The QCD equation of state for two flavor
QCD at non-zero chemical potential
Shinji Ejiri
(University of Tokyo)
C. Allton, S. Hands (Swansea),
M. Döring, O.Kaczmarek, F.Karsch, E.Laermann (Bielefeld),
K.Redlich (Bielefeld & Wroclaw)
Collaborators:
(Phys. Rev. D71, 054508 (2005) +a)
Quark Matter 2005, August 4-9, Budapest
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
1, Perform simulations at m=0.
2, Modify the weight for non-zero m.
Sign problem
for large m
Studies at low density
• Taylor expansion at m=0.
– Calculations of Taylor expansion coefficients: free from the sign problem.
– Interesting regime for heavy-ion collisions is low density.
(mq/T~0.1 for RHIC, mq/T~0.5 for SPS)
 mq 
 mq 
 mq 
p
1

ln Z  c0  c2    c4    c6    
4
3
T
VT
T 
T 
T 
2
4
6
• Calculation of thermodynamic quantities.
– The derivatives of lnZ: basic information in lattice simulations.
Quark number density:
nu,d 
T  ln Z
p

V m u,d m u,d
 
 
2 p
nu  nd   2

Quark number susceptibility: q  
mq
 m u md 
Chiral condensate:
 
T  ln Z
V m
Higher order terms:
natural extension.
Equation of State via Taylor Expansion
Equation of state at low density
• T  Tc ; quark-gluon gas is expected.
Compare to perturbation theory
QGP
T
hadron
color superconductor?
m
• 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 N f =2 at ma=0.1 (mp/mr0.70 at Tc)
and investigate T dependence of Taylor expansion coefficients.
 mq 
 mq 
 mq 
p
p




  




m

0

c

c

c
2
4
6
4
4



T
T
T 
T 
T 
2
Pressure:
q
4
6
 mq 
 mq 




m

2
c

12
c

30
c
2
4
6
Quark number susceptibility: T 2

 T   
T
 
 
2
4
m
m
I
I
I q 
I q 



  


m

2
c

12
c

30
c
2
4
6

T2
T
T
 
 
2
Isospin susceptibility:
N 3  n ln Z
N 3
 n ln Z
I
cn 
, cn 
3
n
n! N   (m q T )
n! N 3  (m I T ) 2 (m q T ) n  2
4
m  m
q
u
 m d  2, m I  m u  m d  2
Moreover, Taylor expansion coefficients of chiral condensate and
static quark-antiquark free energy are calculated.
• Symanzik improved gauge action and p4-improved staggered fermion action
• Lattice size: N site  N 3  N   163  4
Derivatives of pressure and susceptibilities
q
 mq 
 mq 


  


m

2
c

12
c

30
c
2
4
6
2


T
T 
T 
2
4
• Difference between
m
m
I
I
I q 
I q 



  


m

2
c

12
c

30
c
2
4
6
2

T
T 
T 
2
4
T0  Tc at m q  0
 q and  I is small at m=0.
3
– Perturbation theory: The difference is O( g )
• Large spike for c4, the spike is milder for iso-vector.
• c6  0 at T  Tc
– Consistent with the perturbative prediction in O( .g 3 )
Difference of pressure for m>0 from m=0
 mq 
 mq 
 mq 
p
p




m   4 0  c2    c4    c6    O m8q
4
T
T
T 
T 
T 
2
4
6
 
 mq 
 mq 
p
p
6








m

0

c

c

O
m
2
4
q
4
4

T 
T
T
T 
 
2
Chemical potential effect is small. cf. pSB/T4~4 at m=0.
RHIC ( m q / T  0.1) : only ~1% for p.
The effect from O(m6) term is small.
4
 
Quark number susceptibility and Isospin susceptibility
q
 mq 
 mq 
6






m

2
c

12
c

30
c

O
m
q
2
4
6
q
T 
T 
T4
 
 
2
4
 
m
m
I
I
I q 
I q 
mq   2c2  12c4    30c6    O m6q
4
T
T 
T 
2
4
 
q
• Pronounced peak for  q around m q / T  1
Critical endpoint in the (T,m) ?
• No peak for  I
Consistent with the prediction from the sigma model.
Chiral susceptibility
 
T  ln Z
V m
m
m
cs
cs  q 
cs  q 
 c0  c2    c4    O m 6q
T 
T 
2
 
4
 
(disconnected part only)
• Peak height increases as
m q increases.
Consistent with the prediction
from the sigma model.
Comparison to hadron resonance gas model
Hadron resonance gas prediction
 3m q 
p
,
 GT   F T  cosh 
4
T
 T 
q  2  p T 4 
 3m q 
,

 9 F T  cosh 
2
2
T
 m q T 
 T 
Hadron resonance gas
Free QG gas
 mq 
 mq 
 mq 
p






m

c

c

c

c
0
2
4
6


 T   
T4
T
T
 
 
 
2
4
6
c4 c2  3 4 , c6 c4  3 10 ,
• At T  Tc , consistent with hadron resonance gas model.
• At T  Tc , approaches the value of a free quark-gluon gas.
Hadron resonance gas model for Isospin
susceptibility and chiral condensate
 3m q 
I
I
I

 G T   F T cosh
2
T
 T 

m
m
I
I
I q 
I q 



  


m

2
c

12
c

30
c
2
4
6

T2
T
T
 
 

2
4
Hadron resonance gas
T3
T3
 3m q 

 G  T   F  T cosh
 T 
m
m

  q 
  q 



m   c0  c2    c4    
T 
T 
2
Hadron resonance gas
Free QG gas
• At T  Tc , consistent with hadron resonance gas model.
4
Debye screening mass
• QQ free energy from Polyakov loop correlation
Singlet free energy (Coulomb gauge) Averaged free energy
 1
F T , m, r   T ln Re  Tr L(0) L† (r )
 Nc

1
QQ
N
where
L( x)  U 4 x, t  :

 ,


 1

1
†

F T , m, r   T ln Re 
Tr L(0)
Tr L (r ) 
Nc
 Nc

av
QQ
Polyakov loop
at m  0
t 1
• Assumption at T>Tc
FQQ T , m, r   FQQ T , m,    
4 aT , m   m T ,m r
e
3 r
Color-electric screening mass:
mT , m    lim


1
ln FQ(1Q) T , m,    FQ(1Q) T , m, r 
r  r
perturbative prediction (T. Toimela, Phys.Lett.B124(1983)407)
mT , m   m0
 mq 
Nc N f
  , m0  Ag T T
1

2 
2 Nc  N f p  T 
3
6
3N f
O.Kaczmarek and F.Zantow,
Phys.Rev.D71 (2005) 114510
2
m1 
1 av
m
2
Taylor expansion coefficients of screening mass
m
m
m
mT , m   m0 T   m2 T    m4 T    m6 T    
T 
T 
T 
2
4
6
perturbative prediction
m12  m2av 2
m1 T at m  0
m14  0, m4av 2  0
Consistent with
perturbative prediction
m61 , m6av 2
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 mq / T0  1 .
• Iso-spin fluctuations: No peak for mq / T0  1 .
• Chiral susceptibility: peak height becomes larger as mq increases.
This suggests the critical endpoint in (T , m ) plane?
• Debye screening mass at non-zero mq is consistent with the
perturbative result for T  2Tc .
• To find the critical endpoint, further studies for higher order
terms and small quark mass are required.