 poles in the Mandelstam
variables $s$, $u$, and $t$, and class B is the non-polar contact-type current
$M^\mu_{\text{int}}$ originating from the interaction of the photon with the interior of
the vertex. The full current $M^\mu$, therefore, can be written as
\begin{equation}
  M^\mu = M^\mu_s+M^\mu_u+M^\mu_t +M^\mu_{\text{int}}~,
  \label{eq:Msuti}
\end{equation}
as indicated in Fig.~\ref{fig:Msutc}.
This structure is based on topology alone and therefore independent of the details of
the individual current contributions.


These details matter, of course, if one wishes to derive the currents for
practical applications. In general, an electromagnetic current for a hadronic
process is defined by first employing minimal substitution for the connected
part of the hadronic Green's functions and then taking the functional
derivative with respect to the electromagnetic four-potential $A^\mu$, in the
limit of vanishing $A^\mu$. The current is then obtained by removing the
propagators of the external hadron legs from this derivative in an LSZ
reduction procedure~\cite{LSZ55}. The gauge-derivative procedure of
Ref.~\cite{Haberzettl97} provides a formally equivalent method that is much
simpler to handle in practice because it essentially amounts to the simple
recipe of attaching a photon line to any topologically distinct feature of a
hadronic process expressed in terms of Feynman diagrams and summing up the
corresponding contributions to obtain the full current.


For the single-pion photoproduction process at hand, the connected part of the
free $\pi NN$ Green's function is given by $G_0 F S$, where $F$ is the $\pi NN$
vertex shown on the left-hand side of Fig.~\ref{fig:Msutc}, $S$ is the
propagator of the incoming nucleon leg and
\begin{equation}
  G_0 = S(p_N^{}) \circ \Delta(q_\pi^{})
  \label{eq:G0=SoDelta}
\end{equation}
is the product of the outgoing nucleon and pion propagators $S$ and $\Delta$,
respectively, written here with generic momenta for the nucleon and the pion,
where the sum of the respective four-momenta $p_N^{} + q_\pi^{}$ is the fixed available
total momentum. Within a loop integration, this free $\pi N$ propagator would
correspond to a convolution integration of these momenta, as indicated by
``$\circ$'' here. The LSZ expression for the photoproduction current may now be
written as~\cite{Haberzettl97}
\begin{equation}
  M^\mu = -G_0^{-1}\left\{G_0 F S\right\}^\mu S^{-1}~,
  \label{eq:Mcurrentdefined}
\end{equation}
where $\{\cdots\}^\mu$ is the short-hand notation for the gauge derivative
introduced in Ref.~\cite{Haberzettl97}, with $\mu$ indicating the Lorentz index
of the incoming photon. Being a derivative, the product rule applies and we
obtain
\begin{align}
  M^\mu   &=-G_0^{-1}\left\{G_0\right\}^\mu F
  -\left\{F \right\}^\mu -  F\left\{ S\right\}^\mu S^{-1}
  \nonumber\\
  &= d^\mu G_0 F +  M^\mu_{\text{int}} + FSJ_N^\mu~,
  \label{eq:MmuFT}
\end{align}
where in the last step
\begin{subequations}\label{eq:GDdetail}
\begin{align}
-\left\{ S\right\}^\mu &= S J_N^\mu S~,
\\
  -\left\{F\right\}^\mu &= M^\mu_{\text{int}}~, \label{eq:MintGDeriv}
\\
  -\left\{G_0\right\}^\mu &=G_0 d^\mu G_0~,
  \label{eq:dmu}
\end{align}
\end{subequations}
were used~\cite{Haberzettl97}, which relate the corresponding gauge derivatives
to the nucleon current operator $J^\mu_N$ of the incoming nucleon, the
interaction current $M^\mu_{\text{int}}$ for the interior of the vertex $F$, and the
dual-current contribution of the free $\pi N$ system,
\begin{align}
  G_0d^\mu G_0 &= S \circ (\Delta J^\mu_\pi\Delta)+(SJ^\mu_NS) \circ \Delta   ~,
   \label{eq:G0dmuG0}
\end{align}
as depicted in Fig.~\ref{fig:dmu}, which sum up attaching the photon to $G_0$
in terms of the corresponding nucleon current $J^\mu_N$ and the pion current $J_\pi^\mu$.
The three polar currents in Eq.~(\ref{eq:Msuti}) obviously are given here by
\begin{subequations} \label{eq:Msut}
\begin{align}
  d^\mu G_0 F &= M^\mu_t + M^\mu_u~,
\\
  FSJ_N^\mu &= M^\mu_s~,
\end{align}
\end{subequations}
which completes matching the field-theoretical result of Eq.~(\ref{eq:MmuFT})
with the topological one in Eq.~(\ref{eq:Msuti}).


\begin{figure}[!t]
\centering
  \includegraphics[width=.8\columnwidth]{fig2.pdf}
\caption{\label{fig:dmu}%
Graphical representation of the dual-current contribution $G_0d^\mu G_0$ of
Eq.~(\ref{eq:G0dmuG0}) for the photon being coupled to the free $\pi N$
propagator $G_0=S(p_N)\circ \Delta(q_\pi)$ of Eq.~(\ref{eq:G0=SoDelta}).}
\end{figure}


Note here that with the external momenta of the photo-process given as in
Fig.~\ref{fig:Msutc}, namely,
\begin{equation}
  \gamma(k) + N(p) \to \pi(q)+N(p')~,
\end{equation}
it was not necessary to write out the momentum dependence of any of the
elements of the preceding equations because it can easily be found explicitly
by knowing that the photon carries a momentum $k$ into the element to which it
is attached.


\subsection{Gauge invariance}


Gauge invariance as the manifestation of $U(1)$ symmetry is of fundamental
importance for any photo-process because it provides a conserved (on-shell)
current and thus implies charge conservation. The requirement of \emph{local\/}
gauge invariance~\cite{PS}, in particular, implies the very existence of the
electromagnetic field and thus is of fundamental importance for the formulation
of \emph{consistent\/} reaction dynamics of photo-processes, which goes beyond
the mere on-shell constraint of charge conservation.


For single-pion photoproduction, local gauge invariance is formulated in terms
of the generalized Ward-Takahashi identity (WTI)~\cite{Haberzettl97,Kazes59}
\begin{eqnarray}
  k_\mu M^\mu &=& S^{-1}(p') Q_{N_f} S(p'-k) F_u
  \nonumber\\   &&\mbox{}
   +\Delta^{-1}(q) Q_\pi \Delta (q-k) F_t
     \nonumber\\   &&\mbox{}
    -F_s S(p+k)Q_{N_i} S^{-1}(p)~,
   \label{eq:gWTIpiN}
\end{eqnarray}
where the four-momenta are those shown in Fig.~\ref{fig:Msutc} and the vertices
$F_x$ are the $\pi NN$ vertex functions in the specific kinematic situations
described by the Mandelstam variables $x=s,u,t$ in the figure. The charge
operators for the initial and final nucleons are represented by $Q_{N_i}$ and
$Q_{N_f}$, respectively, and $Q_\pi$ is the charge operator for the outgoing
pion. The inverse propagators here ensure that this four-divergence vanishes
for matrix elements with all hadron legs on-shell and thus provides a conserved
current. The generalized  WTI as such, however, is an \emph{off-shell\/}
constraint, thus providing a continuous dynamical link between the transverse
and longitudinal regimes. This is analogous to the usual single-particle
Ward-Takah