Presentation on theme: "A method of finding the critical point in finite density QCD"— Presentation transcript:
1A method of finding the critical point in finite density QCD Shinji Ejiri(Brookhaven National Laboratory)Existence of the critical point in finite density lattice QCDPhysical Review D77 (2008) [arXiv: ]Canonical partition function and finite density phase transition in lattice QCDPhysical Review D 78 (2008) [arXiv: ]Tools for finite density QCD, Bielefeld, November 19-21, 2008
2QCD thermodynamics at m0 Interesting properties of QCDMeasurable in heavy-ion collisionsCritical point at finite densityLocation of the critical point ?Distribution function of plaquette valueDistribution function of quark number densitySimulations:Bielefeld-Swansea Collab., PRD71,054508(2005).2-flavor p4-improved staggered quarks with mp 770MeV163x4 latticeln det M: Taylor expansion up to O(m6)quark-gluon plasma phasehadron phaseRHICSPSAGScolor super conductor?nuclear mattermqT
3Effective potential of plaquette Physical Review D77 (2008) [arXiv: ]
4Effective potential of plaquette Veff(P) Plaquette distribution function (histogram) SU(3) Pure gauge theoryQCDPAX, PRD46, 4657 (1992)First order phase transitionTwo phases coexists at Tce.g. SU(3) Pure gauge theoryGauge actionPartition functionhistogramhistogramEffective potential
5Problem of complex quark determinant at m0 Problem of Complex Determinant at m0Boltzmann weight: complex at m0Monte-Carlo method is not applicable.Configurations cannot be generated.(g5-conjugate)
6Distribution function and Effective potential at m0 (S.E., Phys.Rev.D77, 014508(2008)) Distributions of plaquette P (1x1 Wilson loop for the standard action)(Weight factor at m=0)(Reweight factor)R(P,m): independent of b, R(P,m) can be measured at any b.Effective potential:1st order phase transition?m=0 crossovernon-singular+=?
7Reweighting for b(T) and curvature of –lnW(P) Change: b1(T) b2(T)Weight:Potential:+=Curvature of –lnW(P) does not change.Curvature of –lnW(P,b) : independent of b.
8m-dependence of the effective potential Critical pointCrossover+=m= reweightingCurvature: Zero1st order phase transition+=m= reweightingCurvature: Negative
9Sign problem and phase fluctuations Complex phase of detMTaylor expansion: odd terms of ln det M (Bielefeld-Swansea, PRD66, (2002))|q| > p/2: Sign problem happens.changes its sign.Gaussian distributionResults for p4-improved staggeredTaylor expansion up to O(m5)Dashed line: fit by a Gaussian functionWell approximatedq: NOT in the range of [-p, p]histogram of q
10Effective potential at m0 (S.E., Phys.Rev.D77, 014508(2008)) Results of Nf=2 p4-staggared, mp/mr0.7[data in PRD71,054508(2005)]detM: Taylor expansion up to O(m6)The peak position of W(P) moves left as b increases at m=0.at m=0Solid lines: reweighting factor at finite m/T, R(P,m)Dashed lines: reweighting factor without complex phase factor.
11Truncation error of Taylor expansion Solid line: O(m4)Dashed line: O(m6)The effect from 5th and 6th order term is small for mq/T 2.5.
12Curvature of the effective potential Nf=2 p4-staggared, mp/mr0.7at mq=0+=?Critical point:First order transition for mq/T 2.5Existence of the critical point: suggestedQuark mass dependence: largeStudy near the physical point is important.
13Physical Review D 78 (2008) 074507[arXiv:0804.3227] Canonical approachPhysical Review D 78 (2008) [arXiv: ]
14Canonical approach Canonical partition function Effective potential as a function of the quark number N.At the minimum,First order phase transition: Two phases coexist.
15First order phase transition line In the thermodynamic limit,Mixed state First order transitionInverse Laplace transformation by Glasgow methodKratochvila, de Forcrand, PoS (LAT2005) 167 (2005)Nf=4 staggered fermions, latticeNf=4: First order for all r.
16Canonical partition function Fugacity expansion (Laplace transformation)canonical partition functionInverse Laplace transformationNote: periodicityDerivative of lnZIntegralArbitrary m0Integral path, e.g.1, imaginary m axis2, Saddle point
17Saddle point approximation (S.E., arXiv:0804.3227) IntegralSaddle pointInverse Laplace transformationSaddle point approximation (valid for large V, 1/V expansion)Taylor expansion at the saddle point.At low density: The saddle point and the Taylor expansion coefficients can be estimated from data of Taylor expansion around m=0.Saddle point:
18Saddle point approximation Canonical partition function in a saddle point approximationChemical potentialSaddle point:saddle pointreweighting factorSimilar to the reweighting method(sign problem & overlap problem)
19Saddle point in complex m/T plane Find a saddle point z0 numerically for each conf.Two problemsSign problemOverlap problem
20Technical problem 1: Sign problem Complex phase of detMTaylor expansion (Bielefeld-Swansea, PRD66, (2002))|q| > p/2: Sign problem happens.changes its sign.Gaussian distributionResults for p4-improved staggeredTaylor expansion up to O(m5)Dashed line: fit by a Gaussian functionWell approximatedq: NOT in the range [-p, p]histogram of q
21Technical problem 2: Overlap problem Role of the weight factor exp(F+iq) The weight factor has the same effect as when b (T) increased.m*/T approaches the free quark gas value in the high density limit for all temperature.free quark gasfree quark gasfree quark gas
22Technical problem 2: Overlap problem Density of state methodW(P): plaquette distributionSame effect when b changes.for small Plinear for small Plinear for small P
23Reweighting for b(T)=6g-2 (Data: Nf=2 p4-staggared, mp/mr0.7, m=0)Change: b1(T) b2(T)Distribution:Potential:+=Effective b (temperature) for r0(r increases) (b (T) increases)
24Overlap problem, Multi-b reweighting Ferrenberg-Swendsen, PRL63,1195(1989) When the density increases, the position of the importance sampling changes.Combine all data by multi-b reweightingProblem:Configurations do not cover all region of P.Calculate only when <P> is near the peaks of the distributions.Plaquette value by multi-beta reweightingpeak position of the distribution○ <P> at each bP
25Chemical potential vs density Approximations:Taylor expansion: ln det MGaussian distribution: qSaddle point approximationTwo states at the same mq/TFirst order transition at T/Tc < 0.83, mq/T >2.3m*/T approaches the free quark gas value in the high density limit for all T.Solid line: multi-b reweightingDashed line: spline interpolationDot-dashed line: the free gas limitNf=2 p4-staggered, latticeNumber density
26SummaryEffective potentials as functions of the plaquette value and the quark number density are discussed.Approximation:Taylor expansion of ln det M: up to O(m6)Distribution function of q=Nf Im[ ln det M] : Gaussian type.Saddle point approximation (1/V expansion)Simulations: 2-flavor p4-improved staggered quarks with mp/mr 0.7 on 163x4 latticeExistence of the critical point: suggested.High r limit: m/T approaches the free gas value for all T.First order phase transition for T/Tc < 0.83, mq/T >2.3.Studies near physical quark mass: important.Location of the critical point: sensitive to quark mass