Hadron spectrum : Excited States, Multiquarks and Exotics

1. Hadron spectrum : Excited States, Multiquarks and Exotics Nilmani Mathur Department of Theoretical Physics, TIFR, INDIA

2. Baryons (3-quarks) Mesons (2-quarks) The Particle Zoo HADRON SPECTRUM …PDG

3. Can we explain these (at least)?

4. Proof of E=mc2 !! S.Durr et.al, Science 322, 1224 (2008) e=mc2: 103 years later, Einstein's proven right !! …….Times of India : 21 November 2008

5. A constituent picture of Hadrons M. Peardon’s talk

6. Type of Hadrons • Normal hadrons : • Two quark state (meson) • Three quark state (baryon) • Other Hadrons • Multiquarks • Exotics (hybrids) • Glueballs

7. Quark Jungle Gym quark propagators : Inverse of very large matrix of space-time, spin and color Quark (on Lattice sites) Gluon (on Links)

8. t

9. Pion two point function Nucleon interpolating operator

10. Correlator decays exponentially m1 m1, m2 Analysis (Extraction of Mass) Effective mass :

11. Correlator decays exponentially Analysis (Extraction of Mass) Assume that data has Gaussian distribution Uncorrelated chi2 fitting by minimizing m1 However, data is correlated and it is necessary to use covariance matrix m1, m2 How to extract m2 m3… : excited states? Non linear fitting. Variable projection method

12. Bayesian Fitting Priors

13. the conditional probability of measuring the data D given a set of parameters ρ Posterior probability distribution the conditional probability that ρ is correct given the measured data D Bayes’ theorem : Bayesian prior distribution P(D) prior predictive probability

14. Variational Analysis ψi : gauge invariant fields on a timeslice t that corresponds to Hilbert space operatorψjwhose quantum numbers are also carried by the states|n>. Construct a matrix • Need to find out variational coefficients • such that the overlap to a state is maximum • Variational solution  Generalized eigenvalue problem : • Eigenvalues give spectrum : • Eigenvectors give the optimal operator :

15. Importance of t0 • Basis of operators is only a part of the Hilbert space (n = 1,…N; N≠∝) • The eigenvectors are orthogonal only in full space. • Orthogonality is controlled by the metric C(t0) : • t0 should be chosen such that the NXN correlator matrix is dominated by the lightest N states at t0 • Excited states contribution falls of exponentially  go to large t0 • However, signal/noise ratio increases at large t0 • Choose optimum t0

16. Sommer : arXiv:0902.1265v2

18. Overlap Factor (Z)

20. Dudek et.al

22. What is a resonance particle? • Resonances are simply energies at which differential cross-section of a particle reaches a maximum. • In scattering expt. resonance  dramatic increase in cross-section with a corresponding sudden variation in phase shift. • Unstable particles but they exist long enough to be recognized as having a particular set of quantum numbers. • They are not eigenstates of the Hamiltonian, but has a large overlap onto a single eigenstates. • They may be stable at high quark mass. • Volume dependence of spectrum in finite volume is related to the two-body scattering phase-shift in infinite volume. • Near a resonance energy : phase shift rapidly passes through pi/2, an abrupt rearrangement of the energy levels known as avoided “level crossing” takes place.

23. Identifying a Resonance State • Method 1 : • Study spectrum in a few volumes • Compare those with known multi-hadron decay channels • Resonance states will have no explicit volume dependence whereas scattering states will have inverse volume dependence. • Method 2 : • Relate finite box energy to infinite volume phase shifts by Luscher formula • Calculate energy spectrum for several volumes to evaluate phase shifts for various volumes • Extract resonance parameters from phase shifts • Method 3 : • Collect energies for several volumes into momentum bin in energy histograms that leads to a probability distribution which shows peaks at resonance position. ….V. Bernard et al, JHEP 0808,024 (2008)

24. Multi-particle statesA problem for finite box lattice • Finite box : Momenta are quantized • Lattice Hamiltonian can have both resonance and decay channel states (scattering states) • A  x+y, Spectra of mA and • One needs to separate out resonance states from scattering states

25. Scattering state and its volume dependence Normalization condition requires : Continuum Two point function : Lattice For one particle bound state spectral weight (W) will NOT be explicitly dependent on lattice volume

26. Scattering state and its volume dependence Normalization condition requires : Continuum Two point function : Lattice For two particle scattering state spectral weight (W) WILL be explicitly dependent on lattice volume

27. C. Morningstar, Lat08

28. Solution in a finite box C. Morningstar, Lat08

29. Rho decay

30. ….V. Bernard et al, JHEP 0808,024 (2008)

31. Hybrid boundary condition • Periodic boundary condition on some quark fields while anti-periodic on others • Bound and scattering states will be changing differently.

32. S11(1535) _ _ Δ(1700) Λ(1670) + _ + + Roper (1440) Λ(1405) Δ(1600) + + + Nucleon (938) Λ(1116) Δ(1236) Color-Spin Interaction Excited positive > Negative Flavor-Spin interaction Chiral symmetry plays major role Negative > Excited positive Glozman & Riska Phys. Rep. 268,263 (1996) Hyperfine Interaction of quarks in Baryons ..Isgur N. Mathur et al, Phys. Lett. B605,137 (2005).

33. Roper Resonance for Quenched QCD Compiled by H.W. Lin

34. Mahbub et.al : arXiv:1011.5724v1

35. Symmetries of the lattice Hamiltonian • SU(3) gauge group (colour) • Zn⊗ Zn⊗ Zncyclic translational group (momentum) • SU(2) isospin group (flavour) • OhD crystal point group (spin and parity)

36. Octahedral group and lattice operators Construct operator which transform irreducibly under the symmetries of the lattice Baryon Meson …R.C. Johnson, Phys. Lett.B 113, 147(1982)

37. Lattice operator construction • Construct operator which transform irreducibly under the symmetries of the lattice • Classify operators according to the irreps of Oh: G1g, G1u, G1g, G1u,Hg, Hu • Basic building blocks : smeared, covariant displaced quark fields • Construct translationaly invariant elemental operators • Flavor structure  isospin, color structure  gauge invariance • Group theoretical projections onto irreps of Oh : PRD 72,094506 (2005) A. Lichtl thesis, hep-lat/0609019

38. Radial structure : displacements of different lengths Orbital structure : displacements in different directions …C. Morningstar

39. Pruning • All operators do not overlap equally and it will be very difficult to use all of them. • Need pruning to choose good operator set for each representation. • Diagonal effective mass. • Construct average correlator matrix in each representation and find condition number. • Find a matrix with minimum condition number.

41. Nucleon mass spectrum Hadron spectrum collaboration : Phys. Rev. D79:034505, 2009

43. CASCADE MASSES

44. WIDTHS

46. Mike Peardon’s talk

49. Hadron Spectrum collaboration : Dudek et.al : arXiv:1102.4299v1

50. Hadron spectrum collaboration : Phys. Rev. D 82, 014507 (2010)