Spin 3/2 Penta-quarks in anisotropic lattice QCD
A high-precision mass measurement for the pentaquark (5Q) $\Theta^+$ in $J^{P}=3/2^{\pm}$ channel is performed in anisotropic quenched lattice QCD using a large number of gauge configurations as $N_{conf}=1000$. We employ the standard Wilson gauge action at $\beta=5.75$ and the $O(a)$ improved Wilson (clover) quark action with $\kappa=0.1210(0.0010)0.1240$ on a $12^3 \times 96$ lattice with the renormalized anisotropy as $a_{s}/a_{t} = 4$. % The Rarita-Schwinger formalism is adopted for the interpolating fields. Several types of the interpolating fields with isospin $I=0$ are examined such as (a) the NK$^*$-type, (b) the (color-)twisted NK$^*$-type, (c) a diquark-type. % The chiral extrapolation leads to only massive states, i.e., $m_{5Q} \simeq 2.1-2.2$ GeV in $J^P=3/2^-$ channel, and $m_{5Q} = 2.4-2.6$ GeV in $J^P=3/2^+$ channel. % The analysis with the hybrid boundary condition(HBC) is performed % to investigate whether these states are compact 5Q resonances or not. % No low-lying compact 5Q resonance states are found below 2.1GeV.
penta32.tex
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\begin{document}
\title{
Spin 3/2 Penta-quarks in anisotropic lattice QCD
}
\author{
  N.~Ishii$^1$ \footnote{E-mail : ishii@rarfaxp.riken.jp},
  T.~Doi$^2$ \footnote{E-mail: doi@quark.phy.bnl.gov},
  Y.~Nemoto$^3$ \footnote{E-mail: nemoto@hken.phys.nagoya-u.ac.jp},
  M.~Oka$^1$ \footnote{E-mail: oka@th.phys.titech.ac.jp},
  and H.~Suganuma$^4$ \footnote{E-mail: suganuma@th.phys.titech.ac.jp}
}
\affiliation{$^1$
  Department of Physics, H-27,
  Tokyo Institute of Technology,
  2-12-1 Oh-okayama,
  Meguro, Tokyo 152-8551, Japan
}
\affiliation{$^2$
  RIKEN BNL Research Center, Brookhaven National Laboratory,
  Upton, New York 11973, USA
}
\affiliation{$^3$
  Department of Physics, Nagoya University, Furo, Chikusa,
  Nagoya 464-8602, Japan
}
\affiliation{$^4$
  Department of Physics, Kyoto University, Kitashirakawaoiwake, 
  Kyoto 606-8502, Japan
}
%%%%%%%%%%%%%%%% cut %%%%%%%%%%%%%%%%
%% \date{\bf \today}
%%%%%%%%%%%%%%%% cut %%%%%%%%%%%%%%%%

\begin{abstract}
A high-precision  mass  measurement for the pentaquark (5Q) $\Theta^+$ in $J^{P}=3/2^{\pm}$ channel  
is performed in anisotropic quenched lattice QCD using a large number of gauge configurations as $N_{\rm conf}=1000$.   
We employ the standard Wilson gauge action  at $\beta=5.75$  and  the  $O(a)$ improved  Wilson (clover)  quark
action  with  $\kappa=0.1210(0.0010)0.1240$ on a  $12^3 \times 96$ lattice
with the  renormalized anisotropy  as $a_{\rm s}/a_{\rm t} =  4$.
%
The  Rarita-Schwinger  formalism  is  adopted  for  the  interpolating
fields.  Several types of  the interpolating fields with isospin $I=0$
are  examined  such as  (a)  the  NK$^*$-type,  (b) the  (color-)twisted
NK$^*$-type, (c) a diquark-type.
%
The chiral  extrapolation leads to only massive  states, i.e., $m_{\rm 5Q}
\simeq 2.1-2.2$ GeV in $J^P=3/2^-$ channel, and $m_{\rm 5Q} = 2.4-2.6$ GeV
in $J^P=3/2^+$ channel.
%
The analysis with the hybrid boundary condition(HBC) is performed
%
to investigate whether these states are compact 5Q resonances or not.
%
No low-lying compact 5Q resonance states are found below 2.1GeV.
\end{abstract}

\pacs{
12.38.Gc, % Lattice QCD calculations
12.39.Mk, %Glueball and nonstandard multi-quark/gluon states  
14.20.-c, % Baryons
14.20.Jn   % Hyperons
}% PACS, the Physics and Astronomy
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The main text
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\label{introduction}
The recent discovery of  the manifestly exotic baryon $\Theta^+(1540)$
by the  LEPS group at SPring-8 has  made a great impact  on the exotic
hadron  physics  \cite{nakano}.  Apart  from  other pentaquark  baryon
candidates,   $\Xi^{--}(1862)$    \cite{NA49}   and   $\Theta_c(3099)$
\cite{H1}, several  other candidates of exotic hadrons  have also been
discovered, such as $X(3872)$, $D_s(2317)$, $S_0(3115)$, $X(3940)$ and
$Y(3840)$  \cite{exotics}.  They also  receive an  increasing interest
from theoretical side as well.
%
$\Theta^+(1540)$  is  supposed to  have  baryon  number $B=1$,  charge
$Q=+1$  and strangeness  $S=+1$. Since  $uudd\bar{s}$ is  the simplest
quark content to implement  this quantum number, $\Theta^+(1540)$ is a
manifestly exotic penta-quark(5Q) state.
%
The  penta-quark $\Theta^+$  had  been considered  several times  even
before the experimental discovery \cite{diakonov, jaffe-76, strottman,
weigel,praszalowicz}.
%
In particular,  \Ref{diakonov} provided  the direct motivation  of the
experimental search \cite{nakano}.
%
The  discovered peak  in  the  $nK^+$ invariant  mass  is centered  at
$1.54\pm 0.01$ GeV with a width smaller than 25 MeV.
%
At   the    present   stage,   some   groups    confirmed   the   LEPS
discovery\cite{diana,clas,saphir,experiments},    while   the   others
reported null results\cite{null}.
%
It will still take a while to establish the existence or non-existence
of $\Theta^+(1540)$ experimentally \cite{hicks}.
%
\Ref{saphir}  claims  that  $\Theta^+$  must be  isoscalar,  since  no
$\Theta^{++}$ is observed in the $pK^+$ invariant mass spectrum.

Enormous  theoretical   efforts  have  been  devoted   to  5Q  baryons
\cite{diakonov,jaffe-76,strottman,weigel,      praszalowicz,      oka,
zhu.review,    cohen,   itzhaki,    kim,   hosaka,    jaffe,   lipkin,
carlson-positive,  stancu,  jennings,  glozman,  enyo,  zhu,  matheus,
sugiyama,  carlson,  shinozaki,  huang, maezawa,  shlee,  narodetskii,
sugamoto,  suganuma,  okiharu,  bicudo, oset,  hosaka32,  nishikawa32,
takeuchi32, sugiyama32,  zhu32, inoue32, jaffe32,  huang32, capstic32,
dudek32,hyodo32,nam32}.
%
One of the most challenging problems in understanding its structure is
its extremely  narrow decay width  as $\Gamma \alt 1$  MeV \cite{pdg}.
Several ideas have been proposed:
%
(1) $I=2$  possibility \cite{capstic32},
%
(2) Jaffe-Wilczek's diquark picture \cite{jaffe},
%
(3) $\pi K N$ hepta-quark picture \cite{bicudo,kishimoto},
%
(4) string picture \cite{suganuma,sugamoto},
%
(5) $J^P=3/2^-$ possibility \cite{hosaka,enyo}.
%
Although each  gives a  mechanism to explain  the narrow  decay width,
none of them can satisfy  all the known properties of $\Theta^+(1540)$
simultaneously.
%% Although these ideas provide us with their own mechanisms to implement
%% the narrow decay width, they  hold several drawbacks in their own way.
%% Indeed, no  consistent picture  have been so  far proposed,  which can
%% satisfy all the known properties of $\Theta^+(1540)$ simultaneously.
%

%% In   this  paper,   we  concentrate   ourselves  on   the  $J^P=3/2^-$
%% possibility.
In  this  paper,  we   are  interested  in  the  $J=3/2$  possibility,
$J^P=3/2^-$ in particular.
%
Note that  the spin  of $\Theta^+(1540)$ has  not yet  been determined
experimentally.
%
%% \begin{@@@}
In  the   constituent  quark  picture,  the  narrow   decay  width  of
$J^P=3/2^-$  penta-quarks  can  be  understood in  the  following  way
\cite{hosaka,enyo}.
%%
We expect that  the special configuration $(0s)^5$ is  dominant in the
5Q ground-state in $J^P=3/2^-$ channel.
%
Although $J^P=3/2^-$ penta-quarks  can decay to KN in  the d-wave, the
spectroscopic factor to find d-wave KN states in the dominant $(0s)^5$
configuration vanishes.  Since the  decay is thus allowed only through
its sub-dominant d-wave configuration, the decay width is suppressed.
%
Note that it is further  suppressed by the d-wave centrifugal barrier,
leading  to  the  significantly  narrow  decay  width  of  $J^P=3/2^-$
penta-quarks.
%%
%%
%% Since  $\langle  NK({\rm  d\hbox{-}wave})|(0s)^5\rangle  = 0$  to  the
%% leading order in the quark  model, we expect a significant suppression
%% of the decay width to NK.
%
A possible disadvantage of $J^P=3/2^-$ assignment is that such a state
tends  to be  massive due  to  the color-magnetic  interaction in  the
constituent quark models,
%
which seems to be one of the main reasons why there are only a limited
number  of   effective  model   studies  for  spin   3/2  penta-quarks
\cite{hosaka32,nishikawa32,takeuchi32,sugiyama32,zhu32,
inoue32,jaffe32,huang32,capstic32,hyodo32,nam32,enyo}.
%
However,  it is  not  clear whether  these  conventional framework  is
applicable  to a  new  exotic 5Q  system  as $\Theta^+(1540)$  without
involving any modifications.
%
%% \begin{itemize}
%% \item
Indeed,  a  model  was proposed  where  a  part  of  the role  of  the
color-magnetic   interaction  can   be  played   by   the  flavor-spin
interaction, which  makes the  mass-splitting between the  $1/2^-$ and
the $3/2^-$ states smaller \cite{takeuchi32}.
%% \item
%% Indeed,  the  flavor-spin  interaction   can  play  the  role  of  the
%% color-magnetic interaction in part. As a result, the contribution from
%% the  color-magnetic interaction  becomes reduced,  leading  to smaller
%% mass    splitting   between   the    $1/2^-$   and    $3/2^-$   states
%% \cite{takeuchi32}.
%% \end{itemize}
%% \end{@@@}





There  have been  several lattice  QCD  calculations of  5Q states  by
today\cite{scikor12,sasaki,chiu,kentacky,ishii12,rabbit,
lasscock12,alexandrou12,csikor122,holland,lasscock32}.  However, these
studies  are restricted to  $J^P=1/2^\pm$ channels  except for  a very
recent one \cite{lasscock32}.
%
Enormous efforts are being devoted to
%
more accurate  studies of $J^P=1/2^\pm$ states,  using the variational
technique to  extract multiple excited  states, among which  a compact
resonance state is sought for.
%
Indeed,  quite   large  scale  calculations  are   planned  and  being
performed\cite{fleming}  attempting to  elucidate  some of  the
mysterious natures  of $\Theta^+(1540)$ such as  its diquark structure
and/or non-localities desired in interpolating fields.
%
Here,   we  emphasize  again   that  these   studies  are   aiming  at
$J^P=1/2^\pm$ states, not at $3/2^\pm$ states.

In this paper, we present anisotropic lattice QCD results on 5Q states
in $J^P=3/2^\pm$ channels using a large number of gauge configurations
as  $N_{\rm conf}=1000$
%
as  an attempt to  search for  a low-lying  5Q state  in $J^P=3/2^\pm$
channel.
%% Note that, if  such a state exists at $m_{\rm 5Q} \simeq  1540$ MeV, it is
%% expected  to  have a  significantly  narrow  decay  width due  to  the
%% mechanism  mentioned   above,  which  can  be   then  identified  with
%% $\Theta^+(1540)$.
%
We  adopt the  standard Wilson  gauge  action at  $\beta=5.75$ on  the
$12^3\times 96$ lattice  with the renormalized anisotropy $a_{s}/a_{t}
= 4$.   The anisotropic lattice is  known to serve as  a powerful tool
for    high-precision    measurements    of    temporal    correlators
\cite{klassen,matsufuru,nemoto,ishii-gb}.
%
The large  number of gauge configurations $N_{\rm  conf}=1000$ plays a
key role in our calculation, because 5Q correlators in $J^P=3/2^{\pm}$
channels are found to be quite noisy.
%
For quark action, we adopt $O(a)$-improved Wilson (clover) action with
four  values  of  the  hopping parameter  as  $\kappa=0.1210  (0.0010)
0.1240$.
%
One of  the purpose of our  calculation is to examine  how the results
depend on the choice of interpolating field operators.
%
We  employ   several  types  of   interpolating  fields  as   (a)  the
NK$^*$-type,  (b) the (color-)twisted  NK$^*$-type, (c)  a diquark-type,
and adopt a smeared source to enhance the low-lying spectra.
%

%% \begin{@@@}
%% We see operator dependences in our calculation.
%% %
%% In the 5Q effective mass plot in $J^P=3/2^-$ channel, we find plateaus
%% for the  NK$^*$-type and  the twisted NK$^*$-type  interpolating fields,
%% whereas  no plateau  is  found in  the  diquark-type one,  which is  a
%% considerably noisy correlator.
%% %
%% The  former two  give  almost consistent  results.   After the  chiral
%% extrapolation, these  two lead to  massive 5Q states  as $m_{\rm 5Q}\simeq
%% 2.17$ and $2.11$ GeV, respectively.
%% %
%% %%
%% %%
%% In the 5Q effective mass plot in $J^P=3/2^+$ channel, we find plateaus
%% in  all the  three cases.   However, the  behavior of  the NK$^*$-type
%% effective mass plot shows a clear difference from the other two, i.e.,
%% the twisted  NK$^*$-type and the diquark-type effective mass plots.
%% %
%% After  the chiral  extrapolation, the  latter two  lead to  massive 5Q
%% state as  $m_{\rm 5Q}\simeq 2.48$ and $2.42$ GeV,  respectively, which are
%% considered to be the same state.
%% %%
%% The  chiral extrapolation for  the NK$^*$-type  correlator leads  to a
%% more massive states as $m_{\rm 5Q}\simeq 2.64$ GeV, which is considered to
%% be a different state from the other two.
%% %%
%% In  this way, we  find only  massive 5Q  states in  both $J^P=3/2^\pm$
%% channels  in our  calculation. Note  that all  of these  states appear
%% above the NK threshold, which is raised artificially due to the finite
%% extent of the spatial lattice as $L\simeq 2.15$ fm.
%% \end{@@@}
In  $J^P=3/2^-$ channel,  we obtain  massive states  $m_{\rm 5Q}\simeq
2.1-2.2$ GeV  except for  the diquark-type interpolating  field, which
involves a considerable size  of the statistical error.
%%
%% In  $J^P=3/2^+$  channel,  we   obtain  more  massive  states  $m_{\rm
%% 5Q}\simeq  2.4-2.5$  GeV,  except  for the  NK$^*$-type  interpolating
%% field, which suggest  an even more massive state  as $m_{\rm 5Q}\simeq
%% 2.64$  GeV.
In  $J^P=3/2^+$  channel,  we   obtain  more  massive  states  $m_{\rm
5Q}\simeq 2.4-2.6$ GeV.
%%
%% None of these 5Q states appear below the NK threshold, which is raised
%% up artificially  due to  the finite extent  of the spatial  lattice as
%% $L\simeq 2.15$ fm.
None of these  5Q states appear below the NK  threshold. Note that the
NK threshold  is raised up  by about $200-250$  MeV due to  the finite
extent of the spatial lattice as $L\simeq 2.15$ fm,
%%
from which we expect the  penta-quark signal  to appear  below the
(raised)  NK  threshold  considering  the  empirical  mass  difference
between N+K(1440) and $\Theta^+(1540)$.
%%
To  clarify whether our  5Q states  are compact  resonance or  not, we
perform an analysis with the hybrid boundary condition(HBC), which was
recently  proposed  in \Ref{ishii12}.
%% \begin{@@@}
%% HBC analysis  shows that, in $J^P=3/2^-$ channel,  the observed states
%% are consistent with the s-wave NK$^*$ scattering states.
%% %
%% In $J^P=3/2^+$  channel, HBC does not affect  the two-particle spectra
%% so much. Although it is thus  not easy to investigate our 5Q states in
%% $J^P=3/2^+$ channel into detail, we attempt to perform HBC analysis to
%% elucidate their nature as possible.
%% %% In $J^P=3/2^+$  channel, we  present the 5Q  effective mass  plots for
%% %% completeness, i.e.,  HBC affects the  two-particle scattering spectrum
%% %% so weak that no significant changes are found.
%% %%
%% We  obtained  only  very  massive  5Q  states  in  both  $J^P=3/2^\pm$
%% channels,  among which  no compact  resonance states  are found  to be
%% included.
%% \end{@@@}
HBC analysis  indicates that no  compact 5Q resonance is  contained in
our 5Q states both in $J^P=3/2^\pm$ channels.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The paper is organized as follows.
%
In  \Sect{general.formalisms}, we discuss  the general  formalisms. We
begin   by  introducing   several  types   of   interpolating  fields,
determining their parity  transformation properties.  We next consider
the  temporal correlator  and its  spectral decomposition.  We finally
discuss the two-particle scattering states involved in 5Q spectra, and
introduce  the hybrid boundary  condition (HBC)  to examine  whether a
state of our concern is a compact resonance state or not.
%
\Sect{lattice}  is devoted to  the brief  descriptions of  our lattice
action and parameters.
%
In  \Sect{numerical.results},  we present  our  numerical results  for
$J^P=3/2^{\pm}$   channels   in   the   standard   periodic   boundary
condition(PBC).
%
We  show 5Q  correlators of  various interpolating  fields,  i.e., the
NK$^*$-type, the (color-)twisted NK$^*$-type, the diquark-type.
%
In  \Sect{HBC.analysis},  we attempt  to  determine  whether these  5Q
states  are compact  5Q  resonance states  or two-particle  scattering
states by using the HBC.
%
In \Sect{summary}, we summarize our results.

\section{General formalisms}
\label{general.formalisms}
\subsection{Interpolating fields}
We consider an iso-scalar  interpolating field of {\em NK$^*$-type} in
Rarita-Schwinger form \cite{ioffe,benmerrouche,hemmert} as
\begin{eqnarray}
  \psi_{\mu}
  &\equiv&
  \epsilon_{abc}
  \left( u^T_a C\gamma_5 d_b\right) u_c
  \cdot
  \left( \bar{s}_d \gamma_{\mu} d_d \right)
  \label{nkstar}
  \\\nonumber
  &&
  -
  \epsilon_{abc}
  \left( u^T_a C\gamma_5 d_b\right) d_c
  \cdot
  \left( \bar{s}_d \gamma_{\mu} u_d \right),
\end{eqnarray}
where  $\mu$ denotes  the  Lorentz  index, $a-d$  refer  to the  color
indices,  and  $C=\gamma_4\gamma_2$  denotes  the  charge  conjugation
matrix.
%
Unless otherwise indicated, the  gamma matrices are represented in the
Euclidean form given in \Ref{montvay}.
%
%We will make a comment on the other interpolating fields later.

We  are  also  interested   in  the  {\em  (color-)twisted  NK$^*$-type}
interpolating field as
\begin{eqnarray}
  \psi_{\mu}
  &\equiv&
  \epsilon_{abc}
  \left( u^T_a C\gamma_5 d_b\right) u_d
  \cdot
  \left( \bar{s}_d \gamma_{\mu} d_c \right)
  \label{n8k8star}
  \\\nonumber
  &&
  -
  \epsilon_{abc}
  \left( u^T_a C\gamma_5 d_b\right) d_d
  \cdot
  \left( \bar{s}_d \gamma_{\mu} u_c \right),
\end{eqnarray}
which is an extension to the one originally proposed in \Ref{scikor12}
to study $J^P=1/2^P$ 5Q states.
%\begin{bf}
It  has a  slightly more  elaborate color-structure  than \Eq{nkstar},
suggesting somewhat  stronger coupling  to a genuine  5Q state,  if it
exists, than simple NK$^*$ states.
%\end{bf}

Another   interpolating   fields  of   our   possible  interests   are
diquark-type interpolating fields such as
\begin{equation}
  \psi_{\mu}
  \equiv
  \epsilon_{abc}
  \epsilon_{def}
  \epsilon_{cfg}
  \left( u_a^T C\gamma_5 d_b \right)
  \left( u_d^T C\gamma_5 \gamma_{\mu} d_e \right)
  C\gamma_5 \bar s_g
  \label{diquark-type.SV}
\end{equation}
which    is     an    extension    to    the     one    proposed    in
Refs.~\cite{sasaki,sugiyama}.
%
The first  factor corresponds to  the scalar diquark (color  $\bar {\bf 3}$,
$I=0$, $J^P=0^+$), which is expected to play important roles in hadron
physics \cite{jaffe-exotica}.
%
The second factor  corresponds to the vector diquark  (color $\bar {\bf 3}$,
$I=0$,  $J^P=1^-$).
%% \begin{bf}
%
Note  that, although the  axial-vector diquark  (color $\bar {\bf 3}$, $I=1$,
$J^P=1^+$) is considered to play a more important role than the vector
diquark, it  cannot replace the  vector diquark due to  its iso-vector
nature.
%%\end{bf}
%
Unless otherwise  indicated, we  refer to \Eq{diquark-type.SV}  as the
``{\em diquark-type}'' interpolating field.
%
We can also consider another interpolating field of diquark-type as
\begin{equation}
  \psi_{\mu}
  \equiv
  \epsilon_{abc}
  \epsilon_{def}
  \epsilon_{cfg}
  \left( u_a^T C d_b \right)
  \left( u_d^T C\gamma_5 \gamma_{\mu} d_e \right)
  C \bar s_g,
  \label{diquark-type.PV}
\end{equation}
which consists  of the pseudo-scalar  diquark (color $\bar  3$, $I=0$,
$J^P=0^-$)  and  the  vector  diquark.  However,  actual  lattice  QCD
calculation shows that  its correlator is afflicted with  quite a huge
statistical  error.
%% \begin{bf}
%
A possible reason  could be attributed to the fact  that both of these
diquark fields do not survive the non-relativistic limit.
%%\end{bf}
Hence, we do not consider this interpolating field in this paper.

Under the spatial  reflection of the quark fields  as
\begin{equation}
  q(\tau,\vec x) \to \gamma_4 q(\tau,-\vec x),
\end{equation}
all of these interpolating fields transform as
%% \begin{equation}
%%   \psi_{\mu}(\tau,\vec x)
%%   \to
%%   \gamma_4 \psi^{\mu}(\tau,-\vec x).
%%   \label{parity}
%% \end{equation}
%% Remark that the position of the Lorentz index $\mu$ is changed.
\begin{equation}
%%   \psi_{0}(\tau,\vec x)
%%   &\to&
%%   + \gamma_4 \psi_{0}(\tau,-\vec x)
  \psi_{i}(\tau,\vec x)
  \to
  -\gamma_4 \psi_{i}(\tau,-\vec x),
  \label{parity}
\end{equation}
for $i=1,2,3$.

\subsection{5Q correlators and parity projection}
We consider the Euclidean temporal correlator as
\begin{equation}
  G_{\mu\nu}(\tau)
  \equiv
  \sum_{\vec x}
  \left\langle
  \psi_{\mu}(\tau,\vec x) \bar\psi_{\nu}(0,\vec 0)
  \right\rangle,
  \label{correlator}
\end{equation}
where  $\sum_{\vec x}$  projects the  total 5Q  momentum to  zero.
%
Since  the  spin  3/2  contribution  from the  temporal  component  of
Rarita-Schwinger spinor  vanishes in the  rest frame, we  can restrict
ourselves   to  the   spatial  parts,   i.e.,   $\mu\nu=1,2,3$.   Now,
\Eq{correlator} is decomposed in the following way:
\begin{eqnarray}
  G_{ij}(\tau)
  &=&
  {\bf P}^{(3/2)}_{ij} G^{(3/2)}(\tau)
  +
  {\bf P}^{(1/2)}_{ij} G^{(1/2)}(\tau),
  \label{correlator.1}
\end{eqnarray}
where  $i,j=1,2,3$ denote  the spatial  part of  the  Lorentz indices,
${\bf P}^{(3/2)}$ and ${\bf P}^{(1/2)}$ denote the projection matrices
onto the spin 3/2 and 1/2 subspaces defined as
\begin{eqnarray}
  {\bf P}^{(3/2)}_{ij}
  &\equiv&
  \delta_{ij}   -  (1/3)\gamma_i\gamma_j,
  \\\nonumber
  {\bf P}^{(1/2)}_{ij}
  &\equiv&
  (1/3)\gamma_i\gamma_j.
\end{eqnarray}
They satisfy the following relations as
\begin{eqnarray}
  {\bf P}^{(3/2)}_{ij} {\bf P}^{(3/2)}_{jk} &=& {\bf P}^{(3/2)}_{ik}
  \\\nonumber
  {\bf P}^{(1/2)}_{ij} {\bf P}^{(1/2)}_{jk} &=& {\bf P}^{(1/2)}_{ik}
  \\\nonumber
  {\bf P}^{(1/2)}_{ij} + {\bf P}^{(3/2)}_{ij} &=& \delta_{ij}
  \\\nonumber
  {\bf P}^{(1/2)}_{ij} {\bf P}^{(3/2)}_{jk} &=& {\bf P}^{(3/2)}_{ij} {\bf P}^{(1/2)}_{jk} = 0.
\end{eqnarray}
Here, summations over repeated indices are understood.
%
$G^{(3/2)}(\tau)$ and $G^{(1/2)}(\tau)$  in \Eq{correlator} denote the
spin 3/2  and 1/2  contributions to $G(\tau)$,  respectively,
%
which  can  be  derived  by  operating  ${\bf  P}^{(3/2)}$  and  ${\bf
P}^{(1/2)}$ on $G(\tau)$, respectively.
%%
(In  our   practical  lattice   QCD  calculation,  we   construct  the
Rarita-Schwinger   correlator   $G_{ij}(\tau)$   for   $i=1,2,3$   and
$j=3$(fixed), and  multiply ${\bf P}^{(3/2)}$ from the  left to obtain
$G^{(3/2)}(\tau)$.)


In the  asymptotic region  ($0 \ll \tau  \ll N_t$),  contaminations of
excited states are  suppressed.  Considering the parity transformation
property  \Eq{parity},  $G^{(3/2)}(\tau)$  and  $G^{(1/2)}(\tau)$  are
expressed in this region as\\
%
\begin{eqnarray}
  \lefteqn{G^{(3/2)}(\tau)}
  \label{correlator.2}
  \\\nonumber
  &=&
  P_+
  \left\{
  |\lambda_{3/2^-}|^2 e^{-\tau m_{3/2^-}}
  + |\lambda_{3/2^+}|^2 e^{-(N_t - \tau) m_{3/2^+}}
  \right\}
  \\\nonumber
  &-&
  P_-
  \left\{
  |\lambda_{3/2^+}|^2 e^{-\tau m_{3/2^+}}
  + |\lambda_{3/2^-}|^2 e^{-(N_t - \tau) m_{3/2^-}}
  \right\}
  \\\nonumber
  %%%%%%%%%%%%%
  \lefteqn{G^{(1/2)}(\tau)}
  \\\nonumber
  &=&
  P_+
  \left\{
  |\lambda_{1/2^-}|^2 e^{-\tau m_{1/2^-}}
  + |\lambda_{1/2^+}|^2 e^{-(N_t - \tau) m_{1/2^+}}
  \right\}
  \\\nonumber
  &-&
  P_-
  \left\{
  |\lambda_{1/2^+}|^2 e^{-\tau m_{1/2^+}}
  + |\lambda_{1/2^-}|^2 e^{-(N_t - \tau) m_{1/2^-}}
  \right\},
\end{eqnarray}
where $P_{\pm}\equiv (1\pm\gamma_4)/2$  denote the projection matrices
onto the ``upper'' and ``lower'' Dirac subspaces, respectively.
%
$m_{3/2^\pm}$  and  $m_{1/2^\pm}$ denote  the  lowest-lying masses  in
$J^P=3/2^\pm$ and $1/2^\pm$ channels, respectively.
%
$\lambda_{3/2^\pm}$ and $\lambda_{1/2^\pm}$ represent the couplings to
the  interpolating field  \Eq{nkstar} with  $J^P=3/2^\pm$  and $1/2^\pm$
states, respectively.
%
In  \Eq{correlator.2}, we adopt  the anti-periodic  boundary condition
along the temporal direction.
%
A  brief  derivation  of  \Eq{correlator.1} and  \Eq{correlator.2}  is
presented in \Appendix{spectral.representation}.
%
The  forward propagation  is dominant  in the  region $0  <  \tau \alt
N_t/2$,  while the  backward  propagation is  dominant  in the  region
$N_t/2 \alt \tau < N_t$.
%
To separate  the negative (positive) parity  contribution, we restrict
ourselves  to  the region  $0  < \tau  \alt  N_t/2$,  and examine  the
``upper'' (``lower'') Dirac component.
%
%% \subsection{Scattering states involved in 5Q spectrum(1)}
%% \label{boundary.condition}
%% We  consider the  scattering states involved in 5Q spectrum.
%% %
%% In  the $J^P=3/2^-$ channel,  d-wave NK  and s-wave  NK$^*$ scattering
%% states are  important.  By  neglecting interactions among  two hadrons
%% involved,  which  are  considered  to  be  weak  except  for  possible
%% resonance contribution, these scattering states can be approximated as
%% $J=3/2$ projections of the plane wave states as
%% \begin{eqnarray}
%%   \lefteqn{
%%     |NK(\mbox{d-wave})\rangle
%%   }
%%   \label{nkstar-d-wave}
%%   \\\nonumber
%%   &\simeq&
%%   {\cal P}^{(3/2)}
%%   \left(
%%   |N(\vec p,s) K(-\vec p)\rangle
%%   +
%%   |N(-\vec p,s) K(\vec p)\rangle
%%   \right)
%%   \\\nonumber
%%   \lefteqn{
%%     |NK^*(\mbox{s-wave})\rangle
%%   }
%%   \\\nonumber
%%   &\simeq&
%%   {\cal P}^{(3/2)}
%%   \left(
%%   |N(\vec p,s) K^*_i(-\vec p)\rangle
%%   +
%%   |N(-\vec p,s) K^*_i(\vec p)\rangle
%%   \right),
%% \end{eqnarray}
%% where ${\cal P}^{(3/2)}$ represents the $J=3/2$ projection operator, and
%% $\vec p$ denotes the spatial momentum allowed for a particular spatial
%% boundary condition adopted. For instance, if these hadrons are subject
%% to the spatially periodic BC, their momenta are discretized as
%% \begin{equation}
%%   p_i = 2n_i\pi/L, \Hs n_i \in \ZZ,
%%   \label{pbc}
%% \end{equation}
%% where $L$ denotes  the spatial extent of the  lattice. In contrast, if
%% they are subject to the  spatially anti-periodic BC, their momenta are
%% discretized as
%% \begin{equation}
%%   p_i = (2 n_i + 1)\pi/L, \Hs n_i \in \ZZ.
%%   \label{apbc}
%% \end{equation}
%% %
%% If  we assume  that the  interaction  between N  and K  are weak,  the
%% energies of these states are approximated as
%% \begin{eqnarray}
%%   E_{NK({\rm d-wave})}
%%   &\simeq&
%%   \sqrt{m_N^2 + \vec p^2}
%%   +
%%   \sqrt{m_K^2 + \vec p^2}
%%   \\\nonumber
%%   E_{NK^*({\rm s-wave})}
%%   &\simeq&
%%   \sqrt{m_N^2 + \vec p^2}
%%   +
%%   \sqrt{m_{K^*}^2 + \vec p^2}.
%% \end{eqnarray}
%% Note that,  for $\vec p = \vec  0$, only $|NK^*(\mbox{s-wave})\rangle$
%% survives,     and      $|NK(\mbox{d-wave})\rangle$     vanishes     in
%% \Eq{nkstar-d-wave}.
%% %
%% %% \begin{bf}
%% %% %
%% %% There   are  two   other  possibilities,   i.e.,   N$^*$K(p-wave)  and
%% %% N$^*$K$^*$(p-wave),  which, however,  are considered  to play  only of
%% %% secondary role, because N$^*$ is rather massive.
%% %% \end{bf}

%% In the  $J^P=3/2^+$ channel, the p-wave  NK and the  p-wave NK$^*$ are
%% important. These  states can be  approximated as $J=3/2$  projections of
%% the following states as
%% \begin{eqnarray}
%%   \lefteqn{
%%     |NK(\mbox{p-wave})\rangle
%%   }
%%   \\\nonumber
%%   &\simeq&
%%   {\cal P}^{(3/2)}
%%   \left(
%%   |N(\vec p,s) K(-\vec p)\rangle
%%   -
%%   |N(-\vec p,s) K(\vec p)\rangle
%%   \right)
%%   \\\nonumber
%%   \lefteqn{
%%     |NK^*(\mbox{p-wave})\rangle
%%   }
%%   \\\nonumber
%%   &\simeq&
%%   {\cal P}^{(3/2)}
%%   \left(
%%   |N(\vec p,s) K^*_i(-\vec p)\rangle
%%   -
%%   |N(-\vec p,s) K^*_i(\vec p)\rangle
%%   \right).
%% \end{eqnarray}
%% Note that  these two states  vanish for vanishing spatial  momentum as
%% $\vec p = \vec 0$.
%% %
%% Their energies are approximated as
%% \begin{eqnarray}
%%   E_{NK({\rm p-wave})}
%%   &\simeq&
%%   \sqrt{m_N^2 + \vec p^2}
%%   +
%%   \sqrt{m_K^2 + \vec p^2}
%%   \\\nonumber
%%   E_{NK^*({\rm p-wave})}
%%   &\simeq&
%%   \sqrt{m_N^2 + \vec p^2}
%%   +
%%   \sqrt{m_{K^*}^2 + \vec p^2}.
%% \end{eqnarray}

\subsection{Scattering states involved in 5Q spectrum}
We  consider  the  (two-particle)  scattering states  involved  in  5Q
spectrum.   For $J^P=3/2^\pm$ iso-scalar  penta-quarks, NK  and NK$^*$
scattering states  play an important role. ($\Delta$K  does not couple
to the iso-scalar channel.)  These states are expressed as
\begin{equation}
  |N(\vec p,s)K(-\vec p)\rangle,
  \Hs
  |N(\vec p,s)K^*(-\vec p,i)\rangle,
\end{equation}
where $s$ and $i$ denote the  spin of the nucleon and K$^*$, and $\vec
p$ denotes the spatial momentum  allowed for a particular choice of the
spatial boundary condition adopted. For instance, if these hadrons are
subject to  the spatially  periodic boundary condition,  their momenta
are quantized as
\begin{equation}
  p_i = 2 n_i \pi /L, \Hs n_i \in \ZZ,
  \label{pbc}
\end{equation}
where $L$ denotes  the spatial extent of the  lattice. In contrast, if
they  are subject  to the  spatially anti-periodic  boundary condition,
their momenta are quantized as
\begin{equation}
  p_i = (2 n_i + 1)\pi /L, \Hs n_i \in \ZZ.
  \label{apbc}
\end{equation}
%
We first perform the parity projections. The positive and the negative
parity states are obtained in the following way:
\begin{eqnarray}
  \lefteqn{
    |NK(\pm)\rangle
  }
  \label{parity.projections1}
  \\\nonumber
  &=&
  |N(\vec p,s)K(-\vec p)\rangle
  \mp
  |N(-\vec p,s)K(\vec p)\rangle
  \\
  \lefteqn{
    |NK^*(\pm)\rangle
  }
  \label{parity.projections2}
  \\\nonumber
  &=&
  |N(\vec p,s)K^*(-\vec p,i)\rangle
  \mp
  |N(-\vec p,s)K^*(\vec p,i)\rangle.
\end{eqnarray}
Assuming that the interactions between N and K and between N and K$^*$
are weak, their energies are approximated as
\begin{eqnarray}
  E_{NK}
  &\simeq&
  \sqrt{m_N^2 + \vec p^2} + \sqrt{m_K^2 + \vec p^2}
  \\
  E_{N^*K}
  &\simeq&
  \sqrt{m_N^2 + \vec p^2} + \sqrt{m_{K^*}^2 + \vec p^2},
\end{eqnarray}
respectively.
%
The scattering  states which couple to  $J^P=3/2^\pm$ penta-quarks are
obtained  as  spin-3/2  projections  of  \Eq{parity.projections1}  and
\Eq{parity.projections2}.  The d-wave NK  states and the s-wave NK$^*$
states  can couple  to the  $J^P=3/2^-$ channel,  while the  p-wave NK
states and NK$^*$ states can couple to the $J^P=3/2^+$ channel.
%

The scattering states  with vanishing spatial momentum $\vec  p = \vec
0$ are exceptional in the following sense.
%
On the one hand, the  positive parity states vanish, because the first
terms coincides with the  second terms in \Eq{parity.projections1} and
\Eq{parity.projections2}  in the  right hand side.
%
On the  other hand,  the negative parity  states are  constructed only
from the spin degrees of freedom, i.e., the spin degrees of freedom of
the  nucleon  in \Eq{parity.projections1},  and  the  spin degrees  of
freedoms of the nucleon and K$^*$ in \Eq{parity.projections2}.
%
By   counting  the  degeneracy   of  the   resulting  states,   it  is
straightforward  to see  that no  d-wave states  are  contained, i.e.,
\Eq{parity.projections1}  gives only s-wave  NK states  in $J^P=1/2^-$
channel,  and that \Eq{parity.projections2}  gives only  s-wave NK$^*$
states in $J^P=1/2^-$ and $3/2^-$ channels.
%
%% Note  that, unlike  the formulation  with continuous  spatial momenta,
%% there is no  ambiguity arising from the limiting  procedure as $\vec p
%% \to \vec 0$.

\subsection{Hybrid boundary condition(HBC)}
\begin{table}
\begin{ruledtabular}
\begin{tabular}{lcccccc}
$\kappa$ & & 0.1210 & 0.1220 & 0.1230 & 0.1240 & emp. \\
\hline
NK$^*$(s-wave) &	 PBC  & 2.996  & 2.815  & 2.633  & 2.445  & 1.830  \\
NK$^*$(p-wave) &	 PBC  & 3.222  & 3.052  & 2.883  & 2.710  & 2.163  \\
NK(p/d-wave) &  	 PBC  & 2.987  & 2.806  & 2.624  & 2.438  & 1.865  \\
NK$^*$(s/p-wave) &	 HBC  & 3.167  & 2.995  & 2.823  & 2.647  & 2.084  \\
NK(p/d-wave) &  	 HBC  & 2.924  & 2.739  & 2.553  & 2.363  & 1.770  \\
\end{tabular}
\end{ruledtabular}
\caption{  Numerical  values of  NK  and  NK$^*$  thresholds for  each
hopping parameter  $\kappa$ in  the physical unit  GeV in  the spatial
lattice of the size $L \simeq 2.15$ fm in PBC and HBC.
%
The rightmost column labeled as ``emp.''  shows the thresholds for the
physical  values  of  N,  K,  and K$^*$  as  $m_{N}\simeq  0.94$  GeV,
$m_{K}\simeq 0.5$ GeV, $m_{K^*}\simeq 0.89$ GeV.
%
}
\label{thresholds}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  HBC
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In order to  determine whether a state of our concern  is a compact 5Q
resonance state  or a  scattering state of  two particles, we  use two
distinct  spatial   boundary  conditions(BC),  i.e.,   the  (standard)
periodic BC(PBC) and the hybrid BC(HBC), which is recently proposed in
\Ref{ishii12}.
%
In PBC, one imposes the spatially periodic BC on u,d and s-quarks.  As
a result, all the hadrons are subject to the periodic BC.
%
In  this case, due  to \Eq{pbc},  
all hadrons can take zero-momentum, 
and the smallest  non-vanishing momentum
$\vec p_{\rm min}$ is of the form as
\begin{equation}
  (\pm 2\pi/L,0,0),\Hs
  (0,\pm 2\pi/L,0),\Hs
  (0,0,\pm 2\pi/L),
\end{equation}
which gives 
\begin{equation}
  |\vec p_{\rm min}^{\rm \,PBC}| = 2\pi/L.
  \label{pmin.pbc}
\end{equation}
%
On the other hand, in HBC, we impose the spatially anti-periodic BC on
u  and d-quarks,  whereas  the  spatially periodic  BC  is imposed  on
s-quark.      Since    N($uud,udd$),     K($u\bar{s},d\bar{s}$)    and
K$^*$($u\bar{s},d\bar{s}$) contain odd numbers of u and d quarks, they
are subject to the anti-periodic BC.
%
Therefore, due  to \Eq{apbc}, N, K  and K$^*$ cannot  have a vanishing
momentum in HBC. The smallest  possible momentum $\vec p_{\rm min}$ is
of the form as
\begin{equation}
  (\pm \pi/L,\pm \pi/L, \pm \pi/L).
\end{equation}
Hence, its norm $|\vec p_{\rm min}|$ is expressed as
\begin{equation}
  |\vec p_{\rm min}^{\rm \,HBC}| = \sqrt{3}\pi/L.
  \label{pmin.hbc}
\end{equation}
%
In  contrast,  $\Theta^+$($uudd\bar{s}$) is  subject  to the  spatially
periodic BC,  since it contains even  number of u and  d quarks.
%
Therefore, $\Theta^+$ can have the vanishing momentum.

Switching from PBC, HBC  affects the low-lying two-particle scattering
spectrum.  A drastic change is expected in the s-wave NK$^*$ channel.
%
In PBC, the energy of the lowest NK$^*$ state is given as
\begin{equation}
  E_{\rm min}^{\rm PBC}(NK^*(\mbox{s-wave}))
  \simeq
  m_N + m_{K^*}.
\end{equation}
In  contrast, in  HBC, since  both N  and K$^*$  are required  to have
non-vanishing momenta $|\vec p_{\rm min}| = \sqrt{3}\pi/L$, the energy of
the lowest NK$^*$ state is raised up as
\begin{eqnarray}
  \lefteqn{
    E_{\rm min}^{\rm HBC}(NK^*(\mbox{s-wave}))
  }
  \\\nonumber
  &\simeq&
  \sqrt{m_N^2 + 3\pi^2/L^2} + \sqrt{m_{K^*}^2 + 3\pi^2/L^2}.
\end{eqnarray}
%%%%%%%
Note that the shift amounts typically  to a few hundred MeV for $L\sim
2$ fm.
%%%%%%%

HBC affects  NK(d-wave), NK(p-wave), NK$^*$(p-wave)  as well. However,
these changes  are not as  drastic as that in  NK$^*$(s-wave), because
they  are induced by  the minor  change in  the minimum  momentum from
$|\vec p_{\rm min}|=2\pi/L$ to $\sqrt{3}\pi/L$.
%
In PBC, the energies of  the lowest two-particles states are expressed
as
\begin{eqnarray}
  \lefteqn{
    E_{\rm min}^{\rm PBC}(NK(\mbox{p/d-wave}))
  }
  \\\nonumber
  &\simeq&
  \sqrt{m_N^2 + 4\pi^2/L^2} + \sqrt{m_{K}^2 + 4\pi^2/L^2}
  \\\nonumber
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \lefteqn{
    E_{\rm min}^{\rm PBC}(NK^*(\mbox{p-wave}))
  }
  \\\nonumber
  &\simeq&
  \sqrt{m_N^2 + 4\pi^2/L^2} + \sqrt{m_{K^*}^2 + 4\pi^2/L^2}.
\end{eqnarray}
In HBC, they are shifted down as
\begin{eqnarray}
  \lefteqn{
    E_{\rm min}^{\rm HBC}(NK(\mbox{p/d-wave}))
  }
  \\\nonumber
  &\simeq&
  \sqrt{m_N^2 + 3\pi^2/L^2} + \sqrt{m_{K}^2 + 3\pi^2/L^2}
  \\\nonumber
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \lefteqn{
    E_{\rm min}^{\rm HBC}(NK^*(\mbox{p-wave}))
  }
  \\\nonumber
  &\simeq&
  \sqrt{m_N^2 + 3\pi^2/L^2} + \sqrt{m_{K^*}^2 + 3\pi^2/L^2}.
\end{eqnarray}
Numerical  values  of  NK  and  NK$^*$  thresholds  for  each  hopping
parameter in  spatial lattice of the  size $L\simeq 2.15$  fm for both
PBC and HBC are summarized in \Table{thresholds}.

%\begin{@@@}
%% In contrast to the scattering states,  HBC is not expected to affect a
%% compact 5Q resonance $\Theta^+(uudd\bar{s})$ so much.  This is because
%% HBC  affects  $\Theta^+$ only  through  the  change  in its  intrinsic
%% structure, which is expected to induce only a minor effect.
%% %
%% %
%% Therefore, our way to find a compact 5Q resonance state is to seek for
%% such a state which is not affected by HBC.
In contrast to the scattering states,  HBC is not expected to affect a
compact 5Q resonance $\Theta^+$ so much.
%
Since $\Theta^+(uudd\bar{s})$ can have vanishing momentum also in HBC,
the shift  of the penta-quark  mass $m_{\rm 5Q}$ originates  only from
the change  in its  intrinsic structure.  In  this case, the  shift is
expected  to  be  less  significant  than the  shift  induced  by  the
kinematic reason as is the case in N, K, and K$^*$.
%
%
Now our way to find a compact 5Q resonance state is to seek for such a
state which is not affected by HBC.
%\end{@@@}


\section{Lattice actions and parameters}
\label{lattice}
\begin{table*}
\begin{ruledtabular}
\begin{tabular}{ccccccccccl}
$\beta$ & $\gamma_{\rm  G}$ & $a_{\rm s}/a_{\rm t}$ & $a_{s}^{-1}$  [GeV] & Size &
$N_{\rm conf}$ & $u_{s}$ & $u_{t}$ & $\gamma_F$ & $\kappa_c$ &
Values of $\kappa$ \\
%
\hline
%
5.75 &  3.2552 & 4 &  1.100(6) & $12^3\times  96$ & 1000 &  0.7620(2) &
0.9871(0) & 3.909 & 0.12640(5) & 0.1240, 0.1230, 0.1220, 0.1210
\end{tabular}
\end{ruledtabular}
\caption{Parameters  of the lattice  simulation.  The  spatial lattice
spacing $a_{\sigma}$ is  determined with $r_0^{-1} = 385$  MeV for the
Sommer parameter.  The mean-field values of link variables ($u_\sigma$
and $u_\tau$) are defined in the Landau gauge.  $\kappa_c$ denotes the
critical  value of  $\kappa$. }
\label{table.lattice.parameters}
\end{table*}
%
To generate gauge field  configurations, we use the standard plaquette
action on the anisotropic lattice of the size $12^3\times 96$ as
\begin{eqnarray}
  S_{\rm G}
  &=&
  \frac{\beta}{N_c}
  \frac1{\gamma_{\rm G}}
  \sum_{x,i<j\le3}
  \mbox{Re} \mbox{Tr}
  \left\{ 1 - P_{ij}(x)\right\}
  \\\nonumber
  &+&
  \frac{\beta}{N_c}
  \gamma_{\rm G}
  \sum_{x,i\le 3}
  \mbox{Re} \mbox{Tr}
  \left\{ 1 - P_{i4}(x)\right\},
\end{eqnarray}
where $P_{\mu\nu}(x) \in  \mbox{SU(3)}$ denotes the plaquette operator
in  the  $\mu$-$\nu$-plane.   The   lattice  parameter  and  the  bare
anisotropy parameter are  fixed as $\beta \equiv 2N_c/g^2  = 5.75$ and
$\gamma_{\rm G}=3.2552$, respectively.  These values are determined to
reproduce the  renormalized anisotropy as $\xi\equiv  a_{s}/a_{t} = 4$
\cite{klassen}.
%
Adopting  the  pseudo-heat-bath  algorithm,  we pick  up  gauge  field
configurations every  500 sweeps after skipping 10,000  sweeps for the
thermalization.   We use  totally 1000  gauge field  configurations to
construct the temporal correlators.
%
Note  that  the  high  statistics  of  $N_{\rm  conf}=1000$  is  quite
essential  for our  study, because  the  5Q correlators  for spin  3/2
states are found to be rather noisy.
%
In  fact,   a  preliminary  analysis  with   less  statistics  $N_{\rm
conf}\simeq   500$   leads   to   a  spurious   resonance-like   state
\cite{ishii320}.
%% Note  that correlators  for spin  3/2 states  are found  to  be rather
%% noisy.
%% Hence, the  high statistics of $N_{\rm conf}=1000$  is quite essential
%% for  our study.  In  fact, if  crude analysis  is performed  with less
%% statistics,   say,   with  $N_{\rm   conf}\simeq   500$,  a   spurious
%% resonance-like state will be observed \cite{ishii320}.
%% \begin{itemize}
%% \item
%% In fact, we had observed a resonance-like state at $N_{\rm conf}\simeq
%% 500$ stage \cite{ishii320}, which has been disappeared after the gauge
%% configurations have been increased up to $N_{\rm conf}=1000$.
%% \item
%% In  fact, a  crude analysis  at $N_{\rm  conf}\simeq 500$  level still
%% involves  a  considerable size  of  statistical  error,  leading to  a
%% spurious resonance-like state, which disappears at $N_{\rm conf}=1000$
%% level\cite{ishii320}.
%% \item
%% We note  that high statistics of $N_{\rm conf}=1000$  is quite essential for  this study.
%% In fact,  If crude analysis  had been performed from  less statistics,
%% say,  with  $N_{\rm conf}\simeq  500$, a spurious resonance-like  state
%% would be observed \cite{ishii320}.
%% \end{itemize}
%%
%% We emphasize  that the large statistics  has played a key  role in our
%% calculations.
%
The  lattice spacing  is determined  from the  static  quark potential
adopting the Sommer parameter $r_0^{-1} = 395$ MeV ($r_0 \sim 0.5$ fm)
as $a_{s}^{-1} = 1.100(6)$ GeV ($a_{\rm s} \simeq 0.18$ fm). Note that
the lattice size $12^3\times  96$ amounts to $(2.15\mbox{fm})^3 \times
(4.30\mbox{fm})$ in the physical unit.

%
We adopt the $O(a)$-improved Wilson (clover) action on the anisotropic
lattice for quark fields $\psi$ and $\bar\psi$ as \cite{matsufuru}
%\begin{widetext}
\begin{eqnarray}
  S_{\rm F}
  &\equiv&
  \sum_{x,y} \bar\psi(x) K(x,y) \psi(y),
  \\\nonumber
  %
  %
  %
  K(x,y)
  &\equiv&
  \renewcommand{\arraystretch}{1.8}
  \delta_{x,y}
  -
  \kappa_{\rm t}\left\{\Tate\right.
  \Bs
  \begin{array}[t]{l} \displaystyle
    (1 - \gamma_4)\; U_4(x)\; \delta_{x+\hat 4,y}
    \\\displaystyle
    +
    (1 + \gamma_4)\; U_4^\dagger(x - \hat 4)\; \delta_{x-\hat 4,y}
    \left.\Tate\right\}
  \end{array}
  \\\nonumber
  &-&
  \renewcommand{\arraystretch}{1.8}
  \kappa_{\rm s}
  \sum_i
  \left\{\Tate\right.
  \Bs
  \begin{array}[t]{l} \displaystyle
    (r - \gamma_i)\; U_i(x)\; \delta_{x+\hat i,y}
    \\\displaystyle
    +
    (r + \gamma_i)\; U_i^\dagger(x - \hat i)\; \delta_{x-\hat i,y}
    \left.\Tate \right\}
  \end{array}
  \\\nonumber
  &-&
  \kappa_{\rm s}\; c_E \sum_i \sigma_{i4} F_{i4} \delta_{x,y}
  -
  r\; \kappa_{s}\; c_B \sum_{i<j} \sigma_{ij} F_{ij} \delta_{x,y},
\end{eqnarray}
%\end{widetext}
where  $\kappa_{\rm  s}$  and  $\kappa_{\rm  t}$  denote  the  hopping
parameters for the spatial  and the temporal directions, respectively.
The  field  strength  $F_{\mu\nu}$  is defined  through  the  standard
clover-leaf-type construction.
%
$r$  denotes the  Wilson  parameter. $c_{E}$  and  $c_{B}$ denote  the
clover coefficients.  
%
To achieve the tadpole improvement, the link variables are rescaled as
$U_i(x) \to U_i(x)/u_{\rm s}$ and $U_4(x) \to U_4(x)/u_{\rm t}$, where
$u_{\rm  s}$ and  $u_{\rm  t}$  denote the  mean-field  values of  the
spatial     and      temporal     link     variables,     respectively
\cite{matsufuru,nemoto}.
%
This is  equivalent to the  redefinition of the hopping  parameters as
the  tadpole-improved  ones  (with  tilde), i.e.,  $\kappa_{\rm  s}  =
\tilde\kappa_{\rm   s}   /   u_{\rm   s}$  and   $\kappa_{\rm   t}   =
\tilde\kappa_{\rm t} / u_{\rm t}$.
%
The   anisotropy   parameter    is   defined   as   $\gamma_F   \equiv
\tilde\kappa_{\rm t} / \tilde\kappa_{\rm s}$, which coincides with the
renormalized anisotropy $\xi = a_{\rm s} / a_{\rm t}$ for sufficiently
small quark mass at the tadpole-improved level \cite{matsufuru}.
%
For given $\kappa_{\rm s}$, the  four parameters $r$, $c_E$, $c_B$ and
$\kappa_{\rm s}/\kappa_{\rm t}$ should be, in principle, tuned so that
``Lorentz symmetry'' holds up to discretization errors of $O(a^2)$.
%
Here, $r$, $c_E$ and $c_B$  are fixed by adopting the tadpole improved
tree-level values as
\begin{equation}
  r   = \frac1{\xi},
  \Hs
  c_E =  \frac1{u_{\rm s} u_{\rm t}^2},
  \Hs
  c_B =  \frac1{u_{\rm s}^3}.
\end{equation}
Only the value  of $\kappa_{\rm s} / \kappa_{\rm  t} \left( = \gamma_F
\cdot (u_{\rm s}  / u_{\rm t}) \right)$ is  tuned nonperturbatively by
using the meson dispersion relation \cite{matsufuru}.
%
It is convenient to define $\kappa$ as
\begin{equation}
  \frac1{\kappa}
  \equiv
  \frac1{\tilde\kappa_{\rm s}}
  -
  2\left( \gamma_F - 3r - 4 \right).
\end{equation}
Then the bare  quark mass is expressed as  $m_0 = \frac1{2}(1/\kappa -
8)$ in the spatial lattice unit in the continuum limit.  This $\kappa$
plays the role of the  hopping parameter ``$\kappa$'' in the isotropic
formulation.
%
For  detail,  see  Refs.~\cite{nemoto,matsufuru},  where we  take  the
lattice  parameters.   The  values   of  the  lattice  parameters  are
summarized in \Table{table.lattice.parameters}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}
\begin{ruledtabular}
%% The following is the result with N_{conf}=504
%% \begin{tabular}{llllll}
%% $\kappa$   & 0.1210 & 0.1220 & 0.1230 & 0.1240 & $\kappa_{\rm phys.}$\\
%% \hline
%% $m_{\pi}$  & 1.005(2) & 0.898(2) & 0.784(2) & 0.656(3) & 0.140    \\
%% $m_{\rho}$ & 1.240(3) & 1.161(3) & 1.085(4) & 1.011(5) & 0.850(7) \\
%% $m_{K}$    & 0.845(2) & 0.785(2) & 0.723(2) & 0.656(3) & 0.530(4) \\
%% $m_{K^*}$  & 1.126(4) & 1.087(4) & 1.048(4) & 1.011(5) & 0.930(8) \\
%% $m_{N}$    & 1.878(5) & 1.744(5) & 1.604(5) & 1.460(6) & 1.173(9) \\
%% $m_{N^*}$  & 2.334(26) & 2.209(31)& 2.086(40) & 1.972(62) & 1.715(76)
%% \end{tabular}
%%
%% The following is the result with N_{conf}=1000
\begin{tabular}{llllll}
$\kappa$   & 0.1210 & 0.1220 & 0.1230 & 0.1240 & $\kappa_{\rm phys.}$\\
\hline
$m_{\pi}$  & 1.007(2) & 0.897(1) & 0.785(2) & 0.658(2)  & 0.140      \\
$m_{\rho}$ & 1.240(4) & 1.157(5) & 1.074(7) & 0.991(11) & 0.823(13)  \\
%
$m_{K}$    & 0.846(2) & 0.785(1) & 0.722(2) & 0.658(2)  & 0.476(2)   \\
$m_{K^*}$  & 1.119(6) & 1.076(7) & 1.033(9) & 0.991(11) & 0.902(15)  \\
%
$m_{N}$    & 1.877(4) & 1.739(3) & 1.600(4) & 1.454(5)  & 1.164(8)   \\
$m_{N^*}$  & 2.325(17) & 2.194(21)& 2.059(28) & 1.918(42) & 1.648(53)
\end{tabular}
\end{ruledtabular}
\caption{Masses  of $\pi$,  $\rho$, K,  K$^*$, N,  and N$^*$  for each
hopping  parameter $\kappa$  in the  physical unit  GeV.  $\kappa_{\rm
phys.}\simeq  0.1261$ denotes  the  value of  $\kappa$ which  achieves
$m_{\pi} \simeq 0.14$ GeV.}
\label{table.mass}
\end{table}
We  adopt four  values  of the  hopping  parameter as  $\kappa=0.1210,
0.1220, 0.1230$ and $0.1240$,  which correspond to $m_{\pi}/m_{\rho} =
0.81, 0.78, 0.73$ and $0.66$, respectively. These values roughly cover
the region $m_s \alt m \alt 2 m_s$.
%
For temporal direction, we  impose anti-periodic boundary condition on
all  the quark  fields.  For  spatial directions,  we  impose periodic
boundary condition on all  the quarks, unless otherwise indicated.  We
refer to this boundary condition as ``{periodic BC (PBC)}''.

By  keeping $\kappa_s=0.1240$  fixed for  $s$ quark,  and  by changing
$\kappa=0.1210-0.1240$ for  $u$ and $d$ quarks, we  perform the chiral
extrapolation to the physical quark mass region.
%
In the following part of the paper, we will use
\begin{equation}
  (\kappa_s,\kappa)=(0.1240,0.1220),
\label{typical.set}
\end{equation}
as a typical  set of hopping parameters in  presenting correlators and
effective mass plots.
%
For convenience, we summarize masses of $\pi$, $\rho$, K, K$^*$, N and
N$^*$($J^P=1/2^-$ baryon) for each hopping parameter $\kappa$ together
with their  values at the  physical quark mass  in \Table{table.mass}.
Here, the chiral extrapolations of  these particles are performed by a
linear function in $m_{\pi}^2$.
%
%% The physical  point of  the hopping parameter,  which can  achieve the
%% physical pion mass $m_{\pi} \simeq 0.14$ GeV, is given as follows
%% \begin{equation}
%%   \kappa_{\rm phys.} \simeq 0.1261.
%% %0.1261815864
%% \label{kappa.phys}
%% \end{equation}
%%%
Unless  otherwise indicated,  we adopt  the jackknife  prescription to
estimate the statistical errors.

%%% smearing %%%
We  use a  smeared  source  to enhance  the  low-lying spectra.   More
precisely, we  employ spatially  extended interpolating fields  of the
gaussian size $\rho\simeq 0.4$ fm,  which is obtained by replacing the
quark fields  $q(x)$ in 5Q  interpolating fields by the  smeared quark
fields $q_{\rm smear}(x)$ in the Coulomb gauge as
\begin{equation}
  q_{\rm smear}(\tau, \vec x)
  \equiv
  {\cal N}
  \sum_{\vec y}
  \exp\left\{ - \frac{|\vec x - \vec y|^2}{2\rho^2} \right\}
  q(\tau, \vec y),
\label{eq.gaussian}
\end{equation}
where ${\cal N}$ is  an appropriate normalization factor.
%
%The  smearing procedure  is performed  after Coulomb  gauge  fixing.
%
For a practical use, we extend \Eq{eq.gaussian} appropriately so as to
fit a particular choice of the spatial boundary condition.
%
%
In  this paper, we  present correlators  with a  smeared source  and a
point sink.
%

\section{Numerical results on 5Q spectrum}
\label{numerical.results}
We  present our lattice  QCD results  on 5Q  spectrum in  the standard
periodic boundary condition(PBC) in this section.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% J^P=3/2(-)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{$J^P=3/2^-$ 5Q spectrum in PBC}
We    consider    5Q   spectrum    in    $J^P=3/2^-$   channel.     In
\Fig{fig.three.half.minus.pbc},  we show the  effective mass  plots in
$J^P=3/2^-$  channel for  three  interpolating fields,  i.e., (a)  the
NK$^*$-type, (b) the twisted NK$^*$-type, (c) a diquark-type.
%
%
The  dotted  lines  indicate  the  s-wave NK$^*$  and  the  d-wave  NK
thresholds,    which    happen    to    coincide    accidentally    in
\Fig{fig.three.half.minus.pbc} for  the spatial lattice  size $L\simeq
2.15$ fm.
\begin{figure}
\begin{center}
\includegraphics[height=0.48\textwidth,angle=-90]{NKstar-403_401-minus.eps}  
\\
\includegraphics[height=0.48\textwidth,angle=-90]{N8K8star-403_401-minus.eps}  
\\
\includegraphics[height=0.48\textwidth,angle=-90]{Penta32.SS-403_401-minus.eps}
\end{center}
\caption{The  5Q effective mass  plots in  $J^P=3/2^-$ channel  in the
standard  periodic BC(PBC)  for three  types of  interpolating fields,
i.e.,  (a) the  NK$^*$-type, (b)  the twisted  NK$^*$-type, and  (c) the
diquark-type.
%
\Eq{typical.set}   is   adopted   as   a  typical   set   of   hopping
parameters.  The statistical  error  is estimated  with the  jackknife
prescription.
%
The  dotted  lines  indicate  the  s-wave NK$^*$  and  the  d-wave  NK
threshold in the spatial lattice size $L\simeq 2.15$ fm.
%
Note that they accidentally coincide with each other.
%
The  solid lines  denote  the results  of  the single-exponential  fit
performed in each plateau region.
%
%% (We do  not show the best-fit  result in the last  figure, because its
%% considerably large noise prevent  us from identifying a plateau region
%% unambiguously.)
%
}
\label{fig.three.half.minus.pbc}
\end{figure}

We define the effective mass as a function of $\tau$ by
\begin{equation}
  m_{\rm eff}(\tau)
  \equiv
  \log\left({ G^{(3/2)}(\tau) \over G^{(3/2)}(\tau+1) }\right),
\label{eq.effmass}
\end{equation}
where $G^{(3/2)}(\tau)$ denotes the temporal correlator.
%
At  sufficiently large  $\tau$,  the correlator  is  dominated by  the
lowest-lying state  with energy $m$  as $G^{(3/2)}(\tau) \sim  A e^{-m
\tau}$.  Then  \Eq{eq.effmass} gives a constant  as $m_{\rm eff}(\tau)
\simeq m$.
%
Thus a plateau in $m_{\rm eff}(\tau)$ indicates that the correlator is
saturated  by  a  single-state.   In  such cases,  we  can  perform  a
single-exponential fit in the plateau region.

\Fig{fig.three.half.minus.pbc}~(a) shows  the effective mass  plot for
the NK$^*$-type  interpolating field.  In  the region $0 \le  \tau \le
24$,  the  contamination  of  the higher  spectral  contributions  are
gradually  reduced, which  is indicated  by the  decreases  in $m_{\rm
eff}(\tau)$.
%
There is  a plateau in  the interval $25  \alt \tau \alt 35$,  where a
single-state is expected to  dominate the 5Q correlator.  Beyond $\tau
\sim  36$, the  statistical  error becomes  large.   In addition,  the
effect of the backward  propagation becomes gradually more significant
as $\tau\sim 48$ is approached.  Hence, we simply neglect the data for
$\tau \agt 36$,  and perform the single-exponential fit  in the region
$25  \le \tau  \le  35$.   We obtain  $m_{\rm 5Q}=2.90(2)$  GeV, which  is
denoted by the solid line.
%
One sees  that the 5Q states  appears above the s-wave  NK$^*$ and the
d-wave NK thresholds.

\Fig{fig.three.half.minus.pbc}~(b) shows  the effective mass  plot for
the twisted NK$^*$-type interpolating field.
%
There is a  plateau in the interval $24 \alt \tau  \alt 35$, where the
single-exponential  fit is performed  leading to  $m_{\rm 5Q}=2.89(1)$
GeV.
%
The  5Q state  is again  above  the s-wave  NK$^*$ and  the d-wave  NK
thresholds.

%%% %%% %%% %%%
\Fig{fig.three.half.minus.pbc}~(c) shows  the effective mass  plot for
the diquark-type interpolating field.
%
We see that the statistical error is too large to identify the plateau
unambiguously.  Hence, we do not perform the fit.
%
Note that  this plot  is obtained by  using $N_{\rm  conf}=1000$ gauge
configurations.
%
A possible  reason for  such a large  noise is that  the interpolating
field  \Eq{diquark-type.SV}  does  not  survive  the  non-relativistic
limit due to the vector diquark.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Chiral Extrapolation (J^P=3/2(-))
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
\includegraphics[height=0.48\textwidth,angle=-90]{chiral-minus.eps}
\end{center}
\caption{$m_{\rm 5Q}$ in  $J^P=3/2^-$ channel against  $m_{\pi}^2$ for the
two interpolating  fields, i.e.,  (circle) the NK$^*$-type,  (box) the
twisted NK$^*$-type.
%
Open  symbols denote  the direct  lattice QCD  data, while  the closed
symbols  and the  solid  lines  represent the  results  of the  chiral
extrapolations to the physical quark mass region.
%
%% The  dotted  lines  indicate  the  s-wave NK$^*$  and  the  d-wave  NK
%% thresholds, which coincide  with each other in the  spatial lattice of
%% the size $L\simeq 2.15$ fm.
}
\label{fig.three.half.minus.chiral.pbc}
\end{figure}
\begin{table}
\begin{ruledtabular}
\begin{tabular}{lclllll}
$J^P$   & I.F. &0.1210 & 0.1220 & 0.1230 & 0.1240 & $\kappa_{\rm phys.}$ \\
\hline
$3/2^-$ & (a) & 3.08(1) & 2.90(2) & 2.72(2) & 2.54(3) & 2.17(4)
\\
$3/2^-$ & (b) & 3.08(1) & 2.89(1) & 2.70(2) & 2.49(3) & 2.11(4)
\\
$3/2^-$ & (c) & --       & --       & --       & --       & --
\\
$3/2^+$ &  (a) & 3.52(2) & 3.34(3) & 3.17(11) &3.00(5) & 2.64(7)
\\
$3/2^+$ &  (b) & 3.27(3) & 3.11(4) & 2.95(5) & 2.83(9) & 2.48(10)
\\
$3/2^+$ &  (c) & 3.34(2) & 3.16(2) & 2.98(3) & 2.78(5) & 2.42(6)
\end{tabular}
\end{ruledtabular}
\caption{
%
$m_{\rm  5Q}$ for each  value of  $\kappa$ in the physical unit GeV.
%
The  first column  labeled by  ``$J^P$''  indicates the  spin and  the
parity.
%
The  second column  labeled by  ``I.F.''  indicates  the interpolating
field used, i.e., (a) the  NK$^*$-type, (b) the twisted NK$^*$-type, (c)
the diquark-type.
%
$\kappa_{\rm phys.}\simeq 0.1261$ denotes  the value of $\kappa$ which
achieve $m_{\pi}\simeq 0.14$ GeV.
%
``--'' indicates  that the fitting is  not performed due  to the large
statistical error.  }
\label{table.chiral.extrapolation}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Now, we perform the chiral extrapolation. As mentioned before, we keep
$\kappa=0.1240$ fixed  for $s$-quark, and  vary $\kappa=0.1210-0.1240$
for $u$ and $d$ quarks.
%
\Fig{fig.three.half.minus.chiral.pbc}   shows   the   5Q   masses   in
$J^P=3/2^-$ channel against $m_{\pi}^2$.  Circles and boxes denote the
data  obtained from  the NK$^*$-type  and the  twisted  NK$^*$-type 5Q
correlators, respectively.
%
Note that they agree with each other within the statistical error.
%
%% A possible  reason is  that these two  interpolating fields  share the
%% common non-relativistic limit.
%% \begin{bf}
%% A possible  reason would be  that these two interpolating  fields have
%% the  common  non-relativistic  limit,  i.e.,  the  NK$^*$,  which  may
%% dominate the intermediate state at large Euclidean time.
%% \end{bf}
%
%% We do not show the data of the diquark-type 5Q correlator, because the
%% fit is not  performed due to the large  statistical error as mentioned
%% before.
%
The open symbols refer to the direct lattice QCD data.
%
Since these data  behave almost linearly in $m_{\pi}^2$,  we adopt the
linear chiral  extrapolation in $m_{\pi}^2$ to obtain  $m_{\rm 5Q}$ in
the physical quark mass region.   Note that the ordinary non-PS mesons
and baryons show similar linearity in $m_{\pi}^2$ \cite{nemoto}.
%
The closed symbols denote the results of the chiral extrapolation.
%
%% For  convenience, we also  show the  s-wave NK$^*$  and the  d-wave NK
%% thresholds  with the  dotted lines,  respectively,  which accidentally
%% coincide with each other.
%
We see that  all the 5Q states appear above the  s-wave NK$^*$ and the
d-wave NK thresholds.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
As a  result of  the chiral extrapolation,  we obtain only  massive 5Q
states as  $m_{\rm 5Q}=2.17(4), 2.11(4)$ GeV from the  NK$^*$-type and the
twisted NK$^*$-type correlators, respectively,
%%
which is too  heavy to be identified with  the experimentally observed
$\Theta^+(1540)$.
%% \begin{@@@}
Numerical values  of $m_{\rm 5Q}$ at each hopping  parameter together with
their    chirally    extrapolated    values    are    summarized    in
\Table{table.chiral.extrapolation}.
%
%% \begin{bf}
%% \end{bf}
%% \end{@@@}
%
%% To obtain a low-lying state at  $m_{\rm 5Q}\simeq 1540$ MeV, a 5Q state is
%% required to appear below these  thresholds at least in the light quark
%% mass region.
To obtain a low-lying state at $m_{\rm 5Q}\simeq 1540$ MeV, a 5Q state
should appear below these thresholds  at least in the light quark mass
region. In this case, a significantly large chiral effect is required.
Of course,  this point  can be in  principle clarified by  an explicit
lattice QCD calculation with chiral fermions.
%
%


%% As mentioned  before, the constituent  quark models suggest  that spin
%% $3/2^-$ 5Q  states tend  to become massive  due to  the color-magnetic
%% interaction.
%% %% Since a  5Q system is  a new system,  which we have  never encountered
%% %% before, we  hoped that something new  could appear to  lower its mass.
%% %% However, we do not observe such a new effect in our calculation.
%% In  our calculation,  we  do  not observe  any  inconsistency to  this
%% convensional picture in the quark mass region $m_s \alt m \alt 2m_s$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% J^P=3/2(+)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{$J^P=3/2^+$ 5Q spectrum in PBC}
We consider 5Q spectrum in $J^P=3/2^+$ channel.
%
$J^P=3/2^+$ is  an interesting quantum  number from the view  point of
the  diquark  picture  of  Jaffe  and  Wilczek\cite{jaffe}.   In  this
picture,  the pair  of diquarks  has  angular momentum  one, which  is
combined with the  spin 1/2 of $\bar{s}$ quark.   Hence, there are two
possibilities as  $J^P=1/2^+$ and  $3/2^+$, i.e., the  diquark picture
can support $J^P=3/2^+$ possibility as well.
%
Its mass splits  from the $J^P=1/2^+$ state depending  on a particular
form  of the  LS-interaction.
%
%% The  mass  of  this  state  depends  on a  particular  choice  of  the
%% LS-interaction.
%
If it is massive,  it is expected to have a large  decay width.  If it
is light  enough,
%% the  same mechanism  as in  $J^P=1/2^+$  case works  to implement  the
%% narrow decay width.
its exotic structure  may work to implement the  narrow decay width as
in $J^P=1/2^+$ case.

In \Fig{fig.three.half.plus.pbc}, we show  the 5Q effective mass plots
in PBC  employing three types  of interpolating fields, i.e.,  (a) the
NK$^*$-type, (b) the twisted NK$^*$-type, (c) the diquark-type.
%
The dotted lines indicate the s-wave N$^*$K$^*$, the p-wave NK$^*$ and
the p-wave NK  thresholds in the spatial lattice  of the size $L\simeq
2.15$ fm, respectively.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
\includegraphics[height=0.48\textwidth,angle=-90]{NKstar-403_401-plus.eps}  
\\
\includegraphics[height=0.48\textwidth,angle=-90]{N8K8star-403_401-plus.eps}  
\\
\includegraphics[height=0.48\textwidth,angle=-90]{Penta32.SS-403_401-plus.eps}
\end{center}
\caption{The 5Q effective mass plots in $J^P=3/2^+$ channel in PBC for
three types  of interpolating fields,  i.e., (a) the  NK$^*$-type, (b)
the twisted NK$^*$-type, and (c) the diquark-type.
%
\Eq{typical.set}   is   adopted   as   a  typical   set   of   hopping
parameters.
%
The dotted lines indicate the s-wave N$^*$K$^*$, the p-wave NK$^*$ and
the p-wave  NK thresholds in  the spatial lattice size  $L\simeq 2.15$
fm.
%
The  solid lines  denote  the results  of  the single-exponential  fit
performed in each plateau region.
%
}
\label{fig.three.half.plus.pbc}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
\includegraphics[height=0.48\textwidth,angle=-90]{chiral-plus.eps}
\end{center}
\caption{$m_{\rm 5Q}$ in  $J^P=3/2^+$ channel against  $m_{\pi}^2$ for the
three interpolating fields, i.e.,  (circle) the NK$^*$-type, (box) the
twisted      NK$^*$-type,     and     (triangle)      a     diquark-type
[\Eq{diquark-type.SV}].
%
Open  symbols denote  the direct  lattice QCD  data, while  the closed
symbols  and the  solid  lines  represent the  results  of the  chiral
extrapolations to the physical quark mass region.
%
%% The  dotted  lines  indicate  the  p-wave NK$^*$  and  the  p-wave  NK
%% thresholds in the spatial lattice of the size $L\simeq 2.15$ fm.
%
}
\label{fig.three.half.plus.chiral.pbc}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Fig{fig.three.half.plus.pbc}~(a)  shows the  5Q  effective mass  plot
employing the  NK$^*$-type interpolating  field. In the  region, $0\le
\tau  \alt 17$,  the contaminations  of higher  spectral contributions
become gradually reduced.
%
There is a flat region $18 \alt \tau \le 30$, which is still afflicted
with slightly large statistical errors.
%
%
The single-exponential fit in this region gives $m_{\rm 5Q}=3.34(3)$ GeV.
%
Note  that this  value  agrees with  the  s-wave N$^*$K$^*$  threshold
$E_{\rm th}\simeq 3.27$  GeV.  (See \Table{table.mass} for $m_{N^*}$.)

\Fig{fig.three.half.plus.pbc}~(b)  shows the  5Q  effective mass  plot
corresponding to the twisted  NK$^*$-type interpolating field. We have
a rather  stable plateau in the  interval $21 \le \tau  \le 27$, where
the   single-exponential   fit  is   performed.   We  obtain   $m_{\rm
5Q}=3.11(4)$ GeV.  The result is denoted by the solid line.

\Fig{fig.three.half.plus.pbc}~(c) shows the 5Q effective mass plot for
the  diquark-type  interpolating field.   We  find  a  plateau in  the
interval $19  \le \tau  \le 29$, where  the single-exponential  fit is
performed.  We  obtain $m_{\rm 5Q}=3.16(2)$ GeV,  which is denoted  by the
solid line.


Now,  we  perform  the  chiral  extrapolation.
%
In  \Fig{fig.three.half.plus.chiral.pbc}, $m_{\rm 5Q}$ is  plotted against
$m_{\pi}^2$.  Circles,  boxes and  triangles denote the  data obtained
from the  NK$^*$-type, the twisted  NK$^*$-type and the  diquark-type 5Q
correlators, respectively.
%
Note that the latter two  agree with each other within the statistical
errors.
%
%
%% As before, the open symbols  denote the direct lattice QCD data, which
%% are seen  to behave almost  linearly against $m_{\pi}^2$.   We perform
%% the linear  chiral extrapolation, which  leads to the  results denoted
%% with the closed symbols in the physical quark mass region.
%% %
%% The  dotted lines  indicate  the corresponding  p-wave  NK$^*$ and  NK
%% thresholds, respectively.

As  a   result  of  the   chiral  extrapolation,  we   obtain  $m_{\rm
5Q}=2.64(7)$ GeV from the NK$^*$-type correlator,
%
$m_{\rm 5Q}=2.48(10)$ GeV from  the twisted NK$^*$-type correlator,
%
and $m_{\rm 5Q}=2.42(6)$ GeV from the diquark-type correlator.
%
Numerical values  of $m_{\rm 5Q}$ in  $J^P=3/2^+$ channel at  each hopping
parameter together  with their  chirally extrapolated values  are also
summarized in \Table{table.chiral.extrapolation}.
%
The  two  data  from   the  twisted  NK$^*$-type  and  the  diquark-type
correlators  are considered to  be almost  consistent with  the p-wave
NK$^*$ threshold,
%\begin{bf}
%%\begin{@@@}
while the data from the  NK$^*$-type correlator seems to correspond to
a  more massive  state,  which is  likely  to be  consistent with  the
N$^*$K$^*$(s-wave) threshold.
%%\end{@@@}
%
We see  again that all  of our data  of $m_{\rm 5Q}$ appear  above the
NK$^*$  threshold(p-wave),  which is  located  above the  artificially
raised  NK threshold(p-wave)  due  to the  finiteness  of the  spatial
lattice as $L\simeq 2.15$ fm.  As a result, we are left only with such
massive 5Q states.
%

Now, several comments are in order.
%%\begin{@@@}
(1) \Ref{lasscock32} reported the existence of a low-lying 5Q state in
$J^P=3/2^+$ channel  using NK$^*$-type interpolating  field.  However,
we have not observed such a low-lying 5Q state in our calculation.
%
%%\end{@@@}
%
%%\begin{@@@}
(2) Recall that,  except for a  single calculation\cite{chiu}, lattice
QCD   calculations   indicate   that   $J^P=1/2^+$  state   is   heavy
\cite{scikor12,   sasaki,  kentacky,   ishii12,   rabbit,  lasscock12,
  alexandrou12, csikor122,  holland}, for instance  $m_{\rm 5Q} \simeq
2.25$ GeV in \Ref{ishii12}.
%
From  the viewpoint of  the diquark  picture, it  could be  natural to
obtain such massive 5Q states in $J^P=3/2^+$ channel.
%% \end{@@@}
If there  were a low-lying 5Q  state in $J^P=3/2^+$  channel, then the
diquark picture could suggest also a low-lying 5Q state in $J^P=1/2^+$
channel nearby.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%   HBC ANALYSIS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Analysis with HBC}
\label{HBC.analysis}
\begin{figure}
\begin{center}
\includegraphics[height=0.48\textwidth,angle=-90]{NKstar-513_401-minus.eps}  
\\
\includegraphics[height=0.48\textwidth,angle=-90]{N8K8star-513_401-minus.eps}  
\\
\includegraphics[height=0.48\textwidth,angle=-90]{Penta32.SS-513_401-minus.eps}
\end{center}
\caption{The 5Q effective mass plots in $J^P=3/2^-$ channel in HBC for
three types  of interpolating fields,  i.e., (a) the  NK$^*$-type, (b)
the twisted NK$^*$-type, and (c) the diquark-type.
%
The meanings of  the dotted lines and the solid lines  are the same as
in \Fig{fig.three.half.minus.pbc}.
%
%% (We do  not show the best-fit  result in the last  figure, because its
%% considerably large noise prevents us from identifying a plateau region
%% unambiguously.)
%
}
\label{fig.three.half.minus.hbc}
\vspace{6ex}
\end{figure}
%% The aim of this section is  to determine whether the 5Q states oberved
%% in  the   previous  section  are   compact  5Q  resonance   states  or
%% two-particle scattering  states.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% %
In the  previous section,  we have only  massive 5Q states,  which are
obtained by using the linear chiral extrapolation in $m_{\pi}^2$.
%
However, the chiral  behavior may deviate from a  simple linear one in
the light quark mass region, which could lead to somewhat less massive
states.
%
Considering  this, we  think  it of  worth  at this  stage to  analyze
whether our 5Q states are compact 5Q resonances or not.
%
This  is done  by  switching the  spatial  periodic BC  to the  hybrid
BC(HBC) introduced in \Sect{general.formalisms}.

\subsection{$J^P=3/2^-$ 5Q spectrum in HBC}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Fig{fig.three.half.minus.hbc}  shows the 5Q  effective mass  plots in
HBC employing the  three types of interpolating fields,  i.e., (a) the
NK$^*$-type, (b) the twisted NK$^*$-type, and (c) the diquark type.
%
These  figures  should be  compared  with  their  PBC counterparts  in
\Fig{fig.three.half.minus.pbc}.
%%
The  dotted  lines  denote  the   s-wave  NK$^*$  and  the  d-wave  NK
thresholds.
%% The meanings  of the  dotted  and the
%% solid lines are the same as in \Fig{fig.three.half.minus.pbc}.
%
For the typical set of hopping parameters, i.e., \Eq{typical.set}, the
s-wave NK$^*$ threshold(the  thick dotted line) is raised  up by $\sim
180$ MeV, and the d-wave NK threshold(the thin dotted line) is lowered
down  by $\sim 70$  MeV due  to HBC  in the  finite spatial  extent as
$L\simeq 2.15$ fm.  (See \Table{thresholds}.)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Fig{fig.three.half.minus.hbc}~(a)  shows the  5Q effective  mass plot
for the NK$^*$-type interpolating field in HBC.
%
We find  a plateau  in the interval  $23 \le  \tau \le 35$,  where the
single-exponential fit is performed  leading to $m_{\rm 5Q}= 2.98(1)$ GeV,
which is denoted by the solid line.
%
We see that $m_{\rm 5Q}$ is raised up by 80 MeV due to HBC.
%
%% Although it  appears slightly below  the s-wave NK$^*$  threshold,
The  value  of $m_{\rm  5Q}$  is  consistent  with the  s-wave  NK$^*$
threshold within the statistical error.
%
Therefore, we regard this state as an NK$^*$ scattering state.

\Fig{fig.three.half.minus.hbc}~(b)  shows the  5Q effective  mass plot
for the twisted NK$^*$-type interpolating field.
%
We find  a plateau  in the interval  $24 \le  \tau \le 35$,  where the
single exponential fit is performed leading to $m_{\rm 5Q} = 2.98(1)$ GeV,
which is denoted by the solid line.
%
The situation is similar to the NK$^*$-interpolating field case.
%
We see that $m_{\rm 5Q}$ is raised up  by 90 MeV due to HBC.
%
%which appears slightly below the s-wave NK$^*$ threshold.
%
Since the value is consistent  with the s-wave NK$^*$ threshold within
the statistical error, we regard it as an NK$^*$ scattering state.

\Fig{fig.three.half.minus.hbc}~(c) shows the 5Q effective mass plot
for the diquark-type interpolating field.
%
We  see that  it is  afflicted with  considerable size  of statistical
errors as before, due to which the best-fit is not performed.
%

%% \begin{bf}
%
In this way,  all of our 5Q states in $J^P=3/2^-$  channel turn out to
be NK$^*$  scattering states.  More  precisely, we do not  observe any
compact 5Q  resonance states in  $J^P=3/2^-$ channel below  the raised
s-wave NK$^*$ threshold, i.e., in the following region:
\begin{equation}
  E
  \alt 
  \sqrt{m_N^2 + \vec p_{\rm min}^2}
  +
  \sqrt{m_{K^*}^2 + \vec p_{\rm min}^2},
\end{equation}
with $|\vec p_{\rm min}|\simeq 499$ MeV.
%% \end{bf}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{$J^P=3/2^+$ 5Q spectrum in HBC}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\begin{center}
\includegraphics[height=0.48\textwidth,angle=-90]{NKstar-513_401-plus.eps}  
\\
\includegraphics[height=0.48\textwidth,angle=-90]{N8K8star-513_401-plus.eps}  
\\
\includegraphics[height=0.48\textwidth,angle=-90]{Penta32.SS-513_401-plus.eps}
\end{center}
\caption{5Q  effective mass plots  in $J^P=3/2^+$  channel in  HBC for
three types of interpolating fields (a) the NK$^*$-type, (b) the twisted
NK$^*$-type, and (c) the diquark-type.
%
The meanings  of the  dotted and the  solid lines  are the same  as in
\Fig{fig.three.half.plus.pbc}.
%
}
\label{fig.three.half.plus.hbc}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Fig{fig.three.half.plus.hbc} shows the 5Q effective mass plots in HBC
employing  the three  types  of interpolating  fields,  i.e., (a)  the
NK$^*$-type,  (b) the  twisted  NK$^*$-type, and  (c) the  diquark-type.
These  figures  should be  compared  with  their  PBC counterparts  in
\Fig{fig.three.half.plus.pbc}.  The  meanings of  the  dotted and  the
solid lines are the same as in \Fig{fig.three.half.plus.pbc}.

HBC may  not be useful in  $J^P=3/2^+$ channel, since  it induces only
minor changes in the two-particle spectra.
%
For the typical set of hopping parameters, i.e., \Eq{typical.set}, the
p-wave NK$^*$ threshold is lowered down only by $\sim 60$ MeV, and the
p-wave NK threshold is lowered down only by $\sim 70$ MeV.
%
We see  that these shifts are  rather small. This is  because they are
induced  by the changes  in the  minimum non-vanishing  momentum, i.e.,
$|\vec p_{\rm min}|=2\pi/L\simeq 576$ MeV to $\sqrt{3}\pi/L\simeq 499$
MeV as mentioned before.
%
In  $J^P=3/2^+$ channel, N$^*$K$^*$(s-wave)  threshold shows  the most
drastic change, i.e., the upper  shift by 170 MeV, which however plays
a less significant role, since its location is at rather high energy.

\Fig{fig.three.half.plus.hbc}~(a)  shows the  5Q  effective mass  plot
employing the NK$^*$-type interpolating field.
%
There is a flat region $16 \le \tau \alt 25$, which is still afflicted
with  slightly large  statistical errors.
%% as is the case in its PBC counterpart.
%
The   single-exponential  fit   in  this   region  leads   to  $m_{\rm
5Q}=3.38(2)$ GeV,  which is  denoted by the  solid line.  We  see that
$m_{\rm 5Q}$ is raised up by 40 MeV.
%
Although the  shift of 40 MeV  is rather small, $m_{\rm  5Q}$ is again
almost  consistent  with the  s-wave  N$^*$K$^*$  threshold.
%
Considering  its rather  large  statistical error,  this  5Q state  is
likely to be an s-wave N$^*$K$^*$ scattering state.
%
To draw  a more  solid conclusion  on this state,  it is  necessary to
improve the statistics further more.
%
%% \begin{@@@}
%% Recall      that,     in      its      PBC     counterpart,      i.e.,
%% \Fig{fig.three.half.plus.pbc}.(a), the  flat region appears  at around
%% the  location  of  the  N$^*$K$^*$(s-wave)  threshold.   Due  to  HBC,
%% N$^*$K$^*$(s-wave) threshold is raised above by 178 MeV,
%% %
%% whereas the shift  in $m_{\rm 5Q}$ is rather small as  60 MeV. $m_{\rm 5Q}$ is
%% located below the N$^*$K$^*$(s-wave) threshold by about 80 MeV.
%% %
%% Now,  one of  the  possible  understanding is  that  this flat  region
%% corresponds to  a compact  5Q resonance state.   However, we  think it
%% more probable  that the  state which corresponds  to this  flat region
%% contains only  a negligible amount  of N$^*$K$^*$ contribution,  due to
%% which its location is unaffected so much.
%% %
%% Note  that this state  locates above  the NK$^*$(p-wave)  threshold by
%% more than 600 MeV, which makes it difficult to elucidate the nature of
%% this state.
%% %
%% \end{@@@}

\Fig{fig.three.half.plus.hbc}~(b)  shows the  5Q  effective mass  plot
employing the twisted NK$^*$-type interpolating field.
%
There is  a plateau  in the interval  $23 \le  \tau \le 31$,  where we
perform the single-exponential fit. The result $m_{\rm 5Q}=3.02(3)$ GeV is
denoted by the solid line.  We see that $m_{\rm 5Q}$ is lowered down by 90
MeV,  which is  considered  to be  consistent  with the  shift of  the
NK$^*$(p-wave)  threshold.
%% Therefore, we regard this state as an NK$^*$(p-wave) scattering state.
Therefore, this state is likely to be an NK$^*$(p-wave) scattering state.

\Fig{fig.three.half.plus.hbc}~(c)  shows the  5Q  effective mass  plot
employing the diquark-type interpolating field.
%
Although  the  data is  slightly  noisy, there  is  a  plateau in  the
interval  $23 \le  \tau \le  28$.   A single-exponential  fit in  this
plateau region leads to $m_{\rm 5Q}=3.08(4)$  GeV, which is denoted by the
solid line.
%
$m_{\rm 5Q}$ is lowered down by 80 MeV due to HBC.
%
%% Considering its statistical error and  also the amount of the shift in
%% NK$^*$(p-wave)  threshold,
The situation is  similar to \Fig{fig.three.half.plus.hbc}~(b).
%% and we regard this 5Q state as an NK$^*$(p-wave) scattering state.
This state is likely to be an NK$^*$(p-wave) scattering state.

%% \begin{@@@}
%% We see that the location  of $m_{\rm 5Q}$ remains almost unchanged (within
%% the statistical  error) suggesting an interesting  possibility that it
%% may be a compact 5Q resonance state.
%% %
%% However, considering  that the  shift of the  NK$^*$(p-wave) threshold
%% amounts to only 63 MeV,
%% \begin{itemize}
%% \item
%% we cannot judge its validity with the current statistics.
%% \item
%% we will leave the judgement of its validity for future studies.
%% \end{itemize}
%% \end{@@@}

%% In this  way, all of  our 5Q  states turn out  to be either  an s-wave
%% N$^*$K$^*$ scattering  state or a  p-wave NK$^*$ scattering  state.
In  this  way,  all  of  our   5Q  states  are  likely  to  be  either
N$^*$K$^*$(s-wave)  or NK$^*$(p-wave)  states rather  than  compact 5Q
resonance states.
%
Of course, because  HBC induces only minor changes  in the 5Q spectrum
in $J^P=3/2^+$ channel, and  also because 5Q correlators still involve
considerable size  of statistical error, more  statistics is desirable
to draw a more solid conclusion on the real nature of these 5Q states.
%
Here, we  can at  least state  that these 5Q  states are  all massive,
which locate above NK$^*$(p-wave) threshold.

%% \begin{@@@}
%% In this  way, we  see that HBC  induces only  minor changes in  the 5Q
%% spectrum in $J^P=3/2^+$  channel. As a consequence, it  is not easy so
%% far  to  elucidate  the  natures  of  our  5Q  states  in  $J^P=3/2^+$
%% channel. Here,  all that  we can  state is that  these states  are all
%% massive, which locate above NK$^*$(p-wave) threshold.
%% \end{@@@}

\section{Summary and conclusion}
\label{summary}
We have  studied $J^P=3/2^\pm$ penta-quark(5Q)  baryons in anisotropic
lattice QCD at  the quenched level
%
with  a  large  number   of  gauge  field  configurations  as  $N_{\rm
conf}=1000$ for high precision measurements.
%
We emphasize that the spin  of $\Theta^+(1540)$ has not yet determined
experimentally, and  that the $J^P=3/2^-$ assignment  provides us with
one of the possible solutions to  the puzzle of the narrow decay width
of $\Theta^+(1540)$ \cite{hosaka32}.
%
We have employed  the standard Wilson gauge action  on the anisotropic
lattice of  the size $12^3\times 96$ with  the renormalized anisotropy
$a_{\rm s}/a_{\rm  t} = 4$  at $\beta=5.75$, which  leads to
$a_{\rm s} \simeq 0.18$ fm and $a_{\rm t} \simeq 0.045$ fm.
%
We have found that correlators of 5Q baryons in $J^P=3/2^\pm$ channels
are rather noisy.  Hence,  the large statistics as $N_{\rm conf}=1000$
has played a key role to get a solid result in our calculation.
%
%% The anisotropic lattice  is known to serve as a  powerful tool for the
%% accurate      measurement     of     the      temporal     correlators
%% \cite{matsufuru,nemoto,ishii-gb,ishii12}.
%% %
For the  quark part, we have employed  $O(a)$-improved Wilson (clover)
action   with    four   values   of   the    hopping   parameters   as
$\kappa=0.1210(0.0010)0.1240$,  which  roughly  cover the  quark  mass
region  as $m_s \alt  m \alt  2m_s$.  To  avoid the  contaminations of
higher spectral contributions, we have employed the spatially extended
source in the 5Q correlators.

We have examined  several types of the 5Q  interpolating fields as (a)
the  NK$^*$-type,   (b)  the  (color-)twisted   NK$^*$-type,  (c)  the
diquark-type.
%
%% We have seen operator dependences.
%
% JP = 3/2(-)
%
In $J^P=3/2^-$ channel, there are plateaus in the effective mass plots
for  the NK$^*$-type  and the  twisted NK$^*$-type  interpolating field,
whereas no  plateau has been  identified in that for  the diquark-type
interpolating field due to the significantly large statistical error.
%
The former two give almost identical results.
%
We  have employed  the  linear chiral  extrapolations in  $m_{\pi}^2$,
which  have  lead  to  $m_{\rm 5Q}\simeq  2.17$ and  $2.11$  GeV  for  the
NK$^*$-type and the twisted NK$^*$-type 5Q correlators, respectively.
%
% JP = 3/2(+)
%
In $J^P=3/2^+$ channel,  we have recognized plateaus in  all the three
effective  mass  plots.   However,  the plateau  for  the  NK$^*$-type
interpolating field  is located at  a somewhat higher energy  than the
other two.
%
The chiral extrapolations have lead to $m_{\rm 5Q}\simeq 2.64$ GeV for
the  NK$^*$-type  correlator,  $m_{\rm  5Q}\simeq 2.48$  GeV  for  the
twisted NK$^*$-type  correlator, and  $m_{\rm 5Q}\simeq 2.42$  GeV for
the diquark-type correlator.
%
%% Note  that the  latter two  correspond to  the same  state,  while the
%% former to a more massive one.
%
In this way,  our data have not supported low-lying  5Q states in both
$J^P=3/2^{\pm}$ channels.
%
All the 5Q  states have been observed to appear  above the d/p-wave NK
threshold, which is artificially raised up by a few hundred MeV due to
the finiteness of the spatial  lattice as $L\simeq 2.15$ fm.
%%
Note that, to  obtain low-lying 5Q states in  $J^P=3/2^\pm$ channel, a
5Q  state should  appear below  the raised  NK  threshold(p/d-wave) at
least in the light quark mass region.

In  order  to clarify  whether  the  observed  states are  compact  5Q
resonances or not, we have  performed an analysis with hybrid boundary
condition(HBC), which was recently proposed by \Ref{ishii12}.
%
In $J^P=3/2^-$ channel, our 5Q  states observed in the NK$^*$-type and
the  twisted NK$^*$-type  correlators  have turned  out  to be  s-wave
NK$^*$ scattering states.
%
In  $J^P=3/2^+$   channel,  for   the  twisted  NK$^*$-type   and  the
diquark-type correlators, the observed 5Q states are most likely to be
NK$^*$(p-wave) scattering states.
%%% \begin{@@@}
For the other one, i.e., the NK$^*$-type interpolating field, although
more statistics  is needed to draw  a definite conclusion,  it is most
likely to be an s-wave N$^*$K$^*$ scattering state.
%
Note that,  since HBC  does not affected  the two-particle  spectra so
much in $J^P=3/2^+$  channel, it is not easy  to elucidate the natures
of the 5Q states only with HBC.
%
At any rate, whatever the real  nature of these 5Q states may be, they
result in  a considerably  massive states in  the physical  quark mass
region,  which  cannot   be  identified  as  $\Theta^+(1540)$  without
involving a significantly large chiral contribution.
%
%
%%% \end{@@@}

In this way,  we have not obtained any  relevant signals for low-lying
compact 5Q resonance states in $J^P=3/2^\pm$ channel below 2.1 GeV
in this paper, 
although the  $J^P=3/2^-$ possibility
provides us  with one of the  possible solutions to the  puzzle of the
narrow decay width of $\Theta^+(1540)$.
To get more solid conclusion on the pentaquark, 
it is important to perform the systematic studies of the 5Q states 
with the various  quantum numbers in lattice QCD 
with more sophisticated conditions. For instance, it is desired to use 
(1) unquenched full lattice QCD, 
(2) finer and larger volume lattice,  
(3) chiral fermion with small mass,
(4) more sophisticated interpolating field corresponding to the 
%pentaquark
diquark picture and so on.
%
In any case,  the mysterious exotic hadron of  the pentaquark would be
much clarified  in future  studies of  lattice QCD as  well as  in the
future experiments.


%% (Do you need the following comments? In my impression, it is not necessary...)

%% The following issues are desirable to be checked.

%% (1) The quark-mass effect may be nontrivial for the pentaquark mass especially near the chiral limit.   The  chiral
%%     behavior may deviate  from a simple linear one  in the light quark
%%     mass  region. In  this direction,  it is  desirable to  use chiral
%%     fermions.  However,  this amounts  to a rather  heavy calculation.
%%     Instead, it would be easier  but still important to develop a more
%%     elaborate formalism of the chiral extrapolation of 5Q states based
%%     on the chiral perturbation theory.

%% (2) It is important to develop more elaborate interpolating fields. We
%%     have not observed  a low-lying  5Q state in  this paper. The natural 
%%     understanding  of our  result is  that  there is  no low-lying  5Q
%%     resonance in $J^P=3/2^\pm$ channels.  However, as a possibility, our null-result may
%%     just indicate that  the 5Q state of our concern  has a quite small
%%     overlap  with our  interpolating  fields.  In  this  sense, it  is
%%     important  to apply  the variational  technique  diagonalizing the
%%     cross correlators to obtain a better interpolating field. However,
%%     the  present   calculation  is  already  a  quite   heavy  one  in
%%     $J^P=3/2^\pm$ channels. Hence, such  an extension will also result
%%     in much more heavy calculations.

%% (3) Volume effect may be important. Note that there is no experimental
%%     information on the size of 5Q  state. The 5Q state may be so large
%%     that  our  lattice  of the  spatial  size  $L\sim  2$ fm  may  not
%%     accommodate it.  Hence, it is  desirable to perform the lattice QCD
%%     calculation of a larger spatial size.

%In  this paper, we  have not observed any relevant signals of a low-lying compact 5Q resonance state  in $J^P=3/2^-$
%channel below 2.1GeV. 
%However,  since the  calculation of the  mass is  much easier
%than the calculation of the decay  width, the strategy to search for a
%low-lying  5Q state  in the  quantum  number which  can implement  the
%narrow decay width  may provide an efficient way  to identify the real
%$\Theta^+(1540)$  in lattice QCD.   In  this sense,  we think  it
%important  to study the  5Q states  in the  quantum number  other than
%$J^P=1/2^-$ and $I=0$ as well.  
%In particular, it is also important to
%perform a more systematic calculation of spin 3/2 5Q states in lattice
%QCD.

%% \begin{@@@}
%% \begin{enumerate}
%% \item
%% There are several lattice QCD calcutions, which indicate the existence
%% of  compact 5Q resonance  states in  $J^P=1/2^-$ channels\cite{sasaki,
%% rabbit,  alexandrou}. However,  effective model  calculations indicate
%% that, although  5Q states in $J^P=1/2^-$  channel tend to  be lower in
%% mass, they  tend to  have a  larger decay width.
%% %
%% Hence, it would be desirable to provide an estimate of the decay width
%% for  these  $J^P=1/2^-$ penta-quarks  in  order  to  be identified  as
%% $\Theta^+(1540)$.
%% %
%% The calculation of  the decay width is expected  to be a significantly
%% tough calculation.
%% %
%% Instead, the quantum numbers which achieves the narrow decay width
%% ...
%% \item
%% In  this paper, we  have missed  a low-lying  5Q state  in $J^P=3/2^-$
%% channel. However,  since the  calculation of the  mass is  much easier
%% than the calculation of the decay  width, the strategy to search for a
%% low-lying  5Q state  in the  quantum  number which  can implement  the
%% narrow decay width  may provide an efficient way  to identify the real
%% $\Theta^+(1540)$  in the  lattice QCD.   In  this sense,  we think  it
%% important  to study the  5Q states  in the  quantum number  other than
%% $J^P=1/2^-$ and $I=0$ as well.  In particular, it is also important to
%% perform a more systematic calculation of spin 3/2 5Q states in lattice
%% QCD.
%% \end{enumerate}
%% \end{@@@}




\begin{acknowledgments}
We thank  A.~Hosaka, J.~Sugiyama, T.~Shinozaki  for useful information
and discussions.
%
M.~O and H.~S  are supported in part by  Grant for Scientific Research
((B)  No.   15340072 and  (C)  No.   16540236)  from the  Ministry  of
Education, Culture, Sports, Science and Technology, Japan.
%
T.~D. is supported by Special Postdoctoral Research Program of RIKEN.
%
Y.~N. is supported by 21st Century COE Program of Nagoya University.
%
%
%% M.~O.  is supoorted  in part  by  Grant for  Scientific Research  ((B)
%% No.~15340072) from the Ministry of Education, Culture, Sports, Science
%% and Technology, Japan.
%% %
%% H.~S. is supported by @@@.
%
The  lattice  QCD Monte  Carlo  calculations  have  been performed  on
NEC-SX5 at Osaka University.
\end{acknowledgments}

%% \begin{center}\begin{bf}@@@ Note added in proof @@@\end{bf}\end{center}
%% When  finalizing our  paper, lattice  QCD result  for  spin=3/2 states
%% appears from Adelaide group.

\appendix
\section{Spectral representation}
\label{spectral.representation}
Considering the importance of  the parity determination of $\Theta^+$,
we  present  a brief  derivation  of  the  spectral representation  of
Rarita-Schwinger    correlators,    i.e.,   \Eq{correlator.1},    with
\Eq{correlator.2}.
%\begin{bf}
In this  section, gamma matrices  are represented in  Minkowskian form
(See \Ref{itzykson}).
%\end{bf}
To  avoid  unnecessary complexities,  we  derive only  $J^P=3/2^{\pm}$
parts.   The  $J^P=1/2^{\pm}$  parts  can  be  obtained  as  a  slight
modification,  on which  we will  make  a comment  at the  end of  the
section.

We   first  consider   the  coupling   of  our   interpolating  fields
$\psi_{\mu}$ to $J^P=3/2^{\pm}$ (anti-)baryon states.
%
Due    to    \Eq{parity},     our    interpolating    fields,    i.e.,
Eqs.~(\ref{nkstar}),(\ref{n8k8star})  and (\ref{diquark-type.SV}) have
the negative intrinsic parity.   Hence, their couplings to $J^P=3/2^-$
(anti-)baryons are parameterized in the following way:
\begin{eqnarray}
  \langle 0|
  \psi_{\mu}(0)
  |B_{3/2^-}(k,\alpha)\rangle
  &=&
  \lambda_{3/2^-} u_{\mu}(m_{3/2^-}; k,\alpha)
  \label{coupling.1}
  \\\nonumber
  \langle 0|
  \bar \psi_{\mu}(0)
  |\bar{B}_{3/2^-}(k,\alpha)\rangle
  &=&
  \lambda_{3/2^-}^*
  \bar v_{\mu}(m_{3/2^-}; k,\alpha),
\end{eqnarray}
where         $|B_{3/2^-}(k,\alpha)\rangle$         and         $|\bar
B_{3/2^-}(k,\alpha)\rangle$  denote  $J^P=3/2^-$ (anti-)baryon  states
with momentum $k$, helicity $\alpha$, and mass $m_{3/2^+}$.
%
$u_{\mu}(m;   k,\alpha)$  and   $v_{\mu}(m;   k,\alpha)$  denote   the
Rarita-Schwinger  spinors  for $J=3/2$  particles  with momentum  $k$,
helicity $\alpha$ and mass $m$ \cite{ioffe,benmerrouche,hemmert}.
%
% \Eq{nkstar} and \Eq{n8k8star}
Eqs.~(\ref{nkstar}),(\ref{n8k8star})    and    (\ref{diquark-type.SV})
couples to  $J^P=3/2^+$ (anti-)baryons as  well.  In this  case, their
couplings involve $\gamma_5$ in the following way:
\begin{eqnarray}
  \langle 0|
  \psi_{\mu}(0)
  |B_{3/2^+}(k,\alpha)\rangle
  &=&
  \lambda_{3/2^+} \gamma_5 u_{\mu}(m_{3/2^+}; k,\alpha)
  \label{coupling.2}
  \\\nonumber
  \langle 0|
  \bar \psi_{\mu}(0)
  |\bar{B}_{3/2^+}(k,\alpha)\rangle
  &=&
  -
  \lambda_{3/2^+}^*
  \bar v_{\mu}(m_{3/2^+}; k,\alpha) \gamma_5,
\end{eqnarray}
where
%
$|B_{3/2^+}(k,\alpha)\rangle$  and  $|\bar B_{3/2^+}(k,\alpha)\rangle$
denote $J^P=3/2^+$ (anti-)baryon states with momentum $k$, helicity
$\alpha$ and mass $m_{3/2^+}$.
%
``$-$''  originates  from  the  anti-commutativity of  $\gamma_0$  and
$\gamma_5$.

To  derive the  spectral  representation,  the best  way  would be  to
express it in the operator representation in the following way:
\begin{equation}
  G_{\mu\nu}(\tau,\vec x)
  =
  Z^{-1}
  \mbox{Tr}
  \left(
  e^{-\beta H}
  T_{\tau}[\psi_{\mu'}(\tau,\vec x) \bar\psi_{\mu}(0)]
  \right),
  \label{correlator.a.1}
\end{equation}
where  $\beta$ denotes the  temporal extent  of the  lattice, $H\equiv
H_{\rm QCD}$ denotes the QCD Hamiltonian,
%
$Z\equiv \mbox{Tr}(e^{-\beta H})$ denotes the partition function,
%
and  $T_{\tau}[*]$  represents  the  time-ordered  product  along  the
imaginary time direction.
%
The interpolating fields are  represented in the Heisenberg picture in
imaginary-time,   i.e.,   $\psi_{\mu}(\tau,\vec   x)  =   e^{\tau   H}
\psi_{\mu}(0,\vec x)  e^{-\tau H}$ and  $\bar\psi_{\mu}(\tau,\vec x) =
e^{\tau H} \bar\psi_{\mu}(0,\vec x) e^{-\tau H}$.
%
By  restricting  ourselves to  the  interval  $0  \le \tau  <  \beta$,
\Eq{correlator.a.1} reduces to
\begin{equation}
  G_{\mu\nu}(\tau,\vec x)
  =
  \mbox{Tr}
  \left(
  \frac{e^{-\beta H}}{Z}
  \psi_{\mu'}(\tau,\vec x)
  \bar\psi_{\mu}(0)
  \right).
  \label{expression.1}
\end{equation}
Note that it can be equivalently expressed as
\begin{equation}
  G_{\mu\nu}(\tau,\vec x)
  =
  \mbox{Tr}
  \left(
  \psi_{\mu'}(\tau-\beta,\vec x)
  \frac{e^{-\beta H}}{Z}
  \bar\psi_{\mu}(0)
  \right).
  \label{expression.2}
\end{equation}
In the large  $\beta$ limit, we have $e^{-\beta  H}/Z \simeq |0\rangle
\langle   0|$,   which   is   inserted  into   \Eq{expression.1}   and
\Eq{expression.2}.
%
Note  that   the  resulting  two  expressions   serve  as  independent
contributions  to  the original  ``Tr'',  i.e., \Eq{expression.1}  (or
\Eq{expression.2}).
%
Hence, we keep these two contributions to obtain
\begin{eqnarray}
  \lefteqn{
    G_{\mu\nu}(\tau,\vec x)
  }
  \\\nonumber
  &\simeq&
  \langle 0 |
  \psi_{\mu'}(\tau,\vec x)
  \bar\psi_{\mu}(0)
  | 0 \rangle
  +
  \langle 0 |
  \bar\psi_{\mu}(0) \psi_{\mu'}(\tau-\beta,\vec x)
  | 0\rangle.
\end{eqnarray}
Note that the 1st term corresponds to the forward propagation, whereas
the 2nd term to the backward propagation.
%
By  inserting single-(anti-)baryon intermediate  states, and  by using
\Eq{coupling.1} and \Eq{coupling.2}, we are left with
\begin{widetext}
\begin{eqnarray}
  G_{\mu\nu}(\tau,\vec x)
  &=&
  \sum_{\alpha=1}^4
  \int {d^3k\over(2\pi)^3}
  {m_{3/2^+}\over k_0}
  e^{-\tau k_0}
  \langle 0 | \psi_{\mu'}(\vec x) | B_{3/2^+}(k,\alpha)\rangle
  \langle B_{3/2^+}(k,\alpha) | \bar\psi_{\mu}(0)| 0 \rangle
  +
  \cdots.
  \\\nonumber
  &=&
  |\lambda_{3/2^-}|^2
  \int {d^3 k\over (2\pi)^3}
  \frac{m_{3/2^-}}{k_0}
  (-1) P^{(3/2)}_{\mu\nu}(k)
  \left[
  e^{-\tau k_0}
  e^{i\vec k\cdot\vec x}
  \left( \frac{m_{3/2^-} + \fslash{k}}{2 m_{3/2^-}} \right)
  +
  e^{-(\beta - \tau) k_0}
  e^{-i\vec k\cdot\vec x}
  \left( \frac{m_{3/2^-} - \fslash{k}}{2 m_{3/2^-}} \right)
  \right]
  \\\nonumber
  &-&
  |\lambda_{3/2^+}|^2
  \int {d^3 k\over (2\pi)^3}
  \frac{m_{3/2^+}}{k_0}
  (-1)P^{(3/2)}_{\mu\nu}(k)
  \left[
  e^{-\tau k_0}
  e^{i\vec k\cdot\vec x}
  \left( \frac{m_{3/2^+} - \fslash{k}}{2 m_{3/2^+}} \right)
  +
  e^{-(\beta - \tau) k_0}
  e^{-i\vec k\cdot\vec x}
  \left( \frac{m_{3/2^+} + \fslash{k}}{2 m_{3/2^+}} \right)
  \right],
\end{eqnarray}
\end{widetext}
where  $k_0\equiv\sqrt{m_{3/2^-}^2 + \vec  k^2}$ for  $J^P=3/2^-$,
$k_0\equiv\sqrt{m_{3/2^+}^2 + \vec k^2}$ for $J^P=3/2^+$,
%
and the following identities are used.
\begin{eqnarray}
  \sum_{\alpha=1}^4
  u_{\mu}(m; k,\alpha) \bar u_{\nu}(m; k,\alpha)
  &=&
  - \frac{m + \fslash{k}}{2m}
  P^{(3/2)}_{\mu\nu}(k)
  \\\nonumber
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \sum_{\alpha=1}^4
  v_{\mu}(m; k,\alpha) \bar v_{\nu}(m; k,\alpha)
  &=&
  \frac{m - \fslash{k}}{2m}
  P^{(3/2)}_{\mu\nu}(k),
\end{eqnarray}
where  $P^{(3/2)}_{\mu\nu}(k)$  is the  spin  3/2 projection  operator
defined as
\begin{equation}
  P^{(3/2)}_{\mu\nu}(k)
  \equiv
  g_{\mu\nu} 
  - \frac1{3}\gamma_{\mu}\gamma_{\nu}
  - \frac1{3k^2}
  \left(
  \fslash{k}\gamma_{\mu}k_{\nu}
  + k_{\mu}\gamma_{\nu}\fslash{k}
  \right).
\end{equation}
By  performing  the  integration   over  $\vec  x$  for  zero-momentum
projection, and  by replacing the Minkowskian gamma  matrices by their
Euclidean   counterparts,   we   finally   arrive  at   the   spectral
representation (\Eq{correlator.1} with \Eq{correlator.2}).

The  derivation of  the  spin 1/2  parts  is obtained  by repeating  a
similar  procedure using  the following  parameterizations  instead of
\Eq{coupling.1} and \Eq{coupling.2} as
\begin{eqnarray}
  \lefteqn{
    \langle 0 |
    \psi_{\mu}(0)
    |B_{1/2^-}(k,\alpha)\rangle
  }
  \\\nonumber
  &=&
  \left(
  \lambda_{1/2^-} \gamma_{\mu}
  +
  \lambda'_{1/2^-} k_{\mu}
  \right)
  u(m_{1/2^-}; k,\alpha)
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \\\nonumber
  \lefteqn{
    \langle 0 |
    \bar \psi_{\mu}(0)
    |\bar B_{1/2^-}(k,\alpha)\rangle
  }
  \\\nonumber
  &=&
  \bar v(m_{1/2^-}; k,\alpha)
  \left(
  \lambda_{1/2^-}^* \gamma_{\mu}
  -
  {\lambda'}_{1/2^-}^* k_{\mu}
  \right)
  \\\nonumber
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \lefteqn{
    \langle 0 |
    \psi_{\mu}(0)
    |B_{1/2^+}(k,\alpha)\rangle
  }
  \\\nonumber
  &=&
  \left(
  \lambda_{1/2^+} \gamma_{\mu}
  +
  \lambda'_{1/2^-} k_{\mu}
  \right)
  \gamma_5
  u(m_{1/2^+}; k,\alpha)
  \\\nonumber
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \lefteqn{
    \langle 0 |
    \bar \psi_{\mu}(0)
    |\bar B_{1/2^-}(k,\alpha)\rangle
  }
  \\\nonumber
  &=&
  -
  \bar v(m_{1/2^-}; k,\alpha)
  \gamma_5
  \left(
  \lambda_{1/2^-}^* \gamma_{\mu}
  -
  {\lambda'}_{1/2^-}^* k_{\mu}
  \right),
\end{eqnarray}
where         $|B_{1/2^\pm}(k,\alpha)\rangle$        and        $|\bar
B_{1/2^\pm}(k,\alpha)\rangle$  denote the  $J^P=1/2^\pm$ (anti-)baryon
states with momentum $k$, helicity $\alpha$ and mass $m_{1/2^\pm}$.
%
$u(m;k,\alpha)$  and $v(m;k,\alpha)$  denote the  Dirac  bispinors for
spin 1/2 particles with mass  $m$, momentum $k$ and helicity $\alpha$.
$\lambda_{1/2^\pm}$  and $\lambda'_{1/2^\pm}$ represent  the couplings
to $J^P=1/2^{\pm}$ (anti-)baryons.

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  L.Y.~Glozman,
  Phys. Lett. B{\bf 575}, 18 (2003).
\bibitem{enyo}
  Y.~Kanada-Enyo, O.~Morimatsu, and T.~Nishikawa,
  Phys. Rev. C{\bf 71}, 0452002 (2005).
%%%%% hepta
\bibitem{bicudo}
  P.~Bicudo, and G.M.~Marques,
  Phys. Rev. D{\bf 69}, 011503 (2004).
%%%%% hepta
\bibitem{oset}
  F.J.~Llanes-Estrada, E.~Oset, and V.~Mateu,
  Phys. Rev. C{\bf 69}, 055203 (2004).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%% negative parity %%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% minimul constituent quark model
\bibitem{carlson}
  C.E.~Carlson, C.D.~Carone, H.J.~Kwee and V.~Nazaryan,
  Phys. Lett. B{\bf 573}, 101 (2003).
%%% chiral SU(3) quark model
\bibitem{huang}
  F.~Huang, Z.Y.~Zhang, Y.W.~Yu, and B.S.~Zou,
  Phys. Lett. B{\bf 586}, 69 (2004).
%%%%% bag model
\bibitem{shinozaki} T.~Shinozaki, M.~Oka, and S.~Takeuchi,
  Phys. Rev. D{\bf 71}, 074025 (2005).
%%%%% QCD sum rule
\bibitem{zhu}
  S.-L.~Zhu,
  Phys. Rev. Lett. {\bf 91}, 232002 (2003).
%%%%% QCD sum rule
\bibitem{matheus}
  R.D.~Matheus, F.S.~Navarra, M.~Nielsen, R.~Rodrigues~da~Silva, and S.H.~Lee,
  Phys. Lett. B{\bf 578}, 323 (2004).
%%%%% QCD sum rule
\bibitem{sugiyama}
  J.~Sugiyama, T.~Doi, and M.~Oka,
  Phys. Lett. B {\bf 581}, 167 (2004).
%%% The SU(3) group structure
\bibitem{shlee}
  Y.~Oh, H.~Kim, and S.H.~Lee,
  Phys. Rev. D{\bf 69}, 094009 (2004).
\bibitem{maezawa}
  Y.~Maezawa, T.~Maruyama, N.~Itagaki, and T.~Hatsuda,
  Acta Phys. Hung. A{\bf 22}, 61 (2005).
%%%%% string
\bibitem{narodetskii}
  I.M.~Narodetskii, Yu.A.~Simonov, M.A.~Trusov, and A.I.~Veselov,
  Phys. Lett. B{\bf 578}, 318 (2004).
%%%%% string
\bibitem{sugamoto}
  M.~Bando, T.~Kugo, A.~Sugamoto, and S.~Terunuma,
  Prog. Theor. Phys. {\bf 112}, 325 (2004).
%%%%% string
\bibitem{suganuma}
  H.~Suganuma, T.T.~Takahashi, F.~Okiharu, and H.~Ichie,
  Talk given at International Workshop
  {\it PENTAQUARK04}, July 2004, SPring-8, Harima, Japan.
%%%% penta potential
\bibitem{okiharu}
  F.~Okiharu, H.~Suganuma, and T.T.~Takahashi, 
Phys. Rev. Lett. {\bf 94}, 192001 (2005), hep-lat/0407001; 
hep-lat/0412012;  
%
  H.~Suganuma, T.T.~Takahashi, F.~Okiharu,  and H.~Ichie,
  Proc. of QCD Down Under, Adelaide, March 2004,
  Nucl. Phys. {\bf B}(Proc. Suppl.){\bf 141}, 92 (2005).
%%% Nakano & Hics
%% \bibitem{nakano-hicks}
%%   T.~Nakano, and K.~Hicks,
%%   Mod. Phys. Lett. A{\bf 19}, 645 (2004).
%%%%% parity determination
%% \bibitem{thomas}
%%   A.W.~Thomas, K.~Hicks and A.~Hosaka,
%%   Prog. Theor. Phys. {\bf 111}, 291 (2004).
%%%%% group theoretical construction of interpolating fields
%% \bibitem{morningstar}
%%   LHP Collaboration, S.~Basak {\it et al.},
%%   hep-lat/0312003.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% JP=3/2 Effective model approaches
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{hosaka32}
  A.~Hosaka, M.~Oka, and T.~Shinozaki,
  hep-ph/0409102.
\bibitem{nishikawa32}
  T.~Nishikawa, Y.~Kanada-En'yo, and O.~Morimatsu,
  Phys. Rev. D{\bf 71}, 076004 (2005).
\bibitem{takeuchi32}
  S.~Takeuchi, and K.~Shimizu,
  hep-ph/0410286.
\bibitem{sugiyama32}
  J.~Sugiyama, T.~Doi, and M.~Oka, spin 3/2 penta-quark in preparation.
%%%%%%%%%%
\bibitem{zhu32}
  W.~Wei, P.-Z.~Huang, H.-X.~Chen, and S.-L.~Zhu,
  hep-ph/0503166.
\bibitem{inoue32}
  T.~Inoue, V.E.~Lyubovitskij, Th.~Gutsche, and A.~Faessler,
  hep-ph/0407305.
\bibitem{jaffe32}
  R.~Jaffe and F.~Wilczek,
  Phys.~Rev.~D{\bf 69}, 114017 (2004).
\bibitem{huang32}
  F.~Huang, Z.Y.~Zhang, and Y.W.~Yu,
  hep-ph/0411222.
\bibitem{capstic32}
  S.~Capstick, P.R.~Page, and W.~Roberts,
  Phys. Lett. {\bf B570}, 185 (2003).
\bibitem{dudek32}
  J.J.~Dudek, and F.E.~Close,
  Phys. Lett. {\bf B583} 278 (2004).
\bibitem{hyodo32}
  T.~Hyodo, and A.~Hosaka,
  Phys. Rev. D{\bf 71}, 054017 (2005).
\bibitem{nam32}
  S.I.~Nam, A.~Hosaka, H.-C.~Kim,
  hep-ph/0505134.
%%%%% Particle Data Group
\bibitem{pdg}
  Particle Data Group,
  S.~Eidelman {\it et al.},
  Phys. Lett. B{\bf 592}, 1 (2004).
%%%%% hepta (experiment)
\bibitem{kishimoto}
  T.~Kishimoto, and T.~Sato,
  hep-ex/0312003.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% TSUIKA
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Lattice %%%
\bibitem{scikor12}
  F.~Csikor, Z.~Fodor, S.D.~Katz, and T.G.~Kovacs,
  JHEP {\bf 0311}, 070 (2003).
%%% Lattice %%%
\bibitem{sasaki}
  S.~Sasaki,
  Phys. Rev. Lett. {\bf 93}, 152001 (2004).
%%% Lattice %%%
\bibitem{chiu}
  T.W.~Chiu and T.H.~Hsieh,
  hep-ph/0403020.
%%% Lattice %%%
\bibitem{kentacky}
  N.~Mathur, F.X.~Lee, A.~Alexandru, C.~Bennhold, Y.~Chen, S.J.~Dong,
  T.~Draper, I.~Horv\'ath, K.F.~Liu, S.~Tamhankar, and J.B.~Zang,
  Phys. Rev. D{\bf 70}, 074508 (2004).
%%% Lattice %%%
\bibitem{ishii12}
  N.~Ishii, T.~Doi, H.~Iida, M.~Oka, F.~Okiharu, and H.~Suganuma,
  Phys. Rev. D{\bf 71}, 034001 (2005).
%%% Lattice %%%
\bibitem{rabbit}
  T.T.~Takahashi, T.~Kunihiro, T.~Onogi, and T.~Umeda,
hep-lat/0503019, Phys. Rev. D (2005) in press.
%%% Lattice (NEW) %%%
\bibitem{lasscock12}
  B.G.~Lasscock, J.~Hedditch, D.B.~Leinweber, W.~Melnitchouk,
  A.W.~Thomas, A.G.~Willams, R.D.~Young, and J.M.~Zanotti,
  hep-lat/0503008.
%%% Lattice (NEW) %%%
\bibitem{alexandrou12}
  C.~Alexandrou and A.~Tsapalis,
  hep-lat/0503013.
%%% Lattice (NEW) %%%
\bibitem{csikor122}
  F.~Csikor, Z.~Fodor, S.D.~Katz, T.G.~Kov\'acs, and B.C.~T\'oth,
  hep-lat/0503012.
%%% Lattice (NEW) %%%
\bibitem{holland}
  K.~Holland, and K.J.~Juge,
  hep-lat/0504007.
%%% Lattice (NEW)
\bibitem{lasscock32}
  B.G.~Lasscock, D.B.~Leinweber, W.~Melnitchouk, A.W.~Thomas, A.G.~Williams,
  R.D.~Young, J.M.~Zanotti,
  hep-lat/0504015.
%%% Lattice (PLAN) %%%
\bibitem{fleming}
  G.T.~Fleming,
  hep-lat/0501011
%%% Lattice (PLAN) %%%
%\bibitem{negele}
%  J.~Negele,
%  http://www.fz-juelich.de/exotics/talks/J.W.Negele.pdf
%%% Anisotropic Lattice %%%
\bibitem{klassen}
  T.R.~Klassen,
  Nucl. Phys. B{\bf 533}, 557 (1998).
%%% Anisotropic Lattice %%%
\bibitem{matsufuru}
  H.~Matsufuru, T.~Onogi, and T.~Umeda,
  Phys.~Rev.~D{\bf 64}, 114503 (2001).
%%% Anisotropic Lattice %%%
\bibitem{nemoto}
  Y.~Nemoto, N.~Nakajima, H.~Matsufuru, and H.~Suganuma,
  Phys.~Rev.~D{\bf68}, 094505 (2003).
%%% Anisotropic Lattice %%%
\bibitem{ishii-gb}
  N.~Ishii, H.~Suganuma, and H.~Matsufuru,
  Phys. Rev. D{\bf 66}, 094506 (2002);
  Phys. Rev. D{\bf 66}, 014507 (2002).
%%%%% Rarita-Schwinger
\bibitem{ioffe}
  B.L.~Ioffe,
  Nucl. Phys. {\bf B188}, 317 (1981).
%%%%% Rarita-Schwinger
\bibitem{benmerrouche}
  M.~Benmerrouche, R.M.~Davidson, and N.C.~Mukhopadhyay,
  Phys. Rev. C{\bf 39}, 2339 (1989).
%%%%% Rarita-Schwinger
\bibitem{hemmert}
  T.R.~Hemmert, B.R.~Holstein, and J.~Kambor,
  J. Phys. {\bf G24}, 1831 (1998).
%%% lattice text book %%%
\bibitem{montvay}
  I.~Montvay and G.~M\"unster, ``{\it  Quantum Fields on a Lattice}'',
  (Cambridge University Press, Cambridge, England, 1994), p.~1.
%%% J=3/2 Penta quark %%%
\bibitem{jaffe-exotica}
  R.L.~Jaffe, Nucl. Phys. B(Proc. Suppl.) {\bf 142}, 343 (2004).
%%%%% ILLUSIONARY RESONANCE in J^P=3/2^- channel
\bibitem{ishii320}
  N.~Ishii, T.~Doi, Y.~Nemoto, M.~Oka, H.~Suganuma,
  a  talk   given  at  Japan-US  Workshop   on  {\it Electromagnetic  Meson
  Production  and  Chiral Dynamics},
  April 8--9,  2005, Osaka  Univ., Osaka, Japan.
%%% general text book %%% 
\bibitem{itzykson}
  C.~Itzykson and J.-B.~Zuber,
  ``{\it Quantum Field Theory}'',
  (McGraw Hill, Singapore, 1985), p.~45.
\end{references}
\end{document}

hep-lat0506022.extracted_bib
\bibitem{nakano}
  LEPS Collaboration, T.~Nakano  {\it   et  al}.,
  Phys. Rev. Lett. {\bf 91}, 012002 (2003).

\bibitem{NA49} 
NA49 Collaboration (C. Alt {\it et al.}), 
Phys. Rev. Lett. {\bf 92}, 042003 (2004). 
%

\bibitem{H1} 
H1 Collaboration (A. Aktas {\it et al.}), 
Phys. Lett. B{\bf 588}, 17 (2004).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibitem{exotics}
  %%% X(3872)
  Belle Collaboration,
  S.K.~Choi {\it et al.},
  Phys. Rev. Lett. {\bf 91}, 262001 (2003);
  %
  CDF II Collaboration,
  D.~Acosta {\it et al.},
  Phys. Rev. Lett. {\bf 93}, 072001 (2004);
  %%% X(3940)
  P.~Pakhlov (For the Belle Collaboration),
  hep-ex/0412041;
  %%% Y(3940)
  Belle Collaboration,
  S.-K.~Choi {\it et al.},
  Phys. Rev. Lett. {\bf 94}, 182002 (2005);
  %%% S_0(3115)
  T.~Suzuki {\it et al.},
  Phys. Lett. B{\bf 597}, 263 (2004).
  %%% D_s(2317)
  BABAR Collaboration,
  B.~Aubert {\it et al.},
  Phys. Rev. Lett. {\bf 90}, 242001 (2003);
  %
  BELLE Collaboration,
  S.-K.~Choi {\it et al.},
  Phys. Rev. Lett. {\bf 91}, 262002 (2003);
  %
  CLEO Collaboration,
  D.~Besson {\it et al.},
  Phys. Rev. D{\bf 68}, 032002 (2003).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibitem{diakonov}    D.~Diakonov,   V.~Petrov    and   M.V.~Polyakov,
  Z. Phys. A{\bf 359}, 305 (1997).
%%%%%
%%%%% old days (bag model)

\bibitem{jaffe-76}
  R.L.~Jaffe, SLAC-PUB-1774 (1976).
%%%%% old days (bag model)

\bibitem{strottman}
  D.~Strottman,
  Phys. Rev. D{\bf 20}, 748 (1979).
%%% Skyrme soliton 1998

\bibitem{weigel}
  H.~Weigel,
  Eur. Phys. J. A{\bf 2}, 391, (1998).
%%%%% Skyrme model

\bibitem{praszalowicz}
  M.~Praszalowicz,
  Phys. Lett. B{\bf 575}, 234 (2003);\\
  M.~Praszalowicz, in : M.~Jezabek, M.~Praszalowicz (Eds.),
  Proceedings of the Workshop on Skyrmions and Anomalies,
  World Scientific, Singapore, 1987, p.~112.
%%%%

\bibitem{diana}
  DIANA Collaboration,
  V.V.~Barmin {\it et al.},
  Phys. Atom. Nucl. {\bf 66}, 1715 (2003).

\bibitem{clas}
  CLAS Collaboration,
  S.~Stepanyan {\it et al.},
  Phys. Rev. Lett. {\bf 91}, 252001 (2003).

\bibitem{saphir}
  SAPHIR Collaboration,
  J.~Barth {\it et al.},
  Phys. Lett. B{\bf 572}, 127 (2003).

\bibitem{experiments}
  A.E.~Asratyan, A.G.~Dolgolenko, M.A.~Kubantsev,
  Phys. Atom. Nucl. {\bf 67}, 682 (2004);
  %  \\
  %
  CLAS Collaboration,
  V.~Kubarovsky, {\it et al.},
  Phys. Rev. Lett. {\bf 92}, 032001 (2004);
  % \\
  %
  HERMES Collaboration,
  A.~Airapetian, {\it et al.},
  Phys. Lett. B{\bf 585}, 213 (2004);
  %\\
  %
  SVD Collaboration,
  A.~Aleev, {\it et al.},
  hep-ex/0401024;
  % Submitted to Yad.Fiz. 
  %\\
  %
  COSY-TOF Collaboration,
  M.~Abdel-Bary, {\it et al.},
  Phys. Lett. B{\bf 595}, 127 (2004);
  %\\
  %
  ZEUS Collaboration,
  S.~Chekanov, {\it et al.},
  Phys. Lett. B{\bf 591}, 7 (2004);
  %\\
  %
  %\begin{@@@}{\bf TSUIKA(POSITIVE-EXP)}
\bibitem{null}
  BES Collaboration,
  J.Z.~Bai {\it et al.},
  Phys. Rev. D{\bf 70}, 012004 (2004);
  %\\
  %
  HERA-B Collaboration,
  K.T.~Knoepfle {\it et al.},
  J. Phys. G{\bf 30}, S1363 (2004);
  %\\
  %
  PHENIX Collaboration,
  C.~Pinkenburg {\it et al.},
  J. Phys. G{\bf 30}, S1201 (2004);
  %\\
  %
  HERA-B Collaboration,
  I.~Abt {\it et al.},
  Phys. Rev. Lett {\bf 93}, 212003 (2004);
  %\\
  %
  SPHINX Collaboration,
  Y.-M.~Antipov {\it et al.},
  Eur. Phys. J. A{\bf 21}, 455 (2004);
  %\\
  %
  CDF Collaboration,
  I.V.~Gorelov {\it et al.},
  hep-ex/0408025;
  % Talk given at 12th International Workshop on Deep Inelastic Scattering
  % (DIS 2004), Strbske Pleso, Slovakia, 14-18 Apr 2004.
  % Published in *Strbske Pleso 2004, Deep inelastic scattering* 615-619 
  %
  %\\
  HyperCP Collaboration,
  M.J.~Longo {\it et al.},
  Phys. Rev. D{\bf 70}, 111101 (2004);
  %\\
  %
  BABAR Collaboration,
  B.~Aubert {\it et al.},
  hep-ex/0408064;
  % Submitted to 32nd International Conference on High-Energy Physics
  % (ICHEP 04), Beijing, China, 16-22 Aug 2004. 
  %\\
  %
  %\begin{@@@}{\bf TSUIKA(NULL-EXP)}
\bibitem{hicks}
  For recent review of the experimental status,
  K.H.~Hicks,
  hep-ex/0504027,
  and references therein.
%%% review 1 %%%

\bibitem{oka}
  M.~Oka, Prog. Theor. Phys. {\bf 112}, 1 (2004), and references therein.
%%% review 2 %%%

\bibitem{zhu.review}
  S.L.~Zhu, Int. J. Mod. Phys. {\bf A19}, 3439 (2004), and references therein.
%%%%% chiral soliton, large Nc

\bibitem{cohen}
  T.D.~Cohen,
  Phys. Lett. B{\bf 581}, 175 (2004).
%%%%% chiral quark soliton
%%%%% bound-K skyrmion

\bibitem{itzhaki}
  N.~Itzhaki, I.R.~Klebanov, P.~Ouyang, L.~Rastelli,
  Nucl. Phys. B{\bf 684}, 264 (2004).
%%%%% chiral quark soliton model (magnetic moment)

\bibitem{kim}
  H.-Ch.~Kim,
  Phys. Lett. B{\bf 585}, 99 (2004).
%%%%% chiral bag

\bibitem{hosaka}
  A.~Hosaka,
  Phys. Lett. B{\bf 571}, 55 (2003).
%%%%% diquark

\bibitem{jaffe}
  R.L.~Jaffe and F.~Wilczek,
  Phys. Rev. Lett. {\bf 91}, 232003 (2003).

\bibitem{lipkin}
  M.~Karliner and H.J.~Lipkin,
  Phys. Lett. B{\bf 575}, 249 (2003).
%%% quark model (positive parity)

\bibitem{carlson-positive}
  C.E.~Carlson, C.D.~Carone, H.J.~Kwee, V.~Nazaryan,
  Phys. Lett. B{\bf 579}, 52 (2004).

\bibitem{stancu}
  Fl.~Stancu and D.O.~Riska,
  Phys. Lett. B{\bf 575}, 242 (2003).
%%% quark model (positive parity)

\bibitem{jennings}
  B.K.~Jennings and K.~Maltman,
  Phys. Rev. D{\bf 69}, 094020 (2004).
%%% quark model (positive parity)

\bibitem{glozman}
  L.Y.~Glozman,
  Phys. Lett. B{\bf 575}, 18 (2003).

\bibitem{enyo}
  Y.~Kanada-Enyo, O.~Morimatsu, and T.~Nishikawa,
  Phys. Rev. C{\bf 71}, 0452002 (2005).
%%%%% hepta

\bibitem{bicudo}
  P.~Bicudo, and G.M.~Marques,
  Phys. Rev. D{\bf 69}, 011503 (2004).
%%%%% hepta

\bibitem{oset}
  F.J.~Llanes-Estrada, E.~Oset, and V.~Mateu,
  Phys. Rev. C{\bf 69}, 055203 (2004).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%% negative parity %%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% minimul constituent quark model

\bibitem{carlson}
  C.E.~Carlson, C.D.~Carone, H.J.~Kwee and V.~Nazaryan,
  Phys. Lett. B{\bf 573}, 101 (2003).
%%% chiral SU(3) quark model

\bibitem{huang}
  F.~Huang, Z.Y.~Zhang, Y.W.~Yu, and B.S.~Zou,
  Phys. Lett. B{\bf 586}, 69 (2004).
%%%%% bag model

\bibitem{shinozaki} T.~Shinozaki, M.~Oka, and S.~Takeuchi,
  Phys. Rev. D{\bf 71}, 074025 (2005).
%%%%% QCD sum rule

\bibitem{zhu}
  S.-L.~Zhu,
  Phys. Rev. Lett. {\bf 91}, 232002 (2003).
%%%%% QCD sum rule

\bibitem{matheus}
  R.D.~Matheus, F.S.~Navarra, M.~Nielsen, R.~Rodrigues~da~Silva, and S.H.~Lee,
  Phys. Lett. B{\bf 578}, 323 (2004).
%%%%% QCD sum rule

\bibitem{sugiyama}
  J.~Sugiyama, T.~Doi, and M.~Oka,
  Phys. Lett. B {\bf 581}, 167 (2004).
%%% The SU(3) group structure

\bibitem{shlee}
  Y.~Oh, H.~Kim, and S.H.~Lee,
  Phys. Rev. D{\bf 69}, 094009 (2004).

\bibitem{maezawa}
  Y.~Maezawa, T.~Maruyama, N.~Itagaki, and T.~Hatsuda,
  Acta Phys. Hung. A{\bf 22}, 61 (2005).
%%%%% string

\bibitem{narodetskii}
  I.M.~Narodetskii, Yu.A.~Simonov, M.A.~Trusov, and A.I.~Veselov,
  Phys. Lett. B{\bf 578}, 318 (2004).
%%%%% string

\bibitem{sugamoto}
  M.~Bando, T.~Kugo, A.~Sugamoto, and S.~Terunuma,
  Prog. Theor. Phys. {\bf 112}, 325 (2004).
%%%%% string

\bibitem{suganuma}
  H.~Suganuma, T.T.~Takahashi, F.~Okiharu, and H.~Ichie,
  Talk given at International Workshop
  {\it PENTAQUARK04}, July 2004, SPring-8, Harima, Japan.
%%%% penta potential

\bibitem{okiharu}
  F.~Okiharu, H.~Suganuma, and T.T.~Takahashi, 
Phys. Rev. Lett. {\bf 94}, 192001 (2005), hep-lat/0407001; 
hep-lat/0412012;  
%
  H.~Suganuma, T.T.~Takahashi, F.~Okiharu,  and H.~Ichie,
  Proc. of QCD Down Under, Adelaide, March 2004,
  Nucl. Phys. {\bf B}(Proc. Suppl.){\bf 141}, 92 (2005).
%%% Nakano & Hics
%% 
\bibitem{nakano-hicks}
%%   T.~Nakano, and K.~Hicks,
%%   Mod. Phys. Lett. A{\bf 19}, 645 (2004).
%%%%% parity determination
%% 
\bibitem{thomas}
%%   A.W.~Thomas, K.~Hicks and A.~Hosaka,
%%   Prog. Theor. Phys. {\bf 111}, 291 (2004).
%%%%% group theoretical construction of interpolating fields
%% 
\bibitem{morningstar}
%%   LHP Collaboration, S.~Basak {\it et al.},
%%   hep-lat/0312003.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% JP=3/2 Effective model approaches
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibitem{hosaka32}
  A.~Hosaka, M.~Oka, and T.~Shinozaki,
  hep-ph/0409102.

\bibitem{nishikawa32}
  T.~Nishikawa, Y.~Kanada-En'yo, and O.~Morimatsu,
  Phys. Rev. D{\bf 71}, 076004 (2005).

\bibitem{takeuchi32}
  S.~Takeuchi, and K.~Shimizu,
  hep-ph/0410286.

\bibitem{sugiyama32}
  J.~Sugiyama, T.~Doi, and M.~Oka, spin 3/2 penta-quark in preparation.
%%%%%%%%%%

\bibitem{zhu32}
  W.~Wei, P.-Z.~Huang, H.-X.~Chen, and S.-L.~Zhu,
  hep-ph/0503166.

\bibitem{inoue32}
  T.~Inoue, V.E.~Lyubovitskij, Th.~Gutsche, and A.~Faessler,
  hep-ph/0407305.

\bibitem{jaffe32}
  R.~Jaffe and F.~Wilczek,
  Phys.~Rev.~D{\bf 69}, 114017 (2004).

\bibitem{huang32}
  F.~Huang, Z.Y.~Zhang, and Y.W.~Yu,
  hep-ph/0411222.

\bibitem{capstic32}
  S.~Capstick, P.R.~Page, and W.~Roberts,
  Phys. Lett. {\bf B570}, 185 (2003).

\bibitem{dudek32}
  J.J.~Dudek, and F.E.~Close,
  Phys. Lett. {\bf B583} 278 (2004).

\bibitem{hyodo32}
  T.~Hyodo, and A.~Hosaka,
  Phys. Rev. D{\bf 71}, 054017 (2005).

\bibitem{nam32}
  S.I.~Nam, A.~Hosaka, H.-C.~Kim,
  hep-ph/0505134.
%%%%% Particle Data Group

\bibitem{pdg}
  Particle Data Group,
  S.~Eidelman {\it et al.},
  Phys. Lett. B{\bf 592}, 1 (2004).
%%%%% hepta (experiment)

\bibitem{kishimoto}
  T.~Kishimoto, and T.~Sato,
  hep-ex/0312003.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% TSUIKA
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Lattice %%%

\bibitem{scikor12}
  F.~Csikor, Z.~Fodor, S.D.~Katz, and T.G.~Kovacs,
  JHEP {\bf 0311}, 070 (2003).
%%% Lattice %%%

\bibitem{sasaki}
  S.~Sasaki,
  Phys. Rev. Lett. {\bf 93}, 152001 (2004).
%%% Lattice %%%

\bibitem{chiu}
  T.W.~Chiu and T.H.~Hsieh,
  hep-ph/0403020.
%%% Lattice %%%

\bibitem{kentacky}
  N.~Mathur, F.X.~Lee, A.~Alexandru, C.~Bennhold, Y.~Chen, S.J.~Dong,
  T.~Draper, I.~Horv\'ath, K.F.~Liu, S.~Tamhankar, and J.B.~Zang,
  Phys. Rev. D{\bf 70}, 074508 (2004).
%%% Lattice %%%

\bibitem{ishii12}
  N.~Ishii, T.~Doi, H.~Iida, M.~Oka, F.~Okiharu, and H.~Suganuma,
  Phys. Rev. D{\bf 71}, 034001 (2005).
%%% Lattice %%%

\bibitem{rabbit}
  T.T.~Takahashi, T.~Kunihiro, T.~Onogi, and T.~Umeda,
hep-lat/0503019, Phys. Rev. D (2005) in press.
%%% Lattice (NEW) %%%

\bibitem{lasscock12}
  B.G.~Lasscock, J.~Hedditch, D.B.~Leinweber, W.~Melnitchouk,
  A.W.~Thomas, A.G.~Willams, R.D.~Young, and J.M.~Zanotti,
  hep-lat/0503008.
%%% Lattice (NEW) %%%

\bibitem{alexandrou12}
  C.~Alexandrou and A.~Tsapalis,
  hep-lat/0503013.
%%% Lattice (NEW) %%%

\bibitem{csikor122}
  F.~Csikor, Z.~Fodor, S.D.~Katz, T.G.~Kov\'acs, and B.C.~T\'oth,
  hep-lat/0503012.
%%% Lattice (NEW) %%%

\bibitem{holland}
  K.~Holland, and K.J.~Juge,
  hep-lat/0504007.
%%% Lattice (NEW)

\bibitem{lasscock32}
  B.G.~Lasscock, D.B.~Leinweber, W.~Melnitchouk, A.W.~Thomas, A.G.~Williams,
  R.D.~Young, J.M.~Zanotti,
  hep-lat/0504015.
%%% Lattice (PLAN) %%%

\bibitem{fleming}
  G.T.~Fleming,
  hep-lat/0501011
%%% Lattice (PLAN) %%%
%
\bibitem{negele}
%  J.~Negele,
%  http://www.fz-juelich.de/exotics/talks/J.W.Negele.pdf
%%% Anisotropic Lattice %%%

\bibitem{klassen}
  T.R.~Klassen,
  Nucl. Phys. B{\bf 533}, 557 (1998).
%%% Anisotropic Lattice %%%

\bibitem{matsufuru}
  H.~Matsufuru, T.~Onogi, and T.~Umeda,
  Phys.~Rev.~D{\bf 64}, 114503 (2001).
%%% Anisotropic Lattice %%%

\bibitem{nemoto}
  Y.~Nemoto, N.~Nakajima, H.~Matsufuru, and H.~Suganuma,
  Phys.~Rev.~D{\bf68}, 094505 (2003).
%%% Anisotropic Lattice %%%

\bibitem{ishii-gb}
  N.~Ishii, H.~Suganuma, and H.~Matsufuru,
  Phys. Rev. D{\bf 66}, 094506 (2002);
  Phys. Rev. D{\bf 66}, 014507 (2002).
%%%%% Rarita-Schwinger

\bibitem{ioffe}
  B.L.~Ioffe,
  Nucl. Phys. {\bf B188}, 317 (1981).
%%%%% Rarita-Schwinger

\bibitem{benmerrouche}
  M.~Benmerrouche, R.M.~Davidson, and N.C.~Mukhopadhyay,
  Phys. Rev. C{\bf 39}, 2339 (1989).
%%%%% Rarita-Schwinger

\bibitem{hemmert}
  T.R.~Hemmert, B.R.~Holstein, and J.~Kambor,
  J. Phys. {\bf G24}, 1831 (1998).
%%% lattice text book %%%

\bibitem{montvay}
  I.~Montvay and G.~M\"unster, ``{\it  Quantum Fields on a Lattice}'',
  (Cambridge University Press, Cambridge, England, 1994), p.~1.
%%% J=3/2 Penta quark %%%

\bibitem{jaffe-exotica}
  R.L.~Jaffe, Nucl. Phys. B(Proc. Suppl.) {\bf 142}, 343 (2004).
%%%%% ILLUSIONARY RESONANCE in J^P=3/2^- channel

\bibitem{ishii320}
  N.~Ishii, T.~Doi, Y.~Nemoto, M.~Oka, H.~Suganuma,
  a  talk   given  at  Japan-US  Workshop   on  {\it Electromagnetic  Meson
  Production  and  Chiral Dynamics},
  April 8--9,  2005, Osaka  Univ., Osaka, Japan.
%%% general text book %%% 

\bibitem{itzykson}
  C.~Itzykson and J.-B.~Zuber,
  ``{\it Quantum Field Theory}'',
  (McGraw Hill, Singapore, 1985), p.~45.
