Chiral condensate from lattice QCD with GinspargWilson fermions a mini review

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Chiral condensate from lattice QCD with
Ginsparg-Wilson fermions ? a mini review

• Shoji Hashimoto (KEK)
• _at_ KEK Theory Journal club 2006
• May 22, 2006

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Chiral symmetry breaking

• Chiral symmetry is spontaneously broken in the
  QCD vacuum quarks acquire masses of O(?QCD),
  which is the (dominant) source of masses of
  matters. pion as the Nambu-Goldstone boson, etc
• Can be shown in model calculations, such as the
  Nambu-Jona-Lasinio, but very hard to prove within
  QCD except for the numerical calculation on the
  lattice.
• The value of chiral condensate, order parameter
  of the chiral symmetry breaking, is usually
  estimated phenomenologically need lattice QCD to
  calculate from first-principles.

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Problems on the lattice

• Numerical proof of chiral symmetry breaking
  (and/or the calculation of chiral condensate) is
  not a standard practice in the lattice
  calculations, because
• For many years, there was no lattice fermion
  formulation having chiral symmetry
  (Nielsen-Ninomiya no-go theorem)
• Wilson fermion chiral symmetry is broken at
  finite a expected to recover in the continuum
  limit. But there are power divergences.
• Staggered fermion U(1) chiral symmetry out of
  U(4)xU(4) for four flavors of fermions.
• Ginsparg-Wilson fermion provides the
  breakthrough, but it is numerically costly so
  that the calculation is restricted in the
  quenched approximation.

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Outline

• Chiral condensate
• GMOR, Banks-Casher, etc
• Ginsparg-Wilson fermion
• General properties, implementations
• Extraction of chiral condensate
• pion masses, epsilon regime, eigenvalue
  distribution, ..
• Renormalization
• Non-perturbative renormalization.

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1. Chiral condensate
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Chiral condensate

• Chiral symmetry
• satisfied in the chiral (massless) limit
• Chiral condensate
• zero, perturbatively could become non-zero
  non-perturbatively (spantaneous symmetry
  breaking)
• Pion as the Nambu-Goldstone boson

Gell-Mann-Oakes-Renner (GMOR)
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Spectral density

• Dirac operator D, eigenmode uk(x)
• Spectral density
• Banks-Casher relation
• accumulation of near-zero modes in the
  thermodynamical limit (V?? first, then m?0)

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Chiral effective theory

• Assuming the chiral symmetry breaking, effective
  theory can be derived.
• Dynamics is dominated by pions effects of other
  excitations are encoded in the higher order
  terms.
• Parameters, such as f?, ?, are universal, i.e.
  same value in any environment as far as the
  effective theory is valid. There are many ways to
  extract ?.

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Ginsparg-Wilson fermion
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Ginsparg-Wilson relation

• Remnant of continuum chiral symmetry on the
  lattice (Ginsparg-Wilson, 1982)
• Lattice action is invariant under the modified
  chiral transformation if the Dirac operator
  satisfies the GW relation (Luscher, 1998)
• Exact chiral symmetry is realized at finite
  lattice spacings ( continuum like axial Ward
  identities are obtained).

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Consistent with anomaly

• Fermion measure is not invariant under the
  modified chiral transformation.
• Eigenvectors of the Dirac operator
• The trace of ?5 vanishes unless there are zero
  modes ?k0, which appears on topologically
  non-trivial gauge configurations (Atiyah-Singer
  index theorem)..

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Implementations

• Domain-wall fermion (Kaplan, Shamir)
• Fermion living in 5D (L4xLs) chiral modes appear
  on its 4D surface.
• Exact chiral symmetry in the limit of Ls??.
• Overlap fermion (Neuberger)
• 4D formulation build from the Wilson-Dirac
  kernel HW?5(DW?M) with large negative mass M.
• Need an evaluation of matrix sign function using
  polynomial/rational of HW.

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Extraction of chiral condensate
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Naïve calculation

• Naively, one may simply calculate
• But there are two problems
• UV divergence additive renormalization with
  power divergence a-3 exists with the
  Ginsparg-Wilson fermion, it can be exactly
  subtracted
• Chiral condensate is exactly zero (after the UV
  subtraction), when m?0 limit is taken first.
• Chiral symmetry is restored in the finite
  volume.

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Finite size scaling

• Hernandez, Jansen, Lellouch, PLB469 (1999) 198.
• Consider the ?-regime, which is characterized by
  M?Llt1. The partition function of ChPT is written
  as
• Chiral condensate is calculated for each
  topological sector ? (by Fourier transforming
  from ? to ?).
• ? can be extracted by looking at the m and V
  dependence of the condensate at each topological
  sector.

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Quenched artifact

• 1/m divergence in the non-trivial topology
  sectors due to the fermionic zero modes artifact
  of quenching.
• First observed by the RBC collaboration with the
  domain wall fermions Blum et al., PRD69 (2004)
  074502. But topological sectors are all averaged.

• additional complication due to finite Ls Ls
  dependence seen in the plot (Ls16 (circles), 24
  (squares)).

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Subtracting the zero modes

• Hernandez, Jansen, Lellouch, PLB469 (1999) 198.
• V84, 104, 124 from bottom to top.

• 1/mV term in the ??1 sector can be subtracted by
  taking the fermionic modes away.
• Volume scaling of the remaining term is shown ?
  can then be extracted.

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Low-lying eigenvalues

• Go one step further
• Random matrix theory is equivalent to ChPT in the
  ?-regime provides the distribution of lowest
  lying eigenvalues. (Damgaard, Nishigaki, Wettig,
  )

• Bietenholz, Shcheredin, Jansen, JHEP 07 (2003)
  033.
• Giusti, Luscher, Weisz, Wittig, JHEP 11 (2003)
  023.

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More to measure

• Kenji Ogawa, our former colleague until March,
  did further calculations in this direction
  (Ogawa, SH, PTP 114 (2005) 609.)
• Eigenvalue distribution
• Partition function
• Leutwyler-Smilga sum rules
• Topological suseptibility.
• Work extended to unquenched cases.

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Standard GMOR way

• Essentially, the measurement of pion mass, the
  common practice in the lattice calculation.
• Away from the ?-regime, the zero mode
  contribution is negligible one may subtract it
  by scalar correlater.
• Chiral extrapolation is necessary chiral log
  must be taken into account, otherwise less solid
  method.

• Giusti, Helbling, Rebbi, PRD64, 114508 (2001).
• Babich, Berruto, Garron, Hoelbling, Howard,
  Lellouch, Rebbi, Shoresh, JHEP 01 (2006) 086.

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4. Renormalization
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Lattice to continuum

• The scalar density operator is logarithmically
  divergent it cancels with the mass
  renormalization ZmZS-1.
• Matching is necessary to present the number in
  a given renormalization scheme, say the MSbar
  scheme.
• The matching can be done perturbatively
  (typically at the one-loop level)
• but the convergence is poor, zS1.355.
  Non-perturbative renormalization is desired.

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Non-perturbative renormalization

• Two (major) methods in the market
• RI-MOM (Martinelli et al., 1995)
• calculate quark (or gluon) n-point functions on
  the lattice in a fixed gauge.
• renormalization condition imposed in momentum
  space, then match with continuum perturbation
  theory.
• rely on the window where the perturbative
  calculation is applicable and, at the same time,
  the lattice artifact is small enough.
• Shrodinger functional (Luscher et al., 1991)
• calculate correlation functions in finite volume
  with an carefully chosen boundary condition
• renormalization condition imposed in coordinate
  space at a given point.
• step scaling towards large energy scale, then
  match onto the continuum perturbation theory.

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RG invariant quark mass

• Capitani, Luscher, Sommer, Wittig, NPB544 (1999)
  669.
• As an intermediate step, use the RG invariant
  quark mass (analogue of ?QCD)
• which is regularization and scheme independent.
• Its relation to the MSbar scheme is known to four
  loops

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Schrodinger functional

• Matching to be done
• Use the Schrodinger functional scheme in order to
  control the scaling (step scaling)
• The Z factor is given by

• Calculation is for the non-perturbatively
  O(a)-improved Wilson fermion.

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Once calculated

• Hernandez, Jansen, Lellouch, Wittig, JHEP 07
  (2001) 018.
• Once the Z factor is known in the continuum
  limit, one can use it as
• UM is the value of M at the reference value of
  mPS2.

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State-of-the-art

• Wennekers, Wittig, JHEP 09 (2005) 059.
• Bare condensate determined through the comparison
  with RMT, ??k???V??k?? for several different k
  and ?.
• ZSZm-1 is nonperturbatively determined following
  the previous work.
• Continuum limit scaling is good with the GW
  fermion.
• ?MS(2GeV)(285?9 MeV)3.
• with fK to set the scale.
• Still quenched.

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Summary

• Chiral condensate as an order parameter of chiral
  symmetry breaking in QCD can be unambiguously
  calculated using the GW fermions. Exact chiral
  symmetry is essential.
• numerical proof of chiral symmetry breaking
• Lowest order parameter of the chiral lagrangian
  extracted in various ways.
• Non-perturbative renormalization is possible.
• Next step go beyond the quenched approximation.