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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Lattice QCD
Hadrons (Pion, nucleon etc.)
4
Creation Operator
QCD Vacuum
(Music of the Void)
5
The Particle Zoo
Mesons (2-quarks)
Baryons (3-quarks)
HADRON SPECTRUM
PDG
6
Problems for solving QCD

• Expansion around the limit when particles
  (fields) do not interact
• H(g) H0 gH1 g2H2 Feynman
  diagrams
• However, coupling constant is too strong!

Why so many particles? Why their masses are like
that ? No analytical solution for
nonperturbative hadronic physics.
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Field Theory on Computer
FEYNMAN (1948) Quantum Field Theory is
equivalent to Integration
Probability Amplitudes
from sum over histories
. Rev Mod Phys 20,
2367 (1948) (Ph.D. thesis)

• The probability amplitude for a particle to move
  from
• point x1 at time t1 to point x2 at time t2 is
  given by

? Integration is over the paths (infinitely
many variables) ? Even the ordinary integration
is just a symbol representing the limiting
procedure
8
Euclidean Formulation
? Paths are weighted with an oscillating
function and so is not suitable for numerical
calculation! ? Change real time to imaginary
time (Minkowski space to Euclidean space)
9
Euclidean Green Function for Quantum Field
Theory
Partition Function

• RHS is similar to a statistical ensemble average,
  with a Boltzmann distribution given by e-Sx
• Green function of a Field Theory
  Correlation function of the
• 
  corresponding
  Statistical System
• Paths
  Statistical configurations

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Correspondence between Euclidean Field Theory
and Classical Statistical
Mechanics
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Does this method work for QCD (a gauge theory)??
And the answer is YES!!
15
12
Phys. Rev. D10, 2445 (1974)
One of All time favorites
Nobel Laureate 1982
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Discrete Euclidean space-time
a
Continuous space-time
Quark (on Lattice sites)
Quark Jungle Gym
Violation of Lorentz invariance is controllable
and removable
Gluon (on Links)
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Lattice QCD
Lattice QCD a well suited non-perturbative
method that uses directly the QCD Lagragian
(and therefore no new parameters enter)

• Wilsons paper marks the beginning of a new
  branch of particle physics, called Lattice QCD /
  Lattice Field Theory / Lattice Gauge Theory
• Field has grown considerably.
• e-print archive xxx.lanl.gov hep-lat
• Large grand challenge collaborations with
  dedicated supercomputers around the world (US,
  Japan, Italy, Germany, Australia, India).
• USQCD (www.usqcd.org) under SciDAC project.

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Minkowski (M) to Euclidean (E)
On lattice
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Lattice Formulation of QCD

• Preserve the symmetry on the lattice (QCD has
  gauge symmetry)

• Gauge Symmetry --- Wilson, 1974 (Aµ ? Uµ)
  U Link Variable

33 SU(3) matrices
Violation of Lorentz invariance is controllable
and removable

• Wilson gauge action

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Fermions on the Lattice

• Fermion species doubling
• Naïve free fermion with symmetric derivative
• Free quark propagator
• 2d poles at
  
  
  
• Wilson fermion
• Introduce an irrelevant term to lift the 15 extra
  doublers
• Wilson fermion action

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Nielsen-Ninomiya Theorem

• There is no free lattice fermion action
• which can be written as
• and which simultaneously satisfy
• D(x) is local (bounded by )
• Fourier transform
• There are no massless fermion doubler poles, i.e,
  D(p) is invertible with p?0
• Chiral symmetry

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Chiral fermions
New fermion formulation Ginsparg-Wilson
relation D-1?5 ?5D-1a?5 where D is the
fermion matrix There are two equivalent
approaches in constructing a D that obeys the
Ginsparg-Wilson relation
Narayanan and Neuberger PLB 302 (1998)
1. Overlap fermions
Expensive to simulate ? next stage full QCD
calculations
2. Domain wall fermions Define the Wilson
fermion matrix in five dimensions
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Observables

• Generate a sample of N independent gauge fields U
• Calculate propagator D-1 for each U

For a lattice of size 323 x 64 we have 108
gluonic variables
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Statistical Evaluation -- Monte-Carlo
Method

• Large number of integrations then reduce to an
  ensemble average
• Ui's are the configurations generated in the
  stochastic process called
• Monte-Carlo Method.

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What is Monte-Carlo Method ?
The Monte Carlo method provides approximate
solutions to a variety of mathematical problems
by performing statistical sampling experiments on
a computer. It deals with complex problems
ranging from QCD
to economics to regulating the flow of
traffic. Stochastic method for calculating Pi
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Important Sampling

• All paths (configurations) are not equally
  important
• A good algorithm will find out
• important configurations.

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Lattice QCDWhat we do

• Discretize gauge and fermion actions.
• Generate gauge configurations according to QCD
  Lagrangian.
• Generate quark propagators for each gauge
  configuration.
• Write the lattice form of operators which we want
  to calculate (two point function for
• mass and three point function for form
  factor e.g. scalar, axial, vector currents).
• Calculate those operators (which involve quark
  propagators) for each configuration.
• Analyze data by averaging over various
  configurations.
• Repeat calculation for different quark masses.
• Take chiral limit
• Take lattice spacing a 0

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How to calculate an observable??
From statistical mechanical
correlation functions
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The proton in the quark model
The proton in QCD
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Quark Propagator
Fermion Action
Lattice size 163 ? 24 Dimension of M 163 ?
24 ? 2 ? 3 ? 4 106
? 106 !! Need M-1(x,y), TrM-1(x,y),
DetM(x,y)
Translational invariance need only M-1(x,0)
Different type of algorithms for different
fermionic action Conjugate gradient is the best
for Hermitian positie definite matrix.
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Symmetries
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Pion two point function
Nucleon interpolating operator
Exercise do Wick contarction to get two-point
function
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Analysis (Extraction of Mass)
Effective mass
m1
m1, m2
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Effective mass
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Mass in Euclidean space

• Fourier transform in Euclidean time

Mn location of poles in the propagator of
ngt. pole masses of physical state
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Analysis (Extraction of Mass)
Assume that data has Gaussian distribution Uncorr
elated chi2 fitting by minimizing
m1
However, data is correlated and it is necessary
to use covariance matrix
m1, m2
How to extract m2 m3 excited states? Non
linear fitting.
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Quenched Vs dynamical

• Generate a sample of N independent gauge fields U
• Calculate propagator D-1 for each U

? Easy to simulate since SgU is local use M.
C. methods from Statistical Mechanics like the
Metropolis algorithm
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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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The Strong Coupling Constant . PDG Data
Lattice result 0.115 0.03, Combined
average 0.118 0.002
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Static Quark Potential
MILC collaboration
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Precise Results
C.T.H. Davies et al. Phys. Rev. Lett. 92, 022001
(2004)
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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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Ground state Octet Baryons
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Gell-Mann Okubo Relation
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Boinepalli et al. hep-lat/0604022
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Roper
Roper is seen on the lattice at the right mass
with three quark interpolation field
..Mathur et.al. Phys. Lett. B605, 137 (2005)
Cross over occurs in chiral domain
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What about Hyperons?
The ?(1405)?
different story!!
Preliminary
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Hyperfine Interaction of quarks in
Baryons
..Isgur
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Octet ? Baryons
Parity cross over in chiral domain? Roper like
state?
Preliminary
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Decuplet ? Baryons
Preliminary
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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
be explicitly dependent on lattice volume
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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
• Positive parity channel will have
  unphysical (ghost) states

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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
Our 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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Anisotropic Clover Lattice

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

Expected lattice sizes
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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,
• ?CC (3780) Mathur, Lewis,
  Woloshyn et. al. PRD66, 014502 (2002) PRD64,
• 
  
  
  094509 (2001)
• 
  ?CC 3560(47)(2725)

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MESONS
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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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a la Manke and Liao, hep-lat/0210030
Lattice 06, Tucson
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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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1 - -
With Jo Dudek, R Edwards and D. Richards
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Glueballs and hybrid mesons
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Y. Chen et al. Phys. Rev. D73, 014516 (2006)
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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.

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Reserve Slides
New Topic
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Statistical and Systematic Uncertainties

• Procedure, in principle, is exact after
  statistical and systematic errors are controlled.
• Statistical Uncertainties
• For a finite N configurations,
  statistical errors go like 1/vN,
• Dynamical configurations are in general
  O(100) times more costlier than quenched,
• and so they are limited by a very few
  configurations. On the other side a quenched
• calculation is associated with quenched
  artifacts.
• Systematic Uncertainties
• ? Discretization Error Inherent O(a)
  or O(a2) uncertainties. One thus must
• 
  extrapolate to the continuum limit (a ? 0) to
  recover
• 
  physical quantities.
• ? Finite Volume Lattice box must
  hold a hadron state, typically L 2 fm, or more.
• We thus need
  mpL 4 (several pion Compton wavelength).
• ? Chiral Extrapolation Need chiral
  action. Otherwise will have to extrapolate to
  chiral limit
• 
  which will introduce error. Chiral fermions with
  smaller quark
• 
  masses are still very expensive and one thus
  still need to do some

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Perfect World for Lattice

• Lattice spacing a small enough to have continuum
  physics.
• Quark mass small to reproduce the physical pion
  mass.
• Lattice size large
• Fermion determinant Full QCD

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Cost of computation with lattice volume
Need 1.
Supercomputer
2. Parallel processors
3. Good Algorithms
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Dynamical calculations-status and perspectives
A. Kennedy, Lattice 2004
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Scattering Length and energy shift

• Threshold energies

• Energy shift on the finite lattice

• Experimental scattering lengths

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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

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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Quenched Artifacts

• Chiral log in mp2

Quenched QCD
x
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DsJ(2317) 0(0)
Found in Dsp0 Channel PRL 90, 242001(2003)
BABAR
PRL 92, 012002(2004) BELLE,
PRD 68, 032003(2004)CLEO,
hep-ex/0406044 FOCUS Mass 2317.4 0.9
MeV Width lt 4.6 MeV (90 CL) M(DK)
M(DsJ(2317) 45 MeV Below the DK threshold
Isospin violation decay Masses
much lower than potential model P-level
predictions Quark models could not accommodate
this state
P-level state?
DK molecule? Dspatom?
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DsJ(2460) 0(1)
Found in Dsp0 Channel PRD 68, 032003(2004)
CLEO
PRL 92, 012002(2004) BELLE
PRL91, 262002
(2003) BELLE
hep-ex/0405081 BABAR Mass 2459.3
1.3 MeV Width lt 5.5 MeV (90 CL) M(DK)
M(DsJ(2460)) 45 MeV Below the DK threshold
Isospin violation decay Masses
much lower than potential model P-level
predictions Quark models could not accommodate
this state
P-level state? DK
molecule? Dsp atom?
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DsJ(2632)
Found in Ds? and D0K Channels
hep-ex/0406045 Mass 2632 MeV Width lt 17
MeV Not state
Quark models could not accommodate this state
DsJ(2632) M(Ds?) 116 MeV DsJ(2632)
M(D0K) 274 MeV Not
molecular state What
is it??
102
X(3872)

• Mass 3871.9 0.5 MeV M(D0) M(D0)
• Width lt 2.3 MeV (90 CL)

2 ? , 3 ? radial 1 are narrow below
DD
Mass, width, angular dist etc. are all
inconsistent with
DD Molecule? J/? ?? Vector Glueball?