Can we locate the QCD critical endpoint with a Taylor expansion?

Transcription

1 Can we locate the QCD critical endpoint with a Taylor expansion? Bernd-Jochen Schaefer Karl-Franzens-Universität Graz, Austria 7 th February - 6 th March, Internationale Universitätswochen für Theoretische Physik Schladming, Styria, Austria

2 QCD Phase Transitions QCD: two phase transitions: 1 restoration of chiral symmetry SU L+R(N f ) SU L(N f ) SU R(N f ) order parameter: j > symmetry broken, T < Tc qq = symmetric phase, T > T c associate limit: m q Temperature early universe LHC quark gluon plasma RHIC SPS crossover <ψψ> FAIR <ψψ> > AGS SIS quark matter hadronic fluid crossover superfluid/superconducting phases? n B= n B> SC <ψψ> > CFL vacuum nuclear matter neutron star cores µ chiral transition: spontaneous restoration of global SU L(N f ) SU R(N f ) at high T

3 QCD Phase Transitions QCD: two phase transitions: 1 restoration of chiral symmetry de/confinement (center symmetry) order parameter: j = confined phase, T < Tc Φ > deconfined phase, T > T c Φ = 1 i tr cpe N c Z β associate limit: m q dτa (τ, x) Temperature early universe LHC quark gluon plasma RHIC SPS crossover <ψψ> FAIR <ψψ> > AGS SIS quark matter hadronic fluid crossover superfluid/superconducting phases? n B= n B> SC <ψψ> > CFL vacuum nuclear matter neutron star cores µ related to free energy of a static quark state: Φ = e βfq

4 QCD Phase Transitions QCD: two phase transitions: 1 restoration of chiral symmetry de/confinement (center symmetry) order parameter: j = confined phase, T < Tc Φ > deconfined phase, T > T c Temperature early universe LHC quark gluon plasma RHIC SPS crossover <ψψ> FAIR <ψψ> > AGS SIS quark matter hadronic fluid crossover superfluid/superconducting phases? n B= n B> SC <ψψ> > CFL vacuum nuclear matter neutron star cores µ alternative: dressed Polyakov loop (or dual condensate) it relates chiral and deconfinement transition to spectral properties of Dirac operator

5 The conjectured QCD Phase Diagram Temperature early universe LHC crossover RHIC <ψψ> > vacuum hadronic fluid SPS FAIR n = n > B B AGS SIS nuclear matter µ quark gluon plasma <ψψ> quark matter crossover superfluid/superconducting phases? SC <ψψ> > CFL neutron star cores At densities/temperatures of interest only model calculations available Open issues: related to chiral & deconfinement transition existence of CEP? its location? additional CEPs? How many? coincidence of both transitions at µ =? quarkyonic phase at µ >? chiral CEP/ deconfinement CEP? so far only MFA results effect of fluctuations (e.g. size of crit. reg.)?... effective models: 1 Quark-meson model (renormalizable) or other models e.g. NJL Polyakov quark-meson model or PNJL models

6 Outline Three-Flavor Quark-Meson Model...with Polyakov loop dynamics Finite density extrapolations

7 N f = 3 Quark-Meson (QM) model Model Lagrangian: L qm = L quark + L meson Quark part with Yukawa coupling g: L quark = q(i / g λa (σa + iγ 5π a))q Meson part: scalar σ a and pseudoscalar π a nonet fields: φ = 8X a= λ a (σa + iπa) L meson = tr[ µφ µ φ] m tr[φ φ] λ 1 (tr[φ φ]) λ tr[(φ φ) ] + c[det(φ) + det(φ )] +tr[h(φ + φ )] explicit symmetry breaking matrix: H = X a λ a ha U(1) A symmetry breaking implemented by t Hooft interaction

8 Phase diagram for N f = + 1 (µ µ q = µ s ) Model parameter fitted to (pseudo)scalar meson spectrum: PDG: f (6) mass=(4... 1) MeV broad resonance existence of CEP depends on m σ! Example: m σ = 6 MeV (lower lines), 8 and 9 MeV (here mean-field approximation) with U(1) A [BJS, M. Wagner 9] 5 crossover 1st order CEP T [MeV] µ [MeV]

9 Mass sensitivity Chiral limit: RG arguments for N f 3 first-order [Pisarski, Wilczek 84] variation of m π and m K : m π/m π =.49 (lower line),.6,.8..., 1.36 (upper line) m π = 138 MeV, m K = 496 MeV, fixed ratio mπ/mk = m π /m K with U(1) A, m σ = 8 MeV T [MeV] µ [MeV] crossover 1st order CEP tric m s m s nd order O(4)? phys. point 1st order N f = crossover N f = 3 nd order Z() m u, md Pure Gauge 1st order N f = 1

10 Mass sensitivity (lattice, N f = 3, µ B ) Standard scenario: m c(µ) increasing Nonstandard scenario: m c(µ) decreasing µ Real world Heavy quarks µ Real world Heavy quarks * QCD critical point QCD critical point DISAPPEARED X crossover 1rst X crossover 1rst m u,d m s m u,d m s QGP QGP T T c T T c confined Color superconductor confined Color superconductor µ µ [de Forcrand, Philipsen: hep-lat/6117]

11 Chiral critical surface (m σ = 8 MeV) standard scenario for m σ = 8 MeV (as expected) with U(1) A without U(1) A physical point physical point µ [MeV] µ [MeV] m K [MeV] m π [MeV] m K [MeV] m π [MeV] 5 [BJS, M. Wagner, 9] Note: t Hooft coupling µ-independent PNJL with (unrealistic) large vector int. bending of surface see poster by Mario Mitter

12 Outline Three-Flavor Quark-Meson Model...with Polyakov loop dynamics Finite density extrapolations

13 Polyakov-quark-meson (PQM) model Lagrangian L PQM = L qm + L pol with L pol = qγ A q U(φ, φ) polynomial Polyakov loop potential: Polyakov 1978, Meisinger 1996, Pisarski with U(φ, φ) T 4 = b (T, T ) φ φ b 3 φ 3 + φ 3 + b 4 `φ φ 6 16 b (T, T ) = a + a 1 (T /T) + a (T /T) + a 3 (T /T) 3

14 Polyakov-quark-meson (PQM) model Lagrangian L PQM = L qm + L pol with L pol = qγ A q U(φ, φ) polynomial Polyakov loop potential: Polyakov 1978, Meisinger 1996, Pisarski with U(φ, φ) T 4 = b (T, T ) φ φ b 3 φ 3 + φ 3 + b 4 `φ φ 6 16 b (T, T ) = a + a 1 (T /T) + a (T /T) + a 3 (T /T) 3 logarithmic potential: Rössner et al. 7 with U log T 4 = 1 a(t) φφ h + b(t) ln 1 6 φφ + 4 φ 3 + φ 3 3 ` φφ i a(t) = a + a 1 (T /T) + a (T /T) and b(t) = b 3 (T /T) 3

15 Polyakov-quark-meson (PQM) model Lagrangian L PQM = L qm + L pol with L pol = qγ A q U(φ, φ) polynomial Polyakov loop potential: Polyakov 1978, Meisinger 1996, Pisarski with U(φ, φ) T 4 = b (T, T ) φ φ b 3 φ 3 + φ 3 + b 4 `φ φ 6 16 b (T, T ) = a + a 1 (T /T) + a (T /T) + a 3 (T /T) 3 Fukushima Fukushima 8 n U Fuku = bt 54e a/t φ φ h + ln 1 6 φφ + 4 φ 3 + φ 3 3 ` φφ io with a controls deconfinement b strength of mixing chiral & deconfinement

16 Polyakov-quark-meson (PQM) model Lagrangian L PQM = L qm + L pol with L pol = qγ A q U(φ, φ) polynomial Polyakov loop potential: Polyakov 1978, Meisinger 1996, Pisarski with U(φ, φ) T 4 = b (T, T ) φ φ b 3 φ 3 + φ 3 + b 4 `φ φ 6 16 b (T, T ) = a + a 1 (T /T) + a (T /T) + a 3 (T /T) 3 in presence of dynamical quarks: T = T (N f, µ) BJS, Pawlowski, Wambach, 7 N f T [MeV] µ : φ > φ since φ is related to free energy gain of antiquarks in medium with more quarks antiquarks are more easily screened.

17 Logarithmic Polyakov loop potential Mean-field approximation Phase diagram N f = + 1 [BJS, M. Wagner; in preparation 1] influence of Polyakov loop T = 7 MeV (constant) T (µ) (i.e. with µ corrections) 5 5 T [MeV] 15 1 chiral crossover 5 chiral first order deconfinement CEP µ [MeV] T [MeV] 15 1 chiral crossover 5 chiral first order deconfinement CEP µ [MeV] shrinking of possible quarkyonic phase see poster by Tina K. Herbst

18 Outline Three-Flavor Quark-Meson Model...with Polyakov loop dynamics Finite density extrapolations

19 at nonzero density Finite(I) density extrapolations 1 N f = + 1 ure Taylor expansion: p(t, µ) X «µ n = c n(t) with c n(t) = 1 n `p(t, µ)/t 4 d, µ s ) = c u,d,s µu i µd j µs k i,j,k T 4 T T T n! (µ/t) n i,j,k n= µ= fixed temperature convergence radii: convergence-radius has to T method for be determined nonperturbatively limited by first-order line? small µ/t k µu,d,s= 19MeV hadron gas confined, χ-broken quark-gluon plasma deconfined, χ-symmetric color superconductor ρ n = c 1/(n ) c n nuclear matter density µ B [C. Schmidt 9]

20 Finite density extrapolations N f = + 1 Taylor expansion: p(t, µ) T 4 = first three coefficients: c : corresponds to pressure at µ = X «µ n c n(t) with c n(t) = 1 n `p(t, µ)/t 4 T n! (µ/t) n n= µ= c c : c 4 : T/T c T/T

21 Finite density extrapolations N f = + 1 Taylor expansion: p(t, µ) X «µ n = c n(t) with c n(t) = 1 n `p(t, µ)/t 4 T 4 T n! (µ/t) n n= µ= Non-zero density QCD by the Taylor expansion method Chuan Miao and Christian Schmidt recent lattice results: c u SB.1.8 c 4 u n f =+1, m " = MeV n f =, m " =77 MeV filled: N! = 4 open: N! = n f =+1, m " = MeV SB. n f =, m " =77 MeV.4 filled: N! = 4 SB open: N! = 6. T[MeV] T[MeV] -. T[MeV] c 6 u n f =+1, m " = MeV n f =, m " =77 MeV [Miao et al. 8] Figure 1: Taylor coefficients of the pressure in term of the up-quark chemical potential. Results are obtained with the p4fat3 action on N! = 4 (full) and N! = 6 (open symbols) lattices. We compare preliminary results of (+1)-flavor a pion mass of m " MeV to previous results of -flavor simulations with a corresponding pion mass of m p 77 [].

22 New method: based Finite on algorithmic densitydifferentiation extrapolations[m. NWagner, f = A. + Walther, 1 BJS, CPC 1] Taylor coefficients c n numerically known to high order, e.g. n = c c 6-15 c e+7 1e+6-5 c 1 c 14-1e+6 c 16-5e+7 3e+9 e+9 1e+9-1e+9 -e+9-3e+9-4e+9 c 18,98 1 1, c 5e+1-5e+1,98 1 1, T/T χ c,98 1 1, e+1 -e+1-4e+1

23 Finite density extrapolations N f = e+9 e+9 1e+9-1e+9 -e+9-3e+9-4e+9 c c 6-15 c e+7 1e+6 c 1 c 18,98 1 1, c 14 c -1e+6 5e+1-5e+1,98 1 1, T/T χ c 16 c,98 1 1, -5e+7 e+1 -e+1-4e+1 this technique applied to PQM model investigation of convergence properties of Taylor series properties of c n oscillating increasing amplitude no numerical noise small outside transition region number of roots increasing 6th order [F. Karsch, BJS, M. Wagner, J. Wambach; in preparation 1] Can we locate the QCD critical endpoint with the Taylor expansion?

24 Susceptibility N f = + 1 PQM model µ/t =.8 µ/t = µ c/t c χ/t th 16th 8th PQM χ/t th 16th 8th PQM T/T χ T/T χ µ/t = 3. χ/t th 16th 8th PQM T/T χ

25 Susceptibility N f = + 1 PQM model µ/t =.8 µ/t = µ c/t c χ/t th 16th 8th PQM χ/t th 16th 8th PQM T/T χ T/T χ µ/t = 3. convergence radius χ/t th 16th 8th PQM T/T χ n=16 n= n= T/T χ µ/t χ

26 Susceptibility N f = + 1 PQM model Findings: simply Taylor expansion: slow convergence high orders needed disadvantage for lattice simulations Taylor applicable within convergence radius also for µ/t > 1 but 1st order transition not resolvable expansion around µ =

27 Summary N f = and N f = + 1 chiral (Polyakov)-quark-meson model study Mean-field approximation first FRG results: see posters by Tina K Herbst and by Mario Mitter novel AD technique: high order Taylor coefficients, here: n = 6 Findings: Parameter in Polyakov loop potential: T T (N f, µ) Chiral & deconfinement transition possibly coincide for N f = with T (µ)-corrections but possibly not for N f = + 1 cf. poster by Mario Mitter Mean-field approximation encouraging FRG with PQM truncation see poster by Tina K. Herbst Taylorcoefficient c n(t) high order convergence properties of Taylor expansion Outlook: include glue dynamics with FRG full QCD e+9 e+9 1e+9-1e+9 -e+9-3e+9-4e c 6 c 8-15 c e+7 1e+6 c 1 c 14-1e+6 c 16-5e+7 5e+1 e+1 c 18 c c -e+1-5e+1-4e+1,98 1 1,,98 1 1,,98 1 1, T/T χ