N f = 1. crossover. 2nd order Z(2) m, m

Transcription

1 April 24 QCD Thermodynamics from Imaginary Owe Philipsen (University of Sussex) with Philippe de Forcrand (ETH/CERN) Motivation Imaginary chemical potential "Analyticity" of the pseudo-critical line T c (): N f = results and comparison with other methods Critical endpoint and its quark mass dependence for N f = 3 Outlook on N f = Conclusions

2 Motivation Fermion determinant complex for SU(3), 6= ) no standard Monte Carlo importance sampling Recent numerical methods: Fodor, Katz Allton et al. de Forcrand, O.P. No solution to sign problem but tricks to ease or side-step it )approximations, problems? reweighting: potential overlap problem, error analysis reweighting and/or Taylor expansion: dto., convergence imaginary : limited range of applicability (convergence) )Pursue all, cross check to control errors

3 Lattice QCD at nite temperature Karsch hep-lat/1917 T n f=2 3-avour phase diagram 175 MeV 2nd order O(4)? N f = 2 1st order Pure Gauge T d 27 MeV m crit PS ' 2:5 GeV m tric s m s?? phys. point crossover N f = 3 N f = 1 T n f=3 155 MeV 1st order 2nd order Z(2) m, m u d m crit PS ' 2 MeV The critical temperature pure gauge : T c = (271 2) MeV chiral limit of (171 4) MeV clover-impr. Wilson N f = 2 : T c = (173 8) MeV improved KS N f = 3 : T c = (154 8) MeV improved KS N.B: pure gauge: fermions: cont. limit reached, a :3 fm, still coarse

4 Imaginary / xed baryon density Weiss Miller,Redlich Dagotto et al Hasenfratz,Toussaint Z B (T V ) = 1 2 Z 2 d e ;ib=t Z( = i T V ) measure of Z positive, standard MC possible I. compute Z B by numerical Fourier trafo Alford,Kapustin,Wilczek Hubbard model at large T and small B does not work in thermodynamic limit, QCD? II. analytic continuation of observables h i at real and imag. in strong coupling Lombardo

5 Test: screening masses from dim. red. Hart,Laine,O.P. 3d e. action for Matsubara zero-modes (T > 2T c ): S! S + iz Z d 3 x Tr A 3 z = T N f 3 2 T complex, but sign prob. mild, physical volumes possible 8. channel + 8. channel m/t 4. m/t µ real µ imaginary 2. µ real µ imaginary µ 2 /(πt) µ 2 /(πt) 2 M() analytic in (no massless modes, no transition) M 2 4 T = c + c 1 + c2 + ::: T T t c c 1 c 2 to data from real and imag. works for < 1:5T ) Critical line in 4d?

6 QCD at complex : general properties Z(V T ) = Tr e ;( ^H; ^Q)=T )Taylor expansion is in = =T = R + i I term breaks T C T compensated by! ; ) Z() = Z(;) periodicity: Z(3) transf equivalent to shift in I Roberge, Weiss ) Z( R I ) = Z( R I + 2=N) Z(3) sectors identied by Polyakov loop hp (x)i = jhp (x)ije ih'i I =3 within arc: h'i = ;2=3 h'i = hoi = NX n R c n 2n I Z(3)-transitions: ; c I = 2 3 n pert./strong coupling: 1rst order for deconf. phase crossover for conf. phase ) I ;! i I

7 Analyticity of the (pseudo-) critical line with P. de Forcrand phase transition from maximum of susceptibilites: ( V ) = V N t (O ; hoi) 2 O 2 fplaq jp (x)jg nite volume: suscept. nite and analytic in all cases Location of transition: Critical line c () dened by peak max ( c = c 2 c c < : Implicit function theorem: ( ) analytic ) c () analytic! symmetries: () = (;) ) c () = c (;) c () = P n c n(a) 2n

8 Approaching the thermodynamic limit Def. of c () not unique on nite V = ) dierent denitions: nite V innite V T T Crit. line unique in thermodynamic limit! (not for crossover) For large V it is approached arbitrarily well = Order of transition: nite volume scaling: ( c (V ) ; c (1)) V ; = 1 < 1 = 1st order 2nd order crossover

9 N f = 2 results 8 3 4, KS fermions, m 3 MeV (T c ( = ) 17 MeV, a :3 fm) Z(3) transition: phase of P (x) for (a I ) c = =12 p(ϕ) β=5.26 β=5.286 β=5.3 β=5.313 β=5.34 β=5.353 β= rst order for deconf. phase continuous for conf. phase ϕ/π )schematic phase diagram: (cf. N f = 4, D'Elia, Lombardo) T h'i = h'i = ;2= µ Ι /(2πT)

10 The deconnement line T c (): susceptibilities plaquette Polyakov loop.1.3 8e-5 aµ I =. aµ I =.15 aµ I = aµ I =. aµ I =.15 aµ I =.23 χ plaquette 6e-5 4e-5 χ P e β β Lattice phase diagram for = i I fit O(aµ Ι ) 2 fit O(aµ Ι ) 4 β (aµ Ι ) 2 control over Taylor expansion, no volume restrictions!

11 Comparison: T/MeV N f =2, de Forcrand, Philipsen N f =2+1, Fodor, Katz N f =2, Allton et al. N f =4, D Elia, Lombardo µ B /MeV a :3 fm in all calculations Method N f, m q largest lattice reweighting (FK) 2+1, am u = :25, am s = 8am u rew. +Taylor (A et al.) 2, am = : imag. (FP) 2, am = : imag. (EL) 4, am = :

12 Comparison: N f = 4, imag. vs. reweighting D'Elia, Lombardo = c ; :9 2 = c ; :9 2 +: Fodor - Katz Comparison: analyt. continuation ok in SU(2) Giudice Papa Comparison: real vs. imag. to leading order Allton et al..6.4 β C real imaginary(+) imaginary(-) µ 2

13 N f = 3 results, T c ( m): de Forcrand, O.P. vary quark masses m, much more stats. check for NLO terms in Taylor series: c (a am) = X k l= c kl (a) 2k (am) l 5.15 βc -c 1 (am-am c ()) L=8 L=1 L= (aµ I ) 2 )evidence for 4! no evidence for O(m 2 2 m) T c ( m) m ; mc () = 1 + 1:94(2) T c ( = m c ()) T c 2 +:62(9) + :23(9) T c ( m) T c ( m) 4

14 N f dependence 1.98 T/T c N f =2 N f =3 N f = µ B /T c N.B.: only N f = 3 has 4 correction 1.98 T/T c N f =3 N f = µ B /T c

15 The critical endpoint Phase diag. 3d: (T m) conned/deconned )pseudo-crit. surface T c ( m) On this surface, 1.O./crossover )line of crit.points T () = T c ( m c ()) Project on (T ) : T m>m c () m=m c () m<m c () Project on (m ) : µ 2 crossover m c 2nd order first order -1/9 1/9 (µ/πt) 2 m = )true chiral phase transition Expect: m c () m c (=) = 1 + c 1 ; T 2 ) c 1 9

16 Criticality: cumulant ratios, 3d Ising universality: B 4 (m c c ) = h( ) 4 i h( ) 2 i 2! 1:64 V! 1 (< >) rst-order, crossover) t data to Taylor series about crit. point B 4 (am a) = 1:64 + B ; am ; am c () + A(a) 2 + ::: Chain rule: ;1 dam c = d(a) @am = ;A 2 need VERY LONG MC runs for sucient tunneling statistics rst-order crossover.3 L=8 L=1.3 L=8 L= plaquette plaquette

17 B 4 (am a) = 1:64 + B ; am ; am c () + A(a) B aµ I =. aµ I =.1 aµ Ι =.15 aµ I =.2 aµ I = (am+a(aµ I ) 2 ) m c () in agreement with previous result for = Ising FSS ( = c (m )): / jm ; m c ()j ; )B 4 (L=) = B 4 ; (L=) 1= 7 B = :62(3) (Ising) = : L

18 2.5 2 L=8 L=1 L=12 L=16 B (am-am c ()+A (aµ) 2 )L (1/ν) Critical quark mass, c : ) m c() 2 m c ( = ) = 1 + :84(36) + ::: T tentative comparison Taylor exp. rew., improved action: c 1 29::::25 Allton et al. )our critical point at larger c tiny change in m )huge change in c

19 Outlook for N f = Phase diag. 4d: (T m u d m s ) Two observables: B u d 4 (m u d m s ) B s 4 (m u d m s ) expand about N f = 3 = critical quark mass m c B i 4 (m u d m s ) = X n l k b i nlk (m u d ; m c ) n (m s ; m c ) l () 2k B u d 4 (m m ) = Bs 4 (m m ) = BN f =3 4 (m ) )constraints: 2k -coes. equal in all three! )leading order reduces to = investigation )strategy: I. nd (m u d m s ) c for = II. repeat analysis for (T m s ) or (T m u d )

20 Preliminary results: (m s m u d ) phase-diagram, = :.3 Nf=2+1 crit. line, mu= Nf=3 FK critical points.25.2 m_s m_u,d )strong non-linearities in m u d )qualitatively consistent with our N f = 3 results

21 Conclusions Simulations of small baryon densities possible at nite T )analytic continuation of non-singular observables, control over systematics location of transition line consistent with other approaches (pseudo-) critical line T c (m ) extremely at, small quark mass dependence critical point for physical QCD not yet under control )strong quark mass dependence Hard work (continuum limit!), but high T phase diagram in reach!