Constraints on the QCD phase diagram from imaginary chemical potential

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1 SM+FT 211 Bari, September 211 Constraints on the QCD phase diagram from imaginary chemical potential Owe Philipsen Introduction: summary on QCD phase diagram Taking imaginary µ more seriously Triple, critical and tri-critical structures at µ = i πt 3 Implications for the QCD phase diagram 1

2 The (lattice) calculable region of the phase diagram QGP order of p.t. at zero density depends on Nf, quark mass T T c confined Color superconductor µ Sign problem prohibits direct simulation, circumvented by approximate methods: reweigthing, Taylor expansion, imaginary chem. pot., need µ/t < 1 (µ = µ B /3) Upper region: equation of state, screening masses, quark number susceptibilities etc. under control Here: phase diagram itself, so far based on models, most difficult 2

3 Comparing approaches: the critical line de Forcrand, Kratochvila LAT 5 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 confined <sign> ~.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> ~.45(5) µ/t QGP Agreement for µ/t 1 <sign> ~.1(1) PdF & Kratochvila T/T c uni 3

4 Pseudo-critical temperature T c (µ) T c () =1 κ(n f,m q ) ( µ T ) 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 κ( ψψ) =.59(2)(4) hotqcd 11 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: 4

5 Hard part: 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, RBC-BI 9 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 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 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 transition strengthens m > > m c () transition weakens m > m c () QGP QGP T T c T T c confined Color superconductor confined Color superconductor µ 7

8 Much harder: is there a QCD critical point? 2 1 crit. point from reweighting Fodor,Katz, systematics? 8

9 µ, conservative: 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 9

10 Curvature of the chiral critical surface de Forcrand, O.P. 8,9 1

11 On coarse lattice exotic scenario: no chiral critical point at small density 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 11

12 QCD at complex µ: general properties Z(V, µ, T ) = Tr ( e (Ĥ µ ˆQ)/T ) ; µ = µ r +iµ i ; µ = µ/t exact symmetries: µ-reflection and µ i -periodicity Roberge, Weiss Z( µ) =Z( µ), Z( µ r, µ i )=Z( µ r, µ i +2π/N c ) Z 3 transitions Imaginary µ phase diagram: Z(3)-transitions: µ c i = 2π 3 ( ) n T 1rst order for high T, crossover for low T ordered, k=2 T ordered, k= ordered, k=1 T c (=) analytic continuation within: disordered µ /T π/3 µ B < 55MeV 1/3 2/3 µ /("T) -2/3 -/3 /3 2/3 I /T So far: O = N n c n µ 2n i µ i iµ i 12

13 QCD at complex µ: general properties Z(V, µ, T ) = Tr ( e (Ĥ µ ˆQ)/T ) ; µ = µ r +iµ i ; µ = µ/t exact symmetries: µ-reflection and µ i -periodicity Roberge, Weiss Z( µ) =Z( µ), Z( µ r, µ i )=Z( µ r, µ i +2π/N c ) Z 3 transitions Imaginary µ phase diagram: Z(3)-transitions: µ c i = 2π 3 ( ) n T 1rst order for high T, crossover for low T ordered, k=2 T ordered, k= ordered, k=1 T c (=) analytic continuation within: disordered µ /T π/3 µ B < 55MeV 1/3 2/3 µ /("T) -2/3 -/3 /3 2/3 I /T So far: O = N n c n µ 2n i µ i iµ i chiral/deconf. transition 12

14 QCD at complex µ: general properties Z(V, µ, T ) = Tr ( e (Ĥ µ ˆQ)/T ) ; µ = µ r +iµ i ; µ = µ/t exact symmetries: µ-reflection and µ i -periodicity Roberge, Weiss Z( µ) =Z( µ), Z( µ r, µ i )=Z( µ r, µ i +2π/N c ) Z 3 transitions Imaginary µ phase diagram: Z(3)-transitions: µ c i = 2π 3 ( ) n T 1rst order for high T, crossover for low T ordered, k=2 T ordered, k= ordered, k=1 T c (=) analytic continuation within: disordered µ /T π/3 µ B < 55MeV 1/3 2/3 µ /("T) -2/3 -/3 /3 2/3 I /T So far: O = N n c n µ 2n i µ i iµ i Now: endpoint of Z(N) transition 12

15 The Z(3) transition numerically Sectors characterised by phase of Polyakov loop: Nf=2: de Forcrand, O.P. 2 Nf=4: D Elia, Lombardo 3 L(x) = L(x) e iϕ.5 p() #=5.26 #=5.286 #=5.3 #=5.313 #=5.34 #=5.353 #= /" Low T: crossover High T: first order p.t. 13

16 The nature of the Z(3) end points Nf=4: D Elia, Di Renzo, Lombardo 7 Nf=2: D Elia, Sanfilippo 9 Here: Nf=3 Strategy: fix µ i T = π 3, π, measure Im(L), order parameter at µ i T = π determine order of Z(3) branch/end point as function of m Z 3 transitions T ordered, k=2 ordered, k= ordered, k=1 T c (=) B 4 = δim(l)4 δim(l) 2 2 disordered -2/3 -/3 /3 2/3 I /T 14

17 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 15

18 ν =.33,.5,.63 for 1st order, tri-critical, 3d Ising scaling.65.6 Second order.55.5 Tricritical First order quark mass 16

19 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 16

20 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 17

21 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 18

22 Generalisation: nature of the Z(3) endpoint for Nf=2+1 x known tric. points m_s x 2.O 3d Ising x 1.O triple x pure gauge 1.O. x tricritical triple? m_u,d -Diagram computable with standard Monte Carlo, continuum limit feasible -Benchmarks for PNJL, chiral models etc. 19

23 Connection between zero and 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 2.O 3d Ising 1.O. triple? 1.O triple tricritical m_u,d pure gauge µ = µ = i πt 3 -Connection computable with standard Monte Carlo 2

24 3d, imaginary chemical potential included: 21

25 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 22

26 Deconfinement critical surface: tric. scaling shape determined by tric. scaling tricritical lines 23

27 The Nf=2 backplane in progress with Bonati, D Elia, de Forcrand, Sanfilippo 24

28 Preliminary results 25

29 Conclusions For lattices with a~.3 fm no chiral critical point for µ/t < 1 CEP scenario not yet clear: exploring uncharted territory Z(3) transition at imaginary chem. pot. connects with chiral/deconf. transition Curvature of deconfinement critical surface determined by tri-critical scaling Check if same holds for chiral critical surface, consequences for Nf=2 at zero density 26