1. 11/22/2004Excited states (C. Morningstar)1 Extracting excited-state energies with application to the 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

2. 11/22/2004Excited 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

3. 11/22/2004Excited 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)

4. 11/22/2004Excited 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)

5. 11/22/2004Excited 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

6. 11/22/2004Excited 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

7. 11/22/2004Excited 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

8. 11/22/2004Excited 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

9. 11/22/2004Excited 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!!

10. 11/22/2004Excited 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

11. 11/22/2004Excited 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

12. 11/22/2004Excited 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) ,…doubly degenerate (  doubling) Juge, Kuti, Morningstar, PRL 90, 161601 (2003) several higher levels not shown

13. 11/22/2004Excited states (C. Morningstar)13 Dramatis personae the gluon excitation team Jimmy JugeJulius Kuti CMMike Peardon ITP, BernUC San DiegoCarnegie-Mellon, Pittsburgh Trinity College, Dublin student: Francesca Maresca

14. 11/22/2004Excited 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

15. 11/22/2004Excited 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

16. 11/22/2004Excited states (C. Morningstar)16 String spectrum (N=1,2,3) level orderings for N=1,2,3

17. 11/22/2004Excited states (C. Morningstar)17 String spectrum (N=4) N=4 levels

18. 11/22/2004Excited 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

19. 11/22/2004Excited 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)

20. 11/22/2004Excited states (C. Morningstar)20 Gluon excitation gaps (N=1,2) comparison of gaps with

21. 11/22/2004Excited states (C. Morningstar)21 Gluon excitation gaps (N=3) comparison of gaps with

22. 11/22/2004Excited states (C. Morningstar)22 Gluon excitation gaps (N=4) comparison of gaps with

23. 11/22/2004Excited 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 (N  /R gaps – Goldstone modes)

24. 11/22/2004Excited 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 or u) ground state

25. 11/22/2004Excited states (C. Morningstar)25 3d SU(2) gauge theory first excitation

26. 11/22/2004Excited states (C. Morningstar)26 2d SU(2) gauge theory gap of first excitation above ground state

27. 11/22/2004Excited states (C. Morningstar)27 3d SU(2) gaps comparison of gaps with

28. 11/22/2004Excited 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

29. 11/22/2004Excited 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

30. 11/22/2004Excited 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

31. 11/22/2004Excited 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!

32. 11/22/2004Excited 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

33. 11/22/2004Excited states (C. Morningstar)33 Preliminary results principal effective masses for small set of 10 operators

34. 11/22/2004Excited 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