Orthogonal-invariant scattering channels ($s_0=N{S}_1+S_2+S_3,s_{\pm }=S_2\pm S_3$)

The two-vector tensor product splits into a trace singlet, symmetric traceless tensors and antisymmetric tensors. For an O(N)-invariant S-matrix, the corresponding eigenvalues are $s_0=N{S}_1+S_2+S_3$, $s_{+}=S_2+S_3$ and $s_{-}=S_2-S_3$. For $N\ge 2$, analytic unitarity requires $s_a(\theta )s_a(-\theta )=1$. With Hermitian analyticity of a two-particle S-matrix, each channel has unit modulus on the real rapidity axis.