Off-shell renormalization in Higgs effective field theories

The off-shell one-loop renormalization of a Higgs effective field theory possessing a scalar potential $\sim {({\mathrm{\Phi }}^{†}\mathrm{\Phi }-\frac{v^2}{2})}^N$ with $N$ arbitrary is presented. This is achieved by renormalizing the theory once reformulated in terms of two auxiliary fields $X_{1,2}$, which, due to the invariance under an extended Becchi-Rouet-Stora-Tyutin symmetry, are tightly constrained by functional identities. The latter allow in turn the explicit derivation of the mapping onto the original theory, through which the (divergent) multi-Higgs amplitude are generated in a purely algebraic fashion. We show that, contrary to naive expectations based on the loss of power counting renormalizability, the Higgs field undergoes a linear Standard Model like redefinition, and evaluate the renormalization of the complete set of Higgs self-coupling in the $N\to \mathrm{\infty }$ case.