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,

μ = m_{p} m_{π} / (m_{p} + m_{π})

for the reduced mass, and

k

for the relative wave number. The nonrelativistic relations are

E ≃ \frac{m _{p}}{m _{p} + m _{π}} T, k^{2} = \frac{2 μ E}{ℏ ^{2}} .

(1)

For an isolated particle resonance of spin

J

, averaging over the initial proton spin and using the zero pion spin gives

σ_{el} (E) = \frac{π}{k ^{2}} \frac{2 J + 1}{2} \frac{Γ _{π p}^{2}}{( E - E _{R} ) ^{2} + Γ ^{2} /4} .

(2)

For the nearly elastic

Δ^{++}

state,

J = 3/2

and

Γ_{π p} ≃ Γ

, so

σ_{el} (E) ≃ \frac{8 π}{k ^{2}} \frac{Γ ^{2} /4}{( E - E _{R} ) ^{2} + Γ ^{2} /4} .

(3)

Freezing

k

k_{R}

gives a symmetric Lorentzian centered at

E_{R}

, with full width at half maximum

Γ

and height

8 π / k_{R}^{2}

. This is the local curve to sketch. If the displayed variable is the beam kinetic energy

T

, its width is

Δ T ≃ (m_{p} + m_{π}) Γ/ 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

Γ_{π p} (k) \propto k^{2 l + 1} = k^{3}

near threshold. Inserting this in the elastic numerator makes

σ_{el} \propto k^{4}

away from the pole as

k \to 0

, rather than diverging. Equivalently the elastic scattering phase shift has

δ_{1} = O (k^{3})

, and

sin^{2} δ_{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 _{π} ) ^{2} ] [ s - ( m _{p} - m _{π} ) ^{2} ]}{4 s}, k = \frac{p _{*}}{ℏ},

(4)

with

s = (m_{p} + m_{π})^{2} + 2 m_{p} T

. Relative to the symmetric local estimate, this changes the horizontal conversion, peak normalization and tails. The illustrative

1232 MeV

peak has

p_{*} ≃ 227 MeV / c

, giving

8 π (ℏ c)^{2} / (p_{*} c)^{2} ≃ 190 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.