1. R L Jaffe Frascati May 2005 Multiquark Dynamics R. L. Jaffe LNF Frascati May 2005 R. L. Jaffe LNF Frascati May 2005 The problem of exotics in QCD The aufbau principle for hadrons The scalar mesons Dynamical correlations --- diquarks, a new field for study in QCD? Quark states and scattering The problem of exotics in QCD The aufbau principle for hadrons The scalar mesons Dynamical correlations --- diquarks, a new field for study in QCD? Quark states and scattering

2. R L Jaffe Frascati May 2005 Multiquark Dynamics I.General overview A.Historical overview a.Definition of exotic: flavor and CP b.History of searches for exotics c.State of affairs in 2003: absence of exotics & extra scalar nonet B.The Theta (2003-2005) requiescat in pace? a.Definition, quantum numbers, and significance b.Very brief review of history and experiments c.Summary of negative evidence C.Models of hadrons and exotics --- once over quickly a.Quark models and chiral soliton models b.Implications of the death of the Theta for models I.General overview A.Historical overview a.Definition of exotic: flavor and CP b.History of searches for exotics c.State of affairs in 2003: absence of exotics & extra scalar nonet B.The Theta (2003-2005) requiescat in pace? a.Definition, quantum numbers, and significance b.Very brief review of history and experiments c.Summary of negative evidence C.Models of hadrons and exotics --- once over quickly a.Quark models and chiral soliton models b.Implications of the death of the Theta for models

3. R L Jaffe Frascati May 2005 Multiquark Dynamics II.Quarks and Diquarks A.Introduction a.Naive quark model b.Spectroscopy versus dynamics B.Correlations and spectroscopy: the case for diquarks a.Correlations in QCD b.Spectroscopy, Delta I=1/2, hadron spin splittings c.Distribution and fragmentation function regularities, higher twist d.Defining and extracting the properties of diquarks E.A coherent qualitative picture of multiquark hadrons a.Diquarks and the general absence of exotics b.Where, if anywhere, to find multiquark hadrons c. A program in color non-singlet spectroscopy? II.Quarks and Diquarks A.Introduction a.Naive quark model b.Spectroscopy versus dynamics B.Correlations and spectroscopy: the case for diquarks a.Correlations in QCD b.Spectroscopy, Delta I=1/2, hadron spin splittings c.Distribution and fragmentation function regularities, higher twist d.Defining and extracting the properties of diquarks E.A coherent qualitative picture of multiquark hadrons a.Diquarks and the general absence of exotics b.Where, if anywhere, to find multiquark hadrons c. A program in color non-singlet spectroscopy?

4. R L Jaffe Frascati May 2005 Multiquark Dynamics III.Aspects of Multiquark Dynamics A.General discussion. a.Qualitative expections for conventional hadrons (Zero width decoupling) b.Expectations for multiquark hadrons (continuum subsidence) B. Lessons from Large N c a.Mesons at large N c --- a reminder b.Multiquark states at large N c --- color substructure at leading non-trivial order in 1/N C.Wigner’s boundary conditon method a.The physical foundation for Wigner’s R-matrix b.Why hadrons are not K-matrix poles c.Insight from another boundary condition: the P-matirx c.The P-matrix and hadron resonances D.Feshbach-Fano resonances a.Very low energy scattering b.Potential scattering c.Bound states in the continuum III.Aspects of Multiquark Dynamics A.General discussion. a.Qualitative expections for conventional hadrons (Zero width decoupling) b.Expectations for multiquark hadrons (continuum subsidence) B. Lessons from Large N c a.Mesons at large N c --- a reminder b.Multiquark states at large N c --- color substructure at leading non-trivial order in 1/N C.Wigner’s boundary conditon method a.The physical foundation for Wigner’s R-matrix b.Why hadrons are not K-matrix poles c.Insight from another boundary condition: the P-matirx c.The P-matrix and hadron resonances D.Feshbach-Fano resonances a.Very low energy scattering b.Potential scattering c.Bound states in the continuum

5. R L Jaffe Frascati May 2005 Multiquark Dynamics Rigorous Approaches to QCD Perturbative QCD at High Lattice QCD for Simple Questions (quenched, away from chiral limit...) Lattice QCD for Simple Questions (quenched, away from chiral limit...) VOODOO QCD * * * J D Bjorken 1986 Chiral dynamics at very low energies

6. R L Jaffe Frascati May 2005 Multiquark Dynamics I.General overview

7. R L Jaffe Frascati May 2005 Multiquark Dynamics Exotics at the beginning....

8. R L Jaffe Frascati May 2005 Multiquark Dynamics

9. R L Jaffe Frascati May 2005 Multiquark Dynamics Non-Exotics Nuclei Exotic Mesons Exotic Baryons Mesons And Exotics

10. R L Jaffe Frascati May 2005 Multiquark Dynamics (CP Exotics and other oddities) Certain C & P quantum numbers are excluded to mesons in non-relativistic, two-body, Schroedinger quantum mechanics Violation? Relativistic effects (transformation to center of mass doesn’t exist) or constituent gluons This multiplet does not otherwise occur low in the spectrum Baryon analogues: Center of mass excitations of nucleon octet and decuplet

11. R L Jaffe Frascati May 2005 Multiquark Dynamics Major efforts in ‘60’s and ‘70’s Meson-meson scattering phase shift analyses found resonances in all non-exotic channels and no structure in Meson-baryon partial wave analyses found resonances in all non-exotic channels and no Hadron spectroscopy was a premier field of high energy physics: flagship experiments and very sophisticated analysis

12. R L Jaffe Frascati May 2005 Multiquark Dynamics

13. R L Jaffe Frascati May 2005 Multiquark Dynamics

14. R L Jaffe Frascati May 2005 Multiquark Dynamics Elastic resonance would be full circle in Argand diagram Dots are spaced by 50 MeV lab energy. Hyslop, Ardnt, Roper, Workman 1992 Elastic resonance would be full circle in Argand diagram Dots are spaced by 50 MeV lab energy. Hyslop, Ardnt, Roper, Workman 1992 No KN I=0 resonance resonance would be a state decaying to with same spin and parity as the nucleon: No KN I=1 resonance

15. R L Jaffe Frascati May 2005 Multiquark Dynamics Meanwhile, as SU(3) multiplets of mesons and baryons filled up... It became clear that there is a problem with the scalar mesons: Too many Too light Lightest have peculiar mass spectrum Meanwhile, as SU(3) multiplets of mesons and baryons filled up... It became clear that there is a problem with the scalar mesons: Too many Too light Lightest have peculiar mass spectrum

16. R L Jaffe Frascati May 2005 Multiquark Dynamics Summary through 2002 -- No exotics anywhere A nonet of supernumerary scalar mesons Aufbau principle differs from nuclei & atoms No exotics anywhere A nonet of supernumerary scalar mesons Aufbau principle differs from nuclei & atoms 2003 brought new discoveries and controversy! 2003 brought new discoveries and controversy!

17. R L Jaffe Frascati May 2005 Multiquark Dynamics One manifestly exotic baryon seen by many experiments. Another exotic and two non- exotic partners seen by NA49 Experimental Discoveries Charm-analog seen by H1 but not by Zeus (3/04)

18. R L Jaffe Frascati May 2005 Multiquark Dynamics

19. R L Jaffe Frascati May 2005 Multiquark Dynamics Positive sightings of baryons since 2003 Dzierba, Meyer, Szczepaniak hep-ex/0412077

20. R L Jaffe Frascati May 2005 Multiquark Dynamics Negative non-sightings of baryons since 2003 Dzierba, Meyer, Szczepaniak hep-ex/0412077

21. R L Jaffe Frascati May 2005 Multiquark Dynamics Recent re-examinations in the KN system Nussinovhep- ph/0307357 Arndt, Strakovsky, Workmannucl- th/0311030 Cahn & Trillinghep- ph/0311245 Sibirtsev, Haldenbatter, Krewald, Meissnerhep- ph/0405099 Nussinovhep- ph/0307357 Arndt, Strakovsky, Workmannucl- th/0311030 Cahn & Trillinghep- ph/0311245 Sibirtsev, Haldenbatter, Krewald, Meissnerhep- ph/0405099 No sign of any resonance: interpreted as limits on width -- very stringent: < 1--4 MeV

22. R L Jaffe Frascati May 2005 Multiquark Dynamics Very recent results -- first of a second generation Reported by R. De Vita for CLAS at APS Tampa 4/17/05

23. R L Jaffe Frascati May 2005 Multiquark Dynamics More detail

24. R L Jaffe Frascati May 2005 Multiquark Dynamics Same reaction, Phys. Lett. B572 (2003)

25. R L Jaffe Frascati May 2005 Multiquark Dynamics Mass determinations are inconsistent The is not officially dead yet, but.... More low energy photoproduction experiments to report soon No decisive experimental flaws, but see DMS Width limits are extreme and contradictory Lattice calculations find no positive parity resonance New experiments contradict old, lower statistics sightings ? ?

26. R L Jaffe Frascati May 2005 Multiquark Dynamics Only significant, qualitative failure of naive quark assignments is the existence of an extra nonet of scalar mesons with masses below 1 GeV. neutronsEquivalently, the building of hadrons is radically different than the building of atoms and nuclei -- to make atoms add more electrons in the nuclear Coulomb field; to make nuclei, add protons and neutrons in the nuclear mean field; to make hadrons, stop at QQQ or QQ. Situation post Theta No exotics

27. R L Jaffe Frascati May 2005 Multiquark Dynamics Quark Models Match naturally to DIS degrees of freedom Mesons & Baryons -- spectroscopy & electroweak interactions Never fully consistent with relativity -- relativistic single particle models but not field theoretic Not the basis of a systematic expansion Quark models have got a bad rap over the past 20 years. They are still the most powerful, broad, and heuristic tool for the hadron spectrum, and any hadron must have a quark interpretation... Match naturally to DIS degrees of freedom Mesons & Baryons -- spectroscopy & electroweak interactions Never fully consistent with relativity -- relativistic single particle models but not field theoretic Not the basis of a systematic expansion Quark models have got a bad rap over the past 20 years. They are still the most powerful, broad, and heuristic tool for the hadron spectrum, and any hadron must have a quark interpretation... Models of hadrons and the Death of the Theta

28. R L Jaffe Frascati May 2005 Multiquark Dynamics Never could accommodate narrow width and apparent positive parity No prediction because absolute mass scale of could not be determined Absence of Theta limits the strength of the correlation (diquark) that has other important spectroscopic and dynamical implications ? ? ? ? ? ? Implications for quark models and others...

29. R L Jaffe Frascati May 2005 Multiquark Dynamics Three flavors History: Duality ⇒ Strings ⇒ [N c → ∞ ] ⇒ Skyrme ⇒ CSM There is nothing fundamentally “3-ish” about baryons in the CSM, therefore they are teeming with exotics Two flavors Chiral Soliton Models ??? Three flavour exotics in the chiral soliton model: Manohar, Chemtob (1984/85) Praszalowicz uudds* at 1540 MeV Diakonov, Petrov, Polyakov: Narrow (1997) Weigel: excellent balanced summary (1998)

30. R L Jaffe Frascati May 2005 Multiquark Dynamics Implications of the death of the Theta for CSM SU(3) Chiral Soliton Models (Diakonov, Petrov, Polyakov) Strong prediction asserted: M=1540 MeV, = 15 MeV Apparently the model is unreliable, why? ?Truncation of chiral effective lagrangian? ?Adiabatic (rigid) excitation? ?Perturbative implimentation of SU(3) violation? Dynamical balance Assume that rotational excitations neither deform soliton nor mix with radial excitations. No separation of scales. No justification to ignore ?Questionable relation to QCD in the first place

31. R L Jaffe Frascati May 2005 Multiquark Dynamics II. Quarks and Diquarks

32. R L Jaffe Frascati May 2005 Multiquark Dynamics Naive Quark Model Assume you know the basics Uncorrelated quarks in a mean field: NON-RELATIVISTIC POTENTIAL MODELS RELATIVISTIC BAG MODELS YOUR FAVORITE MODEL Uncorrelated quarks in a mean field: NON-RELATIVISTIC POTENTIAL MODELS RELATIVISTIC BAG MODELS YOUR FAVORITE MODEL Good description of super-multiplets of light meson and baryon states Towers of multiquark states: vast number and no information about widths Widths in the quark model: finessed for QQ* and QQQ, it becomes essential for states that can fall apart into mesons and baryons Good description of super-multiplets of light meson and baryon states Towers of multiquark states: vast number and no information about widths Widths in the quark model: finessed for QQ* and QQQ, it becomes essential for states that can fall apart into mesons and baryons DYNAMICS

33. R L Jaffe Frascati May 2005 Multiquark Dynamics Correlations and classification Confinement Chiral symmetry breaking correlations ? Color, flavor, spin antisymmetry

34. R L Jaffe Frascati May 2005 Multiquark Dynamics Classification Fermi statistics – Parity + Parity Color Leaves only two diquarks in the low energy spectrum

35. R L Jaffe Frascati May 2005 Multiquark Dynamics Diquarks Flavo r ColorSpin -8 8/3 “Good” “Bad” Good Bad

36. R L Jaffe Frascati May 2005 Multiquark Dynamics Condensation in quark matter at high density condenses in flavor antisymmetric channel generating color-flavor locked superconductivity Long history in QCD, but never in the mainstream (D. Lichtenberg) Phenomenological evidence for diquarks Certain regularities in spectroscopy Absence of Systematic analysis of baryon and meson resonances. More later -- A. Selem & F. Wilczek in preparation Systematic analysis of baryon and meson resonances. More later -- A. Selem & F. Wilczek in preparation rule in nonleptonic weak decays dominance gives good description of non- perturbative effects. Systematic study by Neubert, Stech & collaborators in late 1980’s.

37. R L Jaffe Frascati May 2005 Multiquark Dynamics Diquark regularities in DIS Baryon parton distribution function regularities follow from Regularities in fragmentation ratios in to hadrons Fragmentation ratios measured at LEP (Delphi) Fragmentation ratios measured at LEP (Delphi) Suggests dominance of -- favored diquark. Known since 1960’s Recent JLab results nucl-ex/0308011 Recent JLab results nucl-ex/0308011

38. R L Jaffe Frascati May 2005 Multiquark Dynamics Good diquark, strange Formally define color antitriplet diquarks in the presence of an infinitely heavy spectator quark (or Polyakov line) Awaiting lattice calculations, estimate from charm and strange systems: Characterizing diquarks Good diquark, non- strange Bad diquark, strange Bad diquark, non- strange

39. R L Jaffe Frascati May 2005 Multiquark Dynamics Other estimates from charm sector Good - bad mass difference decreases with quark mass. Diquark -- heavy quark spin interaction decreases with heavy quark mass and with light quark mass Conclude: charm baryon masses allow estimate of diquark masses in a heavy quark background. Correlation is ~ 200 MeV. Not huge, but important

40. R L Jaffe Frascati May 2005 Multiquark Dynamics QCD explains the absence of exotics in general, Makes explicit predictions for non-exotic two quark, two antiquark states, Which have been verified by experiment, (Though this is not uncontroversial), And the same ideas suggest the channels in which exotic baryons are most likely... And the key to the dynamics is the concept of DIQUARKS QCD explains the absence of exotics in general, Makes explicit predictions for non-exotic two quark, two antiquark states, Which have been verified by experiment, (Though this is not uncontroversial), And the same ideas suggest the channels in which exotic baryons are most likely... And the key to the dynamics is the concept of DIQUARKS ✺ RLJ (1977), RLJ and F. Low (1979) ✺ ✺ Remarkably: it has been known since 1977 that Spectroscopic consequences of diquark correlations Preview

41. R L Jaffe Frascati May 2005 Multiquark Dynamics Spectra Baryons Fermi statistics kills the singlet and allows only one octet which is a mixture of good and bad diquark except for and Mesons No Exotics ! Lightest multiplet is 0 ++ nonet

42. R L Jaffe Frascati May 2005 Multiquark Dynamics Tetraquark Scalar Nonet

43. R L Jaffe Frascati May 2005 Multiquark Dynamics Scalar mesons: a supernumerary nonet

44. R L Jaffe Frascati May 2005 Multiquark Dynamics Pentaquarks Only possible exotic First combine two diquarks Where subscripts denote symmetry in flavor exchange Then combine with antiquark: Diquarks antisymmetric in flavor Diquarks symmetric in flavor

45. R L Jaffe Frascati May 2005 Multiquark Dynamics (Good diquark is a boson) Non-exotic Negative parity Nonet Non-exotic Negative parity Nonet Case I: Symmetric in spin. Antisymmetric in color and flavor Symmetric in space Symmetric in spin. Antisymmetric in color and flavor Symmetric in space Examine symmetry of diquark-diquark state: has odd parity Lowest mass Even parity Lowest mass Even parity

46. R L Jaffe Frascati May 2005 Multiquark Dynamics Note difference in charm sector SU(3)-flavor triplet of exotic (positive baryon number, negative charm) charmed baryons. Lesson: What is not exotic in one sector may show up as exotic in another. SU(3)-flavor triplet of exotic (positive baryon number, negative charm) charmed baryons. Lesson: What is not exotic in one sector may show up as exotic in another. For c and b quarks the flavor antisymmetric [qq][qq] states are also exotic. Perhaps very light Lipkin Stewart, Wessling, Wise hep-ph/0402076 Lipkin Stewart, Wessling, Wise hep-ph/0402076

47. R L Jaffe Frascati May 2005 Multiquark Dynamics Case II: (Good diquark is a boson) Antiymmetric in space: Symmetric in spin. Antisymmetric in color Symmetric in flavor Symmetric in spin. Antisymmetric in color Symmetric in flavor Examine symmetry of diquark-diquark state: has even parity Heavier(!) Odd parity Heavier(!) Odd parity

48. R L Jaffe Frascati May 2005 Multiquark Dynamics Overview: “Ideal” mixing -- diagonalize strange quark number at lowest order. “nucleon” lighter than perhaps “Roper”

49. R L Jaffe Frascati May 2005 Multiquark Dynamics Summary Pentaquarks Lightest Negative parity, non- exotic, s-wave Positive parity, exotic, p- wave Heavier Charmed Exotic s-wave It’s existence is independent of evidence for Theta Exotic s-wave It’s existence is independent of evidence for Theta Obscure for dynamical reasons -- - see later Would be prominent if light enough, but diquark correlation fights with angular momentum. Not precluded by anti-Theta evidence

50. R L Jaffe Frascati May 2005 Multiquark Dynamics Spectra Summary Generically, no exotics among light (u,d,s) Marginal cases: Pentaquark Dibaryon Scalar mesons: Ground state is a nonet of 0 ++ mesons resembling known light nonet. Heavy quark exotics should be explored more thoroughly [ ]

51. R L Jaffe Frascati May 2005 Multiquark Dynamics In Deep Inelastic Processes Structure function regularities, especially as Fragmentation functions If diquarks are strongly correlated, baryon fragmentation functions should not be dramatically smaller than meson f.f. Data on two particular baryons... Good [u,d] diquark in s-wave Good [u,d] diquark in p- wave

52. R L Jaffe Frascati May 2005 Multiquark Dynamics

53. R L Jaffe Frascati May 2005 Multiquark Dynamics Limits on diquarks from higher twist... A. Vainshteyn & RLJ (unpublished) “But aren’t strong correlations in QCD ruled out by the absence of large twist-four corrections to DIS?” “How pointlike can diquarks be?” “But aren’t strong correlations in QCD ruled out by the absence of large twist-four corrections to DIS?” “How pointlike can diquarks be?” corrections to DIS are known to be small, and limit non-perturbative scales in QCD beyond These limits constrain diquarks because twist-four operators include ones sensitive to diquark correlations...

54. R L Jaffe Frascati May 2005 Multiquark Dynamics Leading twist However: “Good” diquark is spinless and does not contribute at twist four ! Dimension-6, spin-2 ⇒ twist-4, but only if quarks are coupled to maximum spin. Twist four -- diquark operator

55. R L Jaffe Frascati May 2005 Multiquark Dynamics QCD spectroscopy in color non-singlet sectors Neutralize with spectator Wilson line (= infinitely heavy quark) (Or with uniform color background charge (how?)) Compare with bottom hadron spectroscopy And with phenomenological models Neutralize with spectator Wilson line (= infinitely heavy quark) (Or with uniform color background charge (how?)) Compare with bottom hadron spectroscopy And with phenomenological models Mesons Baryon s Tetraquark mesons Pentaquark baryons Even color sextet light quark states Study correlations in QCD by studying color non-singlet spectroscopy! The “ affair” suggests that correlations in QCD can be studied by studying color non- singlet light quark systems neutralized by heavy quark spectator. Study correlations in QCD by studying color non-singlet spectroscopy!

56. R L Jaffe Frascati May 2005 Multiquark Dynamics Example: Classification of ground state “Exotics” with unique SU(3) assignments. “Exotics” mixed by SU(3) violating interactions. Total J=3/2 states with (Ignore for simplicity -- dynamics?)

57. R L Jaffe Frascati May 2005 Multiquark Dynamics “Exotics” with unique SU(3) assignments. “Exotics” mixed by SU(3) violating interactions. Non-exotics, mix with single quark states. Total J=1/2 states with (Ignore for simplicity -- dynamics?)

58. R L Jaffe Frascati May 2005 Multiquark Dynamics Color triplet spectroscopy 1/21+1[15] 1/23/20[15] 1/21[15] + [6] 1/2 -2[15] 1/20+1[6] * This representation only occurs when qq are coupled to 6

59. R L Jaffe Frascati May 2005 Multiquark Dynamics The heavy quark limit and the lattice Return to heavy quark background... Suggests a different perspective on the “brown muck” of the heavy quark effective field theory. Consider ground state of QCD in an infinitely heavy color-3 background. Defines diquarks and other correlations... Examples (assuming isospin symmetry, but not SU(3)): I=1/2 “quark” I=0 “diquark” I=1 “diquark” Heavy meson Heavy baryon

60. R L Jaffe Frascati May 2005 Multiquark Dynamics I=0 “triquark” I=1 “triquark” I=3/2 “triquark” Exotic heavy meson Exotic heavy baryon I=0 “tetraquark” versus two diquarks and many more examples Well defined lattice calculations Motivation for B meson and baryon spectroscopy Well defined lattice calculations Motivation for B meson and baryon spectroscopy

61. R L Jaffe Frascati May 2005 Multiquark Dynamics III.Aspects of multiquark dynamics

62. R L Jaffe Frascati May 2005 Multiquark Dynamics General picture Ordinary mesons (qq*) and baryons (qqq) --- zero width resonances when quark pair creation is suppressed. Properties modified with finite widths. Multiquark mesons (qqq*q*) and baryons (qqqqq*) --- part of the meson-meson or meson-baryon continuum, modified by QCD interactions. “Multiquark states fall apart if they are above threshold”. Should be treated in a dynamical picture where they revert to the non- interacting continuum as QCD interactions are turned off.

63. R L Jaffe Frascati May 2005 Multiquark Dynamics Large N c counting (for mesons and qqq*q*) Ordinary mesons Widths go to zero as 1/N c Meson-meson scattering goes to zero as N goes to infinity except at the narrow resonances qqq*q* RLJ SLAC PUB 951 (1981) by duality

64. R L Jaffe Frascati May 2005 Multiquark Dynamics So “binding” of exotic qqq*q* must vanish as N goes to infinity Study... So “binding” of exotic qqq*q* must vanish as N goes to infinity Study... where color, flavor, and spin couplings are suppressed, normed so Any D(x) can be decomposed in terms of color singlet bilinears However, in a fixed basis, eg. (12)(34) an arbitrary D(x) will include “hidden color”, ie qq*-octet, states

65. R L Jaffe Frascati May 2005 Multiquark Dynamics Large N counting Color adjoint mixing is O(1/N) Conclude In the limit, qqq*q* is unbound. It is merely the meson-meson continuum Conclude In the limit, qqq*q* is unbound. It is merely the meson-meson continuum Disconnected diagrams are O(1) Color singlet exchange is O(1/N 2 ) In order color correlations mix into the (qq*)(qq*) system

66. R L Jaffe Frascati May 2005 Multiquark Dynamics Immediate consequences: qqq*q* states should generically be very broad Exception would be if they are below natural decay thresholds If the N-dependence of meson-meson scattering can be studied --- eg. on the lattice or using chiral dynamics --- then the distinction between qqq*q* and qq* states should be clear: qq* states decouple by vanishing width at large N qqq*q* states subside into the continuum at large N Immediate consequences: qqq*q* states should generically be very broad Exception would be if they are below natural decay thresholds If the N-dependence of meson-meson scattering can be studied --- eg. on the lattice or using chiral dynamics --- then the distinction between qqq*q* and qq* states should be clear: qq* states decouple by vanishing width at large N qqq*q* states subside into the continuum at large N Note --- exactly this analysis has been performed for the scalar mesons in chiral effective theory by J. Peláez & collaborators Phys. Rev. Lett. 92 (2004) 102001, hep-ph/0307018, 0306063, 0411107

67. R L Jaffe Frascati May 2005 Multiquark Dynamics But in general, more phenomenological methods are needed... How should quark model eigenstates be represented in the hadron-hadron S-matrix? “Unitarization” When a state is studied in QCD models, its decays are usually ignored... quark models QCD sum rules chiral soliton models So they are “zero width approximations” When a state is studied in QCD models, its decays are usually ignored... quark models QCD sum rules chiral soliton models So they are “zero width approximations” Two classes of unitarization approaches interest me --- both are qualitative, neither has been developed in a fully relativistic, many channel world. Nevertheless they offer insight into the nature of the problem... Boundary condition approachesBoundary condition approaches Feshbach resonancesFeshbach resonances

68. R L Jaffe Frascati May 2005 Multiquark Dynamics Boundary condition approaches Wigner’s R-matrix Invented for neutron scattering from complex nuclei. At very low energies, (eg. thermal neutrons!) Because the wavelength outside the region of interaction is so long, the amplitude outside is >> inside, and to a good approximation the wavefunction vanishes at r = 0, whence the phase shift is zero, unless... The slope of the wavefunction vanishes at the edge of the nucleus. In that case the amplitudes inside and outside are equal and the phase shift is π/2.

69. R L Jaffe Frascati May 2005 Multiquark Dynamics Typical energy, small scale interaction region Typical energy, large scale Wavefunction outside is free and vanishes at origin, so Special energy where At this energy, the amplitudes of the wavefunctions outside and inside are the same. The phase shift at r = b is π/2:

70. R L Jaffe Frascati May 2005 Multiquark Dynamics View from outside At the energy where the slope of the wavefunction vanishes at the boundary of the interaction region Interior Hamiltonian has an eigenstate at E j obeying View from inside Phase shift is π/2 as measured from r=b at E j

71. R L Jaffe Frascati May 2005 Multiquark Dynamics R-Matrix formalism The boundary condition method connects the inside to the outside avoiding detailed statements about how the internal state couples to the scattering channel Inside: Solve the Hamiltonian eigenvalue equation subject to the funny boundary condition This provide the momenta at which the R- matrix has poles. [The residue of the poles in the R-matrix is related to dE/db.] Outside: the logarithmic derivative of the wavefunction at the endge of the interaction region suffices to construct the scattering state:

72. R L Jaffe Frascati May 2005 Multiquark Dynamics S(E) is unitary At very low energies, S(E) is approximately unity except near poles in R(b,E) S(E) is independent of b as long as interaction vanishes for r > b. This gives a constraint on the b-dependence of R, such that dS/db=0 (Potential scattering can be included for r > b) Poles in S(E) are narrow if poles in R(b,E) have small residue S(E) is unitary At very low energies, S(E) is approximately unity except near poles in R(b,E) S(E) is independent of b as long as interaction vanishes for r > b. This gives a constraint on the b-dependence of R, such that dS/db=0 (Potential scattering can be included for r > b) Poles in S(E) are narrow if poles in R(b,E) have small residue UNITARIZATION: assume a pole and residue... and get a finite width resonance in S(E) UNITARIZATION: assume a pole and residue... and get a finite width resonance in S(E) A beautiful formalism, but what is it’s relation to QCD, multiquark hadrons, meson-meson scattering, etc.?

73. R L Jaffe Frascati May 2005 Multiquark Dynamics The K-matrix... The standard approach to unitarization. Suppose that the interaction region is very small compared to the de Broglie wavelength of the scattering particles: Then, Approximate b = 0, Physics? On resonance the scattering wave appears already phase shifted by π/2 at the origin! A historical accident: Dalitz initiated the study of hadron scattering when hadrons were thought to be pointlike When a resonance is very narrow it doesn’t matter what boundary condition you impose on H-interior, the physics is insensitive A historical accident: Dalitz initiated the study of hadron scattering when hadrons were thought to be pointlike When a resonance is very narrow it doesn’t matter what boundary condition you impose on H-interior, the physics is insensitive That doesn’t mean it has the physics right! Hadrons are not pointlike, so the phase shift extrapolated in to the origin has no significance Quark model calculations do not approximate the boundary condition that the derivative of the wavefunction vanishes at the hadron’s surface. Hadrons are not pointlike, so the phase shift extrapolated in to the origin has no significance Quark model calculations do not approximate the boundary condition that the derivative of the wavefunction vanishes at the hadron’s surface.

74. R L Jaffe Frascati May 2005 Multiquark Dynamics Single channel S-wave example Pole in K-matrix occurs when phase shift is π/2. Certainly a reasonable description of a narrow resonance For example, Pole in K-matrix occurs when phase shift is π/2. Certainly a reasonable description of a narrow resonance For example, Choose any other boundary condition, and associated “matrix” will have a pole at an energy very close to K-matrix pole. So exact treatment is not necessary. Choose any other boundary condition, and associated “matrix” will have a pole at an energy very close to K-matrix pole. So exact treatment is not necessary.

75. R L Jaffe Frascati May 2005 Multiquark Dynamics But the energies at which the K-matrix has poles has no physical relation to the solutions to a QCD-motivated Hamiltonian eigenvalue equation! Particularly poorly motivated for channels where multiquark dynamics may be important But the energies at which the K-matrix has poles has no physical relation to the solutions to a QCD-motivated Hamiltonian eigenvalue equation! Particularly poorly motivated for channels where multiquark dynamics may be important What would be a better physical construction in QCD, where quarks are confined in hadrons of radius ~ 1 fermi?? Standard approach to finding resonances in hadron scattering data: 1.Assume the K-matrix is given by a sum of poles (plus a smooth background,...) II.Fit the resulting S-matrix to the data III.Send the resulting energies and residues of K-matrix poles to the PDG! Standard approach to finding resonances in hadron scattering data: 1.Assume the K-matrix is given by a sum of poles (plus a smooth background,...) II.Fit the resulting S-matrix to the data III.Send the resulting energies and residues of K-matrix poles to the PDG!

76. R L Jaffe Frascati May 2005 Multiquark Dynamics Quark model states resemble zeros in Wigner’s R- matrix (or poles in P=1/R) because they are confined at distance scales of order At a pole in P(b,E) Imagine this is the π π relative wave function RLJ and F. E. Low 1979 At a pole in P(b,E) On the inside Define P as the inverse of R-matrix Knowing P allows one to construct the scattering state This is for 1-channel, s-wave. Generalization is straightforward

77. R L Jaffe Frascati May 2005 Multiquark Dynamics Interpretation of P-matrix poles They occur at the energies at which the scattering wave function (eg. ππ) has a node at r = b, and therefore would match smoothly to internal confined quark state. However: ππ wavefunction has nodes at r = b even when there is no interaction! (Here is where things get interesting). Poles occur at k = nπ/b (in the s-wave). These locations of P- matrix poles correspond to no interaction. Interpretation: Four quark interactions shift P-matrix poles and change their residues. Two quark states (narrow) are added to the P-matrix Interpretation: Four quark interactions shift P-matrix poles and change their residues. Two quark states (narrow) are added to the P-matrix

78. R L Jaffe Frascati May 2005 Multiquark Dynamics In the case of no interaction --- the “reference” or “compensation P-matrix, P 0 Look at effect of shifting first pole in P up or down relative to its “compensation” value, k = π/b, (about 700 Mev for ππ scattering Upward shift in P-pole location corresponds to repulsion! and shows up as negative phase shift. ππ phase carries information about repulsive quark interactions Downward shift in P-pole location corresponds to attraction! and shows up as positive phase shift. ππ phase carries information about attractive quark interactions

79. R L Jaffe Frascati May 2005 Multiquark Dynamics P-matrix dynamics... Multiquark states are interpreted as information about the shifts in the poles in the compensating P-matrix. When a multiquark state is shifted down in mass by QCD interactions, the P-matrix pole energy is moved down, and the result is that the phase shift becomes positive in that region. The QCD attraction that shifted the multiquark mass down is observed as an attractive interaction in hadron-hadron scattering. And vica versa for a repulsive QCD interaction shifting a state up As the interaction goes away, the effects subside into the continuum QQ* or QQQ states are added to the P-matrix since they do not appear in the continuum when the interactions are turned off States like the rho meson are added to the compensating P-matrix and appear as narrow resonances (if their residues are small) For all it’s limitations, the P-matrix method encodes the correct large N behavior and physically links QCD interactions to hadron-hadron phase shifts. Multiquark states are interpreted as information about the shifts in the poles in the compensating P-matrix. When a multiquark state is shifted down in mass by QCD interactions, the P-matrix pole energy is moved down, and the result is that the phase shift becomes positive in that region. The QCD attraction that shifted the multiquark mass down is observed as an attractive interaction in hadron-hadron scattering. And vica versa for a repulsive QCD interaction shifting a state up As the interaction goes away, the effects subside into the continuum QQ* or QQQ states are added to the P-matrix since they do not appear in the continuum when the interactions are turned off States like the rho meson are added to the compensating P-matrix and appear as narrow resonances (if their residues are small) For all it’s limitations, the P-matrix method encodes the correct large N behavior and physically links QCD interactions to hadron-hadron phase shifts.

80. R L Jaffe Frascati May 2005 Multiquark Dynamics One pole shifted up since I=2 π π corresponds to bad-bad diquark P-matrix fits to π π scattering One pole shifted down since I=0 π π corresponds to good-good diquark One pole added since I=1 π π correspond s to a confined qq* state P-matrix fits to π π scattering

81. R L Jaffe Frascati May 2005 Multiquark Dynamics Feshbach resonances and confinement The provides an unique opportunity to consider how a hadron should be represented in the S- matrix at very low energies 1.Quasi-non-relativistic 2.Only one channel open A problem in Schrödinger quantum mechanics How do you get striking effects at low energy in non-relativistic quantum mechanics? Ordinary non-relativistic potential scattering Bound state in a closed (= confined) channel in the continuum... A phenomenon first discovered by U. Fano (1935), now known as a Feshbach resonance

82. R L Jaffe Frascati May 2005 Multiquark Dynamics What sort of effects are created by potential scattering in low partial waves at low energies? Resonance results from interplay of attractive interation and long range angular momentum barrier. No potential resonances in the s-wave Potential scattering resonances are typically narrow only when 〈 kb 〉≪ 1 For example, to make a resonance at the mass of the Θ Resonance results from interplay of attractive interation and long range angular momentum barrier. No potential resonances in the s-wave Potential scattering resonances are typically narrow only when 〈 kb 〉≪ 1 For example, to make a resonance at the mass of the Θ

83. R L Jaffe Frascati May 2005 Multiquark Dynamics How, then, can low energy, narrow hadronic resonances appear in low partial waves? Bound state in the continuum (Feshbach-Fano) As the confined state decouples (zero width) Interaction “off resonance” can be arbitrarily weak without removing resonance. Confined channel is absent from continuum. For a discussion applied to Theta, see RLJ and A. Jain, hep-ph/0408046 For a discussion applied to Theta, see RLJ and A. Jain, hep-ph/0408046

84. R L Jaffe Frascati May 2005 Multiquark Dynamics Conjecture: i.Ordinary (qq* and qqq) hadrons are bound states in confined channels that couple to scattering channel through quark pair creation. In low energy scattering they would manifest as Feshbach resonances. ii.Multiquark hadrons are (generically) modulations of the continuum in open channels. In non-relativistic limit they would influence the open-channel potential. iii.Consistent with P-matrix description. Conjecture: i.Ordinary (qq* and qqq) hadrons are bound states in confined channels that couple to scattering channel through quark pair creation. In low energy scattering they would manifest as Feshbach resonances. ii.Multiquark hadrons are (generically) modulations of the continuum in open channels. In non-relativistic limit they would influence the open-channel potential. iii.Consistent with P-matrix description.

85. R L Jaffe Frascati May 2005 Multiquark Dynamics Conclusions The absence of exotics in the hadron spectrum is now qualitatively understood. The Theta may have died, but it refocused interest in the problem of correlations in QCD. Diquarks are the obvious candidates for spectroscopic correlations The absence of exotics in the hadron spectrum is now qualitatively understood. The Theta may have died, but it refocused interest in the problem of correlations in QCD. Diquarks are the obvious candidates for spectroscopic correlations Lots of phenomenological evidence for diquarks Absence of exotics and scalar mesons Role in DIS --- qualitative --- fragmentation functions and higher twist. More to be done Lots of phenomenological evidence for diquarks Absence of exotics and scalar mesons Role in DIS --- qualitative --- fragmentation functions and higher twist. More to be done A systematic study of correlations in QCD: light quark spectroscopy in the color non-singlet sectors What is the correct dynamical framework for analyzing low energy data to make contact with QCD? A systematic study of correlations in QCD: light quark spectroscopy in the color non-singlet sectors What is the correct dynamical framework for analyzing low energy data to make contact with QCD?