= Partial-wave unitarity
Partial-wave unitarity restricts scattering in each angular-momentum channel because total outgoing probability cannot exceed the incident probability. For a single purely elastic channel, $S_\ell=e^{2i\delta_\ell}$ has unit modulus, and for distinguishable spinless particles
$$
\sigma_\ell=\frac{4\pi}{k^2}(2\ell+1)\sin^2\delta_\ell\leq\frac{4\pi}{k^2}(2\ell+1).
$$
The bound follows directly from $\sin^2\delta\leq1$.
If inelastic channels are open, the elastic diagonal element can have modulus smaller than one; the complete coupled-channel S-matrix remains unitary. A truncated perturbative prediction exceeding the applicable bound signals failure of that approximation or missing dynamics, rather than a physical probability larger than one. Spin, identical-particle symmetry and channel conventions change the appropriate partial-wave normalization.
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