C' B' Lang, Graz

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Determining the masses of excited hadrons in QCD
by ab initio calculations
ÖPG Jahrestagung Wien, 30.9.2005

• C. B. Lang, Graz

Tommy Burch Christof Gattringer Leonid
Glozman Christian Hagen Dieter Hierl Andreas
Schäfer
PR D69 (2004) 094513 NP B PS 129/130 (2004)251
PR D70 (2004)054502 NPB 140 (2005) 284
(LAT04) NPA755 (2005)481 (BAR04) Lattice 2005
PoS(LAT2005)
BernGrazRegensburgQCD collaboration
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Nonperturbative QCD
QCD on Euclidean lattices
quenched approximation
Quark propagators
t
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Challenges

• Chirality
• Lattice chirality Ginsparg-Wilson fermions
• Chirally improved fermions
• How close to the physical limit (or even the
  chiral limit) can we come?
• The pion is special (it is a Goldstone boson)
• Interpolators
• Quantum numbers
• Coupling to the states ground state dominance
• Excited states?
• Role of chiral symmetry breaking and quenching

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What Dirac operator to use?
We use chirally improved fermions
approximate GW-fermions
systematic (truncated) expansion
Gattringer PRD 63 (2001) 114501 Gattringer et
al. Nucl. Phys. B697 (2001) 451
. . .
obeying the Ginsparg-Wilson relations
approximately.
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Hadron masses pion
mres0.002
GMOR
BGR, Nucl.Phys. B677 (2004)
Mp280 MeV
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Interpolators and propagator analysis
Propagator sum of exponential decay terms
ground state (large t)
excited states (smaller t)
such a fit is highly unstable! Previous
attempts biased estimators (Bayesian
analysis), maximum entropy,...
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Variational method
(C. Michael 1985, Lüscher Wolff 1990)

• Use several interpolating operators
• Compute all cross-correlations and then
• solve the (generalized) eigenvalue problem
• The spectral representation reads
• Eigenvectors are dominated by physical states

Eigenvalues masses!
Eigenvectors best overlap with physical states
- wave functions
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Which interpolating fields?
Inspired from heavy quark theory, e.g. for the
nucleon
(plus projections to parity)
But
are not sufficient to identify the Roper (cf.
Brömmel et al. PR D69 (2004) 094513 )
excited states have nodes!
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Smeared quark sources

• Smear the quark sources (Jacobi smearing)
• Allow for different smearing widths
• Combine different quark sources in the hadron
  operators
• Eigensystem analysis chooses best physical
  states

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Summarizing our setting

• Lüscher-Weisz gauge action
• Chirally improved fermions (HYP smearing)
• Lattice size
• 123x24, a0.148 fm, L1.8 fm, mp 300 MeV
• 163x32, a0.148 fm, L2.4 fm, mp 280 MeV
• 203x32, a0.119 fm, L2.4 fm, mp 280 MeV
• 100 configurations each
• Nucleons interpolators c1, c2, c3 each for
  6 quark source combinations (nnn), (nnw), (wnn),
  (nww), (wwn), (www)
• Other Baryons and Mesons analogously

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Mesons type
pseudoscalar
vector
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Mesons type
pseudoscalar
vector
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Nucleon (uud)
?
Roper
Level crossing (from - - to - - )?
Chiral approach affects different states
differently
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Sigma (uus)
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Delta (uuu) and Omega (sss)
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Conclusions

• Excited states are hard to identify
• Variational analysis allows to disentangle mixed
  (physical) states
• Interpolating fields allowing for nodal wave
  function- important for finding the excited
  states
• Contact with ChPT
• ghost adds complications
• quenched chiral logs not (yet) clearly seen
• All these techniques can be used for dynamical
  fermion results

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Exit
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Introduction/motivation the Roper state
Mathur et al. (Phys. Lett. B605 (2005)
137) Overlap action, Constrained Bayesian fits,
Quenched ghost analysis hN Roper / N(1535)
level switching window 0.2..0.4 GeV
Sasaki et al. (Prog.Theor.Phys. Suppl. 151 (2003)
143) Wilson action, maximum entropy method,
Roper / N(1535) level switching window 0.6..0.9
GeV
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Analysis of eigenvalues Operator content
lattice size 163x32
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Can we be sure about the correct identification
Roper and/or N(1710)??

• Roper
• is seen in c1 (like the nucleon) but only when
  different quark sources are combined allowing
  nodal wave function (the ground state nucleon is
  seen)
• is orthogonal to the nucleon as a separate state,
  with smaller amplitude
• is it contaminated by N(1710)?
• Continuing from the heavy quark region where
• Roper 56-plet has positive parity for any two
  quarks (seen in c1, but not seen in c2, which has
  negative parity two-quark content)
• N(1710) is in another multiplet containing
  positive and negative parity two-quark
  subsystems, and it is seen in c1 and in c2
• (n.b. The nucleon is seen in c1, but not in c2)