\section{Introduction}

Four amplitudes are considered to contribute to the one photon exchange Born
approximation model where a single photon $\gamma^*$ is assumed to be
emitted by the scattered electron and couple to the hadronic system:
pion-pole, nucleon-pole, crossed nucleon-pole and contact term.  The first
three amplitudes correspond to Mandelstam $t$, $s$ and $u$-channel processes,
respectively.  The contact term is used when gauge invariance needs to be
restored. The charged pion form factor, $F_{\pi}$, is dominated by $t$-channel
processes, namely the quasielastic scattering of a virtual pion from a photon.
Born amplitude based models ~\cite{Ber70,Gut72} indicate that for values of $W$
above the resonance region and for small values of $-t$ (close to the pion
mass), the longitudinal part $\sigma_{L}$ of the total cross section for pion
electroproduction is dominated by the $t$-channel process.  In forward
kinematics (low $-t$), the other amplitudes (transverse response function
$\sigma_{T}$ and interference terms $\sigma_{LT}$ and $\sigma_{TT}$) are small.
At values of $-t$ approaching the pion mass (so called $t$-pole), the
longitudinal response function becomes directly proportional to the charged
pion form factor $F_{\pi}$
\begin{equation}
\sigma_{L} \approx \frac{-t Q^2}{(t-M^2_{\pi})^2} g^2_{\pi NN}(t)
F^2_{\pi}(Q^2, t).
\end{equation}
Here, the factor $g_{\pi NN}(t)$ comes from the $\pi NN$ vertex and represents
the probability amplitude to have a virtual $\pi^{+}$ meson inside the proton
at a given $t$.

In order to reliably extract $F_{\pi}$, the $t$-pole process should be dominant
in the kinematic region under study.  This dominance can be studied experimentally
through the ratio of longitudinal $\gamma^{*}_L n \to \pi^{-} p$ and
$\gamma^{*}_L p \to \pi^{+} n$ cross sections and further in terms of
contributions from isoscalar $A_S$ and isovector $A_V$ photon amplitudes:
\begin{equation}
R_L = \frac{\gamma_L^{*}n \to \pi^{-} p}{\gamma_L^{*}p \to \pi^{+} n} =
\frac{|A_{V}-A_{S}|^2}{|A_{V}+A_{S}|^2}.
\end{equation}
As the $t$-channel reaction proceeds purely via isovector amplitudes, a
significant deviation of $R_L$ from unity would indicate the presence of
isoscalar processes.  Where the $t$-pole dominates (small $-t$), the ratio
$R_L$ should be close to unity.

%==============================================================================
\begin{figure}[ht]
\begin{center}
\vskip 0.5cm
\includegraphics[angle=0,width=0.6\textwidth]{eps/brauel.eps}
\vskip -0.5cm
\caption[Exclusive $\pi^{-}/\pi^{+}$ ratio versus $-t$.]{\label{fig:Brauel79}
Exclusive $\pi^{-}/\pi^{+}$ upseparated cross section ratio plotted versus $-t$
from Ref.~\cite{Brauel79}. The points depicted by squares represent data taken
at $W=2.19 GeV$ and $Q^2=0.70$ GeV$^2$ while the triangles points represent
data taken at $W=2.19 GeV$ and $Q^2=1.35$ GeV$^2$.}
\end{center}
\end{figure} 
%==============================================================================

Regarding the transverse ratio $R_T = \frac{\gamma_T^{*}n \to \pi^{-}p}
{\gamma_T^{*}p \to \pi^{+} n}$, at low $-t$ the photon couples to the charge
of the pion and so the ratio should approach unity.  With increasing $-t$, the
photon starts to probe quarks rather than pions. In the limit of large $-t$,
$R_T$ is expected to approach 1/4 given by the square of the ratio of the quark
charges involved:
\begin{equation}
R_{T}=\frac{2Q_{d}^{2}}{2Q_{u}^{2}}= \frac{(-1/3)^2}{(+2/3)^2} = 1/4
\end{equation} 
The only previous experimental data on exclusive pion production ratios from
deuterium were obtained by P. Brauel {\it et al.} ~\cite{Brauel79} at DESY.
Although these ratios were computed from unseparated cross section data, they
support the assumption that at low $-t$ $R$ approaches 1, while at higher
$-t$ the data are consistent with 1/4 (see Fig. \ref{fig:Brauel79}).  The data
also suggest a $Q^2$ dependence in the slope of $R$ versus $-t$.

We report the results of a new study where for the first time ratios of
separated response functions ($\sigma_{L}$, $\sigma_{T}$, $\sigma_{LT}$, and
$\sigma_{TT}$) were extracted at several kinematic settings: $Q^2 = 0.6, 1.0,
1.6$ GeV$^2$ and $W=1.95$ GeV, and $Q^2=2.45$ GeV$^2$ and $W=2.22$ GeV,
respectively.
