1. QCD Thermodynamics at fixed lattice scale
Takashi Umeda (Univ. of Tsukuba)
for WHOT-QCD Collaboration
This talk is based on arXiv: [hep-lat]
T. Umeda, S. Ejiri, S. Aoki, T. Hatsuda, K. Kanaya,Y. Maezawa, H. Ohno
(WHOT-QCD Collaboration)
ATHIC2008, Univ. of Tsukuba, Ibaraki, Japan, Oct. 2008
ATHIC2008
T.Umeda (Tsukuba)
/15

2. Introduction Equation of State (EOS)
is important for phenomenological study of QGP, etc.
Methods to calculate the EOS have been established,
e.g. Integral method J. Engels et al. (’90).
Temperature T=1/(Nta) is varied by a(β) at fixed Nt
The EOS calculation requires huge computational cost,
in which T=0 calculations dominate despite T>0 study.
Search for a Line of Constant Physics (LCP)
beta functions at each temperature
T=0 subtraction at each temperature
Most of this talk is not so recent studies
ATHIC2008
T.Umeda (Tsukuba)
/15

3. Recent lattice calculations for Tc
RBC-Bielefeld: Nt=4,6, Staggered (p4) quark
pion mass ≥ 140MeV, Nf=2+1
MILC: Nt=4,6, Staggered (Asqtad) quark
pion mass ≥ 220MeV, Nf=2+1
Wuppertal: Nt=4,6,8,10 Staggered (stout) quark
pion mass ~ 140MeV, Nf=2+1
DIK: Nt=8,10,12 Wilson (NPI Clover) quark
pion mass ≥ 500MeV, Nf=2
WHOT-QCD: Nt=4,6 Wilson (MFI Clover) quark
ATHIC2008
T.Umeda (Tsukuba)
/15

4. Recent lattice calculations for EOS
RBC-Bielefeld: Nt=4,6, Staggered (p4) quark
pion mass ~ 220MeV, Nf=2+1
MILC: Nt=4,6, Staggered (Asqtad) quark
pion mass ~ 220MeV, Nf=2+1
Wuppertal: Nt=4, Staggered (stout) quark
pion mass ~ 140MeV, Nf=2+1
CP-PACS: Nt=4,6 Wilson (MFI Clover) quark
pion mass ~ 500MeV, Nf=2
There are problems in Staggered quark formulations
- Flavor symmetry violation
- Rooted Dirac operator
- etc.
Wilson type quark results are important !!!
ATHIC2008
T.Umeda (Tsukuba)
/15

5. T-integration method to calculate the EOS
We propose a new method (“T-integration method”)
to calculate the EOS at fixed scales (*)
Temperature T=1/(Nta) is varied by Nt at fixed a(β)
Most of this talk is not so recent studies
Our method is based on the trace anomaly (interaction measure),
and the thermodynamic relation.
(*) fixed scale approach has been adopted in L.Levkova et al. (’06)
whose method is based on the derivative method.
ATHIC2008
T.Umeda (Tsukuba)
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6. Notable points in T-integration method
Our method can reduce computational cost at T=0 drastically.
Zero temperature subtraction is performed
using a common T=0 calculation.
Line of Constant Physics (LCP) is trivially exact (even in full QCD).
Only the beta functions at the simulation point are required.
However ...
Temperatures are restricted by integer Nt.
 Sufficiently fine lattice is necessary.
Most of this talk is not so recent studies
Example of Temp. resolution (a=0.07fm)
Integer Nt provides
- higher resolution at T~Tc
- lower resolution at high T
T~Tc is important for EOS
ATHIC2008
T.Umeda (Tsukuba)
/15

7. Simulation parameters (isotropic lattices)
We present results from SU(3) gauge theory as a test of our method
plaquette gauge action on Ns3 x Nt lattices
Jackknife analysis with appropriate bin-size
To study scale- & volume-dependence,
we prepare 3-type of lattices.
Most of this talk is not so recent studies
(1) β=6.0, V=(16a)3
a=0.094fm
(2) β=6.0, V=(24a)3
a=0.094fm
(3) β=6.2, V=(22a)3
a=0.078fm
ATHIC2008
T.Umeda (Tsukuba)
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8. Simulation parameters (anisotropic lattice)
Anisotropic lattice is useful to increase Temp. resolution,
we also test our method on an anisotropic lattice as≠ at
plaquette gauge action on Ns3 x Nt lattices
with anisotropy ξ=as/at=4
Most of this talk is not so recent studies
V=(20as)3
=(1.95fm)3
V=(30as)3
=(2.92fm)3
V=(40as)3
=(3.89fm)3
- critical temp.
β=6.1, ξ=4
V=(20as)3
=(1.95fm)3
as=0.097fm
- EOS calculation
- static quark
free energy
ATHIC2008
T.Umeda (Tsukuba)
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9. Trace anomaly ( e - 3p )/T4 on isotropic lattices
(1) β=6.0, a=0.094fm, V=(1.5fm)3
(2) β=6.0, a=0.094fm, V=(2.2fm)3
(3) β=6.2, a=0.068fm, V=(1.5fm)3
beta function : G.Boyd et al. (’96)
lattice scale r0 : R.Edwards et al. (’98)
Most of this talk is not so recent studies
Excellent agreement
between (1) and (3)
 scale violation is small
a=0.1fm is good
Finite volume effect
appears below & near Tc
 volume size is important
V=(2fm)3 is necessary.
dotted lines : cubic spline
ATHIC2008
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10. Trace anomaly ( e - 3p )/T4 on aniso. lattice
(1) ξ=4, as=0.097fm, V=(2.0fm)3
(2) ξ=1, a=0.094fm, V=(2.2fm)3
beta function : obtained by r0/as fit
r0/asdata H.Matsufuru et al. (’01)
Most of this talk is not so recent studies
is required
in SU(3) gauge theory.
T.R.Klassen (’98)
Anisotropic lattice is useful
to increase Temp. resolution.
dotted lines : cubic spline
ATHIC2008
T.Umeda (Tsukuba)
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11. Pressure & Energy density
Integration
is performed with the cubic
spline of (e-3p)/T4
Cubic spline vs trapezoidal inte.
yields small difference ~ 1σ
Our results are roughly
consistent with previous results.
Unlike the fixed Nt approach,
scale/temp. is not constant.
 Lattice artifacts increase
as temperature increases.
Most of this talk is not so recent studies
ATHIC2008
T.Umeda (Tsukuba)
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12. Transition temperature at fixed scale
T-dependence of
the (rotated) Polyakov loop
and its susceptibility
No renormalization is
required upto overall factor
due to the fixed scale.
Rough estimation of
critical temperature
is possible.
Tc = 280~300 MeV
at β=6.1, ξ=4
(SU(3) gauge theory)
Most of this talk is not so recent studies
ATHIC2008
T.Umeda (Tsukuba)
/15

13. Static quark free energy at fixed scale
Static quark free energies
at fixed scale
color singlet
static quark free energy V(r)
Due to the fixed scale,
no renomalization constant
is required.
 small thermal effects in V(r)
at short distance
(without any matching)
Easy to distinguish
temperature effect of V(r)
from scale & volume effects
Most of this talk is not so recent studies
ATHIC2008
T.Umeda (Tsukuba)
/15

14. Conclusion We studied thermodynamics of SU(3) gauge theory
at fixed lattice scale
Our method ( T-integration method ) works well
to calculate the EOS
Fixed scale approach is also useful for
- critical temperature
- static quark free energy
- etc.
Our method is also available in full QCD !!
Therefore ...
Most of this talk is not so recent studies
ATHIC2008
T.Umeda (Tsukuba)
/15

15. Toward full QCD calculations
Our method is suited for
already performed high statistics full QCD results.
When beta functions are (able to be) known at a simulation point
and T=0 configurations are open to the public,
our method requires no additional T=0 simulation !!
We are pushing forward in this direction
using CP-PACS/JLQCD results in ILDG
(Nf=2+1 Clover+RG, a=0.07fm, pion mass ~ 500MeV)
Our final goal is to study
thermodynamics on the physical point (pion mass ~ 140MeV)
with 2+1 flavors of Wilson quarks
Most of this talk is not so recent studies
ATHIC2008
T.Umeda (Tsukuba)
/15

16. Pressure & Energy density
Most of this talk is not so recent studies
ATHIC2008
T.Umeda (Tsukuba)
/15

17. Pressure & Energy density
G.Boyd et al. (’96)
Most of this talk is not so recent studies
ATHIC2008
T.Umeda (Tsukuba)
/15

18. Simulation parameters (isotropic lattices)
We present results from SU(3) gauge theory as a test of our method
plaquette gauge action on Nσ3 x Nτ lattices
Jackknife analysis with appropriate bin-size
To study scale- & volume-dependence,
we prepare 3-type of lattices.
Most of this talk is not so recent studies
ATHIC2008
T.Umeda (Tsukuba)
/15

19. Pressure & Energy density
Integration
is performed with the cubic
spline of (e-3p)/T4
Our results are roughly
consistent with previous results.
-- mild scale violation
-- Large volume is important
Unlike the fixed Nτ approach,
scale/temp. is not constant.
 Lattice artifacts increase
as temperature increases.
Most of this talk is not so recent studies
ATHIC2008
T.Umeda (Tsukuba)
/15

20. EOS on an anisotropic lattice
beta function : obtained by r0/aσ fit
r0/aσdata H.Matsufuru et al. (’01)
Anisotropic lattice is useful
to increase Temp. resolution.
Results are roughly consistent
with previous & isotropic results
Additional coefficients are
required to calculate (e-3p)/T4
Most of this talk is not so recent studies
is required
in SU(3) gauge theory.
T.R.Klassen (’98)
ATHIC2008
T.Umeda (Tsukuba)
/15

21. EOS on an anisotropic lattice
beta function : obtained by r0/aσ fit
r0/aσdata H.Matsufuru et al. (’01)
G.Boyd et al. (’96)
Anisotropic lattice is useful
to increase Temp. resolution.
Results are roughly consistent
with previous & isotropic results
Additional coefficients are
required to calculate (e-3p)/T4
Most of this talk is not so recent studies
is required
in SU(3) gauge theory.
T.R.Klassen (’98)
ATHIC2008
T.Umeda (Tsukuba)
/15

22. Introduction
Recent calculations for Finite temperature QCD (EOS calculations)
RBC-Bielefeld: Nt=4,6, Staggered (p4) quark
pion mass ~ 220MeV, Nf=2+1
MILC: Nt=4,6, Staggered (Asqtad) quark
pion mass ~ 220MeV, Nf=2+1
Wuppertal: Nt=4, Staggered (stout) quark
pion mass ~ 140MeV, Nf=2+1
DIK: None
WHOT-QCD: Nt=4,6 Wilson (MFI Clover) quark
pion mass ~ 500MeV, Nf=2
ATHIC2008
T.Umeda (Tsukuba)