Baryon Resonance Determination using LQCD Robert Edwards Jefferson Lab Baryons 2013 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAAAA

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

Where are the “Missing” Baryon Resonances? 3 What are collective modes? Is there “freezing” of degrees of freedom? What is the structure of the states? N Δ PDG uncertainty on B-W mass Nucleon & Delta spectrum

Where are the “Missing” Baryon Resonances? 4 What are collective modes? Is there “freezing” of degrees of freedom? What is the structure of the states? PDG uncertainty on B-W mass Nucleon & Delta spectrum QM predictions N Δ

Where are the “Missing” Baryon Resonances? 5 What are collective modes? Is there “freezing” of degrees of freedom? What is the structure of the states? QM predictions ??? ??? N Δ PDG uncertainty on B-W mass Nucleon & Delta spectrum

Where are the “Missing” Baryon Resonances? 6 Where is the glue? ??? Not exotic: Different/conflicting predictions N Δ

Strange Quark Baryon Spectrum Strange quark baryon spectrum even sparser ΞΛ

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

QCD 9 What we need is guidance from QCD

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

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) Observables Lattice QCD: finite difference QCD: Vector potentials now 3x3 complex matrices (SU(3)) Lots of “flops/s” Harness GPU-s

Spectrum from variational method Two-point correlator 13

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

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

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

Spectrum from variational method Matrix of correlators “Rayleigh-Ritz method” Diagonalize: eigenvalues  spectrum eigenvectors  spectral “overlaps” Z i n Benefit: orthogonality for near degenerate states Two-point correlator 17 Each state optimal combination of Φ i

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

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

Spin identified Nucleon & Delta spectrum arXiv: ,

Spin identified Nucleon & Delta spectrum arXiv: , Full non-relativistic quark model counting

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

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

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

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

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

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

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:

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:

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

Scattering in finite volume field theory Solutions Quantization condition when -L/2 < z < L/2 Same physics in 4 dim version (but messier) Provable in a QFT (and relativistic) Two spin-less bosons: ψ(x,y) = f(x-y) -> f(z) The idea: 1 dim quantum mechanics non-int momdynamical shift

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

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

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

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

Coupling in Isospin =1 ππ Scattering of composite objects in non-perturbative field theory Feng, et.al, Extracted coupling: stable in pion mass Stability a generic feature of couplings??

Extension to inelastic scattering Toy model of two channel scattering: arXiv: K-matrix: single pole + polynomial in s Spectrum on a 3.2fm lattice

Extension to inelastic scattering Toy model of two channel scattering: arXiv: K-matrix: single pole + polynomial in s Spectrum on a 3.2fm lattice Lattice calculation must map energy dependence Need multiple excited states     

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

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 40 Path forward: resonance determination! Obviously, lighter pion masses needed Observe significant overlap of hybrid stucture with ground level Could have other consequences… 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?

The details… The end 41

Extension to inelastic scattering Elastic case: method extends to higher partial waves Matrix of known functions (in cubic irreps Λ) 4-momentum from lattice e.g., arXiv:

Interpreting content “Spectral overlaps” give clue as to content of states

Extension to inelastic scattering Elastic case: method extends to higher partial waves Matrix of known functions (in cubic irreps Λ) Inelastic case: can generalize to a scattering t-matrix 4-momentum from lattice Channels labelled by i,j where is the scattering t -matrix and is the phase-space for channel i Underconstrained problem: one energy level – many scatt. amps to determine e.g., arXiv:

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 E Requires excited level transition FF’s: some experience Charmonium E&M transition FF’s ( ) Nucleon 1 st attempt: “Roper”->N ( ) Range: few GeV 2 Limitation: spatial lattice spacing Kinematic factor Phase shift