Phase diagram of QCD-like matter from upgraded PNJL model David - PowerPoint PPT Presentation

Introduction (P)NJL models Equation of state Phase diagram of QCD-like matter from upgraded PNJL model David Fuseau fuseau@subatech.in2p3.fr 3rd year Ph.D student in SUBATECH Supervisor : Joerg Aichelin ZIMÁNYI SCHOOL’19, 4.12.2019 1/36

Introduction (P)NJL models Equation of state QGP and phase diagram Two phases predicted for QCD matter : Hadronic phase : Quarks and gluons are bound into hadrons : confinement This is hadronic matter, we can observe it experimentally QGP phase : Quarks and gluons are free in the medium We don’t directly observe this phase experimentally 2/36

Introduction (P)NJL models Equation of state QGP and phase diagram T (GeV) LHC QGP RHIC SPS, GSI T c ≈ 0 . 15 Critical End Point ? Temperature Hadronic Gas CFL Color Superconductor ? Neutron Star µ (GeV/fm 3 ) Chemical Potential Figure – Phase Diagram of nuclear matter 3/36

Introduction (P)NJL models Equation of state QGP and phase diagram QCD lagrangian : life is tough k γ µ ∂ µ ψ j µ ψ j k ψ j L QCD = i δ ij ¯ k + g s ¯ k γ µ λ a k − m k ¯ k − 1 µν F a µν ψ i ψ i ij A a ψ i 4 F a Perturbative approach pQCD Need of a small coupling constant for convergence of the perturbative series, works at high energy / high T, µ . not working at phase transition, the coupling constant is large. Lattice approach lQCD Space-time discretized on a lattice. Matter on the node, gluons are the lines connecting the nodes Static study, no dynamics on lattice, only thermodynamics Does not work at finite chemical potential, only at finite temperature. 4/36

Introduction (P)NJL models Equation of state QGP and phase diagram A. Schmitt from ect* summer school lectures 5/36

Introduction (P)NJL models Equation of state QGP and phase diagram GSI, FAIR NICA BES program (RHIC) SPS (CERN) Lower temperature and higher density : search for critical end point, phase transitions and neutron star physics. Needs prediction to know where to search. Those predictions can only be made using effective models 6/36

Introduction (P)NJL models Equation of state (P)NJL Model 7/36

Introduction (P)NJL models Equation of state Effective model Works only in a special domain of energy but allows finite chemical potential studies. Contact interaction Static approximation : no gluons propagating the interaction Frozen gluons 1 = − 1 p 2 − ǫ 2 ǫ 2 g g if p << ǫ 2 g 8/36

Introduction (P)NJL models Equation of state Nambu-Jona-Lasinio (NJL) Lagrangian k ) 2 + ’t Hooft term L NJL = δ ij ψ i k ( i γ µ ∂ µ − m ) ψ j k + G ( ψ i k λ ij ψ j Symmetries Chiral symmetry SU L ( 3 ) ⊗ SU R ( 3 ) Free parameters Color symmetry SU c ( 3 ) (but global) m 0 q = 0 . 0055 GeV Flavour symmetry SU f ( 3 ) m 0 s = 0 . 134 GeV Λ = 0 . 569 GeV Problem G = 2 . 3 Λ 2 GeV − 2 Center symmetry is missing K = 11 Λ 5 GeV − 5 Confinement is not described 9/36

Introduction (P)NJL models Equation of state PNJL model Polyakov loop Confinement = effective potential U ( φ , ¯ φ , T ) , φ the Polyakov loop. PNJL = Frozen gluons + Thermal gluons. Polyakov extended NJL Lagrangian D 0 − m ) ψ k + G ( ψ k λ i ψ k ) 2 + ’t Hooft − U ( φ , ¯ L PNJL = ψ k ( i / φ , T ) 10/36

Introduction (P)NJL models Equation of state PNJL model U ( φ , ¯ φ , T ) : - mean field in which quarks propagate, gives a pressure to the medium µν F a µν term in the - It corresponds to the thermodynamics of the 1 4 F a QCD lagrangian. - The parameters are determined by fitting with the P YM of lQCD. U ( φ , ¯ φ 3 + φ 3 ) + b 4 φ , T ) = − b 2 ( T ) φφ − b 3 ¯ 6 ( ¯ 4 ( ¯ φφ ) 2 T 4 2 T ) 2 + a 3 ( T 0 with the parameters : b 2 ( T ) = a 0 + a 1 ( T 0 T ) + a 2 ( T 0 T ) 3 a 0 a 1 a 2 a 3 b 3 b 4 T 0 6.75 -1.95 2.625 -7.44 0.75 7.5 270 MeV 11/36

Introduction (P)NJL models Equation of state PNJL model From quarks to hadrons : mesons 12/36

Introduction (P)NJL models Equation of state PNJL model Quark-antiquark bound states In NJL, degrees of freedom are quarks. Mesons need to be build from quark-antiquarks bound states q q Amplitude meson iU ( k 2 ) = Γ − ig 2 k 2 − m 2 Γ m q ¯ q ¯ Mesons masses The mass is given by the poles : m = k 13/36

Introduction (P)NJL models Equation of state PNJL model Bethe-Salpeter equation 2 ig m iU ( k 2 ) = Γ 1 − 2 g m Π ( k 2 ) Γ ... 1 Mesons masses By analogy, the mass is given by the poles : 1 − 2 G Π ( k 0 = m , � k = 0 ) = 0 14/36

Introduction (P)NJL models Equation of state PNJL model Limitations of the model Good things � Lagrangian which quite shares the symmetries of the QCD lagrangian � Works at finite density and in the phase transition region � Degrees of freedom = quarks but hadronic matter made from bound states Bad things Dynamical gluons do not participate in the interaction : low energy approximation. 4-point interactions are non renormalizable : need of a cut-off. 15/36

Introduction (P)NJL models Equation of state Equation of State 16/36

Introduction (P)NJL models Equation of state Grand potential Partition function As always in statistical physics, we need the partition function : � � β � V d 3 x L NJL Z [ ¯ q , q ] = � 0 d τ � D ¯ q D q Grand potential Using the bosonisation procedure, we obtain the mean field partition function : − � β σ 2 � � 4 G + Tr ln S − 1 0 d τ � Z [ ¯ q , q ] = exp MF MF V Ω NJL ( T , µ ) = − T V ln Z [ ¯ q , q ] 17/36

Introduction (P)NJL models Equation of state Grand potential NJL grand potential Ω NJL = − 2 � Λ d 3 p ( 2 π ) 3 E p 0 + 2 T � ∞ 0 ( ln [ 1 + exp ( − β ( E p − µ ))] + ln [ 1 + exp ( − β ( E p + µ ))] ψ k ψ k > 2 − 4 K Π i < ¯ + 2 G ∑ k < ¯ ψ k ψ j > ) PNJL grand potential Ω PNJL = − 2 � Λ d 3 p ( 2 π ) 3 E p 0 + 2 T � ∞ 0 ( ln [ 1 + L † exp ( − β ( E p − µ ))] + ln [ 1 + L exp ( − β ( E p + µ ))] ψ k ψ k > 2 − 4 K Π i < ¯ + 2 G ∑ k < ¯ ψ k ψ j > + U PNJL ) 18/36

Introduction (P)NJL models Equation of state Grand potential Figure from 3 rd student Laurence Pied 19/36

Introduction (P)NJL models Equation of state 1 Nc expansion ψ A µ ‘t Hooft scaling : g ¯ ψ A µ ψ → gN c ¯ with gN c = cst N c ψ g 2 l N k c ≡ ( gN c ) 2 l N ck − 2 l k is the number of fermion lines and l is the number of interaction lines. We go beyond mean field approximation (orange, red) in the N C expansion. iS Σ ( p ) = iS ( p )( O ( 1 ) O ( N c ) + O (( gN c ) 2 ) O ( 1 ) + O (( gN c ) 2 ) O ( 1 ) + O (( gN c ) 4 ) O ( 1 ) + ... ) N c N c Σ ... 20/36

Introduction (P)NJL models Equation of state Mesonic grand potential Mesonic grand potential � � Ω M = - g M � dpp 2 � ds 1 1 s + p 2 − µ ) − 1 ) + √ exp ( β ( √ exp ( β ( √ δ M 8 π 3 s + p 2 s + p 2 + µ ) − 1 ) Phase shift : the physics The phase shift depends on the mesons masses δ M = − Arg [ 1 − 2 K M Π M ] 21/36

Introduction (P)NJL models Equation of state Effective temperature a 0 a 1 a 2 a 3 b 3 b 4 T 0 6.75 -1.95 2.625 -7.44 0.75 7.5 270 MeV Traditional PNJL - Before One of the parameter is T 0 = 270 MeV , the critical temperature for confinement. This is the pure Yang-Mills critical temperature. Quarks are here too ! - Better Shift in the critical temperature if we gluons can split into q- ¯ q pairs. We use the reduced temperature to quantify it. T eff = T − T c → T eff YM ≃ 0 . 57 T eff rs T c https ://arxiv.org/abs/1302.1993, Haas and al. This rescales the critical temperature to T 0 = 190 MeV 22/36

Introduction (P)NJL models Equation of state Effective temperature Different quark-gluon interaction We include a temperature dependance in the rescaling : τ = 0 . 57 T − T 0 ( T ) T 0 ( T ) where : T 0 = a + bT + cT 2 + dT 3 + e 1 T a 1 a 2 a 3 and : b 2 ( T ) = a 0 + 1 + τ + ( 1 + τ ) 2 + ( 1 + τ ) 3 1.0 0.8 a b c d e 0.6 T reduced [ GeV ] 0.4 0.082 0.36 0.72 -1.6 -0.0002 0.2 0.0 -0.2 DF, T Steinert, J.Aichelin arxiv 1908.08122 -0.4 -0.6 Pure Yang Mills approach -0.8 Hass and al approach Subatech approach -1.0 0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 T [GeV] 23/36

Introduction (P)NJL models Equation of state Equation of state at zero µ DF, T Steinert, J.Aichelin arxiv 1908.08122 0.5 20 P/T 4 Pion 0.45 18 S/T 3 a0, σ E/T 4 Kaon 0.4 16 Overall I/T 4 0.35 PNJL 14 P [GeV] 0.3 12 0.25 10 0.2 8 0.15 6 0.1 4 0.05 2 0.0 0 0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 T [ GeV ] T [GeV] https ://arxiv.org/abs/1407.6387v2, HotQCD Collaboration We reproduce lattice results at 0 µ We have an effective model based on a lagrangian that shares QCD symmetry and match lattice results. This is an effective theory, no sign problem, we can expand to finite chemical potential. 24/36