hybrids (mesons and baryons) JLab Advanced Study Institute

Transcription

1 hybrids (mesons and baryons)

2 the resonance spectrum of QCD or, where are you hiding the scattering amplitudes?

3 real QCD real QCD has very few stable particles : states like ρ, &... are resonances asymptotic states of the theory include multi-pion states e.g. in the CM frame with varying continuously up from zero e.g. ππ scattering in isospin-1 π + π + π 0... π 0 p p 73

4 real QCD real QCD has very few stable particles : states like ρ, &... are resonances asymptotic states of the theory include multi-pion states e.g. in the CM frame with varying continuously up from zero e.g. ππ scattering in isospin-1 π + π + p π 0 for small t p π 0 74

5 real QCD real QCD has very few stable particles : states like ρ, &... are resonances asymptotic states of the theory include multi-pion states e.g. in the CM frame with varying continuously up from zero within the field-theory we have correlators e.g. the (Euclidean) vector correlator spectral function the spectrum is continuous! 75

6 real QCD real QCD has very few stable particles : a lattice calc states like ρ, &... are resonances asymptotic states of the theory include multi-pion states 2500 e.g. in the CM frame 2000 with within the field-theory we have correlators 1500 e.g. the (Euclidean) vector correlator 1000 a discrete spectrum!? spectral function varying continuously up from zero the spectrum is continuous! 76

7 field theory in a finite volume consider the case of one space dimension with a periodic boundary condition this will make the allowed momenta of a free particle discrete : free particle periodic boundary condition for integer 77

8 non-interacting two-particle states in a finite volume in a three-dimensional cubic box non-interacting spectral function the spectrum is discrete and volume-dependent 78

9 two-particle states in a finite volume but hadrons do interact, and sometimes strongly e.g. what is the impact of these interactions on the finite-volume spectrum? 79

10 two-particle states in a finite volume e.g. non-rel quantum mechanics in one-dimension two spinless bosons separated by scattering solutions of interacting through a potential in center-of-momentum frame finite length world with periodic b.c. wavefunction and derivative continuous at boundary 80

11 two-particle states in a finite volume e.g. quantum mechanics in one-dimension discrete & volume-dependent spectrum of scattering states non-interacting momentum shift due to interaction e.g. a weak attraction 81

12 two-particle states in a finite volume the analogous expression for two-particle elastic scattering in a finite cubic volume has been derived by Lüscher somewhat complicated by the lack of full rotational symmetry (a cube) for our purposes, we ll pretend that it s as simple as phase-shift in partial wave at scattering energy known function of energy and box length so we measure on one or more volumes and plug into the formula to determine at discrete energies 82

13 ππ isospin-2 scattering e.g. π + π + π + π empirically: weak & repulsive scattering

14 ππ isospin-2 scattering - field theory calculation no quark-annihilation in these correlators variational in a basis of operators : ππ of various relative momenta e.g. 84

15 ππ isospin-2 scattering Hadron Spectrum Collaboration arxiv: variational in a basis of operators : ππ of various relative momenta e.g extracted energies shifted upward slightly from non-interacting pion pairs

16 ππ isospin-2 scattering S-wave scattering phase-shift Ecm / MeV mπ~400 MeV 86

17 ππ isospin-2 scattering S-wave scattering phase-shift mπ~400 MeV 87

18 ππ isospin-2 scattering S-wave scattering phase-shift mπ~400 MeV 88

19 ππ isospin-2 scattering S-wave scattering phase-shift mπ~400 MeV 89

20 ππ isospin-2 scattering S-wave scattering phase-shift mπ~400 MeV 90

21 ππ isospin-2 scattering D-wave scattering phase-shift mπ~400 MeV 91

22 ππ isospin-2 scattering summary (with effective range fits) mπ~400 MeV 92

23 ππ isospin-2 scattering lattice QCD : mπ~400 MeV experimental compendium

24 ππ isospin-1 scattering more interesting scattering channels are those featuring resonances e.g. the ρ in ππ

25 ππ isospin-1 scattering more interesting scattering channels are those featuring resonances

26 ππ isospin-1 scattering expected finite-volume spectrum given a ρ resonance 1000 mπ=139 MeV mρ=770 MeV Γρ=150 MeV

27 ππ isospin-1 scattering a strongly volume-dependent spectrum

28 ππ isospin-1 scattering expected finite-volume spectrum given a ρ resonance 1000 mπ=139 MeV mρ=770 MeV Γρ=150 MeV

29 ππ isospin-1 scattering expected finite-volume spectrum given a ρ resonance level repulsion

30 ππ isospin-1 scattering expected finite-volume spectrum given a ρ resonance 1000 mπ=139 MeV mρ=770 MeV Γρ=150 MeV

31 ππ isospin-1 scattering expected finite-volume spectrum given a ρ resonance 1000 mπ=139 MeV mρ=770 MeV Γρ=150 MeV strongly volume dependent spectrum

32 the spectrum using local operators - T1 mπ~400 MeV no systematic volume dependence observed in the spectrum L=16 L=20 L=24 102

33 the spectrum using local operators - T1 mπ~400 MeV non-interacting ππ energies L=16 L=20 L=24 103

34 the spectrum using local operators - T1 mπ~400 MeV L=16 L=20 L=24 no sign of any multi-meson physics? 104

35 the spectrum using local operators we suspect that this effect can be understood hypothesise that energy eigenstates are superpositions of a state, call it non-interacting basis states hypothesise that local operators have a suppressed overlap onto by at least a factor of 1/ L 3 105

36 ππ isospin-1 scattering expected finite-volume spectrum given a ρ resonance level repulsion a simple two-state mixing model:

37 the spectrum using local operators a simple two-state mixing model: if we only use operators which overlap well with and not with then a variational solution won t be able to find the orthogonal combinations the principal correlator will behave like

38 ππ isospin-1 scattering - lattice calculation do it properly! operator basis: usual big set of derivative-based fermion bilinears ππ-like operators of definite relative momentum need quark annihilation diagrams e.g. 108

39 [100] A1 - with & without ππ operators local vector operators and ππ operators just local vector operators (5) (6) (5) Lattice spectroscopy overview 109

40 [100] A1 - with & without ππ operators local vector operators and ππ operators just local vector operators (5) (6) (5) Lattice spectroscopy overview

41 ππ isospin-1 scattering - a lattice calculation PRELIMINARY Hadron Spectrum Collaboration unpublished mρ=854(2) MeV Γρ=8.5(5) MeV T1mm D4A1m D4E2m D2A1m D2B1m D2B2m D3A1m D3E2m D4A1m T1mm D4A1m D4E2m D2A1m D2B1m D2B2m D3A1m D3E2m mπ=396 MeV MeV 111

42 ππ isospin-1 scattering - a lattice calculation barrier factors required to get this region right mπ=396 MeV k 20 3 behaviour required here a narrow resonance mρ=854(2) MeV Γρ=8.5(5) MeV R=1.25(15) fm MeV 112

43 the spectrum using local operators - T1 mπ~400 MeV L=16 L=20 L=24 113

44 ππ isospin-1 scattering - pion mass dependence European Twisted Mass Collaboration PRD83 (2011) two-flavour calculation (no strange quarks) four different quark masses setting the lattice scale? computed in relatively few frames mπ=480 MeV mρ=1118(14) MeV Γρ=40(8) MeV

45 ππ isospin-1 scattering - pion mass dependence mπ=480 MeV 100 mπ=420 MeV mρ=1118(14) MeV Γρ=40(8) MeV mρ=1047(15) MeV Γρ=55(11) MeV mπ=330 MeV mρ=1033(31) MeV Γρ=123(43) MeV mπ=290 MeV mρ=980(31) MeV Γρ=156(41) MeV

46 forthcoming resonance calculations computationally challenging: pion-nucleon elastic scattering (& resonance) lots of quark lines lots of matrix multiplication... computer time 116

47 forthcoming resonance calculations computationally challenging: pion-nucleon elastic scattering (& resonance) requires untested formalism: _ meson-meson inelastic scattering (e.g a0 in πη-kk ) e.g. 2-channel inelastic scattering finite-volume formalism three real numbers at each scattering energy measured three unknowns 117

48 forthcoming resonance calculations computationally challenging: pion-nucleon elastic scattering (& resonance) requires untested formalism: _ meson-meson inelastic scattering (e.g a0 in πη-kk ) three-meson decays, e.g. ω,a1,a2... πππ all the complications of building unitary, analytic scattering amplitudes present for experiment! 118