1. Hadron Resonance Determination Robert Edwards Jefferson Lab ECT 2014 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAAAA

2. Resonances Most hadrons are resonances –Formally defined as a pole in a partial-wave projected scattering amplitude Can we predict hadron properties from first principles?

3. Lattice QCD as a computational approach The quantities computed in lattice QCD –Euclidean correlation functions Spectrum of eigenstates of H QCD Hadron matrix elements –On a finite cubic grid Let’s discuss how a field theory in a finite volume is related to observables Cubic lattice

4. Quantum mechanics on a circle One-dimensional motion with periodic boundary conditions A free particle –Periodic boundary condition Discrete energy spectrum

5. Quantum mechanics on a circle Solutions Quantization condition when -L/2 < z < L/2 Two spin-less bosons: ψ(x,y) = f(x-y) -> f(z) The idea: 1 dim quantum mechanics non-int momdynamical shift

6. Quantum mechanics on a circle Solutions Quantization condition when -L/2 < z < L/2 Two spin-less bosons: ψ(x,y) = f(x-y) -> f(z) The idea: 1 dim quantum mechanics non-int momdynamical shift discrete energy spectrum is determined by scattering amplitude (or vice-versa)

7. Field theory in a cubic box In 1-D QM, result for phase-shift was: Previous arguments generalize to a field-theory –In 3-space dimension & for coupled channels - “Luscher” method & extensions Known functions of (actually, in cubic irreps) 4-momentum, e.g. from lattice Ignoring for now the complications using cubic box

8. Field theory in a cubic box In 1-D QM, result for phase-shift was: Previous arguments generalize to a field-theory –In 3-space dimension & for coupled channels - “Luscher” method & extensions Idea: –In whatever formalism, compute discrete energies (4-momentum) –Here, we will use a lattice formalism –From these energies one can obtain scattering amplitudes Known functions of (actually, in cubic irreps) 4-momentum, e.g. from lattice Ignoring for now the complications using cubic box

9. Scattering amplitudes from finite volume Method generalizes to higher partial waves (elastic case) e.g., arXiv:1211.0929 Matrix of known functions (actually, in cubic irreps Λ) 4-momentum from lattice

10. How does it work? Imagine if two pions did not interact with each other –Pions have isospin=1 so two pions can form isospin=2 –Isospin=2 J P =2 spectrum would look like π π CUBIC BOX SPECTRUM

11. How does it work? Experimental ππ I=2 S-wave scattering amp. S-WAVE PHASE SHIFT CUBIC BOX SPECTRUM

12. How does it work? Experimental ππ I=2 S-wave scattering amp. –A “weak” repulsive interaction S-WAVE PHASE SHIFT CUBIC BOX SPECTRUM non-interacting spectrum

13. How does it work? Experimental ππ I=2 S-wave scattering amp. –A “weak” repulsive interaction S-WAVE PHASE SHIFTCUBIC BOX SPECTRUM

14. How does it work? Experimental ππ I=2 S-wave scattering amp. –A “weak” repulsive interaction S-WAVE PHASE SHIFTCUBIC BOX SPECTRUM

15. How does it work? Experimental ππ I=2 S-wave scattering amp. –A “weak” repulsive interaction S-WAVE PHASE SHIFTCUBIC BOX SPECTRUM

16. How does it work (now a resonance)? Experimental ππ I=1 P-wave scattering amp. –Contains the ρ resonance P-WAVE PHASE SHIFT CUBIC BOX SPECTRUM

17. How does it work (now a resonance)? Experimental ππ I=1 P-wave scattering amp. –Contains the ρ resonance P-WAVE PHASE SHIFT CUBIC BOX SPECTRUM non-interacting spectrum

18. How does it work (now a resonance)? Experimental ππ I=1 P-wave scattering amp. –Contains the ρ resonance P-WAVE PHASE SHIFT CUBIC BOX SPECTRUM

19. How does it work (now a resonance)? Experimental ππ I=1 P-wave scattering amp. –Artificially narrow ρ resonance P-WAVE PHASE SHIFT CUBIC BOX SPECTRUM

20. How does it work (now a resonance)? Experimental ππ I=1 P-wave scattering amp. –Artificially narrow ρ resonance P-WAVE PHASE SHIFT CUBIC BOX SPECTRUM

21. Lattice QCD Provides a Monte Carlo estimate of Euclidean time correlation functions –a hadron two-point function Contains information about the spectrum e.g. H = finite-volume QCD Hamiltonian CORRELATION FUNCTION

22. Isospin=2 J P =0 + Finite-volume spectrum with

23. Isospin=2 J P =0 + Finite-volume spectrum non-interacting spectrum with

24. Isospin=2 J P =0 + phase-shift Significant extra information from the spectrum in moving frames arxiv:1203.6041

25. Isospin=2 elastic ππ-scattering Example, non-resonant I=2 ππ in S & D-wave Large number of points come from systems of arXiv:1203.6041

26. Isospin=1 J PC =1 -- In the elastic scattering region threshold arxiv:1212.0830

27. Isospin=1 J PC =1 -- Need energy dependent functional form : use a Breit-Wigner parameterization arxiv:1212.0830 parameters m R and g

28. Isospin=1 J PC =1 -- Breit-Wigner fit to the energy dependence BREIT-WIGNER Reduced width from small phase-space arxiv:1212.0830

29. Coupled-channel case Finite-volume formalism only recently developed –E.g., isospin=0, J P =0 + channels i = (ππ, KK, ηη, …) –E.g., baryon ½, ½ - channels i = (πN, ηN, …) Underconstrained problem: one energy level – many scatt. amps to determine –Already showed you an example approach Parameterize t-matrix »“Energy dependent” analysis e.g., arXiv:1211.0929 phase space for channel i arXiv: 0504019, 1010.6018, 1204.0826, 1204.6256, 1305.4903,…

30. Coupled-channel case Finite-volume formalism only recently developed –E.g., isospin=0, J P =0 + channels i = (ππ, KK, ηη, …) –E.g., baryon ½, ½ - channels i = (πN, ηN, …) Underconstrained problem: one energy level – many scatt. amps to determine –Already showed you an example approach Parameterize t-matrix »“Energy dependent” analysis e.g., arXiv:1211.0929 phase space for channel i arXiv: 0504019, 1010.6018, 1204.0826, 1204.6256, 1305.4903,… Couples channels i,j – diagonal in l Couples partial waves l

31. Coupled-channel case Finite-volume formalism only recently developed –E.g., isospin=0, J P =0 + channels i = (ππ, KK, ηη, …) –E.g., baryon ½, ½ - channels i = (πN, ηN, …) Problem is that this is one equation in multiple unknowns –One approach is to parameterize the t-matrix »“Energy-dependent” analysis Underconstrained problem: one energy level – many scatt. amps to determine –Already showed you an example approach Parameterize t-matrix »“Energy dependent” analysis e.g., arXiv:1211.0929 phase space for channel i arXiv: 0504019, 1010.6018, 1204.0826, 1204.6256, 1305.4903,…

32. Isospin=1/2 πK/ηK scattering Spectrum: arXiv:1406.4158 mostly πK Spectral overlaps: Guide to content Shifted πK-like & ηK-like states mostly ηK “extra” level Interacting πK’ + single- particle overlaps Interacting πK’ + single-particle overlaps Interacting ηK’ + single-particle overlaps

33. Isospin=1/2 πK/ηK scattering Two channel scattering: arXiv:1406.4158

34. Isospin=1/2 πK/ηK scattering Two channel scattering: T-matrix: account of threshold behavior K-matrix: pole + polynomial in s = E cm 2 Ensure unitary: Chew-Mandelstam func arXiv:1406.4158 phase space for channel i

35. Isospin=1/2 πK/ηK scattering Two channel scattering: Rewrite in terms of 2 phase-shifts & inelasticity arXiv:1406.4158 Recall, at one energy, have 1 eqn. but 3 variables

36. Isospin=1/2 πK/ηK scattering Two channel scattering: Rewrite in terms of 2 phase-shifts & inelasticity arXiv:1406.4158 Solve eqn. (quantization condition) – must vary perams. in t (l)

37. Isospin=1/2 πK/ηK scattering Two channel scattering: arXiv:1406.4158 Using only rest-frame data Energies from det. Eqn. must agree with model K-matrix: pole + polynomial in s = E cm 2

38. Isospin=1/2 πK/ηK scattering Two channel scattering: arXiv:1406.4158 Energies from det. Eqn. must agree with model K-matrix: pole + polynomial in s = E cm 2 Using only rest-frame data Next, will use all data

39. Isospin=1/2 πK/ηK scattering Broad resonance in S-wave πK ηK coupling is small 3 sub-threshold points naturally included in energy-level fit Bound state pole in J P = 1 - Coupling consistent with expt & phenomenology Narrow resonance in D-wave πK ηK coupling is small Above ππK – need 3-body formalism arXiv:1406.4158

40. Isospin=1/2 πK/ηK scattering arXiv:1406.4158 t-matrix singularities similar to expt Pole found below threshold on unphysical sheet – virtual bound state Unitarized xPT: κ(800) pole  virtual bound-state  bound-state Pole on physical sheet below threshold in J P =1 - Similar to K * (892) but just bound at mπ=391 MeV Poles on unphysical sheets: S-wave, large width, mostly couples to πK Similar to K 0 * (1430) D-wave, narrow width, mostly couples to πK Similar to K 2 * (1430) RESONANCE POLE POSITION[S]

41. Where’s the big answer for the spectrum? Current reality: Meson results are forth coming However, most baryon results limited to single-particle operator constructions No in principle limitation: However, contraction cost for baryon+multi-meson systems is high Do have issue how to systematically parameterize 3-particle scattering With caveats, will show results restricted to single-particle operator constructions

42. Baryon spectrum Positive parity baryons: counting SU(6)xO(3) arXiv:1201.2349 “Hybrid” excitation ~ 1.3GeV

43. πN thr. ππN thr. Baryon spectrum Positive parity baryons –This is the spectrum using only qqq-styled operators –No operators that look like, e.g., πN … »Definitely not the complete spectrum »First results have appeared [1212.5055] arXiv:1201.2349

44. Need “broad” operator basis For variational method Need operators that overlap well with relevant basis states qqbar-like levels shift within hadronic width

45. Multi-particle operator basis # levels increases with moving frames and more operators qbar-q only ops – levels within hadronic width

46. Multi-particle operator basis Our previous calculations used only qqbar - like operators J P =2 + & 1 - Narrow interaction region: old results within width J P =0 + Very broad: scatter of levels indicative of interaction region

47. Matrix elements “Easy” for stable hadrons, e.g. nucleon form-factors –Compute a 3pt function with a vector current –Extract the desired γN  N matrix element –Easy because the nucleon is the stable ground-state in the (I,J P ) = (½, ½+) channel excited state contribution s

48. Matrix elements: How about the N  Δ transition form-factor? sum over eigenstates in this finite-volume

49. Matrix elements: Should be able to extract these finite-volume matrix elements But what do we do with them? SPECTRUM πN scattering phase-shift finite- volume spectrum

50. Matrix elements: How about the N  Δ transition form-factor? L ∞ Need demonstration of formalism for Q 2 >0 Helicity amplitudes at discrete W, Q 2 values Formalism now exists (1.5 weeks ago!) to relate finite-V matrix elements finite-volume matrix element infinite-volume matrix element arXiv:1406.5965 πN scattering phase-shift finite- volume spectrum

51. Pilot project: ρ  γ  Transition form-factor: compute determine

52. Summary Spectrum of eigenstates of a field theory in a finite-volume can be related to scattering amplitudes Can take advantage of this in lattice QCD –Simple cases have been computed already, e.g., elastic ππ in I=1,2 –First results for coupled-channel scattering with partial waves For the (near?) future: –Simplest baryon resonances, N * ( ½, ½-), Δ, … –Finite-volume formalism for three-body scattering ( ΠΠΠ, ΠΠN, …) under development [Bonn(Rusetsky, Meissner), UWash (Sharpe, Hansen), JLab (Briceno), …] –Compute matrix-elements featuring resonant states –Work (possibly less rigorously) to “understand” resonances at the quark-gluon level (?)

53. The details… The end 53

54. Isospin=2 J P =0 Possible finite-volume operators –Now see the physical motivation for these operators “resemble” ΠΠ scattering states

55. Isospin=1 J PC =1 -- Contains the ρ resonance Possible finite-volume operators And similar constructions at non-zero total momentum c.f. and more complicated fermion bilinears

56. Matrix elements “Easy” for stable hadrons, e.g. nucleon form-factors –Compute a 3pt function with a vector current –Extract the desired γN  N matrix element –Easy because the nucleon is the stable ground-state in the (I,J P ) = (½, ½+) channel excited state contribution s

57. Matrix elements: How about the N  Δ transition form-factor? sum over eigenstates in this finite-volume

58. Matrix elements: Should be able to extract these finite-volume matrix elements But what do we do with them? SPECTRUM πN scattering phase-shift finite- volume spectrum

59. Matrix elements: How about the N  Δ transition form-factor? L ∞ Need demonstration of formalism for Q 2 >0 Helicity amplitudes at discrete W, Q 2 values Should be able to calculate the amplitudes at discrete W, Q 2 values finite-volume matrix element infinite-volume matrix element

60. Spin identified Nucleon & Delta spectrum arXiv:1104.5152, 1201.2349

61. Spin identified Nucleon & Delta spectrum arXiv:1104.5152, 1201.2349 Full non-relativistic quark model counting 4531 23 2 1 221 11

62. Interpreting content “Spectral overlaps” give clue as to content of states Large contribution from gluonic- based operators on states identified as having “hybrid” content

63. Spin identified Nucleon & Delta spectrum arXiv:1104.5152, 1201.2349 Interpretation of level content from “spectral overlaps” 4531 23 2 1 221 11

64. Hybrid baryons 64 Negative parity structure replicated: gluonic components (hybrid baryons) [ 70,1 + ] P-wave [ 70,1 - ] P-wave

65. SU(3) flavor limit SU(3) flavor limit: have exact flavor Octet, Decuplet and Singlet representations Full non-relativistic quark model counting Additional levels with significant gluonic components arXiv:1212.5236

66. Light quarks – SU(3) flavor broken Light quarks - other isospins Full non-relativistic quark model counting Some mixing of SU(3) flavor irreps arXiv:1212.5236

67. Light quarks – SU(3) flavor broken Light quarks - other isospins Full non-relativistic quark model counting Some mixing of SU(3) flavor irreps arXiv:1212.5236

68. Where are the “Missing” Baryon Resonances? 68 N Δ PDG uncertainty on B-W mass Nucleon & Delta spectrum

69. Where are the “Missing” Baryon Resonances? 69 2 2 1 QM predictions 4 5 3 1 ??? 1 1 0 2 3 2 1 ??? N Δ PDG uncertainty on B-W mass Nucleon & Delta spectrum Do not see the expected QM counting

70. Strange Quark Baryon Spectrum Strange quark baryon spectrum even sparser 2 3 2 1 ??? 1 1 0 6 8 5 2 ??? Since SU(3) flavor symmetry broken, expect mixing of 8 F & 10 F 3 3 1 Even less known states in Ξ & Ω ΛΞ

71. Volume dependence: isoscalar mesons Energies determined from single-particle operators: Range of J PC - color indicates light-strange flavor mixing Some volume dependence: Interpretation: energies determined up to a hadronic width arXiv:1309.2608

72. Summary & prospects Spectrum of eigenstates of QCD in a finite-box can be related to scattering amplitudes Using lattice QCD - first steps in this direction: Showed you “simple” (elastic) cases of scattering First glimpses at full excited spectrum, but without scattering studies 72 Path forward: resonance determination! Calculations underway at 230 MeV pion masses Currently investigating multi-channel scattering in different systems Challenges: Must develop reliable 3-body formalism (hard enough in infinite volume) Large number of open channels in physical pion mass limit – it’s the real world! Can QCD allow simplifications (e.g., isobars?)

73. QCD QCD is (probably) underlying theory of hadrons via quarks and gluons –Coupling becomes large at low energy scales –Non-perturbative dynamics QCD coupling

74. Its called Strong interactions for a reason Hadrons composed of quarks and in color singlet states –Color confinement considered to give quark confinement Hadrons interacts via quarks/gluons stuck into color singlets Strong coupling makes perturbation theory problematic N N Σ,π,ρ,…

75. QCD: Quantum Chromdynamics Dirac operator: A (vector potential), m (quark mass), γ (Dirac gamma matrices) Observables

76. QCD: Quantum Chromdynamics Dirac operator: A (vector potential), m (quark mass), γ (Dirac gamma matrices) Observables QCD: Vector potentials now 3x3 complex matrices (SU(3)) Running of coupling u,d quarks are very light theory has another scale

77. QCD: Quantum Chromdynamics Dirac operator: A (vector potential), m (quark mass), γ (Dirac gamma matrices) Observables QCD: Vector potentials now 3x3 complex matrices (SU(3)) Lattice QCD: finite difference Lots of “flops/s” Harness GPU-s

78. Variational method A robust technique to extract the spectrum –Compute a matrix of correlators –Find the linear superposition of operators optimal for each state –Corresponds to solving the linear system –If your basis is “broad” enough, should reliably extract the spectrum

79. Variational method Can construct optimal linear combination from eigenvectors 0 −+ EFFECTIVE MASSES

80. Example: charmonium excited spectrum Large c-cbar operator basis & variational method arxiv:1204.5425

81. Multi-particle operators Quark fields act on vacuum to produce states with some quantum numbers Can have combinations of composite-operators Can form different meson & baryon operator constructions to overlap with desired J PC and J P of interest

82. Isospin=2 0 + spectrum in lattice QCD Need at least four quark fields to construct isospin=2 –Could choose local tetraquark basis –Instead, use a more physically motivated choice (with optimized pion operator) –For zero total momentum, scalar operator

83. Resonances Most hadrons are resonances –E.g., a bump in elastic hadron-hadron scattering

84. We want to determine resonances Most hadrons are resonances –E.g., a bump in elastic hadron-hadron scattering –Formally defined as a pole in a partial-wave projected scattering amplitude –Will appear as a pole in a production amplitude like πN cross section

85. Scattering 85 E.g. just a single elastic resonance e.g. Experimentally - determine amplitudes as function of energy E

86. Scattering - in finite volume! E.g. just a single elastic resonance e.g. At some L, have discrete excited energies 86 Scattering in a periodic cubic box (length L)

87. Isospin=2 elastic ππ-scattering Example, non-resonant I=2 ππ in S & D-wave Large number of points come from systems of arXiv:1203.6041

88. Single channel elastic scattering Isospin=1: ππ arXiv:1212.0830

89. Coupling in Isospin =1 ππ Comparison to other calculations: Feng, et.al, 1011.5288 Extracted coupling: stable in pion mass Stability a generic feature of couplings??

90. Form Factors What is a form-factor off of a resonance? What is a resonance? Spectrum first! Extension of scattering techniques: –Finite volume matrix element modified Requires excited level transition FF’s: some experience –Charmonium E&M transition FF’s (1004.4930) –Nucleon 1 st attempt: “Roper”->N (0803.3020) E Kinematic factor Phase shift

91. Need “broad” operator basis For variational method Need operators that overlap well with relevant basis states

92. Contractions Cost to produce correlators driven by contractions Propagators Operators Many permutations

93. Reminder – scattering in a finite volume E.g. just a single elastic resonance e.g. At some L, have discrete excited energies 93 Scattering in a periodic cubic box (length L)

94. Interpreting content “Spectral overlaps” give clue as to content of states Large contribution from gluonic- based operators on states identified as having “hybrid” content

95. Hybrid meson models With minimal quark content,, gluonic field can in a color singlet or octet `constituent’ gluon in S-wave `constituent’ gluon in P-wave bag model flux-tube model

96. Hybrid meson models With minimal quark content,, gluonic field can in a color singlet or octet `constituent’ gluon in S-wave `constituent’ gluon in P-wave bag model flux-tube model

97. Hybrid baryon models Minimal quark content,, gluonic field can be in color singlet, octet or decuplet bag model flux-tube model Now must take into account permutation symmetry of quarks and gluonic field

98. Hybrid baryon models Minimal quark content,, gluonic field can be in color singlet, octet or decuplet bag model flux-tube model Now must take into account permutation symmetry of quarks and gluonic field

99. Hybrid hadrons “subtract off” the quark mass Appears to be a single scale for gluonic excitations ~ 1.3 GeV Gluonic excitation transforming like a color octet with J PC = 1 +- arXiv:1201.2349

100. SU(3) flavor limit In SU(3) flavor limit – have exact flavor Octet, Decuplet and Singlet representations Full non-relativistic quark model counting Additional levels with significant gluonic components arXiv:1212.5236

101. Spectrum from variational method Matrix of correlators Two-point correlator 101

102. Spectrum from variational method Two-point correlator 102

103. Spectrum from variational method Matrix of correlators “Rayleigh-Ritz method” Diagonalize: eigenvalues  spectrum eigenvectors  spectral “overlaps” Z i n Two-point correlator 103

104. Spectrum from variational method Matrix of correlators “Rayleigh-Ritz method” Diagonalize: eigenvalues  spectrum eigenvectors  spectral “overlaps” Z i n Two-point correlator 104 Each state optimal combination of Φ i

105. Extension to inelastic scattering Can generalize to a scattering t-matrix Underconstrained problem: one energy level – many scatt. amps to determine –Already showed you an example approach Parameterize t-matrix »“Energy dependent” analysis e.g., arXiv:1211.0929 Channels labelled by i,j where is the scattering t -matrix and is the phase-space for channel i E.g.: isospin=0, J P =0 + channels i = (ππ, KK, ηη, …) E.g.: baryon ½ - channels I = (πN, ηN, …)

106. Excited hadrons are resonances Decay thresholds open (even for 400 MeV pions) PRD82 034508 (2010) arXiv:1309.2608 ππ continuum of ππ states ?

107. Excited hadrons are resonances ππ KK _ Decay thresholds open (even for 400 MeV pions) PRD82 034508 (2010) arXiv:1309.2608

108. Patterns in baryon spectrum