  The off-shell one-loop renormalization of a Higgs effective field theory
possessing a scalar potential $\sim\left(\Phi^\dagger\Phi-\frac{v^2}2\right)^N$
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 $N\to\infty$ case.
