Transcript Slide 1

University of Cyprus
Pentaquarks on the Lattice
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
LQCD = -
1 a aμν
Fμν F +ψ  D - m q  ψ
4
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
ChPT
La t t i c e
QCD
pQCD
E
Lattice QCD
Lattice QCD is a discretised version of the QCD Lagrangian with only parameters the
coupling constant and the masses of the quarks
• 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
a
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
• Wick rotation into Euclidean time:
e
i dt d3 x L
xe 
- dd3 x H
limits applicability to lower states
• 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
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: |  >= J + | 0 >
• Evolve in imaginary time:
e-Ht |  >
• Take overlap with trial state:
i.e. assume transfer matrix
< t | e-Ht |  >=< 0 | J(t)J + | 0 >
insert complete set of energy
eigenstates

C(t) = < 0 | J(t)J | 0 > =  < 0 |e Ht J e -Ht | n >< n | J  | 0 >
+
n=0
Correlator / two-point function

=  |<  | n >| e
2
n=0

-(En -E0 )t
=  w n e-(En -E0 )t
n= 0
spectral weights
• Take limit t   : extract E1 measured w.r.t.to vacuum energy provided w0 = <0|φ>=0 and
w1= <1|φ> is non zero
Effective mass:
 w1e-(E1 -E0 )t + w 2e-(E2 -E0 )t + ...  t>>1
 C(t) 
meff (t)  -ln 
  ln 
  E1

-(E1 -E0 )(t-1)
w1e
+ ...
 C(t -1) 


Pion mass:
t>>1
C(t) = < 0|eHt J π (x,0) e-Ht J+π (0,0)|0 > 
|< 0|J π |π >|2 e-mπt
x
Projects to zero momentum
where the operator Jπ = d γ5 u has the pion quantum numbers
Using Wick contractions the correlator can be written in terms of quark propagators


γ5 d(0,0)
|0>=<0|d(x,t) γ5 u(x,t) u(0,0)
 Tr γ5 G(0,0;x,t)γ5 G(x,t;0,0)
Contamination
due
to
x
excited states
G
fit plateau  mπ
Smearing
suppresses
excited states
x


G+(x;0)
bending due to
antiperiodic b.c.
G+
Precision results in the quenched approximation
u
u
d
Included in the quenched
approximation
u
u
d
The quenched light quark spectrum from
CP-PACS, Aoki et al., PRD 67 (2003)
• Lattice spacing a  0
• Chiral extrapolation
• Infinite volume limit
Not included in the
quenched approximation
Excited states?
Construct NxN mass correlation matrix:
Cjk (t) =  < 0 | J j (t, x)J k (0) | 0 >
x
C. Michael, NPB259 (1985) 58
M. Lüscher & U. Wolff, NPB339
(1990) 222
Maximization of ground state overlap leads to the generalized eigenvalue equation
C(t)v(t) = λ(t,t 0 )C(t 0 )v(t)
It can be shown that
t>>1
λn (t,t 0 ) 
e-(En -E0 )(t-t0 ) (1+ e-ΔEn t )
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
Resonances
Consider two interacting particles in a finite box with periodic or
antiperiodic boundary conditions
 discrete momentum leading to discrete energy spectrum E = m2N +p2 + m2K +p2
where p =2πk/L , 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)
Difficult in
practice
Can one distinguish a resonance from two-particle scattering states?
• different volume dependence of energies and spectral weights
M. Lüscher NPB364 (1991)
• resonances show up as extra states with weak volume dependence
Demonstrated in a toy model: O(4) non-linear σ-model
4
ˆ + J  Φ4 (x)
S = -2κ  Φa (x)Φa (x + μ)
x μ=1
x
4
 Φ (x)Φ (x) = 1
a
a=1
a
M. Göckeler et al.,
NPB 425 (1994) 413
Two pion-system in I=2
Correlation matrix
Cjk (t) =  < 0 | J j (x)Jk (0) | 0 >
with J(x) product of pion- and rho-type interpolating fields e.g.
x
total momentum=0
Enh1h2 = m2h1 +n 2π/Ls  + m2h2 +n 2π/Ls 
2
Spacing between
scattering states~1/ Ls2
J1 (x) = J1π (x) J1π (x),
J1π (x) = d(x)γ 5u(x)
J 2 (x) = J ρ0 (x) J ρ0 (x),
J ρ0 (x) = d(x) γ i u(x)
2
3
i=1
Ε12π
2mρ
2mπ
Slower approach to
asymptotic plateau
value
Project to zero relative momentum: π(0)π(0)
s+
Cjk (t,p = 0) =  < 0 | J sj (x)J sj (y)J s+
s = π,ρ
k (0)J k (0) | 0 >
x,y
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 D:scalar diquark s
and q q D
A diquark and an anti-diquark mutually attract making a meson of diquarks
D 3f  D 3f ⇒8f 1f
tetraquarks
A nonet with JPC=0++  if diquarks dominate no exotics in q2q2
Exotic baryons?
D 3f  D 3f  q3f ⇒10f  8f  8f 1f
pentaquarks
Soliton model Diakonov, Petrov and Polyakov in 1997 predicted narrow
Θ+(1530) in antidecuplet
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
qq
linear potential
G. Bali, K. Schilling, C. Schlichter, 1995
Static potential for tetraquarks and pentaquarks
q
q
q
q
q
q
q
q
q
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. and G. Koutsou, PRD 71 (2005)
Can we study non-static diquarks on the Lattice?
Define color antitriplet diquarks in the presence of an infinitely heavy spectator:
3f  3f = 3f  6f
Flavor symmetric
 spin one
Baryon with an infinitely
heavy quark
t
Flavor antisymmetric
 spin zero
0+
1+
color
3
t=0
light quark propagator G(x;0)
R. Jaffe hep-ph/0409065
JP
Static quark
propagator
flavor
diquark structure
3
qTCγ5q, qTCγ5γ0q
Models suggest that scalar diquark
is lighter than the vector
attraction: M0
M1>M0
3
6
qTCγiq, qTCσ0i q
M1
In the quark model, one gluon exchange gives rise to color spin interacion:
Hcs = -αs  Mijσi .σ j λ i .λ j
i,j
M1 –M0 ~ 2/3 (MΔ-MN)= 200 MeV and ΔM 
1
mq2
Mass difference between ``bad`` and ``good`` diquarks
ΔM (GeV)
C.A., Ph. de Forcrand and B. Lucini Lattice 2005
β=6.0 κ=0.153
• 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)
β
5.8
5.8
6.0
mπ(MeV) ΔΜ
(MeV)
1000
800
800
K. Orginos Lattice 2005: unquenched results with lighter light quarks
70 (12)
109 (13)
143 (10)
Diquark distribution
Two-density correlators : provide information on the spatial distribution of quarks inside the
heavy-light baryon
quark propagator G(x;0)
j0(x)
Ccharge(x,y) = <B|j0(x)j0(y)|B>
j0(y)
j0 (x) = : u(x) γ0 u(x) :
u
θ
Study the distribution of d-quark around u-quark. If there is
attraction the distribution will peak at θ=0
d
Diquark distribution
``Good´´ diquark
peaks at θ=0
Pentaquarks?
SPring-8 : γ 12C  Κ+ Κ- n
CLAS at Jlab: γD K+ K- pn
High statistics confirmed the peak
Summary of experimental results
Negative results
Positive results
Experiment Reaction
Experiment
Reaction
Mass
(MeV)
Width
(MeV)
CDF
p pPX
LEPS
γ C12K- K+ n
1540(10)
<25
ALEPH
Hadronic Z decays
DIANA
K+ Xe KS0 pXe’ 1539(2)
L3
γγΘΘ
pA  PX
CLAS
γd
γ p  K- K+ nπ+
HERA-Β
1542(5)
Belle
KN  PX
BaBar
e+ e- Y
SAPHIR
γ pKS0 K+ n
1540(6)
<25
Bes
e+ e- J/ψ
COSY
ppΣ+ KS0 p
1530(5)
<18
HyperCP
(K+,π+,p)CuPX
SVD
pA KS0 pX
1526(3)
<24
SELEX
(p,Σ,π)p  PX
ITEP
νAKS0pX
1533(5)
<20
FOCUS
γp  PX
E690
pp  PX
DELPHI
Hadronic Z decays
COMPASS
μ+(6Li D) PX
ZEUS
ep  PX
SPHINX
pC ΘK0C
PHENIX
AuAuPX
HERMES
ZEUS
K- K+ np
e+ d KS0pX
e pKs0 p X
A. Dzierba et al., hep-ex/0412077
1528(3)
1522(3)
<9
<21
13(9)
8(4)
P=pentaquark state (Θs,Ξ,Θc)
Pentaquark mass
s*
s*
u
d
u
d
Time evolution
u
d
u
d
Initial state with the quantum
numbers of Θ+ at time t=0
Θ at a later time t>0
C(t) ~Correlator:
w1exp(-mKNC(t)
t)+w~2 exp(-m
exp(-mΘΘ t)
t) +…
mass of Θ
mΘ-mKN~100 MeV
Models
Jaffe and Wilczek PRL 91 232003 (2003):
Diquark formation
Antisymmetric color 3c,
spin, s=0 and flavor 3f
u
Karliner and Lipkin, PLB575, 249 (2003) :
Diquark-triquark structure
s
d
JP=1/2+
s
L=1
L=0
-1/2
L=1
Diquark is 3f and triquark in 6f
3f  6f =10f  8f
u
d
Θ+ in the antidecuplet
JP=1/2+
Hyperfine interaction short range  acts only within the clusters
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:
Jdiquark = εabc  u aTCγ5d b   u cTCde    u eTCdc  CseT
• Motivated by KN strucutre:
Diquark structure
J NK = εabc  u aTCγ5d b  u c  sγ5d  -dc  sγ5u  
N
K
Modified NK
JNK = εabc  u aTCγ5d b  u e  seγ5dc  -de  seγ5u c  
Both local and smeared quark fields were considered :
q(x,t) = f(x,y,U(t)) q(y,t)
y
Results should be independent of
the interpolating field if it has
reasonable overlap with our state
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
Lüscher NPB364 (1991)
The energy spectrum of a KN scattering state on the lattice is given by
where p =2πk/L , kx,y,z=0,1,2,.. assuming periodic b.c.
2π
p
=
n
or
L , 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
E = m 2N +p2 + m 2K +p2
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
∞
C(t) =  w j e
-E j t
j=1
• If |n> is a KN scattering state well below resonance energy then w n~ 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
The two lowest KN scattering
states with non-zero momentum
E = m2N +n 2π/Ls  + m2K +n 2π/Ls 
2
n=1
n=2
Θ+
Contributes only in
negative parity channel
S-wave KN
Correlator:
C(t) = w1 e-mKNt +w2 e-mΘt +...
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
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
There is agreement among
lattice groups on the raw data
but the interpretation differs
depending on the criterion
used
Negative parity
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)
From Lassock et al. hep-lat/0503008
All lattice computations done in the quenched theory
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.
• Measure the energies
Csikor et al.
hep-lat/0503012
JHEP 0311 (2003)
JKN and Jdiquark fields are used
Results based on J’KN with a check done
with non-trivial spatial structure
using the correlation matrix with J’KN and
on lattices of size ~2. and 2.4 fm
JKN. In the negative parity channel, Swave KN scattering state is identified as
the lowest state and the next higher in
Negative parity
Positive parity
+
energy as the Θ .
203x36, β=6
n=2
n=1
n=1
KN scattering states
L=0
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
Θ+
E0KN
Θ+
E1KN
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.
ratio of weights
Negative parity
Expected for a
scattering state
mπ (GeV)
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 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
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
Negative parity
3.0
κ=0.121
Spatial size~2.2 fm
κ=0.122
κ=0.123
2.5
κ=0.124
2.0
Standard BC
Hybrid BC
Strange quark mass
• 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
Mass difference between
Δ(1232) and the P-wave Nπ
Positive parity
hep-lat/0504015: maybe a 3/2+ isoscalar pentaquark?
Mass difference
between the
pentaquark and the Swave KN
Mass difference
between the
pentaquark and the Pwave KN
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?
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 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