Determining the masses of excited hadrons in QCD by ab initio calculations

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

2. Nonperturbative QCD QCD on Euclidean lattices: “quenched” approximation Quark propagators t Determining the masses of excited hadrons in QCD by ab initio calculations

3. 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 Determining the masses of excited hadrons in QCD by ab initio calculations

4. + + = + 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. Determining the masses of excited hadrons in QCD by ab initio calculations

5. Hadron masses: pion mres=0.002 GMOR BGR, Nucl.Phys. B677 (2004) Mp=280 MeV Determining the masses of excited hadrons in QCD by ab initio calculations

6. Interpolators and propagator analysis Propagator: sum of exponential decay terms: excited states (smaller t) ground state (large t) …such a fit is highly unstable! Previous attempts: biased estimators (Bayesian analysis), maximum entropy,... Determining the masses of excited hadrons in QCD by ab initio calculations

7. 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” Determining the masses of excited hadrons in QCD by ab initio calculations

8. 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! Determining the masses of excited hadrons in QCD by ab initio calculations

9. - = 2.2 1.2 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” Determining the masses of excited hadrons in QCD by ab initio calculations

10. Summarizing our setting • Lüscher-Weisz gauge action • Chirally improved fermions (+HYP smearing) • Lattice size • 123x24, a=0.148 fm, L~1.8 fm, mp ~ 300 MeV • 163x32, a=0.148 fm, L~2.4 fm, mp ~ 280 MeV • 203x32, a=0.119 fm, L~2.4 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 Determining the masses of excited hadrons in QCD by ab initio calculations

11. Mesons: type pseudoscalar vector Determining the masses of excited hadrons in QCD by ab initio calculations

12. Mesons: type pseudoscalar vector Determining the masses of excited hadrons in QCD by ab initio calculations

13. Nucleon (uud) ? Roper Level crossing (from + - + - to + - - +)? Chiral approach affects different states differently Determining the masses of excited hadrons in QCD by ab initio calculations

14. Sigma (uus) Determining the masses of excited hadrons in QCD by ab initio calculations

15. Delta (uuu) and Omega (sss) Determining the masses of excited hadrons in QCD by ab initio calculations

16. 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 Determining the masses of excited hadrons in QCD by ab initio calculations

17. Exit Determining the masses of excited hadrons in QCD by ab initio calculations