Constructing the general partial wave and renormalization in effective field theory

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We construct the general partial wave amplitude basis for the $N\ensuremath{\rightarrow}M$ scattering, which consists of Poincar\'e Clebsch-Gordan coefficients, in a Lorentz invariant form given in terms of the 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.