Duality in the inelastic structure function and the disappearingΔresonance

We study the dual relationship between the resonances and the scaling curve for the inelastic structure function $\nu W_2$ in the $Q^2$ range 1 to 17 ${(\mathrm{G}\mathrm{e}\mathrm{V}/\mathit{c})}^2$. We focus on an exception to the rule that resonance bumps fall with increasing $Q^2$ at the same rate as the background, namely, the $\Delta \mathrm{}\left(1232\right)\mathrm{}$, for which the resonance falls faster than the background. We show that $\nu W_2$, averaged over suitable ranges of $x$, yields the smooth scaling curve seen at higher values of $Q^2$ at the same values of $x$, for all resonance regions. The fall of the $\Delta \mathrm{}\left(1232\right)\mathrm{}$ and corresponding rise of the background hints at a common dynamics for both.