1. The QCD equation of state for two flavor QCD at non-zero chemical potential Shinji Ejiri (University of Tokyo) Collaborators: C. Allton, S. Hands (Swansea), M. Döring, O.Kaczmarek, F.Karsch, E.Laermann (Bielefeld), K.Redlich (Bielefeld & Wroclaw) (Phys. Rev. D71, 054508 (2005) +  ) Quark Matter 2005, August 4-9, Budapest

2. Numerical Simulations of QCD at finite Baryon Density Boltzmann weight is complex for non-zero . –Monte-Carlo simulations: Configurations are generated with the probability of the Boltzmann weight. –Monte-Carlo method is not applicable directly. Reweighting method Sign problem 1, Perform simulations at  =0. for large  2, Modify the weight for non-zero .

3. Studies at low density Taylor expansion at  =0. –Calculations of Taylor expansion coefficients: free from the sign problem. –Interesting regime for heavy-ion collisions is low density. (  q /T~0.1 for RHIC,  q /T~0.5 for SPS) Calculation of thermodynamic quantities. –The derivatives of lnZ: basic information in lattice simulations. Quark number density: Quark number susceptibility: Chiral condensate: Higher order terms: natural extension.

4. Equation of State via Taylor Expansion Equation of state at low density ; quark-gluon gas is expected. Compare to perturbation theory Near ; singularity at non-zero  (critical endpoint). Prediction from the sigma model ; comparison to the models of free hadron resonance gas. QGP color super- conductor? hadron  T

5. Simulations We perform simulations for =2 at ma=0.1 (m  /m   0.70 at T c ) and investigate T dependence of Taylor expansion coefficients. Moreover, Taylor expansion coefficients of chiral condensate and static quark-antiquark free energy are calculated. Symanzik improved gauge action and p4-improved staggered fermion action Lattice size: Quark number susceptibility: Isospin susceptibility: Pressure:

6. Derivatives of pressure and susceptibilities Difference between and is small at  =0. –Perturbation theory: The difference is Large spike for, the spike is milder for iso-vector. at –Consistent with the perturbative prediction in.

7. Difference of pressure for  >0 from  =0 Chemical potential effect is small. cf. p SB /T 4 ~4 at  =0. RHIC : only ~1% for p. The effect from O(  6 ) term is small.

8. Quark number susceptibility and Isospin susceptibility Pronounced peak for around Critical endpoint in the (T,  ) ? No peak for Consistent with the prediction from the sigma model.

9. Chiral susceptibility Peak height increases as increases. Consistent with the prediction from the sigma model. (disconnected part only)

10. Comparison to hadron resonance gas model At, consistent with hadron resonance gas model. At, approaches the value of a free quark-gluon gas. Hadron resonance gas Free QG gas Hadron resonance gas prediction

11. Hadron resonance gas model for Isospin susceptibility and chiral condensate At, consistent with hadron resonance gas model. Hadron resonance gas Free QG gas Hadron resonance gas

12. Debye screening mass QQ free energy from Polyakov loop correlation Singlet free energy (Coulomb gauge) Averaged free energy where : Polyakov loop Assumption at T>T c Color-electric screening mass: perturbative prediction (T. Toimela, Phys.Lett.B124(1983)407) O.Kaczmarek and F.Zantow, Phys.Rev.D71 (2005) 114510

13. Taylor expansion coefficients of screening mass Consistent with perturbative prediction

14. Summary Derivatives of pressure with respect to  q up to 6 th order are computed. The hadron resonance gas model explains the behavior of pressure and susceptibilities very well at. –Approximation of free hadron gas is good in the wide range. Quark number density fluctuations: A pronounced peak appears for. Iso-spin fluctuations: No peak for. Chiral susceptibility: peak height becomes larger as  q increases. This suggests the critical endpoint in plane? Debye screening mass at non-zero  q is consistent with the perturbative result for. To find the critical endpoint, further studies for higher order terms and small quark mass are required.