Effects of (axial)vector mesons on the chiral phase transition: - PowerPoint PPT Presentation

Introduction The model eLSM at finite T /µ B Summary Effects of (axial)vector mesons on the chiral phase transition: initial results P´ eter Kov´ acs Wigner Research Centre for Physics, Budapest kovacs.peter@wigner.mta.hu May 30, 2014 MESON 2014 Collaborators: Zsolt Sz´ ep ,Gy¨ orgy Wolf

Introduction The model eLSM at finite T /µ B Summary Overview Introduction 1 Motivation QCD’s chiral symmetry, effective models The model 2 Axial(vector) meson extended linear σ -model with constituent quarks and Polyakov-loops eLSM at finite T /µ B 3 Polyakov loop Equations of states Parametrization at T = 0 T dependence Summary 4

Introduction The model eLSM at finite T /µ B Summary Motivation QCD phase diagram Phase diagram in the T − µ B − µ I space At µ B = 0 T c = 151(3) MeV Y. Aoki, et al. , PLB 643 , 46 (2006) Is there a CEP? At T = 0 in µ B where is the phase boundary? Behaviour as a function of µ I /µ S Details of the phase diagram are heavily studied theoretically (Lattice, EFT), and experimentally (RHIC, LHC, FAIR, NICA)

Introduction The model eLSM at finite T /µ B Summary Motivation Previous results (with linear σ -model) Critical surface and the CEP P. Kov´ acs, Zs. Sz´ ep: Phys. Rev. D 75 , 025015 µ B,CEP ✶ µ B [MeV] µ B [MeV] 1000 900 800 700 600 ✶ 500 500 p h y s i c a l p o i n t 400 400 m K [MeV] 300 200 300 100 diagonal 200 0 60 80 100 100 120 140 160 m 180 [ M e V ] π The surface bends towards the physical point = ⇒ The CEP must exist

Introduction The model eLSM at finite T /µ B Summary Motivation Previous results (with linear σ -model) The CEP at the physical point of the mass plane P. Kov´ acs, Zs. Sz´ ep: Phys. Rev. D 75 , 025015 lattice CEP cross-over line 160 1 st order line 140 spinodal freeze-out curve 120 T Ising T [MeV] 100 80 ✶ CEP 60 40 20 0 0 100 200 300 400 500 600 700 800 900 1000 1100 µ B [MeV] lattice effective model T c ( µ B = 0) = 151(3) MeV T c ( µ B = 0) = 154 . 84 MeV ∆ T c ( χ ¯ ψψ ) = 28(5) MeV ∆ T c ( x χ ) = 15 . 5 MeV Y. Aoki, et al. , PLB 643 , 46 (2006) T CEP = 74 . 83 MeV µ B , CEP = 895 . 38 MeV T CEP = 162(2) MeV µ B , CEP = 360(40) MeV d 2 T c � µ B =0 = − 0 . 09 T c � − 0 . 058(2) d µ 2 � B Z. Fodor, et al. , JHEP 0404 (2004) 050

Introduction The model eLSM at finite T /µ B Summary Motivation Addressed problems By adding more degrees of freedom to our model how does the phase boundary change? More specifically adding (axial)vector mesons to the model how does the position of the CEP change? What is the effect of the medium on the various masses? Results will be closer to the Lattice?

Introduction The model eLSM at finite T /µ B Summary QCD’s chiral symmetry, effective models Chiral symmetry If the quark masses are zero (chiral limit) = ⇒ QCD invariant under the following global transformation (chiral symmetry): U (3) L × U (3) R ≃ U (3) V × U (3) A = SU (3) V × SU (3) A × U (1) V × U (1) A U (1) V term − → baryon number conservation U (1) A term − → broken through axial anomaly SU (3) A term − → broken down by any quark mass SU (3) V term − → broken down to SU (2) V if m u = m d � = m s → totally broken if m u � = m d � = m s (realized in nature) − Since QCD is very hard to solve − → low energy effective models can be set up − → reflecting the global symmetries of QCD − → degrees of freedom: observable particles instead of quarks and gluons Linear realization of the symmetry − → linear sigma model (nonlinear representation − → chiral perturbation theory (ChPT))

Introduction The model eLSM at finite T /µ B Summary Axial(vector) meson extended linear σ -model with constituent quarks and Polyakov-loops Lagrangian (2/1) 0 Tr(Φ † Φ) − λ 1 [Tr(Φ † Φ)] 2 − λ 2 Tr(Φ † Φ) 2 L Tot = Tr[( D µ Φ) † ( D µ Φ)] − m 2 �� m 2 � � − 1 4Tr( L 2 µν + R 2 1 ( L 2 µ + R 2 + Tr[ H (Φ + Φ † )] µν ) + Tr 2 + ∆ µ ) + c 2 (det Φ − det Φ † ) 2 + i g 2 2 (Tr { L µν [ L µ , L ν ] } + Tr { R µν [ R µ , R ν ] } ) + h 1 µ ) + h 2 Tr[( L µ Φ) 2 + (Φ R µ ) 2 ] + 2 h 3 Tr( L µ Φ R µ Φ † ) . 2 Tr(Φ † Φ)Tr( L 2 µ + R 2 + g 3 [Tr( L µ L ν L µ L ν ) + Tr( R µ R ν R µ R ν )] + g 4 [Tr ( L µ L µ L ν L ν ) + Tr ( R µ R µ R ν R ν )] + g 5 Tr ( L µ L µ ) Tr ( R ν R ν ) + g 6 [Tr( L µ L µ ) Tr( L ν L ν ) + Tr( R µ R µ ) Tr( R ν R ν )] + ¯ Ψ ( i / ∂ − g F Φ 5 ) Ψ + L Polyakov where D µ Φ = ∂ µ Φ − ig 1 ( L µ Φ − Φ R µ ) − ieA µ [ T 3 , Φ]

Introduction The model eLSM at finite T /µ B Summary Axial(vector) meson extended linear σ -model with constituent quarks and Polyakov-loops Lagrangian (2/2) 8 8 � � Φ = ( σ i + i π i ) T i , H = T i : U (3) generators h i T i i =0 i =0 8 8 R µ = L µ = � ( ρ µ i − b µ � ( ρ µ i + b µ i ) T i , i ) T i i =0 i =0 L µν = ∂ µ L ν − ieA µ [ T 3 , L ν ] − { ∂ ν L µ − ieA ν [ T 3 , L µ ] } R µν = ∂ µ R ν − ieA µ [ T 3 , R ν ] − { ∂ ν R µ − ieA ν [ T 3 , R µ ] } ¯ u , ¯ Ψ = (¯ d , ¯ s ) non strange – strange base: � � ϕ N = 2 / 3 ϕ 0 + 1 / 3 ϕ 8 , � � ϕ S = 1 / 3 ϕ 0 − 2 / 3 ϕ 8 , ϕ ∈ ( σ, π, h ) broken symmetry: non-zero condensates � σ N � , � σ S � ← → φ N , φ S

Introduction The model eLSM at finite T /µ B Summary Axial(vector) meson extended linear σ -model with constituent quarks and Polyakov-loops Included fields - pseudoscalar and scalar meson nonets η N + π 0  π + K +  √ 8 1 2 � η N − π 0 √ Φ PS = π i T i = K 0  ( ∼ ¯ q i γ 5 q j )  π −  √  2 2 i =0 ¯ K − K 0 η S  σ N + a 0  a + K + 0 √ 8 0 S 2 1 �  σ N − a 0  √ Φ S = σ i T i =  ( ∼ ¯ q i q j ) a − K 0 0 √   2 0 S 2  i =0 ¯ K − K 0 σ S S S Particle content: Pseudoscalars: π (138) , K (495) , η (548) , η ′ (958) Scalars: a 0 (980 or 1450) , K ⋆ 0 (800 or 1430) , ( σ N , σ S ) : 2 of f 0 (500 , 980 , 1370 , 1500 , 1710)

Introduction The model eLSM at finite T /µ B Summary Axial(vector) meson extended linear σ -model with constituent quarks and Polyakov-loops Included fields - vector meson nonets µ ω N + ρ 0  ρ + K ⋆ +  √ 8 2 1 ρ µ V µ � ω N − ρ 0 = i T i = √ ρ − K ⋆ 0   √ 2   2 ¯ i =0 K ⋆ − K ⋆ 0 ω S µ  f 1 N + a 0  a + K + 1 √ 8 1 1 2 1 A µ b µ �  f 1 N − a 0  = i T i = √ a − K 0 1  √  V 2 1 1 2   i =0 K − ¯ K 0 f 1 S 1 1 Particle content: Vector mesons: ρ (770) , K ⋆ (894) , ω N = ω (782) , ω S = φ (1020) Axial vectors: a 1 (1230) , K 1 (1270) , f 1 N (1280) , f 1 S (1426)

Introduction The model eLSM at finite T /µ B Summary Polyakov loop Polyakov loops in Polyakov gauge x ) = Tr c ¯ x ) = Tr c L ( � x ) L ( � x ) and ¯ Polyakov loop variables: Φ( � Φ( � with N c N c � β � � L ( x ) = P exp i 0 d τ A 4 ( � x , τ ) − → signals center symmetry ( Z 3 ) breaking at the deconfinement transition � ¯ � low T : confined phase, � Φ( � x ) � , Φ( � x ) = 0 � ¯ � high T : deconfined phase, � Φ( � x ) � , Φ( � x ) � = 0 Polyakov gauge: the temporal component of the gauge field is time independent and can be gauge rotated to a diagonal form in the color space A 4 , d ( � x ) = φ 3 ( � x ) λ 3 + φ 8 ( � x ) λ 8 ; λ 3 , λ 8 : Gell-Mann matrices . In this gauge the Polyakov loop operator is x ) = diag( e i βφ + ( � x ) , e i βφ − ( � x ) , e − i β ( φ + ( � x )+ φ − ( � x )) ) L ( � √ where φ ± ( � x ) = ± φ 3 ( � x ) + φ 8 ( � x ) / 3

Introduction The model eLSM at finite T /µ B Summary Polyakov loop Polyakov loop potential “Color confinement” “Color deconfinement” � Φ � =0 − → no breaking of Z 3 � Φ � � =0 − → spontaneous breaking of Z 3 one minimum minima at 0 , 2 π/ 3, − 2 π/ 3 one of them spontaneously selected 5 U( Φ ) / T 4 U( Φ ) / T 4 0 -0.2 4 -0.4 5 0.5 3 -0.6 4 0 -0.8 3 -0.5 2 2 -1 -1 1 1 -1.5 -1.2 -2 0 -1.4 0 -2.5 -1 -3 -2 1.5 1.5 1 1 0.5 0.5 -1.5 -1.5 0 0 -1 -1 -0.5 Im Φ -0.5 Im Φ -0.5 -0.5 0 0 0.5 -1 0.5 -1 Re Φ 1 Re Φ 1 -1.5 -1.5 1.5 1.5 from H. Hansen et al., PRD75, 065004 (2007)

Introduction The model eLSM at finite T /µ B Summary Polyakov loop Form of the potential I.) Simple polynomial potential invariant under Z 3 and charge conjugation: R.D.Pisarski, PRD 62, 111501 U poly ( Φ , ¯ Φ ) Φ 3 + ¯ � 2 = − b 2 ( T ) ¯ � ¯ ΦΦ − b 3 Φ 3 � + b 4 � ΦΦ T 4 2 6 4 T 2 T 3 b 2 ( T ) = a 0 + a 1 T 0 with T + a 2 T 2 + a 3 0 0 T 3 II.) Logarithmic potential coming from the SU (3) Haar measure of group integration K. Fukushima, Phys. Lett. B591 , 277 (2004) U log (Φ , ¯ � Φ 3 + ¯ � 2 � Φ) = − 1 2 a ( T )Φ¯ 1 − 6Φ¯ Φ¯ � Φ 3 � � Φ + b ( T ) ln Φ + 4 − 3 Φ T 4 T 2 T 3 a ( T ) = a 0 + a 1 T 0 T + a 2 T 2 , 0 b ( T ) = b 3 0 with T 3 Φ , ¯ � � U Φ models the free energy of a pure gauge theory − → the parameters are fitted to the pure gauge lattice data