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At incident wave number $k$, low kinetic energy means $ka\ll 1$: the de Broglie wavelength is much larger than the scattering region. A generic finite-range scatterer can then be described to leading order without resolving its angular structure. In partial-wave scattering, this is leading S-wave scattering. The $\ell =0$ spherical harmonic is constant, whereas the centrifugal term ${\hslash }^2\ell (\ell +1)/(2m{r}^2)$ impedes penetration to the potential for $\ell >0$.

For a central potential with regular threshold behaviour, the finite-range partial-wave threshold law gives ${\delta }_{\ell }=O(k^{2\ell +1})$. Thuswhich is independent of scattering angle. For finite nonzero scattering length $a_s$, $f\to -a_s$ and $d\sigma /d\Omega \to a_s^2$ is isotropic. The potential range $a$ and scattering length $a_s$ are different quantities. The claim describes the generic leading S-wave scattering limit; tuned vanishing S-wave scattering amplitude or exceptional threshold resonances require separate analysis rather than a universal angular conclusion.

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, $\mu =m_p{m}_{\pi }/(m_p+m_{\pi })$ for the reduced mass, and $k$ for the relative wave number. The nonrelativistic relations areFor an isolated particle resonance of spin $J$, averaging over the initial proton spin and using the zero pion spin givesFor the nearly elastic ${\Delta }^{++}$ state, $J=3/2$ and ${\Gamma }_{\pi p}\simeq \Gamma$, soFreezing $k$ 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 ${\sigma }_{\mathrm{el}}\propto k^4$ away from the pole as $k\to 0$, 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 iswith $\sqrt{s}=\sqrt{(m_p+m_{\pi }{)}^2+2m_{p}T}$. Relative to the symmetric local estimate, this changes the horizontal conversion, peak normalization and tails. The illustrative $1232\mathrm{MeV}$ peak has $p_{\ast }\simeq 227\mathrm{MeV}/c$, giving $8\pi (\hslash c{)}^2/(p_{\ast }c{)}^2\simeq 190\mathrm{mb}$ 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.

The partial-wave scattering channel with orbital angular momentum quantum number one. Its angular dependence is built from degree-one spherical harmonics. For a finite-range potential away from a threshold pole, the finite-range partial-wave threshold law gives phase shift ${\delta }_1=O(k^3)$ and a partial elastic scattering cross-section proportional to $k^4$ as the relative wavenumber $k$ tends to zero.