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

2. TIFR, August 25, 2009 The Particle Zoo The Particle Zoo HADRON SPECTRUM Mesons (2-quarks) Mesons (2-quarks) Baryons (3-quarks) Baryons (3-quarks) …PDG

3. TIFR, August 25, 2009 Can we explain these (at least)? Can we explain these (at least)?

4. TIFR, August 25, 2009 Proof of E=mc 2 !! e=mc 2 : 103 years later, Einstein's proven right !! …….Times of India : 21 November 2008 S.Durr et.al, Science 322, 1224 (2008)

5. TIFR, August 25, 2009 A constituent picture of Hadrons M. Peardon’s talk

6. TIFR, August 25, 2009 Type of Hadrons Type of Hadrons Normal hadrons : Normal hadrons :  Two quark state (meson)  Three quark state (baryon) Other Hadrons Other Hadrons  Multiquarks  Exotics (hybrids)  Glueballs

7. TIFR, August 25, 2009 quark propagators : Inverse of very large matrix of space-time, spin and color Quark (on Lattice sites) Gluon (on Links) QuarkJungle Gym Gym

8. TIFR, August 25, 2009 t

9. Pion two point function Nucleon interpolating operator

10. TIFR, August 25, 2009 Analysis (Extraction of Mass) Analysis (Extraction of Mass) Correlator decays exponentially m1m1m1m1 m 1, m 2 Effective mass :

11. TIFR, August 25, 2009 Analysis (Extraction of Mass) Analysis (Extraction of Mass) Correlator decays exponentially How to extract m 2 m 3 … : excited states? Non linear fitting. Variable projection method m1m1m1m1 m 1, m 2 Assume that data has Gaussian distribution Uncorrelated chi 2 fitting by minimizing However, data is correlated and it is necessary to use covariance matrix

12. TIFR, August 25, 2009 Bayesian Fitting Priors

13. TIFR, August 25, 2009 Bayes’ theorem : Bayesian prior distribution Posterior probability distribution prior predictive probability the conditional probability of measuring the data D given a set of parametersρ the conditional probability that ρ is correct given the measured data D P(D)

14. TIFR, August 25, 2009 ψ i : gauge invariant fields on a timeslice t that corresponds to Hilbert space operator ψ j whose 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 such that the overlap to a state is maximum  Variational solution  Generalized eigenvalue problem :  Eigenvalues give spectrum :  Eigenvectors give the optimal operator : Variational Analysis

15. TIFR, August 25, 2009 Importance of t 0 Importance of t 0 Basis of operators is only a part of the Hilbert space (n = 1,…N; N≠ ∝ ) The eigenvectors are orthogonal only in full space. C(t 0 ) : Orthogonality is controlled by the metric C(t 0 ) : t 0 t 0 should be chosen such that the NXN correlator matrix is dominated by the lightest N states at t 0 t 0 Excited states contribution falls of exponentially  go to large t 0 t 0 However, signal/noise ratio increases at large t 0 t 0 Choose optimum t 0

16. TIFR, August 25, 2009 Sommer : arXiv:0902.1265v2

17. TIFR, August 25, 2009

18. Overlap Factor (Z)

19. TIFR, August 25, 2009

20. Dudek et.al

21. TIFR, August 25, 2009

22. What is a resonance particle? 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. TIFR, August 25, 2009 Identifying a Resonance State Identifying a Resonance State Method 1 : 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 : 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 : 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) ….V. Bernard et al, JHEP 0808,024 (2008)

24. TIFR, August 25, 2009 Multi-particle states A problem for finite box lattice Multi-particle states A problem for finite box lattice Finite box : Momenta are quantized Finite box : Momenta are quantized Lattice Hamiltonian can have both Lattice Hamiltonian can have both resonance and decay channel states resonance and decay channel states (scattering states) (scattering states) A  x+y, Spectra of m A and A  x+y, Spectra of m A and One needs to separate out resonance states from scattering states One needs to separate out resonance states from scattering states

25. TIFR, August 25, 2009 Scattering state and its volume dependence Scattering state and its volume dependence Normalization condition requires : Two point function : Lattice Continuum For one particle bound state spectral weight (W) will NOT be explicitly dependent on lattice volume

26. TIFR, August 25, 2009 Scattering state and its volume dependence Scattering state and its volume dependence Normalization condition requires : Two point function : Lattice Continuum For two particle scattering state spectral weight (W) WILL be explicitly dependent on lattice volume

27. TIFR, August 25, 2009 C. Morningstar, Lat08

28. TIFR, August 25, 2009 Solution in a finite box C. Morningstar, Lat08

29. TIFR, August 25, 2009 Rho decay Rho decay

30. TIFR, August 25, 2009 ….V. Bernard et al, JHEP 0808,024 (2008)

31. TIFR, August 25, 2009 Hybrid boundary condition Hybrid boundary condition Periodic boundary condition on some quark fields while anti-periodic on others Periodic boundary condition on some quark fields while anti-periodic on others Bound and scattering states will be changing differently. Bound and scattering states will be changing differently.

32. TIFR, August 25, 2009 Hyperfine Interaction of quarks in Baryons Hyperfine Interaction of quarks in Baryons _ + + Nucleon (938) Roper (1440) S 11 (1535) + + _ Δ(1236) Δ(1700) Δ(1600) + + _ Λ(1116) Λ(1405) Λ(1670) Color-Spin Interaction Excited positive > Negative Glozman & Riska Phys. Rep. 268,263 (1996) Flavor-Spin interaction Chiral symmetry plays major role Negative > Excited positive..Isgur N. Mathur et al, Phys. Lett. B605,137 (2005).

33. TIFR, August 25, 2009 Roper Resonance for Quenched QCD Compiled by H.W. Lin

34. TIFR, August 25, 2009 Mahbub et.al : arXiv:1011.5724v1

35. TIFR, August 25, 2009 Symmetries of the lattice Hamiltonian Symmetries of the lattice Hamiltonian SU(3) gauge group (colour) SU(3) gauge group (colour) Z n ⊗ Z n ⊗ Z n cyclic translational group (momentum) Z n ⊗ Z n ⊗ Z n cyclic translational group (momentum) SU(2) isospin group (flavour) SU(2) isospin group (flavour) O h D crystal point group (spin and parity) O h D crystal point group (spin and parity)

36. TIFR, August 25, 2009 Octahedral group and lattice operators Octahedral group and lattice operators Λ J G 1 G 1 G 2 G 2 H 1/2 ⊕ 7/2 ⊕ 9/2 ⊕ 11/2 … 5/2 ⊕ 7/2 ⊕ 11/2 ⊕ 13/2 … 3/2 ⊕ 5/2 ⊕ 7/2 ⊕ 9/2 … Λ J A1 A1 A2 A2 E E T1 T1 T2 T2 A1 A1 A2 A2 E E T1 T1 T2 T2 E 0⊕4⊕6⊕8 …0⊕4⊕6⊕8 …3⊕6⊕7⊕9 …3⊕6⊕7⊕9 …2⊕4⊕5⊕6 …2⊕4⊕5⊕6 …1⊕3⊕4⊕5 …1⊕3⊕4⊕5 …2⊕3⊕4⊕5 …2⊕3⊕4⊕5 …0⊕4⊕6⊕8 …0⊕4⊕6⊕8 …3⊕6⊕7⊕9 …3⊕6⊕7⊕9 …2⊕4⊕5⊕6 …2⊕4⊕5⊕6 …1⊕3⊕4⊕5 …1⊕3⊕4⊕5 …2⊕3⊕4⊕5 …2⊕3⊕4⊕5 … Baryon Meson …R.C. Johnson, Phys. Lett.B 113, 147(1982) Construct operator which transform irreducibly under the symmetries of the lattice

37. TIFR, August 25, 2009 Lattice operator construction Lattice operator construction Construct operator which transform irreducibly under the symmetries of the lattice Construct operator which transform irreducibly under the symmetries of the lattice Classify operators according to the irreps of O h : Classify operators according to the irreps of O h : G 1g, G 1u, G 1g, G 1u,H g, H u G 1g, G 1u, G 1g, G 1u,H g, H u Basic building blocks : smeared, covariant displaced quark fields Basic building blocks : smeared, covariant displaced quark fields Construct translationaly invariant elemental operators Construct translationaly invariant elemental operators Flavor structure  isospin, color structure  gauge invariance Flavor structure  isospin, color structure  gauge invariance Group theoretical projections onto irreps of O h : Group theoretical projections onto irreps of O h : PRD 72,094506 (2005) A. Lichtl thesis, hep-lat/0609019

38. TIFR, August 25, 2009 Radial structure : displacements of different lengths Orbital structure : displacements in different directions …C. Morningstar

39. TIFR, August 25, 2009 Pruning Pruning All operators do not overlap equally and it will be very difficult to use all of them. 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. Need pruning to choose good operator set for each representation. Diagonal effective mass. Diagonal effective mass. Construct average correlator matrix in each representation and find condition number. Construct average correlator matrix in each representation and find condition number. Find a matrix with minimum condition number. Find a matrix with minimum condition number.

40. TIFR, August 25, 2009

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

42. TIFR, August 25, 2009

43. CASCADE MASSES

44. TIFR, August 25, 2009 WIDTHS

45. TIFR, August 25, 2009

46. Mike Peardon’s talk

47. TIFR, August 25, 2009

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

50. TIFR, August 25, 2009 Hadron spectrum collaboration : Phys. Rev. D 82, 014507 (2010)

51. TIFR, August 25, 2009 Smeared operators (for example) :

52. TIFR, August 25, 2009 Engel et.al : arXiv:1005.1748v2 M π ~ 320 MeV a =0.15 fm 16^3 X 32 Chirally improved f

53. TIFR, August 25, 2009 Prediction : : Ξ’ b = 5955(27) MeV Cohen, Lin, Mathur, Orginos : arXiv:0905.4120v2

54. TIFR, August 25, 2009  Spin identification  Multi-particle states  Isolating resonance states from multi-particle states  Extracting resonance parameters Problems

55. TIFR, August 25, 2009 Hadron Spectrum collaboration

56. TIFR, August 25, 2009