1. Hadron Structure from Lattice QCD
Robert Edwards
Jefferson Lab
DWF Valence on a Asqtad light sea
Nucleon Properties:
Spectrum
Form Factors
Pion Form Factor
Radiative transitions - Hybrid meson decays

2. LHP Collaboration Dru Renner University of Arizona
Richard Brower, James Osborn Boston University
Rebecca Irwin, Michael Ramsey-Musolf CalTech
Jimmy Juge, Adam Lichtl, Colin Morningstar, Carnegie Melon University
Jozef Dudek, Robert Edwards, Nilmani Mathur, David Richards JLab
Kostas Orginos, JLab/William & Mary
Steve Wallace, Univ. Maryland
Bojan Bistrovic, Jonathan Bratt, Patrick Dreher, Oliver Jahn, John Negele, Andrew Pochinsky, Dmitry Sigaev MIT
Matthias Burkardt, Michael Engelhardt New Mexico State University
George Fleming Yale
Constantia Alexandrou, Antonios Tsapalis Cyprus, Wolfram Schroers DESY, Philippe de Forcrand EHT-Zurich, Philipp Hägler Tech.Munich

3. Spectrum - Group Theoretical Operators
JLab Hall B: Baryon spectrum experiments
Group theoretical construction of baryon interpolating fields
Diagonalize cross-correlator matrix: find mass from each ev.
Work in progress:
Use in full QCD to extract multi-hadron states
Asymmetric lattice – long time extent for plateaus and energy scale
Need multi-volume techniques
Effective mass: Nucleon G1g rep. (1/2)+
Small volume quenched Wilson fermion test case
8 Excited states!

4. Spectrum – Domain Wall Spectrum of Octet/Decuplet baryons
Hybrid action - Domain wall fermions over Staggered sea quarks
Only site site operators - no excited states
Need  PT : less effected by chiral logs
Previous on-going work using older (conventional) action will compute spectrum

5. Spectrum – Future Directions
Calculating high lying excited states high priority for JLab lattice group
Extraction of masses requires disentangling decay modes
Requires full QCD
Generation of new dynamical anisotropic configurations
Only group pursuing this approach
Also useful for simple vector/axial-vector current form-factors, e.g. Roper transition FF
Scale of resources:
One lattice spacing, several masses down to m = 200 MeV, Cost ~ 7 TFlop-yr .
This year about 5.2 TFlop allocatable
Doable in U.S. given expected growth in resources

6. Anatomy of a Matrix Element Calculation
Jf,iy : Current with desired quantum numbers of state A,B
Normalize:
Compute ratio:
Need propagator from t ! tf
Want |h 0|Jy|ni|2 » dn,0 for best plateau

7. Proton Electromagnetic Form-Factors
LT separation disagrees with polarization transfer
New exp. at Q2 = 9 GeV2
Does lattice QCD predict the vanishing of GEp(Q2) around Q2 ~ 8 GeV2 ?
Results use (hybrid) mixed action – DWF/staggered
C. Perdrisat (W&M) , JLab Users Group Meeting, June 2005

8. Nucleon Electromagnetic Form-Factors
J.J. Kelly (Maryland), PRC 70
Using covariance matrix (thanks to J.J. Kelly), these fits can be transformed into Dirac-Pauli with isospin decomposition

9. Isospin Decomposition of Nucleon FF

10. Nucleon Isovector F1 (or A10) Form Factor
Dirac charge radius h r2iu-dch using dipole ansatz and Q2 · 1 GeV2
Chiral extrap. Using LNA and LA terms and finite-range regulator (PRL 86, 5011)
Best fit:  ¼ 740 MeV
As the pion mass approaches the physical value, the size approaches the correct value

11. Nucleon Isovector F1 at Higher Q^2
Breit frame at m ¼ 600 MeV
Need more statistics as Q2 increases
Both polarization transfer experiments and lattice data consistent with dipole ansatz up to 4 GeV2
Go to smaller lattice spacing to reach higher Q2

12. Nucleon Isovector GE at Higher Q^2
Fits of experimental data suggests GEu-d vanishes at Q2 ~ 4 GeV2
With m ¼ 600 MeV on 282 configs, the lattice points are consistent with something interesting happening around 4 GeV2
Need more lattice data and/or goto smaller lattice spacing

13. Nucleon Isovector Ratio F2/F1

14. Polarized Nucleon FF’s GA(Q2) and GP(Q2)
Pion electroproduction at threshold
(CCQE  n ! - p not shown)
Pion electroproduction and muon capture
- p !  n at rest
J. Phys. G; Nucl. Part. Phys. 28

15. Nucleon Isovector Ratio GA/F1
Two experimental methods for GA(Q2)
Electroproduction mA=1.069(16)
CCQE mA=1.026(21)
Lattice results favoring GA(Q2) from pion electroproduction at higher Q2

16. Nucleon Isovector Ratio GP/GA

17. Pion Electromagnetic Form Factor: F(Q2)
Transition between perturbative and non-perturbative aspects of QCD
Asymptotic normalization determined (?)
Small Q2, vector-meson dominance provides faithful description
Free of disconnected contributions – straightforward lattice calculation

18. Pion Form Factor: Experimental Results
Existing data fit VMD monopole formulae too well. Where’s perturbative QCD?
Because of simple valence structure, argued that PQCD should apply to lower Q2 than other hadrons
PQCD calc. far off.
Experimental interpretation problematic – must extrapolate to the pion pole
Use lattice QCD!

19. Pion Form Factor –III
Pion Form factor over Q2 commensurate with experiment
Pion GPDs and transition form factors next

20. Spin Physics – Future Directions
Currently, only done 1 lattice spacing of 3
Smaller lattice spacing gives access to higher Q2
Need for chiral fermions:
Comparison to twist expansion requires derivative insertions – need action with good chiral symmetry to avoid mixings
Scale of resources:
3 lattice spacings, several masses down to m = 250 MeV, Cost ~ 7 TFlop-yr .
This year about 5.2 TFlop allocatable
Doable in U.S. given expected growth in resources

21. JLab, GlueX and photocouplings
GlueX plans to photoproduce mesons
Especially exotic JPC mesons
resonance, X
exchange
meson, M 21

22. Photoproduction of Exotics?
Exotic quantum numbers 1-+, 0+-, 2+- may be explicable as hybrid meson states
Photoproduction an untested method
Relies upon reasonably large couplings e.g.
Large in some model estimations (e.g., Close&Dudek PRL91, PRD69) 22

23. Lattice QCD estimation?
Relatively straightforward in principle; evaluate three-point function with a vector current
In practice, not so easy
truly light quarks unfeasible (for now)
transitions involve unstable states
experimental data is limited and imprecise
(even for conventional meson transitions) 23

24. charmonium Pragmatic approach
Try out an untested method in a region where approximations are controllable
and where there is good experimental data to compare with
charmonium 24

25. Charmonium - expt.
Multiple states below DD threshold have narrow widths
Radiative transitions are big branching fractions
Precision measurements 25

26. Charmonium - lattice States are small - small volumes OK
Quenched theory not sick* - just not expt.
Disconnected diagrams perturbatively suppressed
Need a fine lattice spacing a-1 & 3 GeV
* “one heavy flavour QCD”; will only notice non-unitary up near 6 GeV 26

27. First tests: VMD in Charmonium
Many new techniques: test using conventional mesons
Chiral fermions at charm scale – automatic O(a) improvement
Quenched anisotropic lattices – as = 0.1fm, at = fm
Extraction of excited transition form-factors
Warmup result: VMD does not hold in charmonium
VMD ansatz bad fit
Quark model inspired form works well
Lack of VMD probably due to close proximity of (excited state) vector poles

28. V! S +  in Charmonium Form-factor decomposition of matrix-element
Plot of time extracted FF versus current time-insertion
Time plateaus: C1 E1

29. V ! S+ (E1) transition our ‘poster boy’ not used in the fit PDG CLEO
lat.

30. V ! S +  (C1) transition

31. V ! P +  in Charmonium new CLEO-c number soon!
Lattice calculation of V! P transition form factor
Extrapolate using quark-model wavefunc
phase space
physical
Crystal
Ball expt.
lattice
New CLEO-c number soon!
Dudek, Edwards, Richards, PRD, 2006
new CLEO-c number soon!
new CLEO-c number soon!
new CLEO-c number soon!
new CLEO-c number soon!
new CLEO-c number soon!
new CLEO-c number soon!
new CLEO-c number soon!
new CLEO-c number soon!
new CLEO-c number soon!
new CLEO-c number soon!
new CLEO-c number soon!
new CLEO-c number soon!
new CLEO-c number soon!
new CLEO-c number soon!
new CLEO-c number soon!
new CLEO-c number soon!
new CLEO-c number soon!
new CLEO-c number soon!
new CLEO-c number soon!
new CLEO-c number soon!
new CLEO-c number soon!

32. Quark potential model?
Our fitting form inspired by NR potential model with rel. corrections:
C1 params fairly consistent,  = 501(33), 502(38), 545(49) MeV 32

33. Photon Decays
How does one compute a photon in an external state in QCD?
Use LSZ reduction in Euclidean space and perturbation series for QED
Effectively, all excited states of  summed with unknown weights
Integrand is saturated by these sums

34. Photon Decays Decays of c! 2 and c0! 2
c possibly effected by hyperfine splitting. PDQ values very dispersed
c! 2
c0 ! 2
Method can be used for 2 in GDH sum rule!
Can have alternative low Q2 lattice calculation

35. Hybrid Decays – Future Directions
Calculating hybrid photocoupling (also) a high priority for JLab lattice group
Will use same gauge backgrounds as spectrum project
Anisotropic lattice (lattice spacing in time direction compared to space)
Also useful for simple vector/axial-vector current form-factors, e.g. Roper transition FF

36. Conclusions and Perspective
Use of hybrid Asqtad/DWF calculation providing entry into pion-cloud regime for range of physical quantities – spectrum, nucleon and meson form factors and transition form factors, moments of PDF’s and GPD’s…
Computations being extended both to smaller values of lattice spacing (a=0.9, 0.6 fm), and to lighter m = 250 MeV – reveal effects of pion cloud.
Spectrum calculation on-going – high excited states appear possible
First lattice calculation of radiative transistions.
Determination of hybrid photocoupling next.

37. Extra Slides
For the curious…

38. Partially Quenched Chiral Perturbation Theory
Full QCD expensive!
Leverage off cheap(er) valence calcs
Correct low-energy constants, in principle
Must be in domain of validity
Extend partially quenched cPT to include O(a) terms
Mixed actions
Bär, Rupak, Shoresh, 2002, 2003

39. “Decay” in Quenched Approximation
Dramatic behavior in isotriplet scalar particle a0!hp intermediate state
Loss of positivity of a0 propagator from missing bubble insertions
Quenched a0 has double pole in cPT
Also appears in mh0
Bardeen, Duncan, Eichten, Thacker, 2000

40. Partially Quenched Singularity
Non-positivity of a0 correlator
(Partially) Quenched singularity (still) present at mp, valencea = mp, seaa
Suggests not single staggered pion in chiral loops – taste breaking not neglible
Need complete partial cPT
Vary valence and sea masses
Theory under development…

41. Rho!Pion Transition Form-Factor
Electro-disintegration of deuteron intensively studied
Isovector exchange currents identified
Isoscalar exchange currents not clear
h p(pf)|Jm|rk(pi)i » 2 V(Q2) pf pI / (mp+mr)
Extracted coupling rough agreement to expt
Need higher Q2 to discern VMD
First lattice calc!
Maris-Tandy (2002)