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
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
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 5.6 5.4 5.2 5 4.98 4.96 4.94 4.92 4.9 4.88 4.86 4.84 4.82 4.8.1.2.3.4.5 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).5 1 1.5 2 µ/t QGP Agreement for µ/t 1 <sign> ~.1(1) PdF & Kratochvila 1..95.9.85.8.75.7 T/T c uni 3
Pseudo-critical temperature T c (µ) T c () =1 κ(n f,m q ) ( µ T ) 2 +... 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
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.35.3.25.2.15.1.5 Nf=2+1 physical point tric m s - C 2/5 mud.1.2.3.4 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
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 1.64 3d Ising 1 first order 3 crossover µ = : B 4 (m, L) = 1.64 + bl 1/ν (m m c ), ν =.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 1.5 First order -1 -.5.5 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.18.21.24.27.3.33.36 am 6
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
Much harder: is there a QCD critical point? 2 1 crit. point from reweighting Fodor,Katz, systematics? 8
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
Curvature of the chiral critical surface de Forcrand, O.P. 8,9 1
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
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 + 1 2 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
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 + 1 2 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
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 + 1 2 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
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().4.3.2 #=5.26 #=5.286 #=5.3 #=5.313 #=5.34 #=5.353 #=5.366.1-1 -.5.5 1 /" Low T: crossover High T: first order p.t. 13
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
Results: 3 2.5 m=.4 L=8 L=12 L=16 3 2.5 m=.4 L=8 L=12 L=16 ν =.33 B4(Im(L)) 2 B4(Im(L)) 2 1.5 1.5 1 5.216 5.217 5.218 5.219 1 -.6 -.4 -.2.2.4.6 (- 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
ν =.33,.5,.63 for 1st order, tri-critical, 3d Ising scaling.65.6 Second order.55.5 Tricritical.45.4.35 First order.3.1.1 1 quark mass 16
ν =.33,.5,.63 for 1st order, tri-critical, 3d Ising scaling.65.6 Second order.55.5 Tricritical.45.4.35 First order.3.1.1 1 quark mass On infinite volume, this becomes a step function, smoothness due to finite L 16
Details of RW-point: distribution of Im(L).25.2 m=.5 m=.1 m=.2 m=.3.15.1.5 -.2 -.15 -.1 -.5.5.1.15.2 Small+large masses: three-state coexistence Intermediate masses: middle peak disappears triple point Ising distribtion in magn. direction tri-critical point in between 17
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
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
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
3d, imaginary chemical potential included: 21
Heavy quarks: 3d 3-state Potts and strong coupling Potts: QCD, Nt=1, strong coupling series: Langelage, O.P. 9 9.5 9 Imag. mu Real mu 6.6 + 1.6 (mu 2 + (pi/3) 2 ) 2 /5 4 3.5 8.5 3 M/T 8 m c 2.5 2 7.5 1.5 7 1 6.5-1.5-1 -.5.5 1 1.5 2 2.5 3 (mu/t) 2.5-1.5-1 -.5.5 1 (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
Deconfinement critical surface: tric. scaling shape determined by tric. scaling tricritical lines 23
The Nf=2 backplane in progress with Bonati, D Elia, de Forcrand, Sanfilippo 24
Preliminary results 25
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