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/ )
RIKEN, February 2005

2. 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

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

4. 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.

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= 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~ 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
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.

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

19. 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.

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 expect cq 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.