The QCD phase diagram at real and imaginary chemical potential

Transcription

1 Strongnet Meeting Trento, October 211 The QCD phase diagram at real and imaginary chemical potential Owe Philipsen Is there a critical end point in the QCD phase diagram? Is it connected to a chiral phase transition? Imaginary chemical potential: rich phase structure, benchmarks, constraints Original work with Ph. de Forcrand (ETH/CERN) 1

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

3 QCD phase diagram: theorist s conjectures ~17 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 21: no finite density lattice calculations, sign problem Expectation based on models: NJL, NJL+Polyakov loop, linear sigma models, random matrix models,... 3

4 Phase boundary from hadron freeze-out?? 4

5 Theory: how to calculate p.t., critical temperature = crit. exponent 5

6 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 d Ising 1 first order 3 crossover µ = : B 4 (m, L) = bl 1/ν (m m c ), ν = V 1 V 2 > V 1 V 3 > V 2 > V 1 V=$$" Crossover B First order x x c parameter along phase boundary, T = T c (x) 1.5 L=8 1.4 L=12 L= Ising am 6

7 Order of p.t., arbitrary quark masses µ = chiral p.t. m s m s tric 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 Nf=2+1 physical point tric m s - C 2/5 mud am u,d physical point: crossover in the continuum Aoki et al 6 chiral critical line on N t =4,a.3 fm de Forcrand, O.P. 7 consistent with tri-critical point at m u,d =,m tric s 2.8T But: N f =2chiral O(4) vs. 1st still open Di Giacomo et al 5, Kogut, Sinclair 7 U A (1) anomaly Chandrasekharan, Mehta 7 7

8 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 d Ising 1 first order 3positive crossover weights importance sampling requires µ = : B 4 (m, L) = bl 1/ν (m m c ), ν =.63 Dirac operator: D/ (µ) = γ 5 D/ ( µ )γ 5 det(m) complex for SU(3), µ real positive for SU(2), µ = iµ i V 1 V > V 1 V 3 > V 2 > V 1 V=$$" 1.8 Crossover 3 N.B.: all expectation values real, imaginary parts 1.7 cancel, B 4 real positive for but importance sampling config. by config. impossible 1.6 µ u = µ d 1.5 Same problem in many condensed matter systems 1.5 First order 1.5 L=8 1.4 L=12 L= Ising SU(2) -1 Simon -.5 Hands, RMT Jaques.5 Bloch, 1 Langevin Nucu Stamatescu x x c parameter along phase boundary, T = T c (x) am 8

9 Finite density: methods to evade the sign problem Reweighting: Z = ~exp(v) statistics needed, overlap problem DU det M() use for MC det M(µ) det M() e S g calculate integrand S µ= finite µ Optimal: use det in measure, reweight in phase Taylor expansion: O (µ) = O () + k=1 c k ( µ πt ) 2k U coefficients one by one, convergence? Imaginary µ = iµ i : no sign problem, fit by polynomial, then analytically continue O (µ i )= N k= c k ( µi πt ) 2k, µi iµ requires convergence for analytic continuation All require µ/t < 1 9

10 Test of methods: comparing T c (µ) 1

11 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 11

12 The (pseudo-) critical temperature T c (µ) T c () =1 κ(n f,m q ) ( µ T ) Curvature rather small κ N f N c Toublan 5 de Forcrand, O.P. 3 D Elia, Lombardo 3 12

13 Pseudo-critical temperature Curvature of crit. line from Taylor expansion 2+1 flavours, Nt=4, 8 improved staggered Extrapolation to chiral limit assuming O(4),O(2) scaling of magn. EoS hotqcd 11 κ( ψψ) =.59(2)(4) Endrödi et al. 11 Curvature of crit. line from Taylor expansion 2+1 flavours, Nt=6,8,1 improved staggered Observables ψψ r, χ s Continuum extrapolation: 13

14 Comparison with freeze-out curve freeze-out 14

15 Finite density: chiral critical line critical surface Much harder: is there a QCD critical point? 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 phys. point 1st 2nd order Z(2) m, m u N f = 3 crossover d N f = 1 m u,d * QCD critical point X crossover 1rst m s m u,d QCD critical point DISAPPEARED X crossover 1rst m s m c (µ) m c () = 1 + ( µ c k πt k=1 ) 2k m > > m c () m > m c () QGP QGP T T c T T c confined Color superconductor confined Color superconductor µ 15

16 Much harder: is there a QCD critical point? 16

17 Much harder: is there a QCD critical point? 1 16

18 Much harder: is there a QCD critical point?

19 Approach 1a: CEP from reweighting Fodor, Katz 4 Critical point from reweighting Fodor,Katz JHEP 4 N t = 4, N f = physical quark masses, unimproved staggered fermions Lee-Yang zero: abrupt change: physics caused by or baryon problem or of pion the condensation? method? Splittorff 5; Han, Stephanov 8 Splittorf 5, Stephanov 8 17

20 Approach 1b: CEP from Taylor expansion p T 4 = n= 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 c 2n 1 2n Different definitions agree only for not n=1,2,3,... control of systematics? n (p/t 4 ) 4 (p/t 4 ) 2 (" B ) 4 (" B ) C.Schmidt, hotqcd 9 Hadron resonance gas T/T c () Radius of convergence necessary condition for CEP, but can it proof its existence? 18

21 Approach 2: follow chiral critical line surface chiral p.t. chiral p.t. m c (µ) m c () = 1 + ( µ c k πt k=1 ) 2k hard/easy de Forcrand, O.P. 8,9 19

22 Curvature of the chiral critical surface B4/µ i B4(pbp) finite µ i, fit (µ 2 +µ 4 ) fit µ 2 fit (µ 2 +µ 4 ) µ i Nf=3: a) fit to imaginary chemical potential b) calculation of coefficient by finite differences Importance of higher order terms? de Forcrand, O.P. 8,9 2

23 On coarse lattice exotic scenario: no chiral critical point at small density µ Real world Heavy quarks QCD critical point DISAPPEARED X crossover 1rst m u,d m s Weakening of p.t. with chemical potential also for: -Heavy quarks de Forcrand, Kim, Takaishi 5 -Light quarks with finite isospin density Kogut, Sinclair 7 -Electroweak phase transition with finite lepton density Gynther 3 21

24 Towards the continuum: N t =6,a.2 fm 2nd order O(4)? N = 2 f 2nd order Z(2) 1st Pure Gauge 6 4 B4/(aµ i ) 2 m tric s N = 3 f phys. crossover point N = 1 f 2-2 m s 1st Nt=6 2nd order Z(2) m, m u d Nt= B4(pbp) fit µ 2 fit (µ 2 +µ 4 ) (aµ i ) 2 m c π(n t = 4) m c π(n t = 6) 1.77 N f =3 de Forcrand, Kim, O.P. 7 Endrödi et al 7 Physical point deeper in crossover region as a Cut-off effects stronger than finite density effects Preliminary: curvature of chiral crit. surface remains negative de Forcrand, O.P. 11 No chiral critical point at small density 22

25 Same statement with different methods Study suitably defined width of crossover region strengthening of transition Endrödi et al.11 find weakening of crossover continuum extrapolated Nt=6,8,1 23

26 Understanding the curvature from imaginary µ Nf=4: D Elia, Di Renzo, Lombardo 7 Nf=2: D Elia, Sanfilippo 9 Nf=3: de Forcrand, O.P. 1 Strategy: fix µ i T = π 3, π, measure Im(L), order parameter at determine order of Z(3) branch/end point as function of m µ i T = π ordered ordered T disordered disordered µ i T /( π 3 ) 24

27 Results: m=.4 L=8 L=12 L= m=.4 L=8 L=12 L=16 ν =.33 B4(Im(L)) 2 B4(Im(L)) (- c )L 1/" B 4 (β,l)=b 4 (β c, )+C 1 (β β c )L 1/ν + C 2 (β β c ) 2 L 2/ν... B4 at intersection has large finite size corrections (well known), ν more stable 25

28 ν =.33,.5,.63 for 1st order, tri-critical, 3d Ising scaling.65.6 Second order.55.5 Tricritical First order quark mass 26

29 ν =.33,.5,.63 for 1st order, tri-critical, 3d Ising scaling.65.6 Second order.55.5 Tricritical First order quark mass On infinite volume, this becomes a step function, smoothness due to finite L 26

30 Details of RW-point: distribution of Im(L).25.2 m=.5 m=.1 m=.2 m= Small+large masses: three-state coexistence Intermediate masses: middle peak disappears triple point Ising distribtion in magn. direction tri-critical point in between 27

31 Phase diagram at µ = i πt 3 triple line 3d Ising triple line Nf=2, light and intermediate masses:1st and 3d Ising behaviour D Elia, Sanfilippo 9 28

32 Critical lines at imaginary µ m s m s tric 2nd order O(4)? phys. point 1st N = 2 f 2nd order Z(2) m, m u 2nd order Z(2) N = 3 crossover d f 1st Pure Gauge N = 1 f m_s x 2.O 3d Ising 1.O. x triple? x 1.O triple tricritical m_u,d x pure gauge µ = µ = i πt 3 -Connection computable with standard Monte Carlo -Here: heavy quarks in eff. theory 29

33 Heavy quarks: 3d 3-state Potts and strong coupling Potts: QCD, Nt=1, strong coupling series: Langelage, O.P Imag. mu Real mu (mu 2 + (pi/3) 2 ) 2 / M/T 8 m c (mu/t) (µ/t) 2 tri-critical scaling: m c T (µ2 )= m tric T + K [ (π 3 ) 2 + ( µ T ) 2 ] 2/5 exponent universal 3

34 Deconfinement critical surface: tric. scaling shape determined by tric. scaling tricritical lines 31

35 Conclusions Reweighting, Taylor, canonical: indications for critical point on coarse lattices Chiral crit. surface, deconfinement crit. surface: Transitions weaken with chemical potential, decreasing lattice spacing No chiral critical point for µ/t < 1 Still possible: chiral critical point at large chemical potential non-chiral critical point(s)? 32