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Pentaquarks on the Lattice University of Cyprus C. Alexandrou EINN 2005 Workshop “New Hadrons: Facts and Fancy” Milos, 19 September 2005
Pentaquarks on the Lattice
University of Cyprus
C. 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
• Resonances on the Lattice
• Diquarks
• Pentaquarks
• Summary of quenched results on pentaquarks
• Conclusions
Solving QCD
• At large energies, where the coupling constant is small, perturbation theory is applicable has been successful in describing high energy processes
• 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
• At very low energies chiral perturbation theory becomes applicable
pQCDChPT
E
L a t t i c e Q C D
L a μνQCD μ q
aν
1=- F F +ψ D- m ψ
4 coupling constant g
• Wick rotation into Euclidean time:
limits applicability to lower states
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 a0 to recover continuum physics
a
Lattice QCD is a discretised version of the QCD Lagrangian with only parameters the coupling constant and the masses of the quarks
• 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
x
e e3 3i dt d - d d xL H
x
Masses of Hadrons
Energies can be extracted from the time evolution of correlation functions:
+| >= J | 0 >• Create initial trial state with operator J+ that has the quantum numbers of the hadron we want to study:
insert complete set of energy eigenstates
• Take overlap with trial state: -Ht +t< | e | >=< 0 | J(t)J | 0 >
|
n 0 n 0
+ Ht -Ht
n=0
-(E -E )t -(E -E )t2
n=0 n=n
0
C(t) = < 0 | J(t)J | 0 > = < 0 |e J e | n >< n | J | 0 >
= |< n >| e = ew
spectral weights
Correlator / two-point function
-Hte | >• Evolve in imaginary time: i.e. assume transfer matrix
• Take limit : extract E1 measured w.r.t.to vacuum energy provided w0 = <0|φ>=0 and w1= <1|φ> is non zero
t
ln
1 0 2 0
1 0
-(E -E )t -(E -E )t1 2
1-(E -E )(t-1)1
w e + w e + ...E
w e + ...efft>>1C(t)
m (t) - lnC(t- 1)
Effective mass:
π- m tt>>1Ht - Ht + 2π π π
x
C(t) = <0|e J (x,0) e J (0,0)| 0> | <0| J | π>| e
Projects to zero momentum
Pion mass:
Using Wick contractions the correlator can be written in terms of quark propagators
��������������
5 5 5 5
xx<0|d(x,t)γ u(x,t) u(0,0) γ d(0,0)|0>=- Tr γ G(0,0;x,t)γ G(x,t;0,0)
G+(x;0)
where the operator Jπ = d γ5 u has the pion quantum numbers
G
G+
fit plateau mπ
bending due to antiperiodic b.c.
Contamination due to excited states
Smearing suppresses excited states
The quenched light quark spectrum from CP-PACS, Aoki et al., PRD 67 (2003)
• Lattice spacing a 0
• Chiral extrapolation
• Infinite volume limit
Precision results in the quenched approximation
u
u
d
Included in the quenchedapproximation
u
u
d
Not included in thequenched approximation
Excited states?
0 0C(t)v(t) = λ(t, t )C(t )v(t)
n 0 0 n-(E -E )(t-t ) -ΔE tt>>1n 0λ (t, t ) e (1+ e )
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
jk j kx
C (t) = < 0 | J (t, x)J (0) | 0 >
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. Lüscher NPB364 (1991)
Resonances
ˆ 4 4
4a a a a
x μ=1 x a=1
S = -2κ Φ (x)Φ (x + μ)+ J Φ (x) Φ (x)Φ (x) = 1
2 2 2 2N KE = m +p + m +p
Consider two interacting particles in a finite box with periodic or antiperiodic boundary conditions
Difficult in practice
Can one distinguish a resonance from two-particle scattering states?
• different volume dependence of energies and spectral weights
• resonances show up as extra states with weak volume dependence
M. Lüscher NPB364 (1991)
Demonstrated in a toy model: O(4) non-linear σ-modelM. Göckeler et al., NPB 425 (1994) 413
p=2πk/L
Two pion-system in I=2
jk j kx
C (t) = < 0 | J (x)J (0) | 0 >
π π π1 1 1 1 5
3ρ ρ ρ
2 0 0 0 ii=1
J (x) = J (x) J (x), J (x) = d(x)γ u(x)
J (x) = J (x) J (x), J (x) = d(x) γ u(x)
2mπ
2mρ
Ε12π
Slower approach to asymptotic plateau value
Correlation matrix
total momentum=0
with J(x) product of pion- and rho-type interpolating fields e.g.
1 2
nh hE
1 2
2 22 2h s h s= m +n 2π/L + m +n 2π/L
Spacing between scattering states~1/ Ls2
Project to zero relative momentum: π(0)π(0)
s s s+ s+jk j j k k
x,y
C (t,p = 0) = < 0 | J (x)J (y)J (0)J (0) | 0 > s = π,ρ
Check taking p=0 on small lattice (163x32)
Diquarks
Originally proposed by Jaffe in 1977: Attraction between two quarks can produce diquarks:
qq in 3 flavor, 3 color and spin singlet behave like a bosonic antiquark in color and flavor :scalar diquark
andq q
Soliton model Diakonov, Petrov and Polyakov in 1997 predicted narrow Θ+(1530) in antidecuplet
A diquark and an anti-diquark mutually attract making a meson of diquarks
tetraquarks
A nonet with JPC=0++ if diquarks dominate no exotics in q2q2
ff3 3ff⇒ 8 1D D
pentaquarks
Exotic baryons?
8 D Dff f3 3 3f ff fq ⇒ 10 8 1
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.
Flux
tube
forms
between
linear potential
G. Bali, K. Schilling, C. Schlichter, 1995
Static potential for tetraquarks and pentaquarks
C. Α. and G. Koutsou, PRD 71 (2005)
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
q
q
q
q
q
q
q
q
q
Can we study non-static diquarks on the Lattice?
Define color antitriplet diquarks in the presence of an infinitely heavy spectator:
f f f f3 3 = 63
Flavor antisymmetric spin zero
Flavor symmetric spin one
R. Jaffe hep-ph/0409065
JP color flavor diquark structure
0+
1+ 6
qTCγ5q, qTCγ5γ0q
qTCγiq, qTCσ0i q
3
3
3 attraction: M0
M1
M1>M0
2q
1ΔM
m
t t=0
light quark propagator G(x;0)
Static quark propagator
Baryon with an infinitely heavy quark
Models suggest that scalar diquark is lighter than the vector
In the quark model, one gluon exchange gives rise to color spin interacion:
cs s ij i j i ji,j
H = -α M σ .σ λ .λ
M1 –M0 ~ 2/3 (MΔ-MN)= 200 MeV and
Mass difference between ``bad`` and ``good`` diquarksΔ
M (
GeV
) β=6.0 κ=0.153
K. Orginos Lattice 2005: unquenched results with lighter light quarks
• 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=0.1575 at β=5.8 and κ=0.153 at β=6.0)
• heavier mass mπ ~1 GeV to see decrease in mass (κ=0.153 at β=5.8)
C.A., Ph. de Forcrand and B. Lucini Lattice 2005
β mπ(MeV) ΔΜ (MeV)
5.8
5.8
6.0
1000
800
800
70 (12)
109 (13)
143 (10)
Diquark distribution
Two-density correlators : provide information on the spatial distribution of quarks inside the heavy-light baryon
��������������charge 0 0C (x,y)=<B|j (x)j (y)|B>
j0(x)
j0(y)j0 (x) = : u(x) γ0 u(x) :
Study the distribution of d-quark around u-quark. If there is attraction the distribution will peak at θ=0
quark propagator G(x;0)
u
d
θ
Diquark distribution
``Good´´ diquark peaks at θ=0
Pentaquarks?
SPring-8 : γ 12C Κ+ Κ- n
High statistics confirmed the peak
CLAS at Jlab: γD K+ K- pn
Experiment Reaction Mass (MeV)
Width (MeV)
LEPS γ C12K- K+ n 1540(10) <25
DIANA K+ Xe KS0 pXe’ 1539(2) <9
CLAS γ d K- K+ np
γ p K- K+ nπ+
1542(5) <21
SAPHIR γ pKS0 K+ n 1540(6) <25
COSY ppΣ+ KS0 p 1530(5) <18
SVD pA KS0 pX 1526(3) <24
ITEP νAKS0pX 1533(5) <20
HERMES e+ d KS0pX 1528(3) 13(9)
ZEUS e pKs0 p X 1522(3) 8(4)
Experiment Reaction
CDF p pPX
ALEPH Hadronic Z decays
L3 γγΘΘ
HERA-Β pA PX
Belle KN PX
BaBar e+ e- Y
Bes e+ e- J/ψ
HyperCP (K+,π+,p)CuPX
SELEX (p,Σ,π)p PX
FOCUS γp PX
E690 pp PX
DELPHI Hadronic Z decays
COMPASS μ+(6Li D) PX
ZEUS ep PX
SPHINX pC ΘK0C
PHENIX AuAuPX
Summary of experimental results
Positive results Negative results
P=pentaquark state (Θs,Ξ,Θc)A. Dzierba et al., hep-ex/0412077
Pentaquark mass
Time evolution
Correlator: C(t) ~ exp(-mΘ t)
mass of Θ
Initial state with the quantum numbers of Θ+ at time t=0
s*
u d u d
Θ at a later time t>0
s*
u d u d
C(t) ~ w1exp(-mKN t)+w2 exp(-mΘ t) +…
mΘ-mKN~100 MeV
Models
Jaffe and Wilczek PRL 91 232003 (2003): Diquark formation
JP=1/2+
L=0
L=1
u d
u d
s
Karliner and Lipkin, PLB575, 249 (2003) : Diquark-triquark structure
Diquark is 3f and triquark in 6f
fff f3 6 =10 8
Θ+ in the antidecuplet
JP=1/2+
L=1
u d
u d
s
-1/2
Hyperfine interaction short range acts only within the clusters
Antisymmetric color 3c, spin, s=0 and flavor 3f
b Tdi 5
c Tc c equar a bk
a T Te euu Cγ Cd u CddJ Cs=ε
Interpolating fields for pentaquarks
a Te e e e
bb
c5 c 5 caNK 5 u su Cγ γ d - d s γ udJ =ε
Modified NK
What is a good initial |φ> for Θ+? All lattice groups have used one or some combinations of the following isoscalar interpolating fields:
Results should be independent of the interpolating field if it has reasonable overlap with our state
Both local and smeared quark fields were considered :
y
q(x,t)= f(x,y,U(t))q(y,t)
• Motivated by KN strucutre:
N K
a T
Nc
c 5 c 5b
5K ba u sγ d - d sγ uJ u Cγ d=ε
• Motivated by the diquark structure:
Diquark structure
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
The energy spectrum of a KN scattering state on the lattice is given by
2 2 2 2N KE = m +p + m +p 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
Lüscher NPB364 (1991)
The spectral decomposition of the correlator is given by
j
∞- E t
jj=1
C(t)= w e
• 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
p=2πk/L
2πp = n
L
Scattering states
Correlator: ...ΘKN - m t- m t1 2C(t)=w e +w e +
Dominates if w2>>w1 and (mΘ-mKN) t <1 t<10 GeV-1 assuming energy gap~100MeV or t/a<20
If mixing is small w1~L-3 suppressed for large L
S-wave KN
Θ+
Contributes only in negative parity channel
The two lowest KN scattering states with non-zero momentum
2 22 2N s K sE = m +n 2π/L + m +n 2π/L
E (
GeV
) n=1 n=2
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
Works for our test two-pion system provided:
1. Accurate data
2. 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 Θ+ on the Lattice
Alexandrou & Tsapalis (2.9 fm)Lasscock et al. (2.6 fm)
Mathur et al. (2.4 fm)
Ishii et al. (2.15 fm)
Mathur et al. (3.2 fm)
Csikor et al. (1.9 fm)
Sasaki (2.2 fm)
Negative parity
There is agreement among lattice groups on the raw data but the interpretation differs depending on the criterion used
From Lassock et al. hep-lat/0503008
All lattice computations done in the quenched theory
KN scattering states
hep-lat/0503012
JKN and Jdiquark fields are used with non-trivial spatial structure on lattices of size ~2. and 2.4 fm
Negative parity Positive parity
L=0
n=1 n=1
n=2
Θ+
Review of lattice results
All lattice computations are done 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.
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+.
• Measure the energies
203x36, β=6
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
E0KN
E1KN
Negative parity Positive parity
Θ+ Θ+
E x
2.9
GeV
mπ~750 MeV
Double plateau structure is not observed in other similar calculations
• 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.
mπ (GeV)
rati
o of
wei
ghts
Negative parity
Expected for a scattering state
PentaquarksPerform 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
Different from two pion system can not exclude a resonance
Ratio WL1/WL2 ~1 for ti/a up to 26 which is the range available on the small lattices
C.A. and A. Tsapalis, Lattice 2005
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
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
Spatial size~2.2 fm
Strange quark mass
Negative parity
Standard BC Hybrid BC2.0
2.5
3.0
κ=0.124
κ=0.123
κ=0.122
κ=0.121
E (
GeV
)
• Binding
Lasscock et 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
Positive 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
hep-lat/0504015: maybe a 3/2+ isoscalar pentaquark?
Positive parity Θ+
Chiu and Hsieh, hep-ph/0403020
Domain wall fermions Lattice size 1.8 fm
1.554 +/- 0.15 GeV
KN
The lowest state extracted from an 3x3 correlation matrix
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
Summary of lattice computations
Group Method of analysis/criterion Conclusion
Alexandrou and Tsapalis Correlation matrix, Scaling of weights
Can not exclude a resonance state. Mass difference seen in positive channel of right order but mass too large
Chiu et al. Correlation matrix Evidence for resonance in the positive parity channel
Csikor et al. Correlation matrix, scaling of energies
First paper supported a pentaquark , second paper with different interpolating fields produces a negative result
Holland and Juge Correlation matrix Negative result
Ishii et al. Hybrid boundary conditions Negative result in the negative parity channel
Lasscosk et al. Binding energy Negative result
Mathur et al. Scaling of weights Negative result
Sasaki Double plateau Evidence for a resonance state in the negative parity channel.
Takahashi et al. Correlation matrix, scaling of weights
Evidence for a resonance state in the negative parity channel.
J. Negele, Lattice 2005 Correlation matrix, scaling of weights
Maybe evidence for a resonance state?
ConclusioConclusionsns
Diquark dynamics
Studies of exotics and two-body decays
• 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
• 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
