Pentaquarks on the Lattice

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University of Cyprus
Pentaquarks on the Lattice

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

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

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

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Solving QCD
coupling constant g

• 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

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

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

• Create initial trial state with operator J that
  has the quantum numbers of the hadron we want to
  study

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Effective mass
G
G
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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

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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 (?n(t,t0)
/?n(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
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Resonances
Consider two interacting particles in a finite
box with periodic or antiperiodic boundary
conditions

• discrete momentum leading to discrete energy
  spectrum
• where , kx ,ky, kz0,1,2,..
  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)

Difficult in practice
Demonstrated in a toy model O(4) non-linear
s-model
M. Göckeler et al., NPB 425 (1994) 413
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Two pion-system in I2
Correlation matrix
with J(x) product of pion- and rho-type
interpolating fields e.g.
total momentum0
Spacing between scattering states1/ Ls2
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Project to zero relative momentum p(0)p(0)
Check taking p0 on small lattice (163x32)
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Diquarks
Soliton model Diakonov, Petrov and Polyakov in
1997 predicted narrow T(1530) in antidecuplet
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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
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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. ?. and G. Koutsou, PRD 71 (2005)
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Can we study non-static diquarks on the Lattice?

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

Static quark propagator
Baryon with an infinitely heavy quark
Flavor symmetric ? spin one
Flavor antisymmetric ? spin zero
t
t0
light quark propagator G(x0)
R. Jaffe hep-ph/0409065
JP color flavor diquark structure
0 1 6 qTC?5q, qTC?5?0q qTC?iq, qTCs0i q
Models suggest that scalar diquark is lighter
than the vector
attraction M0
M1gtM0
M1
In the quark model, one gluon exchange gives rise
to color spin interacion
M1 M0 2/3 (M?-MN) 200 MeV and
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Mass difference between bad and good
diquarks
C.A., Ph. de Forcrand and B. Lucini Lattice 2005

• First results using 200 quenched configurations
  at ß5.8 (a0.15 fm) ß6.0 (a0.10 fm)
• fix mp800 MeV (?0.1575 at ß5.8 and ?0.153 at
  ß6.0)
• heavier mass mp 1 GeV to see decrease in mass
  (?0.153 at ß5.8)

ß6.0 ?0.153
?M (GeV)
ß mp(MeV) ?? (MeV)
5.8 5.8 6.0 1000 800 800 70 (12) 109 (13) 143 (10)
K. Orginos Lattice 2005 unquenched results with
lighter light quarks
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Diquark distribution
Two-density correlators provide information on
the spatial distribution of quarks inside the
heavy-light baryon
quark propagator G(x0)
Study the distribution of d-quark around u-quark.
If there is attraction the distribution will peak
at ?0
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Diquark distribution
Good diquark peaks at ?0
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Pentaquarks?
SPring-8 ? 12C ? ? ?- n
CLAS at Jlab ?D? K K- pn
High statistics confirmed the peak
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Summary of experimental results
Negative results
Positive results
Experiment Reaction
CDF p p?PX
ALEPH Hadronic Z decays
L3 ???TT
HERA-? pA ? PX
Belle KN ? PX
BaBar e e- ?Y
Bes e e- ?J/?
HyperCP (K,p,p)Cu?PX
SELEX (p,S,p)p ? PX
FOCUS ?p ? PX
E690 pp ? PX
DELPHI Hadronic Z decays
COMPASS µ(6Li D)? PX
ZEUS ep ? PX
SPHINX pC ?TK0C
PHENIX AuAu?PX
Experiment Reaction Mass (MeV) Width (MeV)
LEPS ? C12?K- K n 1540(10) lt25
DIANA K Xe ?KS0 pXe 1539(2) lt9
CLAS ? d ? K- K np ? p ? K- K np 1542(5) lt21
SAPHIR ? p?KS0 K n 1540(6) lt25
COSY pp?S KS0 p 1530(5) lt18
SVD pA? KS0 pX 1526(3) lt24
ITEP ?A?KS0pX 1533(5) lt20
HERMES e d ?KS0pX 1528(3) 13(9)
ZEUS e p?Ks0 p X 1522(3) 8(4)
Ppentaquark state (Ts,?,Tc)
A. Dzierba et al., hep-ex/0412077
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Pentaquark mass
Correlator C(t) exp(-mT t)
C(t) w1exp(-mKN t)w2 exp(-mT t)
mass of T
mT-mKN100 MeV
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Models
Jaffe and Wilczek PRL 91 232003 (2003) Diquark
formation
JP1/2
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Interpolating fields for pentaquarks
What is a good initial fgt for T? 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
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Does lattice QCD support a T?
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

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Energy spectrum
Lüscher NPB364 (1991)
The energy spectrum of a KN scattering state on
the lattice is given by
where , kx,y,z0,1,2,..
assuming periodic b.c. or
, n0,1,2,..
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 ngt 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 wn1
• ? off-resonance states are suppressed relative to
  states around the resonance mass

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Scattering states
The two lowest KN scattering states with non-zero
momentum
n1
n2
E (GeV)
T
Contributes only in negative parity channel
S-wave KN
Correlator
Dominates if w2gtgtw1 and (mT-mKN) t lt1
? tlt10 GeV-1 assuming energy
gap100MeV or t/alt20
If mixing is small w1L-3 ? suppressed for large L
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Does lattice QCD support a T?
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

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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
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Identifying the T 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
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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 I0.

• Measure the energies

Csikor et al. JHEP 0311 (2003) Results based
on JKN with a check done using the correlation
matrix with JKN 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 T.
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S. Sasaki, PRL 93 (2004) Used Jdiquark and fitted
to first plateau to extract the T mass on a
lattice of size 2.2 fm (323x48 ß6.2) with
mp0.6-1 GeV
mp750 MeV
Negative parity
Positive parity
E x 2.9 GeV
T
T
E1KN
E0KN
Double plateau structure is not observed in other
similar calculations
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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.
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Pentaquarks
Perform a similar analysis as in the two-pion
system using Jdiqaurk and JKN
Takahashi et al., Pentaquark04 and
hep-lat/0503019 JKN and JKN on spatial
lattice size 1.4, 1.7, 2.0 and 2.7 with a larger
number of configurations
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Spectral weights for pentaquark
C.A. and A. Tsapalis, Lattice 2005
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Does lattice QCD support a T?
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

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Hybrid boundary conditions
Ishii et al., PRD 71 (2005) Use antiperiodic
boundary conditions for the light quarks and
periodic for the strange quark T 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 p/L and
therefore the lowest KN scattering state is
shifted to larger energy
Negative parity
3.0
Spatial size2.2 fm
?0.121
?0.122
?0.123
E (GeV)
2.5
Strange quark mass
?0.124
2.0
Standard BC
Hybrid BC
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Binding
Lasscock et al., hep-lat/0503008
Interpolating fields JKN, JKN, Jdiquark on a
lattice size2.6 fm. Although a 2x2 correlation
matrix was considered the results for I0 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 ?(1232) and the P-wave Np
Positive parity
hep-lat/0504015 maybe a 3/2 isoscalar
pentaquark?
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Positive parity T
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
KN
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Holland and Juge, hep-lat/0504007
Fixed point action and Dirac operator, 2x2
correlation matrix analysis using JKN and JKN on
a lattice of size 1.8 fm, mp0.550-1.390 GeV
Energies of the two lowest states are consistent
with the energy of the two lowest KN scattering
states
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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?
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Conclusions

• 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 10s 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