Partial-Wave Unitarity Bounds on Higher-Dimensional Operators from 2-to-$N$ Scattering

We present a systematic method for deriving partial-wave unitarity bounds on Wilson coefficients of higher-dimensional operators in effective field theories involving more than four fields, which naturally appear in tree-level 2-to-$N$ scattering processes with $N \geq 3$. Unlike 2-to-2 scattering, 2-to-$N$ scattering with $N \geq 3$ features multiple amplitudes associated with the same total angular momentum. To resolve these degeneracies, we provide a way to construct an orthonormal amplitude basis by parameterizing the phase space manifold of massless particles using spinor-helicity variables, enabling analytical integration over the phase space with arbitrary particle numbers. We provide Mathematica code to analytically evaluate phase space integrals of interference between two local on-shell amplitudes up to four final-state particles, with straightforward generalization to $N$ final-state particles. As practical applications, we demonstrate the use of this tool by deriving unitarity bounds on some dimension-7 and dimension-8 operators in the Standard Model effective field theory involving five and six fields, respectively.