Thermodynamics of a Nonlocal PNJL Model for Two and Three Flavors

Transcription

1 for Two and Three Flavors Thomas Hell, Simon Rößner, Marco Cristoforetti, and Wolfram Weise Physik Department Technische Universität München INT Workshop The QCD Critical Point July 28 August 22, 2008

2 Outline 1 The Nonlocal Nambu Jona-Lasinio Model abandoning the NJL cutoff Effective Action of a Covariant Nonlocal Model for Two Flavors Mean Field Approximation Instantons vs. Perturbative QCD Dynamical Quark Mass 2 Thermodynamics of the Nonlocal PNJL Model Coupling Quarks and Polyakov Loop Gap Equations in Mean-Field Approximation Pressure and Mesonic Corrections Finite Density Case and QCD Phase Diagram 3 The Three-Flavor Case Effective Nonlocal Action and t Hooft Determinant Thermodynamics 4 Summary and Conclusions

3 1 The Nonlocal Nambu Jona-Lasinio Model

4 Formalism of the Nonlocal NJL Model Nonlocal Action in Euclidean Space S E = d 4 x ψ(x)( i/ +m q )ψ(x) G d 4 x d 4 y j µ a 2 (x)c(x y) ja µ (y) Color Currents Effective Interaction j µ a (x)= ψ(x)γ µ t a ψ(x) Fierz transformation j µ a (x)= ψ(x)γ µ a ψ(x) Γ a = (1, iγ 5 τ ) p+ q p Γ µ a G 2 C(q) Γ ν b p + q p G. Ripka, Quarks bound by chiral fields. (1997) [1] G. Dumm, D. Grundfeld and N. N. Scoccola, Phys. Rev. D (2006) [2]

5 Formalism of the Nonlocal NJL Model Nonlocal Action in Euclidean Space S E = d 4 x ψ(x)( i/ +m q )ψ(x) G d 4 x d 4 y j µ a 2 (x)c(x y) ja µ (y) Distribution Function Effective Interaction NJL p+ q G 2 Γ ν b p + q C(q) nonlocal NJL p Γ µ a C(q) p q [GeV] G. Ripka, Quarks bound by chiral fields. (1997) [1] G. Dumm, D. Grundfeld and N. N. Scoccola, Phys. Rev. D (2006) [2]

6 Improvements on the Standard NJL Model Nonlocality arises naturally in the instanton liquid model and in Dyson-Schwinger calculations [3, 4] Underlying Lagrangian preserves SU L (2) SU R (2) chiral symmetry Axial anomalies are incorporated, current algebra relations are preserved [5] f π g πqq = σ+o(m q ) m 2 πf 2 π= m q ψψ +O(m 2 q) (Goldberger Treiman) (Gell-Mann Oakes Renner) No UV-cutoff is needed No unphysical decays (e. g.π qq)

7 Bosonization of the Euclidean Action Bosonized Euclidean Action Introduction of bosonic fieldsϕ a (x)=(σ(x), π(x)) Z= D ψdψ e S E Z= DσD π e Sbos E, where S bos E = ln det Â+ 1 d 4 p 2G (2π) 4φ a(p)φ a(p) with A(p, p )= ( ) ( p+ p /p+ m q (2π) 4 δ (4) (p p ) )+C Γ a φ a (p p ) 2

8 Power Series Expansion of the Action Mean Field (MF) Approximation Expansion ofs bos E about MF-values σ MF = σ and π MF = 0 S bos E =S MF E +S(2) E +...

9 Power Series Expansion of the Action Mean Field (MF) Approximation Expansion ofs bos E about MF-values σ MF = σ and π MF = 0 S bos E =S MF E MF-Approximation of the Action S MF E V (4) = 2N f N c with M(p)=m q +C(p) σ Mesonic Fluctuations S (2) E = 1 2 +S(2) E +... d 4 p (2π) 4 ln[ p 2 + M 2 (p) ] + σ2 2G, d 4 p (2π) 4 [ G + (p 2 )δσ(p)δσ( p)+g (p 2 )δ π(p) δ π( p) ], G ± (p 2 )= 1 G Γ± Γ ±

10 Determination of the Distribution Function C Constituent Quark Mass in MF-Approximation Constituent quark mass M given by pole of fermion propagator ( ) 0=S 1 F, nl (p)= δ 2 S E M(p δ ψ(p) 2 )=m q +C(p) σ δψ(0) p 2 = M 2

11 Determination of the Distribution Function C Constituent Quark Mass in MF-Approximation Constituent quark mass M given by pole of fermion propagator ( ) 0=S 1 F, nl (p)= δ 2 S E M(p δ ψ(p) 2 )=m q +C(p) σ δψ(0) p 2 = M 2 Perturbative QCD Comparison to QCD mass formula H. D. Politzer, Nucl. Phys. B (1976) [7] C(p 2 ) 2π 3 with α s (p 2 )= α s (p 2 ) p 2 for p 2 Γ 2 4π ( β 0 ln p 2 Λ 2 QCD )+...

12 Determination of the Distribution Function C Constituent Quark Mass in MF-Approximation Constituent quark mass M given by pole of fermion propagator ( ) 0=S 1 F, nl (p)= δ 2 S E M(p δ ψ(p) 2 )=m q +C(p) σ δψ(0) p 2 = M 2 Non-perturbative QCD Comparison to the Instanton Liquid Model T. Schäfer, E. V. Shuryak, Rev. Mod. Phys (1998) [6] C(p 2 ) e p2 d 2 /2 for p 2 <Γ 2 with the instanton size d= 0.35 fm=(0.56 GeV) 1 Perturbative QCD Comparison to QCD mass formula H. D. Politzer, Nucl. Phys. B (1976) [7] C(p 2 ) 2π 3 with α s (p 2 4π )= ( β 0 ln α s (p 2 ) p 2 for p 2 Γ 2 p 2 Λ 2 QCD )+...

13 Determination of the Distribution Function C Distribution Function 2πα s (p 2 ) C(p 2 for p 2 Γ 2 ) 3 p 2 e p2 d 2 /2 for p 2 <Γ 2 Distribution Function C C(p) instantons pqcd Γ p [GeV]

14 Gap Equation and Constituent Quark Mass MF-value σ can be found by the principle of least action δsmf E δσ = 0 Gap Equation (MF-Approximation) d 4 p M(p) σ=4n f N c G (2π) 4C(p) p 2 + M 2 (p) with M(p)=m q +C(p) σ

15 Gap Equation and Constituent Quark Mass MF-value σ can be found by the principle of least action δsmf E δσ = 0 Gap Equation (MF-Approximation) d 4 p M(p) σ=4n f N c G (2π) 4C(p) p 2 + M 2 (p) with M(p)=m q +C(p) σ Constituent Quark Mass 0.3 Lattice data from P. O. Bowman et al., Nucl. Phys. Proc. Suppl. (2003) [8] M(p) [GeV] p [GeV]

16 2 Thermodynamics of the Nonlocal PNJL Model

17 Coupling Quarks and Polyakov Loop Definition (Polyakov Loop) Φ( x ) = 1 [ ] tr c L( x ) N c { L( x )=P exp i β 0 dτ A 4 (τ, x ) Polyakov Loop serves as an order parameter for confinement-deconfinement phase transition [9, 10] Φ =e βf q } Confinement-Deconfinement Phase Transition confinement Φ =0 deconfinement Φ 0 Z (3) unbroken Z (3) spontaneously broken

18 Coupling Quarks and Polyakov Loop Introduction of gluons through minimal coupling p µ p µ + A µ 3- and 8-components of gluon fields; no spatial fluctuations A µ =δ 0 µ(a 3 0 λ 3+ A 8 0 λ 8) Integrate out non-diagonal components of SU(3) c Polyakov Potential K. Fukushima, Phys. Lett. B (2004) [11] U T 4= 1 2 b 2(T )Φ Φ+b 4 (T ) ln [ 1 6Φ Φ+4 ( Φ 3 +Φ 3) 3(Φ Φ) 2] with Φ= 1 3 tr [ (( ))] c exp i βa3 λ 3 +βa 8 λ 8, coefficients b 2 (T ), b 4 (T ) and parameters from S. Rößner, C. Ratti and W. Weise, Phys. Rev. D (2007) [12]

19 Polyakov Potential T< T c T> T c Each minimum is populated equiprobably Color neutrality preserved if one sums over all six minima!

20 Thermodynamic Potential Partition Function { Z= Dϕ DA e ln detã exp β 0 dτ } d 3 x ϕ2 (x) 2G

21 Thermodynamic Potential Partition Function { Z= Dϕ DA e ln detã exp β 0 dτ } d 3 x ϕ2 (x) 2G Thermodynamic Potential Ω= T V ln Z= ln det [ β S 1] + σ2 2G +U(Φ,Φ, T ) with the Nambu-Gor kov propagator in Matsubara space S 1 (iω n, p )= ( iωn γ 0 γ p M ia 4 γ 0 0 M= diag c (M(ω n, p ), M(ω+ n, p ), M(ω0 n, p )) ω n = (2n+1)πT,ω ± n=ω n ± A 4,ω 0 n =ω n 0 iω n γ 0 γ p M + ia 4 γ 0 )

22 Gap Equations in MF Approximation Gap Equations (A 4 = 0 S. Rößner, Diploma Thesis, Technische Universität München (2006) [13]) 8 Ω σ = Ω = Ω = 0 A 4 A 4 3 8

23 Gap Equations in MF Approximation Gap Equations (A 4 = 0 S. Rößner, Diploma Thesis, Technische Universität München (2006) [13]) 8 PSfrag Ω σ = Ω = Ω = 0 A 4 A Chiral Condensate and Polyakov Loop (T c = 207 MeV) ψψ / ψψ Lattice data from M. Cheng et al., Phys. Rev. D (2008) [14] T/T c Φ

24 Pressure and Mesonic Corrections Pressure Contributions in Random Phase Approximation J. Hüfner, S. P. Klevansky, P. Zhuang, H. Voss, Annals Phys. (1994) [15] Hartree Fock RPA chains d 3 [ p P MF (T )=2N f T (2π) ln ω i 2 3 n + p 2 + M(ω i n ], p )2 σ2 2G U i=0,± n Z d M d 3 P meson (T )= 2 T p (2π) ln 1 G ν m, p 3 M=π,σ m Z

25 Pressure and Mesonic Corrections Pressure and Mesonic Contributions MF + π,σ (in-medium) MF + π (in-medium) MF MF + π,σ (ideal) P/T mesons quarks T/T c

26 Pressure and Mesonic Corrections m π = 140 MeV m π = 500 MeV MF + π,σ (in-medium) MF + π (in-medium) MF MF + π,σ (ideal) 2.0 MF + π,σ (in-medium) MF + π (in-medium) MF MF + π,σ (ideal) P/T P/T T/T c T/T c Strong suppression of mesonic contributions for high quark masses

27 Related Thermodynamic Quantities Trace anomaly Energy density (ε 3P)/T 4 8 Lattice data from F. Karsch et al., 6 arxiv: (2008) [16] 4 2 MF + π,σ(in-medium) MF ε/t Lattice data from F. Karsch et al., Phys. Lett. B (2000) [17] MF + π,σ(in-medium) MF T/Tc T/Tc p/ε Sound velocity M. Cheng et al., Phys. Rev. D (2008) [14] p/ε MF + π,σ(in-medium) MF v 2 s MF + π,σ(in-medium) MF ε 1/4 [(GeV/fm 3 ) 1/4 ] ε 1/4 [(GeV/fm 3 ) 1/4 ]

28 Finite Density and Phase Diagram Introduction of chemical potential throughω n ω n iµ+a 4

29 Finite Density and Phase Diagram Introduction of chemical potential throughω n ω n iµ+a 4 σ as a function of T andµ Σ TGeV ΜGeV CEP Critical endpoint is found at (T CEP,µ CEP )= (167 MeV, 175 MeV)

30 3 Nonlocal Quark Model for the 3-Flavor Case

31 Derivation of the Nonlocal Lagrangian Chirally invariant nonlocal two-flavor Lagrangian easily generalized to three flavors by the replacementτ i λ a Task: Inclusion of explicit breaking of U A (1) symmetry

32 Derivation of the Nonlocal Lagrangian Chirally invariant nonlocal two-flavor Lagrangian easily generalized to three flavors by the replacementτ i λ a Task: Inclusion of explicit breaking of U A (1) symmetry Solution: Instanton induced quark interactions T. Schäfer, E. V. Shuryak (1998) [6] S E int d 4 x(detj + + detj ) f f with ( J ± (x) ) ij= d 4 y d 4 zψ i (y) 1 ( ) 1 γ5 K(y x, z x)ψj (z) 2 S E int invariant under SU(3) V SU(3) A U(1) V transformations t Hooft determinant not invariant under U(1) A transformations U(1) A problem is solved describing theη mass correctly

33 Bosonized Euclidean Action N. N. Scoccola, (2004) [19] Introduction of 9 scalar and pseudoscalar fieldsσ a (x) andπ a (x) Introduction of 9+9=18 auxiliary fieldsσ a (x) andπ a (x) Bosonization G. Ripka, (1997) [1]; B. O. Kerbikov and Yu. A. Simonov (1995) [18] S E bos = ln det{ ( /p+ ˆm q )(2π) 4 δ (4) (p p ) ( p+ p [ +C )λ a σa (p p )+iγ 5 π a (p p ) ]} 2 { d 4 x σ a Σ a +π a Π a + G 2 (Σ aσ a +Π a Π a ) + H } 4 D abc (Σ a Σ b Σ c 3Σ a Π b Π c ) DetermineΣ a andπ a using the Stationary Phase Approximation (SPA)

34 Thermodynamic Potential Thermodynamic Potential in Mean Field Approximation d 3 p Ω MF = 2T Tr ln [ ω c n2 + p 2 + M 2 (2π) 3 f (ωc n, p )] [ f {u,d,s} c {±,0} f {u,d,s} n Z ( σf Sf + G 2 S 2 f ] ) H + 2 S u Sd Ss +U(Φ,Φ, T ) Chiral Condensate and Loop Pressure SB qq / qq ūu = dd Φ P/T ss T [GeV] T[GeV]

35 Summary and Conclusions Summary Nonlocal model derived from first symmetry principles of QCD Coupling to gluons is possible through introduction of Polyakov LoopΦ Corrections beyond mean field approximation were included in the 2-flavor case Conclusions Poor man s Dyson-Schwinger approach to nonperturbative QCD Momentum dependent effective quark mass Thermodynamics: Entanglement of chiral and confinement/deconfinement transition Model applicable to finite density description of QCD Model leads reasonable results both for the 2- and 3-flavor case

36 Thank you for your attention

37 Literature I Georges Ripka. Quarks bound by chiral Fields. Oxford University Press, Oxford, D. Gomez Dumm, A. G. Grunfeld, and N. N. Scoccola. On covariant nonlocal chiral quark models with separable interactions. Phys. Rev., D74:054026, C. D. Roberts and A. G. Williams. Dyson-Schwinger equations and the application to hadronic physics. Progress in Particle and Nuclear Physics, 33:477, Craig D. Roberts and Sebastian M. Schmidt. Dyson-Schwinger equations: Density, temperature and continuum strong QCD. Prog. Part. Nucl. Phys., 45:S1 S103, 2000.

38 Literature II E. Ruiz Arriola and L. L. Salcedo. Chiral and scale anomalies of non local dirac operators. Phys. Lett., B450: , Thomas Schafer and Edward V. Shuryak. Instantons in QCD. Rev. Mod. Phys., 70: , H. David Politzer. Effective quark masses in the chiral limit. Nucl. Phys., B117:397, Patrick O. Bowman, Urs M. Heller, Derek B. Leinweber, and Anthony G. Williams. Modelling the quark propagator. Nucl. Phys. Proc. Suppl., 119: , 2003.

39 Literature III Larry D. McLerran and Benjamin Svetitsky. A Monte Carlo study of SU(2) Yang-Mills theory at finite temperature. Phys. Lett., B98:195, Larry D. McLerran and Benjamin Svetitsky. Quark liberation at high temperature: A Monte Carlo study of SU(2) gauge theory. Phys. Rev., D24:450, Kenji Fukushima. Chiral effective model with the Polyakov loop. Physics Letters B, 591:277, Claudia Ratti, Michael A. Thaler, and Wolfram Weise. Phases of QCD: Lattice thermodynamics and a field theoretical model. Phys. Rev., D73:014019, 2006.

40 Literature IV Simon Rößner. Field theoretical modelling of the QCD phase diagram. Diploma Thesis, M. Cheng et al. The QCD Equation of State with almost Physical Quark Masses. Phys. Rev., D77:014511, J. Hufner, S. P. Klevansky, P. Zhuang, and H. Voss. Thermodynamics of a quark plasma beyond the mean field: A generalized Beth-Uhlenbeck approach. Annals Phys., 234: , Frithjof Karsch. Equation of state and more from lattice regularized QCD F. Karsch, E. Laermann, and A. Peikert. The pressure in 2, 2+1 and 3 flavour QCD. Phys. Lett., B478: , 2000.

41 Literature V B. O. Kerbikov and Yu. A. Simonov. Path integral bosonization of three flavor quark dynamics A. Scarpettini, D. Gomez Dumm, and N. N. Scoccola. Light pseudoscalar mesons in a nonlocal SU(3) chiral quark model. Phys. Rev., D69:114018, 2004.