Hadron spectrum : Excited States, Multiquarks and Exotics Hadron spectrum : Excited States, Multiquarks and Exotics Nilmani Mathur Department of Theoretical

Hadron spectrum : Hadron spectrum : Excited States, Multiquarks and Excited States, Multiquarks and

Exotics Exotics

Nilmani MathurNilmani MathurDepartment of Theoretical Physics, Department of Theoretical Physics,

TIFR, INDIATIFR, INDIA

TIFR, August 25, 2009TIFR, August 25, 2009

The Particle ZooThe Particle Zoo

HADRON SPECTRUM

Mesons (2-quarks)Mesons (2-quarks) Baryons (3-Baryons (3-quarks)quarks)

……PDGPDG


Can we explain these (at least)?Can we explain these (at least)?


Proof of E=mc2 !!

e=mc2: 103 years later, Einstein's proven right !! …….Times of India : 21 November 2008

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


A constituent picture of Hadrons

M. Peardon’s talk


Type of HadronsType of Hadrons

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

• Other Hadrons Other Hadrons MultiquarksMultiquarks Exotics (hybrids)Exotics (hybrids) Glueballs Glueballs


quark propagators :Inverse of very large matrix of space-time, spin and color

Quark(on Lattice sites)

Gluon(on Links)

QuarkQuarkJungleJungle GymGym


tt

n 1 2 1

)(1

)(

2

0)(

00,

)().(

00,

)().().(

0)'.()(

0,

)'.()().(

0,

).(

010

0

00

000

00000

0

,)(0

0)(,,)(0)(

0)(,,)(0

0)(,,)(0

0)(,,)(0),(

)0()(

ttEt

n

ttEn

n

ttE

qn

ttExqpi

qn

ttExxqi

x

xxpi

xxpittH

qn

xxpittH

x

xxpi

qnx

xxpi

HtHt

nnp

np

nq

nq

eWeW

pnxe

xqnqnxeqp

xqnqnxee

xqnqnexee

xqnqnxeptG

eet


spin andcolor :Tr )],0,(),0,([

),0,())(,0,(

))(0,,())(,0,(

)0,,(),0,()()(

)0()()()0()()(

)())(()0())(0()()0(

)(

,

11

11

51

51

1155

55

55

55

45

xdMxuMTr

xdMxuM

xdMxuM

xdMxuM

dxdxuu

xuxddux

uddu

uudu

425 ,])([ cudcu cbaTabc

Pion two point functionPion two point function

Nucleon interpolating operatorNucleon interpolating operator


Analysis (Extraction of Mass)Analysis (Extraction of Mass)

)( 11

1

mN

i

mi eWeWG i

Correlator decays exponentially

mm11

mm11, , mm22

)||/|(|1[

...||||

...||||ln

)1(

)(ln)(

)1(

)(

/)(21

221

)(22

)(21

22

21

)1(

12

21

21

11

aEE

aEaE

EE

mm

ewwEa

ewew

ewew

G

Gm

eG

G

Effective mass : Effective mass :


Analysis (Extraction of Mass)Analysis (Extraction of Mass)

N

i i

ii

t

tGtf χ

1

2

)(

)()(

Correlator decays exponentially

How to extract How to extract mm22 m m33…… : : excited excited states?states?

Non linear fitting.Non linear fitting.Variable projection methodVariable projection method

mm11

mm11, , mm22

)()()()(

)()( )()(

1

1

1,

2

jjk

N

kii

kij

jjij

N

jiii

tGtGtGtGC

tGtfCtGtf χ

C

Assume that data has Gaussian Assume that data has Gaussian distributiondistribution

Uncorrelated chiUncorrelated chi22 fitting by minimizing fitting by minimizing

However, data is correlated and it is However, data is correlated and it is necessary to use covariance matrixnecessary to use covariance matrix


Bayesian Fitting

Priors


Bayes’ theorem :

Bayesian prior distribution

Posterior probability distribution

prior predictive probability

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

ρ

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

P(D)


ψi : gauge invariant fields on a timeslice t that corresponds to Hilbert space operator ψj whose quantum numbers are also carried by the states |n>.Construct a matrix

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

Variational solution Generalized eigenvalue problem :

Eigenvalues give spectrum :

Eigenvectors give the optimal operator :

Variational Analysis


Importance of tImportance of t00

• 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(tC(t00) : ) :

• t0 should be chosen such that the NXN correlator

matrix is dominated by the lightest N states at tt00

• Excited states contribution falls of exponentially go to large tt00

• However, signal/noise ratio increases at large t t00

• Choose optimum tt00


Sommer : arXiv:0902.1265v2



Overlap Factor (Z)



Dudek et.al



What is a resonance What is a resonance particle?particle? Resonances are simply energies at which differential cross-section of a Resonances are simply energies at which differential cross-section of a

particle reaches a maximum.particle reaches a maximum.

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

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

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

They may be stable at high quark mass.They may be stable at high quark mass.

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

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


Identifying a Resonance Identifying a Resonance StateState• Method 1 : Method 1 :

Study spectrum in a few volumesStudy spectrum in a few volumes Compare those with known multi-hadron decay channelsCompare those with known multi-hadron decay channels Resonance states will have no explicit volume dependence Resonance states will have no explicit volume dependence

whereas scattering states will have inverse volume whereas scattering states will have inverse volume dependence.dependence.

• Method 2 :Method 2 :Relate finite box energy to infinite volume phase shifts by Relate finite box energy to infinite volume phase shifts by

Luscher formula Luscher formulaCalculate energy spectrum for several volumes to evaluate Calculate energy spectrum for several volumes to evaluate

phase shifts for various volumesphase shifts for various volumesExtract resonance parameters from phase shiftsExtract resonance parameters from phase shifts

• Method 3 :Method 3 : Collect energies for several volumes into momentum bin in Collect energies for several volumes into momentum bin in

energy histograms that leads to a probability distribution energy histograms that leads to a probability distribution which shows peaks at resonance position. which shows peaks at resonance position.

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


Multi-particle statesMulti-particle states A problem for finite box latticeA problem for finite box lattice

Finite box : Momenta are quantizedFinite box : Momenta are quantized Lattice Hamiltonian can have bothLattice Hamiltonian can have both

resonance and decay channel states resonance and decay channel states

(scattering states)(scattering states)

A A x+y, Spectra of x+y, Spectra of mmAA and and

One needs to separate out resonance states One needs to separate out resonance states from scattering statesfrom scattering states


Scattering state and its volume dependenceScattering state and its volume dependence ),,|

1,,| spn

Vspn

nn

n

tMn

x n

tM

n

x

M

nW

eW

eVM

n

txTtG

n

n

2

|)0(|0

2

|)0(|0

0|))0(),((|0)(

2

2

Normalization condition requires :

Two point function : Lattice

Continuum

For one particle bound state spectral weight (W) will NOT be explicitly dependent on lattice volume

Vx


Scattering state and its volume dependenceScattering state and its volume dependence ),,|

1,,| spn

Vspn

tEE

nn

nn

tEE

nn nnx

x

nn

nn

eV

WW

eVMVM

nn

txtxTtG

,

,

222

211

2121

11

21

21

11

21 21

2 2

|)0(|0|)0(|0

0|))0()0(),(),((|0)(

Normalization condition requires :

Two point function : Lattice Continuum

For two particle scattering state spectral weight (W) WILL be explicitly dependent on lattice volume

Vx


C. Morningstar, Lat08


Solution in a finite box

nL nLxVxV

LxL

)()(

,2

1

2

1

C. Morningstar, Lat08


Rho decayRho decay


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


Hybrid boundary conditionHybrid boundary condition

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

• Bound and scattering states will be changing Bound and scattering states will be changing differently.differently.


Hyperfine Interaction of quarks in BaryonsHyperfine Interaction of quarks in Baryons

_

+ + Nucleon

(938)

Roper (1440)

S11(1535)

+ + _

Δ(1236)

Δ(1700)

Δ(1600)

+

+

_

Λ(1116)

Λ(1405)

Λ(1670)

2121 .. cc

Color-Spin Interaction Color-Spin Interaction

Excited positive > Excited positive > Negative Negative

Glozman & RiskaPhys. Rep. 268,263 (1996)

2121 .. FF Flavor-Spin interaction Flavor-Spin interaction

Chiral symmetry plays major Chiral symmetry plays major rolerole

Negative > Excited Negative > Excited positivepositive

..Isgur..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


Symmetries of the lattice Symmetries of the lattice HamiltonianHamiltonian

• SU(3) gauge group (colour)SU(3) gauge group (colour)

• ZZnn⊗ ⊗ ZZnn⊗ ⊗ ZZnn cyclic translational group cyclic translational group (momentum)(momentum)

• SU(2) isospin group (flavour)SU(2) isospin group (flavour)

• OOhhDD crystal point group (spin and parity) crystal point group (spin and parity)


Octahedral group and lattice Octahedral group and lattice operatorsoperators

ΛΛ JJ

GG11

GG22

HH

1/21/2⊕⊕7/27/2⊕⊕9/29/2⊕⊕11/211/2 … …

5/25/2⊕⊕7/27/2⊕⊕11/211/2⊕⊕13/213/2 ……

3/23/2⊕⊕5/25/2⊕⊕7/27/2⊕⊕9/29/2 … …ΛΛ JJ

AA11

AA22

EE

TT11

TT22

00⊕⊕44⊕⊕66⊕⊕88 … …

33⊕⊕66⊕⊕77⊕⊕99 … …

22⊕⊕44⊕⊕55⊕⊕66 … …

11⊕⊕33⊕⊕44⊕⊕55 … …

22⊕⊕33⊕⊕44⊕⊕55 … …

BaryonBaryon

MesonMeson

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

Construct operator which transform irreducibly under the symmetries of the latticeConstruct operator which transform irreducibly under the symmetries of the lattice


)210(nt displaceme link :)(~~ )( ..,, j nxDAa

nj

Lattice operator constructionLattice operator construction

• Construct operator which transform irreducibly under the Construct operator which transform irreducibly under the symmetries of the latticesymmetries of the lattice

• Classify operators according to the irreps of Classify operators according to the irreps of OOhh ::

GG1g1g, G, G1u1u, G, G1g1g, G, G1u1u,H,Hgg, H, Huu

• Basic building blocks : smeared, covariant displaced Basic building blocks : smeared, covariant displaced quark fieldsquark fields

• Construct translationaly invariant elemental operators Construct translationaly invariant elemental operators

• Flavor structure Flavor structure isospin, color structure isospin, color structure gauge gauge invarianceinvariance

• Group theoretical projections onto irreps of Group theoretical projections onto irreps of OOh : h :

CcnkBb

njAa

niabc

FABC

F xDxDxDxB )(~~)(~~

)(~~)(

RFiR

ORO

i UtBURDg

dtB

Dh

Dh

F

)()()(

PRD 72,094506 (2005) PRD 72,094506 (2005) A. Lichtl thesis, A. Lichtl thesis, hep-lat/0609019hep-lat/0609019


Radial structure : displacements of different lengthsRadial structure : displacements of different lengthsOrbital structure : displacements in different directionsOrbital structure : displacements in different directions

……C. MorningstarC. Morningstar


PruningPruning

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

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

• Diagonal effective mass.Diagonal effective mass.

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

• Find a matrix with minimum condition number.Find a matrix with minimum condition number.



Nucleon mass Nucleon mass spectrumspectrumHadron spectrum collaboration : Phys. Rev. D79:034505, 2009



CASCADE MASSES


WIDTHS



Mike Peardon’s talk




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

TIFR, August 25, 2009TIFR, August 25, 2009Hadron spectrum collaboration : Phys. Rev. D 82, 014507 (2010)


Smeared operators (for example) :


Engel et.al : arXiv:1005.1748v2

Mπ ~ 320 MeVa =0.15 fm16^3 X 32Chirally 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

Problems

TIFR, August 25, 2009TIFR, August 25, 2009Hadron Spectrum collaboration


Documents

Hadron spectrum : Excited States, Multiquarks and Exotics Hadron spectrum : Excited States, Multiquarks and Exotics Nilmani Mathur Department of Theoretical