hadron resonance determination robert edwards jefferson lab ect 2014 texpoint fonts used in emf....
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Hadron Resonance Determination
Robert Edwards Jefferson Lab
ECT 2014
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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 HQCD
• 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 mom dynamical 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 mom dynamical 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 JP=2 spectrum would look like
ππ
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-interactingspectrum
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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-interactingspectrum
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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 JP=0+
• Finite-volume spectrum
with
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Isospin=2 JP=0+
• Finite-volume spectrum
non-interactingspectrum
with
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Isospin=2 JP=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 JPC=1--
• In the elastic scattering region
threshold
arxiv:1212.0830
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Isospin=1 JPC=1--
• Need energy dependent functional form : use a Breit-Wigner parameterization
arxiv:1212.0830
parameters mR and g
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Isospin=1 JPC=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, JP=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, JP=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, JP=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 = Ecm2
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 = Ecm2
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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 = Ecm2
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 JP = 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 JP=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 K0*(1430)
• D-wave, narrow width, mostly couples to πK• Similar to K2
*(1430)
RESONANCE POLE POSITION[S]
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Where’s the big answer for the spectrum?
Current reality: Meson results are forth comingHowever, 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 spectrumPositive parity baryons: counting SU(6)xO(3) arXiv:1201.2349
“Hybrid” excitation ~ 1.3GeV
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πN thr.
ππN thr.
Baryon spectrumPositive 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 basisFor 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• JP=2+ & 1- Narrow interaction region: old results within width• JP=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,JP) = (½, ½+) channel
excit
ed
state
cont
ribut
ion
s
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Matrix elements: • How about the NΔ 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-volumespectrum
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Matrix elements: • How about the NΔ transition form-factor?
L
∞⤳
Need demonstration of formalism for Q2>0
• Helicity amplitudes at discrete W, Q2 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-volumespectrum
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Pilot project: ργπ • 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 JP=0
• Possible finite-volume operators
– Now see the physical motivation for these operators• “resemble” ΠΠ scattering states
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Isospin=1 JPC=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,JP) = (½, ½+) channel
excit
ed
state
cont
ribut
ion
s
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Matrix elements: • How about the NΔ 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-volumespectrum
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Matrix elements: • How about the NΔ transition form-factor?
L
∞⤳
Need demonstration of formalism for Q2>0
• Helicity amplitudes at discrete W, Q2 values
• Should be able to calculate the amplitudes at discrete W, Q2 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, 1201.2349Full non-relativistic quark model counting
4 5 3 1 2 3 2 1
2 2 1 1 1
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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, 1201.2349Interpretation of level content from “spectral overlaps”
4 5 3 1 2 3 2 1
2 2 1 1 1
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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 countingAdditional 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 02 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 8F & 10F
3 3 1
Even less known states in Ξ & Ω
Λ Ξ
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Volume dependence: isoscalar mesons
Energies determined from single-particle operators:Range of JPC - color indicates light-strange flavor mixing
Some volume dependence:
Interpretation: energies determined up to a hadronic widtharXiv: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 NΣ,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 differenceLots 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 JPC and JP 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 resonancee.g.
Experimentally - determine amplitudes as function of energy E
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Scattering - in finite volume!
E.g. just a single elastic resonancee.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 scatteringIsospin=1: ππ
arXiv:1212.0830
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Coupling in Isospin =1 ππComparison 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 1st attempt: “Roper”->N (0803.3020)
EKinematic factor
Phase shift
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Need “broad” operator basisFor 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 resonancee.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’ gluonin S-wave
`constituent’ gluonin 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’ gluonin S-wave
`constituent’ gluonin 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 JPC= 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” Zi
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” Zi
n
Two-point correlator
104
Each state optimal combination of Φi
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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, JP=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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Patterns in baryon spectrum