How the Higgs potential got its shape

String-localized quantum field theory allows renormalizable couplings involving massive vector bosons, without invoking negative-norm states and compensating ghosts. We analyze the most general coupling of a massive vector boson to a scalar field, and find that the scalar field necessarily comes with a quartic potential which has the precise shape of the shifted Higgs potential. In other words: the shape of the Higgs potential has not to be assumed, but arises as a consistency condition among fundamental principles of QFT: Hilbert space, causality, and covariance. The consistency can be achieved by relaxing the localization properties of auxiliary quantities, including interacting charged fields, while observable fields and the S-matrix are not affected. This is an instance of the "L-V formalism" - a novel model-independent scheme that can be used as a tool to "renormalize the non-renormalizable" by adding a total derivative to the interaction.