Colin Morningstar Carnegie Mellon University Quantum Fields in the Era of Teraflop Computing

1. Extracting excited-state energies with application tothe static quark-antiquark system and hadrons Colin Morningstar Carnegie Mellon University Quantum Fields in the Era of Teraflop Computing ZiF, University of Bielefeld, November 22, 2004 Excited states (C. Morningstar)

2. Outline • spectroscopy is a powerful tool for distilling key degrees of freedom • spectrum determination requires extraction of excited-state energies • will discuss how to extract excited-state energies from Monte Carlo estimates of correlation functions in Euclidean lattice field theory • application: gluonic excitations of the static quark-antiquark system • application: excitations in 3d SU(2) static source-antisource system • application: ongoing efforts of LHPC to determine baryon spectrum with an eye toward future meson, tetraquark, pentaquark systems Excited states (C. Morningstar)

3. Energies from correlation functions • stationary state energies can be extracted from asymptotic decay rate of temporal correlations of the fields (in the imaginary time formalism) • consider an action depending on a real scalar field • evolution in Heisenberg picture ( Hamiltonian) • spectral representation of a simple correlation function • assume transfer matrix, ignore temporal boundary conditions • focus only on one time ordering • extract and as (assuming and ) insert complete set of energy eigenstates (discrete and continuous) Excited states (C. Morningstar)

4. Fitting procedure • extraction of and done by correlated- fit using single exponential where represents the MC estimates of the correlation function with covariance matrix and model function is • uncertainties in fit parameters obtained by jackknife or bootstrap • fit must be done for time range for acceptable • can fit to sum of two exponentials to reduce sensitivity to • second exponential is garbage  discard! • fits using large numbers of exponentials with a Bayesian prior is one way to try to extract excited-state energies (not discussed here) Excited states (C. Morningstar)

5. Effective mass • the “effective mass” is given by • notice that (take ) • the effective mass tends to the actual mass (energy) asymptotically • effective mass plot is convenient visual tool to see signal extraction • seen as a plateau • plateau sets in quickly for good operator • excited-state contamination before plateau Excited states (C. Morningstar)

6. Reducing contamination • statistical noise generally increases with temporal separation • effective masses associated with correlation functions of simple local fields do not reach a plateau before noise swamps the signal • need better operators • better operators have reduced couplings with higher-lying contaminating states • recipe for making better operators • crucial to construct operators using smeared fields • spatially extended operators • large set of operators with variational coefficients Excited states (C. Morningstar)

7. Link variable smearing • link variables: add staples with weight, project onto gauge group • define • common 3-d spatial smearing • APE smearing • or new analytic stout link method (hep-lat/0311018) • iterate Excited states (C. Morningstar)

8. Quark field smearing • quark fields: gauge covariant smearing • tunable parameters • three-dimensional gauge covariant Laplacian defined by • uses the smeared links • square of smeared field is zero, like simple Grassmann field • preserves transformation properties of the quark field Excited states (C. Morningstar)

9. Unleashing the variational method • consider the correlation function of an operator given by a linear superposition of a set of operators • choose coefficients to minimize excited-state contamination • minimize effective mass at some early time separation • simply need to solve an eigenvalue problem • this is essentially a variational method! • yields the “best” operator by the above criterion • added benefit  other eigenvectors yield excited states!! Excited states (C. Morningstar)

10. Principal correlators • application of such variational techniques to extract excited-state energies was first described in Luscher, Wolff, NPB339, 222 (1990) • for a given correlator matrix they defined the principal correlators as the eigenvalues of where (the time defining the “metric”) is small • L-W showed that • so the principal effective masses defined by now tend (plateau) to the lowest-lying stationary-state energies Excited states (C. Morningstar)

11. Principal effective masses • just need to perform single-exponential fit to each principal correlator to extract spectrum! • can again use sum of two-exponentials to minimize sensitivity to • note that principal effective masses (as functions of time) can cross, approach asymptotic behavior from below • final results are independent of , but choosing larger values of this reference time can introduce larger errors Excited states (C. Morningstar)

12. Excitations of static quark potential • gluon field in presence of static quark-antiquark pair can be excited • classification of states: (notation from molecular physics) • magnitude of glue spin projected onto molecular axis • charge conjugation + parity about midpoint • chirality (reflections in plane containing axis) P,D,…doubly degenerate (L doubling) several higher levels not shown Juge, Kuti, Morningstar, PRL 90, 161601 (2003) Excited states (C. Morningstar)

13. Dramatis personae • the gluon excitation team Jimmy Juge Julius Kuti CM Mike Peardon ITP, Bern UC San Diego Carnegie-Mellon, Pittsburgh Trinity College, Dublin student: Francesca Maresca Excited states (C. Morningstar)

14. Initial remarks • viewpoint adopted: • the nature of the confining gluons is best revealed in its excitation spectrum • robust feature of any bosonic string description: • gaps for large quark-antiquark separations • details of underlying string description encoded in the fine structure • study different gauge groups, dimensionalities • several lattice spacings, finite volume checks • very large number of fits to principal correlators  web page interface set up to facilitate scrutining/presenting the results Excited states (C. Morningstar)

15. String spectrum • spectrum expected for a non-interacting bosonic string at large R • standing waves: with circular polarization • occupation numbers: • energies E • string quantum number N • spin projection • CP Excited states (C. Morningstar)

16. String spectrum (N=1,2,3) • level orderings for N=1,2,3 Excited states (C. Morningstar)

17. String spectrum (N=4) • N=4 levels Excited states (C. Morningstar)

18. Generalized Wilson loops • gluonic terms extracted from generalized Wilson loops • large set of gluonic operators  correlation matrix • link variable smearing, blocking • anisotropic lattice, improved actions Excited states (C. Morningstar)

19. Three scales • studied the energies of 16 stationary states of gluons in the presence of static quark-antiquark pair • small quark-antiquark separations R • excitations consistent with states from multipole OPE • crossover region • dramatic level rearrangement • large separations • excitations consistent with expectations from string models Juge, Kuti, Morningstar, PRL 90, 161601 (2003) Excited states (C. Morningstar)

20. Gluon excitation gaps (N=1,2) • comparison of gaps with Excited states (C. Morningstar)

21. Gluon excitation gaps (N=3) • comparison of gaps with Excited states (C. Morningstar)

22. Gluon excitation gaps (N=4) • comparison of gaps with Excited states (C. Morningstar)

23. Possible interpretation • small R • strong E field of -pair repels physical vacuum (dual Meissner effect) creating a bubble • separation of degrees of freedom • gluonic modes inside bubble (low lying) • bubble surface modes (higher lying) • large R • bubble stretches into thin tube of flux • separation of degrees of freedom • collective motion of tube (low lying) • internal gluonic modes (higher lying) • low-lying modes described by an effective string theory (Np/R gaps – Goldstone modes) Excited states (C. Morningstar)

24. 3d SU(2) gauge theory • also studied 11 levels in (2+1)-dimensional SU(2) gauge theory • levels labeled by reflection symmetry (S or A) and CP (g oru) ground state Excited states (C. Morningstar)

25. 3d SU(2) gauge theory • first excitation Excited states (C. Morningstar)

26. 2d SU(2) gauge theory • gap of first excitation above ground state Excited states (C. Morningstar)

27. 3d SU(2) gaps • comparison of gaps with Excited states (C. Morningstar)

28. 3d SU(2) gaps (N=4) • comparison of gaps with • large R results consistent with string spectrum without exception • fine structure less pronounced than 4d SU(3) • no dramatic level rearrangement between small and large separations Excited states (C. Morningstar)

29. Baryon blitz • charge from the late Nathan Isgur to apply these techniques to extract the spectrum of baryons (Hall B at JLab) • formed the Lattice Hadron Physics Collaboration (LHPC) in 2000 • current collaborators: Subhasish Basak, Robert Edwards, George Fleming, David Richards, Ikuro Sato, Steve Wallace • for spectrum, need large sets of extended operators  correlation matrix techniques • since large sets of operators to be used and to facilitate spin identification, we shunned the usual method of operator construction which relies on continuum space-time constructions • focus on constructing operators which transform irreducibly under the symmetries of the lattice Excited states (C. Morningstar)

30. Three stage approach • concentrate on baryons at rest (zero momentum) • operators classified according to the irreps of • (1) basic building blocks: smeared, covariant-displaced quark fields • (2) construct elemental operators (translationally invariant) • flavor structure from isospin, color structure from gauge invariance • (3) group-theoretical projections onto irreps of • wrote Grassmann package in Maple to do these calculations Excited states (C. Morningstar)

31. Incorporating orbital and radial structure • displacements of different lengths build up radial structure • displacements in different directions build up orbital structure • operator design minimizes number of sources for quark propagators • useful for mesons, tetraquarks, pentaquarks even! Excited states (C. Morningstar)

32. Spin identification and other remarks • spin identification possible by pattern matching • total numbers of operators is huge  uncharted territory • ultimately must face two-hadron states total numbers of operators assuming two different displacement lengths Excited states (C. Morningstar)

33. Preliminary results • principal effective masses for small set of 10 operators Excited states (C. Morningstar)

34. Summary • discussed how to extract excited-state energies in lattice field theory simulations • studied energies of 16 stationary states of gluons in presence of static quark-antiquark pair as a function of separation R • unearthed the effective QCD string for R>2 fm for the first time • tantalizing fine structure revealedeffective string action clues • dramatic level rearrangement between small and large separations • showed similar results in 3d SU(2) • outlined our method for extracting the baryon spectrum Excited states (C. Morningstar)