Anisotropic Lattice QCD studies of penta-quark baryons

1
Anisotropic Lattice QCD studies of penta-quark
baryons
N. Ishii (TITECH, Japan) T. Doi
(RIKEN BNL)H. Iida (TITECH, Japan)Y.
Nemoto (Nagoya Univ.) M. Oka
(TITECH, Japan) F. Okiharu (Nihon Univ.,
Japan)H. Suganuma (Kyoto Univ., Japan)

• Plan of the talk
• Introduction
• General Formalism
• Numerical Result on JP1/2()
• A Further Investigation of the Negative parity
  state
• Hybrid Boundary Condition(HBC) method
• Numerical Result II
• Anisotropic Lattice QCD result on JP3/2()
• Summary/Discussion

START
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1. Introduction
Since the first discovery of a manifestly exotic
baryon by LEPS group at
SPring-8, enormous efforts have been devoted to
the studies of penta quarks.

• ? The parity of T(1540) is one of the most
  important topics.
• Experimental determination of the parity of
  T(1540) is difficult.
• Theoretical opinions are divided into two pieces.
• Positive parity is supported bySoliton models,
  Jaffe-Wilczek diquark model, ...
• Negative parity is supported byNaive quark
  models, QCD sum rule, lattice QCD(?)

3
Lattice QCD studies of the penta quarks
? A number of lattice QCD studies of 5Q system
has been increased recently.
However, these results are not reached the
consensus yet.
The aim of this talk is (1) to provide a accurate
data using anisotropic lattice QCD.(2) to
provide a further studies of JP1/2(-) state
using a new method with the Hybrid boundary
condition(HBC).(3) to provide the anisotropic
lattice QCD result on JP3/2() channel with
a large number of gauge configurations as
Nconf1000.
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2.General Formalism (Part I JP1/2())
Interpolating field for T
As adopted in(1) J.Sugiyama et al.,
PLB581,167(2004).(2) S.Sasaki, PRL93,152001
(2004).
A non-NK type operator (I0, J1/2)
To reduce the overlap with NK scattering states
Temporal correlator
(lower component)
(upper component)
Positive parity states dominate.
Negative parity states dominate.
Positive parity contribution cannot become
negligible.
Negative parity contribution cannot become
negligible.
T
T
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3. Numerical Result I

• Lattice Parameter Setup
• Gauge Config by standard Wilson gauge action
• Lattice size 12396 (2.2fm)34.4fm in
  physical unit
• ß 5.75
• Lattice spacing from
  Sommer parameter r0.
• Anisotropic latticeRenormalized anisotropy
  as/at4for accurate measurements of correlators
  and masses
• (gauge config) 504
• The gauge configurations are separated by 500
  pseudo heat-bath sweeps, after skipping 10000
  thermalization sweeps.
• O(a) improved Wilson quark (clover) action.
• Smeared source to reduce higher spectral
  contributions

These values covers
0.1240 0.1230 0.1220 0.1210
656(2) 784(1) 893(1) 1005(1)
1011(5) 1085(4) 1162(3) 1240(3)
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Negative parity channel (JP1/2(-))
Correlator
Effective mass
Single-state saturation is achieved.
Higher spectral contribution is gradually reduced.
best fit in the plateau
Plateau
Effective Mass
negligible !
average mass at time-slice t
If then Existence of the plateau
indicates the single-state saturation of the
correlator G(t).
NK threshold(s-wave)By neglecting the
interaction between N and K
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Effective mass plot
We consider the large t region to suppress
contributions from excited states.
effective mass meff(t) is a useful indicator to
determine whether the single-state saturation is
achieved or not.
average of masses at each time-slice t
After the contribution from the excited states
becomes negligible, i.e.,
negligible !
G(t) can be approximated as a single exponential.
Then we have,
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Positive parity channel JP1/2()
Correlator
Effective mass
Higher spectral contribution is gradually reduced.
Plateau
best fit in the plateau
Single-state saturation is achieved.
L
L
L
NK threshold (p-wave) The quantized spatial
momenta are due to the finiteness of the box.
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Chiral extrapolation
NK threshold (p-wave)
At physical point (1) Positive parity 2.25(11)
GeV(2) Negative parity 1.75(3) GeV
NK threshold (s-wave)

1. Our data does not support the low-lying positive
   parity . To obtain a low-lying state, it
   should appear below the raised NK threshold.
2. For negative parity channel, m1.75 GeV is rather
   close to the empirical value 1.54 GeV. However,
   it should be clarified whether this state is a
   compact 5Q resonance or not.(We will perform a
   further study in this direction from the next
   slide)

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4. Further study of the negative parity
state.(a) NEW METHOD with Hybrid BC(HBC)
Hybrid Boundary Condition(HBC)
u quark spatially anti-periodic BC
d quark spatially anti-periodic BC
s quark spatially periodic BC
Cosequence on hadrons
quark contents spatial BC minimum momentum minimum momentum
N anti-periodic BC
K anti-periodic BC
periodic BC
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Expected consequence on the spectrum

1. NK-threshold is raised up due to finite voluem
   effect(200 MeV if L2fm.)
2. Compact 5Q resonance state is expected to be less
   sensitive to the change of the boundary condition.

200 MeV
S-wave
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An example
Response of a compact resonance state to the
change of boundary condition.
For this purpose, nucleon is not appropriate,
because nucleon is sujbect to the anti-periodic
BC.
spatially periodic BC
A localized resonance is less sensitive to the
change of boundary condition !
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Numerical result II
Hybrid BC(HBC)
Periodic BC(PBC)
NK threshold(s-wave)
NK threshold(s-wave)
The plateau is shifted above by the expected
amount. (1) No compact 5Q resonance exists in
the region as (2) The state observed in the
negative parity channel turns out to be an NK
scattering state.

• The hopping parameter leads to mN1.74
  GeV, mK0.79 GeV
• Expected shift of the NK threshold for L2.15 fm
  is

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Combining the results from the other quark masses

• data points The best fit value on the
  plateau.
• solid lines NK(s-wave) threshold

We have not found a compact 5Q resonance in
JP1/2(-) in our calculation.
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Part II Numerical result on JP3/2() channel

1. Spin of T is also not yet determined
   experimentally.
2. JP3/2(-) possibility can solve the puzzle of the
   narrow decay width.(proposed by A.Hosaka et al.,
   PRD71,074021(2005).)Advantage(a) It allows
   the configuration of (0s)5.(b) It decays into a
   d-wave KN state.Suppressed overlap to d-wave KN
   state The decay width is expected to be
   significantly narrow.Disadvantage(a) The
   color-magnetic interaction makes it massive.If
   some contribution can cancel the color-magnetic
   interaction to make its mass around 1540, we will
   obtain a penta-quark with a significantly narrow
   width.

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Rarita-Schwinger interpolating fields
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JP3/2(-)state (effective mass plot)
This correlator is too noisy !
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NK(JP3/2(-))

1. NK(s-wave) threshold is raised up by 179 MeV.
2. NK(d-wave) threshold is lowered down by 66
   MeV.
3. Best-fit(m5Q) is raised up by 80 MeV.In HBC, the
   5Q state appears below the NK(s-wave) threshold.
   However, its value is almost consistent with
   NK-threshold(s-wave). This state is an NK
   scattering state.A large number of config.
   Nconf1000 has played a crucial role. (At our
   preliminary stage with Nconf500, these two
   states had pretended to be different from each
   other.)

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color-fused NK(JP3/2(-))

• The situation is similar to the NK-correlator.
• NK(s-wave) threshold is raised up by 179 MeV.
• NK(d-wave) threshold is lowered down by 66
  MeV.
• Best-fit(m5Q) is raised up by 90 MeV.In HBC, the
  state appears below the NK-threshold(s-wave).
  However, its value is almost consistent with the
  NK-threshold.
• This state is also an NK scattering
  state.

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Chiral extrapolation (JP3/2(-))
Physical quark mass region
?(circle) from NK-type correlator ?(box)
from color-fused NK-type correlator ?
Results from diquark-type correlator are not
shown due to huge statistical error.
NK scattering states

• In the physical quark mass region
• NK-type m5Q 2.17(4) GeV
• Color-fused NK-type m5Q 2.11(4) GeV
• No evidence for a low-lying 5Q state

To obtain a low-lying 5Q state, it should appear
below the raised NK threshold(d-wave) at least in
the light quark mass region.
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JP3/2() state (effective mass plot)
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NK(JP3/2())

• Changes in the two-particle spectrum are too
  small in JP3/2() channel.
• NK(p-wave) threshold is lowered down by 57
  MeV.
• NK(p-wave) threshold is lowered down by 66
  MeV.
• Best-fit(m5Q) is raised up by 60 MeV.The shift
  is small. But the shifts of NK and NK thresholds
  are also small.To elucidate the nature of this
  state, we need more statistics.
• The nature of this state is not clear so
  far. But it is clearly a very massive state.

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Color-fused NK(JP3/2())

• Changes in two-particle spectrum are too small in
  JP3/2() channel.
• NK(p-wave) threshold is lowered down by 57
  MeV.
• NK(p-wave) threshold is lowered down by 66
  MeV.
• Best-fit(m5Q) is raised up by 90 MeV.
• This state is an NK scattering state.

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Diquark-type(JP3/2())

• Changes in the two-particle spectrum are too
  small in JP3/2() channel.
• NK(p-wave) threshold is lowered down by 57
  MeV.
• NK(p-wave) threshold is lowered down by 66
  MeV.
• Best-fit(m5Q) is raised up by 80 MeV.
• This state is an NK-scattering state.

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Chiral extrapolation (JP3/2())
Physical quark mass region
?(circle) from NK-type correlator ?(box)
from color-fused NK-typecorrelator ?(triangle
) from diquark-type correlator
?

• In the physical quark mass region,
• NK-type m5Q 2.64(7) GeV
• Color-fused NK-type m5Q 2.48(10) GeV
• Diquark-type m5Q2.42(6) GeV
• No evidence for a low-lying 5Q states.

NK scattering states
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6. Summary/discussion

• We have studied T(1540) by using the anisotropic
  lattice QCD. For acuracy,(a) renormalized
  anisotropy as /at 4(b) O(a) improved Wilson
  (clover) action for quarks(c) smeared source(d)
  large number of gauge configurations Ncf1000
  (for JP3/2() states)
• JP1/2()
• Non-NK type interpolating field
• JP1/2() m5Q 2.25(11) GeV --- too massive to
  be identified as T(1540)
• JP1/2(-) m5Q 1.75(4) GeV --- rather close to
  the observed value.
• We have proposed a new method (Hybrid BC
  HBC).HBC analysis showsthe state(1.75 GeV) is
  not a compact 5Q state but an NK scattering
  state.
• JP3/2() A large statistics as Ncf1000
  has played an important role.
• Three-types of interpolating field(NK-type,
  color-fused NK-type, diquark-type)
• Only massive states after the chiral
  extrapolationJP3/2(-)
  JP3/2()
• HBC analysis is performed.Compact 5Q resonances
  are not found in our calculation.
• Following possibilies would be interesting for
  T(1540)(a) small quark mass effects (and/or
  more elaborate chiral extrapolation), (b) large
  spatial volume, (c) dynamical quark(including
  pKN hepta-quark picture), (d) elaborate
  interpolating fields to fit the diquark picture.