= Solution
For $g=0$, the <invariant scattering amplitude> from the <tree-level Feynman diagram> in <2/c/i> has a <pole> at $s=M^2$. The intermediate heavy <antiparticle> can go on shell and live for a time of order its <mean lifetime>. Near this <scattering resonance>, inserting a single <self-energy> is no longer a uniformly small correction. <Dyson resummation> gives, to leading accuracy in a narrow <decay width>,
$$
\boxed{\mathcal M(s)\simeq-\frac{h^2}{s-M^2+iM\Gamma_{\rm phys}}.}
$$
Here $M$ is understood as the <renormalized> resonance mass; the real <self-energy> shifts the original pole position. The imaginary term removes the unphysical divergence of the fixed-order <invariant scattering amplitude>.
For equal incoming and outgoing light masses, the two-body <momentum> ratio cancels in the <cross section>. The angle-independent <invariant scattering amplitude>, integrated over the full solid angle with the <identical final-state symmetry factor> $1/2!$, gives $\sigma=|\mathcal M|^2/(32\pi s)$. Thus the <Breit-Wigner resonance cross section> is
$$
\boxed{\sigma(s)\simeq\frac{h^4}{32\pi s\left[(s-M^2)^2+M^2\Gamma_{\rm phys}^2\right]},\qquad
\Gamma_{\rm phys}=\frac{h^2\beta}{32\pi M}.}
$$
At $s\gg M^2$ it reproduces $h^4/(32\pi s^3)$. At the resonance peak,
$$
\boxed{\sigma(M^2)\simeq\frac{32\pi}{M^2\beta^2}\simeq\frac{32\pi}{M^2}\quad(M\gg m).}
$$
This is also $8\pi/k_*^2$ for $k_*=M\beta/2$, the elastic identical-boson <S wave> <partial-wave unitarity> limit when this is the only open <decay channel>. The approximation assumes $\Gamma_{\rm phys}\ll M$ and a resonance sufficiently far from threshold for the width to vary slowly across it.
If one mechanically uses the unsymmetrized $\Gamma_{\rm printed}=2\Gamma_{\rm phys}$ from <2/d>, the same displayed <Breit-Wigner resonance cross section> would have that doubled width and predict a peak $8\pi/(M^2\beta^2)$. This is the literal-width substitution, rather than the consistent physical result: the given high-energy <cross section> already includes the identical-final-particle factor that the printed <decay width> integral omits.
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