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

Numerical simulations 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 phase hadron phase RHIC early universe SPS AGS mN/3~300MeV color flavor locking? color super conductor? nuclear matter Numerical simulations mq T quark-gluon plasma Tc~170MeV

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)

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

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.

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)

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:

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.

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:

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.

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).

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.

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)

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

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

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

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

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

Singular point at finite density Radius 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.

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.

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

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.