Lattice QCD. QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 1

Transcription

1 Lattice QCD QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 1

2 Lattice QCD : Some Topics QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 1

3 Lattice QCD : Some Topics Basic Lattice Gauge Theory Phase Diagram Quark Number Susceptibility Screening Lengths Summary QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 1

4 Basic Lattice Gauge Theory Discrete space-time : Lattice spacing a UV Cut-off. QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 2

5 Basic Lattice Gauge Theory Discrete space-time : Lattice spacing a UV Cut-off. Matter fields ψ(x), ψ(x) on lattice sites. QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 2

6 Basic Lattice Gauge Theory Discrete space-time : Lattice spacing a UV Cut-off. Matter fields ψ(x), ψ(x) on lattice sites. Gauge transformation : ψ (x) = V x ψ(x), V x SU(3). QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 2

7 Basic Lattice Gauge Theory Discrete space-time : Lattice spacing a UV Cut-off. Matter fields ψ(x), ψ(x) on lattice sites. Gauge transformation : ψ (x) = V x ψ(x), V x SU(3). Gauge Fields on links U µ(x) = V x U µ (x)v 1 x+ˆµ. QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 2

8 Basic Lattice Gauge Theory Discrete space-time : Lattice spacing a UV Cut-off. Matter fields ψ(x), ψ(x) on lattice sites. Gauge transformation : ψ (x) = V x ψ(x), V x SU(3). Gauge Fields on links U µ(x) = V x U µ (x)v 1 x+ˆµ. Gauge invariance Actions from Closed Wilson loops, e.g., plaquette. QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 2

9 Basic Lattice Gauge Theory Discrete space-time : Lattice spacing a UV Cut-off. Matter fields ψ(x), ψ(x) on lattice sites. Gauge transformation : ψ (x) = V x ψ(x), V x SU(3). Gauge Fields on links U µ(x) = V x U µ (x)v 1 x+ˆµ. Gauge invariance Actions from Closed Wilson loops, e.g., plaquette. Fermion Doubling Problem QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 2

10 Basic Lattice Gauge Theory Discrete space-time : Lattice spacing a UV Cut-off. Matter fields ψ(x), ψ(x) on lattice sites. Gauge transformation : ψ (x) = V x ψ(x), V x SU(3). Gauge Fields on links U µ(x) = V x U µ (x)v 1 x+ˆµ. Gauge invariance Actions from Closed Wilson loops, e.g., plaquette. Fermion Doubling Problem Staggered Fermions (partial chiral and flavour symmetry), QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 2

11 Basic Lattice Gauge Theory Discrete space-time : Lattice spacing a UV Cut-off. Matter fields ψ(x), ψ(x) on lattice sites. Gauge transformation : ψ (x) = V x ψ(x), V x SU(3). Gauge Fields on links U µ(x) = V x U µ (x)v 1 x+ˆµ. Gauge invariance Actions from Closed Wilson loops, e.g., plaquette. Fermion Doubling Problem Staggered Fermions (partial chiral and flavour symmetry), Wilson fermions (only flavour symmetry), QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 2

12 Recent Overlap fermions (exact chiral and flavour symmetry). QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 3

13 Recent Overlap fermions (exact chiral and flavour symmetry). Typically, we need to evaluate Θ(m v ) = DU exp( S G )Θ(mv) Det M(ms) DU exp( S G ) Det M(ms), (1) where M is the Dirac matrix in x, color, spin, flavour space for fermions of mass m s, S G is the gluonic action, and the observable Θ may contain fermion propagators of mass m v. Since Θ is computed by averaging over a set of configurations {U µ (x)} which occur with probability exp( S G ) Det M, the complexity of evaluation of Det M = approximations : Quenched ( m s = limit), Partially Quenched ( low m s = m u = m d ), and Full (including a heavier s quark). Q PQ Full Computer time and Precision. QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 3

14 Phase Diagram Lattice details : N 3 s N t Lattice, N s N t for T 0, Spatial Volume V = N 3 s a 3, Temperature T = 1/N t a(β). QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 4

15 Phase Diagram Lattice details : N 3 s N t Lattice, N s N t for T 0, Spatial Volume V = N 3 s a 3, Temperature T = 1/N t a(β). Order Parameters : Chiral condensate ψψ, Polyakov Loop L, where L( x) = 1 3 Nt t=1 tr U 4( x, t) QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 4

16 Phase Diagram Lattice details : N 3 s N t Lattice, N s N t for T 0, Spatial Volume V = N 3 s a 3, Temperature T = 1/N t a(β). Order Parameters : Chiral condensate ψψ, Polyakov Loop L, where L( x) = 1 3 Nt t=1 tr U 4( x, t) Theoretical expectations based on effective models : QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 4

17 QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 5

18 Good News from results (on small N t lattices ) so far : QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 6

19 Good News from results (on small N t lattices ) so far : Transition temperature for 2 light dynamical quarks agree for Wilson quarks (171(4) MeV) and Staggered quarks (173(8) MeV) agree. (CP-PACS Collaboration and Bielefeld Group) QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 6

20 Good News from results (on small N t lattices ) so far : Transition temperature for 2 light dynamical quarks agree for Wilson quarks (171(4) MeV) and Staggered quarks (173(8) MeV) agree. (CP-PACS Collaboration and Bielefeld Group) Transition seems to be continuous in both cases. QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 6

21 Good News from results (on small N t lattices ) so far : Transition temperature for 2 light dynamical quarks agree for Wilson quarks (171(4) MeV) and Staggered quarks (173(8) MeV) agree. (CP-PACS Collaboration and Bielefeld Group) Transition seems to be continuous in both cases. Transition temperature for 3 light flavours : 154 ± 8 MeV. (Bielefeld Group) QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 6

22 Good News from results (on small N t lattices ) so far : Transition temperature for 2 light dynamical quarks agree for Wilson quarks (171(4) MeV) and Staggered quarks (173(8) MeV) agree. (CP-PACS Collaboration and Bielefeld Group) Transition seems to be continuous in both cases. Transition temperature for 3 light flavours : 154 ± 8 MeV. (Bielefeld Group) Theoretical expectations on the Phase diagram work out too. QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 6

23 Lattice 02 : Schmidt et al., hep-lat/ am s two-state, Columbia [7] no two-state, Columbia [7] crossover like, JLQCD [8] 1st order like, JLQCD [8] two-state, JLQCD [8] B 4 =1.604 Rew(0.03) m crit am u,d QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 7

24 Bad News, however, is : QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 8

25 Bad News, however, is : Critical exponents do not match for the two type of quarks; O(4)-like for only Wilson quarks. Thus the order of the phase transition is NOT yet established firmly. QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 8

26 Bad News, however, is : Critical exponents do not match for the two type of quarks; O(4)-like for only Wilson quarks. Thus the order of the phase transition is NOT yet established firmly. Fermion Symmetry Problem? Small Lattices?? What if m u m d? QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 8

27 Bad News, however, is : Critical exponents do not match for the two type of quarks; O(4)-like for only Wilson quarks. Thus the order of the phase transition is NOT yet established firmly. Fermion Symmetry Problem? Small Lattices?? What if m u m d? Recent results show that for 1 m d /m u 2, transition stays put. (Gavai & Gupta, hep-lat/ ; PRD in press.) QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 8

28 Bad News, however, is : Critical exponents do not match for the two type of quarks; O(4)-like for only Wilson quarks. Thus the order of the phase transition is NOT yet established firmly. Fermion Symmetry Problem? Small Lattices?? What if m u m d? Recent results show that for 1 m d /m u 2, transition stays put. (Gavai & Gupta, hep-lat/ ; PRD in press.) Lattice; m d /mu = 1 (dashed), 2(full) and 10 (dotted) <ψψ> <Re L> β β QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 8

29 µ B 0 QCD at nonzero baryon density may Color Superconductivity, (Strange) Quark Stars, etc. Phase Problem : Det M(µ) is complex for µ 0. Exciting results in recent past for small µ. Re-weighting Method (Fodor & Katz, JHEP 02) Imaginary µ (de Forcrand & Philipsen, May 02, D Elia & Lombardo, May 02) Taylor Expansion in µ (Allton et al., April 02) Large µ simulations possible when Det M is real, e.g., 2 colours or µ I3 0. Show agreement with effective chiral theory (Kogut & Sinclair 02, S. Gupta 02) QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 9

30 µ B 0 QCD at nonzero baryon density may Color Superconductivity, (Strange) Quark Stars, etc. Phase Problem : Det M(µ) is complex for µ 0. Exciting results in recent past for small µ. Re-weighting Method (Fodor & Katz, JHEP 02) Imaginary µ (de Forcrand & Philipsen, May 02, D Elia & Lombardo, May 02) Taylor Expansion in µ (Allton et al., April 02) Large µ simulations possible when Det M is real, e.g., 2 colours or µ I3 0. Show agreement with effective chiral theory (Kogut & Sinclair 02, S. Gupta 02) QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 9

31 µ B 0 QCD at nonzero baryon density may Color Superconductivity, (Strange) Quark Stars, etc. Phase Problem : Det M(µ) is complex for µ 0. Exciting results in recent past for small µ. Re-weighting Method (Fodor & Katz, JHEP 02) Imaginary µ (de Forcrand & Philipsen, May 02, D Elia & Lombardo, May 02) Taylor Expansion in µ (Allton et al., April 02) Large µ simulations possible when Det M is real, e.g., 2 colours or µ I3 0. Show agreement with effective chiral theory (Kogut & Sinclair 02, S. Gupta 02) QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 9

32 µ B 0 QCD at nonzero baryon density may Color Superconductivity, (Strange) Quark Stars, etc. Phase Problem : Det M(µ) is complex for µ 0. Exciting results in recent past for small µ. Re-weighting Method (Fodor & Katz, JHEP 02) Imaginary µ (de Forcrand & Philipsen, May 02, D Elia & Lombardo, May 02) Taylor Expansion in µ (Allton et al., April 02) Large µ simulations possible when Det M is real, e.g., 2 colours or µ I3 0. Show agreement with effective chiral theory (Kogut & Sinclair 02, S. Gupta 02) QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 9

33 µ B 0 QCD at nonzero baryon density may Color Superconductivity, (Strange) Quark Stars, etc. Phase Problem : Det M(µ) is complex for µ 0. Exciting results in recent past for small µ. Re-weighting Method (Fodor & Katz, JHEP 02) Imaginary µ (de Forcrand & Philipsen, May 02, D Elia & Lombardo, May 02) Taylor Expansion in µ (Allton et al., April 02) Large µ simulations possible when Det M is real, e.g., 2 colours or µ I3 0. Show agreement with effective chiral theory (Kogut & Sinclair 02, S. Gupta 02) QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 9

34 µ B 0 QCD at nonzero baryon density may Color Superconductivity, (Strange) Quark Stars, etc. Phase Problem : Det M(µ) is complex for µ 0. Exciting results in recent past for small µ. Re-weighting Method (Fodor & Katz, JHEP 02) Imaginary µ (de Forcrand & Philipsen, May 02, D Elia & Lombardo, May 02) Taylor Expansion in µ (Allton et al., April 02) Large µ simulations possible when Det M is real, e.g., 2 colours or µ I3 0. Show agreement with effective chiral theory (Kogut & Sinclair 02, S. Gupta 02) QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 9

35 µ B 0 QCD at nonzero baryon density may Color Superconductivity, (Strange) Quark Stars, etc. Phase Problem : Det M(µ) is complex for µ 0. Exciting results in recent past for small µ. Re-weighting Method (Fodor & Katz, JHEP 02) Imaginary µ (de Forcrand & Philipsen, May 02, D Elia & Lombardo, May 02) Taylor Expansion in µ (Allton et al., April 02) Large µ simulations possible when Det M is real, e.g., 2 colours or µ I3 0. Show agreement with effective chiral theory (Kogut & Sinclair 02, S. Gupta 02) QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 9

36 µ B 0 QCD at nonzero baryon density may Color Superconductivity, (Strange) Quark Stars, etc. Phase Problem : Det M(µ) is complex for µ 0. Exciting results in recent past for small µ. Re-weighting Method (Fodor & Katz, JHEP 02) Imaginary µ (de Forcrand & Philipsen, May 02, D Elia & Lombardo, May 02) Taylor Expansion in µ (Allton et al., April 02) Large µ simulations possible when Det M is real, e.g., 2 colours or µ I3 0. Show agreement with effective chiral theory (Kogut & Sinclair 02, S. Gupta 02) QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 9

37 Fodor-Katz Results N 3 s 4 Lattices, N s = 4,6,8; Bit heavy u,d quarks. Critical Endpoint : T = 160(4) MeV, µ = 725(35) MeV As m ud, does µ E? Larger N t?? QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 10

38 Quark Number Susceptibility Theoretical Checks : Resummed Perturbation expansions; Finite Density Results QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 11

39 Quark Number Susceptibility Theoretical Checks : Resummed Perturbation expansions; Finite Density Results Crucial for QGP Signatures : Q, B Fluctuations; Strangeness production QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 11

40 Quark Number Susceptibility Theoretical Checks : Resummed Perturbation expansions; Finite Density Results Crucial for QGP Signatures : Q, B Fluctuations; Strangeness production Definitions: For u, d, and s quarks, the partition function is Z = DU exp( S G ) f=u,d,s Det M(m f,µ f ), (2) QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 11

41 Quark Number Susceptibility Theoretical Checks : Resummed Perturbation expansions; Finite Density Results Crucial for QGP Signatures : Q, B Fluctuations; Strangeness production Definitions: For u, d, and s quarks, the partition function is Z = DU exp( S G ) f=u,d,s Det M(m f,µ f ), (2) where µ f are corresponding chemical potentials. Defining µ 0 = µ u + µ d + µ s and µ 3 = µ u µ d, baryon and isospin density/susceptibilities can be obtained as : (Gottlieb et al. 87, 96, 97, Gavai et al. 89) QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 11

42 Quark Number Susceptibility Theoretical Checks : Resummed Perturbation expansions; Finite Density Results Crucial for QGP Signatures : Q, B Fluctuations; Strangeness production Definitions: For u, d, and s quarks, the partition function is Z = DU exp( S G ) f=u,d,s Det M(m f,µ f ), (2) where µ f are corresponding chemical potentials. Defining µ 0 = µ u + µ d + µ s and µ 3 = µ u µ d, baryon and isospin density/susceptibilities can be obtained as : (Gottlieb et al. 87, 96, 97, Gavai et al. 89) n i = T V ln Z µ i, χ ij = T V 2 ln Z µ i µ j QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 11

43 Setting µ i = 0, n i =0 but χ ij are nontrivial. Diagonal χ s are χ 0 = 1 2 [O 1(m u ) O 2(m u )] (3) χ 3 = 1 2 O 1(m u ) (4) χ s = 1 4 [O 1(m s ) O 2(m s )] (5) QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 12

44 Setting µ i = 0, n i =0 but χ ij are nontrivial. Diagonal χ s are χ 0 = 1 2 [O 1(m u ) O 2(m u )] (3) χ 3 = 1 2 O 1(m u ) (4) χ s = 1 4 [O 1(m s ) O 2(m s )] (5) Here O i trace of products of M 1, M and M and are estimated by a stochastic method: Tr A = N v i=1 R i AR i/2n v, and (Tr A) 2 = 2 L i>j=1 (Tr A) i(tr A) j /L(L 1), where R i is a complex vector from a set of N v, subdivided in L independent sets. QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 12

45 Setting µ i = 0, n i =0 but χ ij are nontrivial. Diagonal χ s are χ 0 = 1 2 [O 1(m u ) O 2(m u )] (3) χ 3 = 1 2 O 1(m u ) (4) χ s = 1 4 [O 1(m s ) O 2(m s )] (5) Here O i trace of products of M 1, M and M and are estimated by a stochastic method: Tr A = N v i=1 R i AR i/2n v, and (Tr A) 2 = 2 L i>j=1 (Tr A) i(tr A) j /L(L 1), where R i is a complex vector from a set of N v, subdivided in L independent sets. Earlier results : Only close to T c & for fixed ma. Gavai & Gupta PR D 01; Gavai, Gupta & Majumdar, PR D 2002 QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 12

46 χ F F T Ideal gas results for same Lattice. χ /χ 3 FFT m v x 4 Lattice N = 2; m /T = 0.1 f s c N = 0; Quenched f T/Tc QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 13

47 χ F F T Ideal gas results for same Lattice. χ /χ 3 FFT m v x 4 Lattice N = 2; m /T = 0.1 f s c N = 0; Quenched f T/Tc Note that PDG values for strange quark mass = m strange v /T c (N f =0); (N f =2). QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 13

48 Perturbation Theory QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 14

49 Perturbation Theory Weak coupling expansion gives: χ = 1 2( α s π ) + 8 ( N f )( α s χ F F T (Kapusta 1989). π )3 2 QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 14

50 Perturbation Theory Weak coupling expansion gives: χ = 1 2( α s π ) + 8 ( N f )( α s χ F F T (Kapusta 1989). π ) χ/χ FFT T c 1.5T c 3T c 1.5T c 0.85 LO Plasmon Nf=0 Plasmon Nf=2 Plasmon Nf= α s /π Minm (0.986) at 0.03 (0.02) for N f = 0 (2). For 1.5 T/T c 3 pert. theory (1.08=1.03) for N f = 0 (2). QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 14

51 Resummed Perturbation Theory QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 15

52 Resummed Perturbation Theory Hard Thermal Loop & Self-consistent resummation give : (Blaizot, Iancu & Rebhan, PLB 01; Chakraborty, Mustafa & Thoma, EPJC 02). QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 15

53 Resummed Perturbation Theory Hard Thermal Loop & Self-consistent resummation give : (Blaizot, Iancu & Rebhan, PLB 01; Chakraborty, Mustafa & Thoma, EPJC 02). QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 15

54 Resummed Perturbation Theory Hard Thermal Loop & Self-consistent resummation give : (Blaizot, Iancu & Rebhan, PLB 01; Chakraborty, Mustafa & Thoma, EPJC 02). Our results for N t = 4 Lattice artifacts? Check for larger N t and improved actions. QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 15

55 χ ud QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 16

56 χ ud Off-diagonal Susceptibility : χ ud = T V Tr M 1 u M utr M 1 d M d QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 16

57 χ ud Off-diagonal Susceptibility : χ ud = T V Zero within 1 σ O(10 6 ) for T > T c. Tr M 1 u M utr M 1 d M d QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 16

58 χ ud Off-diagonal Susceptibility : χ ud = T V Zero within 1 σ O(10 6 ) for T > T c. Tr M 1 u M utr M 1 d M d Identically zero for Ideal gas but O(α 3 s) in P.T. Using the same scale and α s as for χ 3 χ ud O(10 4 )!! QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 16

59 χ ud Off-diagonal Susceptibility : χ ud = T V Zero within 1 σ O(10 6 ) for T > T c. Tr M 1 u M utr M 1 d M d Identically zero for Ideal gas but O(α 3 s) in P.T. Using the same scale and α s as for χ 3 χ ud O(10 4 )!! NONZERO for T < T c and M 2 π. QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 16

60 χ ud Off-diagonal Susceptibility : χ ud = T V Zero within 1 σ O(10 6 ) for T > T c. Tr M 1 u M utr M 1 d M d Identically zero for Ideal gas but O(α 3 s) in P.T. Using the same scale and α s as for χ 3 χ ud O(10 4 )!! NONZERO for T < T c and M 2 π. 8 χ M 2 /T 4 ud π c M π /T c Lattice; Quenched. T = 0.75T c Gavai, Gupta & Majumdar, PR D 2002 QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 16

61 (Gavai & Gupta, PR D 02 and hep-lat/ ) Taking Continuum Limit QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 17

62 (Gavai & Gupta, PR D 02 and hep-lat/ ) Taking Continuum Limit Investigate larger N t : 6, 8, 10, 12 and 14. QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 17

63 (Gavai & Gupta, PR D 02 and hep-lat/ ) Taking Continuum Limit Investigate larger N t : 6, 8, 10, 12 and 14. Naik action : Improved by O(a) compared to Staggered. Introduction of µ nontrivial but straightforward. (Naik, NP B 1989; Gavai, hep-lat/ ) QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 17

64 (Gavai & Gupta, PR D 02 and hep-lat/ ) Taking Continuum Limit Investigate larger N t : 6, 8, 10, 12 and 14. Naik action : Improved by O(a) compared to Staggered. Introduction of µ nontrivial but straightforward. (Naik, NP B 1989; Gavai, hep-lat/ ) χ/τ Aspect ratio 3 naive naik p4 Does improve the N t -dependence of the free fermions N t QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 17

65 Results at 2T c : /T 2 χ /N 2 t QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 18

66 Results at 2T c : /T 2 χ /N 2 t Nt 2 a 2 extrapolation works and leads to same results within errors for both staggered and Naik fermions. QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 18

67 Results at 2T c : /T 2 χ /N 2 t Nt 2 a 2 extrapolation works and leads to same results within errors for both staggered and Naik fermions. Milder N 2 t a 2 -dependence for Naik fermions. QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 18

68 The continuum susceptibility vs. T therefore is : QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 19

69 The continuum susceptibility vs. T therefore is : 1.0 NL 0.9 HTL χ 3 /Τ T/T c Naik action (Squares) and Staggered action (circles) QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 19

70 The continuum susceptibility vs. T therefore is : 1.0 NL 0.9 HTL χ 3 /Τ T/T c Naik action (Squares) and Staggered action (circles) λ s (T c ) = 2χ s /(χ u + χ d ) vis-a-vis RHIC value 0.47 ± 0.4. QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 19

71 The continuum susceptibility vs. T therefore is : 1.0 NL 0.9 HTL χ 3 /Τ T/T c Naik action (Squares) and Staggered action (circles) λ s (T c ) = 2χ s /(χ u + χ d ) vis-a-vis RHIC value 0.47 ± 0.4. χ ud behaves the same way for ALL N t and both fermions, leading to the same O(10 6 ) values in continuum too. QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 19

72 Screening Lengths QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 20

73 Screening Lengths Obtained from the exponential decay of C Γ (z) = x,y,t M 1 αβ 1 (x, y, z, t)γm (x, y, z, t)γ (6) βα Γ Spin-flavour matrix, α,β colour indices and M 1 quark propagator with source at origin. QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 20

74 Screening Lengths Obtained from the exponential decay of C Γ (z) = x,y,t M 1 αβ 1 (x, y, z, t)γm (x, y, z, t)γ (6) βα Γ Spin-flavour matrix, α,β colour indices and M 1 quark propagator with source at origin. Known results : Degenerate parity partners, FFT results for all except π. (DeTar-Kogut, Boyd et al., Gottlieb et al., Gavai-Gupta, ) QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 20

75 Screening Lengths Obtained from the exponential decay of C Γ (z) = x,y,t M 1 αβ 1 (x, y, z, t)γm (x, y, z, t)γ (6) βα Γ Spin-flavour matrix, α,β colour indices and M 1 quark propagator with source at origin. Known results : Degenerate parity partners, FFT results for all except π. (DeTar-Kogut, Boyd et al., Gottlieb et al., Gavai-Gupta, ) Could χ 3 and M π both have some, perhaps the same, non-perturbative effect? QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 20

76 Screening Lengths Obtained from the exponential decay of C Γ (z) = x,y,t M 1 αβ 1 (x, y, z, t)γm (x, y, z, t)γ (6) βα Γ Spin-flavour matrix, α,β colour indices and M 1 quark propagator with source at origin. Known results : Degenerate parity partners, FFT results for all except π. (DeTar-Kogut, Boyd et al., Gottlieb et al., Gavai-Gupta, ) Could χ 3 and M π both have some, perhaps the same, non-perturbative effect? Summing up the C Γ for pion Pion susceptibility. QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 20

77 N t = 4 Lattices with N z = 16. 4χ 3 /T 2 (open symbols) and χ π /10T 2 (filled) at 2T c (lower set) and 3T c. (Gavai, Gupta & Majumdar, PR D 02) 10 χ/t M S/T QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 21

78 N t = 4 Lattices with N z = 16. 4χ 3 /T 2 (open symbols) and χ π /10T 2 (filled) at 2T c (lower set) and 3T c. (Gavai, Gupta & Majumdar, PR D 02) 10 χ/t M S/T Why? χ 3 propagator of nonlocal vector meson. QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 21

79 Again Taking Continuum Limit QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 22

80 Again Taking Continuum Limit On finer lattices, a = 1/8T 1/12T, Pion screening lengths become degenerate with those of ρ, i.e, also close to FFT!! (Gavai & Gupta, hep-lat/ ) 1.1 µ/µ FFT m v /T c = 0.03, Lattices up to /N2 t QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 22

81 Again Taking Continuum Limit On finer lattices, a = 1/8T 1/12T, Pion screening lengths become degenerate with those of ρ, i.e, also close to FFT!! (Gavai & Gupta, hep-lat/ ) 1.1 µ/µ FFT m v /T c = 0.03, Lattices up to /N2 t However, chiral condensate, ψψ differs from FFT by 2, as do the detailed shapes of the correlators. QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 22

82 Note that both PS and V have SAME fit (green line)with changed normalization. log C(z) PS, V : QCD at 2T c PS, V : FFT 12x26x26x48 Lattice M v = 0.03 T c zt QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 23

83 Summary Phase diagram in T µ plane has firmed up on small N t : Different fermions, different methods, same T c, and (T E, µ E ). QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 24

84 Summary Phase diagram in T µ plane has firmed up on small N t : Different fermions, different methods, same T c, and (T E, µ E ). Quark number susceptibilities RHIC signal physics. QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 24

85 Summary Phase diagram in T µ plane has firmed up on small N t : Different fermions, different methods, same T c, and (T E, µ E ). Quark number susceptibilities RHIC signal physics. Continuum limit of QNS in Quenched QCD obtained. Yields λ s in agreement with RHIC and SPS results. Leaves scope for improvement in resummations. QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 24

86 Summary Phase diagram in T µ plane has firmed up on small N t : Different fermions, different methods, same T c, and (T E, µ E ). Quark number susceptibilities RHIC signal physics. Continuum limit of QNS in Quenched QCD obtained. Yields λ s in agreement with RHIC and SPS results. Leaves scope for improvement in resummations. As N t, M π /M F F T 1, as for ρ QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 24

87 Summary Phase diagram in T µ plane has firmed up on small N t : Different fermions, different methods, same T c, and (T E, µ E ). Quark number susceptibilities RHIC signal physics. Continuum limit of QNS in Quenched QCD obtained. Yields λ s in agreement with RHIC and SPS results. Leaves scope for improvement in resummations. As N t, M π /M F F T 1, as for ρ Many questions still for full 2+1 QCD : Order, Large N t,. QCD 2002, I. I. T. Kanpur, November 19, 2002 R. V. Gavai Top 24