QCD Thermodynamics at fixed lattice scale

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QCD Thermodynamics at fixed lattice scale
Takashi Umeda (Univ. of Tsukuba) for WHOT-QCD
Collaboration
This talk is based on arXiv0809.2842
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,
13-15 Oct. 2008
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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 T1/(Nta) is varied by a(ß) at
  fixed Nt
• The EOS calculation requires huge computational
  cost,
• in which T0 calculations dominate despite
  Tgt0 study.
• Search for a Line of Constant Physics (LCP)
• beta functions at each temperature
• T0 subtraction at each temperature

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Recent lattice calculations for Tc
RBC-Bielefeld Nt4,6,8 Staggered (p4)
quark pion mass 140MeV,
Nf21 MILC Nt4,6,8
Staggered (Asqtad) quark
pion mass 220MeV, Nf21 Wuppertal
Nt4,6,8,10 Staggered (stout) quark
pion mass 140MeV, Nf21 DIK
Nt8,10,12 Wilson (NPI Clover) quark
pion mass 500MeV,
Nf2 WHOT-QCD Nt4,6 Wilson (MFI
Clover) quark pion mass
500MeV, Nf2
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Recent lattice calculations for EOS
RBC-Bielefeld Nt4,6,8 Staggered (p4)
quark pion mass 220MeV,
Nf21 MILC Nt4,6,8
Staggered (Asqtad) quark
pion mass 220MeV, Nf21 Wuppertal
Nt4,6 Staggered (stout) quark
pion mass 140MeV, Nf21 CP-PACS
Nt4,6 Wilson (MFI Clover) quark
pion mass 500MeV,
Nf2 There are problems in Staggered quark
formulations - Flavor symmetry violation
- Rooted Dirac operator - etc. Wilson
types quark results are important !!!
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T-integration method to calculate the EOS
We propose a new method (T-integration method)
to calculate the EOS at fixed scales ()
Temperature T1/(Nta) is varied by Nt at fixed
a(ß)
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.
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Notable points in T-integration method

• Our method can reduce computational cost at T0
  drastically.
• Zero temperature subtraction is performed
• using a common T0 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.

Example of Temp. resolution (a0.07fm)
Integer Nt provides - higher resolution at
TTc - lower resolution at high T TTc is
important for EOS
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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.
(1) ß6.0, V(16a)3 a0.094fm
(2) ß6.0, V(24a)3 a0.094fm
(3) ß6.2, V(22a)3 a0.078fm
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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 ?as/at4

V(20as)3 (1.95fm)3 V(30as)3
(2.92fm)3 V(40as)3 (3.89fm)3 - critical
temp.
ß6.1, ?4 V(20as)3 (1.95fm)3
as0.097fm - EOS calculation - static quark
free energy
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Trace anomaly ( e - 3p )/T4 on isotropic
lattices
(1) ß6.0, a0.094fm, V(1.5fm)3 (2) ß6.0,
a0.094fm, V(2.2fm)3 (3) ß6.2, a0.068fm,
V(1.5fm)3
beta function G.Boyd et al. (96) lattice
scale r0 R.Edwards et al. (98)

• Excellent agreement
• between (1) and (3)
• ? scale violation is small
• a0.1fm is good
• Finite volume effect
• appears below near Tc
• ? volume size is important
• V(2fm)3 is necessary.

dotted lines cubic spline
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Trace anomaly ( e - 3p )/T4 on aniso. lattice
(1) ?4, as0.097fm, V(2.0fm)3 (2) ?1,
a0.094fm, V(2.2fm)3
beta function obtained by r0/as fit
r0/asdata H.Matsufuru et al. (01)

• Anisotropic lattice is useful
• to increase Temp. resolution.

dotted lines cubic spline
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Pressure Energy density

• Integration
• is performed with the cubic
• spline of (e-3p)/T4
• Cubic spline vs trapezoidal inte.
• yields small difference 1s
• Our results are roughly
• consistent with previous results.
• Unlike the fixed Nt approach,
• scale/temp. is not constant.
• ? Lattice artifacts increase
• as temperature increases.

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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 280300 MeV
• at ß6.1, ?4
• (SU(3) gauge theory)

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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

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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 ...

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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 T0 configurations are open to the
  public,
• our method requires no additional T0
  simulation !!
• We are pushing forward in this direction
• using CP-PACS/JLQCD results
  in ILDG
• (Nf21 CloverRG, a0.07fm,
  pion mass 500MeV)
• Our final goal is to study
• thermodynamics on the physical point
  (pion mass 140MeV)
• with 21 flavors of Wilson quarks

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Pressure Energy density
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Pressure Energy density
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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.
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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 Nt approach,
• scale/temp. is not constant.
• ? Lattice artifacts increase
• as temperature increases.

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EOS on an anisotropic lattice
beta function obtained by r0/as fit
r0/asdata 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

is required in SU(3) gauge theory.
T.R.Klassen (98)
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EOS on an anisotropic lattice
beta function obtained by r0/as fit
r0/asdata 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

is required in SU(3) gauge theory.
T.R.Klassen (98)
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