  Neutron stars are perfect candidates to investigate the effects of a modified
gravity theory, since the curvature effects are significant and more
importantly, potentially testable. In most cases studied in the literature in
the context of massive scalar-tensor theories, inflationary models were
examined. The most important of scalar-tensor models is the Higgs model, which,
depending on the values of the scalar field, can be approximated by different
scalar potentials, one of which is the inflationary. Since it is not certain
how large the values of the scalar field will be at the near vicinity and
inside a neutron star, in this work we will answer the question, which
potential form of the Higgs model is more appropriate in order for it to
describe consistently a static neutron star. As we will show numerically, the
non-inflationary Higgs potential, which is valid for certain values of the
scalar field in the Jordan frame, leads to extremely large maximum neutron star
masses, however the model is not self-consistent, because the scalar field
approximation used for the derivation of the potential, is violated both at the
center and at the surface of the star. These results shows the uniqueness of
the inflationary Higgs potential, since it is the only approximation for the
Higgs model, that provides self-consistent results.
