We apply on-shell methods to the bottom-up construction of electroweak amplitudes, allowing for both renormalizable and non-renormalizable interactions. We use the little-group covariant massive-spinor formalism, and flesh out some of its details along the way. Thanks to the compact form of the resulting amplitudes, many of their properties, and in particular the constraints of perturbative unitarity, are easily seen in this formalism. Our approach is purely bottom-up, assuming just the standard-model electroweak spectrum as well as the conservation of electric charge and fermion number. The most general massive three-point amplitudes consistent with these symmetries are derived and studied in detail, as the primary building blocks for the construction of scattering amplitudes. We employ a simple argument, based on tree-level unitarity of four-point amplitudes, to identify the three-point amplitudes that are non-renormalizable at tree level. This bottom-up analysis remarkably reproduces many low-energy relations implied by electroweak symmetry through the standard-model Higgs mechanism and beyond it. We then discuss four-point amplitudes. The gluing of three-point amplitudes into four-point amplitudes in the massive spinor helicity formalism is clarified. As an example, we work out the ψ c ψ Zh amplitude, including also the non-factorizable part. The latter is an all-order expression in the effective-field-theory expansion. Further constraints on the couplings are obtained by requiring perturbative unitarity. In the ψ c ψ Zh example, one for instance obtains the renormalizable-level relations between vector and fermion masses and gauge and Yukawa couplings. We supplement our bottom-up derivations with a matching of three- and fourpoint amplitude coefficients onto the standard-model effective field theory (SMEFT) in the broken electroweak phase. This establishes the correspondence with the usual Lagrangian approach and paves the way for SMEFT computations in the on-shell formalism.

中文翻译：

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来自壳上振幅的电弱有效场理论

我们将壳上方法应用于电弱振幅的自下而上构造，允许可重整化和不可重整化相互作用。我们使用小群协变大规模自旋形式主义，并在此过程中充实其一些细节。由于所得振幅的紧凑形式，它们的许多属性，特别是微扰幺正性的约束，在这种形式主义中很容易看出。我们的方法纯粹是自下而上的，仅假设标准模型电弱谱以及电荷和费米子数守恒。导出并详细研究了与这些对称性一致的最一般的大质量三点振幅，作为构建散射振幅的主要构建块。我们采用一个简单的论证，基于四点幅度的树级单一性，识别在树级不可重整化的三点幅度。这种自下而上的分析通过标准模型希格斯机制和超越它，显着地再现了电弱对称性所暗示的许多低能量关系。然后我们讨论四点振幅。阐明了大自旋量螺旋形式中三点振幅与四点振幅的粘合。例如，我们计算出 ψ c ψ Zh 振幅，也包括不可因式分解的部分。后者是有效场论展开中的全阶表达式。通过要求微扰单一性获得对耦合的进一步约束。在 ψ c ψ Zh 的例子中，例如，获得矢量和费米子质量以及规范和汤川耦合之间的可重整化水平关系。我们通过将三点和四点幅度系数匹配到断电弱相中的标准模型有效场理论 (SMEFT) 来补充我们的自下而上的推导。这与通常的拉格朗日方法建立了对应关系，并为壳上形式主义中的 SMEFT 计算铺平了道路。