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Hadron Resonance Determination Robert Edwards Jefferson Lab ECT 2014 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAAAA
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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?
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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
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Quantum mechanics on a circle One-dimensional motion with periodic boundary conditions A free particle –Periodic boundary condition Discrete energy spectrum
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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
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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)
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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
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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
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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
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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
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How does it work? Experimental ππ I=2 S-wave scattering amp. S-WAVE PHASE SHIFT CUBIC BOX SPECTRUM
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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
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How does it work? Experimental ππ I=2 S-wave scattering amp. –A “weak” repulsive interaction S-WAVE PHASE SHIFTCUBIC BOX SPECTRUM
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How does it work? Experimental ππ I=2 S-wave scattering amp. –A “weak” repulsive interaction S-WAVE PHASE SHIFTCUBIC BOX SPECTRUM
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How does it work? Experimental ππ I=2 S-wave scattering amp. –A “weak” repulsive interaction S-WAVE PHASE SHIFTCUBIC BOX SPECTRUM
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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
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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
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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
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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
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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
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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
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Isospin=2 J P =0 + Finite-volume spectrum with
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Isospin=2 J P =0 + Finite-volume spectrum non-interacting spectrum with
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Isospin=2 J P =0 + phase-shift Significant extra information from the spectrum in moving frames arxiv:1203.6041
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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
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Isospin=1 J PC =1 -- In the elastic scattering region threshold arxiv:1212.0830
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Isospin=1 J PC =1 -- Need energy dependent functional form : use a Breit-Wigner parameterization arxiv:1212.0830 parameters m R and g
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Isospin=1 J PC =1 -- Breit-Wigner fit to the energy dependence BREIT-WIGNER Reduced width from small phase-space arxiv:1212.0830
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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,…
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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
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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,…
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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
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Isospin=1/2 πK/ηK scattering Two channel scattering: arXiv:1406.4158
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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
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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
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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)
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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
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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
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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
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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]
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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
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Baryon spectrum Positive parity baryons: counting SU(6)xO(3) arXiv:1201.2349 “Hybrid” excitation ~ 1.3GeV
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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
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Need “broad” operator basis For variational method Need operators that overlap well with relevant basis states qqbar-like levels shift within hadronic width
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Multi-particle operator basis # levels increases with moving frames and more operators qbar-q only ops – levels within hadronic width
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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
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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
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Matrix elements: How about the N Δ transition form-factor? sum over eigenstates in this finite-volume
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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
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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
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Pilot project: ρ γ Transition form-factor: compute determine
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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 (?)
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The details… The end 53
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Isospin=2 J P =0 Possible finite-volume operators –Now see the physical motivation for these operators “resemble” ΠΠ scattering states
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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
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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
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Matrix elements: How about the N Δ transition form-factor? sum over eigenstates in this finite-volume
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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
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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
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Spin identified Nucleon & Delta spectrum arXiv:1104.5152, 1201.2349
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Spin identified Nucleon & Delta spectrum arXiv:1104.5152, 1201.2349 Full non-relativistic quark model counting 4531 23 2 1 221 11
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Interpreting content “Spectral overlaps” give clue as to content of states Large contribution from gluonic- based operators on states identified as having “hybrid” content
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Spin identified Nucleon & Delta spectrum arXiv:1104.5152, 1201.2349 Interpretation of level content from “spectral overlaps” 4531 23 2 1 221 11
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Hybrid baryons 64 Negative parity structure replicated: gluonic components (hybrid baryons) [ 70,1 + ] P-wave [ 70,1 - ] P-wave
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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
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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
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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
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Where are the “Missing” Baryon Resonances? 68 N Δ PDG uncertainty on B-W mass Nucleon & Delta spectrum
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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
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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
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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
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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?)
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QCD QCD is (probably) underlying theory of hadrons via quarks and gluons –Coupling becomes large at low energy scales –Non-perturbative dynamics QCD coupling
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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,…
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QCD: Quantum Chromdynamics Dirac operator: A (vector potential), m (quark mass), γ (Dirac gamma matrices) Observables
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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
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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
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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
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Variational method Can construct optimal linear combination from eigenvectors 0 −+ EFFECTIVE MASSES
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Example: charmonium excited spectrum Large c-cbar operator basis & variational method arxiv:1204.5425
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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
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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
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Resonances Most hadrons are resonances –E.g., a bump in elastic hadron-hadron scattering
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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
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Scattering 85 E.g. just a single elastic resonance e.g. Experimentally - determine amplitudes as function of energy E
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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)
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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
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Single channel elastic scattering Isospin=1: ππ arXiv:1212.0830
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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??
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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
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Need “broad” operator basis For variational method Need operators that overlap well with relevant basis states
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Contractions Cost to produce correlators driven by contractions Propagators Operators Many permutations
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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)
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Interpreting content “Spectral overlaps” give clue as to content of states Large contribution from gluonic- based operators on states identified as having “hybrid” content
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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
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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
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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
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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
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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
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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
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Spectrum from variational method Matrix of correlators Two-point correlator 101
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Spectrum from variational method Two-point correlator 102
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Spectrum from variational method Matrix of correlators “Rayleigh-Ritz method” Diagonalize: eigenvalues spectrum eigenvectors spectral “overlaps” Z i n Two-point correlator 103
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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
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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, …)
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Excited hadrons are resonances Decay thresholds open (even for 400 MeV pions) PRD82 034508 (2010) arXiv:1309.2608 ππ continuum of ππ states ?
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Excited hadrons are resonances ππ KK _ Decay thresholds open (even for 400 MeV pions) PRD82 034508 (2010) arXiv:1309.2608
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Patterns in baryon spectrum
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