  It is well known that in order to make the path integral of general
relativity converge, one has to perform the Wick rotation over the conformal
factor in addition to the more familiar Wick rotation of the time axis to pass
to the space-time with Euclidean signature. In this article, we will apply this
technique to a scalar field in the conformally invariant scalar-tensor gravity
with a conformally invariant beyond-standard-model (BSM). It is then shown that
a potential term in the conformally invariant potential, which corresponds to
the Higgs mass term in the Higgs potential of the standard model (SM), can have
a negative coefficient. The change of sign of the potential term naturally
induces spontaneous symmetry breakdown of the electroweak gauge symmetry after
symmetry breaking of conformal symmetry (local scale symmetry) via the
Coleman-Weinberg mechanism around the Planck scale. The present study might
shed light on the fact that the existence of a stable vacuum in quantum gravity
is relevant to that in the SM.
