Lattice QCD at finite density Shinji Ejiri (University of Tokyo) Collaborators: C. Allton, S. Hands (U. Wales Swanse a), M. Döring, O.Kaczmarek, F.Karsch, E.Laerman n (U. Bielefeld), K.Redlich (U. Bielefeld & U. Wroclaw) (hep-lat/0501030) RIKEN, February 2005
Lattice QCD at finite density. Shinji Ejiri (University of Tokyo) Collaborators: C. Allton, S. Hands (U. Wales Swansea) , M. D öring, O.Kaczmarek, F.Karsch, E.Laermann (U. Bielefeld), K.Redlich (U. Bielefeld & U. Wroclaw). (hep-lat/0501030) RIKEN, February 2005. - PowerPoint PPT Presentation
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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
phase
hadron phase
RH
ICearly universe
SP
S
AG
S
mN/3~300MeV
color flavor locking?
color super conductor?
nuclear matter
Numerical simulationsNumerical simulations
q
Tquark-gluon plasma
Introduction• High temperature and density QCD
• Low density region– Heavy-ion collisions
Comparison with different density
– Critical endpoint?
Simulation parameter: q/T
• High density region Y. Nishida
Tc~
170M
eV
Chemical freeze out parameter
• Statistical thermal model– Well explains the particle pr
oduction rates(P. Braun-Munzinger et al., n
ucl-th/0304013)
• Relation to the chiral/ confinement phase transition
• Relation to (,p,S,n)
Lattice calculations
qB 3
Lattice(10% error)
Critical endpoint• Various model calculations (M.A. Stephanov, Prog.Theor.Phys.Suppl.153 (2004)139)
qB 3
Baryon fluctuations becomes bigger as large.
Numerical Simulations of QCD at finite Baryon Density
• Boltzmann weight is complex for non-zero .– Monte-Carlo simulations: Configurations are generated with t
he 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 .
Studies at low density
• Reweighting method only at small .– Not very serious for small lattice. (~ Nsite)– Interesting regime for heavy-ion collisions is low density. (q
/T~0.1 for RHIC, q/T~0.5 for SPS)
• Taylor expansion at =0.– Taylor expansion coefficients are free from the sign problem.
(The partition function is a function of q/T)
6
q6
4
q4
2
q244
0T
cT
cT
cT
p
T
p
• Quark number density:
• Quark (Baryon) number susceptibility: diverges at E.
• Iso-vector susceptibility: does not diverge at E.
• Charge susceptibility: important for experiments.
• Chiral susceptibility: order parameter of the chiral phase transition
• We compute the Taylor expansion coefficients of these susceptibilities.
Fluctuations near critical endpoint E
du,du,du,
ln
pZ
V
Tn
2q
2
dudu
q
p
nn
2I
2
dudu
I
p
nn
dudu
C 3
1
3
2
3
1
3
2nn
qdu 2I
2
q
2C
4
1
36
1
TTT
For the case:
2duI
2duq
Z
V
Tp ln
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 (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 (m/m0.70 at Tc)
and investigate T dependence of Taylor expansion coefficients.
• Symanzik improved gauge action and p4-improved staggered fermion action
• Lattice size: 41633site NNN
6
q6
4
q4
2
q244
0T
cT
cT
cT
p
T
p
6
q
6
3
3
64q
4
3
3
42q
2
3
3
2 )(
ln
!6 ,
)(
ln
!4 ,
)(
ln
2 T
Z
N
Nc
T
Z
N
Nc
T
Z
N
Nc
.)()(
ln
!6 ,
)()(
ln
!4 ,
)(
ln
2 4q
2I
6
3
3I62
q2
I
4
3
3I42
I
2
3
3I2 TT
Z
N
Nc
TT
Z
N
Nc
T
Z
N
Nc
4
qI6
2
qI4
I22
I 30122T
cT
ccT
4
q6
2
q422
30122T
cT
ccT
q
Derivatives of pressure and susceptibilities
• Difference between q and I is small at =0.Perturbation theory: The difference is O(g3)
• Large spike for c4, the spike is milder for iso-vector.
0at q0 cTT
Shifting the peak of d2/d2
• c6 changes the sign at Tc.• The peak of d2/d2 moves left, corresponding
to the shift of Tc.
increases
• c6 < 0 at T > Tc.Consistent with the perturbative pr
ediction in O(g3).
Difference of pressure for >0 from =0
Chemical potential effect is small. cf. pSB/T4~4. RHIC (q/T0.1): only ~1% for p.
The effect from O(6) term is small.
8q
6
q6
4
q4
2
q244
0
O
Tc
Tc
Tc
T
p
T
p 6
q
4
q4
2
q244
0
O
Tc
Tc
T
p
T
p
Quark number susceptibility
• We find a pronounced peak for q/T~ 1. Critical endpoint in the (T,)?
• Peak position moves left as increases, corresponds to the shift of Tc()
6q
4
q6
2
q42q4
q 30122
O
Tc
Tcc
T 4
q
2
q42q4
q 122
O
Tcc
T
Iso-vector susceptibility
• No peak is observed.
Consistent with the prediction from the sigma model.
dudu
I nn
6
q
4
qI6
2
qI4
I2q4
I 30122
O
Tc
Tcc
T
(disconnected) chiral susceptibility
• Peak height increases as q increases.
Consistent with the prediction from the sigma model.
133
tr 1
)(
detln1
M
NNma
M
NN
6q
4
q4
2
q20
21213
tr tr 1
OT
cT
cc
MMNN
cscscs
Comparison to the hadron resonance gas• Non-interacting hadron gas: dependence must be
• Taylor expansion:
we get
,06
q6
4
q4
2
q244
Tc
Tc
Tc
T
p
T
p
,3
cosh92
q
T
TFT
q ,3
cosh 4
TTFTG
T
p q
T
TFTGT
qII 3cosh
2I
4
qI6
2
qI4
I22
I 30122T
cT
ccT
,103 ,103 ,43 I4
I64624 cccccc
4
q6
2
q422
30122T
cT
ccT
q
Hadron resonance gas or quark-gluon gas
• At T<Tc, consistent with hadron resonance gas model.
• At T>Tc, approaches the value of a free quark-gluon gas.
4
q6
2
q422
q 30122T
cT
ccT
4
qI6
2
qI4
I22
I 30122T
cT
ccT
Hadron resonance gas
Free QG gasFree QG gas
Hadron resonance gas
Hadron resonance gas for chiral condensate
• At T<Tc, consistent with hadron resonance gas model.
4
q4
2
q20 T
cT
cc Hadron resonance gas
Singular point at finite densityRadius of convergence
•
• We define the radius of convergence
• The SB limit of n for n>4 is
• At high T, n is large and 4 > 2 > 0
– No singular point at high T.
nnnn
nn
n ccTTcTc
2 2
2 qqq
nn
lim
,06
q6
4
q4
2
q244
Tc
Tc
Tc
T
p
T
p
Radius of convergence• The hadron resonance gas prediction
• The radius of convergence should be infinity at T<Tc.
• Near Tc, n is O(1)• It suggests a singular point around
Tc ~ O(1) ??
– However, still consistent with HRGM.– Too early to conclude.
n
nlim
9
)12)(22(
2
nn
nc
nc
n
Mechanical instability
• Unstable point
• We expect q to diverge at the critical endpoint.
Unstable point appears?
• There are no singular points.• Further studies are necessary.
0
q
q
Tq
n
n
p
T
n
n
p q
q
q
T
3tanh
3
1(resonance gas)
5. Summary • Derivatives of pressure with respect to q up to 6th order are compute
d.
• 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 /T0 ~ 1.0.
• Iso-spin fluctuations: No peak for /T0 <1.0.• Chiral susceptibility: peak height becomes larger as q increases.
This suggests the critical endpoint in (T,) 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.