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University of Cyprus. Pentaquarks on the Lattice. Alexandrou EINN 2005 Workshop “New Hadrons: Facts and Fancy” Milos, 19 September 2005. The Storyteller, like a cat slipping in and out of the shadows. Slipping in and out of reality?. Θ +. Outline. Spectroscopy from Lattice QCD

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University of Cyprus

Pentaquarks on the Lattice

  • Alexandrou
  • EINN 2005 Workshop “New Hadrons: Facts and Fancy”
  • Milos, 19 September 2005

The Storyteller, like a cat slipping in and out of the shadows. Slipping in and out of reality?


  • Spectroscopy from Lattice QCD
  • Resonances on the Lattice
  • Diquarks
  • Pentaquarks
  • Summary of quenched results on pentaquarks
  • Conclusions
solving qcd

L a t t i c e Q C D




Solving QCD

coupling constantg

  • At large energies, where the coupling constant is small, perturbation theory is applicable  has been successful in describing high energy processes
  • At very low energies chiral perturbation theory becomes applicable
  • At energies ~ 1 GeV the coupling constant is of order unity  need a non-perturbative approach
  • Present analytical techniques inadequate
  • Numerical evaluation of path integrals on a space-time lattice
  •  Lattice QCD – a well suited non-perturbative method that uses directly the QCD Langragian and therefore no new parameters enter
lattice qcd


  • Finite lattice spacing a: is determined from the coupling constant and gives the length/energy scale with respect to which all physical observables are measured

must take a0 to recover continuum physics

  • specify the bare quark mass mq: is taken much larger than the u and d quark mass extrapolate to the chiral limit
  • must be solved numerically on the computer using similar methods to those used in Statistical Mechanics  Finite volume: must take the spatial volume to infinity
  • Wick rotation into Euclidean time:
  • limits applicability to lower states
Lattice QCD

Lattice QCD is a discretised version of the QCD Lagrangian with only parameters the coupling constant and the masses of the quarks

masses of hadrons

Evolve in imaginary time:

i.e. assume transfer matrix

  • Take overlap with trial state:

insert complete set of energy eigenstates

Correlator / two-point function

spectral weights

  • Take limit : extract E1 measured vacuum energy provided w0 = <0|φ>=0 and w1= <1|φ> is non zero
Masses of Hadrons
  • Create initial trial state with operator J+ that has the quantum numbers of the hadron we want to study:

Energies can be extracted from the time evolution of correlation functions:


Pion mass:

Using Wick contractions the correlator can be written in terms of quark propagators

Projects to zero momentum

where the operator Jπ = d γ5u has the pion quantum numbers


fit plateau  mπ

Contamination due to excited states

bending due to antiperiodic b.c.

Smearing suppresses excited states

Effective mass:



precision results in the quenched approximation
Precision results in the quenched approximation

The quenched light quark spectrum from CP-PACS, Aoki et al., PRD 67 (2003)

  • Lattice spacing a  0
  • Chiral extrapolation
  • Infinite volume limit
excited states
Excited states?

Construct NxN mass correlation matrix:

C. Michael, NPB259 (1985) 58

M. Lüscher & U. Wolff, NPB339 (1990) 222

Maximization of ground state overlap leads to the generalized eigenvalue equation

It can be shown that

The effective masses defined as-ln (λn(t,t0) /λn(t-1,t0)determine N plateaus from which the energies of the N lowest lying stationary states can be extracted

Final result is independent of t0, but for larger t0 values the statistical errors are larger


Can one distinguish a resonance from two-particlescattering states?

  • different volume dependence of energies and spectral weights
  • resonances show up as extra states with weak volume dependence

M. LüscherNPB364 (1991)


Consider two interacting particles in a finite box with periodic or antiperiodic boundary conditions

  • discrete momentum leading to discrete energy spectrum

where , kx ,ky, kz=0,1,2,.. assuming periodic b.c.and therefore E depends on L

 from the discrete energy spectrum one can, in principle deduce scattering phase shifts and widths, M. LüscherNPB364 (1991)

Difficult in practice

Demonstrated in a toy model: O(4) non-linear σ-model

M. Göckeler et al., NPB 425 (1994) 413

two pion system in i 2




Slower approach to asymptotic plateau value

Two pion-system in I=2

Correlation matrix

with J(x) product of pion- and rho-type interpolating fieldse.g.

total momentum=0

Spacing between scattering states~1/ Ls2

project to zero relative momentum 0 0
Project to zero relative momentum: π(0)π(0)

Check taking p=0 on small lattice (163x32)


Originally proposed by Jaffe in 1977: Attraction between two quarks can produce diquarks:

qqin 3 flavor, 3 color and spin singlet  behave like a bosonic antiquark in color and flavorD :scalar diquark s

and D

A diquark and an anti-diquark mutually attract making a meson of diquarks


A nonet with JPC=0++ if diquarks dominate no exotics in q2q2

Exotic baryons?





Soliton model Diakonov, Petrov and Polyakov in 1997 predicted narrow Θ+(1530) in antidecuplet

linear confining potential
Linear confining potential

A tube of chromoelectric flux forms between a quark and an antiquark. The potential between the quarks is linear and therefore the force between them constant.

linear potential

G. Bali, K. Schilling, C. Schlichter, 1995

static potential for tetraquarks and pentaquarks










Static potential for tetraquarks and pentaquarks

Main conclusion:When the distances are such that diquark formation is favored the static potentials become proportional to the minimal length flux tube joining the quarks signaling formation of a genuine multiquark state

C. Α. andG. Koutsou, PRD 71 (2005)

can we study non static diquarks on the lattice
Can we study non-static diquarks on the Lattice?

Static quark propagator

Baryon with an infinitely heavy quark

Flavor symmetric  spin one

Define color antitriplet diquarks in the presence of an infinitely heavy spectator:

Flavor antisymmetric  spin zero



light quark propagator G(x;0)

R. Jaffe hep-ph/0409065

Models suggest that scalar diquark is lighter than the vector

attraction: M0



In the quark model, one gluon exchange gives rise to color spin interacion:

M1 –M0 ~ 2/3 (MΔ-MN)= 200 MeV and

mass difference between bad and good diquarks
Mass difference between ``bad`` and ``good`` diquarks

C.A., Ph. de Forcrand and B. Lucini Lattice 2005

  • First results using 200 quenched configurations at β=5.8 (a~0.15 fm) β=6.0(a~0.10 fm)
  • fix mπ~800 MeV (κ=0.1575at β=5.8 and κ=0.153at β=6.0)
  • heavier mass mπ ~1 GeV to see decrease in mass(κ=0.153at β=5.8)

β=6.0 κ=0.153

ΔM (GeV)

K. Orginos Lattice 2005: unquenched results with lighter light quarks

diquark distribution

j0 (x) = : u(x) γ0 u(x) :






Diquark distribution

Two-density correlators : provide information on the spatial distribution of quarks inside the heavy-light baryon

quark propagator G(x;0)

Study the distribution of d-quark around u-quark. If there is attraction the distribution will peak at θ=0

diquark distribution1
Diquark distribution

``Good´´ diquark peaks at θ=0


SPring-8 : γ12C Κ+ Κ- n

CLAS at Jlab: γD K+ K- pn

High statistics confirmed the peak

summary of experimental results
Summary of experimental results

Negative results

Positive results

A. Dzierba et al., hep-ex/0412077

P=pentaquark state (Θs,Ξ,Θc)

pentaquark mass



Time evolution

u d

u d

u d

u d

Initial state with the quantum numbers of Θ+at time t=0

Θ at a later time t>0

Pentaquark mass

C(t) ~w1exp(-mKN t)+w2exp(-mΘ t)+…

Correlator: C(t) ~exp(-mΘ t)

mass of Θ

mΘ-mKN~100 MeV


Karliner and Lipkin, PLB575, 249 (2003) : Diquark-triquark structure

Antisymmetric color 3c, spin, s=0 and flavor 3f


u d

u d






u d


Diquark is 3f and triquark in 6f

u d

Θ+in the antidecuplet

Hyperfine interaction short range  acts only within the clusters


Jaffe and Wilczek PRL 91 232003 (2003): Diquark formation


interpolating fields for pentaquarks

Motivated by KN strucutre:

Diquark structure



Modified NK

Both local and smeared quark fields were considered :

Interpolating fields for pentaquarks

What is a good initial |φ> for Θ+? All lattice groups have used one or some combinations of the following isoscalar interpolating fields:

  • Motivated by the diquark structure:

Results should be independent of the interpolating field if it has reasonable overlap with our state

does lattice qcd support a
Does lattice QCD support a Θ+?

Objective for lattice calculations: to determine whether quenched QCD supports a five quark resonance state and if it does to predict its parity.

  • Method used:
  • Identify the two lowest states and check for volume dependence of their energy
energy spectrum
Energy spectrum

Lüscher NPB364 (1991)

The energy spectrum of a KN scattering state on the lattice is given by

where , kx,y,z=0,1,2,.. assuming periodic b.c.

or , n=0,1,2,..

depends on the spatial size of the lattice for non-zero value of k whereas for a resonance state the mass should be independent of the volume

Therefore by studying the energy spectrum as function of the spatial volume one can check if the measured energy corresponds to a scattering state

The spectral decomposition of the correlator is given by

  • If |n> is a KN scattering state well below resonance energy then wn~ L-3 because of the normalization of the two plane waves
  • For a resonance state wn~1
  •  off-resonance states are suppressed relative to states around the resonance mass
scattering states
Scattering states

The two lowest KN scattering states with non-zero momentum



E (GeV)


Contributes only in negative parity channel

S-wave KN


If mixing is small w1~L-3  suppressed for large L

Dominates if w2>>w1 and (mΘ-mKN) t <1  t<10 GeV-1 assuming energy gap~100MeV or t/a<20

does lattice qcd support a1
Does lattice QCD support a Θ+?

Objective for lattice calculations: to determine whether quenched QCD supports a five quark resonance state and if it does to predict its parity.

  • Method used:
  • Identify the two lowest states and check for volume dependence of their mass
  • Extract the weights and check their scaling with the spatial volume
volume dependence of spectral weights
Volume dependence of spectral weights
  • Works for our test two-pion system provided:
  • Accurate data
  • Fit within a large time window especially for large spatial volumes to extract the correct amplitude

Cross check needed

Small upper fit range

identifying the on the lattice

Alexandrou & Tsapalis (2.9 fm)

Lasscock et al. (2.6 fm)

Mathur et al. (3.2 fm)

Mathur et al. (2.4 fm)

Csikor et al. (1.9 fm)

Sasaki (2.2 fm)

Ishii et al. (2.15 fm)

Identifying the Θ+ on the Lattice

There is agreement among lattice groups on the raw data but the interpretation differs depending on the criterion used

Negative parity

From Lassock et al. hep-lat/0503008

All lattice computations done in the quenched theory

review of lattice results


JKN and Jdiquark fields are used with non-trivial spatial structure on lattices of size ~2. and 2.4 fm

Negative parity

Positive parity

203x36, β=6





KN scattering states


Review of lattice results

All lattice computations aredone in the quenched theory using Wilson, domain wall or overlap fermions and a number of different actions. All groups but one agree that if the pentaquark exists it has negative parity. Here I will only show results for I=0.

  • Measure the energies

Csikor et al.

JHEP 0311 (2003)

Results based on J’KN with a check done using the correlation matrix with J’KN and JKN. In the negative parity channel,S-wave KN scattering state is identified as the lowest state and the next higher in energy as the Θ+.


S. Sasaki, PRL 93 (2004)

Used Jdiquark and fitted to “first” plateau to extract the Θ+ mass on a lattice of size ~2.2 fm (323x48 β=6.2) with mπ=0.6-1 GeV

mπ~750 MeV

Negative parity

Positive parity

E x 2.9 GeV





Double plateau structure is not observed in other similar calculations

scaling of weights

Negative parity

ratio of weights

Expected for a scattering state


Scaling of weights

Mathur et al. PRD 70 (2004)

Interpolating field JNK for quark masses giving pion mass in the range 1290 to 180 MeV and lattices of size ~2.4 and 3.2 fm. The weights were found to scale with the spatial volume.







Perform a similar analysis as in the two-pion system using Jdiqaurk and JKN

Takahashi et al., Pentaquark04 and hep-lat/0503019 :JKN and J’KN on spatial lattice size ~1.4, 1.7, 2.0 and 2.7 with a larger number of configurations

spectral weights for pentaquark

Ratio WL1/WL2 ~1 for ti/a up to 26 which is the range available on the small lattices

Different from two pion system  can not exclude a resonance

Spectral weights for pentaquark

C.A. and A. Tsapalis, Lattice 2005

does lattice qcd support a2
Does lattice QCD support a Θ+?

Objective for lattice calculations: to determine whether quenched QCD supports a five quark resonance state and if it does to predict its parity.

  • Method used:
  • Identify the two lowest states and check for volume dependence of their mass
  • Extract the weights and check their scaling with the spatial volume
  • Change from periodic to antiperiodic boundary condition in the spatial directions and check if the mass in the negative parity channel changes
  • Check whether the binding increases with the quark mass
hybrid boundary conditions
Hybrid boundary conditions

Ishii et al., PRD 71 (2005)

Use antiperiodic boundary conditions for the light quarks and periodic for the strange quark:

Θ+is unaffected since it has even number of light quarks

N has three light quarks and K one  smallest allowed momentum for each quark is π/L and therefore the lowest KN scattering state is shifted to larger energy

Negative parity



Spatial size~2.2 fm



E (GeV)



Strange quark mass


Standard BC

Hybrid BC


Lasscocket al., hep-lat/0503008

Interpolating fields JKN, J’KN, Jdiquark on a lattice size~2.6 fm. Although a 2x2 correlation matrix was considered the results for I=0 were extracted from a single interpolating field

Negative parity

Mass difference between the pentaquark and the S-wave KN

Mass difference between the pentaquark and the P-wave KN

Mass difference between Δ(1232) and the P-wave Nπ

Positive parity

hep-lat/0504015: maybe a 3/2+ isoscalar pentaquark?

positive parity
Positive parity Θ+

Chiu and Hsieh, hep-ph/0403020

Domain wall fermions Lattice size 1.8 fm

The lowest state extracted from an 3x3 correlation matrix

1.554 +/- 0.15 GeV



Holland and Juge, hep-lat/0504007

Fixed point action and Dirac operator, 2x2 correlation matrix analysis using JKN and J’KN on a lattice of size ~1.8 fm, mπ=0.550-1.390 GeV

Energies of the two lowest states are consistent with the energy of the two lowest KN scattering states

  • State-of-the-art Lattice QCD calculations enable us to obtain with good accuracy observables of direct relevance to experiment
  • A valuable method for understanding hadronic phenomena
  • Diquark dynamics
  • Studies of exotics and two-body decays
  • Computer technology will deliver 10´s of Teraflop/s in the next five years and together with algorithmic developments will make realistic lattice simulations feasible
  • Provide dynamical gauge configurations in the chiral regime
  • Enable the accurate evaluation of more involved matrix elements