Excited Hadrons: Lattice results

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Excited Hadrons Lattice results
Oberwölz, September 2006

• Christian B. Lang
• Inst. F. Physik FB Theoretische Physik
• Universität Graz

In collaboration with T. Burch, C. Gattringer,
L.Y. Glozman, C. Hagen, D. Hierl and A. Schäfer
PR D 73 (2006) 017502 hep-lat/0511054 PR D 73
(2006) 094505 hep-lat/0601026 PR D 74 (2006)
014504 hep-lat/0604019
BernGrazRegensburgQCD collaboration
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Lattice simulation with Chirally Improved Dirac
actions

• Quenched lattice simulation results
• Hadron ground state masses
• p/K decay constants
• fp96(2)(4) MeV), fK106(1)(8) MeV
• Quark masses
• mu,d4.1(2.4) MeV, ms101(8) MeV
• Light quark condensate -(286(4)(31) MeV)3
• Pion form factor
• Excited hadrons
• Dynamical fermions
• First results on small lattices

BGR (2004)
Gattringer/Huber/CBL (2005)
Capitani/Gattringer/Lang (2005)
CBL/Majumdar/Ortner (2006)
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Lattice simulation with Chirally Improved Dirac
actions

• Quenched lattice simulation results
• Hadron ground state masses
• p/K decay constants
• fp96(2)(4) MeV), fK106(1)(8) MeV
• Quark masses
• mu,d4.1(2.4) MeV, ms101(8) MeV
• Light quark condensate -(286(4)(31) MeV)3
• Pion form factor
• Excited hadrons
• Dynamical fermions
• First results on small lattice

BGR (2004)
Gattringer/Huber/CBL (2005)
Capitani/Gattringer/Lang (2005)
CBL/Majumdar/Ortner (2006)
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Motivation

• Little understanding of excited states from
  lattice calculations
• Non-trivial test of QCD
• Classification!
• Role of chiral symmetry?
• Its a challenge

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Quenched Lattice QCD
QCD on Euclidean lattices
Quark propagators
t
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Quenched Lattice QCD
QCD on Euclidean lattices
quenched approximation
Quark propagators
t
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Quenched Lattice QCD
QCD on Euclidean lattices
quenched approximation
Quark propagators
t
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The lattice breaks chiral symmetry

• Nogo theorem Lattice fermions cannot have
  simultaneously
• Locality, chiral symmetry, continuum limit of
  fermion propagator
• Original simple Wilson Dirac operator breaks the
  chiral symmetry badly
• Duplication of fermions, no chiral zero modes,
  spurious small eigenmodes (problems to simulate
  small quark masses)
• But the lattice breaks chiral symmetry only
  locally
• Ginsparg Wilson equation for lattice Dirac
  operators
• Is related to non-linear realization of chiral
  symmetry (Lüscher)
• Leads to chiral zero modes!
• No problems with small quark masses

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The lattice breaks chiral symmetry locally

• Nogo theorem Lattice fermions cannot have
  simultaneously
• Locality, chiral symmetry, continuum limit of
  fermion propagator
• Original simple Wilson Dirac operator breaks the
  chiral symmetry badly
• Duplication of fermions, no chiral zero modes,
  spurious small eigenmodes (problems to simulate
  small quark masses)
• But the lattice breaks chiral symmetry only
  locally
• Ginsparg Wilson equation for lattice Dirac
  operators
• Is related to non-linear realization of chiral
  symmetry (Lüscher)
• Leads to chiral zero modes!
• No problems with small quark masses

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GW-type Dirac operators

• Overlap (Neuberger)
• Perfect (Hasenfratz et al.)
• Domain Wall (Kaplan,)
• We use Chirally Improved fermions

This is a systematic (truncated) expansion
Gattringer PRD 63 (2001) 114501 Gattringer
/Hip/CBL., NP B697 (2001) 451
. . .
obey the Ginsparg-Wilson relations approximately
and have similar circular shaped Dirac operator
spectrum (still some fluctuation!)
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Quenched simulation environment

• Lüscher-Weisz gauge action
• Chirally improved fermions
• Spatial lattice size 2.4 fm
• Two lattice spacings, same volume
• 203x32 at a0.12 fm
• 163x32 at a0.15 fm
• (100 configs. each)
• Two valence quark masses (mumd varying, ms
  fixed)
• Mesons and Baryons

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Usual method Masses from exponential decay
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Hadron masses pion
mres0.002
GMOR
(quenched)
BGR, Nucl.Phys. B677 (2004)
Mp280 MeV
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Interpolators and propagator analysis
Propagator sum of exponential decay terms
excited states (smaller t)
ground state (large t)
Previous attempts biased estimators (Bayesian
analysis), maximum entropy,... Significant
improvement Variational analysis
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Variational method
(Michael Lüscher/Wolff)

• Use several interpolators
• Compute all cross-correlations
• Solve the generalized eigenvalue problem
• Analyse the eigenvalues
• The eigenvectors are fingerprints over
  t-ranges
• For tgtt0 the eigenvectors allow to trace the
  state composition from high to low quark masses
• Allows to cleanly separate ghost contributions
  (cf. Burch et al.)

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Interpolating fields (I)
Inspired from heavy quark theory Baryons
Mesons
i.e., different Dirac structure of interpolating
hadron fields..
(plus projection to parity)
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Interpolating fields (II)
However
are not sufficient to identify the Roper state
excited states have nodes!
? smeared quark sources of different widths
(n,w) using combinations like nw nw, ww nnn,
nwn, nww etc.
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Mesons
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Effective mass examplemesons
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Mesons type
pseudoscalar
vector
4 interpolaters ng5n, ng4g5n, ng4g5w, wg4g5w
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Mesons type
pseudoscalar
vector
4 interpolaters ng5n, ng4g5n, ng4g5w, wg4g5w
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Meson summary (chiral extrapolations)
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Baryons
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Nucleon (uud)
Level crossing (from - - to - - )?
Roper
Positive parity w(ww)(1,3), w(nw)(1,3) ,
n(ww)(1,3) Negative parity w(n,n)(1,2),
n(nn)(1,2)
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Masses (1)
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Masses (2)
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Eigenvectors fingerprints
Nucleon Positive parity states
ground state
1st excitation
2nd excitation
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Mass dependence of eigenvector (at t4)
c1w(nw) c1n(ww) c1w(ww)
c3w(nw) c3n(ww) c3w(ww)
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S (uus)
Positive parity w(ww)(1,3), w(nw)(1,3) ,
n(ww)(1,3) Negative parity w(n,n)(1,2),
n(nn)(1,2)
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X (ssu)
?
?
Positive parity w(ww)(1,3), w(nw)(1,3) ,
n(ww)(1,3) Negative parity w(n,n)(1,2),
n(nn)(1,2)
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L octet (uds )
Positive parity w(ww)(1,3), w(nw)(1,3) ,
n(ww)(1,3) Negative parity w(n,n)(1,2),
n(nn)(1,2)
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D (uuu ), W(sss)
?
?
Positive/Negative parity n(nn), w(nn), n(wn),
w(nw), n(ww), w(ww)
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Baryon summary (chiral extrapolations)
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Baryon summary (chiral extrapolations)
Bold predictions

• W 1st excited state, pos.parity 2300(70) MeV
• W ground state, neg.parity 1970(90) MeV
• X ground state, neg.parity 1780(90) MeV
• X 1st excited stated, neg.parity 1780(110) MeV

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Summary and outlook

• Method works
• Large set of basis operators
• Non-trivial spatial structure
• Ghosts cleanly separated
• Applicable for dynamical quark configurations
• Physics
• Larger cutoff effects for excited states
• Positive parity excited states too high
• Negative parity states quite good
• Chiral limit seems to affect some states strongly
• Further improvements
• Further enlargement of basis, e.g. p-wave sources
  (talk by C. Hagen) and non-fermionic
  interpolators (mesons)
• Studies at smaller quark mass

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Thank you