Lattice QCD at finite density Shinji Ejiri University

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

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

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

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. l Reweighting method 1, Perform simulations at m=0. 2, Modify the weight for non-zero m. Sign problem for large 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 m. E • Quark number density: • Quark (Baryon) number susceptibility: diverges at m. E. • Iso-vector susceptibility: does not diverge at m. E. • Charge susceptibility: important for experiments. For the case: • Chiral susceptibility: order parameter of the chiral phase transition • We compute the Taylor expansion coefficients of these susceptibilities.

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 p 4 -improved staggered fermion action • Lattice size:

Derivatives of pressure and susceptibilities • Difference between cq and c. I is small at m=0. Perturbation theory: The difference is O(g 3) • Large spike for c 4, the spike is milder for iso-vector.

Shifting the peak of d 2 c/dm 2 • c 6 changes the sign at Tc. • The peak of d 2 c/dm 2 moves left, corresponding to the shift of Tc. m increases • c 6 < 0 at T > Tc. Consistent with the perturbative prediction in O(g 3).

Difference of pressure for m>0 from m=0 l Chemical potential effect is small. cf. p. SB/T 4~4. l RHIC (mq/T 0. 1): only ~1% for p. l The effect from O(m 6) 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 Hadron resonance 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 Hadron resonance gas • 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 r 4 > r 2 > r 0 – 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? (resonance gas) • There are no singular points. • Further studies are necessary.

5. Summary • Derivatives of pressure with respect to mq up to 6 th 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/T 0 ~ 1. 0. • Iso-spin fluctuations: No peak for m/T 0 <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.