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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

Hadron spectrum : Excited States, Multiquarks and Exotics

Nilmani Mathur

Department of Theoretical Physics,


The particle zoo

Baryons (3-quarks)

Mesons (2-quarks)

The Particle Zoo



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

Type of hadrons
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

    Analysis extraction of mass

    Correlator decays exponentially


    m1, m2

    Analysis (Extraction of Mass)

    Effective mass :

    Analysis extraction of mass1

    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

    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 set of parameters

    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 t 0
    Importance of t set of parameters0

    • 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 set of parameters : arXiv:0902.1265v2

    Overlap Factor (Z) set of parameters

    Dudek set of parameters et.al

    What is a resonance particle
    What is a resonance particle? set of parameters

    • 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
    Identifying a Resonance State set of parameters

    • 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 states a problem for finite box lattice
    Multi-particle states set of parametersA 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
    Scattering state and its volume dependence set of parameters

    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 dependence1
    Scattering state and its volume dependence set of parameters

    Normalization condition requires :


    Two point function :


    For two particle scattering state

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

    C. Morningstar, Lat08 set of parameters

    Solution in a finite box set of parameters

    C. Morningstar, Lat08

    Rho decay
    Rho decay set of parameters

    Hybrid boundary condition
    Hybrid boundary condition set of parameters

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

    • Bound and scattering states will be changing differently.

    Hyperfine interaction of quarks in baryons

    S set of parameters11(1535)









    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 set of parameters

    Compiled by H.W. Lin

    Mahbub set of parameters et.al : arXiv:1011.5724v1

    Symmetries of the lattice hamiltonian
    Symmetries of the lattice Hamiltonian set of parameters

    • 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
    Octahedral group and lattice operators set of parameters

    Construct operator which transform irreducibly under the symmetries of the lattice



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

    Lattice operator construction
    Lattice operator construction set of parameters

    • 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 set of parameters

    Orbital structure : displacements in different directions

    …C. Morningstar

    Pruning set of parameters

    • 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
    Nucleon mass spectrum set of parameters

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

    CASCADE MASSES set of parameters

    WIDTHS set of parameters

    Mike Peardon’s talk set of parameters

    Engel et.al : (2010)


    Mπ ~ 320 MeV

    a =0.15 fm

    16^3 X 32

    Chirally improved f

    Prediction : (2010): Ξ’b=5955(27) MeV

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

    Problems (2010)

    • Spin identification

    • Multi-particle states

    • Isolating resonance states from multi-particle states

    • Extracting resonance parameters