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Hadron spectrum : Excited States, Multiquarks and Exotics. Nilmani Mathur Department of Theoretical Physics, TIFR, INDIA. Baryons (3-quarks). Mesons (2-quarks). The Particle Zoo. HADRON SPECTRUM. …PDG. Can we explain these (at least)?. Proof of E=mc 2 !!.

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Hadron spectrum : Excited States, Multiquarks and Exotics

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Hadron spectrum : Excited States, Multiquarks and Exotics

Nilmani Mathur

Department of Theoretical Physics,


Baryons (3-quarks)

Mesons (2-quarks)

The Particle Zoo



Can we explain these (at least)?

Proof of E=mc2 !!

S.Durr et.al, Science 322, 1224 (2008)

e=mc2: 103 years later, Einstein's proven right !!

…….Times of India : 21 November 2008

A constituent picture of Hadrons

M. Peardon’s talk

Type of Hadrons

  • Normal hadrons :

    • Two quark state (meson)

    • Three quark state (baryon)

  • Other Hadrons

    • Multiquarks

    • Exotics (hybrids)

    • Glueballs

  • Quark



    quark propagators :

    Inverse of very large

    matrix of space-time,

    spin and color


    (on Lattice sites)


    (on Links)


    Pion two point function

    Nucleon interpolating operator

    Correlator decays exponentially


    m1, m2

    Analysis (Extraction of Mass)

    Effective mass :

    Correlator decays exponentially

    Analysis (Extraction of Mass)

    Assume that data has Gaussian distribution

    Uncorrelated chi2 fitting by minimizing


    However, data is correlated and it is

    necessary to use covariance matrix

    m1, m2

    How to extract m2 m3… : excited states?

    Non linear fitting.

    Variable projection method

    Bayesian Fitting


    the conditional probability of measuring the data D given a set of parameters


    Posterior probability distribution

    the conditional probability that ρ is correct given the measured data D

    Bayes’ theorem :

    Bayesian prior distribution


    prior predictive probability

    Variational Analysis

    ψi : gauge invariant fields on a timeslice t that corresponds to

    Hilbert space operatorψjwhose quantum numbers are also carried

    by the states|n>.

    Construct a matrix

    • Need to find out variational coefficients

    • such that the overlap to a state is maximum

    • Variational solution  Generalized eigenvalue problem :

    • Eigenvalues give spectrum :

    • Eigenvectors give the optimal operator :

    Importance of t0

    • Basis of operators is only a part of the Hilbert space (n = 1,…N; N≠∝)

    • The eigenvectors are orthogonal only in full space.

    • Orthogonality is controlled by the metric C(t0) :

    • t0 should be chosen such that the NXN correlator matrix is dominated by the lightest N states at t0

    • Excited states contribution falls of exponentially  go to large t0

    • However, signal/noise ratio increases at large t0

    • Choose optimum t0

    Sommer : arXiv:0902.1265v2

    Overlap Factor (Z)

    Dudek et.al

    What is a resonance particle?

    • Resonances are simply energies at which differential cross-section of a particle reaches a maximum.

    • In scattering expt. resonance  dramatic increase in cross-section with a corresponding sudden variation in phase shift.

    • Unstable particles but they exist long enough to be recognized as having a particular set of quantum numbers.

    • They are not eigenstates of the Hamiltonian, but has a large overlap onto a single eigenstates.

    • They may be stable at high quark mass.

    • Volume dependence of spectrum in finite volume is related to the two-body scattering phase-shift in infinite volume.

    • Near a resonance energy : phase shift rapidly passes through pi/2, an abrupt rearrangement of the energy levels known as avoided “level crossing” takes place.

    Identifying a Resonance State

    • Method 1 :

      • Study spectrum in a few volumes

      • Compare those with known multi-hadron decay channels

      • Resonance states will have no explicit volume dependence whereas scattering states will have inverse volume dependence.

  • Method 2 :

    • Relate finite box energy to infinite volume phase shifts by Luscher formula

    • Calculate energy spectrum for several volumes to evaluate phase shifts for various volumes

    • Extract resonance parameters from phase shifts

  • Method 3 :

    • Collect energies for several volumes into momentum bin in energy histograms that leads to a probability distribution which shows peaks at resonance position.

      ….V. Bernard et al, JHEP 0808,024 (2008)

  • Multi-particle statesA problem for finite box lattice

    • Finite box : Momenta are quantized

    • Lattice Hamiltonian can have both

      resonance and decay channel states

      (scattering states)

    • A  x+y, Spectra of mA and

    • One needs to separate out resonance states from scattering states

    Scattering state and its volume dependence

    Normalization condition requires :


    Two point function :


    For one particle bound state

    spectral weight (W) will NOT be explicitly dependent on lattice volume

    Scattering state and its volume dependence

    Normalization condition requires :


    Two point function :


    For two particle scattering state

    spectral weight (W) WILL be explicitly dependent on lattice volume

    C. Morningstar, Lat08

    Solution in a finite box

    C. Morningstar, Lat08

    Rho decay

    • ….V. Bernard et al, JHEP 0808,024 (2008)

    Hybrid boundary condition

    • Periodic boundary condition on some quark fields while anti-periodic on others

    • Bound and scattering states will be changing differently.










    Roper (1440)






    Nucleon (938)



    Color-Spin Interaction

    Excited positive > Negative

    Flavor-Spin interaction

    Chiral symmetry plays major role

    Negative > Excited positive

    Glozman & Riska

    Phys. Rep. 268,263 (1996)

    Hyperfine Interaction of quarks in Baryons


    N. Mathur et al, Phys. Lett. B605,137 (2005).

    Roper Resonance for Quenched QCD

    Compiled by H.W. Lin

    Mahbub et.al : arXiv:1011.5724v1

    Symmetries of the lattice Hamiltonian

    • SU(3) gauge group (colour)

    • Zn⊗ Zn⊗ Zncyclic translational group (momentum)

    • SU(2) isospin group (flavour)

    • OhD crystal point group (spin and parity)

    Octahedral group and lattice operators

    Construct operator which transform irreducibly under the symmetries of the lattice



    …R.C. Johnson, Phys. Lett.B 113, 147(1982)

    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, covariant 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

    Radial structure : displacements of different lengths

    Orbital structure : displacements in different directions

    …C. Morningstar


    • All operators do not overlap equally and it will be very difficult to use all of them.

    • Need pruning to choose good operator set for each representation.

    • Diagonal effective mass.

    • Construct average correlator matrix in each representation and find condition number.

    • Find a matrix with minimum condition number.

    Nucleon mass spectrum

    Hadron spectrum collaboration : Phys. Rev. D79:034505, 2009



    Mike Peardon’s talk

    Hadron Spectrum collaboration : Dudek et.al : arXiv:1102.4299v1

    Hadron spectrum collaboration : Phys. Rev. D 82, 014507 (2010)

    Smeared operators (for example) :

    Engel et.al :


    Mπ ~ 320 MeV

    a =0.15 fm

    16^3 X 32

    Chirally improved f

    Prediction :: Ξ’b=5955(27) MeV

    Cohen, Lin, Mathur, Orginos : arXiv:0905.4120v2


    • Spin identification

    • Multi-particle states

    • Isolating resonance states from multi-particle states

    • Extracting resonance parameters

    Hadron Spectrum collaboration

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