Z\pi$. We then have
\begin{equation}
T_{fi} = T_{fi}^{\psi^\prime}+T_{fi}^{J/\psi} = \left(1 +
\frac{g_{ZJ/\psi\pi}}{g_{Z\psi^\prime\pi}}
\frac{f_{\psi^\prime}}{f_{J/\psi}} e^{i\theta} \right)
T_{fi}^{\psi^\prime}
\end{equation}
where we have already set the form factors
\begin{equation}
F_{ZJ/\psi \pi}(q^2) = F_{Z\psi^\prime\pi}(q^2)
\end{equation}
for simplicity, and $e^{i\theta}$ is the relative phase between the
two transition matrix element. Couplings $g_{ZJ/\psi\pi}$ and
$f_{J/\psi}$ can be determined by Eq.~(\ref{gamma-z}) and
Eq.~(\ref{vmd}), respectively. The cross section becomes
\begin{equation}
\sigma^{J/\psi+\psi^\prime}_{\gamma p\to Z^+ n} \simeq
(1+0.75^2x+1.5\sqrt{x} \cos{\theta})\sigma^{\psi^\prime}_{\gamma
p\to Z^+ n} \.
\end{equation}
The worst destructive situation is at $\theta=\pi$. But the
destructive effects will depend on the value of $x$. Note that Belle
has not seen $Z\to J/\psi\pi$ decay, which allows an estimate of
$x\leq 0.1$, and with which the total cross section would be still
sizeable.



If the $Z$ meson is a genuine resonance, the threshold enhancement
will be a signature for its existence. This can be further clarified
by the study of background effects in $\gamma p\to \psi^\prime \pi^+
n$ in the next section.





\section{Background analysis}

Experimentally, the final state particles identified are
$\psi^\prime$, $\pi$ and nucleon. As a result, background
contributions to the  $\psi^\prime\pi N$ channel can interfere with
the transitions we are interested in. As discussed earlier, one of
the major contributions is due to the diffractive production of
$\psi^\prime$ (Fig.~\ref{fig:2}). In this section, we combine
$\gamma p\to Z^+ n\to \psi^\prime \pi^+ n$ and $\gamma p\to
\psi^\prime p\to \psi^\prime \pi^+ n$ together to make an estimate
of the signal and background contributions. This could be useful for
experimental search for the $Z^+$ in this channel. Since the
production rate for all these three spin-parities are similar, we
only consider $J^P=1^-$ in the following analysis.

The schematic transition diagrams are illustrated in
Fig.~\ref{fig:2}. We first extend the formulae of the previous
section to $\gamma p\to Z^+ n\to \psi^\prime \pi^+ n$, and then
include the diffractive contributions from the Pomeron exchanges.






\subsection{Meson exchange terms}



By including the vertex coupling for $Z^+\to \psi^\prime\pi^+$, the
transition amplitude for pion exchange can be explicitly written
down
 \begin{eqnarray}
 T^{\mathcal{M}}_{1fi} &=& \sum\limits_{pol} \Gamma_{\gamma
\psi^\prime}\Gamma_{Z\psi^\prime\pi}^1\Gamma_{Z\psi^\prime\pi}^2
\Gamma_{pn\pi} \times  F_{Z\psi^\prime\pi}(q_1^2)F_{\pi
 NN}(q_1^2)F_{Z\psi^\prime\pi}(q_2^2)\nonumber \\
&=& -i\left[ \sqrt{2}g_{\pi NN} \frac{e}{f_\psi^\prime} \left(
\frac{g_{Z\psi^\prime\pi}}{M_Z} \right)^2 \right] \bar{u}(p_3)
\gamma_5 u(p_1) \epsilon_\mu(k)\epsilon_{\psi^\prime}^{\nu}(q)
\varepsilon_{\mu_1\nu_1\alpha_1\beta_1}k^{\mu_1}q_2^{\alpha_1}
\varepsilon_{\mu_2\nu\alpha_2\beta_2}q^{\mu_2}q_2^{\alpha_2}\nonumber \\
&\times&
\frac{g^{\mu\nu_1}g^{\beta_1\beta_2}}{(q_1^2-m_\pi^2)(q_2^2-M_Z^2+iM_Z\Gamma)}
F_{Z\psi^\prime\pi}(q_1^2)F_{\pi
 NN}(q_1^2)F_{Z\psi^\prime\pi}(q_2^2) \,
 \end{eqnarray}
where the vertex functions are
 \begin{eqnarray}
 \Gamma_{\gamma \psi^\prime} &=&
 \frac{eM_{\psi^\prime}^2}{f_{\psi^\prime}}\epsilon_\mu(k)\epsilon_{\psi^\prime}^{\mu}(k)\\
 \Gamma_{Z\psi^\prime\pi}^1 &=&
 \frac{g_{Z\psi^\prime\pi}}{M_Z}\varepsilon_{\mu_1\nu_1\alpha_1\beta_1}k^{\mu_1}\epsilon_{\psi^\prime
^{\nu_1}(k)
 q_2^{\alpha_1}\epsilon_Z^{\beta_1}(q_2) \\
\Gamma_{Z\psi^\prime\pi}^2 &=&
 \frac{g_{Z\psi^\prime\pi}}{M_Z}\varepsilon_{\mu_2\nu\alpha_2\beta_2}q^{\mu_2}\epsilon_{\psi^\prime}^
\nu}(q)
 q_2^{\alpha_2}\epsilon_Z^{\beta_2}(q_2) \\
 \Gamma_{pn\pi} &=& -i\sqrt{2}g_{\pi NN}\bar{u}(p_3)\gamma_5
 u(p_1) \,
 \end{eqnarray}
 and the form factor is
 \begin{equation}
F_{Z\psi^\prime\pi}(q_2^2) =\frac{\Lambda_Z^2-M_Z^2}{\Lambda_Z^2 -
q_2^2}.
 \end{equation}
Considering a moderate modification of the cutoff will not change
the results significantly, we take $\Lambda_Z=M_{\psi^\prime}$ as an
input value. The kinematic variables are denoted in
Fig.~\ref{fig:2}.



\subsection{Pomeron exchange term}

Pomeron as a phenomenology accounting for the diffractive
transitions has been widely studied in the
literature~\cite{donnachie,pichowsky,laget}. In this approach, the
Pomeron mediates the long range interaction between a confined quark
and a nucleon, and has been shown to behave just like a $C=+1$
isoscalar photon.



The Pomeron-nucleon coupling is determined by the vertex:
\begin{equation}
F_\mu(t)=3\beta_0\gamma_\mu f(t)
\end{equation}
where $t$ is the Pomeron momentum squared, $\beta_0$ represents the
coupling constants between a single Pomeron and a light constituent
quark, and $f(t)$ is the isoscalar nucleon electromagnetic form
factor:
\begin{equation}
f(t)=\frac{4M_N^2-2.8t}{(4M_N^2-t)(1-t/0.7)^2} \.
\end{equation}
For the $\gamma V \mathcal{P}$ vertex ($V$ represents the
corresponding vector meson, here is $\psi^\prime$), we take the
on-shell approximation for restoring gauge invariance
as~\cite{zhaoq}. Therefore the equivalent vertex for the $\gamma
\psi^\prime \mathcal{P}$ is reduced to a loop tensor as follows:
\begin{equation}
\frac{2\beta_c \times 4\mu_0^2}{(M_{\psi^\prime}^2
-t)(2\mu_0^2+M_{\psi^\prime}^2 -t)} T^{\mu\alpha\nu} \,
\end{equation}
where
\begin{equation}
T^{\mu\alpha\nu} = (k+q)^\alpha g^{\mu\nu} -2k^\nu g^{\alpha\mu}
\end{equation}
and $\beta_c$ is the effective coupling constant between Pomeron and
a charm quark within $\psi^\prime$; $\mu_0$ is a cutoff of the form
factor related to the Pomeron.


The amplitudes for Fig.~\ref{fig:2}(a)