PNJL Model and QCD Phase Transitions

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1 PNJL Model and QCD Phase Transitions Hiromichi Nishimura Washington University in St. Louis INT Workshop, Feb. 25, 2010

2 Phase Transitions in Quantum Chromodynamics This Talk Low Temperature Lattice and Phenomenology say it is confined. High Temperature Perturbation theory is reliable and shows deconfinement. Want to make a phenomenological model which describe both low and high temperature QCD behavior in a single picture. PNJL

3 Outline Confinement and Chiral Symmetry Breaking Nambu-Jona Lasinio Model (NJL) Polyakov-Nambu-Jona Lasinio Model (PNJL) Fundamental Fermions (Finite-Temperature QCD) Adjoint Fermions - Antiperiodic Boundary Conditions (ABC) - Periodic Boundary Conditions (PBC)

4 Confinement Consider pure gauge theory (i.e. only gauge bosons and no fermions.) Order Parameter: Polyakov Loop P ( x) =Pe i R β 0 dta 0(x) The world line of massive quark at x. T rp ( x) = e βf q β=1/t P ( x) g SU(N c ) z Z(N c ) : P ( x) zp( x) T rp ( x) =0 F q = Confinement T rp ( x) 0 F q = F inite Deconfinement χsb Confinement Deconfinement Condensate of bound quark-antiquark pairs, almost like Cooper pairs in BCS. m 0 ψψ 0 ψψ =0 Order Parameter: ψψ Dynamically generated quark mass (constituent mass) <K. Fukushima, PRD69, 2004>

5 NJL Model (1/2): Effective Lagrangian How does low energy QCD lagrangian look like? L QCD = ψ(iγ µ µ m 0 )ψ 1 4 F a µνf µν a + g ψγ µ A µ ψ 1. Effective Quark Theory. Guess 2. Integrate out high energy degrees of freedom. 3. Chiral Symmetry. L NJL = ψ (iγ m 0 ) ψ + g S 2 <Y. Nambu, and G. Jona-Lasinio, PR122 & PR124, 1961> [ ( ψλ a ψ ) 2 + ( ψiγ5 λ a ψ ) 2 ] + g D [ det ψ (1 γ5 ) ψ + h.c. ] 4 Fermion Interaction U A (1) Breaking - Similar to the microscopic theory of superconductivity. - Non-renormalization theory in three dimension. 2 Flavors: Set g D =0. L NJL ψ (iγ m 0 ) ψ + g S 2 [ ( ψλ a ψ ) 2 + ( ψiγ5 λ a ψ ) 2 ]

6 NJL Model (2/2): Gap Equation ψψ = σ S eff = i N f j=1 ln [det (iγ m 0j +2g S σ j )] + d 4 x N f j=1 g S σj 2 Effective mass: m = m 0 2g s σ V eff =2g S σ 2 4N c Λ d 3 p (2π) 3 p2 + m 2 dv eff m = m 0 +4g S N c Λ d 3 p (2π) 3 dm =0 Gap Equation m p2 + m 2 If g S Λ 2 > π2 8, then σ 0 and chiral symmetry is broken. There is a great deal of work on the model. A downside is NJL model lacks confinement.

7 S eff = i N f j=1 PNJL Model (1/3): Setup ln [det (iγ ( ia) m 0j +2g S σ j )] + d 4 x N f j=1 g S σj 2 - The covariant derivative couples fermions to a background Polyakov loop. A 0 - is constant and diagonal. ( Polyakov Gauge ) - Interplay between Polyakov loop and σ is now easy to see. V P NJL = V cond + V zero + V quark,t + V gluon,t V cond + V zero =2g S σ 2 4N c Λ V quark,t = 2N f Tr R [T d 3 p (2π) 3 p2 + m 2 d 3 p (2π) 3 ln(1 ± Pe ω p/t )+h.c.] = 4m2 T 2 π 2 + Antiperiodic B.C. - Periodic B.C. n=1 ( 1) n Tr R P n n 2 K 2 (nm/t ) <P. Meisinger, and M. Ogilvie, PRD65, 2002> V gluon,t Need a phenomenological model.

8 V pert = 2Tr A [ PNJL Model (2/3): Gluonic Part d 3 p (2π) 3 ln(1 Pe ω p/t )] T 4 Pure Gauge ω p = p Valid at high T m 0 Add subleading term (T 2 ) by hand to get correct behavior at low T. Introduce a mass term, M: ω p = p 2 + M 2 M is just a parameter, not mass of gluon. <P. Meisinger, T. Miller, and M. Ogilvie, PRD65, 2002> V gluon,t = f(p )T 4 + g(p )M 2 T TrP T 3 M = 596 MeV T d = 270 MeV T MeV Pure Gauge

9 PNJL Model (3/3): Results Now we have a whole potential: V P NJL = V cond + V zero + V quark,t + V gluon,t 1.0 m T m 0 m 0 =5.5 MeV 0.8 Λ = 631 MeV 0.6 Confined Deconfined g S = GeV ψψ 0 ψψ =0 <K. Fukushima,PLB59, 2004> 0.2 <HN and M. Ogilvie, PRD81, 2010> 0.0 TrP T T MeV Fundamental Fermions (Nf =2) PNJL model works! m 0 =5.5 MeV m0

10 Why Adjoint QCD? SUSY - N f =1/2 Conformal Theory <E. Poppitz and M. Unsal, JHEP0909, 2009> - 2 <N f <N is in the conformal window. Eguchi Kawai Model - Volume independence of SU(N) gauge theories in the large N limit. Confinement - Confinement at high temperature or small L.

11 Confinement with Small Radius Large β Confined Large L Confined Phase Transitions? Small β Deconfined Small L Confined! <M. Unsal and L. Yaffe, PRD78, 2008> Z E = DA DΨ D Ψ e R β 0 dτ R d 3 xl E = T r e βh ABC: Temporal Component β =1/T PBC: Spatial Component β = L Adjoint Fermions with PBC - Confinement at high T or small radius. <P. Kovtun, M. Unsal, and L. Yaffe, JHEP, 2007> - New confinement is analytically tractable. <M. Unsal, PRD, 2009>

12 Adjoint fermions with ABC (finite T) in PNJL Lattice shows that scale of deconfinement and chiral T χ /T d 7.8 symmetry restoration are different! <J. Engels, S. Holtmann, and T. Schulzel, NPB724, 2005> Choose the parameters; 1.0 m 0 =0 0.8 Λ = GeV 0.6 g S Λ 2 = TrP T 3 m T m 0 such that T χ /T d T MeV <HN and M. Ogilvie, PRD81, 2010>

13 Adjoint fermions with PBC in PNJL (1/2) m L 1 m TrP L 1 3 C D S R - No chiral symmetry restoration up to 10 GeV. - 3 distinct 1st order phase transitions for TrP L 1 MeV <HN and M. Ogilvie, PRD81, 2010> 2.0 m 0 =0 Λ = GeV κ = g S Λ 2 = C = Confined D = Deconfined S = Skewed Phase Κ 1.4 D S 0 C? R = Reconfined L 1

14 Adjoint fermions with PBC in PNJL (2/2) 1800 S 1750 m L D C L 1 MeV <G. Cossu and M. D Elia, JHEP0907, 2009> Remaining Puzzle: Casher s Argument D S C - Unsal s scenario predicts confinement without chiral symmetry breaking at small L. 1.6 Κ We need more lattice results! L 1?

15 Conclusions Two fundamental fermions with ABC. The PNJL model can reproduce two crossover transitions at T 170 MeV. Two adjoint fermions with ABC. The PNJL model can reproduce T χ /T d 7.8. PNJL works for finite-temperature cases very well. Two adjoint fermions with PBC. The PNJL model shows that two confining regions are connected. We need more lattice results.