  We construct the general partial wave amplitude basis for the $N\to M$
scattering, which consists of Poincar\'e Clebsch-Gordan coefficients, with
Lorentz invariant forms given in terms of spinor-helicity variables. The inner
product of the Clebsch-Gordan coefficients is defined, which converts on-shell
phase space integration into an algebraic problem. We also develop the
technique of partial wave expansions of arbitrary amplitudes, including those
with infrared divergence. These are applied to the computation of anomalous
dimension matrix for general effective operators, where unitarity cuts for the
loop amplitudes, with an arbitrary number of external particles, are obtained
via partial wave expansion.
