We present the general form of the renormalizable four-point interactions of a complex scalar field furnishing an irreducible representation of SU(2), and derive a set of algebraic identities that facilitates the calculation of higher-order radiative corrections. As an application, we calculate the two-loop beta function for the SM extended by a scalar multiplet, and provide the result explicitly in terms of the group invariants. Our results include the evolution of the Higgs-portal couplings, as well as scalar “minimal dark matter”. We present numerical results for the two-loop evolution of the various couplings.

中文翻译：

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复标量电弱多重态的双环 beta 函数

我们提出了复杂标量场的可重整化四点相互作用的一般形式，提供了 SU(2) 的不可约表示，并推导出了一组代数恒等式，以促进高阶辐射校正的计算。作为一个应用，我们计算了由标量多重扩展扩展的 SM 的双循环 beta 函数，并根据群不变量明确提供结果。我们的结果包括希格斯门户耦合的演变，以及标量“最小暗物质”。我们提供了各种耦合的双回路演化的数值结果。