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

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

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

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 :

Lattice results for 1-+ Where is SIGNAL?

Decay states

Decay states

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)

Dudek, Edwards, Mathur, Richards : Phys. Rev. D77, 034501 (2008)

Dudek, Edwards, Mathur, Richards : Phys. Rev. D77, 034501 (2008)

Overlap Factor (Z)

Renaissance in Charmonium physics S. Olsen arXiv:0801.1153v1 (hep-ex)

Z+(4430) However, more recent Belle analysis still claims it S-K. Choi et al, PRL 100, 142001 (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:0711.4556v2 Recent Babar data do not support it However, more recent Belle analysis still claims it

Preliminary Dudek et. al

Hadron spectrum Collaboration Dudek et. al.

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, 262001 (2009)

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.

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

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/9901004 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.

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

SU(3) Spectrum In current PDG Exotic Glueball Oddballs! Y. Chen…N.Mathur.. et al. Phys. Rev. D73, 014516 (2006) In current PDG

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

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)

Our results shows scalar mass around 1400-1500 MeV, suggesting a0(1450) is a two quark state … hep-ph/0607110

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

R. L. Jaffe

ππ four quark operator (I=0)

d π u N. Mathur et. al.. u d u d Evidence for a tetraquark state? Phys.Rev.D76, 114505 (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?

Volume dependence of spectral weights N. Mathur et.al…Phys.Rev.D76,114505 (2007) W0 W1 Volume independence suggests the observed state is an one particle state

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

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

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

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, 114505 (2007)

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

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

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, 074507 (2005). Study of charm pentaquarks as predicted by models.

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

G. Bali : arXiv:0911.1238

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 http://www-panda.gsi.de The Panda Experiment is collaboration of more then 45 institutes in 15 countries with more than 300 collaborators.

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

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.

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 :

t

Source : N. Christ

Source : N. Christ

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

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.

C. Morningstar, Lat08

Solution in a finite box C. Morningstar, Lat08

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)

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

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

t

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

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