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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,

TIFR, INDIA

Baryons (3-quarks)

Mesons (2-quarks)

HADRON SPECTRUM

…PDG

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

- Normal hadrons :
- Two quark state (meson)
- Three quark state (baryon)

- Multiquarks
- Exotics (hybrids)
- Glueballs

Quark

Jungle

Gym

quark propagators :

Inverse of very large

matrix of space-time,

spin and color

Quark

(on Lattice sites)

Gluon

(on Links)

t

Pion two point function

Nucleon interpolating operator

Correlator decays exponentially

m1

m1, m2

Effective mass :

Correlator decays exponentially

Assume that data has Gaussian distribution

Uncorrelated 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.

Variable projection method

Bayesian Fitting

Priors

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

ρ

Posterior probability distribution

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

Bayes’ theorem :

Bayesian prior distribution

P(D)

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 :

- 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

- 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.

- 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.

- 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

- 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)

- 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

Normalization condition requires :

Continuum

Two point function :

Lattice

For one particle bound state

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

Normalization condition requires :

Continuum

Two point function :

Lattice

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

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

- Periodic boundary condition on some quark fields while anti-periodic on others
- Bound and scattering states will be changing differently.

S11(1535)

_

_

Δ(1700)

Λ(1670)

+

_

+

+

Roper (1440)

Λ(1405)

Δ(1600)

+

+

+

Nucleon (938)

Λ(1116)

Δ(1236)

Color-Spin Interaction

Excited positive > Negative

Flavor-Spin interaction

Chiral symmetry plays major role

Negative > Excited positive

Glozman & Riska

Phys. Rep. 268,263 (1996)

..Isgur

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

- SU(3) gauge group (colour)
- Zn⊗ Zn⊗ Zncyclic translational group (momentum)
- SU(2) isospin group (flavour)
- OhD crystal point group (spin and parity)

Construct operator which transform irreducibly under the symmetries of the lattice

Baryon

Meson

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

- 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.

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

CASCADE MASSES

WIDTHS

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 :

arXiv:1005.1748v2

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

Problems

- Spin identification
- Multi-particle states
- Isolating resonance states from multi-particle states
- Extracting resonance parameters

Hadron Spectrum collaboration