1. Study of the Finite Density State based on SU(2) Lattice QCD
Lattice QCD at Finite Density - Personal View of Recent Progress -
Study of the Finite Density State based on SU(2) Lattice QCD
RIKEN,
*IMC at Hiroshima University
and **Tokuyama Woman’s College
Chiho NONAKA,
*Atsushi NAKAMURA and **Shin MUROYA
I would like talk about ‘’Study of the Finite Density State based on SU(2) Lattice
QCD’’.
August 9, 2002

2. Contents Introduction Recent studies of lattice QCD at finite density
Summary
High T, Low m
Low T, High m
Derivatives of m
Reweighting
Taylor expansion
Imaginary m
Two-color QCD
Finite Isospin
NJL

3. Introduction (1) Phase diagram Tricritical point
location of critical end point ?
Interesting phenomena
at finite density
color superconductivity
super fluid
color- flavor locking phase
Experiment
RHIC, GSI, JHF
Critical end point
2SC
CFL T m
RHIC
GSI, JHF
From here I move to my main topics.
Interesting phenomena such as color superconductivity an color flavor locked phase at finite density is analyzed from theoretical calculation.
Now the analyses at finite region is one of the hot topics.
From the experimental side, RHIC experiment has been working since
But in this experiment the high temperature and low density matter can be
created after heavy ion collision.
On the other hand, in these experiments, GSI, JHF the low temperature
High density matter will be created.
Though in even these experiment it will be difficult to hit this region, recently
The existence of the precursor of color super conductivity is discussed.
Now, how about lattice QCD analyses in finite density?
In three-color QCD, the calculation under finite chemical potential is very difficult
l because the action becomes complex and we can not performance the naïve
Monte Carlo simulation.
There are many trial calculation such as Glasgow method and so on, but most of them did not
work well.
Now we wait the break-through.
Ejiri –san discuss the phase diagram in the high temperature and low density region
which correspond to RHIC experiment.
And Fodor et al. discussed the phase diagram even in finite density.
In order to analyze in finite density with lattice calculation
QCD like theory
In SU(2) lattice QCD there is no difficulty in evaluating the fermion
determinant.
Because in SU(2) lattice QCD or finite isospin
the fermion action becomes real.
By using QCD like theory we expect the information of SU(3) physics at
Finite density.
We argue
Concerning SU(2) lattice QCD, Kogut and Sinclair work activitily.
1GeV
Neutron star core

4. Critical End Point Signature of the tricritical point at heavy ion
collisions
Stephanov, Rajagopal, and Shuryak, PRL81(1998)4816
Experiment ?
Critical end point
Tricritical point
First order transition
Second order transition
Fluctuation
Correlation length

5. Kitazawa et al (nucl-th/0111022) (PRD65(2002)091504)
Precursor
of CSC
This figure is taken from this reference.
They discussed the precursor of color superconductivity based on NJL model.
Even in GSI and JHF experiment it is difficult to hit just the color superconductivity in
phase-diagram.
However they proposed the existense of the precursor of color superconductivity in this
region.
Therefore we may be able to catch the signature of color superconductivity even if
we cannot reach the extreme condition where the color superconductivity realize.
Kitazawa et al (nucl-th/ ) (PRD65(2002)091504)
NJL model

6. k：hopping parameter quark mass
Introduction (2)
Introduction of chemical potential
Wilson fermion
Phase (sign) problem
k：hopping parameter quark mass
Next I explain the introduction of chemical potential in lattice QCD.
There are two types of fermion action, Kogut-Susskind action and Wilson fermion.
Here I explain it about Wilson fermion.
Partition function is given like this.
In the case of Wilson action the fermion action is written like this.
Here \kappa is called hopping parameter and is relation to quark mass.
The hopping parameter is smaller, the quark mass is larger.
At finite density
chemical potential is introduced here and here.

7. Introduction (3) Phase (Sign) problem High T, Low m
Derivative with respect to m ( m=0)
Reweighting
Taylor expansion
Imaginary m
Naive Monte Carlo Calculation
does not work !
Critical end point
2SC
CFL T m
RHIC
GSI, JHF
Neutron star core
High m, Low T
Finite isospin
Two-color QCD
NJL

8. PS meson screening mass
Low m (1)
Hadron mass response to chemical potential
S.Choe et al. (QCD-Taro collaboration PRD65:054501,2002)
Standard gauge + Staggered fermion T
1st response very small m
2nd response
PS meson screening mass

9. Allton et al. (Bielefeld-Swansea)
Low m (2)
Reweighting
Fodor and Katz
Allton et al. (Bielefeld-Swansea)
Taylor expansion at high T
and low m
Multi-parameter reweighting
technique

10. Low m (3) Fodor-Katz, JHEP03(2002)014 Standard gauge
+ Staggered fermion

11. Low m (4) Allton et al. (Bielefeld-Swansea) hep-lat/0204010 170 MeV
Improved action
+ Improved staggered fermion
ma=0.29

12. Low m (5) Imaginary Chemical Potential At small m Z(3) symmetry
deForcrand and Philipsen hep-lat/
(D’Elia and Lombardo hep-lat/ )
At small m
Z(3) symmetry
Standard gauge
+ Staggered fermion

13. deForcrand-Philipsen
Summary 1 (Low m)
Fodor-Katz
Allton et al.
deForcrand-Philipsen
Consistent !?

14. Allton et al. (Bielefeld-Swansea)
Summary 2 (Low m)
Limitation of method
Taylor expansion at small m
Phase
They do not work in
high m and low T region.
Allton et al. (Bielefeld-Swansea)
hep-lat/

15. Limitation of calculation
Recent studies of lattice QCD at finite density
Derivatives of m
Reweighting
Taylor expansion
Imaginary m
High T, Low m
Two-color QCD
Finite Isospin
Low T, High m
Limitation of calculation

16. Monte Carlo Calculations work !
High m (1)
Effective theory
Finite Isospin
Two-color QCD
Monte Carlo Calculations work !

17. High m (2) Finite Isospin High T, Low m: QCD –Taro PRD65:054501,2002
Kogut and Sinclair hep-lat/
Quench:Kogut and Sinclair hep-lat/ , Gupta hap-lat/
High T, Low m: QCD –Taro PRD65:054501,2002
Allton et al. (Bielefeld-Swansea) hep-lat/
Pion condensation Neutron star core
Charged pion condensate:
Standard gauge
+ Staggered fermion
m=0.025, 0.05

18. High m (3) Finite isospin T 1st m 1st T=0, 2nd
Kogut and Sinclair hep-lat/
Son and Stephanov,PRL86(2001)592
Wilson Line T m
1st
1st
T=0, 2nd

19. Analyses for Two-color QCD
SU(2) lattice gauge theory at
Nakamura (PLB140(1984)391)
The first calculation, Pseudo-Fermion Method
Hands,Kogut,Lombardo and Morrison (NPB558(’99)327)
Staggered fermion, HMC and Molecular dynamics
Hands,Montvay,Morrison,Oevers,Scorzato and
Skullerud , Eur.Phys.J. C17 (2000) 285 (hep-lat/ )
Staggered fermion, HMC and Two-Step Multi-Boson algorithm
Kogut, Toublan and Sinclair
PLB514 (2001) 77 (hep-lat/ )
Kogut, Sinclair, Hands and Morrison
PRD64(2001) (hep-lat/ )
Kogut, Toublan, and Sinclair
hep-lat/
Our group (hep-lat/001007, hep-lat/ , hep-lat/ )
Wilson fermion, Link-by-Link update
Now I present the analysis for SU(2) lattice QCD.
First calculation under SU(2) is done this one, uging, pseudo-fermion method.
Recently Hands, Kogut, Siclair work

20. SU(2) (1) The phase diagram of Four Flavor SU(2)
Kogut et al. PRD64(2001)094505(hep-lat/ )
Diquark condensation
Standard gauge
+ Staggered fermion
Kogut et al. did many works about SU(2) lattice QCD and finite isosipn density.
But here I introduce the result about diquark condensation.
In SU(2) QCD diquark is singlet.
Therefore they introduce this source term in fermion action in order to
evaluate the diquark condensation.
They measure this value by changing \lambda and extrapolate from finite
Value to zero.
Then they evaluate the diquark condensation.
Please see this reference in detail.

21. SU(2) (2)
Kogut et al. PLB514 (2001) 77 (hep-lat/ ), hep-lat/ 1
m=0.4 T
0.5
Diquark condensation 1 2
0.4
m=0.3 m T m
Standard gauge
+ Staggered fermion 1 b
1.5

22. SU(2) (3) Staggered fermion almost studies Wilson fermion
Muroya, Nakamura and Nonaka
hep-lat/001007, hep-lat/ , hep-lat/

23. Algorithm Evaluation of det W
Full QCD and Lattice QCD at finite density
Evaluation of det W needs high computer performance.
(quench approximation detW = 1 )
Exact Algorithm (det W)
１）　 by CG (Conjugate Gradient)
２）・Links in H(Hypercube) are updated by
Metropolis
　　・　　　 are obtained by Woodbury’s formula
３）go to the next Hypercube　　
Here I would like explain the algorithm, though this is slightly
In this calculation we use the exact algorithm in which fermion determinant
is evaluated exactly.
The clow of calculation is as follows,
First we focus on the a part of whole link, here. We call this hypercube.
In this region links in hypercube are updated by Metropolis.
And the new value of inverse of fermion matrix is obtaind by Woodbury’s
Formula.
Third, we go to next hypercube.
Using this procedure the whole links are updated link by link.
Therefore this calculation needs very high computer performance.
However we can easily achange the number of flavor of fermion.
Here I report the result of two flavor and three flavor.
whole link

24. Numerical Results hep-lat/0208006
Anti periodic
(b=0.7,Nf=2)
Iwasaki action + Wilson fermion
0.2
Anti periodic, periodic (spatial direction)
Lattice size
Nf=2,3
Phase survey
Baryon number density
Gluon energy density
Polyakov line
Polyakov line susceptibility
Meson, diquark propagator
qq,qq (confinement-deconfinement)
Polyakov line correlation
Gluon Propagator
0.18 k
mc ? 4
0.16 4
0.14 ma
periodic
(b=0.7,Nf=2)
0.2
Here I talk about the numerical results.
We use Iqasaki action for gauge action and Wilson fermion for fermion
action.
The physical phenomena should not change for the lattice size and boundary
condition.
Therefore we perform the calculation under anti-periodic boundary condition
And periodic boundary condition in spatial direction.
First we did phase survey in (\beta, \kappa, \mu) parameter space with
4 to forth lattise size.
Here we focus on the critical chemical potential from the thermodynamical quantities, such as baryon number density, gluon energy density and polyakov line.
This figure shows the calculation parameter in the case of the anti-periodic boundary condition.
And this figure shows the calculation parameter in the case of the periodic
boundary condition.
From this result we analyze the meson and diquark propagator and
force between quark and quark and anti-quark and quark.
0.18 k
0.16
0.14 ma

25. Thermodynamical Quantities
Baryon number density
Gluon energy density
This figure indicates the thermodinamical quantities such as
Baryon number density, gluon energy density and polyakov line
as a function of chemical quantities.
These values increase with chemical potential, respectivily.
We can not see the sharp increase in thease quqntities because
The lattice size is small.
Form this result we are nere to confinement deconfinement
Phase transition, these phisical quantities are not order parameter in the strict sense.

26. Thermodynamical Quantities4
Nf=2, 4
b = 0.7
Baryon number density
Gluon energy density ma k k ma
This figure shows thermodinamical quantities as a function of \kappa and \mu.
These physical quantities increase with chemical potential. But the dependence
on \kappa is small without the expectation value of Polyakov line.
From these results we determine the parameter set (\kappa, \mu) under
which we calculate the meson and diquark propagator, gluon propagator and
so on.
Form these figure we can not see the clear critical chamical potential.
Next I show the Polyakov line susceptibility.
Polyakov line k ma

27. Polyakov line Susceptibility
4 X8 3
Anti periodic (spatial direction) periodic (spatial direction)
k=0.160
0.0002
0.0001
This figure indicates the Polyakof line susceptibility as a function of
chemical potential in both case of anti-periodic boundary condition and
Periodic boundary condition.
Form this figure we can see the sharp peak.
For example, if the hopping parameter is equal to the peak is located
Around 0.8 and if the hopping parameter is equal to the peak is located
Around 0.7.
Because the large hopping parameter means small quark mass and
In the case of small quark mass the effect of chemical potential is large.
And the right part of figure shows the result under periodic boundary
condition. In this case we can see the similar peak.
This results does not depend on boundary condition.
0.4
0.8 ma

28. Propagator 4 X8 Anti-periodic meson Pseudoscalar(p) : Vector(r) :3
Anti-periodic
meson
Pseudoscalar(p) :
Vector(r) :
diquark
t: Pauli matrix
Next I show the meson and diquark propagator which show the
Pseudoscalar (b1b):
Scalar (b5b) :

29. Propagator
4 X8 3
Anti-periodic
From this figure we can see the

30. r at Finite Density 4 X8 Periodic , k = 0.1603
Periodic
, k = 0.160
Clear evidence of r meson decrease
at finite chemical potential !

31. r at Finite Chemical Potential
Peculiar behavior of r at finite density
Anti-perodic boundary condition a a
Mass of r becomes small !

32. r at Finite Chemical Potential
Peculiar behavior of r at finite density
Periodic boundary condition a a
Mass of r becomes small !
Comparison with effective theory & experiment

33. Effective Theory In-medium properties of hadrons in dense matter
G. E. Brown and M. Rho, Phys. Rev. Lett. (1991) 2720
Brown-Rho scaling
Yokokawa et al.(hep-ph/ )
Simultaneous softening of s and r mesons
associated with chiral restoration
Harada et al.(hep-ph/ )
Vector meson mass vanishing at m
effective field theory with hidden local symmetry
Hatsuda and Lee (PRC46,R34(1992))
QCD sum rule c
SU(3)
SU(2)

34. Larger enhancement at 40 AGeV compared to 158 AGeV
Compressed Baryonic Matter Workshop, May 13-16, 2002, GSI Darmstadt:
H. Appelshaeuser, Dileptons from Pb-Au Collisions at 40 AGeV

35. qq/qq Potential at Finite Density
Color averaged potential
qq potential
preliminary
No difference SU(2)
m=0.0
m=0.4
m=0.6
m=0.7
1.4
1.2 1 4 R 8

36. Polyakov Line Correlation
Color dependent potential
qq potential
SU(2) ; V1=Va, V3=Vs

37. Polyakov Line Correlation
Singlet
Triplet
preliminary V1 V3
0.1
m=0.0
m=0.4
m=0.6
m=0.7
0.1
m=0.0
m=0.4
m=0.6
m=0.7
-0.1
-0.1 4 8 4 8 R R

38. Gluon Propagator 8X4x12X4, Nf=3, k=0.170 preliminary Screening mass 1
Free
m=0.0
m=0.7
Confinement Deconfinement
Phase transition? 6 12 Z

39. Gluon Propagator 8X4X12X4, Nf=3, k=0.170 preliminary 10 1 m=0.0 m=0.4
0.01 6 12 Z

40. Medium effect on qq potential
Results
Phase survey in
Various lattice sizes, boundary conditions, Nf=2, 3
Thermodynamical quantities
Baryon number density, Gluon energy density,
Polyakov line, Polyakov line susceptibility
Confinement/deconfinement phase transition
Meson & diquark propagators
Meson & diquark spectroscopy
Clear evidence of the vector meson decrease at finite chemical potential
qq,qq (preliminary)
Polyakov line correlation
Gluon Propagator confinement – deconfinement
(k,m) parameter space
mc ?
Medium effect on qq potential

41. Summary High T, Low m Low T, High m Derivatives of m Reweighting
Taylor expansion
Imaginary m
Phase diagram
EoS
Effective theory
Limitation of
calculation T m
QGP
Hadron
Superfluid/
superconducting
critical point x
Two-color QCD
Finite Isospin
NJL
Phase diagram
Diquark condensation
Pion condensation
Behavior of vector meson

42. new algorithm, new method
Break - through
new algorithm, new method

43. 2003 年奈良へ !
世話人：国広　悌二，中村　純，初田　哲男

44. Kitazawa et al. hep-lat/020207255
vector interaction on
chiral and color superconducting
NJL model

45. Ratio of det W (1)
Numerical instability
k = 0.156, m = 0.8

46. Eigen Values of W
Phase transition ?

47. c c
m=0.2
m=0.2 L c
m=0.1

48. Algorithm (2) For a sub-set H(Hypercube) Woodbury’s formula Then
whole link

49. Screening of hot gluon A.Nakamura, I.Pushkina, T.Saito, S. Sakai
20X20X32X6
quench approximation

50. Polyakov Loop
Iwasaki action
k=0.150