  We study the renormalization flow of the Higgs potential as a function of
both field amplitude and energy scale. This overcomes limitations of
conventional techniques that rely, e.g., on an identification of field
amplitude and RG scale, or on local field expansions. Using a Higgs-Yukawa
model with discrete chiral symmetry as an example, our global flows in field
space clarify the origin of possible meta-stabilities, the fate of the
pseudo-stable phase, and provide new information about the renormalization of
the tunnel barrier. Our results confirm the relaxation of the lower bound for
the Higgs mass in the presence of more general microscopic interactions
(higher-dimensional operators) to a high quantitative accuracy.
