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We present the general form of the renormalizable four-point interactions ofa complex scalar field furnishing an irreducible representation of SU(2), andderive a set of algebraic identities that facilitates the calculation ofhigher-order radiative corrections. As an application, we calculate thetwo-loop beta function for the SM extended by a scalar multiplet, and providethe result explicitly in terms of the group invariants. Our results include theevolution of the Higgs-portal couplings, as well as scalar "minimal darkmatter". We present numerical results for the two-loop evolution of the variouscouplings.

Two-loop Beta Function for Complex Scalar Electroweak Multiplets