If the exotic baryon $\Theta^+(1540)$ is a correlated $udud\bar{s}$ with $J^P = \frac{1}{2}^+$, then there should exist an exotic meson, $J^P = 1^-$ $\vartheta^+ (S=+2)$ $\to K^+K^0$ $\sim 1.6$GeV with width $O(10-100)$MeV. The $\pi_1(1400;1600)$ may be broad members of {10} $\pm$ \10bar in such a picture. Vector mesons in the 1.4 - 1.7GeV mass range are also compared with this picture.
hep-ph0411160.tex

\documentclass[10pt]{article}  
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\newcommand{\eea}{\end{eqnarray}}  
\newcommand{\bean}{\begin{eqnarray*}}  
\newcommand{\eean}{\end{eqnarray*}}  
\newcommand{\kkpi}{\mbox{$K^{0}_{S} K^{\pm} \pi^{\mp}$}}  
\newcommand{\etapipi}{\mbox{$\eta \pi \pi$} }  
\newcommand{\kskbar}{\mbox{$K^* \overline K$} }  
\newcommand{\eqn}[1]{(\ref{#1})}  
\newcommand{\gapproxeq}{\lower  
.7ex\hbox{$\;\stackrel{\textstyle >}{\sim}\;$}}  
\newcommand{\lapproxeq}{\lower  
.7ex\hbox{$\;\stackrel{\textstyle <}{\sim}\;$}}  
\newcommand{\pipipipi}{\mbox{$\pi^+\pi^-\pi^+\pi^-$ }}  
\newcommand{\qbar}{\mbox{$\mathrm{\overline{q}}$} }  
\newcommand{\JPC}{\mbox{J$\mathrm{^{PC}}$}}  
\newcommand{\ubar}{\mbox{$\mathrm{\overline{u}}$} }  
\newcommand{\Smeson}{\mbox{$\mathrm{S^{*}/f}_{0}(980)$} }  
\newcommand{\dbar}{\mbox{$\mathrm{\overline{d}}$}}  
\newcommand{\sbar}{\mbox{$\mathrm{\overline{s}}$}}  
\newcommand{\la}{\langle}  
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\newcommand{\lra}{\leftrightarrow}  
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\def\question#1{{{\marginpar{\small \sc #1}}}}  
\def\qq{$ q\bar q $}  
\def\ss{$ s\bar s $}  
\def\nn{$n\bar{n}$}  
\def\uu{$u\bar u$}  
\def\dd{$d\bar d$}  
\def\cc{$c\bar c$}
\def\10bar{$\mathbf{\overline{10}}$}
\def\octet{$\mathbf{8}$}
\def\3bar{$\bar {\hbox{\bf 3}}$} 
\def\Ts{$ud\bar{s}$} 
\def\TQ{$ud\bar{Q}$}
\def\Tq{$ud\bar{q}$}
\def\QT{$\bar{u}\bar{d}Q$ } 
%four quark combinations
\def\uuud{$uu\bar{u}\bar{d}$}
\def\dudd{$du\bar{d}\bar{d}$}
\def\susd{$su\bar{s}\bar{d}$}
\def\usds{$us\bar{d}\bar{s}$}
\def\uudu{$uu\bar{d}\bar{u}$}
\def\udss{$ud\bar{s}\bar{s}$}

%%%%%
\parindent=0pt  
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\begin{document}  
\normalem  
\begin{titlepage}  
  
  \baselineskip=18pt \vskip 0.9in
\begin{center}  
  {\bf \Large Pentaquark implications for Exotic mesons}\\
  \vspace*{0.3in} {\large T.Burns}\footnote{\tt{e-mail:
      burns@thphys.ox.ac.uk}}
  {\large and F.E.Close}\footnote{\tt{e-mail:f.close@physics.ox.ac.uk}}\\
  \vspace{.1in} {\it Peierls Centre for Theoretical Physics, University of Oxford,\\
    1 Keble Rd., Oxford OX1 3NP, UK} \\
  \vspace{0.1in}
  {\large and J.J.Dudek}\footnote{\tt{e-mail:dudek@jlab.org}}\\
  \vspace{.1in}
%{\large xxx}\footnote{\tt{e-mail: xxx}} \\  
  {\it Thomas Jefferson National
          Accelerator Facility, Newport News, VA 23606, USA}\\
%{\it address}\\  
%\vspace{0.1in}  
%{\large xxx}\footnote{\tt{e-mail: xxx}} \\  
%{\it xx}\\   

%{\it xx}\\  
\end{center}  
  
\begin{abstract}  
  If the exotic baryon $\Theta^+(1540)$ is a correlated $udud\bar{s}$
  with $J^P = \frac{1}{2}^+$, then there should exist an exotic meson,
  $J^P =  1^-$ $\vartheta^+ (S=+2)$ $\to K^+K^0$ $\sim 1.6$GeV with width $O(10-100)$MeV.  The
  $\pi_1(1400;1600)$ may be broad members of {\bf 10} $\pm$ \10bar in
  such a picture. Vector mesons in the 1.4 - 1.7GeV mass range are also compared with
  this picture.

\end{abstract}  
\end{titlepage}  
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{Introduction}

If the narrow $\Theta^+$ seen in $nK^+$ and $pK^0$ is confirmed as a pentaquark, then
correlations among quarks in Strong QCD play an essential role. There
is considerable literature recognising that $ud$ in colour \3bar
with net spin 0 feel a strong attraction\cite{scalardiq,dgg}.  We
denote this $([ud]^{\bar{3}_c}_0)$, the subscript denoting its spin,
the superscript being the colour and the ( ) denoting the 
quasiparticle.  Ref.\cite{jw} considers the following subcluster for
the pentaquark: $([ud]_0)([ud]_0)\bar{s}$ with a $P$-wave between the
assumed bosonic $(ud)$ correlations.  By contrast ref.\cite{kl1}
assumes that the $([ud]_0)$ seed is attracted in $P$-wave to a
strongly-bound ``triquark" $([ud]^{6_c}_1\bar{s})$. These models
assign the $\Theta^+$ to \10bar of flavour.

An essential feature of these dynamics is that in $S$-wave the chromomagnetic
repulsion of like flavours destabilises the configuration such that decay
to meson + baryon in $S$-wave has such a large width as to be effectively 
nonexistent\cite{jafexotic}. It is in the $P$-wave that potentially interesting
pentaquarks emerge.

Mixing between the two configurations $([ud]^{\bar{3}_c}_0)\bar{s}$ 
and $([ud]^{6_c}_1\bar{s})$ has been shown to lead to an eigenstate of low mass,
which may be identified with the $\Theta(1540)$\cite{maltman,hog,vento}.
Further, this mixing potentially stabilises the $ud\bar{s}$ configuration,
underpinning the metstability of the $\Theta$\cite{vento}.
  
The point of departure for this paper is to note that if either of
these correlations is realised empirically for the $P$-wave, then on model independent
grounds one can replace $([ud]^{\bar{3}_c}_0)$ by $\bar{q}$, which
implies the existence of \10bar and {\bf 10} exotic mesons. A specific
example is the analogue of $\Theta^+ \equiv$ (\Ts)$(ud)$ $\to$
$\vartheta^+ \equiv $ (\Ts)$\bar{s}$.  
While many of these may be broad and unmeasurable, we shall suggest
that if the mixing that lowered and stabilised the $\Theta$ configuration in ref\cite{vento}
applies, then this leads to an observable $J^P = 1^-$ $\vartheta$ with strangeness
=+2, together with other $J^{PC} = 1^{-\pm}$ states with rather characteristic signatures.
  If the $\Theta^+$ should survive
high statistics data and with a width of $\sim 1$MeV, then the
observation or otherwise of such mesons may help to discriminate among
models for the dynamical origin of that metastability. 
Recent proposals \cite{vento} to explain the anomalously narrow 
width of the $\Theta^+$ ought to carry over to the meson world, 
although due to greater phase space we expect that the meson analogue
 $\vartheta^+$ will have width of order $10-100$MeV. Analogues
 with $J^P = 0^-,2^-,3^-$ also arise but are expected to be broad and
 unobservable.
 
 The possibility that the $\pi_1(1400)$ and $\pi_1(1600)$\cite{pdg04} could belong
 to these multiplets is discussed. In addition, the pattern of vector mesons in the
 1.4 - 1.7GeV mass region\cite{pdg04} may also have some overlap with these ideas.
 
 \subsection*{Exotic Mesons}


 
The idea that mesons beyond \qq~ exist is not new. Jaffe\cite{jaf77}
noted that the attractive forces alluded to above lead to a low lying
$qq\bar{q}\bar{q}$ $S$-wave  nonet. There is good evidence that
the scalar mesons below 1 GeV are intimately related to such a
picture\cite{ct02} and the idea of these as correlated diquarks has been
resurrected by ref\cite{maiani}. Models of multiquark mesons typically predict a 
large number of
  states, in various flavour multiplets and spin states. This usually
  includes exotics, whose flavour or $J^{PC}$ cannot be
  constructed from $q\bar{q}$. Any model of exotics must face the fact that
   there is at most only a handful
  of such candidates. 
  The full set of flavour representations from 
  $\mathbf{3\otimes 3\otimes \bar{3}\otimes\bar{3}=(\bar{3}+6)\otimes(3+\bar{6})}$ is:
  \\
(i) $\mathbf{\bar{3}\otimes 3=8\oplus 1}$, Jaffe's original nonet\\
(ii)$(\mathbf{6\otimes 3) \oplus (\bar{3}\otimes\bar{6})
=10\oplus 8_a \oplus\overline{10}\oplus 8_b
=18\oplus \overline{18}}$, the decuplets into which ref.\cite{chung} assign the 
$\pi_1(1400)$ state\cite{pdg04}, and accompanying octets, and\\
(iii) $\mathbf{6\otimes\bar{6}=27+8+1}$

All such states were included in the original study of
$[qq][\bar{q}\bar{q}]$ in $S$-wave\cite{jaf77}.  Within the dynamical
assumptions made there, all states other than the nonet (i) were predominantly expected to be
very broad and effectively unobservable. With $L=1$ in the system there are many multiplets with %%@
negative parity,
$J=0,1,2,3$ and $C=\pm$.  Either all the states are broad and
unobservable or some organising principle is required if one wishes to
identify one or two states as members, as e.g.\cite{chung}, and explain away the remainder. 
We examine the  $qq\bar{q}\bar{q}$ systems in $L=1$  in light of
 the recent interest in diquark correlations, focussing on the supposed dominance
 of ``good" diquarks over ``bad" \cite{jafexotic}. We shall see that the
 correlation of triquark-antiquark (and charge conjugate) leads to a limited spectrum of $J^{PC}$ %%@
states, in particular suggesting   a  lowering of the $1^{-\pm}$ multiplets
 to mass scales akin to those of the $\pi_1(1400;1600)$ claimed in  
 $\pi \eta$ and $\pi\eta'$ 
respectively\footnote{We refer the reader to \cite{indiana} for an analysis of the experimental data that disputes a resonant interpretation for the $\pi_1(1400)$}\cite{pdg04}.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{The $\vartheta^+$: an exotic meson analogue of the $\Theta^+$}
  

If the $\Theta^+$ is
confirmed  as a resonant state with $\Gamma \sim 1$MeV,
then the stability of the correlations, at least in $P$-wave, raises
interesting questions for the existence of observable meson analogues in 
representations {\bf 18} and $\overline{\bf 18}$.
In the (\TQ)$\bar{q}$ correlation, at least, the same
dynamics that lead to  $\Theta^+$ in \10bar imply a \10bar of mesons
which will include a  $\vartheta^+$ with strangeness $+2$,
and its {\bf 10} antiparticle with strangeness $-2$. In a later section we will show that a %%@
dynamical picture in which a triquark-antiquark (and charge conjugate)  are in $L=1$ suggests that %%@
the $1^{-(\pm)}$  multiplet lies lowest. 

 Neither of the correlations of \cite{jw} or \cite{kl1} alone derives
 either the low mass or width of the $\Theta^+(1540)$
 readily\cite{fc04ichep}. The triquark correlation in ref\cite{kl1}, with the $[ud]$ in the
 $(ud\bar{s})$ being in the configuration $[ud]^6_1$ was
 chosen for its maximal attraction. However, it has been widely noted
 \cite{hog,vento,maltman} that mixing with the $[ud]^{\bar{3}}_0$ via either
 one-gluon exchange, instanton forces or the effect of virtual $KN$ loops, leads to 
 one eigenstate that
 is lower in energy than either of the unmixed states. That this could cause a decoupling of the %%@
$\Theta^+ \to KN$, along the
 lines suggested in ref\cite{klmix}, is possible but has not been
 demonstrated (and  this mechanism has problems if virtual $K^* N$ loops are included). 

The energy levels in such a situation are interesting.
 Ref\cite{kl1} consider the \TQ $ $ correlation to be $([ud]^6_1\bar{s})$
 for which the chromomagnetic forces are highly attractive, the attractive interaction 
 between  $[ud]^6_1$ diquark and antiquark compensating for the reduced diquark attraction 
 relative to  $[ud]^{\bar{3}}_0$. The
 correlation $([ud]^{\bar{3}}_0\bar{s})$, where there is no such compensating diquark-antiquark 
 hyperfine interaction, is relatively disfavoured.  However,
 the same chromomagnetic forces cause a mixing between these two
 configurations and the emergence of a light and heavy eigenstate.
 Ref\cite{vento} argues that including both gluon exchange and
 instanton forces in the mixing analysis, leads to effective masses
 for the light eigenstate \Ts $ \sim 750$MeV and $m[ud] \sim 450$MeV.
 The reason for this being some 350 MeV more bound than the lightest
 $uds$ $\Lambda$ state is that the gluon and instanton forces are
 twice as attractive in the \qq~ channel than in the $qq$ case.
 Further, as m(\Ts)$ \leq m(K)+m(d/u)$, for constituent quark mass $m(q)$, the pentaquark cannot
 dissociate into the $Ku(d)$. It is proposed\cite{vento} that
 rearrangement is suppressed in $L=1$ with the result that the $\Theta^+$
 is metastable.
  
  
 The reduced mass of the 750 MeV triquark and 450 MeV diquark is 280
 MeV. The similarity of this to the strange quark in the $\phi$, which
 is $\sim m(\phi)/4 =255$MeV, suggests a similar price for
 $L$-excitation in the two systems. Using $m(f_2(1525)/f_1(1420)) -
 m(\phi) \sim 400- 500$MeV as a measure of the orbital excitation
 energy\footnote{ Note that this is more reasonable than the $207$MeV claimed
  by KL \cite{kl1} 
 on the basis of an analogy to the $D_s$ spectrum, see \cite{jjdFermilab} 
 for a critique.} giving the mass scale for $\Theta^+$ $\sim 1600-1700$MeV. Spin
 orbit splitting might reduce this to 1540MeV\cite{fcdudekls}.

 
But now consider the $[ud]$ accompanying the \Ts $ $ in the $P$-wave: we can replace this
 by any of $\bar{u}, \bar{d}$ or $\bar{s}$ to form a meson. Were we to
 do so for any combination $q_iq_j\bar{q_k}$ and $\bar{q_l}$, we would
 have a {\bf 10} and \10bar of mesons with a $P$-wave internally. 
 The $\pi_1(1400;1600)$ could be
 members of such a multiplet (this was originally proposed on symmetry
 grounds in ref\cite{chung}); however, an inescapable consequence of
 such a proposal is that there exists an exotic \Ts$\bar{s}$ meson with
 strangeness +2. With $m(\bar{s}) \sim m[ud]$ we predict this to be
 at $\sim 1600$MeV. Assuming that the $\Gamma(\Theta^+(1540))$ is narrow due to a mixing between
 $([ud]^{6_c}_1\bar{s})$ and $[ud]^{3_c}_0\bar{s}$ such that the low mass eigenstate
 decouples from $KN$, then the $\vartheta^+$ should exist with a ``normal" width.
 With a mass even at 1700MeV the phase space ratio for $\vartheta \to K\bar{K}$ and
 $\Theta^+ \to KN$ is $\sim 16$. The $KK^*$ channel is open; the phase space
 enhancement in this case is  $\sim 5$ but the spin counting will elevate this so that
 we may expect a similar branching ratio to that of the $KK$ mode. The three body $KK\pi$
 mode will also contribute but uncorrelated three body modes are not expected to dominate over the %%@
two
 body ones. The net result is that we expect $\Gamma(\vartheta^+) \lesssim 100$MeV, such that the 
$\vartheta$ should be detectable  (likewise for the equivalent $S=-2$ member of the $\mathbf{10}$).


Surprisingly it is not
immediately possible to exclude such an exotic state in $K^+K^0$ if its
mass is $\sim 1.6 -1.7$ GeV, and a dedicated search is suggested in
e.g. $K^+N \to K^+K^0\Sigma$\cite{oldexpt}.
 Other exotic members of the multiplets are probably broad
and unobservable if the dynamics of ref\cite{vento} underpins the formation of
triquarks (see later).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


 \subsection*{Other members of the meson $\mathbf{10\oplus\overline{10}}$}


 A unified convention for constructing the symmetry states for multiquarks is
 given in Table 1 of \cite{cd2}, reproduced here as Table 1. This gives the combinations of three labels
 for the symmetric and mixed ($\lambda$) states; the mixed ($\rho$) and antisymmetric
 follow trivially. The labels $A,B,C$ are defined 
 $A \equiv [ud] = \bar{s}; B \equiv [ds] = \bar{u}; C \equiv [su] = \bar{d}$; note that
  $[ud] \equiv (ud-du)/\sqrt{2}$, etc., the sign of the antisymmetric combination being important. 
%re understood to mean% $A_1 \equiv [qq]; (B_2C_3) \equiv ([qq]\bar{q})$.

\begin{table}
\hspace{-3.0cm} \begin{tabular}{l|c|c}
  & \10bar & ${\bf 8}_5$ \\
\hline
$\Theta^+$ & $AAA$ &\\
\hline
$p$ & $-(ACA + CAA +AAC)/\sqrt{3}$ & $-(ACA +CAA -2AAC)/\sqrt{6}$\\
$n$ & $(ABA +BAA +AAB)/\sqrt{3}$ & $(ABA + BAA -2AAB)/\sqrt{6}$\\
\hline
$\Sigma^+$ & $(CAC+ACC+CCA)/\sqrt{3}$ & $(CAC+ACC -2CCA)/\sqrt{6}$\\
$\Sigma^0$ & $-(ABC+BAC+ACB+CAB+BCA+CBA)/\sqrt{6}$ & $-(ABC+BAC +ACB+CAB-2BCA-2CBA)/\sqrt{12}$\\
$\Lambda^0$ & & $-(ABC-ACB+BAC-CAB)/2$\\
$\Sigma^-$ & $(BAB+ABB+BBA)/\sqrt{3}$ & $(BAB+ABB-2BBA)/\sqrt{6}$\\
\hline
$\Xi^+$ & $-CCC$ &\\
$\Xi^0$ & $(CBC+BCC+CCB)/\sqrt{3}$ & $(CBC+BCC-2CBB)/\sqrt{6}$\\
$\Xi^-$ & $-(CBB+BCB+BBC)/\sqrt{3}$ & $-(CBB+BCB -2 BBC)/\sqrt{6}$\\
$\Xi^{--}$ & $BBB$ &
\end{tabular}
\caption{Pentaquark wavefunctions where $ABC$ are defined in the text. Note that consistency requires the
meson octet to be defined with each $q\bar{q}$ positive except for $\pi^+ = -u\bar{d}$; $\bar{K}^0 = - s\bar{d}$ and then $\pi^0 = (u\bar{u} - d \bar{d})/\sqrt{2}$. In this convention $\eta_8 = (2s\bar{s} - u\bar{u} - d\bar{d})/\sqrt{6}$}
\end{table}

In the meson case it is useful to adopt the order  $A_1 B_2\equiv ([qq]\bar{q}),C_3 \equiv \bar{q}$    %%@
and the $(..)$ shows which are understood to be in the triquark correlation. 
 % In the meson cases of interest here the ordering mirrors the baryon case, such that it is
 % $\bar{q}(\bar{q}[qq])$ and the $(..)$ shows which are understood to be in the triquark.
 % The anti-triquark is defined by charge conjugating each constituent.

Triquarks  $[qq]\bar{q}$ are in flavour $\mathbf{\bar{3}\otimes \bar{3}=\bar{6}\oplus 3}$. To make %%@
a $\mathbf{\overline{10}\oplus{8}}$ of mesons (or pentaquarks) the triquark must be in flavour %%@
$\mathbf{\bar{6}}$, which is composed of the following members: \begin{center}
 $AA=[ud]\bar{s}$               \\
\vspace{1.5ex} 
 $\{AB\}=([ud]\bar{u}+[ds]\bar{s})/\sqrt{2}$\qquad\qquad %%@
$\{CA\}=([ud]\bar{d}+[su]\bar{s})/\sqrt{2}$\\
\vspace{1.5ex}
$BB=[ds]\bar{u}$\qquad\qquad\qquad $\{BC\}=([ds]\bar{d}+[su]\bar{u})/\sqrt{2}$\qquad\qquad\qquad %%@
$CC=[su]\bar{d}$\\
\end{center}

\par
The nonexotic combinations      in which $q_iq_j\bar{q_j}$ are in flavour $\mathbf{3}$  allow the %%@
possibility of $q_j\bar{q_j} \to $
  gluons and hence mixing with conventional hadrons; such combinations have no advantages in
  forming metastable states      and comprise the $P$-wave excited version of Jaffe's nonet %%@
\cite{scalardiq}. The combinations $\{AB\},\{BC\}$ and $\{CA\}$ are at least stable against  %%@
$q_j\bar{q_j} \to $ gluons (by $SU(3)_F$) although in the mechanism of Vento \cite{vento}, it is %%@
only the $AA$ triquark that is fully stable, all others being unstable against decay into $\pi$ or %%@
$\eta_s$.

Using the conventions of \cite{cd2}, a full set of meson representations can be constructed:

\begin{tabular}{ll}
$\vartheta^+ = AAA$ & \\ %[ud]\bar{s}\bar{s}$& \\
$K^+_{\mathbf{\overline{10}}} = -(ACA+CAA+AAC)/\sqrt{3}$ & $K^+_{\mathbf{8b}} = -(ACA+CAA-2AAC)/\sqrt{6}$\\
$\pi^+_{\mathbf{\overline{10}}} = -(CAC+ACC+CCA)/\sqrt{3}$ & $\pi^+_{\mathbf{8b}} = -(CAC+ACC-2CCA)/\sqrt{6}$\\
$``K^+"= CCC$ & \\
\end{tabular}

The other charge combinations follow by acting on these with $I_-$ accordingly.
Charge conjugate analogues of these correspond to a  $\mathbf{10\oplus 8}$ .   


 Ref.\cite{kl1} considered only the exotic states at the corner of the $\mathbf{\overline{10}}$.
The dynamics of ref.\cite{vento}, discussed earlier, suggest that
other configurations, such as $([su]\bar{d})$ (and  $([ds]\bar{u})$) are energetically unfavoured, 
 due to the presence of $[us]$ in place of
 $[ud]$, for which the instanton forces are less attractive, and unstable
 because $m([us]\bar{d}) \geq m(\pi) + m(s)$, which enables decay.  One consequence could appear to %%@
be that only the $\Theta^+$
 will be narrow in such a dynamics: the $\Xi^{+,--}$ contain
 $([su]\bar{d})$ or $([ds]\bar{u})$ which are unstable against decay into
 $\pi + s$, while the other states mix with {\bf 8}. 
  
  The absence of prominent signals  other than the $\Theta^+$
 in the \10bar, in particular the $\Sigma_5$\cite{fcqz03}, may thus be
 explained: if such correlations occur, then they only create
 metastable configurations if either $[ud]$ and/or $([ud]\bar{s})$ are 
 involved. 
 We argue that the same situation arises in the  $\mathbf{\overline{10}}$ and  $\mathbf{10}$ of %%@
mesons: the only exotic combinations with a chance of 
suppressed widths are the $\vartheta^+$ and $\vartheta^-$, containing the triquarks $AA$ and %%@
$\bar{A}\bar{A}$ respectively. 


The exotic $S=-1$  $K^+$ $([su]\bar{d})\bar{d}$ and $K^{--}$  $([ds]\bar{u})\bar{u}$ contain %%@
triquarks
 that are  unstable against $\pi$ emission in the model of ref\cite{vento}. Thus we expect their %%@
widths to be at least 
300MeV.
 Identifying such states will be a challenge. 
 %Likewise for the equivalent members of the $\mathbf{10}$. 
 The mass gap between the $\vartheta^+$ and the $K^{+,0,-,--}$ states will be $\sim 2m(s - d) - %%@
m([us] - [ud])$. As the $m([us] - [ud])$ is likely
to be at least as large as $m(s-d)$  the
spread is likely to be only $\sim 100$ MeV.  


The other combinations occur in nonexotic multiplets and in general will mix. Consider for example 
the $J^P =1^-$ states, $\rho^0$ or $\pi^0_1$. 
%Relative to a base mass typified by $ud\bar{u}\bar{d}$ the mass gap as multiples of
  %$m(s)-m(d)$ will be
  %$m$(\10bar) = $4/3$; $m$({\bf 8}$^{\lambda}$) = 1/3, hence  
 %$m$(\10bar) - $m$({\bf 8}$^{\lambda}$) =  $m(s)-m(d)$
Depending on the mixing angle between the \10bar and {\bf 8 } basis states, the
mass eigenstates can be degenerate   
  or separated by up to $2(m(s)-m(d))$. In this latter case the physical
  states are the ideal mass eigenstates $[qq]\bar{q}\bar{q}$ and $[sq]\bar{q}\bar{s}$.
  This small mass range for the states suggests that the hidden flavour mass basis is more 
  representative than an $SU(3)_F$ multiplet basis.

%The $\pi_1(1600;1400)$ will occur as mixed linear combinations of $\pi_{10}\pm\pi_{\overline{10}}$ %%@
%and  $\pi_{8_a}\pm\pi_{8_b}$.
%Each of these are formed from ``unstable" triquarks ($CA, AC, CC$), triggering their broad decay %%@
%widths
 %into $\pi \eta$ or $\pi \eta'$ as observed ($\geq 300$MeV\cite{pdg04}). The subtleties of these %%@
%states with be discussed in a later section.

Thus we need to count the number of states.
 For the neutral states with $I=1$ and $I=0$ there are six permutations of the distinct $ABC$
  labels. In the $SU(3)_F$ symmetry basis these correspond to
  
 \begin{tabular}{ll}
  \10bar: $I=1$; & {\bf 8}$^{\lambda}$:$(I=1) + (I=0)$;\\
  {\bf 8}$^{\rho}$:$(I=1) + (I=0)$; & {\bf 1}: $I=0$.
  \end{tabular}
  
  The latter trio correspond to the familiar $\rho; \omega_8; \omega_1$ combinations
  in the case of $1^-$ and as such are indistinguishable from radially excited \qq~ nonets.
  The former would correspond to a pair of $\rho$s and a single $\omega_8$, and hence would be %%@
novel.
  Thus counting the population of $I=1$ and $I=0$ vector mesons within an energy range of $\sim %%@
300$MeV
  can hint at which underlying multiplets are present. 
  
  While the $\rho(1700),\omega(1650),\phi(1680)$ form a candidate nonet, their masses are
  somewhat unnatural. The $\rho(1450)$ and $\omega(1420)$ are missing a partner to complete
  the set. Depending on whether this is $I=0$ or $I=1$ could be novel. The $K^*(1410)$ appears
  to be anomalously low in mass for \qq~ systems but fits naturally into the \10bar configuration.
  The widths of most of these states are hundreds of MeV.
 

The general feature is that for a given $J^{P(C)}$ six neutral members are expected within a few %%@
hundred MeV. If any of the plethora of observed states\cite{pdg04} is associated with
these, such that their widths of $\sim 300$MeV give a scale for their (in)stability, then a %%@
$\vartheta^+$ seems an unavoidable consequence.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Dynamics and $J^{PC}$ in $L=1$ $qq\bar{q}\bar{q}$ mesons}

An $L=1$ $qq\bar{q}\bar{q}$ system has a variety of $J^{PC}$ combinations. The relative masses
and potential stability of these can depend upon the correlations of strong QCD.
We now investigate the different dynamical arrangements for an $L=1$ $qq\bar{q}\bar{q}$ system, %%@
distinguished by the configuaration of the orbital angular momentum. We will see that dynamics may %%@
favour a triquark configuration for the  $\mathbf{18\oplus \overline{18}}$ of mesons, and that such %%@
a configuration has a limited spectum of $J^{PC}$ states. The same dynamical assumptions, namely %%@
the prevalence of ``good'' diquarks over ``bad'', explains the absence of higher representations %%@
such as $\mathbf{27}$.

In the diquark-diquark correlation, the  $qq$ and $\bar{q}\bar{q}$ systems are each in $L=0$ with %%@
an $L=1$ between them. For quark pairs which are symmetric in space, the remaining degrees of %%@
freedom must be antisymmetric. The allowed combinations are as follows, for total quark spin $S$,  %%@
with $\{\quad \}$ and $[\quad ]$ denoting flavour  $\mathbf{6}$ ($\mathbf{\overline{6}}$) and  %%@
$\mathbf{3}$ ($\mathbf{\overline{3}}$) respectively, and superscripts and subscripts denoting %%@
colour 
and spin:\\
\begin{tabular}{lccl}

&$\mathbf{ 10+8 }$                                      &       $\mathbf{ \overline{10}+8 }$                                    
& \\

(i)&${\{ qq \} }^{\bar{3}}_1{[\bar{q}\bar{q}]}^3_0$     &       ${[qq]}^{\bar{3}}_0  {\{ 
\bar{q}\bar{q} \} }^3_1 $       & $S=1,J^P=0^-,1^-,2^-$ \\
(ii)&${\{ qq \} }^6_0  {[\bar{q}\bar{q}]}^{\bar{6}}_1$   &       ${[qq]}^6_1      
  {\{ \bar{q}\bar{q} \} }^{\bar{6}}_0$& $S=1,J^P=0^-,1^-,2^-$ \\

\end{tabular}\par


A different set of configurations arises if the $L=1$ is \emph{between} a 
pair of quarks or antiquarks and the concept of a ``diquark'' dissolves. The quark pairs are now %%@
spatially antisymmetric, so that to satisfy the Pauli 
principle, the same flavour and colour correlations as above will have spin symmetry flipped
from 0 (antisymmetric) to 1 (symmetric), or vice versa. The resulting 
combinations, where $q|q$ denotes a pair of quarks in $L=1$, are:
\par
\begin{tabular}{lccl}

&$\mathbf{ 10+8 }$                                      &$\mathbf{ 
\overline{10}+8 }$                              & \\

(iii)&${\{ q|q \} }^{\bar{3}}_0  [{\bar{q}\bar{q}]}^3_0 $ &${[qq]}^{\bar{3}}_0  
{\{ \bar{q}|\bar{q} \} }^3_0 $  &$S=0,J^P=1^-$ \\
(iv)&${\{ q|q \} }^6_1   {[\bar{q}\bar{q}]}^{\bar{6}}_1$ &${[qq]}^6_1 {\{ 
\bar{q}|\bar{q} \} }^{\bar{6}}_1$       &$S=0,J^P=1^-$ \\
                                                        &                               
        &               &$S=1,J^P=0^-,1^-,2^-$ \\
                                                        &                               
        &               &$S=2,J^P=0^-,1^-,2^-,3^-$ \\
(v)&${\{ qq \} }^6_0  [{\bar{q}|\bar{q}]}^{\bar{6}}_0   $       &${[q|q]}^{\bar{3}}_1    
  {\{ \bar{q}\bar{q} \} }^3_1$ &$S=0,J^P=1^-$ \\
(vi)&${\{ qq \} }^{\bar{3}}_1    {[\bar{q}|\bar{q}]}^3_1$&${[q|q]}^6_0  {\{ 
\bar{q}\bar{q} \} }^{\bar{6}}_0 $       &$S=0,J^P=1^-$ \\

                                                        &                               
                &       &$S=1,J^P=0^-,1^-,2^-$ \\
                                                        &                               
                &       &$S=2,J^P=0^-,1^-,2^-,3^-$ \\

\\
\end{tabular}\par

The first thing to notice is that the two different pictures have different $J$ 
couplings. If the diquark-diquark picture describes the $\pi_1(1400;1600)$
mesons, then we would also expect $0^{-+}$ and $2^{-+}$ partners at comparable 
mass, shifted by spin orbit splittings. Conversely, the latter picture 
allows for $S=0,1$ or $2$. If we supposed that the dynamics were such that 
the $S=0$ state was favoured (and we will argue that this could be so), then the apparent absence %%@
of $0^{-+}$ and $2^{-+}$ 
siblings to the $\pi_1$ mesons is natural.

 In the diquark-diquark picture we see that a meson in 
$\mathbf{18}$ or $\mathbf{\overline{18}}$ is made of a ``good'' 
and ``bad'' diquark: 
in Jaffe's original paper\cite{scalardiq}, the absence of $S$-wave mesons in this flavour %%@
representation was due to these repulsive colourmagnetic interactions. By contrast, we see that in %%@
the second picture, in which the  $L=1$ is ``within'' a diquark, the spatial antisymmetry annuls %%@
these repulsive forces, turning
 a ``bad'' diquark of a given flavour (colour-spin symmetric) into a ``good'' 
diquark (colour-spin symmetric), as in configurations (iii) and (iv) (or conversely turns a ``good" 
diquark into a ``bad'' diquark in configurations (v) and (vi)). However, 
once the orbital angular momentum separates quarks, the short-range 
hyperfine interaction is heavily suppressed, so it is better to describe these
 $L=1$ diquarks as neither ``good'', nor ``bad'', but ``null''. 

What then can we say about the dynamics of a $[qq]\{\bar{q}|\bar{q}\}$ system? On the one hand, we %%@
could consider this sytem a direct mirror image of the JW correlation\cite{jw} for the $\Theta^+$, %%@
$(ud)\ |\ (ud)\quad \bar{s} \to \bar{q}\ |\ \bar{q}\quad [qq]$. 
%In the JW picture of the $\Theta^+$, the Bose statistics argument that forces the $P$-wave %%@
%prevents $(ud)$ and $\bar{s}$ from coalescing into a triquark. However, in the meson picture, %%@
%there is no such problem. 
However, the $[q q]^{\bar{3}}_0\bar{q}$ and $[q q] ^6_1 \bar{q}$ 
  systems in $S=1/2$ are very light due to attractive colourmagnetic and instanton forces %%@
\cite{hog}\cite{vento}, so it is natural to consider the configurations  (iii)  and  (iv)  as a %%@
tightly bound $S$-wave $qq\bar{q}$ triquark in a 
  relative $P$-wave with a $\bar{q}$, (along with their charge conjugates)\footnote{ 
  Provided the triquark is compact spatially, the difference between a $P$-wave between 
  the antiquarks and a $P$-wave between the free antiquark and the triquark can be neglected}. Each %%@
of these systems has a ``good'' diquark,
  and the presence of a $\bar{q}$ in $S$-wave with these diquarks lowers 
  the energy further. 

Refs\cite{hog} and \cite{maltman} noted that the colourmagnetic interaction 
mixes the $S=1/2$ $[q q]^{\bar{3}}_0\bar{q}$ and $[q q] ^6_1 \bar{q}$ triquarks (configurations %%@
(iii) and (iv)) to give 
the lowest eigenvalue $-21.88C$ in the case of full  $SU(3)_F$ symmetry, a considerably stronger 
attraction than the diquark-diquark system, for which the lightest configuration has 
an energy shift of $-16/3C$.
Ref.\cite{vento} noted that instanton forces cause the same mixing and lower the energy
of the triquark correlation further.
%Before considering triquark-antiquark interactions, or the difference in 
%the $P$-wave excitations of the two systems, it appears as though the triquark  arrangement is 
%dynamically preferred.

Note also that the mixing of the (iii) and (iv) configurations, with their
downward energy shifts, occur only in the $S=0, J^P=1^-$ state. Thus if this
correlation is dominant, the lowest lying multiplets can be those with $J^P =1^-$, all other $J^P$ %%@
states being higher 
in energy. In any event, this possible dynamical picture limits the spectrum of $J^{PC}$ that %%@
could be expected, so that this picture is less readily falsified by the dearth of experimental %%@
evidence of mesons in  $\mathbf{10}$ or $\mathbf{\overline{10}}$.

The same dynamics suggests that higher representations such as $\mathbf{6\otimes 6=27\oplus 8\oplus %%@
1}$ are not dynamically favoured. $\{qq\}\{\bar{q}\bar{q}\}$ systems are made of pairs of ``bad'' %%@
diquarks, and even moving to the triquark picture cannot annul these repulsive forces.

Thus we suggest that although tetraquark states in general will tend to be broad, the $J^P = 1^-$ %%@
\10bar and {\bf 10}, possibly mixed with {\bf 8}, may have the best chance of being observable. 
There are subtleties involved with
forming the $C = \pm$ eigenstates associated with  $\mathbf{18\oplus \overline{18}}$; these are
discussed
in an appendix.   We shall see later that the multiquark dynamics leading to the above also imply %%@
that there
  exists a ``mirror" multiplet with $J^{PC} = 1^{--}$ neutral members.

The phenomonology of the  controversial $\pi_1(1400)$ and $\pi_1(1600)$ is not
inconsistent with the triquark correlation. The dynamically preferred arrangement in triquark %%@
language has $S=0$, which fits in neatly with the absence of clear $0^-$ and $2^-$ partners to the %%@
$\pi_1(1400)$ and $\pi_1(1600)$. It is plausible that the mixed $[q q]^{\bar{3}}_0\bar{q}$ and $[q %%@
q] ^6_1 \bar{q}$ system could be the only stable triquark correlation, and that states which do not %%@
benefit from this mixing ($S=1,2$) are not dynamically preferred.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\subsection*{$J^{PC} = 1^{-\pm}$ multiplets in correlated quark models}



 In the search for evidence of gluonic degrees of freedom in Strong QCD, attention
 has focussed in particular on the prediction of exotic  quantum numbers  such as $J^{PC} = %%@
1^{-+}$,
 which are forbidden for \qq~  in a potential but allowed if the gluonic degrees of freedom are
 excited. The masses of the lightest such hybrids are predicted from lattice QCD
 to be above 1.8GeV\cite{lattice}. Thus
 the appearance of $\pi_1(1600) \to \pi \eta'$ is intriguing as this was long ago
 suggested as a selection rule by Lipkin (see
 citation to unpublished remarks by Lipkin in \cite{lee}) and then in a modern context
 in\cite{fclipkin1} where the decay of a hybrid $1^{-+}$ was predicted to have $\pi\eta' > %%@
\pi\eta$. A problem is that such a state could also occur from $qq\bar{q}\bar{q}$ and the mass %%@
pattern
 of the other members of
 the nonet would need to be identified in order to distinguish this from a canonical
 $q\bar{q}g$ hybrid.  In this context there is also reported a companion state $\pi_1(1400)$, which %%@
is seen in $\pi \eta$
 but not $\pi\eta'$.  If such a decay
 conserves flavour, and if $\eta \equiv \eta_{\mathbf{8}}$, then such a state cannot belong to an {\bf 8}. 
 Motivated by this state, ref\cite{chung} proposed that $\pi_1(1400)$ belong to
 a {\bf 10} $\oplus$ \10bar $ qq\bar{q}\bar{q}$ multiplet, and thus is not
 a hybrid \qq$g$ state in {\bf 8}.
 

As shown in the appendix, we can take a linear combinations of $\pi^+_{\mathbf{10}}$ and $\pi^+_{\mathbf{\overline{10}}}$ %%@
to give a $I^G(J^{PC}) = 1^-(1^{-+})$ state overlapping to $\pi^+\eta_{\mathbf{8}}$, and likewise a %%@
combinations of  $\pi^+_{\mathbf{8_a}}$ and $\pi^+_{\mathbf{8_b}}$ overlapping to  $\pi^+\eta_{\mathbf{1}}$. The combinations %%@
$CAC$ and $CCA$ (and charge conjugates) have 2 strange masses, while $ACC$ has 0 strange masses, so %%@
we can express the $SU(3)_F$ symmetry basis states $X$ and $Y$ for the  $1^-(1^{-+})$ states as:
\bea
X=(\pi^+_{\mathbf{10}}\oplus\pi^+_{\mathbf{\overline{10}}})/\sqrt{2}&=&[\pi^0\eta_{\mathbf{8}}]=(\sqrt{2}qs\bar{q}\bar{s}+
qq\bar{q}\bar{q})/\sqrt{3} \nonumber\\
Y=(\pi^+_{\mathbf{8_a}}\oplus\pi^+_{\mathbf{8_b}})/\sqrt{2}&=&[\pi^0\eta_{\mathbf{1}}]=(-\sqrt{2}qs\bar{q}\bar{s}+2qq\bar{q}\bar%%@
%%@
{q})/\sqrt{6} \nonumber
\eea
where $qs\bar{q}\bar{s}$ and $qq\bar{q}\bar{q}$ denote the (normalised) parts of the wavefunction %%@
with 2 and 0 strange quarks respectively. The orthogonal linear combinations give an 
$I^G(J^{PC}) = 1^+(1^{--})$ $\rho$ state overlapping to meson states $K\overline{K}$ and $\pi\pi$. %%@
For ref.\cite{chung}, who assume $\eta \equiv \eta_{\mathbf{8}}$ and $\eta' \equiv \eta_\mathbf{1}$, it is 
states $X,Y$ 
respectively that decay to $\pi\eta$ and $\pi\eta'$:
the physical $\eta$ and $\eta'$ are not pure {\bf 8} and {\bf 1}. One could choose the mixing of %%@
the decuplet and octet states  $X$ and $Y$
to enforce decays to the $\pi\eta$ and $\pi\eta'$ respectively; this would require the mixing angle
of $X,Y$ to be the same as the $\eta-\eta'$ mixing.  The eigenstates mixed in this way would then %%@
correspond to $\pi_1(1400)\to \eta \pi$ and  $\pi_1(1600)\to \eta' \pi$. Conversely, mass %%@
eigenstates that are ideal
 
\bea
X_H=qq\bar{q}\bar{q}\nonumber\\
X_L=qs\bar{q}\bar{s}\nonumber
\eea
will decay to $\pi\eta_n$ and $\pi\eta_s$ respectively.

In order to go from mass eigenstates to symmetry basis states, 
it is necessary to
have a mixing amplitude $A$[(\dd$u)\to$ (\ss$u$)]  $>m$(\dd$u)- m$(\ss$u$). The stability
of the \ss$u$ in contrast to the \dd$u$ seems to argue against that but a definite answer is 
beyond our ability to determine without further assumptions. If $\pi_1(1400;1600)$ are 
identified with these states, then the  quark mass eigenstates appear to be nearly realised: the %%@
masses based on the simplest flavour
counting are consistent with $\pi^+_1(1400)$ and $\pi^+_1(1600)$ as  the (dominantly)  %%@
$qq\bar{q}\bar{q}$
and $qs\bar{q}\bar{s}$ states. 
This agrees with our earlier prediction that $m(\vartheta)$, the \udss~ state is also $ \sim 1600$MeV,
which in turn is consistent with the $\Theta^+(1540)$. A pair of $\rho$ states with similar
masses to those of the two $\pi_1$s are required. There are known problems with identifying the
$\rho(1460;1600)$ as simply \qq~ states\cite{cdk} and the existence of $\rho(1250)$
remains uncertain. If the latter exists as a \qq~ candidate, then the other pair may be related to %%@
the  $qq\bar{q}\bar{q}$ and $qs\bar{q}\bar{s}$ states. Conversely, if the $\rho(1460)$ is the %%@
lightest such resonance, then the
$K^*(1410)$ mass is more in tune with the pattern of interest here than it would
be for a \qq~ nonet. A test will be the presence or absence of isoscalar partners to these 
states. A {\bf 10} or \10bar have no such $I=0$ $\eta_1$ or $\omega$ states whereas the {\bf 8}
does; a canonical nonet would have the $\rho; \omega; \phi$ analogues. 

 In any case, the $\sim 300$MeV width of the $\pi_1(1400)$ and $\pi_1(1600)$ is consistent with a %%@
triquark dynamics. The wavefunction of a $\pi$ state, in any decuplet-octet mixing scenario, is %%@
composed of triquarks that can decay to $\pi+s, \pi+q$ or $\eta_s+q$ (or charge conjugates, for the %%@
antitriquark component).

In \cite{chung} it is shown that if the $1^{-+}$ $\pi_1(1400) \to \eta\pi$ is a member of a
\10bar $\oplus$ {\bf 10} then there must also be a partner state $\rho \to \pi\pi$ and $K\bar{K}$.
 Using symmetry arguments, ref.\cite{chung} show that this ``supermultiplet'' of  %%@
$\mathbf{10\oplus\overline{10}}$ decaying to two pseudoscalar mesons (PP) ought to be accompanied %%@
by another supermultiplet decaying to a pseudoscalar and vector meson (PV), a $I^G(J^{PC}) = %%@
1^+(1^{--})\to \omega\pi$ and a $1^-(1^{-+})\to \rho\pi,K^* \bar{K}$. Note, however, that the %%@
$1^{-\pm}$ in the PV supermultiplet mentioned by CKK in ref.\cite{chung}  must be accompanied by %%@
siblings $0^{-\pm}$ and $2^{-\pm}$. By visualising the system as a triquark-antiquark in a %%@
$P$-wave, we can immediately understand the origin of the two supermultiplets and their $J^{PC}$ %%@
quantum numbers.



In normal $q\bar{q}$ mesons, $q$ and $\bar{q}$ have opposite parity, so that $P$-wave states have %%@
$P=+$. Conversely, in the triquark-antiquark picture, the triquark $Q$ and antiquark $\bar{q}$ have %%@
the $same$ parity, so that $P$-wave states have $P=-$. In the appendix, we show that the %%@
wavefunction of a $qq\bar{q}\bar{q}$ meson has an extra degree of freedom compared to a $q\bar{q}$ %%@
meson,  manifested in the freedom to take $\mathbf{10+\overline{10}}$ versus %%@
$\mathbf{10-\overline{10}}$ (or equivalently $\mathbf{8_a+8_b}$ versus $\mathbf{8_a-8_b}$), giving %%@
$C=+$ and $C=-$ for each $J$. So if we take the $q\bar{q}$ spectrum of states,
\bea
        S=0, J=1^{+-}   & &      S=1,J=0^{++},1^{++},2^{++}
\eea
flip the parity, and take  $C=\pm$, we acquire precisely the spectrum of \cite{chung}:
\bea
S=0, J=1^{--}   & &      S=1,J=0^{--},1^{--},2^{--}\\
 S=0, J=1^{-+}  & &      S=1,J=0^{-+},1^{-+},2^{-+}
\eea


 In the work of Chung et al \cite{chung} there is no dynamical picture that distinguishes the two %%@
supermultiplets, and hence no suggestion 
as to why the PV supermultiplet has not been experimentally observed. We show
in the appendix that there are different dynamics underpinning the supermultiplets. The PP %%@
supermultiplet, to which the observed $\pi_1\to \eta\pi$ belongs, has $S=0$, and we saw earlier %%@
that this spin configuration is the only one in which  mixing from one gluon exchange and instanton %%@
forces allows a light and metastable triquark. On the contrary, the PV multiplet has $S=1$, a %%@
configuration in which mixing cannot occur, resulting in heavier, possibly unstable triquarks. This  %%@
might account for the experimental elusiveness of the PV supermultiplet.


Earlier we noted that the diquark-diquark picture can only have $S=1, J=0^-,1^-,2^-$. In the %%@
appendix we show that for the diquark-diquark configuration $\zeta=(-1)^l$, so that such a system %%@
in $P$-wave can exist in only $one$ supermultiplet $(\zeta=-1)$. Due to  angular momentum %%@
conservation in the decays of the
 $0^-$ and $2^-$, this supermultiplet can only be PV. Thus, provided it is valid to treat diquarks %%@
as effective bosons, the $1^{-+}$ $\pi_1(1400)\to\eta\pi$ and $\pi_1(1600)\to\eta'\pi$ cannot be in %%@
the diquark-diquark arrangement.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




\subsection*{Conclusion}

In conclusion, confirmation of a narrow $\Theta^+(1540)$ and the absence of other narrow members of %%@
a \10bar can be explained by correlations
that suggest there should be a $\vartheta$ with canonical width $\lesssim 100$MeV
together with
a family of broad partners. This particular dynamics is exceptional as 
received wisdom has been that all these states should ``fall-apart".  If the $\Theta^+$ is an %%@
artefact, or if its narrow width 
is due to  some mechanism other than the mixing among correlations as discussed here, then the 
$1^{-(\pm)}$ tetraquark mesons will be all broad as in \cite{jaf77,jafexotic}. In any event, if
the existence of a resonant $\Theta^+$ with narrow width survives further scrutiny, then
a \10bar of mesons with moderate widths, of which the $\vartheta$ may have a canonical width,
merits investigation.

Models which consider four-quark mesons produce a considerable multiplicity of flavour and spin %%@
states. Using a triquark-quark correlation
for tetraquarks there can be a exception to this rule, with only a reduced set of states appearing
and the possibility that among these the $J^P =1^-$ \10bar and {\bf 10} may contain
observable states. It seems possible to associate certain otherwise peculiar states in the meson %%@
spectrum with those predicted here.

This model is trivially falsifiable by comparision of its prediction of flavour exotic states with %%@
experiment. We have noted that while the $\rho(1700),\omega(1650),\phi(1680)$ form a candidate %%@
nonet, their masses are
  somewhat unnatural. The $\rho(1450)$ and $\omega(1420)$ are missing a partner to complete
  the set and determine whether they are in a nonet or \10bar. The $K^*(1410)$ appears
  to be anomalously low in mass for \qq~ systems but fits naturally into the \10bar configuration.
  The widths of most of these states are hundreds of MeV.
 

The general feature is that for a given $J^{P(C)}$ six neutral members are expected within a few %%@
hundred MeV. If any of the plethora of observed states\cite{pdg04} is associated with
these, such that their widths of $\sim 300$MeV give a scale for their (in)stability, then a %%@
$\vartheta^+$ with a canonical width seems an unavoidable consequence.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


 \subsection*{Appendix} 

 
To obtain charge conjugation eigenstates for tetraquark mesons we need to write their wavefunctions %%@
in $qq\bar{q}\bar{q}$ and $\bar{q}\bar{q}qq$ form. We demonstrate the procedure for the nonstange %%@
neutral members of the $\mathbf{10}$ and $\mathbf{\overline{10}}$, noting that we can easily %%@
generalise to the $I_3=\pm 1$ members with the usual $G$ parity operator. Note also that the %%@
following analyses work for the  $\mathbf{8_a}$ and  $\mathbf{8_b}$. 



From the wavefunctions given earlier , we find
\bea
\pi_{\mathbf{10}}^0=\mathbf{6\otimes 3}&=&-( \{ds\}[\bar{d}\bar{s}]
+\{su\}[\bar{s}\bar{u}]
+\{ud\}[\bar{u}\bar{d}])/\sqrt{3} \\
\widetilde{\pi}_{\mathbf{10}}^0=\mathbf{3\otimes 6}&=&-
([\bar{d}\bar{s}]\{ds\}
+[\bar{s}\bar{u}]\{su\}
+[\bar{u}\bar{d}]\{ud\})/\sqrt{3} \\
\pi_{\mathbf{\overline{10}}}^0=\mathbf{\bar{3}\otimes\bar{6}}&=&-
([ds]\{\bar{d}\bar{s}\}
+[su]\{\bar{s}\bar{u}\}
+[ud]\{\bar{u}\bar{d}\})/\sqrt{3}\\
\widetilde{\pi}_{\mathbf{\overline{10}}}^0=\mathbf{\bar{6}\otimes \bar{3}}&=&-
(\{\bar{d}\bar{s}\}[ds]
+\{\bar{s}\bar{u}\}[su]
+\{\bar{u}\bar{d}\}[ud])/\sqrt{3}
\eea
Notice $C\pi_{\mathbf{10}}^0=\widetilde{\pi}_{\mathbf{\overline{10}}}^0$ and  %%@
$C\pi_{\mathbf{\overline{10}}}^0=\widetilde{\pi}_{\mathbf{10}}^0$, so if we denote the %%@
$\mathbf{10}$ and $\mathbf{\overline{10}}$ wavefunctions
\bea
 \chi(\mathbf{10},\zeta)=\pi_{\mathbf{10}}^0+\zeta\widetilde{\pi}_{\mathbf{10}}^0 & \qquad &
\chi(\mathbf{\overline{10}},\zeta)=\pi_{\mathbf{\overline{10}}}^0+
\zeta\widetilde{\pi}_{\mathbf{\overline{10}}}^0
\eea
then our $\zeta$ is precisely that defined by equation (2) in \cite{chung}:\bea
C\chi(\mathbf{10},\zeta)=\zeta\chi(\mathbf{\overline{10}},\zeta)&\qquad&
C\chi(\mathbf{\overline{10}},\zeta)=\zeta\chi(\mathbf{10},\zeta)
\eea
We have the freedom to choose the phase between the $\mathbf{10}$ and $\overline{\mathbf{10}}$ , %%@
which we denote by $a$, so that the full
 wavefunctions of a four quark $\mathbf{10\oplus\overline{10}}$ have two degrees of freedom, $a$ %%@
and $\zeta$,
\bea
\pi^0(\zeta,a)&=&\pi_{\mathbf{10}}^0+\zeta\widetilde{\pi}_{\mathbf{10}}^0
+a\left(\pi_{\mathbf{\overline{10}}}^0+\zeta\widetilde{\pi}_{\mathbf{\overline{10}}}^0)\right)\eea
We see that
 $C\pi^0(\zeta,a)=\zeta a\pi^0(\zeta,a)$
 and the doubling of states follows: in the $\zeta=-1$ multiplet we have a $1^{--}$  $(\zeta a=-+)$ %%@
and $1^{-+}$ $(\zeta a=--)$, and in the $\zeta=+1$ multiplet we have a $1^{--}$  $(\zeta a=+-)$ and %%@
$1^{-+}$ $(\zeta a=++)$. Looking at the $qq\bar{q}\bar{q}$ part of the wavefunction only, we see %%@
that under interchange of quarks $2\leftrightarrow 3$ linear combinations of $\pi_{\mathbf{10}}^0$ %%@
and $\pi_{\mathbf{\overline{10}}}^0$ correspond to CKK's flavour wavefunctions in %%@
$q\bar{q}q\bar{q}$ form:  the $1^{-+}$ state is %%@
$\pi_{\mathbf{10}}^0+\pi_{\mathbf{\overline{10}}}^0\equiv [\pi^0\eta_8]$, the  $1^{--}$ state is %%@
$\pi_{\mathbf{10}}^0-\pi_{\mathbf{\overline{10}}}^0\equiv\sqrt{1\over %%@
3}\left([\pi^-\pi^+]+[\overline{K^0}K^0]+[K^-K^+] \right)$ (our $\pi^+$ is defined as %%@
$-u\bar{d}$\cite{cd2}, hence the phase difference compared to CKK). Likewise for linear %%@
combinations of the wavefunctions $\mathbf{8_a}$ and $\mathbf{8_b}$.

We can think of a $P$-wave tetraquark meson as a system of two quasi particles in $L=1$: in the %%@
diquark-diquark picture, two bosons of spin 0 and 1, in the triquark picture, two fermions of spin %%@
1/2 (we assume the lightest triquark is spin 1/2 following \cite{chung}\cite{vento}).
 In the JW picture of the $\Theta^+$ \cite{jw}, the $(ud)$ pairs are treated as effective bosons so %%@
that bose statistics forces a $P$-wave. However, this picture suffers from the fact that the $uudd$ %%@
wavefunction is not fully antisymmetrised, only the $(ud)$ pairs individually. In the meson world %%@
there is no such problem. We can treat a $qq\bar{q}\bar{q}$ system as a pair of bosons or pair of %%@
fermions at the same time ensuring their fermionic wavefunctions are fully antisymmetrised.

\par
Let us look firstly at the diquark-diquark correlation. Denoting $\{qq\}$ by $b$ and $[qq]$ by $B$, %%@
in shorthand notation we can write $\pi_{\mathbf{10}}^0=b\bar{B}$,
$\widetilde{\pi}_{\mathbf{10}}^0=\bar{B}b$,
$\pi_{\mathbf{\overline{10}}}^0=B\bar{b}$,
$\widetilde{\pi}_{\mathbf{\overline{10}}}^0=\bar{b}B$,so that the full wavefunctions can be %%@
expressed
\bea
\pi^0(\zeta,a)&=&b_1\bar{B}_2+\zeta \bar{B_1}b_2
+a\left(B_1\bar{b_2}+\zeta\bar{b_1}B_2\right)\eea
where the labels 1 and 2 denote the spin and space degrees of freedom of the bosons. %%@
$C\pi^0(\zeta,a)=\zeta a \pi^0(\zeta,a)$ in this shorthand notation, as expected. Interchanging the %%@
space and spin labels in the wavefunction brings a factor $(-1)^l$
\bea
\pi^0(\zeta,a) & = &(-1)^l[b_2\bar{B}_1+\zeta \bar{B_2}b_1
+a(B_2\bar{b_1}+\zeta\bar{b_2}B_1)]
\eea
and since bosons are commuting variables ($[b_1,\bar{B}_2]=0$),
\bea
\pi^0(\zeta,a) & = &(-1)^l[\bar{B}_1 b_2+\zeta b_1 \bar{B_2}
+a(\bar{b_1}B_2+\zeta B_1\bar{b_2})]\\
&=&\zeta(-1)^l\pi_1^0(\zeta,a)
\eea
so in the diquark-diquark picture $\zeta=(-1)^l$.
\par

In the triquark-antiquark  configuration there are two spin 1/2 fermions in $L=1$. Denoting the %%@
triquarks $\{AB\}$, $\{CA\}$ and $\{BC\}$ by $U,D$ and $S$, 
\bea
\pi_{\mathbf{10}}^0&=&(-u\bar{U}+d\bar{D}+s\bar{S})/\sqrt{3}=q\bar{Q} \\
\widetilde{\pi}_{\mathbf{10}}^0&=&(-\bar{U}u+\bar{D}d+\bar{S}s)/\sqrt{3}=\bar{Q}q \\
\pi_{\mathbf{\overline{10}}}^0&=&(-U\bar{u}+D\bar{d}+S\bar{s})/\sqrt{3}=Q\bar{q} \\
\widetilde{\pi}_{\mathbf{\overline{10}}}^0&=&(-\bar{u}U+\bar{d}D+\bar{s}S)/\sqrt{3}=\bar{q}Q 
\eea
and the full wavefunction is
\bea
\pi^0(\zeta,a)&=&q_1\bar{Q}_2+\zeta \bar{Q_1}q_2
+a\left(Q_1\bar{q_2}+\zeta\bar{q_1}Q_2\right)\\
&=& (-1)^{s+1}(-1)^l [  q_2\bar{Q}_1+\zeta \bar{Q_2}q_1
+a\left(Q_2\bar{q_1}+\zeta\bar{q_2}Q_1\right) ]\\
&=&  (-1)^{l+s} [  \bar{Q}_1q_2+\zeta q_1\bar{Q_2}
+a\left(\bar{q_1}Q_2+\zeta Q_1\bar{q_2}\right) ]\\
&=&(-1)^{l+s}\zeta\pi_1^0(\zeta,a)
\eea
since interchanging the labels $1$ and $2$ brings factors $(-1)^{s+1}$
and $(-1)^l$ for the spin and space labels respectively, and for fermions 
$\{q_1,\bar{Q}_2\}=0$. Thus in the triquark antiquark picture $\zeta=(-1)^{l+s}$. Once again %%@
$C\pi^0(\zeta,a)=\zeta a \pi^0(\zeta,a)$ in this shorthand notation, as expected.
\par

In the work of
Chung et al \cite{chung}, a quantum number $\zeta$ distinguishes the two supermultiplets, being %%@
labelled by $\zeta=+1$ (PP) and $\zeta=-1$ (PV), or vice versa (the overall phase of $\zeta$ is not %%@
important). At first glance, it appears as though the  total spin $S$ of the quarks distinguishes %%@
the two supermultiplets: the $S=0$ states being PP and the $S=1$ states being PV, but some caution %%@
is needed. By  angular momentum conservation for $P$-wave decays, 
$0^-,2^-\to$ PV but
not to PP,
so we can immediately assign the $0^-$ and $2^-$ states to the PV multiplet. However,
since we can have both
$1^-\to$ PV  and 
$\to$ PP in P wave, there is nothing, \emph{a priori}, that tells us to which supermultiplet a %%@
$1^-$ must belong. In the appendix, though, we use Fermi-Dirac statistics to show that for a %%@
$P$-wave triquark-antiquark system, the supermultiplet label $\zeta$ is given by %%@
$\zeta=(-1)^{s+1}$. We can confirm, then, what we had already suspected: that the total spin $S$ of %%@
the quarks determines precisely to which supermultiplet a state belongs. The states with
$S=0$ belong to the PP supermultiplet ($\zeta=-1$), and those with $S=1$ to the PV supermultiplet %%@
($\zeta=+1$).

As a brief digression, it's worth considering  the $P$-wave excited version of Jaffe's original %%@
nonet in the triquark-antiquark correlation.
In this case, we must have a flavour $\mathbf{3}$ of triquarks, where $SU(3)_F$ does not protect %%@
the annihilation of $q\bar q$ pairs, so stable triquarks cannot form. Annihilation of $q_i %%@
q\bar{q}\to q_i$ is equivalent to rewriting $U\to u$, $D\to d$ and  $S\to s$, or simply $Q\to q$ in %%@
our shorthand notation. In that case, the wavefunction is clearly zero for $a=-1$. The extra degree %%@
of freedom has been stripped away, and we recover the ordinary spectrum of $q\bar{q}$ states:
\bea
\zeta a = -+ & & S=0, J=1^{+-}\\
\zeta a = ++ & & S=1, J=0^{++},1^{++},2^{++}
\eea
Thus in the triquark-antiquark picture, a nonet of $P=-$ tetraquarks will not form. On the %%@
contrary, in the diquark-diquark picture the $q\bar{q}$ annihilation is inhibited by the $P$-wave %%@
barrier so it is possible that a nonet should be seen.





\bc {\bf Acknowledgments} \ec
  
This work is supported, in part, by grants from the Particle Physics
and Astronomy Research Council, the EU-TMR program ``Euridice''
HPRN-CT-2002-00311, a Clarendon Fund Bursary and by the U.S. Department of Energy under
contract DE-AC05-84ER40150. 


\begin{thebibliography}{99}  

\bibitem{scalardiq} R.L.Jaffe, Phys.Rev. D {\bf 15} 267 (1977);
  F.E.Close and N.Tornqvist, J.Phys.G. {\bf 28} R249 (2002)
\bibitem{dgg} A.De Rujula, H.Georgi and S.L.Glashow, Phys. Rev. D{\bf
    12}, 147 (1975);Ya.B. Zeldovich and A.D. Sakharov, Yad. Fiz {\bf
    4}(1966)395; Sov. J. Nucl. Phys. {\bf 4}(1967)283.
%\bibitem{wilc} K.Rajagopal and F.Wilczek hep-ph/0011333
%\bibitem{saphir2} SAPHIR report, unpublished, private communication from K Hicks October 2003
 \bibitem{jw} R.L.Jaffe and F.Wilczek, hep-ph/0307341
\bibitem{kl1} M.Karliner and H.J.Lipkin, hep-ph/0307243
 \bibitem{jafexotic} R.L. Jaffe, hep-ph/0409065
\bibitem{maltman} B.Jennings and K.Maltman, hep-ph/0308286
\bibitem{hog} H.Hogaasen and P.Sorba, hep-ph/0406078
 \bibitem{vento} N.I.Kochelev, H.J.Lee and V.Vento, hep-ph/0404065 
\bibitem{pdg04} Particle Data Group, Phys Letters B592 (2004) 1
\bibitem{jaf77} R.L.Jaffe, Phys.Rev. D {\bf 15} 267 (1977)
\bibitem{ct02} F.E.Close and N.Tornqvist, J.Phys.G. {\bf 28} R249 (2002)
\bibitem{maiani} L.Maiani, F.Piccinini, A.D.Polosa, V.Riquer,  hep-ph/0407017 
\bibitem{chung} S.U.Chung, E.Klempt and J G Korner, hep-ph/0211100; S.U.Chung and E.Klempt,
hep-ph/0306018
\bibitem{indiana}  A.~R.~Dzierba {\it et al.}, Phys.\ Rev.\ D {\bf 67} (2003) 094015
\bibitem{fc04ichep} F.E.Close, Rapporteur talk at ICHEP04
\bibitem{klmix} M.Karliner and H.J.Lipkin, Phys.Lett.B {\bf 586} 303, 2004 
\bibitem{jjdFermilab} J.J.Dudek talk at the First Meeting of the APS Topical Group on Hadronic Physics, Fermilab November 2004 (Proceedings to appear in J.Phys.G)
\bibitem{fcdudekls} F.E.Close and J.J.Dudek, Phys.Lett.B {\bf 583}
  278, 2004 
\bibitem{oldexpt} W.P.Dodd, T.Joldersma, R.B.Palmer and N.P.Samios, Phys.\ Rev.\  {\bf 177} (1969) %%@
1991.
\bibitem{cd2} F.E.Close and J J Dudek, hep-ph/0401192
\bibitem{fcqz03} F.E.Close and Qiang Zhao, hep-ph/0404075 
\bibitem{lattice}   
P. Lacock, C. Michael, P. Boyle and P. Rowland, Phys.Lett. B {\bf 401}, 308   
(1997);   
C. Bernard {\it et al.}, Phys.Rev. D {\bf 56}, 7039 (1997);   
P. Lacock and K. Schilling, Nucl. Phys. Proc. Suppl. {\bf 73}, 261 (1999);   
C. McNeile, hep-lat/9904013;   
C. Morningstar, Nucl. Phys. Proc. Suppl. {\bf 90}, 214 (2000)   
%\bibitem{experiments} 1440/1600 experiments
\bibitem{lee} B W Lee, S Okubo and J Schecter Phys Rev. 135 (1964) B219
\bibitem{fclipkin1} F.E.Close and H.J.Lipkin,Phys Lett 196 (1987) 245;




%%CITATION = HEP-EX 0304002;%%








%symmetry paper


%\bibitem{bag} bag, fluxtube
%\bibitem{lattice} lattice

%\bibitem{bcm} T Barnes, F E Close and S Monaghan
%\bibitem{gi} S.Godfrey and N.Isgur, Phys.Rev. D{\bf 32}, 189 (1985)
\bibitem{cdk}   
F.E. Close, A. Donnachie and Yu.S. Kalashnikova, Phys.Rev. D {\bf 65},   
092003 (2002)  
  
\end{thebibliography}  
  
               

 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




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\end{document}  





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

hep-ph0411160.extracted_bib
\begin{thebibliography}{99}  

\bibitem{scalardiq} R.L.Jaffe, Phys.Rev. D {\bf 15} 267 (1977);
  F.E.Close and N.Tornqvist, J.Phys.G. {\bf 28} R249 (2002)
\bibitem{dgg} A.De Rujula, H.Georgi and S.L.Glashow, Phys. Rev. D{\bf
    12}, 147 (1975);Ya.B. Zeldovich and A.D. Sakharov, Yad. Fiz {\bf
    4}(1966)395; Sov. J. Nucl. Phys. {\bf 4}(1967)283.
%\bibitem{wilc} K.Rajagopal and F.Wilczek hep-ph/0011333
%\bibitem{saphir2} SAPHIR report, unpublished, private communication from K Hicks October 2003
 \bibitem{jw} R.L.Jaffe and F.Wilczek, hep-ph/0307341
\bibitem{kl1} M.Karliner and H.J.Lipkin, hep-ph/0307243
 \bibitem{jafexotic} R.L. Jaffe, hep-ph/0409065
\bibitem{maltman} B.Jennings and K.Maltman, hep-ph/0308286
\bibitem{hog} H.Hogaasen and P.Sorba, hep-ph/0406078
 \bibitem{vento} N.I.Kochelev, H.J.Lee and V.Vento, hep-ph/0404065 
\bibitem{pdg04} Particle Data Group, Phys Letters B592 (2004) 1
\bibitem{jaf77} R.L.Jaffe, Phys.Rev. D {\bf 15} 267 (1977)
\bibitem{ct02} F.E.Close and N.Tornqvist, J.Phys.G. {\bf 28} R249 (2002)
\bibitem{maiani} L.Maiani, F.Piccinini, A.D.Polosa, V.Riquer,  hep-ph/0407017 
\bibitem{chung} S.U.Chung, E.Klempt and J G Korner, hep-ph/0211100; S.U.Chung and E.Klempt,
hep-ph/0306018
\bibitem{indiana}  A.~R.~Dzierba {\it et al.}, Phys.\ Rev.\ D {\bf 67} (2003) 094015
\bibitem{fc04ichep} F.E.Close, Rapporteur talk at ICHEP04
\bibitem{klmix} M.Karliner and H.J.Lipkin, Phys.Lett.B {\bf 586} 303, 2004 
\bibitem{jjdFermilab} J.J.Dudek talk at the First Meeting of the APS Topical Group on Hadronic Physics, Fermilab November 2004 (Proceedings to appear in J.Phys.G)
\bibitem{fcdudekls} F.E.Close and J.J.Dudek, Phys.Lett.B {\bf 583}
  278, 2004 
\bibitem{oldexpt} W.P.Dodd, T.Joldersma, R.B.Palmer and N.P.Samios, Phys.\ Rev.\  {\bf 177} (1969) %%@
1991.
\bibitem{cd2} F.E.Close and J J Dudek, hep-ph/0401192
\bibitem{fcqz03} F.E.Close and Qiang Zhao, hep-ph/0404075 
\bibitem{lattice}   
P. Lacock, C. Michael, P. Boyle and P. Rowland, Phys.Lett. B {\bf 401}, 308   
(1997);   
C. Bernard {\it et al.}, Phys.Rev. D {\bf 56}, 7039 (1997);   
P. Lacock and K. Schilling, Nucl. Phys. Proc. Suppl. {\bf 73}, 261 (1999);   
C. McNeile, hep-lat/9904013;   
C. Morningstar, Nucl. Phys. Proc. Suppl. {\bf 90}, 214 (2000)   
%\bibitem{experiments} 1440/1600 experiments
\bibitem{lee} B W Lee, S Okubo and J Schecter Phys Rev. 135 (1964) B219
\bibitem{fclipkin1} F.E.Close and H.J.Lipkin,Phys Lett 196 (1987) 245;




%%CITATION = HEP-EX 0304002;%%








%symmetry paper


%\bibitem{bag} bag, fluxtube
%\bibitem{lattice} lattice

%\bibitem{bcm} T Barnes, F E Close and S Monaghan
%\bibitem{gi} S.Godfrey and N.Isgur, Phys.Rev. D{\bf 32}, 189 (1985)
\bibitem{cdk}   
F.E. Close, A. Donnachie and Yu.S. Kalashnikova, Phys.Rev. D {\bf 65},   
092003 (2002)  
  
\end{thebibliography}