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Excited Hadrons: Lattice results

Oberwölz, September 2006. Excited Hadrons: Lattice results. 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]

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Excited Hadrons: Lattice results

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  1. Oberwölz, September 2006 Excited Hadrons: Lattice results 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

  2. Lattice simulation with Chirally Improved Dirac actions • Quenched lattice simulation results: • Hadron ground state masses • p/K decay constants: fp=96(2)(4) MeV), fK=106(1)(8) MeV • Quark masses: mu,d=4.1(2.4) MeV, ms=101(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)

  3. Lattice simulation with Chirally Improved Dirac actions • Quenched lattice simulation results: • Hadron ground state masses • p/K decay constants: fp=96(2)(4) MeV), fK=106(1)(8) MeV • Quark masses: mu,d=4.1(2.4) MeV, ms=101(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)

  4. Motivation • Little understanding of excited states from lattice calculations • Non-trivial test of QCD • Classification! • Role of chiral symmetry? • It‘s a challenge…

  5. Quenched Lattice QCD QCD on Euclidean lattices: Quark propagators t

  6. Quenched Lattice QCD QCD on Euclidean lattices: “quenched” approximation Quark propagators t

  7. Quenched Lattice QCD QCD on Euclidean lattices: “quenched” approximation Quark propagators t

  8. 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

  9. 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

  10. + + = + 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!)

  11. Quenched simulation environment • Lüscher-Weisz gauge action • Chirally improved fermions • Spatial lattice size 2.4 fm • Two lattice spacings, same volume: • 203x32 at a=0.12 fm • 163x32 at a=0.15 fm • (100 configs. each) • Two valence quark masses (mu=md varying, ms fixed) • Mesons and Baryons

  12. Usual method: Masses from exponential decay

  13. 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

  14. 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 t>t0 the eigenvectors allow to trace the state composition from high to low quark masses • Allows to cleanly separate ghost contributions (cf. Burch et al.)

  15. Interpolating fields (I) Inspired from heavy quark theory: Baryons: Mesons: • i.e., different Dirac structure of interpolating hadron fields….. (plus projection to parity)

  16. 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.

  17. Mesons

  18. „Effective mass“ example:mesons

  19. Mesons: type pseudoscalar vector 4 interpolaters: ng5n, ng4g5n, ng4g5w, wg4g5w

  20. Mesons: type pseudoscalar vector 4 interpolaters: ng5n, ng4g5n, ng4g5w, wg4g5w

  21. Meson summary (chiral extrapolations)

  22. Baryons

  23. 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)

  24. Masses (1)

  25. Masses (2)

  26. Mass dependence of eigenvector (at t=4) c1[w(nw)] c1[n(ww)] c1[w(ww)] c3[w(nw)] c3[n(ww)] c3[w(ww)]

  27. 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)

  28. 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)

  29. 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)

  30. D (uuu ), W(sss) ? ? Positive/Negative parity: n(nn), w(nn), n(wn), w(nw), n(ww), w(ww)

  31. Baryon summary (chiral extrapolations)

  32. Baryon summary (chiral extrapolations) • 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 Bold predictions:

  33. 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

  34. Thank you

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