1. Excited Hadrons: Lattice results Christian B. Lang Inst. F. Physik – FB Theoretische Physik Universität Graz Oberwölz, September 2006 B ern G raz R egensburg QCD collaboration PR D 73 (2006) 017502 ;[hep-lat/0511054] PR D 73 (2006) 094505 [ hep-lat/0601026] PR D 74 (2006) 014504; [hep-lat/0604019] In collaboration with T. Burch, C. Gattringer, L.Y. Glozman, C. Hagen, D. Hierl and A. Schäfer

2. C. B. Lang © 2006 Lattice simulation with Chirally Improved Dirac actions Quenched lattice simulation results:  Hadron ground state masses  p/K decay constants: f p =96(2)(4) MeV), f K =106(1)(8) MeV  Quark masses: m u,d =4.1(2.4) MeV, m s =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. C. B. Lang © 2006 Lattice simulation with Chirally Improved Dirac actions Quenched lattice simulation results:  Hadron ground state masses  p/K decay constants: f p =96(2)(4) MeV), f K =106(1)(8) MeV  Quark masses: m u,d =4.1(2.4) MeV, m s =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. C. B. Lang © 2006 Motivation Little understanding of excited states from lattice calculations Non-trivial test of QCD Classification! Role of chiral symmetry? It‘s a challenge…

5. C. B. Lang © 2006 Quenched Lattice QCD QCD on Euclidean lattices: Quark propagators t

6. C. B. Lang © 2006 Quenched Lattice QCD QCD on Euclidean lattices: Quark propagators t “quenched” approximation

7. C. B. Lang © 2006 Quenched Lattice QCD QCD on Euclidean lattices: Quark propagators t “quenched” approximation

8. C. B. Lang © 2006 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. C. B. Lang © 2006 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. C. B. Lang © 2006 GW-type Dirac operators Overlap (Neuberger) „Perfect“ (Hasenfratz et al.) Domain Wall (Kaplan,…) We use „Chirally Improved“ fermions Gattringer PRD 63 (2001) 114501 Gattringer /Hip/CBL., NP B697 (2001) 451 This is a systematic (truncated) expansion …obey the Ginsparg-Wilson relations approximately and have similar circular shaped Dirac operator spectrum (still some fluctuation!) +... + + = +

11. C. B. Lang © 2006 Quenched simulation environment Lüscher-Weisz gauge action Chirally improved fermions Spatial lattice size 2.4 fm Two lattice spacings, same volume:  20 3 x32 at a=0.12 fm  16 3 x32 at a=0.15 fm  (100 configs. each) Two valence quark masses (m u =m d varying, m s fixed) Mesons and Baryons

12. C. B. Lang © 2006 Usual method: Masses from exponential decay

13. C. B. Lang © 2006 Hadron masses: pion m res =0.002 M  =280 MeV GMOR BGR, Nucl.Phys. B677 (2004) (quenched)

14. C. B. Lang © 2006 Interpolators and propagator analysis Propagator: sum of exponential decay terms: Previous attempts: biased estimators (Bayesian analysis), maximum entropy,... Significant improvement: Variational analysis ground state (large t) excited states (smaller t)

15. C. B. Lang © 2006 Variational method Use several interpolators Compute all cross-correlations Solve the generalized eigenvalue problem Analyse the eigenvalues The eigenvectors are „fingerprints“ over t-ranges: For t>t 0 the eigenvectors allow to trace the state composition from high to low quark masses Allows to cleanly separate ghost contributions (cf. Burch et al.) (Michael Lüscher/Wolff)

16. C. B. Lang © 2006 Interpolating fields (I) Inspired from heavy quark theory: Baryons: (plus projection to parity) Mesons: i.e., different Dirac structure of interpolating hadron fields…..

17. C. B. Lang © 2006 Interpolating fields (II) are not sufficient to identify the Roper state However: …excited states have nodes! → smeared quark sources of different widths (n,w) using combinations like: nw nw, ww nnn, nwn, nww etc.

18. C. B. Lang © 2006 Mesons

19. C. B. Lang © 2006 „Effective mass“ example:mesons

20. C. B. Lang © 2006 Mesons: type pseudoscalarvector 4 interpolaters: ng 5 n, ng 4 g 5 n, ng 4 g 5 w, wg 4 g 5 w

21. C. B. Lang © 2006 Mesons: type pseudoscalarvector 4 interpolaters: ng 5 n, ng 4 g 5 n, ng 4 g 5 w, wg 4 g 5 w

22. C. B. Lang © 2006 Meson summary (chiral extrapolations)

23. C. B. Lang © 2006 Baryons

24. C. B. Lang © 2006 Nucleon (uud) Roper Level crossing (from + - + - to + - - +)? 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)

25. C. B. Lang © 2006 Masses (1)

26. C. B. Lang © 2006 Masses (2)

27. C. B. Lang © 2006 Eigenvectors: fingerprints Nucleon: Positive parity states ground state 1st excitation 2nd excitation

28. C. B. Lang © 2006 Mass dependence of eigenvector (at t=4) c 1 [w(nw)] c 1 [n(ww)] c 1 [w(ww)] c 3 [w(nw)] c 3 [n(ww)] c 3 [w(ww)]

29. C. B. Lang © 2006 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)

30. C. B. Lang © 2006 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) ? ?

31. C. B. Lang © 2006 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)

32. C. B. Lang © 2006 D (uuu ), W (sss) Positive/Negative parity: n(nn), w(nn), n(wn), w(nw), n(ww), w(ww) ? ?

33. C. B. Lang © 2006 Baryon summary (chiral extrapolations)

34. C. B. Lang © 2006 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:

35. C. B. Lang © 2006 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

36. C. B. Lang © 2006 Thank you