= Solution
\b[A constant-width <Breit-Wigner resonance cross section> is the simplest nonrelativistic curve; a <P-wave scattering> width and relativistic kinematics explain important expected distortions.] A comparison to specific observed deviations cannot be made without the missing plot.
Write $E$ for the <centre-of-mass frame> <kinetic energy>, $\mu=m_pm_\pi/(m_p+m_\pi)$ for the <reduced mass>, and $k$ for the relative wave number. The nonrelativistic relations are
$$
E\simeq\frac{m_p}{m_p+m_\pi}T,\qquad
k^2=\frac{2\mu E}{\hbar^2}.
$$
For an isolated <particle resonance> of <spin> $J$, averaging over the initial <proton> <spin> and using the zero <pion> <spin> gives
$$
\sigma_{\rm el}(E)=\frac{\pi}{k^2}\frac{2J+1}{2}
\frac{\Gamma_{\pi p}^2}{(E-E_R)^2+\Gamma^2/4}.
$$
For the nearly elastic $\Delta^{++}$ state, $J=3/2$ and $\Gamma_{\pi p}\simeq\Gamma$, so
$$
\boxed{\sigma_{\rm el}(E)\simeq\frac{8\pi}{k^2}
\frac{\Gamma^2/4}{(E-E_R)^2+\Gamma^2/4}.}
$$
Freezing $k$ at $k_R$ gives a symmetric Lorentzian centered at $E_R$, with <full width at half maximum> $\Gamma$ and height $8\pi/k_R^2$. This is the local curve to sketch. If the displayed variable is the beam <kinetic energy> $T$, its width is $\Delta T\simeq(m_p+m_\pi)\Gamma/m_p$ within this nonrelativistic approximation. Keeping $k(E)$ changes the prefactor and hence the symmetry. Extending a constant-width approximation all the way to $E=0$ produces a spurious divergence, so it is not a valid threshold model.
The <pion> has <intrinsic parity> $-1$ and the <proton> has $+1$. The positive <parity> and $J=3/2$ of the <Delta baryon> require odd <orbital angular momentum>, with the lowest choice $l=1$. The <finite-range partial-wave threshold law> gives $\Gamma_{\pi p}(k)\propto k^{2l+1}=k^3$ near threshold. Inserting this in the elastic numerator makes $\sigma_{\rm el}\propto k^4$ away from the pole as $k\to0$, rather than diverging. Equivalently the elastic <scattering phase shift> has $\delta_1=O(k^3)$, and $\sin^2\delta_1/k^2=O(k^4)$. Away from threshold, form factors and other channels modify the simple cubic width.
At the illustrative peak of part (d), the incident <pion> <kinetic energy> is comparable to its rest <energy>, so a nonrelativistic <pion> treatment is already inaccurate. The exact relative <momentum> is
$$
p_*^2=\frac{[s-(m_p+m_\pi)^2][s-(m_p-m_\pi)^2]}{4s},\qquad k=\frac{p_*}{\hbar},
$$
with $\sqrt{s}=\sqrt{(m_p+m_\pi)^2+2m_pT}$. Relative to the symmetric local estimate, this changes the horizontal conversion, peak normalization and tails. The illustrative $1232\,\mathrm{MeV}$ peak has $p_*\simeq227\,\mathrm{MeV}/c$, giving $8\pi(\hbar c)^2/(p_*c)^2\simeq190\,\mathrm{mb}$ before small inelastic effects.
Finally, nonresonant <partial waves> can interfere with the resonant <scattering amplitude>, higher <particle resonances> and inelastic <scattering channels> can enter, and instrumental resolution can broaden a measured peak. These are physically distinct possible deviations; the missing data do not justify claiming any particular one was actually observed. Both the missing graphical overlay and its data-specific explanation remain source gaps.
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