Lattice QCD at finite density

1. Lattice QCD at finite density Shinji Ejiri (University of Tokyo) Collaborators: C. Allton, S. Hands (U. Wales Swansea), M. Döring, O.Kaczmarek, F.Karsch, E.Laermann (U. Bielefeld), K.Redlich (U. Bielefeld & U. Wroclaw) (hep-lat/0501030) RIKEN, February 2005

2. Numerical simulations T early universe quark-gluon plasma RHIC phase SPS AGS hadron phase color super conductor? nuclear matter color flavor locking? mq mN/3~300MeV Introduction • High temperature and density QCD • Low density region • Heavy-ion collisions Comparison with different density • Critical endpoint? Simulation parameter: mq/T • High density region Y. Nishida Tc~170MeV

3. Chemical freeze out parameter • Statistical thermal model • Well explains the particle production rates (P. Braun-Munzinger et al., nucl-th/0304013) • Relation to the chiral/ confinement phase transition • Relation to (e,p,S,n) Lattice calculations Lattice (10% error)

4. Critical endpoint • Various model calculations(M.A. Stephanov, Prog.Theor.Phys.Suppl.153 (2004)139) Baryon fluctuations becomes bigger as m large.

5. Numerical Simulations of QCD at finite Baryon Density • Boltzmann weight is complex for non-zero m. • 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 m=0. for large m 2, Modify the weight for non-zero m.

6. Studies at low density • Reweighting method only at small m. • Not very serious for small lattice. (~ Nsite) • Interesting regime for heavy-ion collisions is low density. (mq/T~0.1 for RHIC, mq/T~0.5 for SPS) • Taylor expansion at m=0. • Taylor expansion coefficients are free from the sign problem. (The partition function is a function of mq/T)

7. Fluctuations near critical endpoint mE • Quark number density: • Quark (Baryon) number susceptibility: diverges at mE. • Iso-vector susceptibility: does not diverge at mE. • Charge susceptibility: important for experiments. • Chiral susceptibility: order parameter of the chiral phase transition • We compute the Taylor expansion coefficients of these susceptibilities. For the case:

8. Equation of State via Taylor Expansion Equation of state at low density • T>Tc; quark-gluon gas is expected. Compare to perturbation theory • Near Tc; singularity at non-zero m (critical endpoint). Prediction from the sigma model • T<Tc; comparison to the models of free hadron resonance gas.

9. Simulations • We perform simulations for Nf=2 at ma=0.1 (mp/mr0.70 at Tc) and investigate T dependence of Taylor expansion coefficients. • Symanzik improved gauge action and p4-improved staggered fermion action • Lattice size:

10. Derivatives of pressure and susceptibilities • Difference between cq and cI is small at m=0. Perturbation theory: The difference is O(g3) • Large spike for c4, the spike is milder for iso-vector.

11. Shifting the peak of d2c/dm2 • c6 changes the sign at Tc. • The peak of d2c/dm2 moves left, corresponding to the shift of Tc. m increases • c6 < 0 at T > Tc. • Consistent with the perturbative prediction in O(g3).

12. Difference of pressure for m>0 from m=0 • Chemical potential effect is small. cf. pSB/T4~4. • RHIC (mq/T0.1): only ~1% for p. • The effect from O(m6) term is small.

13. Quark number susceptibility • We find a pronounced peak for mq/T~ 1. Critical endpoint in the (T,m)? • Peak position moves left as m increases, corresponds to the shift of Tc(m)

14. Iso-vector susceptibility • No peak is observed. Consistent with the prediction from the sigma model.

15. (disconnected) chiral susceptibility • Peak height increases as mq increases. Consistent with the prediction from the sigma model.

16. Comparison to the hadron resonance gas • Non-interacting hadron gas: m dependence must be • Taylor expansion: we get

17. Hadron resonance gas or quark-gluon gas Hadron resonance gas Hadron resonance gas • At T<Tc,consistent with hadron resonance gas model. • At T>Tc, approaches the value of a free quark-gluon gas. Free QG gas Free QG gas

18. Hadron resonance gas for chiral condensate Hadron resonance gas • At T<Tc,consistent with hadron resonance gas model.

19. Singular point at finite densityRadius of convergence • We define the radius of convergence • The SB limit of rn for n>4 is  • At high T, rn is large and r4 > r2 > r0 • No singular point at high T.

20. Radius of convergence • The hadron resonance gas prediction • The radius of convergence should be infinity at T<Tc. • Near Tc, rn is O(1) • It suggests a singular point around m/Tc ~ O(1) ?? • However, still consistent with HRGM. • Too early to conclude.

21. Mechanical instability • Unstable point • We expectcq to diverge at the critical endpoint. Unstable point appears? • There are no singular points. • Further studies are necessary. (resonance gas)

22. 5. Summary • Derivatives of pressure with respect to mq up to 6th order are computed. • The hadron resonance gas model explains the behavior of pressure and susceptibilities very well at T<Tc. • Approximation of free hadron gas is good in the wide range. • Quark number density fluctuations: A pronounced peak appears for m/T0 ~ 1.0. • Iso-spin fluctuations: No peak for m/T0 <1.0. • Chiral susceptibility: peak height becomes larger as mq increases. This suggests the critical endpoint in (T,m) plane? • To find the critical endpoint, further studies for higher order terms and small quark mass are required. • Also the extrapolation to the physical quark mass value and the continuum limit is important for experiments.