# How the Higgs potential got its shape

September 13, 2022

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. We derive this result as a consequence of
the "Principle of String-Independence" (PSI) for renormalized perturbation
theory with string-localized interactions: While the renormalizable interaction
density may be localized along an auxiliary "string", the S-matrix (and also
the local observables of the theory) must not depend on it. Along the way, we
develop a model-independent scheme how the PSI is implemented in
string-localized perturbation theory, and how it can be used as a tool to
"renormalize the non-renormalizable".

Keywords:

causal perturbation theory, string-localized fields, Abelian Higgs model