A method of finding the critical point in finite density QCD Existence of the critical point in finite density lattice QCD Physical Review D77 (2008) 014508 [arXiv:0706.3549] Canonical partition function and finite density phase transition in lattice QCD Physical Review D 78 (2008) 074507[arXiv:0804.3227] Tools for finite density QCD, Bielefeld, November 19-21, 2008 Shinji Ejiri (Brookhaven National Laboratory)
T quark-gluon plasma phase RHIC SPS AGS hadron phase color super conductor? nuclear matter mq QCD thermodynamics at m0 • Interesting properties of QCD Measurable in heavy-ion collisions Critical point at finite density • Location of the critical point ? • Distribution function of plaquette value • Distribution function of quark number density • Simulations: • Bielefeld-Swansea Collab., PRD71,054508(2005). • 2-flavor p4-improved staggered quarks with mp 770MeV • 163x4 lattice • ln det M: Taylor expansion up to O(m6)
Effective potential of plaquette Physical Review D77 (2008) 014508 [arXiv:0706.3549]
Effective potential of plaquette Veff(P)Plaquette distribution function (histogram) SU(3) Pure gauge theory QCDPAX, PRD46, 4657 (1992) • First order phase transition Two phases coexists at Tc e.g. SU(3) Pure gauge theory • Gauge action • Partition function histogram histogram • Effective potential
Problem of complex quark determinant at m0 (g5-conjugate) • Problem of Complex Determinant at m0 • Boltzmann weight: complex at m0 • Monte-Carlo method is not applicable. • Configurations cannot be generated.
? ? + = Distribution function and Effective potential at m0 (S.E., Phys.Rev.D77, 014508(2008)) • Distributions of plaquette P(1x1 Wilson loopfor 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 crossover non-singular
+ = 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.
+ = + = m-dependence of the effective potential Critical point Crossover m=0 reweighting Curvature: Zero 1st order phase transition m=0 reweighting Curvature: Negative
Sign problem and phase fluctuations • Complex phase of detM • Taylor expansion: odd terms of ln det M (Bielefeld-Swansea, PRD66, 014507 (2002)) • |q| > p/2: Sign problem happens. changes its sign. • Gaussian distribution • Results for p4-improved staggered • Taylor expansion up to O(m5) • Dashed line: fit by a Gaussian function Well approximated q: NOT in the range of [-p, p] histogram of q
Effective potential at m0(S.E., Phys.Rev.D77, 014508(2008)) at m=0 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. Solid lines: reweighting factor at finite m/T, R(P,m) Dashed lines: reweighting factor without complex phase factor.
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.
? + = Curvature of the effective potential Nf=2 p4-staggared, mp/mr0.7 at mq=0 • First order transition for mq/T 2.5 • Existence of the critical point: suggested • Quark mass dependence: large • Study near the physical point is important. Critical point:
Canonical approach Physical Review D 78 (2008) 074507[arXiv:0804.3227]
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.
First order phase transition line In the thermodynamic limit, • Mixed state First order transition • Inverse Laplace transformation by Glasgow method Kratochvila, de Forcrand, PoS (LAT2005) 167 (2005) Nf=4 staggered fermions, lattice • Nf=4: First order for all r.
Canonical partition function • Fugacity expansion (Laplace transformation) canonical partition function • Inverse Laplace transformation • Note: periodicity • Derivative of lnZ Integral Arbitrary m0 Integral path, e.g. 1, imaginary m axis 2, Saddle point
Integral Saddle point Saddle point approximation(S.E., arXiv:0804.3227) • Inverse Laplace transformation • Saddle 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:
Saddle point approximation • Canonical partition function in a saddle point approximation • Chemical potential Saddle point: saddle point reweighting factor Similar to the reweighting method (sign problem & overlap problem)
Saddle point in complex m/T plane • Find a saddle point z0 numerically for each conf. • Two problems • Sign problem • Overlap problem
Technical problem 1:Sign problem • Complex phase of detM • Taylor expansion (Bielefeld-Swansea, PRD66, 014507 (2002)) • |q| > p/2: Sign problem happens. changes its sign. • Gaussian distribution • Results for p4-improved staggered • Taylor expansion up to O(m5) • Dashed line: fit by a Gaussian function Well approximated q: NOT in the range [-p, p] histogram of q
Technical problem 2: Overlap problemRole 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 gas free quark gas free quark gas
Technical problem 2: Overlap problem • Density of state method W(P): plaquette distribution Same effect when b changes. for small P linear for small P linear for small P
(Data: Nf=2 p4-staggared, mp/mr0.7, m=0) Reweighting for b(T)=6g-2 Change: b1(T) b2(T) Distribution: Potential: + = Effective b (temperature) for r0 (r increases) (b (T) increases)
Overlap problem, Multi-b reweightingFerrenberg-Swendsen, PRL63,1195(1989) Plaquette value by multi-beta reweighting peak position of the distribution ○ <P> at each b • When the density increases, the position of the importance sampling changes. • Combine all data by multi-b reweighting Problem: • Configurations do not cover all region of P. • Calculate only when <P> is near the peaks of the distributions. P
Chemical potential vs density Nf=2 p4-staggered, lattice • Approximations: • Taylor expansion: ln det M • Gaussian distribution: q • Saddle point approximation • Two states at the same mq/T • First order transition at T/Tc < 0.83, mq/T >2.3 • m*/T approaches the free quark gas value in the high density limit for all T. • Solid line: multi-b reweighting • Dashed line: spline interpolation • Dot-dashed line: the free gas limit Number density
Summary • Effective 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 lattice • Existence 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