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\line{}
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\Fourteenpoint 

{\centerline{\bf The production model of Ishida {\it et al.}}}
{\centerline{\bf and unitarity}}
\vskip 1cm
%\Fourteenpoint

{\centerline{M.R. Pennington}}
\vskip 1mm

\Twelvepoint  
\centerline{Centre for Particle Theory,
University of Durham,}

\centerline{Durham DH1 3LE, U.K.}
\vskip 1cm
%\baselineskip=14 pt
\Twelvepoint
\centerline{ABSTRACT}
\vskip 5mm
\baselineskip=5mm   
\parskip=0mm  
{\leftskip 1.5cm\rightskip 1.5cm{\noindent The relation between scattering and
production amplitudes imposed by unitarity and analyticity, recently
criticised by Ishida {\it et al.}~$^{1),2)}$, is explained.}\par}
\vskip 1cm


\Twelvepoint
\baselineskip=7mm
\parskip=2mm
In a contribution to the Seventh International Conference on Hadron Spectroscopy
at Brookhaven,
Ishida { \it et al.}~$^{1),2)}$ have questioned one of the orthodox methods of 
implementing the final state interaction theorem
in production processes.  Here we show that this criticism is incorrect, being based on a misunderstanding of the method.
The case considered by Ishida {\it et al.}~$^{1),2),3)}$ is particularly simple, it is that of
a single channel,
for instance $\pi\pi\to\pi\pi$. It is then well-known that each unitary
 partial wave amplitude  can, for real $s$ ---
the square of the c.m. energy, be represented by
$${\cal T}(s)\,=\, {K(s)\over{1-i\rho(s) K(s)}}\quad ,\eqno(1)$$
where $\rho(s)$ is the standard phase-space factor of $\, 2p/\sqrt{s}\,$ ($p$ being the
c.m. 3-momentum) and $K(s)$ is real.  Importantly,
$K(s)$ embodies any real zeros of the amplitude ${\cal T}(s)$.

Now Watson's final state interaction theorem~$^{4)}$ requires that any other (non-strongly interacting)
process producing the same final state must have its corresponding partial wave
${\cal F}(s)$ having the same phase.
To implement this within the $K$--matrix formalism, Aitchison~$^{5)}$ proposed representing
${\cal F}(s)$ by
$${\cal F}(s)\,=\,{P(s)\over{1-i\rho(s) K(s)}}\quad ,\eqno(2)$$
where the function $P(s)$, like $K(s)$, is real for real values of $s$. 
The complex denominator, $1-i\rho K$, not only ensures the production
amplitude has the same phase as the elastic one, Eq.~(1) for $s$ real, but also
ensures  that physical states, which are poles in the complex $s$--plane on 
the nearby unphysical sheets, transmit from one process to another
 through this universal denominator.
 
  It is common practice to parameterise the $K$--matrix in
terms of poles.
However, these introduce artificial zeros, Eq.~(2), in the production amplitude, unless the function $P(s)$ has the very same poles.  A simple method of implementing
this constraint has been proposed by AMP~$^{6)}$.  This is to express the $P$--vector as
$$P(s)\,=\,\alpha(s)\, {\hat K}(s)\quad ,\eqno(3)$$ 
where ${\hat K}$ is the reduced $K$--matrix, which contains the poles of $K(s)$,
but with its zeros divided out.  In this simple
representation
$${\cal F}(s)\,=\,\alpha(s)\, {\hat{\cal T}}(s) \eqno(4)$$
with ${\hat{\cal T}}(s)$ being the $T$-matrix with its zeros removed~$^{6)}$.
In general, then analyticity requires $\alpha(s)$ to be a {\it smooth} function for
$s > s_{threshold}$~$^{6),7)}$.  In physical $\pi\pi$ scattering, the only such zeros
to divide out are the Adler zeros below threshold for the $S$--waves
 and the kinematic zeros at threshold for higher partial waves.
 Such zeros are divided out, since zeros of the amplitudes
${\cal T}(s)$ will not, in general, transmit to production amplitudes, ${\cal F}(s)$, though of course poles do.

Ishida {\it et al.}~$^{2)}$ construct a simple one-channel model in which
the $K$--matrix has two poles, so that
$$K(s)\,=\, {g_1^2\over{s-m_1^2}}\,+\,{g_2^2\over{s-m_2^2}}\; .\eqno(5)$$
Clearly, this example has a zero at $s=s_0$ between $s=m_1^2$ and $s=m_2^2$, where
$$s_0\,=\,\left(g_1^2 m_2^2 + g_2^2 m_1^2\right)/(g_1^2 + g_2^2)\quad .\eqno(6)$$
This zero in ${\cal T}$ will not, in general, occur in 
production processes.  Hence, it is necessary to define a reduced 
${\hat{\cal T}}$ or equivalently
${\hat K}$--matrix.  Then any production process can be expressed as
$${\cal F}(s)\,=\,\alpha(s)\, {{\hat K(s)}\over {1-i\rho(s) K(s)}}\quad ,\eqno(7) $$
where
$${\hat K}(s)\,=\,{(g_1^2+g_2^2)\over{(s-m_1^2)(s-m_2^2)}}\eqno(8)$$
and $\alpha(s)$ having only a left hand cut is expected to be a smooth function
for $s > s_{threshold}$.  If the reduced ${\hat K}$--matrix is not used, but instead $K$ itself, as in the example of
Ishida {\it et al.}~$^{2)}$, a spurious zero transmits to the production
process, unphysically shackling its description.  

This discussion is readily generalised to $n$ coupled channels, when ${\cal T}$
and $K$ are $n\times n$ matrices and ${\cal F}$ and $\alpha$ are 
$n$--component vectors. However, when there is only one channel, the reason for introducing the
reduced ${\hat K}$--matrix is particularly transparent. The hadronic amplitude ${\cal T}$ can be written
in terms of the phase-shift $\delta$ as
$${\cal T}(s)\,=\,{1\over{\rho}}\, \sin \delta\,e^{i\delta}\quad. \eqno(9)$$
The $K$--matrix element is then $\tan\delta/\rho$.  Clearly, the
$K$--matrix has poles when $\delta = (2n+1)\pi/2$ (with $n=1,2,...$)
and the amplitude ${\cal T}$ has zeros when $\delta = n\pi$ (again with $n$ an integer).
In terms of the phase-shift, the production amplitude
${\cal F}$ of Eq.~(2) becomes 
$${\cal F}(s)\;=\;{1\over \rho}\, P \cos \delta \,e^{i\delta}\quad . \eqno(10)$$
It is then obvious that unless $P(s)$ has the poles of the $K$--matrix,
${\cal F}(s)$ will be zero exactly where resonances are expected to show up,
 {\it i.e.}  when $\delta = (2n+1)\pi/2$ making
$\cos \delta =0$. Choosing
$P(s)$ to be simply proportional to $K(s)$ replaces $\cos \delta$ in Eq.~(10) by 
$\sin \delta$. However, then  the zeros of ${\cal T}$,
Eq.~(9),
at $\delta = n\pi$, unnecessarily  transmit to the
production amplitude.  Thus these zeros must be removed by defining
the reduced $K$--matrix, ${\hat K}$, as
$${\hat K}(s)\;=\;K(s)/\prod_n (s-s_n)\; ,\eqno(11)$$
where $\delta(s_n)\,=\,n\pi$,  so that Eq.~(7) follows.
Then Eq.~(3) relates $P(s)$ to $\alpha(s)$, where analyticity requires $\alpha(s)$ to be smooth.


In the example of Ishida {\it et al.}, the phase-shift $\delta = \pi$ at
$s=s_0$ of Eq.~(6) and it is essential that this zero is divided out
before constructing the production amplitude, as in Eqs.~(7,8).
In the case of physical $\pi\pi$ scattering,
inelasticity has set in before any phase-shift reaches $\pi$, consequently this example has no relevance
beyond this model of Ishida {\it et al.}.  However, for physical
$\pi\pi$ scattering the determinant of the $T$--matrix does vanish close to
$K{\overline K}$ threshold and defining a reduced ${\hat K}$--matrix eliminates this zero.
This is the multi-channel generalization of the above example discussed 
in Ref.~7.
%\vfil\eject
\vskip 2.5cm

\centerline{\bf References}
\vskip 5mm

\item{   1.  } M.Y. Ishida, S. Ishida and T. Ishida, {\it Relation between scattering
and production amplitudes --- case of intermediate $\sigma$--particle
 in $\pi\pi$ system},
contribution to Hadron'97, Brookhaven,  August 1997.

\item{   2.  } T. Ishida, K. Takamatsu, T. Tsuru, M.Y. Ishida and S. Ishida,
{\it $\sigma$--particle in production processes}, 
contribution to Hadron'97, Brookhaven,  August 1997.


\item{   3.  } S. Ishida, M.Y. Ishida, H. Takahashi, T. Ishida,
K. Takamatsu and T. Tsuru, Prog. Theor. Phys. {\bf 95} (1996), 745.
 
\item{   4.  } K.M. Watson, Phys. Rev. {\bf 88} (1952), 1163.

\item{   5.  } I.J.R. Aitchison, Nucl. Phys. {\bf A189} (1972), 417.

\item{   6.  } K.L. Au, D. Morgan and M.R. Pennington, Phys. Rev. {\bf D35}
(1987), 1633.

\item{   7.  } D. Morgan and M.R. Pennington, Z. Phys. {\bf C48} (1990), 623.




\bye

