Baryon Resonance Determination using LQCD. Robert Edwards Jefferson Lab. Baryons PDF Free Download

Baryon Resonance Determination using LQCD Robert Edwards Jefferson Lab Baryons 2013

Where are the Missing Baryon Resonances? What are collective modes? Is there freezing of degrees of freedom? What is the structure of the states? 2

Where are the Missing Baryon Resonances? What are collective modes? Is there freezing of degrees of freedom? What is the structure of the states? Nucleon & Delta spectrum PDG uncertainty on B-W mass Nucleon (Exp): 4*, 3*, some 2* Delta (Exp): 4*, 3*, some 2* 2400 N 2400 Δ 2200 2200 2000 2000 Mass (MeV) 1800 1600 Mass (MeV) 1800 1600 1400 1400 1200 1200 1000 1000 800 1/2 + 3/2 + 5/2 + 7/2 + 9/2 + 1/2-3/2-5/2-7/2-9/2-800 1/2 + 3/2 + 5/2 + 7/2 + 1/2-3/2-5/2-7/2-3

Where are the Missing Baryon Resonances? What are collective modes? Is there freezing of degrees of freedom? What is the structure of the states? QM predictions Nucleon & Delta spectrum PDG uncertainty on B-W mass Nucleon (Exp): 4*, 3*, some 2* Delta (Exp): 4*, 3*, some 2* 2400 N 2400 Δ 2200 2200 2000 2000 Mass (MeV) 1800 1600 Mass (MeV) 1800 1600 1400 1400 1200 2 2 1 1200 1 1 0 1000 1000 800 1/2 + 3/2 + 5/2 + 7/2 + 9/2 + 1/2-3/2-5/2-7/2-9/2-800 1/2 + 3/2 + 5/2 + 7/2 + 1/2-3/2-5/2-7/2-4

Where are the Missing Baryon Resonances? What are collective modes? Is there freezing of degrees of freedom? What is the structure of the states? QM predictions Nucleon & Delta spectrum PDG uncertainty on B-W mass Nucleon (Exp): 4*, 3*, some 2* Delta (Exp): 4*, 3*, some 2* 2400 N 2400 Δ 2200 2000??? 2200 2000??? Mass (MeV) 1800 1600 Mass (MeV) 1800 1600 1400 1200 4 5 3 1 2 2 1 1400 1200 2 3 2 1 1 1 0 1000 1000 800 1/2 + 3/2 + 5/2 + 7/2 + 9/2 + 1/2-3/2-5/2-7/2-9/2-800 1/2 + 3/2 + 5/2 + 7/2 + 1/2-3/2-5/2-7/2-5

Where are the Missing Baryon Resonances? Where is the glue? qqq G Not exotic: Different/conflicting predictions 2400 2200 Nucleon (Exp): 4*, 3*, some 2* N??? 2400??? 2200 Delta (Exp): 4*, 3*, some 2* Δ 2000 2000 Mass (MeV) 1800 1600 1400??? Mass (MeV) 1800 1600 1400 1200 1200 1000 1000 800 1/2 + 3/2 + 5/2 + 7/2 + 9/2 + 1/2-3/2-5/2-7/2-9/2-800 1/2 + 3/2 + 5/2 + 7/2 + 1/2-3/2-5/2-7/2-6

Strange Quark Baryon Spectrum Strange quark baryon spectrum even sparser 2400 Lambda Mass Spectrum (Exp): 4*, 3* Λ 2400 Sigma (Exp): 4*, 3* Ξ 2200 2200 2000 2000 Mass (MeV) 1800 1600 Mass (MeV) 1800 1600 1400 1400 1200 1200 1000 1000 800 1/2 + 3/2 + 5/2 + 7/2 + 1/2-3/2-5/2-7/2-800 1/2 + 3/2 + 5/2 + 7/2 + 1/2-3/2-5/2-7/2 -

Strange Quark Baryon Spectrum Strange quark baryon spectrum even sparser Since SU(3) flavor symmetry broken, expect mixing of 8 F & 10 F 2400 2200??? Lambda Mass Spectrum (Exp): 4*, 3* Λ 2400 2200??? Sigma (Exp): 4*, 3* Ξ 2000 2000 Mass (MeV) 1800 1600 1400 1200 2 3 2 1 Mass (MeV) 1800 1600 1400 1 1 0 6 8 5 2 1200 3 3 1 1000 1000 800 1/2 + 3/2 + 5/2 + 7/2 + 1/2-3/2-5/2-7/2-800 1/2 + 3/2 + 5/2 + 7/2 + 1/2-3/2-5/2-7/2 - Even less known states in Ξ & Ω

QCD What we need is guidance from QCD 9

QCD What we need is guidance from QCD Several efforts internationally to compute excited spectrum via lattice QCD: Adelaide Graz/Ljubljana/FNAL/TRIUMF Hadron Spectrum Collab. (Jlab/Trinity College/CMU/Maryland/Tata/Cambridge) Kentucky BMW (Bielefeld/Marseille/Wuppertal) low lying spectrum 10

What is a gauge theory? Maxwell s eqns: field strength tensor and vector potentials Action

QCD Dirac operator: A (vector potential), m (mass), γ (4x4 matrices) D(A, m) = X @ µ + iga µ (x) + m @x µ µ Observables QCD: Vector potentials now 3x3 complex matrices (SU(3)) Lattice QCD: finite difference Lots of flops/s Harness GPU-s

Spectrum from variational method Two-point correlator 13

Spectrum from variational method Two-point correlator Matrix of correlators 14

Spectrum from variational method Two-point correlator Matrix of correlators Rayleigh-Ritz method Diagonalize: eigenvalues è spectrum eigenvectors è spectral overlaps Z i n 15

Spectrum from variational method Two-point correlator Matrix of correlators Rayleigh-Ritz method Diagonalize: eigenvalues è spectrum eigenvectors è spectral overlaps Z i n Each state optimal combination of Φ i 16

Operators Mesons: fermion bi-linears J = 0, 1 J = 0, 1, 2 gauge-covariant derivatives ~ 1 -- J = 0, 1, 2, 3 J = 0, 1, 2, 3, 4 2 derivatives can give chromo B field 1 +-

Operators Mesons: fermion bi-linears J = 0, 1 J = 0, 1, 2 gauge-covariant derivatives ~ 1 -- J = 0, 1, 2, 3 J = 0, 1, 2, 3, 4 2 derivatives can give chromo B field 1 +- Baryons: three quarks J,j = h1l 1 ;1l 2 LlihLl; Ss Jji ~ D l1 ~ Dl2 [ ] s

Spin identified Nucleon & Delta spectrum arxiv:1104.5152, 1201.2349

Spin identified Nucleon & Delta spectrum Full non-relativistic quark model counting arxiv:1104.5152, 1201.2349 4 5 3 1 2 3 2 1 2 2 1 1 1

Interpreting content Spectral overlaps give clue as to content of states Large contribution from gluonicbased operators on states identified as having hybrid content

Hybrid baryons Negative parity structure replicated: gluonic components (hybrid baryons) [70,1 + ] P-wave [70,1 - ] P-wave 23

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

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

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

SU(3) flavor limit In SU(3) flavor limit have exact flavor Octet, Decuplet and Singlet representations 8 F 10 F 1 F Full non-relativistic quark model counting m 700 MeV arxiv:1212.5236 Additional levels with significant gluonic components

Light quarks SU(3) flavor broken Light quarks - other isospins m 391 MeV Full non-relativistic quark model counting Some mixing of SU(3) flavor irreps arxiv:1212.5236

Scattering in finite volume field theory The idea: 1 dim quantum mechanics Two spin-less bosons: ψ(x,y) = f(x-y) -> f(z) Solutions f(p)! cos[p z + (p)], E = p 2 /m Quantization condition when -L/2 < z < L/2 pl +2 (p) =n mod 2 Same physics in 4 dim version (but messier) Provable in a QFT (and relativistic) p = 2 L n 2 L (p) non-int mom dynamical shift

Scattering Experimentally - determine amplitudes as function of energy E E.g. just a single elastic resonance e.g. 32

Scattering (in finite volume!) Scattering in a periodic cubic box (length L) E.g. just a single elastic resonance e.g. At some L, have discrete excited energies 33

Scattering (in finite volume!) Scattering in a periodic cubic box (length L) E.g. just a single elastic resonance e.g. At some L, have discrete excited energies Scattering matrix amplitudes in partial waves Finite volume energy levels E(L) -> δ(e) 34

Single channel elastic scattering Isospin=1: ππ!! arxiv:1212.0830 Progress: now move on to the interesting cases!

Coupling in Isospin =1 ππ Scattering of composite objects in non-perturbative field theory m = 481 MeV Extracted coupling: stable in pion mass m = 421 MeV g m = 330 MeV m 2 GeV 2 m = 290 MeV Feng, et.al, 1011.5288 Stability a generic feature of couplings??

Extension to inelastic scattering Toy model of two channel scattering: K-matrix: single pole + polynomial in s, Spectrum on a 3.2fm lattice 180 150 100 50 0 0.4 0.6 0.8 1.0 1.2 1.4 1.0 0.9 0.8 arxiv:1211.0929

Extension to inelastic scattering Toy model of two channel scattering: K-matrix: single pole + polynomial in s 180, Spectrum on a 3.2fm lattice w 150 100 50 0 0.4 0.6 0.8 1.0 1.2 1.4 1.0 0.9 0.8 w w w w w w Lattice calculation must map energy dependence Need multiple excited states arxiv:1211.0929

Some candidates: determine phase shift Somewhat elastic Hadronic Decays Δè [Nπ] P S 11 è [Nπ] S +[Nη] S First study in of S 11 arxiv:1304.4114 (Graz) 39

Summary & prospects First picture of highly excited spectrum from lattice QCD Broadly consistent with non-relativistic quark model Extra bits interpreted as hybrid states with color octet (magnetic) structure Electric field structure higher in energy Add multi-particle ops - spectrum becomes denser Observe significant overlap of hybrid stucture with ground level Could have other consequences Path forward: resonance determination! Obviously, lighter pion masses needed 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? 40

The details The end 41

Interpreting content Spectral overlaps give clue as to content of states

Extension to inelastic scattering Elastic case: method extends to higher partial waves 20 cot 1 (p) 6B 0 = det 4@ 1 cot 3 (p) C A... 3 pl 7 M 5 2 Matrix of known functions (in cubic irreps Λ) 4-momentum from lattice Inelastic case: can generalize to a scattering t-matrix where and apple 0 = det t (l) ij (E cm) i (E cm )=2 p i(e cm ) E cm t 1 ij + i i ij ijm pi L 2 is the scattering t-matrix Im(t 1 ij )= is the phase-space for channel i ij i Channels labelled by i,j E cm E (i) threshold Underconstrained problem: one energy level many scatt. amps to determine e.g., arxiv:1211.0929

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 hn J µ N i 1 (Q 2,E) [ 0 (E)+ 0 (E)] hn J µ N i volume Phase shift Kinematic factor E Requires excited level transition FF s: some experience Charmonium E&M transition FF s (1004.4930) Nucleon 1 st attempt: Roper ->N (0803.3020) Range: few GeV 2 Limitation: spatial lattice spacing

Baryon Resonance Determination using LQCD. Robert Edwards Jefferson Lab. Baryons 2013