Hadron spectrum from Lattice QCD

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Hadron spectrum from Lattice QCD

• Nilmani Mathur

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Outline

• Introduction to Lattice QCD
• Path integral, Euclidean field theory, Monte
  Carlo method
• How to calculate an observable?
• Mass Two point correlation function
• Form factors Three point correlation function
• Quench Vs Dynamical
• Quenching artifacts -Unphysical ghost states
• Baryons
• Some recent results on baryon spectrum
• Excited states Parity ordering in baryon
  sector (Roper, S11 etc)
• Group theoretical baryon operators (spectrum
  project)
• Multiquark states
• Mesons
• Exotics-Gluex expt
• Glueball

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Particle Spectrum
The quenched light quark spectrum from
CP-PACS, Aoki et al., PRD 67 (2003)
The computed quenched light hadron spectrum is
within 7 of the experiment. The remaining
discrepancy is attributed to the quenched
approximation.
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Quenched Vs dynamical
C.T.H. Davies et al. Phys. Rev. Lett. 92, 022001
(2004)
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The ?' ghost in quenched QCD
Quenched QCD
Full QCD
(hairpin)

• It becomes a light degree of freedom
• with a mass degenerate with the pion mass.
• It is present in all hadron correlators G(t).
• It gives a negative contribution to G(t).
• It is unphysical (thus the name ghost).

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Quenched Artifacts

• Chiral log in mp2

Quenched QCD
x
d 0.2 0.03
Phys. Rev. D70, 034502 (2004)
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Scalar 0 correlation function
Correlation function for Scalar channel
Ground state p?' ghost state, Excited state
0
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Ghost States in Quenched Hadron Spectrum
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MILC Collaboration
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Ground state Octet Baryons
With overlap action
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Gell-Mann Okubo Relation
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Boinepalli et al. hep-lat/0604022
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Hyperfine Interaction of quarks in
Baryons
..Isgur
A conclusive identification of Roper resonance at
right mass region with dynamical fermion will be
very helpful to model builders
Mathur et.al. Phys. Lett. B605, 137 (2005)
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Multiquark
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Multi-particle states A
problem for finite box lattice

• Finite box Momenta are quantized
• Lattice Hamiltonian can have both
• resonance and decay channels states
  (scattering states)
• A ? xy, Spectra of mA and
• One needs to separate out resonance states from
  scattering states
• Need multiple volumes, stochastic propagators

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Scattering state and its volume dependence
Normalization condition requires
Continuum
Two point function
Lattice
For one particle
bound state Spectral weight (W) will NOT be
explicitly dependent on lattice volume
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Scattering state and its volume dependence
Normalization condition requires
Continuum
Two point function
Lattice
For two particle
scattering state Spectral weight (W) will
have different volume dependent than that
of the resonance state.
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Ratio of scalar meson correlator at two volumes
and at two different quark mases
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Problem of studying T on the Lattice
Quark content
Two possible states
Two-particle NK scattering state S-wave mK
mN 1432 MeV P-wave
T bound state m(T) 1540 MeV
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mK mN 1432 MeV
m(T) 1540MeV
C(t) w0exp(-mKN t)w1 exp(-mT t)

• To separate out nearby states
• ? Multi-exponential better fitting
  algorithm with high statistics
• ? Multi-operator cross correlator fitting
  with high statistics

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Volume Dependence in 1/2- channel

• For bound state, fitted weight will not show any
  volume
• dependence.
• For two particle scattering state, fitted weight
  will show
• inverse volume dependence

Phys.Rev.D70074508,2004
Observed ground state is S-wave scattering
state
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Spectrum Project
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Octahedral group and lattice operators
Baryon
Meson
R.C. Johnson, Phys. Lett.B 113, 147(1982)
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Lattice operator construction

• Construct operator which transform irreducibly
  under the symmetries of the lattice
• Classify operators according to the irreps of Oh
  
• G1g, G1u, G1g,
  G1u,Hg, Hu
• Basic building blocks smeared, coariant
  displaced quark fields
• Construct translationaly invariant elemental
  operators
• Flavor structure ? isospin, color structure ?
  gauge invariance
• Group theoretical projections onto irreps of Oh

PRD 72,094506 (2005) A. Lichtl thesis,
hep-lat/0609019
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Lattice operator construction
Three quark elemental operators
With covariant displacement
C. Morningstar
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Radial structure displacements of different
lengths Orbital structure displacements in
different directions
C. Morningstar
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Variational Method Luscher and Wolf, NPB 339,
220 (1990)

• Each operator fa(t) can project to any quantum
  state

• Need to find out variational coefficients
• such that the overlap to a state is
• maximum

• In practice diagonalize the variational matrix

• Construct the optimal operator

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A. Lichtl hep-lat/0609019
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Anisotropic Clover Lattice

• Gauge Action Wilson
• Fermion Action Clover
• Anisotropy (finer temporal lattice spacing)
• Stout smearing

Expected lattice sizes
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280 nodes of Intel Pentium D (dual-core) 3.0GHz
with 800 MHz front side bus, 1 GB memory and 80
GB SATA disk
384 nodes arranged as a 6x8x23 mesh
(torus). single processor 2.8 GHz Xeon, 800 MHz
front side bus, 512 MB memory, and 36 GB disk
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MESONS
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MILC Collaboration
Many ground state mesons are well reproduced
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Recently Observed Hadrons
Hadrons Experiments

• DsJ (2317) ? DSp0 PRL 90,
  242001(2003) BABAR
• DsJ(2460) ? DSp0 PRD68, 032002
  (2003) CLEO
• X(3872) ? J/?pp-
  hep-ex/0308029, SELEX
• Y(3940), Y(4260)
  hep-ex/0507019, 0507033, 0506081
• ?CC (3460)
• ?CC (3520) PRL89,11
  2001(2002) SELEX (not confirmed)
• ?CC (3780) Mathur, Lewis,
  Woloshyn et. al. PRD66, 014502 (2002) PRD64,
• 
  
  
  094509 (2001)
• 
  ?CC 3560(47)(2725)

Lattice calculation is necessary to understand
these states
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Hybrids
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S 0, 1 L 0, 1, 2, 3 J L S
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Paul Eugenio Lattice 2006
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Paul Eugenio Lattice 2006
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Charmonium is our test bed
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CharmoniumLaboratory to test hybrid technology

• Light quark hybrids and higher spin mesons are
  problematic so far
• Charmonium needs less constraints
• -- Chiral extrapolation
• -- Quenching
• Experimental data exists to test photo-couplings

Lattice 06, Tucson
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Radiative transition in Charmonium
Phys.Rev.D73, 074507 (2006),
Would be a good testing ground before going to
light quark (GlueX observables)
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Glueball

• A glueball is a purely gluonic bound state.
• In the theory of QCD glueball self coupling
  admits
• the existence of such a state.
• Problems in glueball calculations
• ? Glueballs are heavy correlation
  functions die rapidly at
• large time seperations.
• ? Glueball operators have large vacuum
  fluctuations
• ? Signal to noise ratio is very bad

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Glueball

• On Lattice, continuum rotational symmetry becomes
  discrete cubic symmetry
• with representation A1, A2 , E, T1 ,T2 etc. of
  different quantum numbers.
• Typical gluon operators

Moringstar and Peardon . hep-lat/9901004
Fuzzed operator Try to make the overlap of the
ground state of the operator to the glueball as
large as possible by killing excited state
contributions.
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Glueball Spectrum
Moringstar and Peardon .hep-lat/9901004
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Y. Chen et al. Phys. Rev. D73, 014516 (2006)
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Glueballs and hybrid mesons
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Suggested Reading Materials

• Books 1. R. J. Rothe
  Lattice Gauge Theory An Introdction
• 2. M. Creutz
  Quarks Gluons and Lattices
• 3. I. Montvay and G. Munster
  Quantum Fields on a Lattice
• 4. J. Smit
  Introduction to Quantum Fields on a
• 
  
  Lattice
• Review Papers hep-lat/9807028,
  hep-ph/0312241, hep-lat/0007032,
• 
  hep-lat/0702020, arXiv0705.4356
• 
  hep-lat/9802029, Rev.Mod.Phys.55775,1983,
• Popular articles Science 300 (May
  16)1076-1077 (2003)
• 
  Physics Today 57(February 2004)45-52
• 
  Science News, Aug. 2004, p. 90.
• Websites http//www.usqcd.org,
  http//www.lqcd.org,
• http//phys.columbi
  a.edu/cqft/
• http//www.rccp.tsu
  kuba.ac.jp
• http//www.physics.
  indiana.edu/sg/milc.html
• http//apegate.roma
  1.infn.it/APE/

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Reserve Slides
New Topic
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Conclusion

• Lattice QCD is entering an era where it can make
  significant
• contributions in nuclear and particle physics.
• Particle Masses Understanding the Structure and
  Interaction of Hadrons.
• Quenched lattice calculations can reproduce
  masses for many ground
• state hadrons within 10 of experimental
  numbers. Qualitatively spectrum
• ordering may well be understood by quench
  calculations.
• However, excited state masses are still not
  accessible comprehensively.
• Data analysis becomes increasingly difficult
  as we go towards chiral
• limit due to the appearance of unphysical
  ghost states. In low quark
• mass region one should consider these states
  in fitting function.
• For full QCD one needs to consider multiquark
  decay channels along with resonance states.
  Multivolume and posssibly stochastic propagators
  are necessary to carry out a reliable study
• A very comprehensive program is ongoing by
  Spectrum group by using group theoretical
  multi-operator variational method in order to
  extract resonance states both for baryons and
  mesons including hybrid states.
• Multiquark (gt3) and hybrid states
• Multiquark and hybrid states may exist in nature
  and lattice QCD can contribute significantly in
  this area.
• One need to be careful to distinguish a bound
  state from a scattering state by volume
  dependence or other methods.