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Lattice QCD at Non-Zero Temperature and Density with Wilson and Neuberger Quarks. Xiang-Qian Luo (with H.S. Chen, L.K. Wu, X.L. Yu) Zhongshan University, Guangzhou, China. Outline. Introduction Lattice Formulation First Results from lattice QCD with Wilson and Neuberger Quarks Conclusion. - PowerPoint PPT Presentation
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Lattice QCD at Non-Zero Temperature and Density
with Wilson and Neuberger Quarks
Xiang-Qian Luo(with H.S. Chen, L.K. Wu, X.L. Yu)
Zhongshan University, Guangzhou, China
X.Q. Luo 2
Outline
• Introduction
• Lattice Formulation
• First Results from lattice QCD with Wilson and Neuberger Quarks
• Conclusion
X.Q. Luo 3
I. Introduction
According to the big bang model in cosmology, the early universe underwent a series of drastic changes. For some time it was a hot and dense quark-gluon plasma (QGP), where quarks and gluons were deconfined. Today it is in a low temperature and low density hadronic phase, where quarks are confined.
X.Q. Luo 4
•RHIC (Relativistic Heavy Ion Collider)
•LHC (Large Hadron Collider)
is to create the QGP phase, and replay the birth and evolution of the Universe.
The ultimate goal of machines such as
X.Q. Luo 5
Satz’s and Aoki’s talks
Phase diagram of QCD at zero-density
X.Q. Luo 6
QCD Phase Diagram
Four fermion model: Alford, Wilczek, et al.,
X.Q. Luo 7
Plenary talks at this conference• June 18 Morning; Heavy-Ion & QCD Phases
8:30-9:05 H. Satz, Bielefeld Critical Behavior in QCD (35')
9:05-9:40 S. Aoki, University of Tsukuba QCD Phases in Lattice QCD (35')
9:40-10:15 T. Hatsuda, University of TokyoSignatures of Deconfinement and Chiral-Symmetry Restoration (35')
10:35-11:10 X. N. Wang, Lawrence Berkeley National Lab Probing the Strongly Interacting Quark-Gluon Plasma via Jet Quenching (35')
11:10-11:45 L. Mclarren, Brookhaven National Lab RHIC and New Forms of Matter (35')
11:45-12:20 J. W. Qiu, Iowa State University QCD Quantum Coherence in High-Energy Nuclear Collisions (35')
X.Q. Luo 8
Parallel talks at this conference• June 18 Afternoon (Lattice) 2:30---3:00 J.Verbaarschot (Stony Brook)
Chiral symmetry breaking at nonzero chemical potential • June 18 Afternoon (RHIC) 2:00---2:30 N. Xu (LBL)
Charm Production at RHIC 4:00—4:30 M. Huang (Tokyo U.)
Resolving instabilities in gapless color superconductor
X.Q. Luo 9
Quark
Gluon
II. Lattice Formulation
Lattice gauge theory (LGT) proposed by Wilson in 1974, is the most reliable technique for the investigation of phase transitions, from first principles.
X.Q. Luo 10
•Continuum Yang-Mills action
with β=6/g2
•replaced by the Wilson gluon action
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•Continuum quark action
•replaced by the discretized quark action
where M is the discretized fermionic matrix.
X.Q. Luo 12
Naïve fermions:
)(2
)()( 2
aOa
axax
dx
d
species doubling of fermion modes in the dispersion relation.
Continuum fermions Naiver fermions: wrong
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•No Go theorem: in any Local lattice theory with Chiral Symmetry, there exists species doubling of fermions.
•Any Solutions to No Go theorem must violate Locality or Chiral Symmetry.
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Kogut-Susskind (staggered) fermions: • doubling reduced by ¼. • flavor symmetry ×
•chiral symmetry (only partially)√ • local √ , but might be problematic in
Wilson fermions: •no doubling
•flavor symmetry√ •chiral symmetry × fine-tuning of the mass parameter has to be done•local √
Ginsparg-Wilson (e.g. Overlap fermions proposed by Neuberger): •no doubling
•flavor symmetry √
•chiral symmetry √
• locality × to expensive for dynamical fermions
4/det fNM
X.Q. Luo 15
X.Q. Luo 16
X.Q. Luo 17
III. QCD at Finite Temperature and Chemical Potential
In the Hamiltonian formulation of lattice QCD, this is well defined.
Greogry, Guo, Kroger, X.Q. Luo, Phys. Rev. D62 (2000) 054508.
Y. Fang, X.Q. Luo, Phys. Rev. D69 (2004) 114501.
X.Q. Luo, Phys. Rev. D70 ( 2004 ) 091504 (Rapid Commun.)
X.Q. Luo 18
In the Lagrangian formulation, this does not work. The vacuum energy density is divergent!
X.Q. Luo 19
5
5
† †5
† †5
( ) , ( 0)
( ) , ( 0)
M M
M M
So the fermionic determinant DetM is complex for any non-zero .
This avoids Monte Carlo simulation with importance sampling: another No Go theorem.
Unfortunately
X.Q. Luo 20
The recent years have seen enormous efforts on solving the complex action problem, and some very interesting information on the phase diagram for QCD with Kogut-Susskind (KS) fermions at large T and small μ has been obtained.
Improved reweighting
Imaginary chemical potential
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Lattice QCD with Imaginary Chemical Potential
X.Q. Luo 22
X.Q. Luo 23
Nf=2 of KS fermions
Nf=4 of KS fermions
Deconfinement phase transition
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First Results from four flavors of Wilson fermions
Wilson fermions: no doublingflavor symmetry√ chiral symmetry × fine-tuning of the mass parameter has to be donelocal √
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Polyakov loop
Chiral condensate
1
0
( ) [ ( )]tN
tt
P x Tr U x
( )
1
1[ ][ ][ ]
1[ ] ( )( ( ))
G F
f G
S S
N S
t
dU d d eZ
dU M U DetM U eZVN
First Results from four flavors of Wilson fermions
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First Results from four flavors of Wilson fermions: at TE<T
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The results above indicate that at higher T, there is Z(3) first order phase transition for QCD with Wilson quarks at imaginary chemical potential.
First Results from four flavors of Wilson fermions: at TE<T
Results above were obtained by scanning in this direction
Now we scan in this direction
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First Results from four flavors of Wilson fermions:
at intermediate quark mass and T<TE
X.Q. Luo 29
First Results from four flavors of Wilson fermions:
at intermediate quark mass and T<TE
X.Q. Luo 30
First Results from four flavors of Wilson fermions:
at intermediate quark mass, finite T and real chemical potential
X.Q. Luo 31
First Results from four flavors of Wilson fermions:
at intermediate quark mass, finite T and real chemical potential
X.Q. Luo 32
First Results from four flavors of Wilson fermions:
at intermediate quark mass, finite T and real chemical potential
Nature of the transition
X.Q. Luo 33
First Results from four flavors of Wilson fermions:
at small or large quark mass and T<TE
X.Q. Luo 34
First Results from four flavors of Wilson fermions:
at finite T and real chemical potential
X.Q. Luo 35
First Results from two flavors of Wilson fermions:
at small quark mass and T<TE
X.Q. Luo 36
First Results from lattice QCD
with two flavors of Neuberger (Overlap) fermions
at finite temperature, real chemical potential and strong coupling
Ginsparg-Wilson (e.g. Overlap fermions proposed by Neuberger): no doublingflavor symmetry √chiral symmetry √ locality ×
X.Q. Luo 37
First Results from lattice QCD
with two flavors of Neuberger (Overlap) fermions
at finite temperature, real chemical potential and strong coupling
X.Q. Luo 38
IV. Conclusion
Four Flavors
H.S. Chen, X.Q. Luo, "Phase diagram of QCD at finite temperature and chemical potential from lattice simulations with dynamical Wilson quarks,"
[hep-lat/0411023], to appear in Phys. Rev. D (2005).
X.Q. Luo 39
First results for QCD phase diagram from lattice QCD with two flavors of overlap (Neuberger) quarks at strong coupling:
Second order phase transition at large T and small μ
First order phase transition at large T and small μ
X.L. Yu, X.Q. Luo, to be submitted.
Two Flavors:
First and Preliminary results from MC simulations of lattice QCD for two flavor QCD with Wilson quarks at imaginary chemical potential: second order at small quark mass, first order at large quark mass.
H.S. Chen, X.Q. Luo, L.K. Wu, to be submitted.
X.Q. Luo 40
QCD Phase Diagram on the (T,μ) plane
from lattice QCD
Multi-dimensional reweighting: Fodor and Katz, …
Hamiltonian lattice QCD with Wilson quarks
X.Q. Luo, Phys. Rev. D70 ( 2004 ) 091504 (Rapid Commun.)
X.L. Yu, X.Q. Luo, Lagrangian lattice QCD with Overlap (Neuberger) quarks
Hamiltonian lattice QCD
Greogry, Guo, Kroger, X.Q. Luo, Phys. Rev. D62 (2000) 054508.Y. Fang, X.Q. Luo, Phys. Rev. D69 (2004) 114501.
Lagrangian Lattice QCD from Imaginary chemical potential method:
de Forcrand, Lombardo, H. Chen, X.Q. Luo, L.K. Wu, …
CPPACS
Bielefeld
X.Q. Luo et al, making efforts