The QCD phase diagram at low baryon density from lattice simulations

ICHEP 2010 Paris, July 2010 The QCD phase diagram at low baryon density from lattice simulations Owe Philipsen Introduction Lattice techniques for finite temperature and density The phase diagram: the confusion before clarity? Reviews: Eur.Phys.J.ST 152 (2007) 29; PoS LAT05: 016 (2006) Original work with Ph. de Forcrand (ETH/CERN): JHEP 0811:012

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

QCD phase diagram: theorist s view ~170 MeV T early universe T c heavy ion collisions QGP confined compact stars? Color superconductor µ B ~1 GeV? QGP and colour SC at asymptotic T and densities by asymptotic freedom Until 2001: no finite density lattice calculations, sign problem Expectation based on models: NJL, NJL+Polyakov loop, linear sigma models, random matrix models,...

Phase boundary to QGP from hadron freeze-out in heavy ion collisions?

The Monte Carlo method, zero density QCD partition fcn: Z = DU f det M(µ f,m f ; U) e S gauge(β;u) links=gauge fields det M e S gauge lattice spacing a<< hadron << L thermodynamic behaviour, large V Monte Carlo by importance sampling U T = 1 Continuum limit: N t,a 0 an t Here: N t =4, 6 a 0.3, 0.2 fm

How to measure p.t., critical temperature

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 B 4 ( ψψ) (δ ψψ) 4 (δ ψψ) 2 2 V 1.604 3d Ising 1 first order 3 crossover µ = 0 : B 4 (m, L) = 1.604 + bl 1/ν (m m c 0), ν = 0.63 4 3.5 3 2.5 V 1 V 2 > V 1 V 3 > V 2 > V 1 V=$$" Crossover 1.9 1.8 1.7 2 B 4 1.6 1.5 1 0.5 First order 0-1 -0.5 0 0.5 1 x x c parameter along phase boundary, T = T c (x) 1.5 L=8 1.4 L=12 L=16 1.3 Ising 0.018 0.021 0.024 0.027 0.03 0.033 0.036 am

Results: order of p.t., arbitrary quark masses µ =0 chiral p.t. m s m s tric 0 0 2nd order O(4)? N = 2 f N = 3 f phys. crossover point 1st 2nd order Z(2) m, m u 2nd order Z(2) d 1st Pure Gauge deconf. p.t. N = 1 f chiral critical line am s 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 Nf=2+1 physical point tric m s - C 2/5 mud 0 0.01 0.02 0.03 0.04 am u,d physical point: crossover in the continuum Aoki et al 06 chiral critical line on N t =4,a 0.3 fm de Forcrand, O.P. 07 consistent with tri-critical point at m u,d =0,m tric s 2.8T But: N f =2chiral O(4) vs. 1st still open Di Giacomo et al 05, Kogut, Sinclair 07 U A (1) anomaly Chandrasekharan, Mehta 07

The QCD transition at zero density Aoki et al. 06 The nature of the phase transition at the physical point Fodor et al. 06...in the staggered approximation...in the continuum...is a crossover

How to The identify sign the problem critical is surface: a phase Binder problemcumulant How to identify the order of the phase transition Z = B 4 ( ψψ) (δ ψψ) 4 DU [det M(µ)] f e S g[u] (δ ψψ) 2 2 V 1.604 3d Ising 1 first order 3positive crossover weights importance sampling requires µ = 0 : B 4 (m, L) = 1.604 + bl 1/ν (m m c 0), ν = 0.63 Dirac operator: D/ (µ) = γ 5 D/ ( µ )γ 5 det(m) complex for SU(3), µ 0 real positive for SU(2), µ = iµ i 4 1.9 V 1 V 3.5 2 > V 1 V 3 > V 2 > V 1 V=$$" 1.8 Crossover 3 N.B.: all expectation values real, imaginary parts 1.7 cancel, 2.5 2 B 4 real positive for but importance sampling config. by config. impossible 1.6 µ u = µ d 1.5 1 0.5 First order 0 Same -1 problem -0.5 in many 0 condensed 0.5 matter 1 systems x x c parameter along phase boundary, T = T c (x) 1.5 L=8 1.4 L=12 L=16 1.3 Ising 0.018 0.021 0.024 0.027 0.03 0.033 0.036 am

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

Comparing The good news: approaches: comparing the critical T c (µ) line de Forcrand, Kratochvila 05 de Forcrand, Kratochvila LAT 05 N t = 4, N f = 4 ; same actions (unimproved staggered), same mass imaginary µ 2 param. imag. µ dble reweighting, LY zeros Same, susceptibilities canonical 5.06 5.04 5.02 5 4.98 4.96 4.94 4.92 4.9 4.88 4.86 4.84 4.82 4.8 0 0.1 0.2 0.3 0.4 0.5 confined <sign> ~ 0.85(1) D Elia, Lombardo 16 3 Azcoiti et al., 8 3 Fodor, Katz, 6 3 Our reweighting, 6 3 deforcrand, Kratochvila, 6 3 a µ <sign> ~ 0.45(5) 0 0.5 1 1.5 2 µ/t QGP Agreement for µ/t 1 <sign> ~ 0.1(1) PdF & Kratochvila 1.0 0.95 0.90 0.85 0.80 0.75 0.70 T/T c uni

The calculable region of the phase diagram QGP T c T confined Color superconductor µ need µ/t < 1 (µ = µ B /3) Upper region: equation of state, screening masses, quark number susceptibilities etc. under control

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

Comparison with freeze-out curve so far freeze-out

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? 2 1

Approach 1a: CEP from reweighting Critical point from reweighting Fodor,Katz 04 JHEP 04 N t = 4, N f = 2 + 1 physical quark masses, unimproved staggered fermions 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 p T 4 = n=0 Nearest singularity=radius of convergence ( µ ) 2n c 2n (T ) T µ E T E = lim n c 2n c 2n+2, lim n c 0 c 2n 1 2n Different definitions agree only for n not n=1-4 CEP may not be nearest singularity, estimator no upper nor lower bound, Control of systematics? Gavai, Gupta 1.2 1.15 1.1 1.05 1 0.95 T / T c (0) FK n f =2+1, m " =220 MeV n f =2, m " =770 MeV CEP from [5] CEP from [6] Bielefeld-Swansea-RBC improved staggered N t =4 N f =2 0.9 0.85 0.8 0.75 2 4 µ B / T c (0) 0 1 2 3 4 5 6

Check of predictivity Bielefeld-RBC lattice data de Forcrand-Herrigel analytic toy model without crit. point Can radius of convergence predict the existence of a CEP?

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

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

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

Curvature of the chiral critical surface de Forcrand, O.P. 08,09

The chiral critical surface on a coarse lattice: 08 Higher order terms? Convergence? Cut-off effects? Critical point not related to chiral p.t.? In any case: even qualitative QCD phase diagram not as clear as anticipated...

The deconfinement transition weakens as well Eff. heavy quark theory: 3d 3-state Potts model, same universality class de Forcrand, Kim, Takaishi 05 The chiral transition weakens also with finite isospin chemical potential Kogut, Sinclair 07

Towards the continuum: N t =6,a 0.2 fm 2nd order O(4)? N = 2 f 2nd order Z(2) 1st Pure Gauge m tric s N = 3 f phys. crossover point N = 1 f m s 0 0 1st Nt=6 Nt=4 2nd order Z(2) m, m u d m c π(n t = 4) m c π(n t = 6) 1.77 N f =3 de Forcrand, Kim, O.P. 07 Endrodi et al 07 Physical point deeper in crossover region as a 0 Cut-off effects stronger than finite density effects Preliminary: curvature of chiral crit. surface remains negative de Forcrand, O.P. 10 No chiral critical point at small density, other crit. points possible

Conclusions Working lattice methods available for µ<t T c (µ), EoS under control at small density µ<t On coarse lattices a~0.3, 0.2 fm no chiral critical point for µ B < 600 MeV Both chiral and deconfinement transitions weaken at finite density Large cut-off and quark mass effects, long way to final numbers Exploring uncharted territory: do QCD critical points exist?