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**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 QCD Physical Review D77 (2008) [arXiv: ] Canonical partition function and finite density phase transition in lattice QCD Physical Review D 78 (2008) [arXiv: ] Tools for finite density QCD, Bielefeld, November 19-21, 2008

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**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) quark-gluon plasma phase hadron phase RHIC SPS AGS color super conductor? nuclear matter mq T

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**Effective potential of plaquette**

Physical Review D77 (2008) [arXiv: ]

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**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

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**Problem of complex quark determinant at m0**

Problem of Complex Determinant at m0 Boltzmann weight: complex at m0 Monte-Carlo method is not applicable. Configurations cannot be generated. (g5-conjugate)

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**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 crossover non-singular + = ?

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**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.

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**m-dependence of the effective potential**

Critical point Crossover + = m= reweighting Curvature: Zero 1st order phase transition + = m= reweighting Curvature: Negative

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**Sign problem and phase fluctuations**

Complex phase of detM Taylor expansion: odd terms of ln det M (Bielefeld-Swansea, PRD66, (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

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**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=0 Solid lines: reweighting factor at finite m/T, R(P,m) Dashed lines: reweighting factor without complex phase factor.

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**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.

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**Curvature of the effective potential**

Nf=2 p4-staggared, mp/mr0.7 at mq=0 + = ? Critical point: 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.

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**Physical Review D 78 (2008) 074507[arXiv:0804.3227]**

Canonical approach Physical Review D 78 (2008) [arXiv: ]

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**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.

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**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.

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**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

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**Saddle point approximation (S.E., arXiv:0804.3227)**

Integral Saddle point 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:

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**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)

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**Saddle point in complex m/T plane**

Find a saddle point z0 numerically for each conf. Two problems Sign problem Overlap problem

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**Technical problem 1: Sign problem**

Complex phase of detM Taylor expansion (Bielefeld-Swansea, PRD66, (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

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**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 gas free quark gas free quark gas

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**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

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**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)

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**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 reweighting Problem: Configurations do not cover all region of P. Calculate only when <P> is near the peaks of the distributions. Plaquette value by multi-beta reweighting peak position of the distribution ○ <P> at each b P

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**Chemical potential vs density**

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 Nf=2 p4-staggered, lattice Number density

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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

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