Detection of resonance particles why we detect an increment in the number of events of the reactions for the energy of resonance particles?
I read that cross section of this new unstable particle must be added to the cross section of the target particle but  i cannot understand the reason. A little example of concepts i cannot understand, we have a process like
\begin{equation}\mathrm{e}^{-}+\mathrm{p} \longrightarrow \mathrm{e}^{-}+\Delta^{+}\end{equation}
where after $\approx 10^{-23}$ s we have
\begin{equation}\mathrm{e}^{-}+\mathrm{p} \longrightarrow \mathrm{e}^{-}+\pi^{+}+\mathrm{n}\end{equation}
via
\begin{equation}\Delta^{+} \longrightarrow \pi^{+}+\mathrm{n}\end{equation}
the compatible collisions must satisfy
\begin{equation}m_{\Delta} c^{2}=\sqrt{\left(E_{\pi}+E_{\mathrm{n}}\right)^{2}-\left(\mathbf{p}_{\pi}+\mathbf{p}_{\mathrm{n}}\right)^{2} c^{2}}\end{equation}
so if we detect and increment in the number of events in such collisions we can say that we have a resonant particle.

where
\begin{equation}Z=\sqrt{\left(E_{\pi}+E_{\mathrm{n}}\right)^{2}-\left(\mathbf{p}_{\pi}+\mathbf{p}_{\mathrm{n}}\right)^{2} c^{2}}\end{equation}
I don't understand what is being displayed, do our detectors differentiate between a simple collision and one where the resonant particle intervenes? Is it possible that the peak detected is due to the fact that the detector is counting both processes if they occur? And finally, should we associate only the local maxima when representing number of events versus energy with resonant particles or can we also associate the valleys?
 A: 
I don't understand what is being displayed, do our detectors differentiate between a simple collision and one where the resonant particle intervenes?

The experiment in your given plot, starts with a fixed number/Delta(E) which for some (confusing) reason they call Z. Usually the protons are stationary and the energy of the electrons is increased step by step and the number of events with three identified  particles, e , pion, neutron are counted per bin.
The gray is the measured plot. The black line is the theoretical curve one would fit if there were no extra attraction ( resonance) between the electron and the proton but just the electromagnetic interaction. Then one counts how many standard deviations the peaks and valleys are from the theoretical smooth curve. If the difference is more than four standard deviations from the theoretical curve , one has a definite signal of an extra attractive force.
As the y axis is not given numerically, one trusts they have done a good job in their analysis and the blue curve shows the fit with the resonances and the expected background added.
