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

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Outline

• Introduction

• Lattice Formulation

• First Results from lattice QCD with Wilson and Neuberger Quarks

• Conclusion

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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.

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•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

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Satz’s and Aoki’s talks

Phase diagram of QCD at zero-density

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QCD Phase Diagram

Four fermion model: Alford, Wilczek, et al.,

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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')

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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

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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.

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•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.

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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

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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.)

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In the Lagrangian formulation, this does not work. The vacuum energy density is divergent!

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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

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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

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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

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First Results from four flavors of Wilson fermions:

at intermediate quark mass and T<TE

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First Results from four flavors of Wilson fermions:

at intermediate quark mass, finite T and real chemical potential

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First Results from four flavors of Wilson fermions:

at intermediate quark mass, finite T and real chemical potential

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First Results from four flavors of Wilson fermions:

at intermediate quark mass, finite T and real chemical potential

Nature of the transition

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First Results from four flavors of Wilson fermions:

at small or large quark mass and T<TE

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First Results from four flavors of Wilson fermions:

at finite T and real chemical potential

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First Results from two flavors of Wilson fermions:

at small quark mass and T<TE

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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 ×

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First Results from lattice QCD

with two flavors of Neuberger (Overlap) fermions

at finite temperature, real chemical potential and strong coupling

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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).

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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.

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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