Study of the critical point in lattice QCD at high temperature and density

1. Study of the critical point in lattice QCD at high temperature and density arXiv:0706.3549 [hep-lat] Lattice 2007 @ Regensburg, July 30 – August 4, 2007 Shinji Ejiri (Brookhaven National Laboratory)

2. Study of QCD phase structure at finite density Interesting topic: • Endpoint of the first order phase transition • Monte-Carlo simulations generate configurations • Distribution function (histogram) of plaquette P. • First order phase transition: Two states coexist. Double peak. • Effective potental: • First order: Double well Curvature: negative

3. Study of QCD phase structure at finite densityReweighting method for m • Boltzmann weight: Complex for m>0 • Monte-Carlo method is not applicable directly. • Cannot generate configurations with the probability • Reweighting method • Perform Simulation at m=0. • Modify the weight factor by (expectation value at m=0.)

4. Weight factor, Effective potential • Classify by plaquette P (1x1 Wilson loop for the standard action) (Weight factor at m=0) Expectation value of O[P] Ex) 1st order phase transition Weight factor at finite m Effective potential: We estimate the effective potential using the data in PRD71,054508(2005).

5. Reweighting for m/T, Taylor expansion Problem 1, Direct calculation of detM : difficult • Taylor expansion in mq(Bielefeld-Swansea Collab., PRD66, 014507 (2002)) • Random noise method Reduce CPU time. • Approximation: up to • Taylor expansion of R(mq) no errortruncation error • The application range can be estimated, e.g., Valid, if is small.

6. histogram Reweighting for m/T, Sign problem Problem 2, detM(m): complex number. Sign problem happens when detM changes its sign frequently. • Assumption: Distribution functionW(P,|F|,q) Gaussian Weight: real positive (no sign problem) When the correlation between eigenvalues of M is weak. • Complex phase: Valid for large volume (except on the critical point) Central limit theorem q: Gaussian distribution

7. Reweighting for m/T, sign problem • Taylor expansion: odd terms of ln det M(Bielefeld-Swansea, PRD66, 014507 (2002)) • Estimation of a(P, |F|): For fixed (P, |F|), • Results for p4-improved staggered • Taylor expansion up to O(m5) • Dashed line: fit by a Gaussian function Well approximated histogram of q

8. (Data: Nf=2 p4-staggared, mp/mr0.7, m=0, without reweighting) Reweighting for b(T)=6g-2 Change: b1(T) b2(T) Weight: Potential: + = Assumption: W(P) is of Gaussian. (Curvature of V(P) at mq=0)

9. Effective potential at finite m/T at m=0 + = • Normal • First order ? + = Solid lines: reweighting factor at finite m/T, R(P,m) Dashed lines: reweighting factor without complex phase factor.

10. Slope and curvature of the effective potential • First order at mq/T  2.5 at mq=0 Critical point:

11. QCD with isospin chemical potential mI • Dashed line: QCD with non-zero mI (mq=0) • The slope and curvature of lnR for isospin chemical potential is small. • The critical mI is larger than that in QCD with non-zero mq.

12. 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 2.5.

13. Canonical approach Nf=2 p4-staggered, lattice • Approximations: • Taylor expansion: ln det M • Gaussian distribution: q • Canonical partition function • Inverse Laplace transformation • Saddle point approximation • Chemical potential • Two states at the same mq/T • First order transition at mq/T >2.5 • Similar to S. Kratochvila, Ph. de Forcrand, hep-lat/0509143 (Nf=4) Number density

14. Summary and outlook • An effective potential as a function of the plaquette is studied. • Approximation: • Taylor expansion of ln det M: up to O(m6) • Distribution function of q=Nf Im[ ln det M] : Gaussian type. • Simulations: 2-flavor p4-improved staggered quarks with m/T=0.4, 163x4 lattice • First order phase transition for mq/T 2.5. • The critical m is larger for QCD with finite mI (isospin) than with finite mq (baryon number). • Studies neat physical quark mass: important.