansatz: µ 2 q = A 3 m 2 π + 2m 2 K 3 mcrit π (0) we obtain A = 0.109(36) crossover m π [MeV] The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 27/28

58 3) Investigation of the Phase Diagram Approximation of the critical surface in the vicinity of the 3-flavor critical point µ crit u,d [MeV] first order n f =3 n f =2+1 p4-action phys. line interpolation ansatz: µ 2 q = A 3 m 2 π + 2m 2 K 3 mcrit π (0) we obtain A = 0.109(36) crossover m π [MeV] = the physical critical point: µ crit q = 135(21) MeV The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 27/28

59 3) Investigation of the Phase Diagram Approximation of the critical surface in the vicinity of the 3-flavor critical point µ crit u,d [MeV] first order n f =3 n f =2+1 p4-action phys. line interpolation ansatz: µ 2 q = A 3 m 2 π + 2m 2 K 3 mcrit π (0) we obtain A = 0.109(36) crossover Fodor & Katz: µ q 120(13) m π [MeV] (standardaction) = the physical critical point: µ crit q = 135(21) MeV The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 27/28

60 3) Investigation of the Phase Diagram Approximation of the critical surface in the vicinity of the 3-flavor critical point µ crit u,d [MeV] first order n f =3 n f =2+1 p4-action phys. line 300 interpolation ansatz: 250 µ 2 q 200 = A 3 m 2 π + 2m 2 K 3 mcrit π (0) we obtain A = 0.109(36) 150 T [MeV] Pure Gauge n f =3, m=0.005 n f =3, m= crossover Fodor & Katz: µ q 120(13) m π [MeV] (standardaction) Forcrand, Philipsen Nucl. Phys. B642 (2002) 290 = the physical critical point: µ crit µ B [GeV] q Fodor, Katz JHEP 03 (2002) 14 = 135(21) MeV The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 27/28

61 Summary Improved actions are important for QCD thermodynamics on the lattice. Improvement is substantial for N τ 6 Reweighting in the chemical potential is possible up to µ B (80 250) MeV dependent of T, V, m Taylor-expansion in µ/t is in principle not limited. truncation error The 3-flavor chical point is Ising-universal. Binder cumulants Large cut-off -dependence of the mass scale at the 3-flavor critical point. m π 290 MeV (standard-action) m π 267 MeV (p4-action) The physical point at µ = 0 is most likely in the crossover region. The line of second order phase transitions in the quark mass plane is rather straight. a 1 only a 1% effect Approximation of the critical surface with 2 critical points in 3-flavor QCD: (µ crit q ) 2 = 0.109(36) 1 3 (m2 π + 2m2 K 3(mcrit π (0)) 2 ) µ crit q = 135(21) MeV cut-off -dependence The behavior of the critical temperature T c (µ) is steeper for m 0. T c (µ q ) < T c (µ I ) for µ small. The chiral critical point in 3 flavor QCD from Taylor expansion at non-zero density, INT, Seattle, 28. April 2004 p. 28/28

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