Lattice 2005 Dublin, July 2005 The QCD Phase Diagram at zero and small Baryon Densities Owe Philipsen Universität Münster • The expected qualitative picture and computational methods • The activities in the last year ⇒systematics ! • A non-standard scenario • Conclusions The expected phase diagram: QGP T, GeV E critical point 0.1 nuclear matter vacuum CFL quark matter 1 0 Non-pert. problem ⇒Lattice quark matter 1975-2001: µ B , GeV µ 6= 0 impossible ⇒sign problem! Where does this picture come from? • Simulations on T -axis (light quarks only now) • models for T = 0, µ 6= 0 Take more general view ⇒parameter space {mu,d , ms , T, µ} Pisarski, Wilczek Nf = 2: Rajagopal, Wilczek ... µ, m = 0 : SU (2)V × SU (2)A → SU (2)V true order parameter hψ̄ψi ⇒separate phases if second order then ⇒O(4) QGP T, GeV QGP T, GeV symmetric (disordered) phase mq = 0 m q non-0 2nd order transition 3cr. point cr. point 0.1 0.1 1st order (models: MIT bag, NJL, RM, ...) broken (ordered) phase vacuum 0 N.B. m=0: nuclear matter 1 1st order (models: MIT bag, NJL, RM, ...) quark matter µ B , GeV nuclear matter vacuum 1 0 quark matter µ B , GeV 1.O. all the way also possible Nf = 3: µ = 0, m < mc first order ⇒standard scenario: µc increases with m µ = 0, m > mc crossover (µ = 0, T = 0 transitions connected) Full phase diagram 3d: {m, T, µ} e.g. Nf = 3 : 0.6 0.5 m 0.4 0.3 T 1.O. 0.2 mu 0.1 mq/T = 0.08 0 5.2 5.3 m 5.4 0 •confined/deconfined ⇒pseudo-crit. surface T0 (µ, m) (from susceptibilities) •1.O./crossover ⇒line of crit.points T E (µ) = T0 (µ, mc (µ)) (from FSS) mu 1.O. Projection onto (pseudo-) critical surface: ⇒µc (m) or mc (µ) crossover 0 m_c m The case Nf = 2 + 1, µ = 0: Nf = 2 Pure Gauge ∞ 1st order 2nd order O(4) ? m ⇒mc (µ = 0) (unimproved KS) Bielefeld; Columbia; de Forcrand, O.P. tric s ? phys. ms universality: 3d Ising Nf = 3 ? • N.B: 2nd order Z(2) mc has cut-off effects! (factor 1/4?) 0 0 Nf = 1 crossover point 1st order Bielefeld m u , md Finite density, µ Bielefeld, MILC ∞ 6= 0: µ phys. line ∞ 0 ms mu,d ∞ 0 Lattice QCD at finite temperature and density Difficult (impossible?): sign problem of lattice QCD Z= Z f −Sg [U ] DU [det M (µ)] e , Sf = X ψ̄M ψ f det(M ) complex for SU(3), µ = µB /3 6= 0 ⇒ no Monte Carlo importance sampling Evading the sign problem: I. Two-parameter reweighting in (µ, β) Z= e −Sg (β) det(M (µ)) e−Sg (β0 ) det(M (µ = 0)) idea: simulate at µ=0,β0 β0 = βc (0), better overlap by sampling both phases; errors? ovlp.? I.a Reweighting + density of states ⇒larger µ, low T Fodor, Katz, Schmidt, Lat05 Fodor, Katz MILC II. Taylor expansion Bielefeld-Swansea Gavai, Gupta idea: for small µ/T , compute coeffs. of Taylor series ⇒local ops. ⇒gain V convergence? de Forcrand, O.P. III.a Imaginary µ + analytic continuation fermion determinant positive ⇒no sign problem D’Elia, Lombardo Azcoiti et al. Chen, Luo idea: for small µ/T , fit full simulation results of imag. µ by Taylor series • vary two parameters (µ, T ) ⇒controlled continuation? III.b Imaginary µ + Fourier transformation de Forcrand, Kratochvila Alexandru et al idea: canonical partition function at fixed baryon density •no analytic continuation, but determinant needed ⇒thermodynamic limit? IV. Theories without sign prob.: finite isospin density Kogut, Sinclair idea: no sign problem, reproduces baryon chem. pot. for small µ/T ? All agree on T0 (m, µ)!!! (µ/T < ∼ 1) Nf = 2, µ = 0, m = 0: is it O(4)? (or O(2) for finite a) •previous evidence inconclusive cf. old proceedings New investigation: FSS on L3 × 4, L D’Elia, Di Giacomo, Pica = 16 − 32, standard staggered Fermions, R-algorithm, m/T > ∼ 0.055 In critical region: α/ν 1/ν yh τ = 1 − T /Tc ⇒spec. heat : CV − C0 ' L fc τ L , am L ⇒suscept. of order param. : χ ' Lγ/ν fχ τ L1/ν , am Lyh and others... Strategy: • Fix yh to O(4) value • choose L, am to keep am Lyh fixed⇒one variable only, inf. V ⇒am = 0. ⇒check consistency with scaling 0.075 0.4 0.07 O(4) scaling O(2) scaling 0.35 O(4) scaling O(2) scaling 0.065 0.06 0.055 Run 1 Run 1 0.05 γ/ν 0.25 χm/Ls (CV-C0)/Ls α/ν 0.3 0.2 0.15 0.045 0.04 0.035 0.03 0.025 0.02 0.1 0.015 0.01 0.05 0.005 0 12 14 16 18 20 22 24 26 28 30 0 10 32 12 14 16 18 Ls 20 22 24 26 28 30 32 Ls similar: keeping τ L1/ν fixed, check of 1st order: 1 /d.o.f. = 4.4 2 χ /d.o.f. = 2.1 0.9 (χm-χ0)/amq 2 1 Order 0.95 −γ/(νyh) χ −γ/(νyh) (χm-χ0)/amq st O(4) 2.5 2 0.85 0.8 0.75 0.7 1.5 0.65 0 0.01 0.02 0.03 0.04 amq 0.05 0.06 0.07 0.08 0.6 0 0.01 0.02 0.03 0.04 amq 0.05 0.06 0.07 0.08 34 But no metastability in plaquette distributions ⇒still not conclusive! • scaling window very small ? O(4) • fermion formulation? (Wilson sees O(4)) • cut-off effects ? Z(2) ⇒Nt = 6 study promised; Suggestions: • FSS at fixed am m_s Z(2) • Binder cumulants • exact algorithm m_u,d Finite density by Taylor expansion, Nf Finite V : = 2: Z(m > 0, µ, T ) analytic in entire parameter space CP-Symmetry: Z(µ) = Z(−µ) ⇒all observables have Taylor series in (µ/T )2 pressure density: p(T, µ) = T V log Z(T, µ), Coefficients: generalized quark number suscept. at µ ∞ µ 2n X p = c2n (T ) 4 T T n=0 = 0 ⇒measurable! V → ∞: phase transitions ⇒singularities in pressure radius of convergence ≡ distance to nearest singularity; c0 1/2n e.g.: ρn = , c2n c2n 1/2 , rn = c2n+2 ρ, r = lim ρn , rn n→∞ if singularity on real axis (asymptotically all coeffs. positive) ⇒critical point Technicalities: det M = tr M −1 ∂M µ-dependence in det M ⇒ ∂ ln∂µ ∂µ etc. ⇒traces of composite local ops., Gaussian noise vectors ∼ O(10 − 100) non-trivial FSS of susceptibilities ⇒delicate cancellations for large V Quark number susceptibility to order µ6 Lattices L3 Gavai, Gupta × 4, L = 8 − 24, standard staggered, R-algorithm, m/T0 = 0.1 Strategy: compute coeffs. at different T /T0 = 0.75 − 2.15⇒extrapolate to finite µ T V 2 V1 µ T /T0 = 0.95, convergence radius vs. order n: 5 7 6 4 5 3 µ /T B µB/T 4 3 2 2 1 03 1 4 5 6 n 7 8 9 ρn on 83 , 243 02 3 4 5 n 6 7 8 rn ond 83 , 243 ⇒strong volume dependence! need Lmπ > ∼ 5 − 6 (L > ∼ 16 − 18) ⇒need more terms for larger V ⇒quoted result: µcB /T = 1.1 ± 0.2 T /T0 = 0.95 The pressure to order µ6 Bielefeld-Swansea Lattice 163 × 4, Symanzik improved Wilson, p4-improved staggered fermions, R-algorithm, m/T0 ≈ 0.4 Comparisons with (µ/T )-expansions of • high T pert.theory Vuorinen • hadron resonance gas (HRG) model Karsch, Redlich, Tawfik coefficients in different orders: c6 c4 c2 ⇒coeffs.∼ O(1) for ”natural” expansion parameter 4 µq/T=1.0 µq/T=0.8 µq/T=0.6 µq/T=0.4 µq/T=0.2 µq/T=0.0 2 χq/T 3 2 1 T/T0 0 0.8 1 1.2 1.4 1.6 1.8 2 µ πT de Forcrand, O.P. nq /χq = radius of convergence: 1.2 c4/c2 c6/c4 1 ∂p ∂nq (= 0 for 2.O.) nq/(µqχq) 1 0.8 0.5 0.6 0 µq/T=0.4 0.4 µq/T=0.8 0.2 T/T0 -0.5 0.8 1 1.2 0 1.4 T/Tc lines: coeffs. from HRG model: c4 /c2 0.8 1 1.2 1.4 = 3/4 • consistency with hadron resonance gas • no evidence for critical point • Not in conflict with Gavai, Gupta (different action, larger m) • but different conclusion than at µ4 ! • previous experience (spin models, strong coupling series), use many terms (10-20) some work, some fail ⇒experiment e.g.Itzykson, Drouffe Systematics of reweighting and Taylor expansion hOi(β,µ) = hO e he nf 4 nf 4 ∆ ln detM −∆Sg e ∆ ln detM e−∆Sg i det M = | det M |eiθ ⇒Breakdown of reweighting for hcos θi 1 i(β0 ,0) ∼ e−V ∆F (β0 ,0) Contours of σ(θ): (fixed V = 163 ) 2.5 2 = hθ 2 i − hθi2 > π/2 ⇒compute variance from Taylor coeffs. 1.5 µq/T or σ(θ) p 1 2π 7π/4 3π/2 5π/4 π 3π/4 π/2 π/4 Bielefeld-Swansea θ (n) n nf X µ2j−1 ∂ 2j−1 ln detM = Im 4 j=1 (2j − 1)! ∂µ2j−1 0.5 0 0.8 1 1.2 1.4 T/T0 1.6 1.8 2 Lee-Yang-zeroes and reweighting Ejiri Fodor, Katz: Observation: LYZ competing with noise of det M D 6iβ N ∆P iθ (N /4)(ln det M (µ)−ln det M (0)) E Im site f Znorm (βRe , βIm , µ) = e e e (βRe ,0,0) ⇒doesn’t work at infinite V (not surprising...) In practice: •enough statistics for a given volume? •errors reliable? QCD at finite isospin density Son, Stephanov Opposite chemical potentials for u,d-quarks: µu = −µd = µ ⇒µI ≡ (µu − µd ) = 2µ ⇒phases of determinants cancel Z Z = DU | det M (µ)|Nf e−Sg [U ] Pion condensate 6= QCD, but for µI heiθ OiµI =2µ hOiµ = heiθ iµI =2µ Numerically verified for T0 (µ, m), < mπ recover QCD: consider O probing p.t. for hcos θiµI Nf = 2, 3: •in Taylor expansion (Bielefeld-Swansea) •simulations at µI ( Kogut, Sinclair) and µi (de Forcrand, O.P.) ∼ 1, O0 = eiθ O shows same transition! Phase of the determinant and phase-quenched QCD Splittorff σ(θ) and the transition to a pion condensate at finite isospin: ⇒related, since phase of det M ”destroys” pion condensate Fodor, Katz endpoints, different m boundary for reweighting? 1.3 1.2 1.2 1 1.1 T/T0 T/T0 0.8 1 0.6 0.9 0.4 0.8 0.2 0 0.7 0 0.5 1 1.5 2 2µ/mπ observation: FK-points both lie on boundary ⇒...? similarly for Taylor exp. 0 0.5 1 2µ/mπ 1.5 2 QCD at complex µ: general properties Z(V, µ, T ) = Tr e−(Ĥ−µQ̂)/T ; µ = µr + iµi ; µ̄ = µ/T exact symmetries: µ-reflection and µi -periodicity Z(µ̄) = Z(−µ̄), Roberge, Weiss Z(µ̄r , µ̄i ) = Z(µ̄r , µ̄i + 2π/Nc ) Z(3)-transitions: µ̄ci = 2π 3 n+ 1 2 1rst order for high T, crossover for low T T Imaginary µ phase diagram: analytic continuation within arc: |µ|/T ≤ π/3 ⇒µB < ∼ 550MeV hOi = N X n cn µ̄2n i ⇒ µi −→ iµi 0 1/3 µΙ/(πT) 2/3 Pade approximants? (Lombardo, Lat05) A generalized imaginary µ-approach Azcoiti, Di Carlo, Galante, Laliena Reformulation of staggered action, chemical potential term 1X † ψ̄n η0 (n) eµa Un,0 ψn+0 − e−µa Un−0,0 ψn−0 2 n 1X † x ψ̄n η0 (n) Un,0 ψn+0 − Un−0,0 ψn−0 2 n → 1X † + y ψ̄n η0 (n) Un,0 ψn+0 + Un−0,0 ψn−0 , 2 n y Hasenfratz, Karsch x = cosh(aµ), y = sinh(aµ) solid line: phase transitions dotted line: physical line x2 x − y2 = 1 Sign problem ⇒simulations at imaginary y = iȳ possible improvement: simulations at different lower temperatures better control of continuation? x, y , Numerical results, Nf =4 83 × 4, standard staggered, R-algorithm, m/T ≈ 0.2 Fitting to βc (ȳ) = β0 + β1 ȳ 2 plus continuation: • works at least as well as standard imag. µ, more expensive • can it be made to work better? control of errors beyond µ/T > ∼1 ? • Nf = 2 in progress, preliminary evidence for critical point • point of departure for reweighting, improved range? Laliena, Lat05 Roberge, Weiss Hasenfratz,Toussaint QCD at fixed baryon number Alford,Kapustin, Wilczek de Forcrand, Kratochvila ⇒δ(3B − d3 x ψ̄γ0 ψ) Z π µ 1 µi −i3B Ti ZC (B) = e ZGC (µ = iµi ) d 2π −π T R Fix baryon number B : canonical ensemble • Sample ZGC (µ = iµM C ) ZC (B) = ⇒ ZGC (iµM C ) 1 det(iµM C ) Z µi det(iµi ) dµi exp i3B T • Fourier transform each determinant exactly (work ∝ L9s Lt ) ⇐= low T • Sign problem ↔ noise in ZC (B) ⇐= governed by B (not V ) • Study few-baryon system at low temperature: Nuclear Physics! preliminary results, Wilson fermions: Alexandru et al, Lat05 Numerical results, Nf =4 de Forcrand, Kratochvila 63 × 4, standard staggered, HMC-algorithm, m/T ≈ 0.2 (p.t. first order ∀µ ) Chemical potential versus baryon density F (B) → µ transformation via saddle point approximation: R ZGC (µ) = dρ exp(− VT (f (ρ)) + µρ)) =⇒ µ ≈ f 0 (ρ) ≈ F (B+1)−F (B) 3 Comparison of methods: aµ 6 x4 0 2 1 β µ/T 1.5 T/Tc = 0.89 T/Tc = 0.92 T/Tc = 0.95 T/Tc = 0.98 T/Tc = 1.02 Weakly interacting massless gas Hadron Resonance Gas 0.5 0 0 5 10 15 20 Baryon number 25 30 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 0.1 0.2 0.3 0.4 0.5 1.0 QGP <sign> ~ 0.85(1) <sign> ~ 0.45(5) 0.95 <sign> ~ 0.1(1) 0.90 0.85 confined 0.80 D’Elia, Lombardo 163 Azcoiti et al., 833 Fodor, Katz, 6 Our reweighting, 63 This work, 63 0 0.5 1 µ/T 0.75 0.70 1.5 2 T/Tc 3 The critical endpoint and its quark mass dependence in Nf Nf = 2 Pure Gauge ∞ 1st order 2nd order O(4) ? tric s Nf = 3 T m ? phys. ms m>mc(0) m=mc(0) m<mc(0) ? Nf = 1 crossover point 1st order • 2nd order Z(2) 0 0 =3 0 m u , md 2 ∞ µ µ Expect: mc (µ) mc (µ=0) = 1 + c1 µ πT 2 phys. line +... Inverted: curvature of critical surface µc (m) ∞ 0 ms mu,d ∞ 0 Criticality: cumulant ratios 3d Ising universality (B4 h(δ ψ̄ψ)4 i → 1.604, V → ∞ B4 (mc , µc ) = 2 2 h(δ ψ̄ψ) i : = 1 (first-order), 3 (crossover) for V = ∞) B4 (am, aµ) = 1.604 + B am − amc (0) + A(aµ) 2 + ... de Forcrand, O.P. 70 1.8 aµI=0.0 aµI=0.1 aµΙ=0.15 aµI=0.2 aµI=0.25 1.6 1.5 1.4 0.025 0.03 0.035 2 (am+A(aµI) ) 0.04 FSS: 50 ν = 0.62(3) ν(Ising) = 0.63 B B4 1.7 60 40 30 20 10 8 10 12 L 14 16 µ 2 mc (µ) ⇒ = 1 + 0.84(36) + ... mc (µ = 0) πT Nf = 3 at finite isospin density Lattices L3 Kogut, Sinclair × 4, L = 8, 12, 16, standard staggered, R-algorithm, m > ∼ mc (0) Finite stepsize effects change order: Extrapolated results: •dB4 /dµ2I > ∼ 0, more crossover with µ?? Redoing Nf = 3 with an exact algorithm Extrapolations dt2 → 0 prohibitively expensive for small m ⇒use exact algorithm here: Rational Hybrid MC Testing RHMC against dt2 − 1.75 1.85 1.8 1.75 1.6 1.7 1.55 1.65 B4 1.65 1.5 1.6 1.45 1.55 1.4 1.5 1.35 1.45 1.3 0 0.2 0.4 0.6 0.8 1 2 (dt/ma) Clark, Kennedy, Lat05 The critical mass mc (0) with RHMC: > 0: Nf3, m=0.034 Nf3, m=0.030 Nf3, m=0.025 1.7 B4 de Forcrand, O.P. 1.2 muI=0.0 muI=0.2 ising 1.4 0.015 R-alg. 0.02 0.025 0.03 am leftmost data: RHMC •25% change in mc (0) •dB4 /dµ2I ≈ 0 •consistent with finite isospin! 0.035 0.04 A non-standard scenario: no critical point at all? continuum conversion: µ 2 mc (µ) ⇒ = 1 − 0.6(2) mc (µ = 0) πT 0 mu m_u m_s Positive or negative curvature: very high quark mass sensitivity of µc !! Can one expect a critical point at “small” µ? µcB ∼ 360 MeV (FK) requires µ 1< phys. line ∞ 0 ms mu,d ∞ 0 m < 1.05 ∼ mc (µ = 0) fine tuning of quark masses!! Outlook: Nf = 2 + 1 de Forcrand, O.P. Two-step procedure: I.(ms , mu,d ) phase-diagram at µ II.repeat for µ =0 ⇒mcs (mu,d ) 6= 0 Nf = 2 Pure Gauge ∞ 0.7 1st order 2nd order O(4) ? NEW Nf=2+1 Nf=3 OLD Nf=2+1 F critical *point a ms =0.7 0.6 0.5 tric ms ? phys. ms ? a ms Nf = 3 crossover Nf = 1 point 0.4 0.3 0.2 1st order 2nd order Z(2) 0.1 0 0 0 m u , md If there is a tricritical point: ∞ 0 0.005 mtric s /T0 ≈ 2.8 0.01 0.015 0.02 a mu,d 0.025 0.03 0.035 The quenched limit: Potts model m → ∞: QCD→ theory of Polyakov lines → universality class of 3d 3-state Potts model (3d Ising) large µ: cluster algorithms Alford, Chandrasekharan, Cox, Wiese small µ/T : sign problem mild, doable for real µ! ⇒Testing ground for real vs. imaginary µ: de Forcrand, Kim, Kratochvila, Takaishi Information on upper right corner: Potts, 72^3 Nf = 2 12 ∞ 8 M/T 1st order 2nd order O(4) ? first order transition 10 from µ_Ι from µ M_infinity limit m s Nf = 3 ? phys. cross-over ms 4 • tric 6 ? 2nd order Z(2) 0 -2 -1 0 1 2 3 4 5 0 Nf = 1 crossover point 1st order 2 0 -3 Pure Gauge m u , md (µ/Τ)^2 ⇒First order region shrinks with µ ! ∞ Conclusions Physics: • Nf = 2 : O(4) vs. 1st order remains open • several groups tackling critical endpoint at finite µ, no convergence yet • strong quark mass sensitivity of µc Systematics: • more methods available, more cross checks • beginning to understand phase of det, finite V • use exact algorithms !!! ⇒qualitative picture in reach ⇒physics in the continuum still a long way.......(we are at a ∼ 0.3f m) ⇒surprises?