Hadron Resonance Determination Robert Edwards Jefferson Lab ECT 2014 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.:

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


– A “weak” repulsive interaction

S-WAVE PHASE SHIFT

CUBIC BOX SPECTRUM

non-interactingspectrum



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



P-WAVE PHASE SHIFT

CUBIC BOX SPECTRUM




P-WAVE PHASE SHIFT

CUBIC BOX SPECTRUM


• Experimental ππ I=1 P-wave scattering amp.– Artificially narrow ρ resonance

P-WAVE PHASE SHIFT

CUBIC BOX SPECTRUM


• 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


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,…





e.g., arXiv:1211.0929


arXiv: 0504019, 1010.6018, 1204.0826, 1204.6256, 1305.4903,…

Couples channels i,j – diagonal in l

Couples partial waves l



• Problem is that this is one equation in multiple unknowns– One approach is to parameterize the t-matrix

» “Energy-dependent” analysis



e.g., arXiv:1211.0929


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


T-matrix: account of threshold behavior

K-matrix: pole + polynomial in s = Ecm2

Ensure unitary:

Chew-Mandelstam func

arXiv:1406.4158



Rewrite in terms of 2 phase-shifts & inelasticity

arXiv:1406.4158

Recall, at one energy, have 1 eqn. but 3 variables


Rewrite in terms of 2 phase-shifts & inelasticity

arXiv:1406.4158

Solve eqn. (quantization condition) – must vary perams. in t(l)


arXiv:1406.4158

Using only rest-frame data

Energies from det. Eqn. must agree with model



arXiv:1406.4158

Energies from det. Eqn. must agree with model


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


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



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


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




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


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


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


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


matrices)

• Observables

• QCD: Vector potentials now 3x3 complex matrices (SU(3))

Running of coupling

u,d quarks are very light

theory has another scale


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!


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


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


Additional levels with significant gluonic components arXiv:1212.5236

Spectrum from variational method

Matrix of correlators

Two-point correlator

101



102



“Rayleigh-Ritz method”Diagonalize: eigenvalues spectrum eigenvectors spectral “overlaps” Zi

n


103



“Rayleigh-Ritz method”Diagonalize: eigenvalues spectrum eigenvectors spectral “overlaps” Zi

n


104

Each state optimal combination of Φi

Extension to inelastic scattering• Can generalize to a scattering t-matrix



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

Hadron Resonance Determination Robert Edwards Jefferson Lab ECT 2014 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.:

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