A method of finding the critical
point in finite density QCD
Shinji Ejiri
(Brookhaven National Laboratory)
Existence of the critical point in finite density lattice QCD
Physical Review D77 (2008) 014508 [arXiv:0706.3549]
Canonical partition function and finite density phase transition in lattice QCD
Physical Review D 78 (2008) 074507[arXiv:0804.3227]
Tools for finite density QCD, Bielefeld, November 19-21, 2008
QCD thermodynamics at m0
• Interesting properties of QCD
T
Critical point at finite density
SPS
RHIC
Measurable in heavy-ion collisions
AGS
• Location of the critical point ?
– Distribution function of plaquette value
– Distribution function of quark number density
quark-gluon
plasma phase
hadron
phase
• Simulations:
– Bielefeld-Swansea Collab., PRD71,054508(2005).
– 2-flavor p4-improved staggered quarks with mp 770MeV
– 163x4 lattice
– ln det M: Taylor expansion up to O(m6)
color super
nuclear
matterconductor?
mq
Effective potential of plaquette
Physical Review D77 (2008) 014508 [arXiv:0706.3549]
Effective potential of plaquette Veff(P)
Plaquette distribution function (histogram)
• First order phase transition
Two phases coexists at Tc
e.g. SU(3) Pure gauge theory
• Gauge action
SU(3) Pure gauge theory
QCDPAX, PRD46, 4657 (1992)
histogram
S g  6 N siteP
• Partition function
(  6 g 2 )
Z , m    dP W P, , m 
histogram
Effective potential
W P' ,    DU detM  f e
N
Veff ( P)   ln W P 
Sg
 P  P ' 
Problem of complex quark determinant at m0

Problem of Complex Determinant at m0
M (m)

  5 M (m) 5
(5-conjugate)
detM (m) 
*

 detM (m)  detM (m)
Boltzmann weight: complex at m0
 Monte-Carlo method is not applicable.
 Configurations cannot be generated.
Distribution function and Effective potential at m0
(S.E., Phys.Rev.D77, 014508(2008))
• Distributions of plaquette P (1x1 Wilson loop for the standard action)
Z m    dP RP, m W P, 
W P ,    DU P-P  detM 0 f e
N
R P , m  
 DU P-P  detM m 
Nf
 DU P-P  detM 0
Nf

S g  6 N siteP
Sg
(Weight factor at m0
 detM m  
P-P  

 detM 0 
Nf
P-P  (,m 0)
(  ,m  0 )
(Reweight factor
R(P,m): independent of ,  R(P,m) can be measured at any .
Effective potential:
m=0 crossover
non-singular
Veff ( P)   ln RP, mW P,  
1st order phase transition?
+ ? =
 ln W P,    ln RP, m 
?
Reweighting for T and curvature of –lnW(P)
Z m    dP RP, m W P, 
Change: 1(T)
N
2(T)
W 1   W  2   e
Weight:
W P' ,    DU detM 0 f e
W 1 
 S g  2  S g 1 
S g ( 2 )  S g (1 )  6 N site  2  1 P
Potential:
 ln W 1   6 Nsite2  1 P   ln W 2 
+
=
Curvature of –lnW(P) does not change.
Curvature of –lnW(P,) : independent of .
Sg
 P  P ' 
m-dependence of the effective potential
Critical point
 ln W P,    ln RP, m 
Crossover
 ln W P,  
+
m=0
=
reweighting
Curvature: Zero
T
QGP
hadron
1st order phase transition
 ln W P,    ln RP, m 
CSC
+
m
m=0
=
reweighting Curvature: Negative
Sign problem and phase fluctuations
q  Nf Imln det M (m)
– Taylor expansion: odd terms of ln det M (Bielefeld-Swansea, PRD66, 014507 (2002))
• Complex phase of detM
 m d ln det M 1  m 3 d 3 ln det M 1  m 5 d 5 ln det M

q  N f Im 
  
  
 
3
5






T
d
m
T
3
!
T
d
m
T
5
!
T
d
m
T
 
 


• |q| > p/2: Sign problem happens.
e iq changes its sign.
 det M (m) 


 det M (0) 
q: NOT in the range of [-p, p]
histogram of q
T  TC
Nf
 eiq F  (statistic al error)
• Gaussian distribution
– Results for p4-improved staggered
– Taylor expansion up to O(m5)
– Dashed line: fit by a Gaussian function
Well approximated
iq
e F
 e
 q2
F
2
F
Effective potential at m0
(S.E., Phys.Rev.D77, 014508(2008))
Results of Nf=2 p4-staggared,
 ln W
mp/mr0.7
[data in PRD71,054508(2005)]
• detM: Taylor expansion up
to O(m6)
• The peak position of W(P)
moves left as  increases at
m=0.
Solid lines: reweighting factor at
finite m/T, R(P,m)
Dashed lines: reweighting factor
without complex phase factor.
at m=0
ln R
Veff P, , m   ln W P,   ln RP, m
Truncation error of Taylor expansion
 1  m  n d n ln det M 
N f ln det M (m)  N f    
n 
n 0 
 n!  T  dm T  
N
• Solid line: O(m4)
• Dashed line: O(m6)
• The effect from 5th and
6th order term is small
for mq/T  2.5.
Curvature of the effective potential
Nf=2 p4-staggared, mp/mr0.7
d 2 ln W

dP 2
at mq=0
+
Critical
=
?
d 2Veff P, , m 
d 2 ln W P,  d 2 ln RP, m 


0
point:
2
2
2
dP
dP
dP
• First order transition for mq/T  2.5
• Existence of the critical point: suggested
– Quark mass dependence: large
– Study near the physical point is important.
Canonical approach
Physical Review D 78 (2008) 074507[arXiv:0804.3227]
Canonical approach
• Canonical partition function
Z GC T , m    Z C T , N  exp Nm T   W N 
N
N
• Effective potential as a function of the quark number N.
Veff ( N )   ln W ( N )   ln Z C (T , N )  N m T
• At the minimum,
Veff ( N )
 ln Z C (T , N ) m
 ln W ( N )


 0
N
N
N
T
• First order phase transition: Two phases coexist.

Veff ( N )
 ln Z C ( N )
N
m T
N
N
First order phase transition line
 ln Z C (T , N )
m*

T
N
V ( N )
 0,
In the thermodynamic limit, eff
N
m*
m

T
T
T  Tcp
T  Tcp
• Mixed state
First order transition
• Inverse Laplace transformation by Glasgow method
Kratochvila, de Forcrand, PoS (LAT2005) 167 (2005)
Nf=4 staggered fermions, 63  4 lattice
– Nf=4: First order for all r.
N
3
s


Canonical partition function
• Fugacity expansion (Laplace transformation)
Z GC T , m    Z C T , N  exp Nm T 
r  N /V
N
mI
canonical partition function
• Inverse Laplace transformation
3 p3
 N m 0 T  i m I T 


Z C T , N  
d
m
T
e
Z GC T , m 0  im I 
I

2p  p 3
N


Z GC m 
1
det
M
(
m
)
S
N



DU
det
M
(m )  e
 

Z GC 0  Z GC 0 
 det M (0) 
m 0
f
f
mR
Integral
g
– Note: periodicity ZGC T , m  2piT 3  ZGC T , m
• Derivative of lnZ
m0
 ln Z C (T , N )
m*

T
N
Arbitrary m0
Integral path, e.g.
1, imaginary m axis
2, Saddle point
Saddle point approximation
mI
(S.E., arXiv:0804.3227)
• Inverse Laplace transformation
3 p3
 N m T  i m


Z C T , N  
d
m
T
e
I
2p  p 3
0
3Z GC (0)

2p

p3
p 3
d m I T  e
I
T
Saddle point
m0
Z GC T , m 0  im I 
 N m 0 T  i m I T 
 det M m 0  im I  


det M 0 


Nf
Integral
• Saddle point approximation (valid for large V, 1/V expansion)
– Taylor expansion at the saddle point.
Saddle point:
m 0 T  z0
 N f ln det M 

 r
0
V


m T

 m  z0
z0
r  N /V
V  N s3
T
• At low density: The saddle point and the Taylor expansion coefficients
can be estimated from data of Taylor expansion around m=0.
n
 
 1  m  n d n ln det M 
m 
N f ln det M (m)  N f    
 VNf N t   Dn   
n 
n 0 
n 0 
 n!  T  dm T  
  T  

mR
Saddle point approximation
• Canonical partition function in a saddle point approximation
Z C T , r

Z GC T ,0 


 i  2
 det M ( z0 ) 
3
1
  Vrz0  e
exp  N f ln 
V R z0 
2p
 det M (0) 


3
exp F  iq
2p
Saddle point:
z0
( T ,m  0 )
(T ,m  0 )
m  N f  2 ln det M 

i


R  

R
e
2
 m T 
T  V
• Chemical potential
z0 exp F  iq (T ,m 0)
m* (r)  1  ln Z C (T , r)


T
V
r
exp F  iq (T ,m 0)
saddle point
reweighting factor
Similar to the reweighting method
(sign problem & overlap problem)
Saddle point in complex m/T plane
• Find a saddle point z0
numerically for each conf.
 N f ln det M 

 r
0
V
m T 

 m  z0
T
• Two problems
– Sign problem
– Overlap problem
Technical problem 1: Sign problem
• Complex phase of detM
(phase)  Nf Imln det M (m)
– Taylor expansion (Bielefeld-Swansea, PRD66, 014507 (2002))

 
 
q  Im V  N f N t  Dn z0  rz0  
n 1
 2
 
q: NOT in the range [-p, p]
• |q| > p/2: Sign problem happens.
r T 3  2.0
e iq changes its sign.
• Gaussian distribution
– Results for p4-improved staggered
– Taylor expansion up to O(m5)
– Dashed line: fit by a Gaussian function
Well approximated
W q 
 q2
e
p
iq F
e e
 e
 q2
F
2 F
e
histogram of q
Technical problem 2: Overlap problem
Role of the weight factor exp(F+iq)
• The weight factor has the same effect as when  (T) increased.
• m*/T approaches the free quark gas value in the high density
limit for all temperature.
free quark gas
3

r
m 1 m 
 Nf   2   
3
T
 T p  T  
free quark gas
free quark gas
Technical problem 2: Overlap problem
m* (r)

T
• Density of state method
W(P): plaquette distribution

exp F  iq P W ( P)  exp F
P

z0 exp F  iq W ( P)dP

 q2
P
exp F  iq PW ( P)dP
P
Same effect when  changes.  exp eff P W ( P)
linear for small P

2   W ( P)
for small P
linear for small P
Reweighting for T=6g-2
(Data: Nf=2 p4-staggared, mp/mr0.7, m=0)
W P' ,    DU detM  e
Nf
 S g ( )
Change: 1(T)
 P  P ' 
2(T)
Distribution:
 S   S  
W 1   W  2   e
W 1 
g
2
g
1
S g ( 2 )  S g (1 )  6 N site  2  1 P
Potential:
 ln W 1   6 Nsite2  1 P   ln W 2 
+
=
Effective  (temperature) for r0
eff
2
d F
d
q
1

P


 dP
2 dP

 1
P 
 N site

(r increases)  ( (T) increases)
Overlap problem, Multi- reweighting
Ferrenberg-Swendsen, PRL63,1195(1989)
• When the density increases,
the position of the importance
sampling changes.
• Combine all data by
multi- reweighting
Problem:
• Configurations do not cover all
region of P.
• Calculate only when <P> is near
the peaks of the distributions.
P
P 
P exp F  iq
exp F  iq
( T ,m  0 )
( T ,m  0 )
Plaquette value by multi-beta reweighting
peak position of the distribution
○ <P> at each 
Chemical potential vs density
• Approximations:
Nf=2 p4-staggered, 16 3  4 lattice
– Taylor expansion: ln det M
– Gaussian distribution: q
– Saddle point approximation
• Two states at the same mq/T
– First order transition at
T/Tc < 0.83, mq/T >2.3
•
•
•
•
m*/T approaches the free quark
gas value in the high density limit
for all T.
Solid line: multi-b reweighting
Dashed line: spline interpolation
Dot-dashed line: the free gas limit
Number density
Summary
• Effective potentials as functions of the plaquette value and the
quark number density are discussed.
• Approximation:
– Taylor expansion of ln det M: up to O(m6)
– Distribution function of q=Nf Im[ ln det M] : Gaussian type.
– Saddle point approximation (1/V expansion)
• Simulations: 2-flavor p4-improved staggered quarks with mp/mr
 0.7 on 163x4 lattice
– Existence of the critical point: suggested.
– High r limit: m/T approaches the free gas value for all T.
– First order phase transition for T/Tc < 0.83, mq/T >2.3.
• Studies near physical quark mass: important.
– Location of the critical point: sensitive to quark mass