1. Exotics and hybrids from lattice QCD
Nilmani Mathur
Department of Theoretical Physics,
TIFR, INDIA

2. Non-quark model states
States with excited gluon
Hybrid mesons ( meson + excited glue)
Hybrid baryons ( baryon + excited glue)
Glue balls (consitutent glue)
Multi-quark states
Tetraquark, Pentaquark and higher number of quark states
So far there is no conclusive experimental evidence
of such a state
These states are not well understood
Quark model fails to explain these states
Lack of understanding makes experimental identification difficult.
Lattice QCD calculations can provide crucial information of such a state

3. S = 0, 1
L = 0, 1, 2, 3…
Allowed :
Forbidden (Exotics) :

4. Example of an Exotic States with quantum number : 1-+
It is not possible to write an interpolating field for this state with a form :
Possible operators :

6. Lattice results for 1-+
Where is SIGNAL?

7. Decay states

8. Decay states

9. Octahedral group and lattice operators
Construct operator which transform irreducibly under the symmetries of the lattice Λ J A1 A2 E T1 T2
0⊕4⊕6⊕8 …
3⊕6⊕7⊕9 …
2⊕4⊕5⊕6 …
1⊕3⊕4⊕5 …
2⊕3⊕4⊕5 … Λ J G1 G2 H
1/2⊕7/2⊕9/2⊕11/2 …
5/2⊕7/2⊕11/2⊕13/2 …
3/2⊕5/2⊕7/2⊕9/2 …
Baryon
Meson
…R.C. Johnson, Phys. Lett.B 113, 147(1982)

11. Dudek, Edwards, Mathur, Richards :
Phys. Rev. D77, (2008)

12. Dudek, Edwards, Mathur, Richards :
Phys. Rev. D77, (2008)

13. Overlap Factor (Z)

14. Renaissance in Charmonium physics
S. Olsen arXiv: v1 (hep-ex)

15. Z+(4430) However, more recent Belle analysis still claims it
S-K. Choi et al, PRL 100, (2008)
Since charged, it cannot be either charmonium or hybrid
So, tetraquark?
Molecule?
Small Charm state
inside a large
light quark hadron?
Study in baryon is needed.
…….M. Voloshyn, arXiv: v2
Recent Babar data do not support it
However, more recent Belle analysis still claims it

16. Preliminary
Dudek et. al

17. Hadron spectrum Collaboration
Dudek et. al.

18. A more recent calculation
Mπ = 700 MeV Nf = 3, as = 0.12 fm, at-1 = 5.6 GeV, L = 2 fm
Dudek et al, Phys.Rev.Lett.103, (2009)

19. Status of exotic and hybrids
There is suggestive, though not conclusive, evidence of the existence of exotics and hybrid both in light and charm quark sectors.
Lattice calculations find evidence of existence of exotics and hybrid at non-realistic quark masses. More detail calculations with controlled systematic and proper understanding of decay channels are necessary.

20. 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 separations.
٭ Glueball operators have large vacuum fluctuations
٭ Signal to noise ratio is very bad

21. Glueball Typical gluon operators : Fuzzed operator :
On Lattice, continuum rotational symmetry becomes discrete cubic symmetry
with representation A1, A2 , E, T1 ,T2 etc. of different quantum numbers.
Typical gluon operators :
Moringstar and Peardon ….
hep-lat/
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.

22. Generalized Wilson loops
Gluonic terms required for glueballs and hybrids can be extracted from generalized Wilson loops

23. SU(3) Spectrum In current PDG Exotic Glueball Oddballs!
Y. Chen…N.Mathur.. et al. Phys. Rev. D73, (2006)
In current PDG

24. E.B. Gregory et al. PoS LATTICE2008:286,2008

25. π(137) ρ(770) σ(600) a0(980) a0(1450) a1(1230) JPG(I) M (MeV) a2(1320)
0¯ ¯(1)
1¯+(1)
0++(0)
0+ ¯(1)
1+ ¯(1)
π(137)
0+ (1/2)
ρ(770)
σ(600)
f0(980)
f0(1370)
f0(1500)
a0(980)
a0(1450)
a1(1230)
K0*(1430)
JPG(I)
M (MeV)
a2(1320)
2+ ¯(1)
f0(1710)
κ(800)

26. Our results shows scalar mass around 1400-1500 MeV, suggesting
a0(1450) is a two quark state … hep-ph/

27. C. Mcneile : Nucl.Phys.Proc.Suppl.186:264-267,2009

28. R. L. Jaffe

29. ππ four quark operator (I=0)

31. d π u N. Mathur et. al.. u d u d Evidence for a tetraquark state?
Phys.Rev.D76, (2007) u d
Scattering states
Possible BOUND state
σ(600)? π u d
Scattering states
(Negative scattering
length) d u
Evidence for a tetraquark state?

32. Volume dependence of spectral weights
N. Mathur et.al…Phys.Rev.D76, (2007) W0 W1
Volume independence suggests the observed state is an one particle state

33. Interpolating fields for Tetraquarks
Prelovsek….NM …et. al (2010)

34. Tetraquark States
Prelovsek….NM …
et. al (2010)

35. Tetraquark States
Prelovsek….NM …
et. al (2010)

36. κ(800) π(137) ρ(770) σ(600) a0(980) a0(1450) a1(1230) JPG(I)) M (MeV)
0¯ ¯(1)
1¯+(1)
0++(0)
0+ ¯(1)
1+ ¯(1)
π(137)
0+ (1/2)
ρ(770)
σ(600)
f0(980)
f0(1370)
f0(1500)
a0(980)
a0(1450)
a1(1230)
K0*(1430)
JPG(I))
M (MeV)
a2(1320)
2+ ¯(1)
f0(1710)
κ(800)
Kπ Mesonium?
ππ Mesonium u d
N. Mathur et. al..
Phys.Rev.D76, (2007)

37. Θ+ on the Lattice Quark content : Two possible states N Kd u d
Two-particle NK scattering state
S-wave : mK+ mN ~ 1432 MeV
P-wave :
Θ+ bound state
m(Θ+) ~ 1540 MeV

38. Θ+ on the Lattice
Most of the experiments do not see any evidence of a pentaquark state N K u d u d
Almost all detail quenched lattice calculations do not see
any evidence of Θ+ pentaquark state

39. Future study on lattice
Future calculations with dynamical fermions and realistic quark masses.
Calculations for spin-3/2 pentaquarks as claimed by a lattice study  Phys. ReV D72, (2005).
Study of charm pentaquarks as predicted by models.

40. How do you distinguished various states?
Local :
Tetraquark :
Molecular :
Hybrid :
Glueball :
Solve generalized eigenvalue problem including all operators and find contribution from each state

41. G. Bali : arXiv:

43. PANDA Antiproton Annihilation at Darmstadt at the High Energy Storage Ring at GSI An Experiment at FAIR “Facility of Antiproton and Ion Research"
Special detector, high luminosity (L~ 2x1032 cm-2s-1) and phase space cooled antiproton beam.
Energy resolution ~50 keV
Physics :
Charonium spectroscopy
Excited Glue (Glueballs and Hybrids)
Charm in Nuclei
Charmonium Hypernuclei
D and DS-Physics 
Other Topics
The Panda Experiment is
collaboration of more then 45
institutes in 15 countries with
more than 300 collaborators.

44. Physics Strangeness nuclear physics, hypernuclei, kaonic nuclei
Exotic hadron search, Chiral dynamics and meson properties in nuclear medium
Structure function, hard exclusive processes, spin structure of the nucleon with target polarization
Hadron physics in neutrino scattering
Nuclei with strangeness

46. Conclusion
Lattice QCD is entering an era where it can make significant contributions in nuclear and particle physics.
Exotic and Multiquark (>3) states :
Exotic and Multiquark states may exist in nature and lattice QCD can contribute significantly by predicting masses and quantum numbers.
One need to be careful to distinguish a bound state from a scattering state.
Various laboratories is (will be) searching for these
states.

47. Variational Analysis
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
such that the overlap to a state is maximum
Variational solution  Generalized eigenvalue problem :
Eigenvalues give spectrum :
Eigenvectors give the optimal operator :

48. t

49. Source : N. Christ

50. Source : N. Christ

51. Mass in Euclidean space
Fourier transform in Euclidean time
Mn : location of poles in the propagator of |n>.
pole masses of physical state

53. 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.

54. C. Morningstar, Lat08

55. Solution in a finite box
C. Morningstar, Lat08

56. 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)

57. Observables Parameters : gauge coupling and quark masses
Jungle
Gym
Integrating out the Grassmann variables is possible since
pQCD
ChPT E
L a t t i c e Q C D
Parameters : gauge coupling
and quark masses

58. Quark Gluon quark propagators : Inverse of very large
(on Lattice sites)
Quark
Jungle
Gym
Gluon
(on Links)
quark propagators :
Inverse of very large
matrix of space-time,
spin and color

59. t

60. Analysis (Extraction of Mass)
Correlator decays exponentially m1
m1, m2
Effective mass :

61. a (a2) L Continuum Limit M M Finite volume Effect M
Physical pion mass mq
Input parameter
Chiral extrapolation M
Continuum Limit
a (a2) M L
Finite volume Effect