Prospects for Lattice Calculations of N Properties

1
Prospects for Lattice Calculations of N
Properties

• David Richards (JLAB)

• Lattice QCD
• Spectroscopy Review
• Identifying excited states the variational
  method
• Lattice symmetries and Lattice Partial Waves
• Lattice properties transition form factors
• Conclusions

2
LHP Collaboration

• S. Basak, J. Dudek, R. Edwards, G. Fleming, U.
  Heller, J. Juge, A. Lichtl, N. Mathur, C.
  Morningstar, DGR, I. Sato, S. Wallace

3
Lattice QCD

• Enables ab initio solution of low-energy
  properties of QCD
• Systematically improvable
• V ! 1
• a !0
• m? ! m?phys

Bornyakov et al., PRD70, 054506
4
Spectroscopy - I

• Classic tool for gleaning information about
  degrees of freedom of QCD
• Experimental and ab initio N and Hybrid programs
  aim at discovering effective degrees of freedom
  of QCD, and resolving competing low-energy
  models

Capstick and Roberts, PRD58 (1998) 074011
Timely opportunity for lattice QCD in concert
with EBAC
5
Spectroscopy - II

• Low-lying states lowest state in lattice
  partial wave benchmark calculations of lattice
  QCD.
• LHPC, full QCD calculations at single value of
  lattice spacing, fixed volume

6
Spectroscopy - III

• Benchmark calculation of lattice QCD

• CP-PACS
• Quenched, quark mass around that of strange.
• Discrepancy in meson sector eliminated in full
  QCD

• Precise results for comparison with experiment
  require control of systematic an statistical
  uncertainties
• V ! 1
• a ! 0
• m? ! 0

7
Resonances - I
Comparison of results by Adelaide group, Zanotti
et al PRD67, 114506 (2003).
Bad nucleon operator vanishes in NR limit.
Need to extract radial and higher spin excitations
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Resonance physics variational method

• Given N N correlator matrix C??(t) h 0 j
  O?(t) O?(0) j 0 i, one defines the N principal
  correlators ?i(t,t0) as the eigenvalues of
• where t0 (the time defining the metric) is
  small
• Principal effective masses defined from
  correlators plateau to lowest-lying energies

min ( En Ei)

• Eigenvalues ! Energies Real!
• Eigenvectors ! wave functions

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Principal effective masses

• Perform single-exponential fit to each principal
  correlator to extract spectrum! Can use
  multi-exponentials to minimize sensitivity to
  excited states
• Note that principal effective
  masses (as
  functions of time)
  can cross,
  approach asymptotic
  behavior from below
• Final results are independent
  
  of t0, but choosing larger values
  
  of this reference time can
  introduce
  larger errors

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Variational methods glueballs

• Morningstar and Peardon
• PRD60, 034509
• Quenched glueball calculations provide road-map
• Variational methods
• Large matrix of correlators
• Method more demanding for particles containing
  quarks

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Variational Method Unstable Particles

• Computations done in a periodic box
• momenta quantized
• discrete energy spectrum of stationary states ?
  single hadron, 2 hadron,
• Scattering phase shifts ? resonance masses,
  widths (in principle) deduced from finite-box
  spectrum
• B. DeWitt, PR 103, 1565 (1956) (sphere)
• M. Luscher, NPB364, 237 (1991) (cube)
• Two-particle states and resonances identified by
  examining behaviour of energies in finite volume
• Resonances with milder volume dependence

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Resonance or scattering state - I
Mathur et al, Phys.Rev. D70 (2004) 074508
Fit to ground and first-excited states of I1
pentaquark correlator, together with scattering
energies
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Resonance or scattering state - II
Two-particle amplitude expected to have
different volume dependence to single-particle
amplitude Amplitudes more consistent with
two-particle state
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Lattice Partial Waves

• States are classified according to
• Flavor structure, F
• Parity
• Total Spin, J, and Helicity, or third component
  of spin, Jz.
• In lattice computation, need to construct
  operators Bi? ? F(t, x) transforming irreducibly
  under lattice symmetries ?, ? and form
  correlation matrix
• Begin by considering flavor and orbital radial
  structure

S. Basak et al., PRD72074501,2005 PRD72094506,20
05
Good lattice QN.
Minimize Propagator Inversions
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Flavour, Orbital and Radial Structure

• Classify states according to SU(2) Flavour
• Build up radial structure using displacement
  operators D
• Provide our elemental BiF(t,x) operators having
  correct flavour and colour structure

Quark-diquark
Spatial structure of operators
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Lattice Symmetries

• Rotations restricted to isotropic cubic lattice
  form Octohedral group O
• O has 24 elements
• O has five conjugacy classes, hence five
  single-valued representations, A1(1), A2(1),
  E(2), T1(3), T2(3)
• Under elements R of O, operator lying in ?
  irreducible representation transforms as (? is
  row)
• Spatial inversions Is ! Oh, with 48 elements
  parities labelled g or u

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Double Octohedral Group

• Has 48 elements
• Contains irreducible representations of O,
  together with 3 spinor irreps G1, G2, H
  R.C.Johnson, PLB114, 147 (82)

Note that states with J gt 5/2 lie in
representations with lower spins.
mH
Spins identified from degeneracies in contiuum
limit
M5/2
mG2
a
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Glimpsing nucleon spectrum - I

• Apply variational method in nucleon sector at m?
  ' 700 MeV
• Classify states according to their Lattice irreps

Adam Lichtl, PhD 2006
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Glimpsing nucleon spectrum - II
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Pion cloud physics

• Roper resonance at light quark masses
• S.J. Dong et al,
• Bayesian statistics and constrained curve fitting
• Used simple three-quark operator
• Roper predominantly a three-quark state?

• Physics at physical values of the pion mass very
  different from the heavy-quark regime chiral
  perturbation theory

a 0.2 fm
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Resonance Spectroscopy - I
New thrust generation of full QCD lattices using
anisotropic clover action for spectroscopy
Good scaling behaviour

• Tadpole-improved tree-level Symanzik gauge
• Anistropic Clover fermion action

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Resonance Spectroscopy - II
Jansen et al, POS (LAT2005) 118
S and T review, 2006
Major effort at Resonance spectroscopy of both
Hybrids and baryons National Program

• Several volumes multiparticle states
• Several spacings identify spins

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Resonance Properties - I

• For well-established measurement, can study
  electromagnetic properties

q
EM Current
p
p q
t

• Prerequesite good interpolating operators for
  N, N

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Resonance Properties - II
One baryon resonance studied for many years..
Lattice points from Leinweber (1992)
Cloet,Leinweber,Thomas (2003) Lee et.al. (2004)
Chiral calculations
Pascalutsa, Vanderhaeghen Thomas, Young ()
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Photocouplings and Transition FF ? N ! ?
RSMGC/GM
REM-GE/GM
Alexandrou et al. (2005)
Pascalutsa

• Computations in full QCD in progress
• Deformation in nucleon or Delta

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N to Delta Axial-transition Form Factors
Alexandrou et al., hep-lat/0610107

• Enable study of off-diagonal Goldberger-Treiman
  relation

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Hybrid Photocouplings - I
Dudek, Edwards, Richards, PRD73, 074507

• Recent study of transitions between conventional
  mesons, e.g. S ! ? V

Not used in the fit
PDG
Lattice
CLEO
lat.
Expt.
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Hyrbid Photocouplings - II

• Q2-dependence inspired by NR potential model with
  rel. corrections

First computation of exotic meson resonance
spectrum, and of 1- photocouplings, at pion
masses down to 220 MeV
28
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Decays and Scattering States

• Scattering lengths related to energy shifts in
  two-particle spectrum in finite volume
• First dynamical calculation of scattering lengths
  in 1S0 channel and 3S1 - 3D1 coupled channels by
  NPLQCD (Orginos et al.).

Current lattice data
ChPT
I2 ?-?
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Decays and Scattering States - II
CP-PACS, hep-lat/0610020
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Conclusions and Summary

• Techniques developed to extract many states for
  each lattice partial wave
• Program of spectroscopy calculations now underway
  at several lattice spacings, several volumes
• Delineate multiparticle and scattering states
• Vital part of both JLab and National LQCD
  Program needs support
• For well-established states, extraction of
  transition form factors straightforward
• Scattering lengths/phase shifts accessible..

32
Spectroscopy - III
Application of Variational Method in trial
calculation allows extraction of up to eight
excited states
LHPC, hep-lat/0609052
Quenched QCD calculation of nucleon spectrum
New thrust generation of full QCD lattices using
anisotropic clover action for spectroscopy