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