Spin-Locality of ${\eta }^2$ and ${\bar{\eta }}^2$ Quartic Higher-Spin Vertices

Abstract

Higher-spin theory contains a complex coupling parameter $\eta$. Different higher-spin vertices are associated with different powers of $\eta$ and its complex conjugate $\bar{\eta }$. 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 ${\eta }^2$ and ${\bar{\eta }}^2$ sectors of the higher-spin equations. These vertices include, in particular, the ${\eta }^2$ and ${\bar{\eta }}^2$ parts of the ${\varphi }^4$ scalar field vertex.

Section 3 extended, clarifications and references added, typos corrected; matches the published version