1. 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) [arXiv: ]
Canonical partition function and finite density phase transition in lattice QCD
Physical Review D 78 (2008) [arXiv: ]
Tools for finite density QCD, Bielefeld, November 19-21, 2008

2. QCD thermodynamics at m0
Interesting properties of QCD
Measurable in heavy-ion collisions
Critical point at finite density
Location of the critical point ?
Distribution function of plaquette value
Distribution function of quark number density
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)
quark-gluon plasma phase
hadron phase
RHIC
SPS
AGS
color super conductor?
nuclear matter mq T

3. Effective potential of plaquette
Physical Review D77 (2008) [arXiv: ]

4. Effective potential of plaquette Veff(P) Plaquette distribution function (histogram)
SU(3) Pure gauge theory
QCDPAX, PRD46, 4657 (1992)
First order phase transition
Two phases coexists at Tc
e.g. SU(3) Pure gauge theory
Gauge action
Partition function
histogram
histogram
Effective potential

5. Problem of complex quark determinant at m0
Problem of Complex Determinant at m0
Boltzmann weight: complex at m0
Monte-Carlo method is not applicable.
Configurations cannot be generated.
(g5-conjugate)

6. 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)
(Weight factor at m=0)
(Reweight factor)
R(P,m): independent of b,  R(P,m) can be measured at any b.
Effective potential:
1st order phase transition?
m=0 crossover
non-singular + = ?

7. Reweighting for b(T) and curvature of –lnW(P)
Change: b1(T) b2(T)
Weight:
Potential: + =
Curvature of –lnW(P) does not change.
Curvature of –lnW(P,b) : independent of b.

8. m-dependence of the effective potential
Critical point
Crossover + =
m= reweighting
Curvature: Zero
1st order phase transition + =
m= reweighting
Curvature: Negative

9. Sign problem and phase fluctuations
Complex phase of detM
Taylor expansion: odd terms of ln det M (Bielefeld-Swansea, PRD66, (2002))
|q| > p/2: Sign problem happens.
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
q: NOT in the range of [-p, p]
histogram of q

10. Effective potential at m0 (S.E., Phys.Rev.D77, 014508(2008))
Results of Nf=2 p4-staggared, 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 b increases at m=0.
at m=0
Solid lines: reweighting factor at finite m/T, R(P,m)
Dashed lines: reweighting factor without complex phase factor.

11. Truncation error of Taylor expansion
Solid line: O(m4)
Dashed line: O(m6)
The effect from 5th and 6th order term is small for mq/T  2.5.

12. Curvature of the effective potential
Nf=2 p4-staggared, mp/mr0.7
at mq=0 + = ?
Critical point:
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.

13. Physical Review D 78 (2008) 074507[arXiv:0804.3227]
Canonical approach
Physical Review D 78 (2008) [arXiv: ]

14. Canonical approach Canonical partition function
Effective potential as a function of the quark number N.
At the minimum,
First order phase transition: Two phases coexist.

15. First order phase transition line
In the thermodynamic limit,
Mixed state First order transition
Inverse Laplace transformation by Glasgow method
Kratochvila, de Forcrand, PoS (LAT2005) 167 (2005)
Nf=4 staggered fermions, lattice
Nf=4: First order for all r.

16. Canonical partition function
Fugacity expansion (Laplace transformation)
canonical partition function
Inverse Laplace transformation
Note: periodicity
Derivative of lnZ
Integral
Arbitrary m0
Integral path, e.g.
1, imaginary m axis
2, Saddle point

17. Saddle point approximation (S.E., arXiv:0804.3227)
Integral
Saddle point
Inverse Laplace transformation
Saddle point approximation (valid for large V, 1/V expansion)
Taylor expansion at the saddle point.
At low density: The saddle point and the Taylor expansion coefficients can be estimated from data of Taylor expansion around m=0.
Saddle point:

18. Saddle point approximation
Canonical partition function in a saddle point approximation
Chemical potential
Saddle point:
saddle point
reweighting factor
Similar to the reweighting method
(sign problem & overlap problem)

19. Saddle point in complex m/T plane
Find a saddle point z0 numerically for each conf.
Two problems
Sign problem
Overlap problem

20. Technical problem 1: Sign problem
Complex phase of detM
Taylor expansion (Bielefeld-Swansea, PRD66, (2002))
|q| > p/2: Sign problem happens.
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
q: NOT in the range [-p, p]
histogram of q

21. Technical problem 2: Overlap problem Role of the weight factor exp(F+iq)
The weight factor has the same effect as when b (T) increased.
m*/T approaches the free quark gas value in the high density limit for all temperature.
free quark gas
free quark gas
free quark gas

22. Technical problem 2: Overlap problem
Density of state method
W(P): plaquette distribution
Same effect when b changes.
for small P
linear for small P
linear for small P

23. Reweighting for b(T)=6g-2
(Data: Nf=2 p4-staggared, mp/mr0.7, m=0)
Change: b1(T) b2(T)
Distribution:
Potential: + =
Effective b (temperature) for r0
(r increases)  (b (T) increases)

24. Overlap problem, Multi-b reweighting Ferrenberg-Swendsen, PRL63,1195(1989)
When the density increases, the position of the importance sampling changes.
Combine all data by multi-b reweighting
Problem:
Configurations do not cover all region of P.
Calculate only when <P> is near the peaks of the distributions.
Plaquette value by multi-beta reweighting
peak position of the distribution
○ <P> at each b P

25. Chemical potential vs density
Approximations:
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
Nf=2 p4-staggered, lattice
Number density

26. 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