The QCD phase diagram at non-zero baryon density Owe Philipsen - PowerPoint PPT Presentation

Durham, 2009 Annual Theory Meeting The QCD phase diagram at non-zero baryon density Owe Philipsen Introduction Lattice techniques for finite temperature and density The phase diagram: the confusion before clarity? Original work with Ph. de Forcrand (ETH/CERN)

QCD at high temperature/density: change of dynamics 0.5 Measured � s QCD 0.4 0.3 0.2 0.1 1 10 100 Energy in GeV Chiral symmetry: broken (nearly) restored chiral condensate , Cooper pairs

QCD at high temperature/density: change of dynamics 0.5 Measured � s QCD 0.4 0.3 0.2 0.1 1 10 100 Energy in GeV Phase transitions? Chiral symmetry: broken (nearly) restored chiral condensate , Cooper pairs

The QCD phase diagram established by experiment: B Nuclear liquid gas transition, Z(2) end point

QCD phase diagram: theorist’s view early universe QGP s n ~170 MeV o i T c s i l l o c ! n T o i y v a e h compact stars ? confined Color superconductor µ ~1 GeV? B Until 2001: no finite density lattice calculations, sign problem! Expectation based on models: NJL, NJL+Polyakov loop, linear sigma models, random matrix models, ...

Model predictions for critical end point (CEP)

How to get funding for heavy ion programs:

How to get funding for heavy ion programs:

Thermal QCD in experiment heavy ion collision experiments at RHIC, LHC, GSI.... have finite baryon density!

Phase boundary from hadron freeze-out? ?

Theory: The Monte Carlo method � QCD partition fcn: � det M ( µ f , m f ; U ) e − S gauge ( β ; U ) Z = DU f det M e − S gauge links=gauge fields lattice spacing a<< hadron << L ! thermodynamic behaviour, large V ! typically dim. integral > 10 8 − 10 10 U Monte Carlo, importance sampling 1 Here: Continuum limit: N t = 4 , 6 T = N t → ∞ , a → 0 aN t a ∼ 0 . 3 , 0 . 2 fm Light fermions expensive, , here staggered fermions

How to measure p.t., critical temperature Lee,Yang:

The order of the p.t., arbitrary quark masses µ = 0 deconfinement p.t.: breaking of global Z (3) chiral p.t. restoration of global SU (2) L × SU (2) R × U (1) A anomalous

How to identify the critical surface: Binder cumulant How to identify the order of the phase transition  1 . 604 3d Ising ψψ ) ≡ � ( δ ¯ ψψ ) 4 �  V →∞ B 4 ( ¯ 1 first − order − → � ( δ ¯ ψψ ) 2 � 2 3 crossover  � µ = 0 : B 4 ( m, L ) = 1 . 604 + bL 1 / ν ( m − m c � 0 ) , ν = 0 . 63 � 4 1.9 V 1 V 2 > V 1 3.5 V 3 > V 2 > V 1 1.8 V= �� " Crossover 3 1.7 2.5 B 4 1.6 2 1.5 1.5 L=8 1 L=12 First order 1.4 L=16 0.5 Ising 1.3 0 0.018 0.021 0.024 0.027 0.03 0.033 0.036 -1 -0.5 0 0.5 1 x − x c am parameter along phase boundary, T = T c ( x )

Hard part: order of p.t., arbitrary quark masses µ = 0 N = 2 Pure f Gauge 0.35 ! 1st Nf=2+1 deconf. p.t. 2nd order 0.3 O(4) ? physical point 2nd order Z(2) 0.25 tric - C m ud 2/5 m s tric m s 0.2 N = 3 f am s crossover N = 1 phys. 0.15 f point chiral p.t. m s 0.1 2nd order Z(2) 0.05 1st 0 0 0 0.01 0.02 0.03 0.04 ! 0 m , m u d am u,d chiral critical line physical point: crossover in the continuum Aoki et al 06 chiral critical line on de Forcrand, O.P. 07 N t = 4 , a ∼ 0 . 3 fm m u,d = 0 , m tric consistent with tri-critical point at ∼ 2 . 8 T s But: chiral O(4) vs. 1st still open Di Giacomo et al 05, Kogut, Sinclair 07 N f = 2 anomaly! Chandrasekharan, Mehta 07 U A (1)

How to identify the critical surface: Binder cumulant How to identify the order of the phase transition The ‘sign problem’ is a phase problem  1 . 604 3d Ising ψψ ) ≡ � ( δ ¯ ψψ ) 4 � importance sampling requires �  V →∞ B 4 ( ¯ DU [det M ( µ )] f e − S g [ U ] 1 first − order − → Z = � ( δ ¯ positive weights ψψ ) 2 � 2 3 crossover  � ⇒ det( M ) complex for SU(3), µ � = 0 µ = 0 : B 4 ( m, L ) = 1 . 604 + bL 1 / ν ( m − m c � 0 ) , ν = 0 . 63 / ( µ ) † = γ 5 D Dirac operator: D / ( − µ ∗ ) γ 5 � ⇒ real positive for SU(2), µ = iµ i ⇒ real positive for µ u = − µ d 4 1.9 V 1 V 2 > V 1 3.5 V 3 > V 2 > V 1 1.8 V= �� " Crossover 3 N.B.: all expectation values real, imaginary parts cancel, 1.7 2.5 but importance sampling config. by config. impossible! B 4 1.6 2 1.5 1.5 L=8 1 L=12 First order 1.4 L=16 0.5 Ising 1.3 Same problem in many condensed matter systems! 0 0.018 0.021 0.024 0.027 0.03 0.033 0.036 -1 -0.5 0 0.5 1 x − x c am parameter along phase boundary, T = T c ( x )

1dim. illustration

Finite density: methods to evade the sign problem integrand DU det M (0)det M ( µ ) � Reweighting: det M (0) e − S g Z = S finite µ µ=0 ~exp(V) statistics needed, overlap problem use for MC calculate U Taylor expansion: � µ coeffs. one by one, � 2 k � � O � ( µ ) = � O � (0) + c k convergence? π T k =1 Imaginary : no sign problem, fit by polynomial, then analytically continue µ = iµ i � µ i N � 2 k � requires convergence � O � ( µ i ) = µ i → − iµ c k , π T for anal. continuation k =0 All require ! µ/T < 1

The good news: comparing T c ( µ ) Comparing approaches: the critical line de Forcrand, Kratochvila LAT 05 de Forcrand, Kratochvila 05 ; same actions (unimproved staggered), same mass N t = 4 , N f = 4 PdF & Kratochvila a µ 0 0.1 0.2 0.3 0.4 0.5 5.06 1.0 QGP 5.04 <sign> ~ 0.85(1) 5.02 <sign> ~ 0.45(5) 0.95 5 4.98 <sign> ~ 0.1(1) 0.90 4.96 4.94 T/T c 0.85 � confined 4.92 4.9 0.80 3 4.88 imaginary µ D’Elia, Lombardo 16 3 2 param. imag. µ Azcoiti et al., 8 4.86 3 0.75 dble reweighting, LY zeros Fodor, Katz, 6 3 4.84 Our reweighting, 6 Same, susceptibilities 3 deForcrand, Kratochvila, 6 canonical 4.82 0.70 4.8 0 0.5 1 1.5 2 µ /T Agreement for µ / T � 1 uni

The (pseudo-) critical temperature very flat, but not yet physical masses, coarse lattices indications that curvature does not grow towards continuum de Forcrand, O.P. 07 extrapolation to physical masses and continuum is feasible! Budapest-Wuppertal 08

Comparison with freeze-out curve so far freeze-out

The calculable region of the phase diagram QGP T c ! T confined Color superconductor µ Upper region: equation of state, screening masses, quark number susceptibilities etc. under control Here: phase diagram itself, most difficult!

Much harder: is there a QCD critical point?

Much harder: is there a QCD critical point? 1

Much harder: is there a QCD critical point? 1 2

Approach 1a: CEP from reweighting Critical point from reweighting Fodor,Katz JHEP 04 Fodor, Katz 04 physical quark masses, unimproved staggered fermions N t = 4 , N f = 2 + 1 Lee-Yang zero: abrupt change: physics or problem of the method? Splittorff 05; Han, Stephanov 08 (entire curve generated from one point!) Splittorf 05, Stephanov 08

Approach 1b: CEP from Taylor expansion � µ ∞ p � 2 n � T 4 = c 2 n ( T ) T n =0 �� 1 � � � µ E c 2 n c 0 2 n � � � � Nearest singularity=radius of convergence = lim � , lim � � � � T E c 2 n +2 c 2 n n →∞ n →∞ � � � Different definitions agree only for not n=1,2,3 n → ∞ CEP may not be nearest singularity, control of systematics? 1.2 Bielefeld-Swansea-RBC T / T c (0) n f =2+1, m " =220 MeV 1.15 n f =2, m " =770 MeV improved staggered 1.1 CEP from [5] N t = 4 CEP from [6] FK 1.05 1 Gavai, Gupta 0.95 0.9 N f = 2 0.85 0.8 ! 2 ! 4 µ B / T c (0) 0.75 0 1 2 3 4 5 6

Approach 2: follow chiral critical line surface chiral p.t. chiral p.t. � µ m c ( µ ) � 2 k � m c (0) = 1 + c k π T k =1 hard/easy de Forcrand, O.P. 08,09

Approach 2: imaginary rather than imagined (?) CEP chiral p.t. chiral p.t. � µ m c ( µ ) � 2 k � m c (0) = 1 + c k π T k =1 hard/easy de Forcrand, O.P. 08,09

Finite density: chiral critical line critical surface Real world Real world Heavy quarks ! N = 2 Pure Heavy quarks f ! Gauge ! 1st 2nd order O(4) ? 2nd order Z(2) QCD critical point DISAPPEARED tric m * QCD critical point s N = 3 f crossover N = 1 phys. f point m crossover s crossover X 1rst ! X 1rst ! 0 2nd order 0 Z(2) 1st m s m s m u,d m u,d 0 ! 0 m , m u d � µ m c ( µ ) � 2 k � m c (0) = 1 + c k π T k =1 m > > m c (0) m > m c (0) QGP QGP T c T c ! T T confined confined Color superconductor Color superconductor ! µ