QCD Phase Transitions and Green s Functions Heidelberg, 7 May 2010 Lorenz von Smekal

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1 QCD Phase Transitions and Green s Functions Heidelberg, 7 May 2010 Lorenz von Smekal 1

2 Special Credit and Thanks to: Reinhard Alkofer Jens Braun Christian Fischer Holger Gies Lisa Haas Axel Maas Kim Maltman Florian Marhauser Jens Müller Jan Pawlowski Bernd-Jochen Schaefer Daniel Spielmann Nucu Stamatescu Andre Sternbeck Jochen Wambach Helmholtz Young Investigator Group 'Nonperturbative Phenomena in QCD' ExtreMe Matter Institute Helmholtz Alliance 'Cosmic Matter in the Laboratory' European Commission 2

3 Contents Introduction: why Green s functions? Functional methods What do we know? (zero temperature and density) Applications to QCD phase transitions (deconfinement and chiral symmetry restoration) Summary and Outlook: what may we expect? 2

4 Introduction Why QCD Green s Functions? quantum field theoretic building blocks for hadron phenomenology hadron masses, decay constants, form factors, magnetic moments, charge radii etc. from covariant bound state equations (Bethe-Salpeter and Faddeev) need QCD quark and gluon Green s functions as input covariant Faddeev equation for baryons G. Eichmann, PhD Thesis, Graz (2009) [arxiv: [hep-ph]] G E Proton Electric Form Factor m q =0.43 GeV, m sc =0.60 GeV, m ax =0.83 GeV Hoehler et al., NPB 114(1976), 505 Bosted et al., PRL 68(1992), 3841 All contributions 2 loop contributions Q 2 [GeV 2 ] µ p G E /G M The Ratio µ p G E /G M JLab Hall A Coll, PRL 84 (2000), 1398 m q =0.36, m sc =0.63, m ax =0.68 m q =0.43, m sc =0.60, m ax = Q 2 [GeV 2 ] Oettel, Pichowsky, LvS, EPJA 8 (2000) 251 Oettel, Alkofer, LvS, EPJA 8 (2000) 553 3

5 Why QCD Green s Functions? Introduction confinement, dynamical mass generation, chiral symmetry breaking and Goldstone s theorem Alkofer & LvS, Phys. Rept. 353/5-6 (2001) 281 Fischer, J. Phys. G 32 (2006) 253 TOPCITE 500+ Quark Mass Function: M(p 2 ) Bottom quark Charm quark Strange quark Up/Down quark Chiral limit p 2 [GeV 2 ] M " [MeV] N f =3, DSE N f =2, JLQCD(2003) N f =2, CP-Pacs (2002/4) M! [MeV] Fischer, Alkofer, PRD 67 (2003) Fischer, Watson, Cassing, PRD 72 (2005)

6 Why QCD Green s Functions? Introduction confinement, dynamical mass generation, chiral symmetry breaking and Goldstone s theorem Alkofer & LvS, Phys. Rept. 353/5-6 (2001) 281 Fischer, J. Phys. G 32 (2006) 253 TOPCITE 500+ Quark Mass Function: M(p 2 ) Bottom quark Charm quark Strange quark Up/Down quark Chiral limit p 2 [GeV 2 ] M " [MeV] N f =3, DSE N f =2, JLQCD(2003) N f =2, CP-Pacs (2002/4) M! [MeV] Fischer, Alkofer, PRD 67 (2003) Fischer, Watson, Cassing, PRD 72 (2005) Here: chiral symmetry restoration and deconfinement transition... 4

7 Generalities Partition Function Z(T,V,µ) = D[A, c, ψ] δ( A) exp 1/T 0 dt V d 3 x 1 4 F 2 + c Dc + ψ( D/ + m µγ 0 )ψ Landau gauge: Lorenz condition, µ A µ =0 Faddeev-Popov ghosts and determinant: c, c Det( µ D µ (A)) Field strengths: F µν = µ A ν ν A µ + ig[a µ,a ν ] Quark fields with a.p. b.c. s: ψ(1/t, x) = ψ(0, x) 5

8 Note on Gribov Copies A l Gribov, Singer (1978): gauge copies unavoidable. A tr Fujikawa, Hirschfeld (1979): average over copies (sign-weighted). A tr Sharpe (1984), Neuberger (1987), Schaden (1998): perfect cancellation, produces 0! A = 0 for A = A tr, many copies! LvS (2008): can be avoided non-perturbative Becchi-Rouet-Stora-Tyutin (BRST) symmetry (on the lattice, but with sign problem). 6

9 Non-Perturbative Tools Functional Methods: (a) Dyson-Schwinger Equations (DSEs) Partition Function Generating Functional Equations of Motion for Green s Functions (b) Functional Renormalisation Group (FRG) Effective Action Energy-Momentum Cutoff RG Flow Wetterich Equation + chiral fermions + no sign problem + dynamical hadronisation Lattice Gauge Theory + no truncations + manifest gauge invariance 7

10 Non-Perturbative Tools Functional Methods: (a) Dyson-Schwinger Equations (DSEs) Partition Function Generating Functional Equations of Motion for Green s Functions (b) Functional Renormalisation Group (FRG) Effective Action Energy-Momentum Cutoff RG Flow Wetterich Equation + chiral fermions + no sign problem + dynamical hadronisation Lattice Gauge Theory + no truncations + manifest gauge invariance combine & compare 7

11 Historical Note Historically: infrared enhanced gluon exchange, V (r) D gluon σ p 4, V (r) σr but no it s ghosts that dominate the long-range/infrared correlations! LvS, Hauck, Alkofer (1997) r 2 conditions for confinement: Kugo, Ojima (1979) (a) avoid Higgs mechanism (b) mass gap 5 Gluon Ghost all physical states are color singlets p 2 [(GeV) 2 ] 8

12 Historical Note Historically: infrared enhanced gluon exchange, V (r) D gluon σ p 4, V (r) σr but no it s ghosts that dominate the long-range/infrared correlations! LvS, Hauck, Alkofer (1997) r 2 conditions for confinement: Kugo, Ojima (1979) (a) avoid Higgs mechanism (b) mass gap 5 Gluon Ghost all physical states are color singlets. F. Coester: p 2 [(GeV) 2 ] Farbloser Quark zusammengeleimt von Gespenstern mit farbigem Leim 8

13 Gluon Propagator Pure Yang-Mills, T =0 4 3 Sternbeck et al., PoS LAT2006, 76 Z(p 2 )!"!! #"$ %&!! #"$ %&!! '""!"!! '"( '#!! '"( p [GeV] A µ (x)a ν (y) : D µν (p) = Z(p2 ) p 2 δ µν p µp ν p 2 9

14 Gluon Propagator 4 3 Sternbeck et al., PoS LAT2006, 76 Z(p 2 )!"!! #"$ %&!! #"$ %&!! '""!"!! '"( '#!! '"( LvS, Hauck, Alkofer, Phys. Rev. Lett. 79 (1997) 3591 p [GeV] 9

15 Gluon Propagator 4 3 Sternbeck et al., PoS LAT2006, 76 Z(p 2 )!"!! #"$ %&!! #"$ %&!! '""!"!! '"( '#!! '"( LvS, Hauck, Alkofer, Phys. Rev. Lett. 79 (1997) 3591 Lerche, LvS, Phys. Rev. D 65 (2002) Fischer, Alkofer, Phys. Rev. D 65 (2002) p [GeV] 9

16 Gluon Propagator 4 3 Sternbeck et al., PoS LAT2006, 76 Z(p 2 )!"!! #"$ %&!! #"$ %&!! '""!"!! '"( '#!! '"( p [GeV] LvS, Hauck, Alkofer, Phys. Rev. Lett. 79 (1997) 3591 Lerche, LvS, Phys. Rev. D 65 (2002) Fischer, Alkofer, Phys. Rev. D 65 (2002) Pawlowski, Litim, Nedelko, LvS, Phys. Rev. Lett. 93 (2004) ; Pawlowski (2006), unp. 9

17 Gluon Propagator 4 3 Sternbeck et al., PoS LAT2006, 76 Z(p 2 )!"!! #"$ %&!! #"$ %&!! '""!"!! '"( '#!! '"( p [GeV] LvS, Hauck, Alkofer, Phys. Rev. Lett. 79 (1997) 3591 Lerche, LvS, Phys. Rev. D 65 (2002) Fischer, Alkofer, Phys. Rev. D 65 (2002) Pawlowski, Litim, Nedelko, LvS, Phys. Rev. Lett. 93 (2004) ; Pawlowski (2006), unp. Fischer, Maas, Pawlowski, Annals Phys. 324 (2009) 2408; Pawlowski, in prep. 9

18 Landau gauge QCD Propagators Since 1997, DSEs, FRGEs, Stochastic Quantisation, Lattice Simulations: Alkofer, Aguilar, Binosi, Bicudo, Bloch, Boucaud, Bogolubsky, Bornyakov, Bowman, Braun, Cucchieri, De Soto, Dudal, Fischer, Gies, Gracey, Huber, Ilgenfritz, Langfeld, Leinweber, Leroy, Litim, Llanes-Estrada, Nakamura, Natale, Nedelko, Maas, Mendes, Micheli, Mitrjushkin, Müller-Preußker, Oliveira, Papavassiliou, Pawlowski, Pene, Petreczky, Quandt, Reinhardt, Rodriguez-Quintero, Schwenzer, Silva, Skullerud, Sorella, Stamatescu, Sternbeck, Vandersickel, Verschelde, LvS, Wambach, Williams, Zwanziger,... Numerical solutions & analytic infrared asymptotics (infrared-scaling, confinement criteria, Gribov ambiguity...). 10

19 Gluon Propagator Gribov ambiguity, of fundamental interest (mass gap, Kugo-Ojima), but phenomenologically not relevant! Z(p 2 )!"!! #"$ %&!! #"$ %&!! '""!"!! '"( '#!! '"( 1 non-perturbative and relevant perturbative p [GeV] 11

20 Gluon Propagator Gribov ambiguity, of fundamental interest (mass gap, Kugo-Ojima), but phenomenologically not relevant! Z(p 2 )!"!! #"$ %&!! #"$ %&!! '""!"!! '"( '#!! '"( 1 non-perturbative and relevant perturbative Combine functional methods (FRGEs + DSEs). Compare with lattice simulations where possible. p [GeV] 11

21 Running Coupling From Landau gauge gluon and ghost propagators: αs MM (p 2 )= g2 4π Z(p2 )G 2 (p 2 ) LvS, Hauck, Alkofer, Annals Phys. 267 (1998) 1 p (infrared-scaling) Lerche, LvS, Phys. Rev. D 65 (2002)

22 Running Coupling From Landau gauge gluon and ghost propagators: αs MM (p 2 )= g2 4π Z(p2 )G 2 (p 2 ) LvS, Hauck, Alkofer, Annals Phys. 267 (1998) 1 p (infrared-scaling) Lerche, LvS, Phys. Rev. D 65 (2002) MiniMOM scheme precise definition, known to 4 loops LvS, Maltman, Sternbeck, Phys. Lett. B 681 (2009)

23 Running Coupling From Landau gauge gluon and ghost propagators: αs MM (p 2 )= g2 4π Z(p2 )G 2 (p 2 ) LvS, Hauck, Alkofer, Annals Phys. 267 (1998) 1 p (infrared-scaling) Lerche, LvS, Phys. Rev. D 65 (2002) MiniMOM scheme precise definition, known to 4 loops LvS, Maltman, Sternbeck, Phys. Lett. B 681 (2009) 336 Lattice determination of α s Sternbeck et al. (LvS), PoS LAT2009, 210 r 0 Λ (0) MS =0.62(1), r 0Λ (2) MS =0.60(3)(2) α MM L (p2 ) N f =0 (β, N 4 )=(6.0, 32 4 ) (6.2, 32 4 ) (6.4, 32 4 ) (6.6, 32 4 ) (6.9, 48 4 ) (7.2, 64 4 ) (7.5, 64 4 ) (8.5, 48 4 ) not a fit! p 2 [GeV 2 ] α MM s (p2 ) 12

24 Applications I Deconfinement Transition Pure gauge theory with static quarks: and (global) Z3 center symmetry: P (x) =P exp ig P (x) ZP(x), 1/T 0 dta 0 (t, x) Order parameter: Φ = 1 3 tr P (x) e F q /T Z {1, exp 2πi/3, exp 4πi/3} Φ =0, confined, (center) symmetric, disordered Φ = 0, deconfined, spontan. broken Z 3, ordered 13

25 Effective Potential Braun, Gies, Pawlowski, Phys. Lett. B 684 (2010) 262 Constant background: V [a] = t Γ k [φ] = 1 2 k a = 1 T ga3 0, Φ[a] = cos a 2, Z(p 2 ) G(p 2 ) p [GeV] p [GeV] with: p 0 =2πT (n ν a/4π), ν = {0, ±1, ±2} 14

26 Effective Potential Braun, Gies, Pawlowski, Phys. Lett. B 684 (2010) 262 Constant background: V [a] = t Γ k [φ] = 1 2 k a = 1 T ga3 0, Φ[a] = cos a 2, Z(p 2 ) G(p 2 ) p [GeV] p [GeV] with: p 0 =2πT (n ν a/4π), ν = {0, ±1, ±2} Up to terms (suppressed at T c ) O(V [a]) 14

27 T c = ± 10 MeV, Effective Potential T c / σ =0.658 ± (consistent with lattice average) Braun, Gies, Pawlowski, Phys. Lett. B 684 (2010) MeV SU(3) V [a]/t MeV 300 MeV 295 MeV 289.5MeV 285 MeV Φ[a] =0 a 2π 15

28 T c = ± 10 MeV, Effective Potential T c / σ =0.658 ± (consistent with lattice average) Braun, Gies, Pawlowski, Phys. Lett. B 684 (2010) MeV SU(3) V [a]/t MeV Φ[a min ] 300 MeV 295 MeV 289.5MeV T/T 0.0 c MeV Φ[a] =0 a 2π 15

29 Order Parameter Polyakov gauge: T c = MeV Marhauser, Pawlowski, arxiv: Φ[a min ] V [a] = k flow V [a], αs MM (k) = t +1 critical exponent from screening mass: V [a min ] t 2ν, ν =0.65(2) [ν Ising =0.63] 16

30 Finite Temperature Propagators Pure (Landau) gauge: Lattice & DSEs: D µν (p) = Z T(p 2 ) p 2 P T µν + Z L(p 2 ) p 2 P L µν Cucchieri, Karsch, Petreczky, Phys. Lett. B 497 (2001) 80 Cucchieri, Karsch, Petreczky, Phys. Rev. D 64 (2001) Maas, Wambach, Grüter, Alkofer, EPJC 37 (2004) 335 Maas, Wambach, Alkofer, EPJC 42 (2005) 93 Cucchieri, Maas, Mendes, Phys. Rev. D 75 (2007) Maas, arxiv: T =0: T>0: Z T = Z L = Z Z T Z Running coupling, FRG: Braun, Gies, JHEP 06 (2006) 024 Braun, Gies, Phys. Lett. B 645 (2007) 53 Marhauser, Pawlowski, arxiv: dim. fixed-point behavior for p<t, only weakly T -dependent otherwise, N f dependence 17

31 Electric Screening Mass 1.4 D L (0) 1/2 (T/T c 1) ν, 1.2 ν =0.60(1) [ν Ising =0.63] T T c (1 + δ), finite volume Gribov copies 0.6 critical behavior? (well established) 0.4 ν =0.62(2) Maas, LvS, in progress 0.2 SU(2) SU(3) Data: Maas, private communication Fischer, Maas, Müller, arxiv: T/T c 18

32 Applications II Deconfinement & Chiral Symmetry Restoration quenched 19

33 Applications II Deconfinement & Chiral Symmetry Restoration quenched Allow quarks with b.c. s: ψ(1/t, x) = e 2πiθ ψ(0, x) in observables: O(θ) = O[ψ θ ] e.g., θ-dependent quark condensate, mass function, density... 19

34 Applications II Deconfinement & Chiral Symmetry Restoration quenched Allow quarks with b.c. s: ψ(1/t, x) = e 2πiθ ψ(0, x) in observables: O(θ) = O[ψ θ ] e.g., θ-dependent quark condensate, mass function, density... observe: O(θ + 1) = O(θ) O(θ + 1/3) = O(θ), always Fourier series,inz 3 -symmetric confined phase 19

35 Dual Order Parameters Order parameter for confinement: O = 1 0 dθ O(θ) e 2πiθ Z3 -symmetry, confinement O =0, dual quark condensate, dual mass, density... Originally from lattice Gattringer, Phys. Rev. Lett. 97 (2006) Synatschke, Wipf, Wozar, Phys. Rev. D 75 (2007) Synatschke, Wipf, Langfeld, Phys. Rev. D 77 (2008) Bilgici, Bruckmann, Gattringer, Hagen, Phys. Rev. D 77 (2008)

36 T-dependent Gluon Propagator Input for quark DSE: Fischer, Phys. Rev. Lett. 103 (2009) Fischer, Müller, Phys. Rev. D 80 (2009) T=0 T=0.6 T c T=0.99 T c T=2.2 T c T=0 T=0.6 T c T=0.99 T c T=2.2 T c Z T (p 2 ) 1.5 Z L (p 2 ) p 2 [Gev 2 ] p 2 [Gev 2 ] Fischer, Maas, Müller, arxiv:

37 Chiral & Deconfinement Transition(s) 0.10 SU(3), quenched dual Polyakov condensate loop (dressed Polyakov Condensate loop) chiral condensate (chiral limit) T/T c T chiral T conf 277 MeV Christian Fischer s talk Fischer, Maas, Müller, arxiv:

38 Applications III Full Dynamical QCD at zero & imaginary chemical potential 23

39 Applications III Full Dynamical QCD at zero & imaginary chemical potential Quarks with b.c. s: ψ(1/t, x) = e 2πiθ ψ(0, x) but observables: O θ (θ) = O[ψ θ ] θ observables in different theories, built on θ-dependent ground states 23

40 Applications III Full Dynamical QCD at zero & imaginary chemical potential Quarks with b.c. s: ψ(1/t, x) = e 2πiθ ψ(0, x) but observables: O θ (θ) = O[ψ θ ] θ observables in different theories, built on θ-dependent ground states observe: O(θ + 1/3) = O(θ) now always! Roberge, Weiss, Nucl. Phys. B 257 (1986) 734 imaginary chemical potential: µ =2πiTθ 23

41 Full Dynamical QCD Landau gauge Yang-Mills with background, compare: (a) a 0 = 1 T ga3 0 θ=0, breaks Roberge-Weiss symmetry, dual observables, deconfinement transition (dual condensate, dual density... ) (b) a θ = 1 T ga3 0 θ, include Nf = 2 dynamical quarks massless, chiral limit quark condensate, mass, density, f π,..., order parameters for chiral phase transition in QCD θ (but no dual order parameters) coupled to mesons dynamical hadronisation c.f., quark meson model: Berges, Tetradis, Wetterich, Phys. Rept. 363 (2002) 223 Schaefer, Wambach, Phys. Part. Nucl. 39 (2008)

42 Full Dynamical QCD, Nf = 2, chiral limit (a) a 0 = 1 T ga3 0 θ=0, µ =0 Braun, Haas, Marhauser, Pawlowski, arxiv: f π (T)/f π (0) Dual density 0.8 Polyakov Loop ±20 MeV T [MeV] T chiral T conf 180 MeV Lisa Haas talk χ L,dual 25

43 Full Dynamical QCD, Nf = 2, chiral limit free energy difference: Braun, Haas, Marhauser, Pawlowski, arxiv: f(t,θ) f(t,0) θ change boundary conditions of confined quarks at no cost 26

44 QCDθ imaginary chemical potential Originally from lattice de Forcrand, Philipsen, Nucl. Phys. B 642 (2002) 290 no sign poblem Here (b) QCD θ, µ =2πiTθ D Elia, Lombardo, Phys. Rev. D 67 (2003) Kratochvila, de Forcrand, Phys. Rev. D 73 (2006) Wu, Luo, Chen, Phys. Rev. D 76 (2007) Braun, Haas, Marhauser, Pawlowski, arxiv: T [MeV] T conf T χ 0 0 π/3 2π/3 π 4π/3 2πθ Lisa Haas talk 27

45 Dual Observables vs. Chiral Transition in QCDθ Quark mass parameter (a) dual mass, confinement 1 = dθ M (θ) e 2πiθ M (b) mass in QCDθ, chiral symmetry breaking Mθ (θ + 1/3) = Mθ (θ) 0 Mθ (θ) M (θ) θ θ = 1/3 28

46 Summary & Outlook Propagators and Running Coupling of Landau Gauge QCD very well understood at T=0, 0.40 increasingly well at finite T α MM L (p2 ) (β, N 4 )=(6.0, 32 4 ) (6.2, 32 4 ) (6.4, 32 4 ) (6.6, 32 4 ) (6.9, 48 4 ) (7.2, 64 4 ) (7.5, 64 4 ) (8.5, 48 4 ) α MM s (p2 ) 4 3 Z(p 2 )!"!! #"$ %&!! #"$ %&!! '""!"!! '"( '#!! '"( N f = p 2 [GeV 2 ] p [GeV] 29

47 Summary & Outlook Propagators and Running Coupling of Landau Gauge QCD very well understood at T=0, increasingly well at finite T 1 st Applications of range from: deconfinement phase transition of the pure gauge theory The hard part done! deconfinement & chiral symmetry restoration in full QCD at zero and imaginary chemical potential 289.5MeV 310 MeV SU(3) f π (T)/f π (0) Dual density Polyakov Loop MeV 295 MeV 289.5MeV 285 MeV χ L,dual T [MeV]

48 Summary & Outlook Propagators and Running Coupling of Landau Gauge QCD very well understood at T=0, increasingly well at finite T 1 st Applications of range from: deconfinement phase transition of the pure gauge theory Tested against Lattice Results The hard part done! deconfinement & chiral symmetry restoration in full QCD at zero and imaginary chemical potential 29

49 Summary & Outlook Propagators and Running Coupling of Landau Gauge QCD very well understood at T=0, increasingly well at finite T 1 st Applications of range from: deconfinement phase transition of the pure gauge theory deconfinement & chiral symmetry restoration in full QCD at zero and imaginary chemical potential Tested against Lattice Results Continue to Combine Methods towards: dynamical quark flavors - real chemical potential: critical end point, triple point...? - obtain model independent information on QCD phase diagram - refine and constrain effective hadronic theories 29

50 Summary & Outlook Propagators and Running Coupling of Landau Gauge QCD very well understood at T=0, increasingly well at finite T 1 st Applications of range from: deconfinement phase transition of the pure gauge theory deconfinement & chiral symmetry restoration in full QCD at zero and imaginary chemical potential Tested against Lattice Results Continue to Combine Methods towards: dynamical quark flavors - real chemical potential: critical end point, triple point...? - obtain model independent information on QCD phase diagram - refine and constrain effective hadronic theories Thank You! 29

51 Extra Sides Some Additional Material 38

52 Dyson-Schwinger Equations Generating functional: Z[j] = exp(j, φ) glue: -1 = -1-1/2-1/2 Green s functions (T =0here) - 1/6-1/2 A µ (x)a ν (y) : D µν (p) = Z(p2 ) p 2 δ µν p µp ν p c(x) c(y) : ψ(x) ψ(y) : D G (p) = G(p2 ) p 2 S(p) = Z ψ(p 2 ) ip/ + M(p 2 ) ghost: quark: -1 = -1-1 = -1 39

53 Functional RG (Flow) Equations Effective action: Legendre transform Γ[φ j ]=(j, φ j ) ln Z[j] k 0 k S IR k- k k UV k= 1PI vertex functions Γ (n) (x 1,...x n ) free energy with φ j = φ j k t = 1 k Γ k [φ] 2 Wetterich, Phys. Lett. B 301 (1993) 90 40

54 Functional RG (Flow) Equations Landau gauge QCD propagators, glue: k k 1 = ghost: k k 1 =

55 Generating Functional and Effective Action D II (Definition) 2 Fields: φ = {A, c, c, ψ, ψ} Introduce sources: Z[j] = exp(j, φ) π G2 (p 2 ) Z (p 2 ) Effective action: Legendre transform [Γ 3g (p 2 )] 2 Z 3 (p 2 ) Γ[φ j ]=(j, φ j ) ln Z[j] Green s functions φ(x)φ(y), φ(x)φ(y)φ(z),... Laufende 1PI vertex functions Kopplun Γ (n) (x 1,...x n ) free energy with φ j = φ j Γ qg (p 2 ) (p 2 ) 1/ Γ ca c Γ 3A Γ 4A ψa ψ Γ [Γ 4g (p 2 )] Z 2 (p 2 ) α qg (p 2 )=α µ [Γ r, C.F., F. Llanes-Estrada, Phys.Lett.B611: , !

56 Ghost Propagator Bogolubsky et al., Phys. Lett. B 676 (2009) scaling decoupling 4 3 β = (14 conf.) 80 4 (11 conf.) 80 4 (5 conf.) L = 13.6 fm G(p 2 ) p 2 [GeV 2 ] p 2 [GeV 2 ] Fischer, Maas, Pawlowski, Annals Phys. 324 (2009) 2408 Maas, arxiv:

57 Imaginary Chemical Potential Polyakov-NJL model: Sakai, Kashiwa, Kouno, Matsuzaki, Yahiro, Phys. Rev. D 79 (2009) Lattice data: Wu, Luo, Chen, Phys. Rev. D 76 (2007)