## The electroweak effective field theory the on-shell way

Yaniv Weiss [1]  ,  Gauthier Durieux [1]  ,  Teppei Kitahara [1,2,3]  ,  Yael Shadmi [1]
[1] Technion---Israel Institute of Technology, Technion city, Haifa 3200003, Israel
[2] Institute for Advanced Research, Nagoya University, Furo-cho Chikusa-ku, Nagoya 464-8601, Japan
[3] Kobayashi-Maskawa Institute for the Origin of Particles and the Universe, Nagoya University, Furo-cho Chikusa-ku, Nagoya 464-8602, Japan

We use on-shell methods to construct the renormalizable and effective electroweak amplitudes, with no reference to an effective field theory (EFT) Lagrangian.
This bottom-up construction of the amplitudes relies on Lorentz symmetry, unitarity and spin statistics.
The spinor helicity formalism used in this approach results in compact forms for the amplitudes.
Thus, many properties of the amplitudes become manifest, especially, the constraints arrising from perturbative unitarity.
We assume the Standard Model (SM) electroweak spectrum and only work from the bottom-up.
The first step is to construct the three-point amplitudes, which are totally determined by Lorentz symmetry.
A simple argument, relying on tree-level perturbative unitarity of the four-point function, determines which three-point amplitudes are renormalizable.
As an example for a four-point function we construct the $\psi^c \psi Z h$ amplitude, including non-factorizable and factrizable contributions.
The expression we obtain is an all-order result in the EFT expansion.
Imposing perturbative unitarity constrains the couplings, and in particular we derive the renormalizable-level relations between vector and fermion masses and gauge and Yukawa couplings.