  A formulation of the linear $\sigma$ model with derivative interactions is
studied. The classical theory is on-shell equivalent to the $\sigma$ model with
the standard quartic Higgs potential. The mass of the scalar mode only appears
in the quadratic part and not in the interaction vertices, unlike in the
ordinary formulation of the theory. Renormalization of the model is discussed.
A non power-counting renormalizable extension, obeying the defining functional
identities of the theory, is presented. This extension is physically equivalent
to the tree-level inclusion of a dimension six effective operator $\partial_\mu
(\Phi^\dagger \Phi) \partial^\mu (\Phi^\dagger \Phi)$. The resulting UV
divergences are arranged in a perturbation series around the power-counting
renormalizable theory. The application of the formalism to the Standard Model
in the presence of the dimension-six operator $\partial_\mu (\Phi^\dagger \Phi)
\partial^\mu (\Phi^\dagger \Phi)$ is discussed.
