 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 particlesThe bound follows directly from ${\sin }^2\delta \le 1$.

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.