Lattice Calculation ofLattice Calculation of Pentaquark Baryons Pentaquark Baryons

Nilmani MathurNilmani MathurDepartment of Physics and AstronomyDepartment of Physics and Astronomy

University of KentuckyUniversity of Kentucky

Collaborators :Collaborators :Kentucky Lattice QCD Group Members, Kentucky Lattice QCD Group Members,

F. Lee, J.B. Zhang and C. BennholdF. Lee, J.B. Zhang and C. Bennhold

OutlineOutline

• Multi-quark. How many of them are Multi-quark. How many of them are together?together?

• Pentaquark on LatticePentaquark on Lattice

• Overlap Fermion and Particle SpectrumOverlap Fermion and Particle Spectrum

• Lattice Calculation for PentaquarkLattice Calculation for Pentaquark

• ResultsResults

• Conclusions Conclusions

Quarks : Six FlavorsQuarks : Six Flavors

Multi-Quark Multi-Quark Two, Three or More? Two, Three or More?

q Single quark has not been observed yet. QCD tells it cannot be observed. All naturally occurring particles are colorless. Each quark and anti-quark has three different colors.

Two quarks : One quark + One anti-quark. Possible mesons : (uu,ud,dd,us,sd,cd,cc,bb etc.)Example : Pion, Rho, Eta, Omega etc Meson

u

d

q

q

Three quarks :Possible three quarks : uud, udd, uds, uus, uds, uss, uuu, sss ,dss, dds, ddd, uss etc.Example : Proton, neutron etc. Baryons

Multi-Quark Multi-Quark Two, Three or More? Two, Three or More?

Four quarks : Two quark + two anti- quark. Like molecular state.Example : σ (500-600 MeV : ππ) : a0(980), f0(980) (KK) : ρρ (I=2) [γγ ρ+ρ¯, ρ0ρ0] : DS (Babar) (CS or DK ?) : B± K+π¯π¯ J/ψ (DD*?)

q1 q2

q2 q1

Five quarks : Four same or different quarks + one antiquark Possible configuration : colorless baryon + colorless meson

q1 q2

q3

q2q1

Possible Pentaquark candidatesPossible Pentaquark candidates

u u

d

Near to Nπ threshold. Decay by strong interaction.

Possible candidate. Can be observed in KN scattering (Signal observed recently).

u

uu

dd

d

s

u u

s d

c

True Pentaquark, not seen so far.Heavier particle, experiment will be difficult

Need :• Weak force between them. • Non-zero overlap between initial wave-function (threshold state) and final state

• DDss

+ + (2313)(2313) BABAR BABAR (PRL 90(2003) 242001)(PRL 90(2003) 242001)

• DDss+ + (2463)(2463) CLEO,CLEO, hep-ex/0305100hep-ex/0305100

• DD00*0 *0 (2308)(2308) BELLEBELLE

• DD00'0 '0 (2427)(2427) hep-ex/0307021hep-ex/0307021

• ΨΨ(3871)/DD*(3817)(3871)/DD*(3817) BELLE, hep-ex/0308029BELLE, hep-ex/0308029

• ΞΞCCCC++ ++ (3460)(3460)

• ΞΞCCCC+ + (3520)(3520) SELEX, hep-ex/0212029SELEX, hep-ex/0212029

• ΞΞCCCC++ ++ (3780) ((3780) ( lattice results before lattice results before experimentsexperiments

……PRD66, 014502 (2002); PRD64, 094509 (2001))PRD66, 014502 (2002); PRD64, 094509 (2001)) • ΘΘ++(1540)(1540) T. Nakano et. al (LEPS)T. Nakano et. al (LEPS) CLAS, DIANA, SAPHIR, ZEUS, CLAS, DIANA, SAPHIR, ZEUS,

HERMESHERMES• ΞΞ¯ ¯(1862)¯ ¯(1862) NA49/CERNNA49/CERN

Recently Observed Recently Observed HadronsHadrons Hadrons Experiments

Experimental evidence for Pentaquarks Experimental evidence for Pentaquarks (summary)(summary)

ExperimentsExperiments Mass Width SignificanceMass Width Significance

(MeV) (MeV) ((MeV) (MeV) (σσ))

SPRING-8 SPRING-8 γγn n KK¯̄(K(K++n)n)

DIANA DIANA KK00P (541 events)P (541 events)CLAS (JLab) CLAS (JLab) γγd d K K+ + KK¯̄n (p)n (p)

SAPHIRSAPHIR

ITEP (ITEP (νν’s)’s)

HERMESHERMES

R. Arndt et al. (KR. Arndt et al. (K++N Scattering)N Scattering)

1540±10±5 1540±10±5 гг < 25 4.6±0.1 < 25 4.6±0.1

1539±2±’’few’’ 1539±2±’’few’’ гг < 8 4.4 < 8 4.4

1542±2±5 1542±2±5 гг < 21 5.3±0.5 < 21 5.3±0.5

1540±4±2 1540±4±2 гг < 25 4.8 < 25 4.8

1535±5 1535±5 гг < 29 6.7 < 29 6.7

1526±2±2.5 1526±2±2.5 гг < 20 5.6 < 20 5.6

гг < 1 (if exists) < 1 (if exists)

World AverageWorld Average 1535±2.51535±2.5

θ+

ExperimentsExperiments ResultsResults NA49 (CERN)NA49 (CERN)Ξ-- Ξ-π-, Ξ 0 Ξ-π+

M = 1862 MeVM = 1862 MeV

гг < 29 MeV < 29 MeV

Ξ¯ ¯

Prediction from different modelsPrediction from different models

ModelModel PredictionPrediction• Chiral Soliton Model (D. Diakonov et al.)Chiral Soliton Model (D. Diakonov et al.)

• Naïve Quark ModelNaïve Quark Model

• Isotensor Formulation (S. Capstick et al.)Isotensor Formulation (S. Capstick et al.)

• qq with qq with ππ interaction (Stanscu, Riska) interaction (Stanscu, Riska)

• Chiral potential (A. Hosaka)Chiral potential (A. Hosaka)

• qq –qqq Model (M. Karliner et al.)qq –qqq Model (M. Karliner et al.)

• Di-quark Model (Jaffe and Wilzcek)Di-quark Model (Jaffe and Wilzcek)

• QCD sum rule (Sugiyama et al.)QCD sum rule (Sugiyama et al.)

• Lattice QCD (F. Csikor et al.)Lattice QCD (F. Csikor et al.)

• Lattice QCD (S. Sasaki)Lattice QCD (S. Sasaki)

• 1/21/2++,, I = 0I = 0

• 1/2¯1/2¯

• 1/2¯, 3/2¯ 5/2¯,1/2¯, 3/2¯ 5/2¯, I = 2 I = 2

• 1/21/2++

• 1/21/2++

• 1/21/2++,, I = 0 I = 0

• 1/21/2++,, I = 0 I = 0

• 1/2¯,1/2¯, I = 0 I = 0

• 1/21/2++ 1/2¯1/2¯

• 1/2¯1/2¯

Quantum Chromodynamics Quantum Chromodynamics ((QQCCDD))

The Fundamental Theory of the The Fundamental Theory of the Strong Interaction Strong Interaction

qmDqFFL qQCD )(Tr 21

],[ :ensorstrength t Field AAgAAF

gAD :derivativeCovariant

• Chiral symmetryChiral symmetry and its spontaneous breaking and its spontaneous breaking

• At high energy, perturbative (At high energy, perturbative (asymptotic asymptotic freedomfreedom))

• At low energy, non-perturbative (At low energy, non-perturbative (confinementconfinement))

The proton in the quark The proton in the quark model:model:

t

yz

u

d

u

u

d

u

The proton in QCD:The proton in QCD:

How good is the quenched How good is the quenched approximation?approximation?

•Light hadron spectrum from CP-PACS, heplat/0206090.•Lattices: 323x56 to 643x128•Spacing 0.1 fm to 0.05 fm•M/ M is 0.75 to 0.4•1 to 3 % statistical error •2% systematic error•Took more than a year of running on a dedicated computer sustaining 300 Gflops.

The computed quenched light hadron spectrum is within 7% of the experiment. The remaining discrepancy is attributed to the quenched approximation.

Overlap FermionOverlap Fermion

• Exact chiral symmetry.Exact chiral symmetry.• No exceptional configurations.No exceptional configurations.• No No OO((aa) error, ) error, OO((aa22) is also small.) is also small.• Critical slowing down is gentle all the way to pion mass Critical slowing down is gentle all the way to pion mass

~180 MeV.~180 MeV.• Numerically checked that there is no addative quark mass Numerically checked that there is no addative quark mass

renormalization.renormalization.• 16163 3 XX 28, a = 0.200(3) 28, a = 0.200(3) fm. fm. La = 3.2La = 3.2 fm fm (80 configurations)(80 configurations)• 121233 XX 28, a = 0.200(3) 28, a = 0.200(3) fm. fm. La = 2.4 La = 2.4 fmfm (80 configurations) (80 configurations)• 202033 X X 32, a ~ 0.171 32, a ~ 0.171 fm. fm. La ~ 3.4 La ~ 3.4 fmfm (100 configurations, (100 configurations, not analyzed yet).not analyzed yet).

SomeSome Lattice Results : Kentucky GroupLattice Results : Kentucky Group

Pentaquark on the Lattice Pentaquark on the Lattice

Interpolating Field :Combination of colorless meson + baryonFor θ+ :Interpolating field with I=0 and J=1/2

Color structure is not unique

Pentaquark Correlation Pentaquark Correlation FunctionFunction

KN scattering

state is part of this

correlation function

Correlation Function for Correlation Function for PentaquarkPentaquark

Time

0 5 10 15 20 25 30

Co

rre

lati

on

Fu

nc

tio

n

-8e-21

-6e-21

-4e-21

-2e-21

0

2e-21

4e-21

6e-21

8e-21

1/2+1/2¯

1/2¯ 1/2+

Anti-periodic boundary condition

)()( 00 22)( ttmttNmL eAeAtG t

)()( 00 22)( ttNmttmU

teAeAtG

Time

0 5 10 15 20 25 30

Co

rre

lati

on

Fu

nc

tio

n

-30

-20

-10

0

10

20

30

1/2¯

1/2¯

1/2+

1/2+

Correlation Function (1/2¯, 3.2 Correlation Function (1/2¯, 3.2 fm)fm)

Correlation Function (1/2Correlation Function (1/2++, 3.2 , 3.2 fm)fm)

Correlation Function (1/2Correlation Function (1/2++, 2.4 , 2.4 fm)fm)

The The ′′ ghost in quenched QCD ghost in quenched QCDQuenched QCDFull QCD

(hairpin)

… ....

tE Netw

)E(1 Modeled as part of G(t) as: • weight w is negative • prefactor (1+Et) preserves the double-pole structure of the hairpin diagram• E ′ N is treated as fit parameter to account for interactions between ′ and N

• It becomes a light degree of freedomIt becomes a light degree of freedom– with a mass degenerate with the pion mass.with a mass degenerate with the pion mass.

• It is present in all hadron correlators G(t).It is present in all hadron correlators G(t).• It gives a negative contribution to G(t). It gives a negative contribution to G(t).

– It is unphysical (thus the name ghost).It is unphysical (thus the name ghost).

Evidence of Evidence of ηη’N GHOST State in S’N GHOST State in S11 11 (1535)(1535) ChannelChannel

-- --η η

W > 0

W<0

Effect of ghost statedecreases as pion mass increases

Effectofghoststateis firsttime seen inbaryonchannel

Ghost States in Pentaquark channelGhost States in Pentaquark channel

Ghost states in PentaquarkGhost states in Pentaquark

• 1/2¯ : 1/2¯ : Parity Negative. S-waveParity Negative. S-wave NKNKππ –parity : (+)(-)(-) = + –parity : (+)(-)(-) = + Total parity : (-1)Total parity : (-1)LL P(NK P(NKππ)) L = 1, therefore, L = 1, therefore, ghost state will be in P-waveghost state will be in P-wave.. Ground state is KN scattering state Ground state is KN scattering state or pentaquark state.or pentaquark state.• 1/21/2+ : + : Parity Positive, P-waveParity Positive, P-wave NKNKππ –parity : (+)(-)(-) = + –parity : (+)(-)(-) = + Total parity : (-1)Total parity : (-1)LL P(NK P(NKππ)) L = 0, therefore, L = 0, therefore, ghost will be in S-wave ghost will be in S-wave

(mass m(mass mππ+m+mKK+m+mNN)) Ground state is Ground state is KNKNππ ghost state (for our lattice) ghost state (for our lattice)

0.5

1.0

1.5

2.5

2.0

Mas

s (G

eV)

N(938) 1/2+

P11(1440) 1/2+

S11(1535) 1/2-

What is the nature of the Roper (P11(1440) 1/2+) resonance?

N(938)1/2+

N(1440)1/2+

N(1535)1/2-

Naïve quark model gives thewrong ordering

ħ

ħ

- Hybrid state (qqqg)? - Dynamical meson-baryon state?

Radial excitationRadial excitation? q? q44qq state?state?

0.5

1.0

1.5

2.5

2.0

Mas

s (G

eV)

N(938) 1/2+

P11(1440) 1/2+

S11(1535) 1/2-

Roper is seen on the lattice at the right mass with three quark interpolation field ..hep ph/0306199

Crossoveroccursinchiraldoman

Radial excitationRadial excitation? ? • Roper is seen on the lattice with Roper is seen on the lattice with three-quarkthree-quark interpolation interpolation

field.field.• Weight :Weight :

|<|<O|O|OONN||R R >|>|2 2 > > |<|<O|O|OONN||NN>|>|2 2 > 0 > 0 (point source, point sink)(point source, point sink)

∑∑ψψ(x)(x) ∑∑OONN(x(x) ) ∑∑ψψ(x)(x)

∑∑ψψ(x)(x) Point sink Wall sourcePoint sink Wall source

<<O|O|∑O∑ONN(x)|N(x)|N><><N| ∑N| ∑ψψ(x) | ∑(x) | ∑ψψ(x) | ∑(x) | ∑ψψ(x)|(x)|OO > > > > 00 However,However, <<O|O|∑O∑ONN(x)|(x)|RR><><RR| ∑| ∑ψψ(x) | ∑(x) | ∑ψψ(x) | ∑(x) | ∑ψψ(x)|(x)|OO > > < < 00

2S

q4q State?

1S

(1232) 3/2+

(1600) 3/2+

(1700) 3/2-

Cross-over in DeltasCross-over in Deltas

0.5

1.0

1.5

2.5

2.0

Mas

s (G

eV)

(1232) 3/2+

(1600) 3/2+

(1700) 3/2-

What about Hyperons? What about Hyperons? The The (1405)?(1405)?……different different story!!story!!

0.5

1.0

1.5

2.5

2.0

Mas

s (G

eV)

(1115) 1/2+

(1405) 1/2 -

(1600) 1/2+

(1600) 1/2+

(1115) 1/2+

(1405) 1/2 -

Hyperfine Interaction of quarks in BaryonsHyperfine Interaction of quarks in Baryons

?..

?..

2121

2121

or

FF

cc

• Flavor spin interaction dominates

Goldstone boson exchange

• No spin-orbit potential

_

+

+ +

+

+

+ _ _

Nucleon (938)

Roper (1440)

S11(1535)

Δ(1236)

Δ(1620-1700)

Δ(1600)

Λ(1116)

Λ(1450-1520)

Λ(1600)

Glozman & Riska Phys. Rep. 268,263 (1996)

Is aIs a00 (1450) a two quark (1450) a two quark state?state?

Ground state : Ground state : ghost stateghost state..

First excited state : First excited state : aa00

Preliminary results shows mass around 1400-1500 MeV, Preliminary results shows mass around 1400-1500 MeV, suggesting suggesting aa00(1450)(1450) is a two quark state.is a two quark state.

CorrelationCorrelationfunctionfunctionfor for Scalar Scalar channelchannel

Scattering Length and energy Scattering Length and energy shiftshift• Threshold energies :Threshold energies :

2 2 2 2

3 3

(1/ 2 ) ,

(1/ 2 ) ((2 / )sin( / 2)) ((1/ )sin( )) ,

2 / 387 / 517 (16 28 /12 28 )

K N

K L N L

L

E m m

E m a p m a p

p L MeV lattice

260 0 0

1 23 2

2( ) 1 ( )KN K N

KN

a a aE m m C C O L

L L L

• Energy shift on the finite latticeEnergy shift on the finite lattice : :

• Experimental scattering lengthsExperimental scattering lengths : : WaveWave I = 0 I = 1I = 0 I = 1

SS PP

0.0±0.03 0.0±0.03 fm -0.32± 0.02 (~8 MeV 3.2 fm Lattice)fm -0.32± 0.02 (~8 MeV 3.2 fm Lattice)

(~18 MeV 2.4 fm Lattice)(~18 MeV 2.4 fm Lattice)

0.08±0.01 -0.16±0.10.08±0.01 -0.16±0.1

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

VE

mspn

...........

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

0|))0()((|0

,

..

.

EV

m

EV

m

sqsqxee

xTeG

x

sq

xqi

x

xpix

xpiNN

x

Normalization condition requires :

Two point function :

VFor one particle bound state there will be no volume dependence.

For two particle state :

Fitting function :

Therefore, fitted weight (Wi) should be proportional to 1/V for two particle scattering state.

V

EV

m

EV

m

EV

m

EV

mG

x

1

,........22

2

22

2

11

1

11

112

i

tmi

ieWtG )(

And,And,

Lattice Continuum

S-wave (1/2¯)S-wave (1/2¯)

•No need to consider ghost state (propagators are positive).•Lowest states in 2.4 fm are higher then those in 3.2 fm which reflect the volume dependence of the energy shift.•The first excited state is

also not the θ+ candidate as it is several hundred MeV higher near EK(p=pL) + EN(p=pL).

•Ratio of spectral weight for two non-interacting particles W(12)/W(16) = V3(16)/V3(12) = 2.37

P-wave (1/2P-wave (1/2++))

• Propagators turns negative. Ground state is S-wave KNη' ghost state. In fitting function this ghost state, pentaquark and KN-P- wave scattering state are the first three states. We find ghost and scattering state.

• The volume dependence in

EK(p=pL) + EN(p=pL) due to

the P-wave nature is seen for medium and high quark masses. Near The chiral limit the scattering length is close to zero which is consistent with the experiment.

Volume Dependence in 1/2Volume Dependence in 1/2++ channel channel

• For bound state, fitted weight will not show any volume dependence.

• For two particle scattering state, fitted weight will show inverse volume dependence

Our observed ground state is p-wave scattering state

Comparison of Lattice ResultsComparison of Lattice Results

22

22

2222

)sin(1

)2

sin(2

,)1(,)0(

apa

map

am

pmpmpEmmpE

LNL

K

NK

NK

LatticeLattice 1/2¯1/2¯ 1/21/2++

SasakiSasaki

Csikor Csikor et. alet. al

This WorkThis Work

EEthth=1.57GeV (threshold state)=1.57GeV (threshold state)

E(p=1) = 1.96 GeVE(p=1) = 1.96 GeV

Observed state :Observed state :

1. Scattering state1. Scattering state

mass (Emass (E00) = ? ) = ?

2. E2. E11~1.76 GeV (~1.76 GeV (θθ++) )

EE00/E(p=0) ~ 0.99 (E/E(p=0) ~ 0.99 (Ethth))

EE11/E(p=0) ~ 1.074 (/E(p=0) ~ 1.074 (θθ++) )

EE00/E(p=0) ~ 1/E(p=0) ~ 1

Weight shows characteristic Weight shows characteristic volume dependence of scattering volume dependence of scattering state.state.

EE1 1 : coincides with E(p=1) state.: coincides with E(p=1) state.

EEth th == E(p=1) = 1.96 GeVE(p=1) = 1.96 GeV

(threshold state)(threshold state)

Observed state :Observed state :

1. 2.62GeV1. 2.62GeV

No overlap with scattering state!No overlap with scattering state!

EE0 0 ~ 2.9 GeV~ 2.9 GeV

Threshold scattering state ??Threshold scattering state ??

EE00 : ghost state : ghost state

EE11/E(p=1) ~ 1/E(p=1) ~ 1

Weight shows characteristic Weight shows characteristic volume dependence of volume dependence of scattering state.scattering state.

Interpolating field should have overlap Interpolating field should have overlap with threshold scattering state unless with threshold scattering state unless one can show that the used one can show that the used interpolating field cannot be interpolating field cannot be transformed to usual KN interpolating transformed to usual KN interpolating field by Fierz transformationfield by Fierz transformation

Comments on hep-lat/0309090 (Csikor Comments on hep-lat/0309090 (Csikor et.alet.al))

Correlation function from one

interpolating field <η1η1>

Cross-correlator : <η1η1> + α <η1η2> +α<η2η1> + α2 <η2η2>

Claim : One peak for each channel. One is θ+(1/2¯) corresponding to I=0.Observed θ+ peak is not sharp enough and it still could be consistent with the threshold scattering state.Also, 1/2+(I=0) is quite large. Where is the P-wave scattering state?? m(1/2+)/m(1/2-) ~ 2 ~1.5 (Sasaki)

Peaks

.2/3for

),(

,2/3,2/3,

55

2/3

z

eecbaT

abc

z

Idu

SduSCu

II

• Diagonal and cross correlators have been calculatedDiagonal and cross correlators have been calculated for three lattices.for three lattices.

• Analysis will be completed very soon.Analysis will be completed very soon.

Ξ¯ ¯

ConclusionsConclusions

• Several experiments reported the discovery of Several experiments reported the discovery of θθ+ + (1540).(1540). One experiment One experiment reported the discovery of reported the discovery of ΞΞ¯ ¯(1860).¯ ¯(1860). However, their existences have not However, their existences have not

been absolutely established yet. We only know their strangeness. Other been absolutely established yet. We only know their strangeness. Other important quantum numbers, like spin, parity, isospin need to be important quantum numbers, like spin, parity, isospin need to be established. More experiments (particularly direct KN scattering) and established. More experiments (particularly direct KN scattering) and careful analysis are needed. More experiments will be performed soon in careful analysis are needed. More experiments will be performed soon in various Laboratories (including JLab) around the world.various Laboratories (including JLab) around the world.

• Width of Width of θθ++(1540)(1540) found to be very very small (even may be < 1 MeV) found to be very very small (even may be < 1 MeV) which is very different than any other resonance particle. If which is very different than any other resonance particle. If θθ++(1540)(1540) exists, exists, theorists must find out new way to explain its width. Its existence will open theorists must find out new way to explain its width. Its existence will open up entirely new (and richer) hadron spectrum and bring new information up entirely new (and richer) hadron spectrum and bring new information about nature of short distance interactions between quarks.about nature of short distance interactions between quarks.

• Different model predicts different quantum numbers andDifferent model predicts different quantum numbers and masses for masses for

θθ+ + (1540)(1540). . They all predict nearby other additional states.They all predict nearby other additional states.

• Lattice QCD can help to find out quantum numbers of pentaquark states.Lattice QCD can help to find out quantum numbers of pentaquark states.

ConclusionsConclusions

• We have not seen We have not seen θθ+ + state on our lattice calculation. We see only scattering states both state on our lattice calculation. We see only scattering states both in positive and negative parity channel.in positive and negative parity channel.

• To claim convincing evidence for To claim convincing evidence for θθ++ from lattice calculation, one must see volume from lattice calculation, one must see volume dependent scattering states along with the volume insensitive dependent scattering states along with the volume insensitive θθ++ bound state. For bound state. For quenched lattice calculation one must consider ghost states in low quark mass region.quenched lattice calculation one must consider ghost states in low quark mass region.

• Our lattice study for Our lattice study for ΞΞ pentaquark is going on (correlators have already been calculated pentaquark is going on (correlators have already been calculated for three lattices). Analysis will be completed soon. Also study of pentaquark by cross-for three lattices). Analysis will be completed soon. Also study of pentaquark by cross-correlators (correlators (a laa la Csikor et al.) will also be completed soon. Csikor et al.) will also be completed soon.

• In future, we will carry out similar study using bigger lattices and many more In future, we will carry out similar study using bigger lattices and many more configurations. Furthermore, we will study other exotic states involving four quarks-configurations. Furthermore, we will study other exotic states involving four quarks-antiquarks (like, antiquarks (like, ππππ, K, KKK, D, DSS).).

Bottom-line : It will be an exciting time for experimentalists, theorists andBottom-line : It will be an exciting time for experimentalists, theorists and Lattice community, and we are fully involved in this game.Lattice community, and we are fully involved in this game.