Spin-Locality of $η^2$ and $\barη^2$ Quartic Higher-Spin Vertices

Higher-spin theory contains a complex coupling parameter $η$. Different higher-spin vertices are associated with different powers of $η$ and its complex conjugate $\bar η$. Using $Z$-dominance Lemma, that controls spin-locality of the higher-spin equations, we show that the third-order contribution to the zero-form $B(Z;Y;K)$ admits a $Z$-dominated form that leads to spin-local vertices in the $η^2$ and $\bar η^2$ sectors of the higher-spin equations. These vertices include, in particular, the $η^2$ and $\bar η^2$ parts of the $ϕ^4$ scalar field vertex.

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