Presentation on theme: "A method of finding the critical point in finite density QCD"— Presentation transcript:
1 A 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
2 QCD 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
4 Effective 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
5 Problem 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)
6 Distribution 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+=?
7 Reweighting 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.
8 m-dependence of the effective potential Critical pointCrossover+=m= reweightingCurvature: Zero1st order phase transition+=m= reweightingCurvature: Negative
9 Sign 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
10 Effective 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.
11 Truncation 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.
12 Curvature 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.
13 Physical Review D 78 (2008) 074507[arXiv:0804.3227] Canonical approachPhysical Review D 78 (2008) [arXiv: ]
14 Canonical approach Canonical partition function Effective potential as a function of the quark number N.At the minimum,First order phase transition: Two phases coexist.
15 First 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.
16 Canonical partition function Fugacity expansion (Laplace transformation)canonical partition functionInverse Laplace transformationNote: periodicityDerivative of lnZIntegralArbitrary m0Integral path, e.g.1, imaginary m axis2, Saddle point
17 Saddle 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:
18 Saddle point approximation Canonical partition function in a saddle point approximationChemical potentialSaddle point:saddle pointreweighting factorSimilar to the reweighting method(sign problem & overlap problem)
19 Saddle point in complex m/T plane Find a saddle point z0 numerically for each conf.Two problemsSign problemOverlap problem
20 Technical 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
21 Technical 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
22 Technical problem 2: Overlap problem Density of state methodW(P): plaquette distributionSame effect when b changes.for small Plinear for small Plinear for small P
23 Reweighting 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)
24 Overlap 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
25 Chemical 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
26 SummaryEffective 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