Excited State Spectroscopy from Lattice QCD

Robert Edwards Jefferson Lab

CERN 2010

Collaborators:J. Dudek, B. Joo, M. Peardon, D. Richards, C. Thomas, S. WallaceAuspices of the Hadron Spectrum Collaboration

Spectroscopy

Spectroscopy reveals fundamental aspects of hadronic physics– Essential degrees of freedom?– Gluonic excitations in mesons - exotic states of

matter?

• New spectroscopy programs world-wide– E.g., BES III (Beijing), GSI/Panda (Darmstadt)– Crucial complement to 12 GeV program at JLab.

• Excited nucleon spectroscopy (JLab)• JLab GlueX: search for gluonic excitations.

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Nuclear Physics & Jefferson Lab

• Lab doubling beam energy to 12GeV• Adding new experimental Hall

JLab undergoing a major upgrade

Future Hall D

Excited states: anisotropy+operators+variational

• Anisotropic lattices with Nf=2+1 dynamical (clover)fermions– Temporal lattice spacing at < as (spatial lattice spacing)– High temporal resolution ! Resolve noisy & excited states– Major project within USQCD – Hadron Spectrum Collab.

• Extended operators– Sufficient derivatives ! nonzero overlap at origin

• Variational method:– Matrix of correlators ! project onto excited states

PRD 78 (2008) & PRD 79 (2009)

PRD 76 (2007), PRD 77 (2008), 0909.0200

PRD 72 (2005), PRD 72 (2005), 0907.4516 (PRD), 0909.0200

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Nf=2+1 Anisotropic Clover

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Nf = 2 + 1 (u,d + s)

Using at m & »=3.5 : as = 0.1227(3)fm, (at)-1 ~ 5.640 GeV

0803.3960 & 0810.3588

Spin and Operator Construction

0905.2160 (PRD), 0909.0200 (PRL), 1004.4930 (PRD)

Gamma matrices and derivatives in circular basis:Couple to build any J,M via usual CG

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Construct all possible operators up to 3 derivatives (3 units orbital angular momentum)

Only using symmetries of continuum QCD

Subduce onto lattice irreps: “remembers” J

Distillation

0905.2160 (PRD), 1002.0818, Bulava (thesis 2010)

Replace smearing with low rank approximation

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Gauge covariant, mom. conservation (source & sink) ! reduced “noise”

Matrix elements of vk(t) ! propagators, mesons, baryons, etc. Make correlators

Stochastic variants (lower cost)

Spectrum from variational method

Matrix of correlators

Diagonalize: eigenvalues ! spectrum eigenvectors ! wave function overlaps

Benefit: orthogonality for near degenerate states

Two-point correlator

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Determining spin on a cubic lattice?

Spin reducible on lattice

Might be dynamical degeneracies

T2 E T2 E T2 E

mass

T1 T2 E A2

Spin 1, 2, 3, or 4 ?

Spin reduction & (re)identification

Variational solution:

Continuum

Lattice

Method: Check if converse is true

Spin (re)identification

Z values: spin 3Nf=3

Well identified: suggests small rotational breaking

Isovector Meson Spectrum

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Nf=3

1004.4930

Isovector Meson Spectrum

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Nf=3

1D2P

1F

1004.4930

1S

2S1P

3S 2D 1G

??

Isovector Meson Spectrum

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Nf=2+1, m¼ ~ 520MeV, L~2fm

1004.4930

Similar to Nf=3

Exotic matterExotics: world summary

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Exotic matter

Suggests (many) exotics within range of JLab Hall D

Previous work: charmonium photo-production rates high

Current work: (strong) decays

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Vector isoscalars (I=0)

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Nf=2+1, m¼ ~ 400MeV, L~2fm

light-strange (ls) basis

I = 0 Must include all disconnected diagrams

Baryon Spectrum

“Missing resonance problem”• What are collective modes?• What is the structure of the states?

– Major focus of (and motivation for) JLab Hall B– Not resolved experimentally @ 6GeV

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PDG uncertainty on B-W mass

Nucleon spectrum

Strange Quark Baryons

Strange quark baryon spectrum poorly known

Future:• Narrow widths: easy(er) to extract (?)

¥ & : unknown spin & parities Widths are small

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Light quark baryons in SU(6)

Conventional non-relativistic construction:

6 quark states in SU(6)

Baryons

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Relativistic operator construction: SU(12)

Relativistic construction: 3 Flavors with upper/lower components

Times space (derivatives)

Color contraction is Antisymmetric ! Totally antisymmetric operators

Dirac

More operators than SU(6): mixes orbital ang. momentum & Dirac spin

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Orbital angular momentum via derivatives

0905.2160 (PRD), 0909.0200 (PRL), 1004.4930

Couple derivatives onto single-site spinors:Enough D’s – build any J,M

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Use all possible operators up to 2 derivatives (2 units orbital angular momentum)

Only using symmetries of continuum QCD

Nucleon & Delta Spectrum

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Nucleon & Delta Spectrum

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[56,2+]D-wave [70,1-]

P-wave

[70,1-]P-wave

[56,2+]D-wave

Change at lighter quark mass? Decays!

Suggests spectrum at least as dense as quark model

[70,2+]D-wave

[20,1+]P-wave

“Roper”?

Hadronic decays

Plot the non-interacting meson levels as a guide

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Current spectrum calculations:no evidence of multi-particle levels

Require multi-particle operators • (lattice) helicity construction • annihilation diagrams

Extract δ(E) at discrete E

Phase Shifts: demonstration¼¼ isospin=2 Extract δ0(E) at discrete E

No discernible pion mass dependence

Phase Shifts: demonstration¼¼ isospin=2 δ2(E)

Hardware: JLab GPU ClustersGPU clusters: ~530 cards

Quads 2.4 GHz Nehalem 48 GB memory / node 117 nodes x 4 GPUs -> 468 GPUs

Singles 2.4 GHz Nehalem 24 GB memory / node 64 nodes x 1 GPU -> 64 GPUs

Inverter Strong Scaling: V=323x256

Local volume on GPU too small (I/O bottleneck)

3 Tflops

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Prospects

• Strong effort in excited state spectroscopy– New operator & correlator constructions ! high lying states– Finite volume extraction of resonance parameters – promising– Progress! Still much more to do

• Initial results for excited state spectrum:– Suggests baryon spectrum at least as dense as quark model– Suggests multiple exotic mesons within range of Jlab’s Hall D

• Resonance determination:– Start at heavy masses: have some “elastic scattering”– Use larger volumes & smaller pion masses (m¼ ~230MeV)

– Now: multi-particle operators & annihilation diagrams (gpu-s)– Need multi-channel finite-volume analysis for (in)elastic

scattering

• Future:– Transition FF-s, photo-couplings (0803.3020, 0902.2214)

Backup slides

• The end

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Interpretation of Meson Spectrum

Future: probe with photon decays

Distillation: mesons

• Smearing in correlator: use low-rank approximation

• Correlator

• Factorizes: operators and perambulators

• A posteriori evaluation of C(2) after ¿arxiv:0905.2160

Distillation: annihilation diagrams

• Two-meson creation op

• Correlator

arxiv:0905.2160

Perambulators

• Current calculation (243x128, m¼ ~ 230MeV)• Solve: [all space sites]• Perambulator

• High angular momentum (J = 4), want N = 128• All time-sources -> 64K inversions / config• Can construct all possible source/sink multi-particles

• Ideally suited for GPU-s: – 300 configs on 180 GPU-s -> 4 months @ 185

GF/gpu

Hadronic & Nuclear Physics with LQCD

• Hadronic spectroscopy– Hadron resonance determinations– Exotic meson spectrum (JLab 12GeV )

• Hadronic structure– 3-D picture of hadrons from gluon & quark spin+flavor distributions– Ground & excited E&M transition form-factors (JLab 6GeV+12GeV+Mainz)– E&M polarizabilities of hadrons (Duke+CERN+Lund)

• Nuclear interactions– Nuclear processes relevant for stellar evolution– Hyperon-hyperon scattering– 3 & 4 nucleon interaction properties [Collab. w/LLNL] (JLab+LLNL)

• Beyond the Standard Model– Neutron decay constraints on BSM from Ultra Cold Neutron source (LANL)

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Bridges in Nuclear PhysicsNP Exascale

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Gauge Generation: Cost Scaling• Cost: reasonable statistics, box size and “physical” pion

mass• Extrapolate in lattice spacings: 10 ~ 100 PF-yr

PF-years

State-of-Art

Today, 10TF-yr

2011 (100TF-yr)

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Computational Requirements

Gauge generation : Analysis

Current calculations• Weak matrix elements: 1 : 1• Baryon spectroscopy: 1 : 10• Nuclear structure: 1 : 4

Computational Requirements: Gauge Generation : Analysis 10 : 1 (2005) 1 : 3 (2010)

Core work: Dirac inverters - use GPU-s

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Modern GPU Characteristics• Hundreds of simple cores: high flop rate• SIMD architecture (single instruction, multiple data)• Complex (high bandwidth) memory hierarchy

• Gaming cards: no memory Error-Correction (ECC) – reliability issue

• I/O bandwidth << Memory bandwidth

Commodity Processors x86 CPU NVIDIA GT200 New Fermi GPU

#cores 8 240 480

Clock speed 3.2 GHz 1.4 GHz 1.4 GHz

Main memory bandwidth 20 GB/s 160 GB/s(gaming card)

180 GB/s(gaming card)

I/O bandwidth 7 GB/s(dual QDR IB)

3 GB/s 4 GB/s

Power 80 watts 200 watts 250 watts

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Science / Dollar for (Some) LQCD Capacity Apps

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Isovector Meson Spectrum

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1004.4930

Determining spin on a cubic lattice?

Spin reducible on lattice

Might be dynamical degeneracies

H G2 H G2H G2

mass

G1 H G2

Spin 1/2, 3/2, 5/2, or 7/2 ?

Spin reduction & (re)identification

Variational solution:

Continuum

Lattice

Method: Check if converse is true

Spectrum of finite volume field theory

Missing states: “continuum” of multi-particle scattering states

Infinite volume: continuous spectrum

2mπ 2mπ

Finite volume: discrete spectrum

2mπ

Deviation from (discrete) free energies depends upon interaction - contains information about scattering phase shift

ΔE(L) ↔ δ(E) : Lüscher method

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Finite volume scattering

Reverse engineer Use known phase shift - anticipate spectrum

E.g. just a single elastic resonancee.g.

Lüscher method - essentially scattering in a periodic cubic box (length L)- finite volume energy levels E(δ,L)

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Finite volume scattering: Lϋscher method

Excited state spectrum at a single volume

Discrete points on the phase shift curve

Do more volumes, get more points

energylevelsL ~ 2.9 fm

e.g. L ~ 2.9 fm

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The interpretation

“non-interacting basis states”

DOTS:Finite volume QCD energy eigenvalues

LINES:Non-interacting two-particle states have known energies

Level repulsion - just like quantum mechanical pert. theory

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The interpretation

energylevels

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Where are the Form Factors??

• Previous efforts– Charmonium: excited state E&M transition FF-s (0909.0200)

– Nucleon: 1st attempt: E&M Roper->N FF-s (0803.3020)

• Spectrum first!– Basically have to address “what is a resonance’’ up front– (Simplistic example): FF for a strongly decaying state:

linear combination of states

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energylevels

Phase Shifts: demonstration¼¼ isospin=2 Extract δ0(E) at discrete E

Phase Shifts: demonstration

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Spin identified Nucleon spectrum

m¼ ~ 520MeV

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Statistical errors < 2%

Spin identified Nucleon spectrum

Where is the “Roper”? Thresholds & decays: need multi-particle ops

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m¼ ~ 520MeV

Phenomenology: Nucleon spectrum

Looks like quark model?

[70,1-]P-wave

[56,2+]D-wave

[70,2+]D-wave

[20,1+]P-wave

m¼ ~ 520MeV

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Discern structure: wave-function overlaps

Phenomenology: Nucleon spectrum

[70,1-]P-wave

Looks like quark model?

[56,2+]D-wave

[70,2+]D-wave

[20,1+]P-wave

m¼ ~ 520MeV

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Discern structure: wave-function overlaps

Spin identified ¢ spectrum

[70,1-]P-wave

[56,2+]D-wave

Spectrum slightly higher than nucleon

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