\section{Polarization requirements}

From the point of view of physics of \hd the main difference between
photon and hadron beams has to do with their quark content. With a good
accuracy, photons may be viewed as a superposition of spin-1 
$q\bar q$-pairs while a typical hadronic beam of pions 
or kaons is a source of  valence spin-0 $q\bar q$ pairs. Thus, as compared
to hadroproduction, photons may lead to different final states or 
  similar final states may be reached via different production
 mechanisms. This  unique quark structure of the photon might be of 
 particular relevance for production of exotic states. This is because
 for low lying exotics theory predicts their valence 
 $q\bar q$ pair to be in the spin-1 
configuration~\cite{IP}.

 The basic  mechanisms of
 meson photoproduction are through diffraction, $t$-channel meson
 exchange and a decay of an excited baryonic resonance. From these
 three,  at \hd energies  $t$-channel exchanges dominate. 
 The hierarchy of exchanges suppresses those which require a change of
  several quantum numbers at once. Thus from the possible $t$-channel
 exchange the Pomeron exchange followed by a 
 charge exchange are expected to dominate. Nevertheless at energies 
 available at \hd an extensive coverage of various production
 mechanisms is 
expected~\cite{Barnes8Plus}.
Real photons have two, transverse degrees of polarization and
therefore are ``richer'' than $\pi$ or $K$ beams. This is reflected in
the number of independent spin-density matrix elements. For example
 in  photoproduction of vector mesons on an unpolarized
 proton target, $\gamma p \to V N$ there are $17$
  independent real spin-density matrix elements. For comparison, 
 an  analogous reaction with pion beams,  $\pi p \to V N$ 
 has only 4 
 independent real parameters in the  spin-density matrix. 
 Dynamical content of photoproduction is clearly larger. 
  The crucial point however is how many of the spin-density matrix
  elements can actually be measured. In general this number depends on
 the type of  photon polarization.  
 In the above example 
  if the vector meson is analyzed 
  via its two body, (pseudo)scalar decay, {\it e.g.}
    $V=\rho\to 2\pi$ or $V=\phi \to K\bar K$ then linearly, circularly,
    and unpolarized photons access 
  9, 5, and 3 out of the 17 spin matrix elements, 
respectively~\cite{Schilling}. 
 In general linear polarization
  enables to extract the largest number of the spin-density matrix
  elements. 

 When $t$-channel exchanges dominate, linear
  polarization is even more important since it enables to isolate 
  natural from unnatural parity exchanges. Naturality, $\tau$ of a state with
  intrinsic parity, $\eta$ and spin $J$   is
  defined as a product $\tau = \eta(-)^{J}$. Thus, for example the 
$\pi(J^{PC}=0^{-+})$ and $a_{1}(J^{PC}=1^{++})$ are unnatural, while
  the   $\rho(J^{PC}=1^{--})$ or the exotic $\hat\rho(J^{PC}=1^{-+})$ are
  natural. Measurement of naturality of the exchanged particle
  together with its $t$-dependence and partial wave analysis are
  central for identification of produced mesonic resonances and
  understanding of production mechanisms. 
 To see the connection between photon polarization and naturality in
  the $t$-channel  
 consider a reaction 
  $a+b \to c+d$ where $a,b$ refer to the beam and target particle 
 respectively and  $c,d$ to the produced meson and the recoiling
  baryon. We will assume that 
  $t$-channel exchange of a particle, $e$ or more generally of a
  Reggion dominates.
 This type of processes are best analyzed in the Gottfried-Jackson frame
  defined as  the rest frame of the produced meson, 
 (particle $c$) with the spin quantization, $z$-axis chosen in the
  direction of the beam (particle $a$) in that frame. 
 Consider the case of  pion beam first. It is possible to show 
  that symmetry of the corresponding production amplitude
  $A(m_c) = 
 A_{a+b\to c+d}$ under interchange $m_c  \to -m_c$,  where  $m_c$ is the spin
  projection of the produced meson 
 (target and the recoiling baryon spins are
  implicit since they
  are irrelevant in the following discussion) 
  is  determined  by the 
 product of naturalities of the exchanged particle
 ($\tau_e$) and that of the produced meson ($\tau_c$). Thus, for a 
 final state of given naturality, (say $\tau_c=+1$) 
 determined by a partial wave
 $A(m_c)$ the two combinations 

\begin{equation}
A^{U,N}(m_c) = {1\over 2}(A(m_c) \pm A(-m_c) )
\end{equation}
 isolate processes of opposite naturalities in the $t$-channel (if
 $\tau_c=-1$, $A^U$ and $A^N$ should be exchanged). 
 For a photon beam an analogous partial wave, 
 $A(m_a,m_c)=A_{a+b\to c+d}$ depends 
 on the photon polarization, $m_a=\pm 1$. In this case when defining 
 amplitudes which project onto definite
 naturality of the  exchanged particle will also require flipping photon
 polarization 
(which is
 identical to photon helicity in the GJ frame). For $\tau_c=+1$ the
 corresponding  two amplitudes are 

\begin{equation}
A^{N,U}(m_c) = {1\over 2}(A(1,m_c) \pm A(-1,-m_c))
\end{equation}
with the definitions being reversed for $\tau_c=-1$. 
It thus follows that amplitudes with defined photon helicity {\it
 i.e.} corresponding  to circularly polarized beams 
 mix exchanges of different naturalities, 

\begin{equation}
A(\pm 1,\pm m_c) = (A^N(m_c) \pm A^U(m_c) ), 
\end{equation}
 and to separate naturality of the exchanged particle it is
necessary to measure coherent superpositions of the two helicity
amplitudes. This can be achieved using linear 
polarization~\cite{Schilling}.

 Finally lets focus on photoproduction of a $\hat\rho (J^{PC}=1^{-+})$ 
exotic in a
 charge exchange reaction, 
$\gamma p \to \hat\rho n$~\cite{AfanasevPage}.
As discussed earlier, at
 \hd energies one would expect diffractive photoproduction to
 dominate the total $\gamma p$ cross section. However, since diffraction
 corresponds to a $C=+$ exchange the $1^{-+}$ exotic would not be
 produced this way. Charge exchange is the next, least suppressed 
  production mechanism, and since 
  $\gamma\hat\rho\pi$ coupling is 
 not ruled out by any symmetry, at low $t$ pion exchange is expected to
 dominate. This is good news since OPE is reasonably well understood
 at high energies. In this case linear polarization has another
  advantage. One can show that in the GJ frame 
  the photoproduction amplitude $A(m_a,m_c)$ is proportional to either
 $\delta_{m_a m_c}$ or 
 $\tau^3_{m_a m_c}$ depending on the naturality of
 the produced 
meson~\cite{AfanasevSzczepaniak}.
Thus in this 
  case linear polarization selects naturality of the produced mesonic
 resonance. For a {\it natural} exotic, $1^{-+}$ state with mass around 
 1.5-1.6 ${\rm{GeV/c^2}}$, as seen in the E852 data~\cite{e852rhopi}, 
the  
 dominant backgrounds are
  expected from the {\it unnatural} $a_1(1260)$ and $\pi_2(1670)$ 
($a_2(1320)$ is too narrow to significantly contribute in this mass
 region). Thus direction of linear polarization of the exotic signal
 is then expected to be at $90^o$ with respect to the backgrounds. 
  This has obvious implications for the partial wave analysis. 

