Lattice QCD, Random Matrix Theory and chiral condensates

1
Lattice QCD, Random Matrix Theory and chiral
condensates

• JLQCD and TWQCD collaboration, Phys.Rev.Lett.98,17
  2001(2007) (hep-lat/0702003), arXiv0705.3322
  hep-lat to appear in Phys.Rev.D.
• Asahi shimbun, 25Apr07, Nikkei shimbun,30Apr07,
  Yomiuri shimbun 28May07
• Nature 47, 118 (10 May 2007), CERN COURIER,
  Vol47, number 5
• Hidenori Fukaya (RIKEN Wako)
• with S.Aoki, T.W.Chiu, S.Hashimoto, T.Kaneko,
  H.Matsufuru, J.Noaki, K.Ogawa,M.Okamoto,T.Onogi
  and N.Yamada
• JLQCD collaboration

2
1. Introduction

• Chiral symmetry
• and its spontaneous breaking are important !
• Mass gap between pion and the other hadrons
• pion as (pseudo) Nambu-Goldstone boson
• while the other hadrons acquire the mass ?QCD.
• Soft pion theorem
• Chiral phase transition at finite temperature
• Banks-Casher relation Banks Casher, 1980
• Chiral SSB is caused by Dirac zero-modes.

3
1. Introduction

• Chiral Random Matrix Theory (ChRMT)
• is an equivalent description of the moduli
  integrals of chiral perturbation theory.
• The spectrum of ChRMT is expected to match with
  the QCD Dirac spectrum ChRMT predicts
• Distribution of individual Dirac eigenvalues
• Spectral density
• Spectral correlation functions
• as functions of
  m, S and V,
• which is helpful to analyze lattice
  data.
• Shuryak Verbaarschot,1993,
• Verbaarschot Zahed,
  1993,Damgaard Nishigaki, 2001

4
1. Introduction

• Chiral Random Matrix Theory (ChRMT)
• For example, ChRMT knows finite V and m
  corrections in Banks-Casher relation
• In m-gt0 limit with V8
• ?(?) is flat and the height
• gives S (Banks-Casher).
• Finite m and V corrections
• are analytically known.
• Note the same S
• with finite V and m !

5
1. Introduction

• Lattice QCD
• is the most promising approach to confirm
  chiral SSB
• from 1-st principle calculation of QCD. But
• Chiral symmetry is difficult. Nielsen Ninomiya
  1981
• m ? 0 is difficult (large numerical cost).
• V ? 8 is difficult (large numerical cost).
• Therefore, the previous works were limited, with
• Dirac operator which breaks chiral symmetry.
  (Wilson or staggered fermions, Domain-wall
  fermion is better but still has the breaking
  effects 5MeV. )
• Heavier u-d quark masses 20-50MeV than real
  value a few MeV.
• ? needs unwanted operator mixing with opposite
  chirality, and m-gt0 extrapolations.
• ? large systematic errors.

6
1. Introduction

• This work
• We achieved 2-flavor lattice QCD simulations with
  exact chiral symmetry.
• The Ginsparg-Wilson relation -gt exact chiral
  symmetry.
• Lueschers admissibility condition -gt smooth
  gauge fields.
• On (1.8fm)4 lattice, achieved m3MeV !
• Finite V effects evaluated by ChRMT.
• m, V, Q dependences of QCD Dirac spectrum are
  calculated.
• A good agreement of Dirac spectrum with ChRMT.
• Strong evidence of chiral SSB from 1st principle.
• obtained

7
Contents

1. Introduction
2. QCD Dirac spectrum ChRMT
3. Lattice QCD with exact chiral symmetry
4. Numerical results
5. Conclusion

8
2. QCD Dirac spectrum ChRMT

• Banks-Casher relation
• In the free theory,
• ?(?) is given by the surface of S3 with the
  radius ?
• With the strong coupling
• The eigenvalues feel the repulsive force
  from each other?becoming non-degenerate? flowing
  to the low-density region around zero? results in
  the chiral condensate.

Banks Casher 1980
9
2. QCD Dirac spectrum ChRMT

• Chiral Random Matrix Theory (ChRMT)
• Consider the QCD partition function at a
  fixed topology Q,
• High modes (? gtgt ?QCD) -gt weak coupling
• Low modes (?ltlt ?QCD) -gt strong coupling
• ? Let us make an assumption For low-lying
  modes,
• with an unknown action V(?) ?
  ChRMT.

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2. QCD Dirac spectrum ChRMT

• Chiral Random Matrix Theory (ChRMT)
• Namely, we consider the partition function
  (for low-modes)
• Universality of RMT Akemann et al. 1997
• IF V(?) is in a certain universality class, in
  large n limit (n size of matrices) the low-mode
  spectrum is proven to be equivalent, or
  independent of the details of V(?) (up to a
  scale factor) !
• From the symmetry, QCD should be in the same
  universality class with the chiral unitary
  gaussian ensemble,
• and share the same
  spectrum, up to a overall

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2. QCD Dirac spectrum ChRMT

• Chiral Random Matrix Theory (ChRMT)
• In fact, one can show that the ChRMT is
  equivalent to the moduli integrals of chiral
  perturbation theory.
• The second term in the exponential is written as
• where
• Let us introduce Nf x Nf real matrix s1 and s2
  as

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2. QCD Dirac spectrum ChRMT

• Chiral Random Matrix Theory (ChRMT)
• Then the partition function becomes
• where is a NfxNf complex matrix.
• With large n, the integrals around the suddle
  point, which satisfies
• leaves the integrals over U(Nf) as
• equivalent to the ChPT modulis integral
  in the eregime.
• ?

13
2. QCD Dirac spectrum ChRMT

• Eigenvalue distribution of ChRMT
• Damgaard Nishigaki 2001 analytically derived
  the distribution of each eigenvalue of ChRMT.
• For example, in Nf2 and Q0 case, it is
• where and
• where
• -gt spectral density or correlation can be
  calculated, too.

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2. QCD Dirac spectrum ChRMT
Note We made a stronger assumption QCD -gt
ChRMT ChPT than usual, QCD
-gt ChPT

• Summary of QCD Dirac spectrum
• IF QCD dynamically breaks the chiral symmetry,
• the Dirac spectrum in finite V should look like

?
Banks-Casher S
Note Analytic solution is not known -gt
lets study lattice QCD!
?
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3. Lattice QCD with exact chiral symmetry

• The overlap Dirac operator
• We use Neubergers overlap Dirac operator
  Neuberger 1998
• (we take m0a1.6) satisfies the
  Ginsparg-Wilson 1982 relation
• realizes modified exact chiral symmetry on
  the lattice
• the action is invariant under
  Luescher 1998

• However, Hw?0 ( topology boundary ) is
  dangerous.
• D is theoretically ill-defined. Hernandez et al.
  1998
• Numerical cost is suddenly enhanced. Fodor et
  al. 2004

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3. Lattice QCD with exact chiral symmetry

• Lueschers admissibility condition Luescher
  1999
• In order to achieve Hw gt 0 Lueschers
  admissibility condition,
• we add topology stabilizing term Vranas
  2006, HF et al(JLQCD), 2006
• with µ0.2. Note Stop ?8 when Hw?0 and Stop?0
  when a?0.
• ( Note
• is extra Wilson fermion and twisted mass
  bosonic spinor with a cut-off
  scale mass. )

• With Stop, topological charge , or the index of
  D, is fixed along
• the hybrid Monte Carlo simulations -gt ChRMT at
  fixed Q.
• Ergodicity in a fixed topological sector ? -gt
  O.K.
• (Local fluctuation of topology is consistent
  with ChPT.)
• JLQCD, in preparation

17
3. Lattice QCD with exact chiral symmetry

• Sexton-Weingarten method
• Sexton Weingarten 1992, Hasenbusch, 2001
• We divide the overlap fermion determinant as
• with heavy m and performed finer (coarser)
  hybrid Monte Carlo step
• for the former (latter) determinant -gt factor
  4-5 faster.
• Other algorithmic efforts
• Zolotarev expansion of D -gt 10 -(7-8) accuracy.
• Relaxed conjugate gradient algorithm to invert D.
• Multishift conjugate gradient for the 1/Hw2.
• Low-mode projections of Hw.

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3. Lattice QCD with exact chiral symmetry

• Numerical cost
• Simulation of overlap fermion was thought to be
  impossible
• D_ov is a O(100) degree polynomial of D_wilson.
• The non-smooth determinant on topology boundaries
  requires extra factor 10 numerical cost.
• ? The cost of D_ov 1000 times of
  D_wilsons .
• However,
• Stop can cut the latter numerical cost 10
  times faster
• Stop can reduce the degree of polynomial 2-3
  times
• New supercomputer at KEK 60TFLOPS 50 times
• Many algorithmic improvements
  5-10 times
• We can overcome this difficulty !

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3. Lattice QCD with exact chiral symmetry

• Simulation summary
• On a 163 32 lattice with a 1.6-1.9GeV (L
  1.8-2fm), we
• achieved 2-flavor QCD simulations with the
  overlap quarks with
• the quark mass down to 3MeV.
  e-regime
• Note m gt50MeV with Wilson fermions in previous
  JLQCD works.
• Iwasaki (beta2.3,2.35) Stop(µ0.2) gauge
  action
• Quark masses ma0.002(3MeV) 0.1.
• 1 samples per 10 trj of Hybrid Monte Carlo
  algorithm.
• 5000 trj for each m are performed.
• Q0 topological sector (No topology change.)
• The lattice spacings a is calculated from quark
  potential(consistent with rho meson mass input).
• Eigenvalues are calculated by Lanzcos algorithm.
• (and projected to imaginary axis.)

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4. Numerical results

• In the following, we mainly focus on the data
  with m3MeV.
• Bulk spectrum
• Almost consistent with the Banks-Cashers
  scenario !
• Low-modes
• accumulation.
• The height
• suggests
• S (240MeV)3.
• gap from 0.
• ? need ChRMT analysis
• for the precise
• measurement of S !

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4. Numerical results

• Low-mode spectrum
• Lowest eigenvalues qualitatively agree with
  ChRMT.

22
4. Numerical results

• Low-mode spectrum
• Cumulative histogram
• is useful to compare the shape of the
  distribution.
• The width agrees with RMT within 2s.

23
4. Numerical results

• Heavier quark masses
• For heavier quark masses, 30160MeV, the
  good agreement
• with RMT is not expected, due to finite m effects
  
• of non-zero modes.
• But our data of the ratio of the eigenvalues
  still show a qualitative
• agreement.
• NOTE
• massless Nf2 Q0 gives
• the same spectrum with
• Nf0, Q2. (flavor-topology
• duality)
• m -gt large limit is
• consistent with QChRMT.

24
4. Numerical results

• Heavier quark masses
• However, the value of S, determined by the
  lowest-eigenvalue,
• significantly depend on the quark mass.
• But, the chiral limit is still consistent with
  the data with 3MeV.

25
4. Numerical results

• Renormalization
• Since S240(2)(6)3 is the lattice bare
  value, it should be
• renormalized.
• We calculated
• the renormalization factor in a non-perturbative
  RI/MOM scheme on the lattice,
• match with MS bar scheme, with the perturbation
  theory,
• and obtained

(non-perturbative)
(tree)
26
4. Numerical results

• Systematic errors
• finite m -gt small.
• As seen in the chiral extrapolation
• of S, m3MeV is very close to
• the chiral limit.
• finite lattice spacing a -gt O(a2) -gt (probably)
  small.
• the observables with overlap Dirac operator are
  automatically free from O(a) error,
• NLO finite V effects -gt 5-10.
• Higher eigenvalue feel pressure from bulk modes.
• higher k data are smaller than RMT. (5-10)
• 1-loop ChPT calculation also suggests 10 .

systematic
statistical
27
5. Conclusion

• We achieved lattice QCD simulations with
• exactly chiral symmetric Dirac operator,
• On (2fm)4 lattice, simulated Nf2 dynamical
  quarks with m3MeV,
• found a good consistency with Banks-Cashers
  scenario,
• compared with ChRMT where finite V and m effects
  are taken into account,
• found a good agreement with ChRMT,
• Strong evidence of chiral SSB from 1st principle.
• obtained

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5. Conclusion

• Future works
• Reduce the NLO V effects (or 1/N effects) of S.
• Larger lattices (prepared).
• NLO calculations of meson correlators in
  (partially quenched) ChPT. analytic part is
  done.
• P.H.Damgaard HF, arXiv0707.3740
• Hadron spectrum
• Test of ChPT (chiral log)
• Pion form factor
• pp0 difference
• BK
• Topological susceptibility
• 21 flavor simulations
• Finer and larger lattices

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6. NLO V effects (preliminary)

• Meson correlators compared with ChPT
• With a direct comparison of meson correlators
  with
• (partially quenched) ChPT, we obtain
• P.H.Damgaard HF, arXiv0707.3740
• where NLO V correction is taken into account.
• JLQCD, in preparation.

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6. NLO V effects (preliminary)

• Meson correlators compared with ChPT