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Exploring Hadron Physics: Lattice Simulation Results in QCD

In this study from September 2006, Christian B. Lang presents excited hadron results utilizing lattice simulation with Chirally Improved Dirac actions. The research covers topics such as hadron masses, quark properties, pion interactions, and more.

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Exploring Hadron Physics: Lattice Simulation Results in QCD

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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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